REVIEWS in MINERALOGY and GEOCHEMISTRY Volume 46 2002
MICAS: CRYSTAL CHEMISTRY AND METAMORPHIC PETROLOGY Editors Annibale Mottana Francesco Paolo Sassi James B. Thompson, Jr. Stephen Guggenheim
Università degli Studi Roma Tre Università di Padova Harvard University University of Illinois at Chicago
FRONT COVER: Perspective view of TOT layers in Biotite down [100] ([001] is vertical), produced by CrystalMaker, Red tetrahedra contain Si and A1, green and white octahedra contain Mg and Fe, respectively, and yellow spheres represents the K interlayer cations. Courtesy of Mickey Gunter, University of Idaho, Moscow. [Data: S.R. Bohlen et al. (1980) Crystal chemistry of a metamorphic biotite and its significance in water barometry. Am Mineral 65: 55-62] BACK COVER: A view down [001] of lepitdolite-2M2, showing tetrahedrally coordinated Si,A1 (blue) joined with bridging oxygens (red thermal ellipsoids) in the T-Layer and ordered, octahedrally coordinated A1 (gray) and Li (yellow) in the O-layer. The interlayer cation I s12-coordinator K (green). Courtesy of Bob Downs, University of Arizona, Tucson. [Data: S. Guggenheim (1981) Cation ordering in lepidolite. Am Mineral 66: 1221-1232]
Series Editor for MSA: Paul H. Ribbe Virginia Polytechnic Institute and State University
MINERALOGICAL SOCIETY of AMERICA Washington, D.C. ACCADEMIA NATIONALE dei LINCEI Roma, Italia
COPYRIGHT 2002
MINERALOGICAL SOCIETY OF AMERICA The appearance of the code at the bottom of the first page of each chapter in this volume indicates the copyright owner’s consent that copies of the article can be made for personal use or internal use or for the personal use or internal use of specific clients, provided the original publication is cited. The consent is given on the condition, however, that the copier pay the stated per-copy fee through the Copyright Clearance Center, Inc. for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law. This consent does not extend to other types of copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. For permission to reprint entire articles in these cases and the like, consult the Administrator of the Mineralogical Society of America as to the royalty due to the Society.
REVIEWS IN MINERALOGY AND GEOCHEMISTRY ( Formerly: REVIEWS IN MINERALOGY )
ISSN 1529-6466
Volume 46
MICAS: Crystal Chemistry and Metamorphic Petrology ISBN 0-939950-58-8 ** This volume is the eighth of a series of review volumes published jointly under the banner of the Mineralogical Society of America and the Geochemical Society. The newly titled Reviews in Mineralogy and Geochemistry has been numbered contiguously with the previous series, Reviews in Mineralogy. Additional copies of this volume as well as others in this series may be obtained at moderate cost from: THE MINERALOGICAL SOCIETY OF AMERICA 1015 EIGHTEENTH STREET, NW, SUITE 601 WASHINGTON, DC 20036 U.S.A.
MICAS: Crystal Chemistry and Metamorphic Petrology Reviews in Mineralogy and Geochemistry Volume 46 2002 FORWARD The editors and contributing editors of this volume participated in a short course on micas in Rome late in the year 2000. It was organized by Prof. Annibale Mottana and several colleagues (details in the Preface below) and underwritten by the Italian National Acadmey, Accademai Nationale dei Lincei (ANL). The Academy subsequently joined with the Mineralogical Society of America (MSA) in publishing this volume. MSA is grateful for their generous involvement. I am particularly thankful to Prof. Mottana for Herculean efforts in supervising the editing of twelve manuscripts from six countries and submitting a single package containing everything needed to compile this volume! This was a uniquely positive experience fro me as Series editor for MSA. Assembling this volume was made tolerable by the exceptional efforts of Steve Guggenheim. During recovery from spinal surgery he spent three weeks painstakingly (no pun) correcting grammar and wording of the many authors from whom English is not their first language. Special thanks to him and the gracious and patient authors who suffered the extra work of assimilating both Steve’s suggestions and mine, above and beyond those of their reviewers and the editors. MSA’s Executive Director, Alex Speer, made all the contractual arrangements with ANL. This is the second of what we hope will be many co-operative projects with international colleagues and members of MSA. The first was in the year 2000: “Transformation Processes in Minerals,” RiMG 39, the proceedings of a short course at Cambridge University in partnership with four European scientific societies. Paul H. Ribbe, Series editor Blacksburg, Virginia April 20, 2002
PREFACE Micas are among the most common minerals in the Earth crust: 4.5% by volume. They are widespread in most if not all metamorphic rocks (abundance: 11%), and common also in sediment and sedimentary and igneous rocks. Characteristically, micas form in the uppermost greenschist facies and remain stable to the lower crust, including anatectic rocks (the only exception: granulite facies racks). Moreover, some micas are stable in sediments and diagenetic rocks and crystallize in many types of lavas. In contrast, they are also present in association with minerals originating from the very deepest parts of the mantle—they are the most common minerals accompanying diamond in kimberlites. The number of research papers dedicated to micas is enormous, but knowledge of them is limited and not as extensive as that of other rock-forming minerals, for reasons mostly relating to their complex layer texture that makes obtaining crystals suitable for careful studies with the modern methods time-consuming, painstaking work. Micas were reviewed extensively in 1984 (Reviews in Mineralogy 13, S.W. Bailey, editor). At that time, “Micas” volume covered most if not all aspects of mica knowledge, thus producing a long shelf-life for this book. Yet, or perhaps because of that iii 1529-6466/02/0046-0000$05.00
DOI: 10.2138/rmg.2002.46.0
excellent review, mica research was vigorously renewed, and a vast array of new data has been gathered over the past 15 years. These data now need to be organized and reviewed. Furthermore, a Committee nominated by the International Mineralogical Association in the late 1970s concluded its long-lasting work (Rieder et al. 1998) by suggesting a new classification scheme which has stimulated a new chemical and structural research on micas. To make a very long story short: -
-
-
the extraordinarily large, but intrinsically vague, micas nomenclature developed during the past two centuries has been reduced from >300 to just 37 species names and 6 series (see page xiii, preceding Chapter 1); the new nomenclature shows wide gaps that require data involving new chemical and structural work; the suggestion of using adjectival modifiers for those varieties that deviate away from end-member compositions requires the need fro new and accurate measurements, particularly fro certain light elements and volatiles; the use of polytype suffixes based on the modified Gard symbolism created better ways of determining precise stacking sequences. This resulted in new polytypes being discovered.
Indeed, all this has happened over the past few years in an almost tumultuous way. It was on the basis of these developments that four scientists (B. Zanettin, A. Mottana, F.P. Sassi and C. Cipriani) applied to Accademia Nazionale dei Lincei—the Italian National Academy—for a meeting on micas. An international meeting was convened in Rome on November 2-3, 2000 with the title Advances on Micas (Problems, Methods, Applications in Geodynamics). The topics of this meeting were the crystalchemical, petrological, and historical aspects if the micas. The organizers were both Academy members (C. Cipriani, A. Mottana, F.P. Sassi, W. Schreyer, J.B. Thompson Jr., and B. Zanettin) and Italian scientist well-known for their studies on layer silicates (Professors M.F. Brigatti and G. Ferraris). Financial support in addition to that by the Academy was provided by C.N.R. (the Italian National Research Council), M.U.R.S.T. (the Italian Ministry for University, Scientific Research and Technology) and the University of Rome III. Approximately 200 scientists attended the meeting, most of them Italians, but, with a sizeable international participation. Thirteen invited plenary lectures and six oral presentations were given, and fourteen posters were displayed. The amount of information presented was large, although the organizers made it very clear that the meeting was to be limited to only a few of the major topics of micas studies. Other studies are promised for a later meeting. Oral and poster presentations on novel aspects of mica research are being printed in the European Journal of Mineralogy, as apart of an individual thematic issue: indeed thirteen papers have appeared in the November 2001 issue. The plenary lectures, which consisted mostly of reviews, are presented in expanded detail in this volume. This book is the first a co-operative project between Accademia Nazionale dei Lincei and Mineralogical Society of America. Hopefully, future projects will involve reviews of the remaining aspects of mica research, and other aspects of mineralogy and geochemistry. The entire meeting was made successful through a co-operative effort. The editing of this book was achieved by a co-operative effort of two Italian Academy members from one side, and by two American scientists from the other side, one of them (JBT) being also a member of Lincei Academy. The entire editing process benefited from the goodwill of many referees, both from those attending Rome meeting and from several who did not. In all the reviewers were distinguished expert of the international iv
community of mica scholars. Their work, as well as our editing work, were aided greatly by RiMG Series Editor, Professor Paul Ribbe, who continuously supported the efforts with all his professional experience and friendly advice. We, the co-editors, thank them all very warmly, but take upon ourselves all remaining shortcomings: we are aware that some shortcomings may be present in spite of all our efforts to avoid them Moreover, we are aware that there are puzzling aspects of micas that are unresolved. Please consider all these possible avenues for future research! Annibale Mottana (Rome) Francesco Paolo Sassi (Padua) James B. Thompson, Jr. (Cambridge, Mass.) Stephen Guggenheim (Chicago)
v
Nomenclature of Micas MICA SIMPLIFIED FORMULA: I M2-3 1-0 T4 O10 A2 where I M T A
= Cs, K, Na, NH4, Rb, Ba, Ca = Li, Fe (2+, 3+), Mg, Mn, Zn, Al, Cr, V, Ti = vacancy = Be, Al, B, Fe(3+), Si = Cl, F, OH, O, S
MICA SERIES NAMES:
biotite glauconite illite lepidolite phengite zinnwaldite
TRUE MICAS
BRITTLE MICAS
INTERLAYERDEFICIENT MICAS
Dioctahedral
Trioctahedral
Dioctahedral
Trioctahedral
Dioctahedral
Trioctahedral
muscovite aluminoceladonite ferro-aluminoceladonite celadonite ferroceladonite roscoelite chromphyllite boromuscovite paragonite nanpingite tobelite
annite phlogopite siderophyllite
margarite chernykhite
clintonite bityite anandite
illite glauconite brammallite
wonesite
eastonite hendricksite montdorite tainiolite polylithionite trilithionite masutomilite norrishite tetra-ferri-annite tetra-ferriphlogopite aspidolite preiswerkite ephesite
kinoshitalite
FORWARD The editors and contributing editors of this volume participated in a short course on micas in Rome late in the year 2000. It was organized by Prof. Annibale Mottana and several colleagues (details in the Preface below) and underwritten by the Italian National Acadmey, Accademai Nationale dei Lincei (ANL). The Academy subsequently joined with the Mineralogical Society of America (MSA) in publishing this volume. MSA is grateful for their generous involvement. I am particularly thankful to Prof. Mottana for Herculean efforts in supervising the editing of twelve manuscripts from six countries and submitting a single package containing everything needed to compile this volume! This was a uniquely positive experience fro me as Series editor for MSA. Assembling this volume was made tolerable by the exceptional efforts of Steve Guggenheim. During recovery from spinal surgery he spent three weeks painstakingly (no pun) correcting grammar and wording of the many authors from whom English is not their first language. Special thanks to him and the gracious and patient authors who suffered the extra work of assimilating both Steve’s suggestions and mine, above and beyond those of their reviewers and the editors. MSA’s Executive Director, Alex Speer, made all the contractual arrangements with ANL. This is the second of what we hope will be many co-operative projects with international colleagues and members of MSA. The first was in the year 2000: “Transformation Processes in Minerals,” RiMG 39, the proceedings of a short course at Cambridge University in partnership with four European scientific societies. Paul H. Ribbe, Series editor Blacksburg, Virginia April 20, 2002
iii 1529-6466/02/0046-0000$05.00
DOI: 10.2138/rmg.2002.46.0f
PREFACE Micas are among the most common minerals in the Earth crust: 4.5% by volume. They are widespread in most if not all metamorphic rocks (abundance: 11%), and common also in sediment and sedimentary and igneous rocks. Characteristically, micas form in the uppermost greenschist facies and remain stable to the lower crust, including anatectic rocks (the only exception: granulite facies racks). Moreover, some micas are stable in sediments and diagenetic rocks and crystallize in many types of lavas. In contrast, they are also present in association with minerals originating from the very deepest parts of the mantle—they are the most common minerals accompanying diamond in kimberlites. The number of research papers dedicated to micas is enormous, but knowledge of them is limited and not as extensive as that of other rock-forming minerals, for reasons mostly relating to their complex layer texture that makes obtaining crystals suitable for careful studies with the modern methods time-consuming, painstaking work. Micas were reviewed extensively in 1984 (Reviews in Mineralogy 13, S.W. Bailey, editor). At that time, “Micas” volume covered most if not all aspects of mica knowledge, thus producing a long shelf-life for this book. Yet, or perhaps because of that excellent review, mica research was vigorously renewed, and a vast array of new data has been gathered over the past 15 years. These data now need to be organized and reviewed. Furthermore, a Committee nominated by the International Mineralogical Association in the late 1970s concluded its long-lasting work (Rieder et al. 1998) by suggesting a new classification scheme which has stimulated a new chemical and structural research on micas. To make a very long story short: -
-
-
the extraordinarily large, but intrinsically vague, micas nomenclature developed during the past two centuries has been reduced from >300 to just 37 species names and 6 series (see page xiii, preceding Chapter 1); the new nomenclature shows wide gaps that require data involving new chemical and structural work; the suggestion of using adjectival modifiers for those varieties that deviate away from end-member compositions requires the need fro new and accurate measurements, particularly fro certain light elements and volatiles; the use of polytype suffixes based on the modified Gard symbolism created better ways of determining precise stacking sequences. This resulted in new polytypes being discovered.
Indeed, all this has happened over the past few years in an almost tumultuous way. It was on the basis of these developments that four scientists (B. Zanettin, A. Mottana, F.P. Sassi and C. Cipriani) applied to Accademia Nazionale dei Lincei—the Italian National Academy—for a meeting on micas. An international meeting was convened in Rome on November 2-3, 2000 with the title Advances on Micas (Problems, Methods, Applications in Geodynamics). The topics of this meeting were the crystalchemical, petrological, and historical aspects if the micas. The organizers were both Academy members (C. Cipriani, A. Mottana, F.P. Sassi, W. Schreyer, J.B. Thompson Jr., and B. Zanettin) and Italian scientist well-known for their studies on layer silicates (Professors M.F. Brigatti and G. Ferraris). Financial support in addition to that by the iii 1529-6466/02/0046-0000$05.00
DOI: 10.2138/rmg.2002.46.0p
Academy was provided by C.N.R. (the Italian National Research Council), M.U.R.S.T. (the Italian Ministry for University, Scientific Research and Technology) and the University of Rome III. Approximately 200 scientists attended the meeting, most of them Italians, but, with a sizeable international participation. Thirteen invited plenary lectures and six oral presentations were given, and fourteen posters were displayed. The amount of information presented was large, although the organizers made it very clear that the meeting was to be limited to only a few of the major topics of micas studies. Other studies are promised for a later meeting. Oral and poster presentations on novel aspects of mica research are being printed in the European Journal of Mineralogy, as apart of an individual thematic issue: indeed thirteen papers have appeared in the November 2001 issue. The plenary lectures, which consisted mostly of reviews, are presented in expanded detail in this volume. This book is the first a co-operative project between Accademia Nazionale dei Lincei and Mineralogical Society of America. Hopefully, future projects will involve reviews of the remaining aspects of mica research, and other aspects of mineralogy and geochemistry. The entire meeting was made successful through a co-operative effort. The editing of this book was achieved by a co-operative effort of two Italian Academy members from one side, and by two American scientists from the other side, one of them (JBT) being also a member of Lincei Academy. The entire editing process benefited from the goodwill of many referees, both from those attending Rome meeting and from several who did not. In all the reviewers were distinguished expert of the international community of mica scholars. Their work, as well as our editing work, were aided greatly by RiMG Series Editor, Professor Paul Ribbe, who continuously supported the efforts with all his professional experience and friendly advice. We, the co-editors, thank them all very warmly, but take upon ourselves all remaining shortcomings: we are aware that some shortcomings may be present in spite of all our efforts to avoid them Moreover, we are aware that there are puzzling aspects of micas that are unresolved. Please consider all these possible avenues for future research! Annibale Mottana (Rome) Francesco Paolo Sassi (Padua) James B. Thompson, Jr. (Cambridge, Mass.) Stephen Guggenheim (Chicago)
iv
MICAS: CRYSTAL CHEMISTRY and METAMORPHIC PETROLOGY Editors: A Mottana, F P Sassi, J B Thompson, Jr & S Guggenheim
Table of Contents
1
Mica Crystal Chemistry and the Influence of Pressure, Temperature, and Solid Solution on Atonlistic Models Maria Franca Brigatti, Stephen Guggenheim
OVERVIEW Treatment of the data and definition of the parameters used End-member crystal chemistry: New end members and new data since 1984 Synthetic micas with unusual properties EFFECT OF COMPOSITION ON STRUCTURE ., Tetrahedral sheet Tetrahedral rotation and interlayer region Tetrahedral cation ordering Octahedral coordination and long-range octahedral ordering Crystal chemistry of micas in plutonic rocks ATOMISTIC MODELS INVOLVING HIGH-TEMPERATURE STUDIES OF THE MICAS Studies of samples having undergone heat treatment Dehydroxylation process for dioctahedral phyllosilicates Dehydroxylation models for trans-vacant 2: 1 layers Dehydroxy lation models for cis-vacant 2: 1 layers Compalison of Na-rich vs. K-rich dioctahedral forms Heat-treated trioctahedral samples: The O,OH,F site and in situ high-temperature studies Heat-treated trioctahedral samples: Polytype comparisons ACKNOWLEDGMENTS APPE~DIX I: DERIV ATIONS Derivation of "tetrahedral cation displacement", T di sp Derivation of f1E 1, f1E 2 , f1E 3 Derivation of ex Explanation of O[eor Explanation of E M - o(4) APPENDIX II: TABLES 1-4 Table 1a. Structural details of trioctahedral true micas-l M, space group C2/m Table 1b. Structural details of trioctahedral true micas-1M, space group C2 Table Ie. Structural details of trioctahedral true micas-2M], space group C2/c Table Id. Structural details oftrioctahedral true micas-2M J , space groups Ce. Cl Table Ie. Structural details of trioctahedral true micas-2M 2 , space group C2!c Table I f. Structural details of trioctahedral true micas-3T, space group P3,12 Table 2a. Structural details of trioctahedral true micas-I M, Mspace groups C2/m and C2 Table 2b. Structural details of trioctahedral true micas-1M, space group C2/c Table 2c. Structural details of trioctahedral true micas-2M, space group C2/e Table 2d. Structural details of trioctahedral true micas-3T, space group P3 J 12 Table 3a. Structural details of trioctahedral brittle micas Table 3b. Structural details of dioctahedral brittle micas Table 4. Structural details of boromuscovite-I M and -2M) calculated from the Rietveld structure refinement by Liang et al. (1995) REFERENCES
Vll
1 3 .4 11 1I 11 19 25 27 37 39 39 .41 43 44 .49 50 51 51 52 52 52 53 54 54 55
55 70 72 74 74 74 76 78 84 84 86 88 88 90
2
Behavior of Micas at High Pressure and High Temperature Pier Francesco Zanazzi, Alessandro Pavese
INTRODUCTION Investigati ve techniques for the study of the thennoelastic behav ior of mi cas p- V and P- V- T equations of state Dioc tahed ral micas Tri oc tahedral mi cas ACKNOWLEDGMENTS REFERENCES
3
99 100 10 1 103 108 ] 14 114
Structural Features of Micas Giovanni Ferraris, Gabriella Ivaldi
INTRODUCTION NOMENCLATURE AND NOTATION MODULARITY OF MICA STRUCTURE The mica module CLOSEST-PACKING aspects Closest-packing and polytypism COMPOSITIONAL ASPECTS SYMMETRY ASPECTS Metric (lattice) symmetry Structural symmetry Symmetry and cation sites Two kinds of mica layer: Ml and M2Iayers The interlayer configuration Possible ordering schemes in the MDO polytypes The phengite case DISTORTIONS The misfit Geo metric parameters describing distortions Ditngonal rotation Other distortions Effects of the distortions on the stacking mode FURTHER STRUCTURAL MODIFICATION Pressure, temperature and chemical influence Thickness of the mica module Ditrigonal rotation and interlayer coordination Effective coordination number (ECoN) CONCLUSIONS APPENDIX I: MICA STRUCTURE AND POLYSOMATIC SERIES Layer silicates as members of modular series ? Mica modules in polysomatic series The heterophyllosicate polysomatic series The palysepiole polysomatic series Conclusions APPENDIX II : OBLIQUE TEXTURE ELECTRON DIFFRACTION (OTED) ACKNOWLEDGMENTS REFERENCES
Vlll
117 1] 7 118 118 ]20 121 122 124 ] 24 124 125 127 128 129 130 130 130 131 131 132 133 135 135 135 137 13 8 138 140 140 140 140 142 143 144 148 148
4
Crystallographic Basis of Polytypism and Twinning in Micas Massimo Nespolo, SlavomiJ Durovic
IN1RODUCTION NOTATION AND DEFINITIONS The mica layer and its constituents Axial settings, indices and lattice parameters Symbols Symmetry and symmetry operations THE UNIT LAYERS OF MICA Alternative unit layers MICA POLYTYPES AND THEIR CHARACTERIZATION Micas as 0D structures SYMBOLIC DESCRIPTION OF MICA POLYTYPES Orientational symbols Rotational symbols RETICULAR CLASSIFICATION OF POLYTYPES: SPACE ORIENTATION AND SYMBOL DEFINITION LOCAL AND GLOBAL SYMME1RY OF MICA POLYTYPES FROM THEIR STACKING SyMBOLS Derivation of MDO polytypes The symmetry analysis from a polytype symbol RELATIONS OF HOMOMORPHY AND CLASSIFICATION OF MDO POLYTYPES BASIC S1RUCTURES AND POLYTYPOIDS. SIZE LIMIT FOR THE DEFINITION OF "POLYTYPE" Abstract polytypes Basic structures _ H1REM observations and some implications IDEAL SPACE-GROUP TYPES OF MICA POLYTYPES AND DESYMME1RIZATION OF LAYERS IN POLYTYPES CHOICE OF THE AXIAL SETTING GEOME1RICAL CLASSIFICATION OF RECIPROCAL LATTICE ROWS SUPERPOSITION S1RUCTURES, FAMILY S1RUCTURE AND FAMILY REFLECTIONS Family structure and family reflections of mica polytypes REFLECTION CONDITIONS NON-FAMILY REFLECTIONS AND ORTHOGONAL PLANES HIDDEN SYMME1RY OF THE MICAS: THE RHOMBOHEDRAL LATTICE TWINNING OF MICAS: THEORY Choice of the twin elements Effect of twinning by selective merohedry on the diffraction pattern Diffraction patterns from twins Allotwinning Tessellation of the hp lattice Plesiotwinning TWINNING OF MICAS. ANALYSIS OF THE GEOME1RY OF THE DIFFRACTION PATTERN Symbolic description of orientation of twinned mica individuals. Limiting symmetry Derivation of twin diffraction patterns Derivation of allotwin diffraction patterns IDENTIFICATION OF MDO POLYTYPES FROM THEIR DIFFRACTION PATTERNS Theoretical background Identification procedure IDENTIFICATION OF NON-MOO POLYTYPES: THE PERIODIC INTENSITY DISTRIBUTION FUNCTION PID in tenns of TS unit layers Derivation of PID from the diffraction pattern
ix
155 156 157 158 158 159 159 160 164 164 172 172 175 178 178 180 180 184 189 191 192 193 193 193 204 206 209 212 213 214 216 217 219 220 223 224 224 230 233 235 237 243 244 244 245 247 249 251
EXPERIMENTAL INVESTIGATION OF MICA SINGLE CRYSTALS FOR TWIN I POLYTYPE IDENTIFICATION Morphological study Surface microtopography Two-dimensional XRD study Diffractometer study APPLICATIONS AND EXAMPLES 24-layer subfamily: A Series I Class b biotite from Ambulawa, Ceylon 8A 2 (subfamily ~ Series O.Class a3) oxybiotit~ from Ruiz Peak, .New Mexico 1M-2M] oxyblOtlte allotwm ZT = 4 from RUiZ Peak, New Mexlco {3,6}[7 {3,6}] biotite plesiotwin from Sambagawa, Japan APPENDIX A. TWINNING: DEFINITION AND CLASSIFICATION APPENDIX B. COMPUTATION OF THE PID FROM A STACKING SEQUENCE CANDIDATE Symlnetry of the PID ACKNOWLEDGMENTS REFERENCES
5
252 252 252 254 256 257 257 258 262 262 267 270 271 272 272
Investigations of Micas Using Advanced Transmission Electron Microscopy Toshihiro Kogure
INTRODUCTION TEMS AND RELATED TECHNIQUES FOR THE INVESTIGATION OF MICA Transmission electron microscopy New recording media for beam-sensitive specimens Sample preparation techniques Image processing and simulation ANALYSES OF POLYTYPES , DEFECT STRUCTURES CONCLUSION ACKNOWLEDGMENTS REFERENCES
6
,
,.281 281 281 286 ,.287 ,288 289 299 310 31 0 310
Optical and Mossbauer Spectroscopy of Iron in Micas M. Darby Dyar
INTRODUCTION OPTICAL SPECTROSCOPY Current instrumentation Review of existing work Sunlmary MOSSBAUER SPECTROSCOPY (MS) Current instrumentation Recoil-free fraction effects Thickness effects Texture effects and other sources of error Techniques for fitting Mossbauer spectra Review of existing Mossbauer data Sumlnary COMPARISON OF TECHNIQUES CONCLUSIONS ACKNOWLEDGMENTS APPENDIX: Other techniques for measurement of Fe 3+/LFe in Micas X-ray ray photoelectron spectroscopy (XPS) Electron energy-loss spectroscopy (EELS) X-Ray absorption spectroscopy (XAS) REFERENCES ,
x
313 315 315 316 320 320 320 320 321 322 323 325 333 334 336 337 337 337 338 338 340
7
Infrared Spectroscopy of Micas Anton Beran
INTRODUCTION LATTICE VIBRATIONS Far-IR region Mid-IR regi on OH STRETCHING VIERATIONS Polarized measurements Quantitative water determination Hydrogen bonding Cation ordering OH-F replacement Dehydroxylati on mechanisms Excess hydroxyl NH4 groups ACKNOWLEDGMENTS REFERENCES
8
351 352 352 353 359 359 360 360 362 365 366 367 367 367 367
X-Ray Absorption Spectroscopy of the Micas Annibale Mottana, Augusto Marcelli, Giannantonio Cibin, and M. Darby Dyar
INTRODUCTION OVERVIEW OF THE XAS METHOD EXAFS XANES Experimental spectra recording Optimizati on of spectra Systematics AC KNOWLEDGMENTS REFERENCES
371 373 375 37 6 384 387 395 .404 .405
9 Constraints on Studies of Metamorphic K-Na White Micas Charles V. Guidotti, Francesco P. Sassi INTRODUCTION EFFECTS OF PETROLOGIC FACTORS ON WHITE MICA CHEMISTRy Important compOSitional vari ations Controls of mica composition by petrologic factors MAXIMIZING INFORMATION FROM MICA STUDIES : SAMPLE SELECTION CONSTRAINTS Petrologic studies Mine ralogic studies DISCUSSION Common failing s in petrology studies Common failings in mineralogy studies "Standard starting points" for the compositional variations of rock-forming dioctahedral and trioctahedral micas ACKNOWLEDGMENTS REFERENCES
Xl
41 3 .41 4 41 4 .41 8 .4 23 4 24 .42 8 440 .44 0 44I 44 1 443 444
10
Modal Spaces for Pelitic Schists James B. Thompson, Jr.
INTRODUCTION NOTATIONS AND CONVENTIONS THE ASSEMBLAGE QUARTZ-MUSCOVITE-BIOTITE-CHLORITE-GARNET. THE ASSEMBLAGE QUARTZ-MUSCOVITE-CHLORITEGARNET-CHLORITOID ASSEMBLAGES CONTAINING CHLORITOID AND BIOTITE OTHER MODAL SPACES ACKNOWLEDGMENTS APPENDIX : INDEPENDENT NET-TRANSFER REACTIONS REFERENCES
11
449 .450 .451 4 54 .455 .458 .458 .460 462
Phyllosilicates in Very Low-Grade Metamorphism: Transformation to Micas Peter Arkai
I.NTRODUCTION MAIN METHODS OF STUDYING LOW-TEMPERATURE TRANS FORMATIONS OF PHYLLOSILICATES XRD techniques TEM techniques ~AIN TRENDS OF PHYLLOSILICATE EVOLUTION AT LOW TEMPERATURE CURRENT PROBLEMS IN STUDYING PHYLLOSILICATE EVOLUTION AT THE LOWER CRYSTALLITE-SIZE LIMITS OF MINERALS REACTION PROGRESS OF PHYLLOSILICATES THROUGH SERIES OF METASTABLE STAGES CONCLUDING REMARKS ACKNOWLEDGMENTS REFERENCES
12
463 464 465 .466 .467 .469 472 .473 .474 .474
Micas: Historical Perspective Curzio Cipriani
INTRODUCTION PRESCIENTIFIC ERA THE EIGHTEENTH CENTURy THE NINETEENTH CENTURy Physical properties Crystallography Chemical composition THE TWENTIETH CENTURY Crystal chemistry Synthesis POLYTYPES SYSTEMATICS CONCLUSIONS REFERENCES APPENDIX I Present-day nomenclature of the mica group and its derivation APPENDIX II Other works consulted in preparation of this historical review XlI
4 79 4 79 .480 .483 4 83 485 .486 491 491 494 494 49 5 .496 497 .498 .499
1
Mica Crystal Chemistry and the Influence of Pressure, Temperature, and Solid Solution on Atomistic Models Maria Franca Brigatti Dipartimento di Scienze della Terra Università di Modena e Reggio Emilia, Via S. Eufemia, 19 I-41100 Modena, Italy
[email protected] Stephen Guggenheim Department of Earth and Environmental Sciences University of Illinois at Chicago 845 West Taylor Street, M/C 186 Chicago, Illinois 60607
[email protected] OVERVIEW The 2:1 mica layer is composed of two opposing tetrahedral (T) sheets with an octahedral (M) sheet between to form a “TMT” layer (Fig. 1a). The mica structure has a general formula of A M2-3 T4 O10 X2 [in natural micas: A = interlayer cations, usually K, Na, Ca, Ba, and rarely Rb, Cs, NH4, H3O, and Sr; M = octahedral cations, generally Mg, Fe2+, Al, and Fe3+, but other cations such as Li, Ti, V, Cr, Mn, Co, Ni, Cu, and Zn can occur also in mica species; T = tetrahedral cations, generally Si, Al and Fe3+ and rarely B and Be; X = (OH), F, Cl, O, S]. Vacant positions (symbol: ) are also common in the mica structure (Rieder et al. 1998). In the tetrahedral sheet, individual TO4 tetrahedra are linked with neighboring TO4 by sharing three corners each (i.e., the basal oxygen atoms) to form an infinite two-dimensional “hex agonal” mesh pattern (Fig. 1b). The fourth oxygen atom (i.e., the apical oxygen atom) forms a corner of the octahedral coordination unit around the M cations. Thus, each octahedral anion atom-coordination unit is comprised of four apical oxygen atoms (two from the upper and two from the lower tetrahedral sheet) and by two (OH) or F, Cl, O and S anions [the X anions, usually indicated as the OH or O(4) site]. The OH site is at the same level as the apical oxygen but not shared with tetrahedra. In the octahedral sheet, individual octahedra are linked laterally by sharing octahedral edges (Fig. 1c). The smallest structural unit contains three octahedral sites. Structures with all three sites occupied are known as trioctahedral, whereas, if only two octahedra are occupied [usually M(2)] and one is vacant [usually M(1)], the structure is defined as dioctahedral. The 2:1 layers, which are negatively charged, are compensated and bonded together by positively charged interlayer cations of the A site. The layer charge ideally is -1.0 for true micas and -2.0 for brittle micas. Thus, in true micas, the layer charge is compensated by monovalent A cations, whereas in brittle micas it is compensated primarily by divalent A cations. In this section, we consider and discuss the structural and chemical features of more than 200 micas. Most are true micas (146 trioctahedral and 55 dioctahedral). Brittle-mica crystal-structure refinements number about twenty, of which only three are dioctahedral (Tables 1-4, at the end of the chapter). Of the six simple polytypes first derived by Smith and Yoder (1956) and reported by Bailey (1984a, p. 7), only five (i.e., 1M, 2M1, 3T, 2M2, and 2O) have been found and studied by three-dimensional crystal-structure refinements. 1529-6466/02/0046-0001$10.00
DOI: 10.2138/rmg.2002.46.01
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Figure 1. (a) The 2:1 layer; (b) the “hexagonal” tetrahedral ring; (c) the octahedral sheet. For site nomenclature see text. a and b are unit cell parameters.
Most of the trioctahedral true-mica structures are 1M polytypes and a few are 2M1, 2M2, and 3T polytypes. In dioctahedral micas, the 2M1 sequence dominates, although 3T and 1M structures have been found. Brittle mica crystal-structure refinements indicate that the 1M polytype is generally trioctahedral whereas the 2M1 polytype is dioctahedral. The 2O structure has been found for the trioctahedral brittle mica, anandite (Giuseppetti and Tadini 1972; Filut et al. 1985) and recently was reported for a phlogopite from Kola Peninsula (Ferraris et al. 2000). The greatest number of the reported structures were refined from single-crystal X-ray diffraction data, with only a few obtained from electron and neutron diffraction experiments. Subsequent sections of this paper present short reviews pertaining to the description of phyllosilicates, an emphasis of the literature since the publication of MICAS, Reviews in Mineralogy, Volume 13, edited by S.W. Bailey (1984a), and a new analysis of the crystal chemistry of the micas. New formulae are presented to clarify how crystal chemistry affects the mica structure. Derivations of these formulae are provided in Appendix I. Also, please refer to other chapters in this volume that cover related topics. For example, see Zanazzi and Pavese for the behavior of micas at high pressure and high temperature.
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Treatment of the data and definition of the parameters used To achieve standardization, all data in Tables 1-4 (Appendix II) were re-calculated from unit-cell parameters and atomic coordinates reported by the authors of the original articles. Information concerning rock type and sample composition was obtained from the original works as well. Suspect refinements are discussed separately or not reported. Of more than 200 reported crystal-structure refinements, about twenty refinements show an agreement factor, R, greater that 9.0%. These structures are considered of poor quality and are not considered further. Several authors used symbols and orientations that differ from convention to describe geometric arrangements of the layer and the stacking sequence of mica polytypes (e.g., Radoslovich 1961; Durovíc 1994; Dornberger-Schiff et al. 1982). To make inter-structure comparisons of features easier, however, it is advantageous to define briefly the site nomenclature adopted and the parameters used to describe and characterize layer geometry. The direction defined by the stacking of 2:1 units defines the [001] direction (i.e., the c axis), whereas the periodicity of the infinite two-dimensional sheets defines [100] and [010] directions (i.e., a and b translations). The actual value of the repeat distance in the [001] direction, as well as lateral a and b parameters, depends on several factors, such as the layer stereochemistry and polytypism (i.e., c ∼ 10 Å × n, where n identifies the number of layers involved in the stacking sequence). The sitenomenclature scheme adopted here starts from the nomenclature generally used for the 2:1 layer of the 1M polytype in C2/m symmetry: T denotes the four-coordinated site, M(1) and M(2) indicate six-coordinated sites with (OH) groups in trans- and cisorientation, respectively, A refers to the interlayer cation, O(1) and O(2) represent the basal tetrahedral oxygen atoms, O(3) is the apical oxygen atom, and O(4) refers to the (OH), F, Cl, S or O anions (Fig. 1a). The number of sites per unit cell is: T = 8; M(1) = 2; M(2) = 4; A = 2; O(1) = 8; O(2) = 4; O(3) = 8; O(4) = 4. The site nomenclature for other structural variants can be derived from this nomenclature by changes that relate to spacegroup differences and to the number of 2:1 layers per unit cell. The definition of parameters reported in Tables 1-4 (Appendix II) follows. For a more extensive review on definition and structural significance of these parameters, see Bailey (1984b) and references therein. Cation-anion bond lengths: (i) tetrahedral 〈T–O〉; (ii) octahedral 〈M–O,OH,F,Cl,S〉 for both M(1) and M(2) sites; and (iii) interlayer 〈A–O〉. Mean bond lengths were compared to those of the original papers and vacant-site distances determined (i.e., vacancy-to-anion distances). The tetrahedral Oapical–T–Obasal angles were used to obtain the tetrahedral flattening angle, τ = ∑3i=1 Oapical–T–Obasal)i/3. The internal angles of the tetrahedral ring were used to determine the tetrahedral rotation angle, α = ∑6i=1 α i / 6 where αi = |120° – φi|/2 and φi is the angle between basal edges of neighboring tetrahedra articulated in the ring. Basal oxygen-plane corrugation, Δz, was determined by Δz = (zObasal(max) – zObasal(min)) × c sinβ. The thickness of the tetrahedral and octahedral sheets was calculated from oxygen z coordinates of each polyhedron, including the OH group (or other X anions). The interlayer separation was obtained by considering the tetrahedral basal oxygen z coordinates of adjacent 2:1 layers. The octahedral flattening angle ψ was calculated from
4
Brigatti & Guggenheim ⎛ octahedral thickness ⎞ ψ = cos −1 ⎜ ⎟ ⎝ 2 × M − O, OH, F, Cl, S ⎠
Tetrahedral cation atomic coordinates, taken from the original reference, were transformed from fractional to Cartesian to calculate the Layer Offset, the Intralayer Shift, and the Overall Shift. The Layer Offset is based on the displacement of the tetrahedral sheet across the interlayer from one 2:1 layer to the next, which should be equal to zero in the ideal mica structure. The Intralayer Shift is the over-shift of the upper tetrahedral sheet relative to the lower tetrahedral sheet of the same 2:1 layer. The Overall Shift relates to both effects. In true micas, the tetrahedral mean bond distance varies from 1.57(1) Å in boromuscovite-2M1 (Liang et al. 1995; Table 4) to 1.750(2) Å in an ordered (Al vs. Si) ephesite-2M1 (Slade et al. 1987; Table 1d); in brittle micas, the 〈T–O〉 mean bond distance varies from 1.620(2) to 1.799(2) Å, both values are from anandite-2O (Filut et al. 1985; Table 3a). Octahedral mean bond length ranges from about 1.882(1) Å in an ordered ferroan polylithionite-1M (Guggenheim and Bailey 1977; Brigatti et al. 2000b; Table 1b) to 2.236(1) Å in anandite 2O (Filut et al. 1985; Table 3a). The radius of the vacant M(1) site in dioctahedral micas (〈M(1)–O〉) varies from 2.190 to 2.259 Å. The shortest 〈A–O〉inner distance occurs in clintonite (〈A–O〉inner = 2.397(2) Å; Alietti et al. 1997, Table 3a), whereas the longest distance occurs in nanpingite and synthetic Cs-tetra-ferri-annite (〈A– O〉 inner of ∼ 3.370 Å; Ni and Hughes 1996 and Mellini et al. 1996, Tables 1c and 1a, respectively). These data show the great variability in bond distances which may be ascribed not only to the local composition but also to the constraints of closest packing within the layer and the confinement of the octahedra between two opposing tetrahedral sheets. We consider the compositional and topological relationships in the following analysis. End-member crystal chemistry: New end members and new data since 1984 Boromuscovite. Boromuscovite was first reported by Foord et al. (1991). The mineral, precipitated from a late-stage hydrothermal fluid (T: 350-400°C; P: 1-2 kbar), occurred in the New Spaulding Pocket, Little Three Mine pegmatite dike (Ramona district, San Diego County, California), as a fine-grained coating of quartz, polylithionite, microcline and topaz. The mineral was found also in elbaite pegmatite at Recice near Mové Mesto na Morave, western Moravia, Czech Republic (Liang et al. 1995; Novák et al. 1999). Relatively high B contents were also reported for muscovite and polylithionite from polylithionite-rich pegmatites of Rozná and Dobrá Voda, Czech Republic (Cerny et al. 1995), for polylithionite-2M1 from Recice (Novák et al. 1999), and for muscovite from metapegmatite at Stoffhütte, Koralpe, Austria (Ertl and Brandstätter 1998). Boromuscovite (Foord et al. 1991) has the general structural formula of KAl2 (Si3B) O10(OH)2, in which [4]Al is replaced by [4]B relative to muscovite. The composition of Little Three Mine boromuscovite is (K0.89Rb0.02Ca0.01)(Al1.93Li0.01Mg0.01)(Si3.06B0.77Al0.17) O9.82F0.16(OH)2.02, whereas the composition of Recice boromuscovite shows a slightly lower [4]B content: (K0.89Na0.01)(Al1.99Li0.01)(Si3.10B0.68Al0.22)O10F0.02(OH)1.98. The unit cell parameters, very similar in natural and synthetic crystals (Schreyer and Jung 1997), are significantly smaller than those reported for muscovite (a = 5.075(1), b = 8.794(4), c = 19.82(3) Å, β = 95.59(3)° and a = 5.077(1), b = 8.775(3), c = 10.061(2) Å, β = 101.31(2)° for Little Three Mine boromuscovite-2M1 and boromuscovite-1M, respectively). A boromuscovite structure determination is complicated by the fine-grained nature
Mica Crystal Chemistry and Influence of P-T-X on Atomistic Models
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of the mineral and by the presence of the mixture of 1M and 2M1 polytypes. Nonetheless the crystal-structure determination of a mixture of 83 wt % boromuscovite-2M1 and 17 wt % boromuscovite-1M from Recice was attempted using a coupled Rietveld-staticstructure energy-minimization method (Liang et al. 1995). Although the high standard deviation for calculated parameters suggests caution in the analysis of crystal chemical details, Liang et al. (1995) indicated that: (i) boron is uniformly distributed between the two polytypes, (ii) 〈T–O〉 distances correspond well with the B-content at the corresponding T-sites, namely 〈T–O〉 distances linearly decrease as B occupancy increases, and (iii) in the 2M1 polytype, slight differences between 〈T(1)–O〉 and 〈T(2)– O〉 distances may imply a B preference for the T(1) site (Table 4). The 11B MAS NMR spectra showed a single, symmetric and narrow line (about 150 Hz wide) at 20.7 ppm. The width was interpreted as possibly relating to the coordination for B with a nearsymmetrical disposition of anions (Novák et al. 1999). Clintonite. Clintonite is the trioctahedral brittle mica with ideal composition of Ca(Mg2Al)(SiAl3)O10(OH)2. This structure violates the Al-avoidance principle of Loewenstein (1954). It crystallizes in H2O-saturated Ca-, Al-rich, Si-poor systems under wide P-T conditions. Clintonite, usually found in metasomatic aureoles of carbonate rocks, is rare in nature because crystallization is limited to environments characterized by both alumina-rich and silica-poor bulk-rock chemistry and very low CO2 and K activities (Bucher-Nurminen 1976; Olesch and Seifert 1976; Kato et al. 1997; Grew et al. 1999). The 1M polytype and 1Md sequences are the most common forms. The 2M1 form is rare (Akhundov et al. 1961) and no 3T forms have been reported. Many 1M crystals are twinned by ±120° rotation about the normal to the {001} cleavage. Such twinning causes extra spots on precession photographs that simulate an apparent three-layer periodicity (MacKinney et al. 1988). Subsequent to an extensive review of brittle micas (Guggenheim 1984), additional crystal-chemical details of clintonite-1M (space group C2/m) were reported by MacKinney et al. (1988) and Alietti et al. (1997). These studies confirmed that natural clintonite crystals do not vary extensively in composition: (i) the octahedral sites contain predominant Mg and Al with Fe2+ to ≤7% of the octahedral-site occupancy; (ii) the extent of the substitution [4]Al-1 [6]Mg-2 [4]Si [6](Al, ), which involves the solid solution of trioctahedral with dioctahedral Ca-bearing brittle micas, is very limited; (iii) Fe3+ content involves tetrahedral site occupancy, but at low ( NH4 and with (001) spacing values intermediate between illite and tobelite are referred to as “NH4-rich illite.” They occur in hydrothermal environments (Sterne et al. 1982; Higashi 1982; Von Damm et al. 1985; Wilson et al. 1992; Bobos and Ghergari 1999); in black-shales (Sterne et al. 1984); in regionally metamorphosed carbonaceous pelites (Juster et al. 1987; Daniels et al. 1996; Liu et al. 1996) and in diagenetic environments (Duit et al. 1986; Lindgreen et al. 1991; Drits et al. 1997). Tobelite-like layers are often found in interstratified dioctahedral minerals having non-expandable (mica-like) and expandable (smectite-like and/or vermiculite-like) layers. Drits et al. (1997) demonstrated that, in interstratified illite-smectite minerals from North
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Sea oil-source rocks, the mica-like component contains both K-rich end-member (illite) and NH4-rich end-member (tobelite) layers. The amount and the distribution of fixed K and NH4 was determined by a peak profile-fitting procedure on experimental powder (X-ray) diffraction features (Drits et al. 1997; Sakharov et al. 1999). Synthetic micas with unusual properties Cesian tetra-ferri-annite and cesian annite. Fe-rich micas have the capacity to contain radioisotopes, such as 135Cs and 137Cs. The study of these materials has been a promising direction of mica research over the last few years; see, for example, Mellini et al. (1996), Drábek et al. (1998), and Comodi et al. (1999). The cesian-tetra-ferri-annite crystal structure was studied by Mellini et al. (1996) and by Comodi et al. (1999) at ambient conditions and at high P-T conditions. Cs-tetra-ferri-annite crystallizes in the 1M polytype (C2/m space group). It has the largest unit-cell volume reported to date for 1M micas and coordination polyhedra are undistorted (Table 1a). The tetrahedral rotation angle (α = 0.2°), and the octahedral-distortion parameter, δ, involved with the counterrotation of upper and lower oxygen triads are near 0° (δM(1) = 0; δM(2) = 0.2°), thus suggesting a nearly undistorted layer with limited internal strain. No detectable internal strain based on such parameters (e.g., α and δ) was observed at high pressure (to 47 Kbar) and temperature (to 582°C). Above 450°C, in air, the reduction of the unit cell volume is related to the loss of H atoms required to balance the layer charge after oxidation of octahedral iron in the M(2)-cis site. Li for K exchange in interlayer sites. Volfiger and Robert (1979, 1980) and Robert et al. (1983) suggested that, in synthetic trioctahedral micas, anhydrous Li can exchange for K in interlayer sites. Although the crystal quality obtained from the run products did not allow a complete crystal structure determination, they indicated, on the basis of the results obtained by infrared and powder X-ray analyses, that Li is located in the interlayer in a pseudo-octahedral cavity. This cavity is partly defined by the hexagonal ring of one layer and by the basal oxygen atoms of two tetrahedra in the adjacent layer. The Li solubility limit was estimated to be a function of the relation: Li/(Li+K)max = 2 [4][Al/(Al+Si)]2. Tetrahedral Al for Si substitution is essential to minimize the electrostatic repulsion between tetrahedral cations and Li, and therefore to create favorable cavities to host Li. Robert et al. (1983) found that the unit cell parameter, c, decreases with K for Li substitution whereas the b parameter slightly increases. In Li-exchanged synthetic paragonite-2M1 and muscovite-2M1, repulsive forces between O atoms across the interlayer region cause an interlayer overshift, resulting in an anomalously high basal spacing and smaller monoclinic β angle (Keppler 1990). Complete and rapid Li exchange in the interlayer sites was obtained for natural phlogopite, ferroan phlogopite and muscovite using “cryptand [222]” as a complexing agent, and dioxane as a solvent (Bracke et al. 1995). Powder X-ray diffraction suggests that the interlayer spacing changes with replacement of K by Li + H2O. The original reflection at 9.93 Å loses intensity progressively and an additional reflection at 11.78 Å appears. EFFECT OF COMPOSITION ON STRUCTURE Tetrahedral sheet In some naturally occurring true micas, Si nearly fills all the tetrahedral sites (e.g., polylithionite, tainiolite, norrishite, and celadonite), whereas in the most common mica species (i.e., muscovite and phlogopite) Al substitutes for Si in a ratio near 1:3. In some true micas and brittle micas, the Al for Si substitution corresponds to a ratio of Al:Si = 1:1 (e.g., ephesite, preiswerkite, siderophyllite, margarite, and kinoshitalite), whereas the
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trioctahedral brittle mica, clintonite, has an unusually high Al content with a ratio of Al:Si of 3:1 (Bailey 1984a,b). Evidence of Fe3+ tetrahedral substitution was reported on the basis of optical observations (e.g., Farmer and Boettcher 1981; Neal and Taylor 1989), spectroscopic studies (e.g., Dyar 1990; Rancourt et al. 1992; Cruciani et al. 1995) and crystal-structure refinement (Guggenheim and Kato 1984; Joswig et al. 1986; Cruciani and Zanazzi 1994; Brigatti et al 1996a, 1999; Medici 1996). However, only in tetra-ferriphlogopite, tetra-ferri-annite and anandite is Fe3+ the only Si-substituting cation, with a Fe3+:Si ratio near 1:3 (e.g., Giuseppetti and Tadini 1972; Semenova et al. 1977; Hazen et al. 1981; Filut et al. 1985; Brigatti et al.1996a,b, 1999; Mellini et al. 1996). Thus, the 1:3 ratio appears to be the greatest Fe3+ tetrahedral substitution possible for the micas. Two mica end-members contain B (boromuscovite; Liang et al. 1995) and Be (bityite; Lin and Guggenheim 1983), and some synthetic micas contain Ge in the T site (Toraya and Marumo 1981; Toraya et al. 1978a,c). Most mica structures display a disordered distribution of tetrahedral cations, with the exception of some brittle mica species, such as margarite (Guggenheim and Bailey 1975, 1978; Kassner et al. 1993), anandite (Giuseppetti and Tadini 1972; Filut et al. 1985) and bityite (Lin and Guggenheim 1983) and a few true micas (e.g., polylithionite-3T, Brown, 1978; muscovite-3T, Güven and Burnham 1967). Some true micas with an apparent ordered distribution of cations in the tetrahedra are those with a high R value and therefore these structures should be considered tentative. Hazen and Burnham (1973) related 〈T–O〉 distances of trisilicic micas to tetrahedral composition by the linear relationship (xAl and xSi represent Al and Si apfu, respectively) ⎛ x Al 〈T − O〉 ( A ) = 0.163 ⋅ ⎜ ⎝ x Al + xSi
⎞ ⎟ + 1.608 ⎠
A more general relationship derived here including both trioctahedral and dioctahedral true and brittle micas (Tables 1-4, Appendix II) between tetrahedral mean bond distances 〈T–O〉 and tetrahedral chemistry (in apfu) is: T − O (Å) = 1.607 + 4.201 ⋅ 10 −2
[4]
Al + 7.68 ⋅10−2[4] Fe
(correlation coefficient, r = 0.965) In the regression analysis, structures containing B, Be, and Ge in tetrahedral sites were not considered, as well as structures with symmetry lower than ideal owing to tetrahedral cation ordering (differences in 〈T–O〉 values greater than 5σ). Only structures containing tetrahedral Si, Al, and Fe were examined. Geometrical considerations of tetrahedral distortion parameters have been considered earlier (e.g., Drits 1969, 1975; Takéuchi 1975; Appelo 1978; Lee and Guggenheim 1981; Weiss et al. 1992). We further discuss these relationships here and relate them to layer composition on the basis of data from a large number of structure determinations. A crystal chemical study of the τ parameter is complex. In an ideal tetrahedron τ is equal to arcos (-1/3) ≅ 109.47°. For non-ideal cases, however, τ was found to be affected by tetrahedral content, increasing as Si increases (Takéuchi 1975) relative to Al. The τ value can deviate from its ideal value as a function of the relative position along c for the basal oxygen atoms with respect to the tetrahedral cation and with respect to the mean basal-edge length and the mean tetrahedral-edge value. These conclusions are based on the linearized topology of the tetrahedron. Several simple models of deformation are considered here (Fig. 2) and only modes (3) and (4) were found to affect the τ value. All dependences (over displacement from an ideal undeformed configuration) of order
Mica Crystal Chemistry and Influence of P-T-X on Atomistic Models
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Figure 2. Geometrical considerations over the dependence of τ from tetrahedron vertex and center displacement. The relationships in the legend have been obtained from a linearized geometrical model. k and e indicate the displacement and the tetrahedron edge length, respectively.
greater than one are ignored. The model, thus, provides results in good agreement with structural data only if displacements are small relative to the characteristic length of the system (i.e., the tetrahedral edge). Figure 3 shows the variations of τ vs. [4]Si content. Although the increase of τ with Si is confirmed, there are two different linear trends, one trend for true and one trend for brittle micas. Brittle micas show τ values greater than expected if just the composition of the tetrahedron is considered. Although this simple model ignores cation ordering, on the basis of geometrical considerations derived before (Fig. 2), the higher τ values may be explained by the increase in the electrostatic attraction of basal oxygen atoms by the high-charge interlayer cation and by the concomitant increase in repulsion between the interlayer cation and the tetrahedral cation. Note, for example, that kinoshitalite usually tends to approach true micas in composition. Samples of kinoshitalite and ferrokinoshitalite (Guggenheim and Kato 1984; Brigatti and Poppi 1993; Guggenheim and Frimmel 1999) contain significant amounts of monovalent K in substitution for Ba, whereas, kinoshitalite refined by Gnos and Armbruster (2000), marked by an arrow in Figure 3, has nearly complete interlayer Ba occupancy and a larger τ value. [4]
To better relate how the interlayer cation affects τ, we have developed a simple electrostatic model. The model is comprised of four tetrahedral oxygen atoms, with the tetrahedral and the interlayer cations located at the center of the tetrahedron and in the
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Figure 3. Relationships between the tetrahedral flattening angle, τ, and Si content in tetrahedral coordination as determined by microprobe analysis. Symbols used: filled circle = annite; filled circle, x-hair = magnesian annite; open circle = phlogopite; open circle, x-hair = ferroan phlogopite; filled circle, dotted = tetra-ferri-annite; open circle, dotted = tetra-ferriphlogopite; open square = polylithionite; filled square = trilithionite; filled square, x-hair = siderophyllite; open square, x-hair = ferroan polylithionite; filled hexagon, x-hair = norrishite; crosses = preiswerkite; open diamond = muscovite; open diamond, xhair = nanpingite; filled diamond = paragonite; filled diamond x-hair = boromuscovite; open triangle up = clintonite; filled triangle up, x-hair = ferrokinoshitalite; filled triangle up = kinoshitalite. The sample arrowed is kinoshitalite by Gnos and Armbruster (2000). For details see text.
middle of the interlayer, respectively. The oxygen atoms were placed at the vertices of an undistorted tetrahedron with a tetrahedral volume equal to that as considered above. A uniform displacement along the [001] direction was then imposed on the basal oxygen atom plane and the electrostatic energy associated with the system was then derived as a function of this displacement. Finally, the displacement which minimizes the electrostatic energy of the system was calculated and compared with the value obtained for a system identical to that described, but differing in the formal charge of the interlayer cation which was arbitrarily set equal to one. Therefore, the model takes into consideration the differences in energy between the two configurations described, not the total energy. The displacement obtained was used to “isolate” the τ value from the influence of the divalent interlayer cation. The τ values of tetrahedrally disordered brittle micas which was thus “isolated” (i.e., τ*) follow the same trend defined for true micas, confirming the influence of interlayer cations on τ (Fig. 4). Unlike other models reported in the literature (e.g., Giese 1984), our model introduces only the Coulombic term and does not consider the repulsive energy or van der Waals interactions. This simplification, as Giese (1984) correctly noted, does not produce correct energy values. For this reason, energy differences between structural systems, which are characterized by the same repulsive energy, were considered. The charge at each position was determined from chemical data and from structural constraints.
Mica Crystal Chemistry and Influence of P-T-X on Atomistic Models
Figure 4. Relationship between τ* and Si tetrahedral content . τ* refers to the τ value “isolated” from the influence of the interlayer cation for the brittle micas clintonite and kinoshitalite. Regression equation: τ* (°) = 2.920 × [4]Si + 101.98, r = 0.950. Symbols and samples as in Figure 3.
Figure 5. Bond energy between tetrahedral cation and tetrahedral basal oxygen atoms compared with the bond energy between interlayer cation and tetrahedral basal oxygen atoms. Symbols and samples as in Figure 3.
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Figure 5 relates the bond energy between the tetrahedral basal oxygen atoms vs. tetrahedral cations (〈T–Obasal〉) and between the basal oxygen atoms vs. interlayer cations (〈A–Obasal〉), respectively. In brittle mica species, the distance between the tetrahedral cation and the basal oxygen atom plane increases, owing to the interaction with the interlayer cation. In this way the increase in 〈T–Obasal〉 bond energy is partly compensated by a decrease in bond energy between the cation and the oxygen atoms of the basal plane. The displacement of the tetrahedral cation from its ideal position can be evaluated (see Appendix I for derivation) from the tetrahedral displacement parameter, Tdisp.: Tdisp. =
T − Obasal
2
O − Obasal − 3
2
−
(T − O ) apical
3
Tdisp. was calculated for all structures starting from observed distances, and then plotted against the τ value observed (Fig. 6).
Figure 6. Mean τ value vs. the displacement of the T cation from the center of the tetrahedron mass (Tdisp.). Symbols and samples as in Figure 3.
The plane of basal oxygen atoms approaches the tetrahedral cation in flattened tetrahedra (the distance between the tetrahedral cation and the basal oxygen-atom plane decreases with respect to the T–Oapical distance), whereas the tetrahedral cation shifts toward the tetrahedral apex (the distance between the tetrahedral cation and basal-oxygen atom plane increases with respect to the T–Oapical distance) in elongated tetrahedra. In preiswerkite and in boromuscovite the tetrahedral cation shifts from its ideal position toward the plane of basal oxygen atoms (τ < 109.47°). In the brittle mica clintonite, the tetrahedral cation more closely approaches the center of the tetrahedron (τ ≈ 109.47°), whereas in other micas the cation shifts toward the tetrahedral apex (τ > 109.47°). The maximum shift was observed in norrishite (Tyrna and Guggenheim 1991) and in polylithionite (Takeda and Burnham 1969).
Mica Crystal Chemistry and Influence of P-T-X on Atomistic Models
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Figure 7. Plot of τ vs. 〈O-O〉basal. Symbols and samples as in Figure 3.
In addition, τ reflects an adjustment for the misfit between the tetrahedral sheet and the octahedral sheet (the regression coefficient, r, of τ vs. the difference between mean basal tetrahedral edges and mean octahedral triads is r = 0.92). Furthermore, as the mean 〈O–O〉 basal distance decreases, the tetrahedral cation moves away from the basal oxygen-atom plane. Thus, τ increases in value (Fig. 7). The deviation of the parameters for clintonite and kinoshitalite from the trend for true micas further suggests that there is a significant influence of the interlayer cation on the value of τ. In conclusion (i) τ increases as the distance between the tetrahedral cation and the basal oxygen-atom plane increases from its ideal value; (ii) τ increases as 〈O–O〉basal decreases, thus reflecting a dimensional adjustment between the tetrahedral sheet and octahedral sheet; and (iii) τ increases with [4]Si content. Differences between τ values of brittle micas from the true micas are related in part to electrostatic features. It is useful to understand why the tetrahedral cation moves from its ideal position. Drits (1969) stated that “the position of the tetrahedral cation depends not only on the degree of substitution of Si by Al in the tetrahedra (Brown and Bailey 1963), but also in the position and distribution in compensating positive charges.” This assumption is related to electrostatic forces in the following way (see Appendix I for derivation): ΔE1 =
−3 ⋅ q T ⎛ −9⋅ q T ⎞ −⎜ ⎟ d TπOb ⎝ T − O apical ⎠
⎛ ΔE 2 = ⎜⎜ ⎜ ⎝
ΔE3 =
q T ⋅ (q A / 4 )
(IS / 2 + d TπOb )2 +
O apical − Oapical
2
⎞ ⎛ ⎟ −⎜ ⎟ ⎜ ⎟ ⎜ ⎠ ⎝
⎛ q 2T q 2T −⎜ IS + 2 ⋅ d TπOb ⎜⎝ IS + 2 / 3 ⋅ T − Oapical
(
⎞ ⎟ ⎟ ⎠
)
q T ⋅(q A / 4)
(IS / 2 + (T − O )/ 3) + O 2
apical
apical
− Oapical
2
⎞ ⎟ ⎟ ⎟ ⎠
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Brigatti & Guggenheim
where qT and qA are the tetrahedral and interlayer charges, respectively; IS is the interlayer separation; dTπOb is the distance of the tetrahedral cation from the basal oxygen atom plane; 〈Oapical–Oapical〉 is the distance between apical oxygen atoms; and T–Oapical is the distance between the tetrahedral cation and apical oxygen atom. E1 relates the electrostatic energy between the tetrahedral cation and the basal oxygen atoms. E2 is the electrostatic energy between the tetrahedral cation and interlayer cation. E3 considers the repulsion between tetrahedral cations of two opposing tetrahedra across the interlayer (Fig. 8). ΔE1 (ΔE2, ΔE3) is the variation of E1 (E2, E3) values in the actual structure and in an ideal structure with the tetrahedral cation ideally spaced from the basal and apical oxygen atoms. ΔE1, ΔE2, and ΔE3 were derived by considering the set of charges represented in Figure 9. This specific arrangement of charges was developed to describe the electrostatic interactions between the basal oxygen atoms of the tetrahedron and interlayer cation. All planes of atoms (i.e., the plane of interlayer cations, the plane of basal oxygen atoms and the plane of tetrahedral cations) can be described through a rigid displacement of the simple charge distribution in Figure 9, thus the energy involving the oxygen-atom plane differs, to a first approximation, from the energy related to the distribution in Figure 9 by just a scale factor. The objective of our model is to describe the factors influencing the interlayer cation displacement from its “ideal” position. However, we consider the difference in energy between the actual structure configuration and that characterized by a tetrahedral cation-basal oxygen atom plane distance, which is equal to (T–Oapical)/3. All terms in energy which do not include that distance, are therefore excluded in this derivation because they must be equal in both the configurations considered. In conclusion, differences in energy among configurations which vary for very small displacements of charge can be very useful. Our model considers van der Waals and repulsion energies equal in both configurations to simplify the calculation.
Figure 8. Relationship between ΔE2 + ΔE3 vs. ΔE1. For the definition of energy E1, E2, and E3, see text. Regression equation [(ΔΕ2 + ΔΕ3) = -1.099 ΔΕ1 + 1.26 × 10-3 ; r = 0.997). Symbols and samples as in Figure 3.
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Figure 9. The set of charges used to derive ΔE1, ΔE2, and ΔE3.
Figure 8 clearly shows that an increase in the electrostatic energy associated with an increase in the tetrahedral cation-basal oxygen atom distance is compensated by a reduction in the repulsion between the interlayer cation and the tetrahedral cation and the tetrahedral-tetrahedral cations (sited in adjacent layers). Given the high correlation coefficient (r = 0.997), the relation may be useful as a predictive tool. The basal oxygen atom plane corrugation effect (Δz) produces an out-of-plane twisting of tetrahedra about the bridging basal oxygen atom in the [110] tetrahedral chain and a shortening of the distance between apical oxygens along the octahedral edge parallel to the (001) plane. Lee and Guggenheim (1981) demonstrated that the corrugation of the basal oxygen atom plane reflects differences in distance between apical oxygen atoms linked to octahedra of different size. Thus Δz is limited in trioctahedral micas with M(1) ≈ M(2) in size, whereas it shows higher values in dioctahedral micas with M(1) >> M(2) in size. Differences in Δz are related to the linkage of the tetrahedral sheet by apical oxygen with octahedral sites different in size. A strong relationship between Δz and ΔM [ΔM = 〈M–O〉max – 〈M–O〉min] for a structure is evident in Figure 10. This result confirms that differences in octahedral site dimensions play an important role over tetrahedral basal oxygen-plane corrugation [regression equation: Δz (Å) = 0.647 × ΔM; r = 0.984]. Figure 11 shows the effect of Al octahedral content ([6]Al) on Δz. Where [6]Al occupancy is less than 1 apfu, Δz is approximately zero (trioctahedral true and trioctahedral brittle micas). In trioctahedral Li-rich micas (polylithionite, trilithionite and siderophillite) and in preiswerkite, [6]Al occupancy is nearly 1 apfu and Δz is as large as 0.15 Å. A Δz of ≤0.24 Å is observed for dioctahedral micas for which [6]Al occupancy reaches 2 apfu. Al is a cation of relatively small size. For micas with significant amounts of octahedral Al and where Al ordering occurs, differences in size between octahedral sites are enhanced and the value of Δz increases. Such differences also occur for micas with a low charge cation (e.g., Li+ in trioctahedral polylithionite) or by vacancies (i.e., in dioctahedral micas), where charge balance occurs within the octahedral sheet only. Tetrahedral rotation and interlayer region The dimensions of an ideal octahedral sheet in the (001) plane are commonly less than those of an ideal and unconstrained tetrahedral sheet. Thus, to obtain congruence, the difference in size of the tetrahedral and octahedral sheets must be adjusted by any one or more of the following: (i) in-plane rotation of adjacent tetrahedra in opposite directions about c* (parameter α); (ii) thickening of the tetrahedra (parameter τ), and (iii) a flat-
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Figure 10. Relationship between the tilting of the basal oxygen plane, Δz and ΔM [ = 〈M–O〉max – 〈M–O〉min ]. Regression equation: Δz (Å) = 0.647 × ΔM; r = 0.984. Symbols and samples as in Figure 3.
Figure 11. Δz (Å) vs. the octahedral Al content determined by microprobe analysis. The arrow indicates the dioctahedral chromiumrich mica (Evsyunin et al. 1997) which presents an unusual chemical composition characterized by an important [6]Cr for [6]Al substitution. Symbols and samples as in Figure 3.
tening of the octahedra (parameter ψ) to lengthen the octahedral edges (Mathieson and Walker 1954, Newnham and Brindley 1956; Zvyagin 1957; Bradley 1959; Radoslovich 1961; Radoslovich and Norrish 1962; Brown and Bailey 1963; Donnay et al. 1964, Bailey 1984b, Lee and Guggenheim 1981). McCauley and Newnham (1971) specified by multiple regression analysis that, although the α value is largely controlled by the tetrahedral-octahedral sheet lateral misfit (∼90%), it also reflects the field strength of the interlayer cation. Toraya (1981) observed
Mica Crystal Chemistry and Influence of P-T-X on Atomistic Models
21
two linear relationships, between α and the difference in length of the octahedral and tetrahedral sheets along the b axis, i.e., α = c1 (2√3eb -3√2do) + c2 (where do is the mean octahedral cation-anion distance, eb is the mean basal edge length of a tetrahedron, 2√3eb and 3√2do are the lengths along the b axis, in the idealized form, of the tetrahedral and octahedral sheet, respectively; c1 and c2 are the regression coefficients, c1 = 35.44 and 12.58, c2 = -11.09 and 4.30 for silicate and germanate micas, respectively). Weiss et al. (1992) used a different dataset and the same assumption of Toraya and found, for Si-rich micas, different values for c1 and c2 (c1 = 25.9 and c2 = -5.0).
Figure 12. α determined by structure refinement vs. α calculated by regression equation α = 25.9 (2√3eb -3√2do) – 5.0 (Weiss et al. 1992). Symbols as in Figure 2. The plot reports only structures published after 1992, i.e., structures not considered in the predictive equation of Weiss et al. (1992).
The calculated α values using the equation of Weiss et al. (1992) and data published after 1992 (i.e., not used to derive the equation) vs. observed α values are consistent mostly in the range of 7-9°, whereas the correspondence is lower for smaller and larger angles (Fig. 12). Weiss et al. (1992) also predicted the α value from sheet composition using a vector-representation grid. They calculated a mean tetrahedral distance, d (T–O), and a mean octahedral bond distances, d (M–A) (where A is any anion), from equations d (T–O) = Σ di (T–O)calc × xI d (M–O) = Σ di (M–O)calc × xi d (M–OH) = Σ dI (M–OH)calc × xi where di (T–O), di (M–O) and di (M–OH) are the calculated mean bond lengths for cations in tetrahedral and octahedral coordination, respectively, and xi represents the atomic fraction of each cation. To better understand the role of tetrahedral-octahedral lateral misfit for 1M, 2M1,
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2M2 and 3T polytypes, we have developed a geometric model (see Appendix I). According to this model, α is equal to ⎛1 / 3 + k ⋅ 4 / 3 − k 2 ⎞ α = tan −1 ⎜ ⎟ − 60 k 2 −1 ⎝ ⎠
and simplifying ⎛ 3 ⎞ α = cos −1 ⎜ ⋅ k⎟ ⎝ 2 ⎠
where k is the ratio between the 〈O–O〉 octahedral triads (〈O–O〉unshared) and 〈O–O〉 tetrahedral basal edges, 〈O–O〉basal. 〈O–O〉unshared very closely corresponds to b/3, b√3/3 and a/3 for trioctahedral-1M (and -2M1), -3T, and -2M2 polytypes, respectively, thus indicating that the “rigid” octahedral sheet primarily determines the unit-cell lateral dimensions of trioctahedral micas. This relationship is obtained with “rigid” tetrahedra and deformation involves only the “hexagonal” silicate ring. Therefore, the deformation of the octahedron and tetrahedron influences the α value only by affecting the tetrahedral and octahedral lengths as given in the formula above. Figure 13 reports α values observed vs. α values thus calculated. The correspondence appears excellent (r = 0.994), although the model could be improved by calculating all the six-ring tetrahedral angles and then averaging. The relationship between αcalculated and k is not linear and that structures which primarily deviate are Li-rich micas with octahedral ordering in the M(2) and M(3) sites. This geometric relationship is useful also to evaluate the influence of composition over α. The mean basal tetrahedral edge depends on tetrahedral cation stereochemistry
Figure 13. α determined by structure refinement vs. α calculated from the equation: α = cos-1 ( 3 2 ⋅k ) where the k is the ratio between octahedral triads (〈O–O〉unshared) and tetrahedral basal edges(〈O–O〉basal). Symbols and samples as in Figure 3.
Mica Crystal Chemistry and Influence of P-T-X on Atomistic Models
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([4]Al and [4]Fe in apfu, r = 0.970) by: 〈O − O〉 basal = 2.581+ 8.836 ⋅10 −2 ×
[4 ]
Al + 0.164
[4 ]
Fe
whereas the mean length of the octahedral triads is well fitted by the following expression ([6]Al, [6]Fe2+ in apfu, r = 0.940): 〈O − O〉 unshared = 3.072 − 4.24 ×10 −2 ×
[ 6]
Al + 2.14 ×10 −2 ×
[ 6]
Fe 2+ − 3.88 ×10 −2 × Ifs
where Ifs is the increase of the interlayer cation field strength (i.e., the charge of the interlayer cation divided by radius) in brittle micas with respect to true micas (both regression equations were obtained using chemical data reported in Tables 1-4, Appendix II). Note that the octahedral site composition is represented in a less accurate way than the tetrahedral composition because of the greater variability in the chemical composition of the octahedron. The relationship between α-observed and that calculated from composition is shown in Figure 14. The fit is fair (r = 0.922) and this low correlation is related to the influence of octahedral, tetrahedral, and interlayer composition on α. In particular, the interlayer composition appears to affect the mean value of the octahedral triads (unshared O–O distances). This result confirms the influence of the interlayer site composition on the tetrahedral in-plane rotation.
Figure 14. α determined by single crystal structure refinement vs. α calculated by the formula α = cos-1 ( 3 2 ⋅k ) where k value was obtained by calculating 〈O–O〉 octahedral unshared edges (〈O–O〉unshared) and 〈O–O〉 tetrahedral basal edges (〈O–O〉basal) from chemical composition (see text). Symbols and samples as in Figure 3.
For large cations such as Cs and Rb (e.g., Cs-tetra-ferri-annite, Rb-,Cs-rich phlogopite, and nanpingite) the small α value corresponds to a large interlayer separation, whereas for small cations such as Na and Ca (e.g., preiswerkite, paragonite, and
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Figure 15. Relationships between α and interlayer separation. Symbols and samples as in Figure 3.
clintonite) large α values correspond to small interlayer separations (Fig. 15). Thus, as previously noted (Radoslovich and Norrish 1962), the shape of the interlayer-cation cavity reflects the field strength of the interlayer cation. The cavity adjusts in size by tetrahedral rotation or by a shift in the cation toward or away from the plane defined by the three basal oxygen atoms (i.e., the basal plane). In K-rich trioctahedral micas, both α and interlayer separation increase from norrishite to tetra-ferriphlogopite (and aluminian phlogopite) toward values for Fe-rich polylithionite, Fe-rich phlogopite, Mg-rich annite, and phlogopite. Annite deviates from the trend of trioctahedral true micas owing to a larger interlayer separation. In the Ba-rich brittle mica, ferrokinoshitalite (M sites mainly occupied by Fe2+), α- and interlayerseparation values are smaller with respect to those of kinoshitalite (M sites mostly occupied by Mg). With respect to trioctahedral micas, the interlayer separation in both muscovite and celadonitic muscovite is smaller, but α values are similar. To explain this behavior, the octahedral, tetrahedral, and O(4) site chemistry must be considered. Compared to kinoshitalite, ferrokinoshitalite shows an enlargement of the octahedral sheet produced by the relatively large size of Fe2+ with respect to Mg and by F for OH substitution on O(4). Therefore, the smaller α value is attributed to the large size of Fe in the octahedra, which allows a better fit to the Al-rich tetrahedral sheet. Less rotation of the tetrahedra produces a larger size of the silicate ring, which allows Ba to better fit within the ring, thus reducing interlayer separation (Guggenheim and Frimmel 1999). In norrishite, the combined effects of a Si-rich tetrahedral sheet, which produces smaller individual tetrahedra within (001), and octahedral flattening owing to the relatively large Li and Mn3+, reduce the tetrahedral-octahedral sheet misfit, thus requiring limited tetrahedral rotation. In addition, the narrow interlayer region is partly related to the increase in the Coulombic interactions of O2- [in the O(4) site] and the interlayer K (Tyrna and Guggenheim 1991). In tetra-ferriphlogopite, the lateral extension of the
Mica Crystal Chemistry and Influence of P-T-X on Atomistic Models
25
tetrahedral sheet is related to Fe3+ and involves a large α value to fit the Mg-rich octahedra. The O(4) site is mostly occupied by OH-, which produces H+–K+ repulsion, thus requiring a greater interlayer separation (Brigatti et al. 1996a). A similar adjustment occurs also in aluminian phlogopite (Alietti et al. 1995) because the composition 3+ [ 4] involving an exchange vector of [6 ] Al 3+ Mn1.96 Al 3+ [4 ] Si4− +1 creates larger tetrahedral-sheet and smaller octahedral-sheet dimensions with respect to phlogopite. With respect to trioctahedral micas, dioctahedral muscovite and celadonitic muscovite have smaller interlayer separations but similar α values. In dioctahedral micas, the proton position results in part from repulsion by the interlayer cation and the cations in the M(2) sites. Thus, the proton is located in that portion of the structure with minimal local positive-charge concentration, near the M(1) site (Radoslovich 1960; Guggenheim et al. 1987). The six-fold coordination of the interlayer cation with the basal inner O atom is distorted and elongated parallel to c*. Both effects (i.e., the distorted coordination of the interlayer cation and the smaller H+–K+ repulsion) thus control the interlayer separation. McCauley and Newman (1971) and Weiss et al. (1992) related α to the coordination of the interlayer cation. In an ideal structure α = 0° and the interlayer cation is in 12-fold coordination. In non-ideal structures, α values of greater than 0° reduce the interlayercation coordination number from 12 to 6. Weiss et al. (1992) determined the coordination number of the interlayer cation using the equation of Hoppe (1979): ECoN = ∑ j=12 j=1 C j , where ECoN is the Effective Coordination Number; (C j = exp[1.0 −( FIR j / MEFIR)6 ]); FIRj was calculated by dividing the A–Oj distance by the sum of anion and cation radii and then multiplying by the cation radii; MEFIR is a weighted mean of FIR, i.e., MEFIR =
j =12 ∑ j =1 w jFIR j ; j =12 ∑j=1 w j
w j = exp(1 − (FIR j FIR min ) 6 ;
FIRmin is the smallest FIRj in the interlayer cation coordination. They found that ECoN is close to 12 in tainiolite and annite, usually varies from 11 to 9 in polylithionite, ferroan polylithionite, phlogopite, and ferroan phlogopite, and is between 9 and 8 in muscovite and celadonitic muscovite, whereas paragonite and most brittle micas have the lowest ECoN (ECoN = 6). In addition to tetrahedral and octahedral site composition, the coordination of the interlayer cation was found to be affected by the stacking of the layers. In the most common polytypes (e.g., 1M, 2M1 and 3T, the polytypes of subfamily A, as defined by Backhaus and Durovíc 1984 and Durovíc et al. 1984) the coordination polyhedron of the interlayer cation varies from ditrigonal antiprism to octahedral, whereas in polytypes of subfamily B (e.g., 2M2 and 6H polytypes) it varies from ditrigonal to trigonal prismatic. Tetrahedral cation ordering Ordering of tetrahedral cations is quite unusual in the common mica species such as muscovite-2M1, phlogopite-1M and annite-1M (Bailey 1975, 1984c), whereas it is common in brittle micas. Margarite, bityite and anandite are examples of minerals with Si,Al (or Fe3+) tetrahedral ordering (Guggenheim 1984). Bailey (1984b) concluded that ordering of tetrahedral cations is favored for 3T structures (Güven and Burnham 1967; Brown 1978, Sidorenko et al. 1977b), for Si:Al ratios near to 1:1 (Guggenheim and Bailey 1975, 1978; Joswig et al. 1983; Lin and Guggenheim 1983) and for muscovite-1M, -2M1 and -2M2 crystals with a significant
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celadonite component (Güven 1971b; Zhoukhlistov et al. 1973; Sidorenko et al. 1975). In contrast, Amisano-Canesi et al. (1994) suggested that no long-range ordering of tetrahedral cations is present in muscovite-3T crystals and concluded that the tetrahedral cation ordering previously found by Güven and Burnham (1967) may be an artifact produced by the small number of independent reflections used in the crystal-structure refinement. It is very unusual to obtain increased cation order at high temperature, where, in most cases increasing disorder is the norm, however the results of neutron powder diffraction studies suggest tetrahedral Si-Al ordering for celadonitic muscovite (referred as “phengite”) at high temperature (Pavese et al. 1997, 1999, 2000). Guggenheim (1984) noted the importance of two factors in determining the degree of Si,Al ordering in crystals with Si:Al ratios of 1:1 that relate to octahedral- and the interlayer-site composition: (i) the charge of an apical oxygen that coordinates two Al3+ octahedral cations is undersaturated with respect to positive charge if the tetrahedral cation is Al3+, whereas it is balanced if the tetrahedral cation is Si4+; (ii) large cations in interlayer sites prop apart two adjacent 2:1 layers, thus minimizing electrostatic repulsions across the interlayer. Therefore Si,Al tetrahedral ordering seems to be favored in species with small, high-charged octahedral cations and small cations in interlayer sites. To date, long-range tetrahedral ordering has not been determined for preiswerkite and clintonite, but was found for ephesite, margarite, bityite and anandite. Although Raman spectra suggest the presence of strong short-range ordering in preiswerkite-1M [NaMg2AlAl2Si2O10(OH)2], long range ordering in tetrahedral sites was not found by crystal-structure refinement (Oberti et al. 1993). In contrast, ephesite [NaLiAl2Al2Si2O10(OH)2], which differs in composition from preiswerkite only for octahedral composition, is strongly ordered in space group C1 (Slade et al. 1987). In the latter case, perhaps, the presence of monovalent and trivalent octahedral cations requires ordering of the tetrahedral cations to achieve a suitable local charge balance on shared apical oxygen atoms. The possibility of tetrahedral cation ordering in kinoshitalite, characterized by a Si:Al ratio close to 1, was addressed by Guggenheim and Kato (1984), Guggenheim (1984), and Gnos and Armbruster (2000). Guggenheim (1984) related the lack of tetrahedral Si,Al ordering in kinoshitalite to the large interlayer separation (3.328 Å; Guggenheim and Kato 1984) caused by the large Ba interlayer cation which increases the separation between adjacent 2:1 layers, thus reducing any T–T electrostatic interactions across the interlayer. Lack of tetrahedral Si,Al ordering was also confirmed for ferrokinoshitalite (3.129 Å; Guggenheim and Frimmel 1999) which also has large interlayer separation but less than that of kinoshitalite. According to Gnos and Armbruster (2000), Si,Al ordering in kinoshitalite may be masked by twinning. They assumed different twin models to explain the average structure of this brittle mica in space group C2/m starting from complete Si,Al tetrahedral ordering in C2 and C 1 symmetries. The C2-space group model assumes that each Si tetrahedron is surrounded by three Al tetrahedra and vice-versa as consistent with Loewenstein’s (1954) Al-avoidance rule. The tetrahedral sheets of two adjacent 2:1 layers are arranged above and below the interlayer to produce the pattern along the c-axis for which Si is always adjacent to Si and Al adjacent to Al tetrahedra. The C 1 -space group model maintains the same Si,Al distribution within the tetrahedral sheets (i.e., one Si atom surrounded by three Al atoms and vice-versa), but differs for the Si,Al distribution along the c-axis. In this latter model, each Si tetrahedron is always opposed to an Al tetrahedron. Gnos and Armbruster (2000) concluded that the crystal-structure refinement is inconsistent with the twinning models that involve completely ordered Si,Al sheets. In contrast, the crystal-structure refinements of two disordered models to
Mica Crystal Chemistry and Influence of P-T-X on Atomistic Models
27
produce C2/m symmetry (i.e., three-dimensional Si,Al disorder and one-dimensional disorder along the c axis) suggested a pattern of one-dimensional disorder along the [001] direction of completely Si,Al ordered tetrahedral sheets. In margarite [CaAl2 Al2Si2O10(OH)2], Al preferentially occupies two of the four symmetrically independent tetrahedra (Guggenheim and Bailey 1975, 1978; Joswig et al. 1983; Kassner et al. 1993). Kassner et al. (1993) found that mean tetrahedral Al–O and Si-O distances are identical in the two crystallographically independent tetrahedral sheets. Thus there is no asymmetry in the distribution of tetrahedral Al in these sheets as indicated by Guggenheim and Bailey (1975, 1978) based on an incompletely refined model. An ordering pattern similar to that of margarite occurs for tetrahedral sites of bityite with nearly complete ordering of Al, Be relative to Si (Lin and Guggenheim 1983). Octahedral coordination and long-range octahedral ordering Three translationally independent octahedral cation sites characterize the 2:1 layer. One site is trans coordinated by OH (or by F and/or Cl, and rarely by S) and is called M(1), the remaining two sites are cis-coordinated and are referred to as M(2) where the layer contains a symmetry plane which relates the two M(2) sites. Otherwise, the two cissites are labeled M(2) and M(3), respectively. M(1) is usually vacant in dioctahedral micas, whereas all three octahedral sites are occupied in trioctahedral micas. The cation distribution in the octahedral sites may be summarized as: (i) all the octahedra are occupied by the same kind of “crystallographic entity” (i.e., the same kind of ion or by a statistical average of different kinds of ions, including voids, referred to as homooctahedral micas by Durovíc 1981, 1994), (ii) two octahedra are occupied by the same kind of “crystallographic entity” and the third by a different entity in an ordered way (meso-octahedral micas), or (iii) each of the three sites is occupied by a different “crystallographic entity” in an ordered way (hetero-octahedral micas). The location of the origin of the octahedral sheet corresponds to: (i) the M(1) site for homo-octahedral micas; (ii) the site with different occupation for meso-octahedral micas; and (iii) the site with the smallest electron density for hetero-octahedral micas (Durovíc et al. 1984). As a consequence, two kinds of layers can be defined, namely, the “M(1) layer” and the “M(2) layer,” the first with the origin of the octahedral sheet in M(1), the latter in either the M(2) or M(3) site (Zvyagin 1967). The “M(1) layer” is the more common. Weiss et al. (1992) identified eight possible geometries of the octahedral sheet based on the size of octahedral sites. In particular, they derived four- and three-different geometries for mesooctahedral and for hetero-octahedral micas, respectively. Toraya (1981) noted that the M(1) site is usually occupied by a cation of lower charge or by a vacancy. He explained this characteristic by considering the effect on the linkages of the polyhedra. An increase in the size of M(2) is energetically unfavorable because the O–O shared edge between two adjacent M(2) cations would be enlarged [increasing the repulsion between octahedral M(2) cations], the O–OH,F edge between M(1) and M(2) would be reduced [thus decreasing repulsion between octahedral M(1) and M(2) cations], and the increased repulsion between oxygen atoms on the unshared lateral edges of M(1) would occur owing to the smaller size of this site. In contrast, the only unfavorable factor created by an increase in M(1) would be an increase in repulsion between M(1) and M(2) cations, which would be mitigated by the decrease in charge of M(1). However, examples where three sites are all equal or each site differs are not unusual in trioctahedral micas. Several phlogopite and tetra-ferriphlogopite crystals (space group C2/m) show the same kind of cations (or a disordered cation distribution) in M(1) and M(2) octahedra, i.e., the difference between the mean bond lengths and mean electron counts (m.e.c.) of M(1) and M(2) sites are equal within the standard deviations
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(Δ 〈M–O〉 = |〈M(1)–O〉 – 〈M(2)–O〉| < 0.004 Å; Δ m.e.c = |m.e.c.M(1) – m.e.c.M(2)| < 1.0 e-; see, for example, in Tables 1-3 (end of chapter) the data by Semenova et al. 1977; Hazen et al. 1981; Brigatti et al. 1996a; Gnos and Armbruster 2000), whereas some Li-rich micas (space group C2) have different cation ordering in M(1), M(2) and M(3) sites; e.g., zinnwaldite-1M (Guggenheim and Bailey 1977), lepidolite-1M (Backhaus 1983), zinnwaldite-2M1 (Rieder et al. 1996), ferroan polylithionite-1M and lithian “siderophyllite”-1M (Brigatti et al. 2000b). The octahedral sheet may show different cation distributions in M(1), M(2), and M(3), but the size of each octahedron need not differ. For example, the zinnwaldite-1M (polylithionite-siderophyllite intermediate) structure refined by Guggenheim and Bailey (1977) shows M(1) ≠ M(2) ≠ M(3) on the basis of the site scattering power, whereas M(1) = M(3) ≠ M(2) on the basis of size of the polyhedra. Verification of ordering requires not only the analysis of cation-anion bond length but also the refinement of octahedral-site occupancies because mean bond lengths of octahedra with different occupancies may be similar. Therefore, this discussion of octahedral ordering is based only on samples for which the m.e.c. of each octahedral site is available. For trioctahedral true micas of the phlogopite-annite join, the m.e.c. of both M(1) and M(2) sites increases from phlogopite to annite through ferroan phlogopite and magnesian annite. This suggests that an increase in the Mg-1 Fe exchange occurs and that Fe occupies both the M(1) and M(2) sites (Fig. 16). However, ferroan-phlogopite and magnesian-annite samples (Tables 1a and 1b) have differences in mean bond lengths (to 0.036 Å) and in m.e.c. (to 2.5 e-) for the M(1) and M(2) sites. Thus, a slight preference for cations with larger radii and atomic numbers for M(1) occurs. The greatest differences between M(1) and M(2) octahedral mean bond distances occur in Al-bearing magnesian annite from peraluminous granites where Al is ordered in M(2) (Brigatti et al. 2000a).
Figure 16. Mean electron count (m.e.c.) of M(2) [M(2) = M(3)] vs. m.e.c. of M(1) site. Symbols: filled circles = annite; filled circles, xhair =magnesian annite; open circle, x-hair = ferroan phlogopite; open circles = phlogopite. Estimated average standard deviation: ±0.3 e-.
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Figure 17. Difference between M(1) and M(2) site mean bond distance in mica crystals of the phlogopite-annite join. Symbols and samples as in Figure 16. The average standard deviation on 〈M(1)–O〉 and 〈M(2)– O〉 bond distances was evaluated as ±0.002 Å.
Thus, ordering along this join seems to be enhanced where, in addition to Mg2+ and Fe2+, cations of different size and charge occur in octahedral coordination (Fig. 17). Although the m.e.c. of M(1) and M(2) increases with the exchange vector of Mg-1Fe (Fig. 16), annite crystals have Δ 〈M–O〉 values much smaller than those for magnesian annite. For compositions intermediate between those of phlogopite and annite, exchange vectors that introduce cations of different charge (or vacancies) in octahedral sites significantly affect the layer topology. In fact, in phlogopite the octahedral sites are equal in size and m.e.c., annite shows octahedral sites with similar m.e.c. (Δ m.e.c. < 0.4 e ) and differences in Δ 〈M–O〉 bond lengths (Δ 〈M–O〉 < 0.02), whereas crystals of phlogopiteannite with intermediate compositions always have one larger octahedron and two smaller octahedra and usually differences in m.e.c. for M(1) and M(2). The reduction of interlayer separation with Ti content (Fig. 18) is related to the decrease in the K-O(4) [ O(4) = OH, O, F, Cl ] distance, which is ascribed to “Ti-oxy” 1− substitution, Ti-oxy = [6] Ti4 +O22 − [6] Mg 2+ −1 (OH) −2 ). The interlayer cation is shifted deeper into the interlayer cavity owing to the deprotonation of the O(4) site; thus, the K–O(4) distance decreases with a decrease in interlayer separation. Cruciani and Zanazzi (1994) observed that the off-center shift of the cation at M(2) is associated with an increase in the proportion of [6]Ti, and this reveals a [6]Ti preference for the M(2) site. Li-rich micas in the siderophlyllite-polylithionite join (Fig. 19) show different patterns of octahedral order. For example, in a synthetic polylithionite (space group C2/m) with octahedral composition Li2Al (Takeda and Burnham 1969) the ordering pattern results in a large M(1) site of composition Li0.89Al0.11 and two equivalent M(2) sites of composition (Li0.55Al0.45). A similar ordering pattern was observed in natural
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Figure 18. Interlayer separation vs. octahedral Ti4+ content for mica crystals along the phlogopite–annite join. Symbols and samples as in Figure 16. The average standard deviation on the interlayer separation was evaluated as ±0.004 Å.
Figure 19. Ternary [6]Al3+ – [6]Li+ -[6]Fe2+ diagram showing compositional data for Li-rich micas. Symbols: filled circles = Li-containing annite crystals; open squares, x-hair = ferroan polylithionite and crystals with composition intermediate between polylithionite and siderophyllite; open squares=polylithionite; filled squares, x-hair = siderophyllite; filled square = trilithionite. The open circles indicate the composition of the end members (from Brigatti et al. 2000b).
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trilithionite-1M (Sartori 1976; Guggenheim 1981) and in polylithionite-2M1 and 2M2 (Takeda et al. 1971; Sartori et al. 1973; Swanson and Bailey 1981). In some Li-rich micas, the ideal layer symmetry is reduced from C2/m to C2, as a result of a different pattern in octahedral ordering in the cis-octahedral sites (Guggenheim and Bailey 1977; Guggenheim 1981; Backhaus 1983; Mizota et al. 1986; Rieder et al. 1996; Brigatti et al. 2000b). These minerals have 〈M(1)–O〉 ≅ 〈M(3)–O〉 > 〈M(2)–O〉 and occasionally 〈M(1)–O〉 ≅ 〈M(2)–O〉 > 〈M(3)–O〉 (Backhaus 1983; Brigatti et al. 2000b). The scattering efficiency for the M(1), M(2) and M(3) sites implies ordering with M(1) ≠ M(2) ≠ M(3), M(1) = M(3) <M(2); or M(2) = M(3) < M(1). Where all sites have different occupancies in Li-rich crystals, both “M(1) layers” and “M(2)- layers” are present. Brigatti et al. (2000b) studied the crystal structure of 1M micas with composition in the polylithionite-siderophyllite-annite field. They showed that the variation in composition follows a near-continuous trend between polylithionite and siderophyllite. They defined the ordering parameter QM(2),M(3) as: M(3) − O − M(2) − O QM ( 2),M (3) = 1 [ M(2) − O + M(3) − O 2
]
Using the ordering parameter, QM(2),M(3), differences in bond lengths between the M(2) and M(3) sites are nearly constant for polylithionite and ferroan polylithionite in the XSid-Pl range between 1.0 and 0.7 [XSid-Pl = [6](Li + Al) / [6](Li+Al+Fe2+)]. However, the difference rapidly decreases starting at XSid-Pl ≅ 0.6 and the layer symmetry approaches that of space group C2/m rather than C2. The ordering parameter EM(2),M(3) is a measure of the m.e.c. in the M(2) and M(3) sites, and is defined as: e− M(2) − e− M(3) E M (2 ),M ( 3) = 1 [e − M(2) + e − M(3)] 2
The ordering parameter, EM(2),M(3) indicates ordering in M(2) and M(3) for 0.6 ≥ XSid≥ 0.4. Therefore, the M(2) and M(3) sites appear to be completely disordered only when Pl
XSid-Pl < 0.4. Although examples of polylithionite with disordered cation distributions for M(2) and M(3) are not unusual, Brigatti et al. (2000b) concluded that Fe2+ controls the cis-site cation distribution and that order-disorder between M(2) and M(3) sites occurs in a narrow compositional interval (Fig. 20). The ideal space group of polylithionite-3T allows all three octahedra to be of different composition. This occurs for polylithionite-3T (Brown 1978) and for lithian siderophyllite-3T (Weiss et al. 1993), which shows a different ordering pattern for each octahedron. Masutomilite (ideally KLiAlMn2+AlSi3O10F2), the Mn analogue of ferroan polylithionite (Mizota et al. 1986), has M(1) and M(3) sites which are nearly equal in size and scattering power. Both the M(1) and M(3) sites contain Li and Mn2+. The greater scattering efficiency and smaller size of the M(2) site indicate that M(2) contains Al and Fe. In contrast, the octahedral ordering pattern in norrishite (KLiMn3+2Si4O12) is such that low-valence cations occur in M(1) (Li+) and trivalent cations (Mn3+) in M(2) [M(2) = M(3)] sites. Several micas were described with total octahedral occupancy midway between dioctahedral and trioctahedral. However, in many cases, intermediate compositions may represent interstratified mixtures of dioctahedral and trioctahedral layers (or species). Lirich muscovite crystals (Brigatti et al. 2001b) have a total octahedral occupancy of 2.24 apfu. The volume of the M(2) site increases sharply with an increase in Li. The M(1) cation site is partially occupied. These results seem to indicate a partial dioctahedraltrioctahedral solid solution. In trioctahedral brittle micas, octahedral cation ordering was found for bityite and for anandite. No evidence of ordering, except for the usual mica ordering with M(1)
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Figure 20. Variation of (a) QM(2),M(3) and (b) EM(2),M(3) parameters as a function of composition. The data points at QM(2),M(3) = 0 and EM(2),M(3) = 0 correspond to C2/m symmetry. Symbols as in Figure 19 (from Brigatti et al. 2000b).
occupied by a cation whose average charge is smaller and whose average size is larger than that found in M(2), was detected for kinoshitalite, ferrokinoshitalite and clintonite (see the “new species and new data” section). Anandite-2O, (Ba 0.96 K0.003Na 0.01 ) (Fe 2+2.02 Fe3+0.31 Mg 0.45 Mn 2+0.04 Mn 3+0.04 ) (Fe3+1.38 Si 2.62 )O10S0.84Cl0.16F0.04(OH)0.96, has octahedral Fe-Mg ordering with two Fe-poor octahedra near the cell corners and two Fe-rich octahedra near the C-face center (Filut et al. 1985). The Fe-Mg ordering requires that hydroxyl groups are associated with the Fe-poor octahedra, whereas S replaces (OH) in the Fe-rich octahedra. Bityite-2M1, (Ca0.95Na0.02)(Li0.550.45Al2.04 Fe 3+ 0.01 )(Al1.34Si2.02Be0.64)O10(OH)2, has the trans-M(1) site occupied by Li and vacancies, and the two cis-M(2) sites are occupied by Al cations (Lin and Guggenheim 1983). Coexistence of dioctahedral [with M(1)-vacant sites] and trioctahedral [with Li-filled M(1) sites] sheets was suggested to explain the two patterns of O–H vector orientation. The topology of each octahedron is influenced not only by local composition but also by the constraints of closest packing within the sheet. Several authors (e.g., Toraya 1981; Lin and Guggenheim 1983; Weiss et al. 1985, 1992) examined the relationships between composition and the octahedral topology (i.e., variations in the octahedral dimensions and in octahedral distortions). In agreement with the observation of Hazen and Wones (1972), who suggested that octahedral flattening is controlled by the octahedral cation radius, Toraya (1981) suggested for 1M silicate and germanate micas, that the octahedral flattening angle, ψ, gradually decreases with decreasing misfit between the tetrahedral and octahedral sheets. He noted also that the tetrahedral lateral dimensions remain constant. Toraya also showed that the degree of octahedral flattening,
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which reflects variations in octahedral thickness and lateral octahedral dimensions, is related to lateral misfit between the sheets of tetrahedra and octahedra and, therefore, by α. Lin and Guggenheim (1983) related the counter-rotation of upper and lower octahedral oxygen triads (the ω angle of Appelo 1978) to the difference between the 〈M(1)–O〉 and 〈M(2)–O〉 distances. They demonstrated that ψ is significantly affected by the field strength of adjacent octahedral cations and less affected either by the octahedral cation size or by the misfit between the tetrahedral and octahedral sheets. They also observed that octahedral flattening and octahedral counter-rotation produce opposing effects. Octahedral flattening increases mean values of the upper and lower triads, and thus increases lateral dimensions of the octahedra, whereas the overall effect of counterrotation is the reduction of the lateral octahedral size. Weiss et al. (1985) also showed that octahedral flattening and the counter-rotation of the upper and lower anion triads are related to the interaction in the “whole” sheet rather than an individual octahedron, and suggested geometrical models to predict the octahedral topology by composition.
Figure 21. Variation of ψM(1) vs. 〈M(1)–O〉 bond distance. Symbols and samples as in Figure 3.
Figure 21 shows the variation of ψ vs. the mean bond distance for the trans M(1) site. Both ψ and 〈M(1)–O〉 increase from trioctahedral micas to dioctahedral micas. As noted previously, the distortion of an octahedral site is not a simple function of the size of the cation residing in the octahedron. In fact, distortions in the vacant site in dioctahedral micas and in the M(1) site in Li-rich micas are caused by the decrease in length of the shared edges of the M(1) octahedron with respect to the mean edge value of M(1). M(1) is required to share edges with smaller adjacent octahedra containing cations with high field strength. In Figure 21, plotted values for the trioctahedral micas (excluding Li-rich) show scatter, but the general trend suggests that ψ decreases as the site size increases. From a geometrical point of view (in space group C2/m), a displacement of O(4) along [001] affects each octahedron in the same way, i.e., 〈M(1)–O〉 and 〈M(2)–O〉 mean bond
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Figure 22. (a) ψ = cos-1 toct / 2〈M–O〉; (b) deformation induced on M(1) by a displacement of the O(4) oxygen atom along the [001] direction; (c) deformation induced on M(2) by a displacement of the O(4) oxygen atom along [001] direction; (d) deformation produced on octahedra by a displacement in (001) plane which produces the C2/m symmetry requirement.
distances decrease equally, whereas ψM(1) and ψM(2) values increase equally [modes (b) and (c), Fig. 22]. Differences between M(1) and M(2) are explained by mode (d) (Fig. 22), with a displacement of the O(4) atom toward M(2). In this way, the 〈M(1)–O〉 distance, and the ψM(1) value increase at nearly two times the rate at which 〈M(2)–O〉 and ψM(2) decrease. Mode (d), therefore, does not affect 〈M–O〉 [〈M–O〉 = (〈M(1)–O〉 + 2 × 〈M(2)–O〉)/3] and 〈ψ〉 [〈ψ〉 = (ψM(1) + 2× ψM(2))/3] mean values. The results provided by the geometrical model (see Appendix I) can be compared to the trend observed for the structures in Figure 23. The parameter otcor (i.e., the difference between the value of the observed octahedral thickness and the thickness of an ideal octahedron whose edge is equal to Σ〈O–O〉unshared) is defined here as: ot cor = ot −
6 ⋅ 〈O − O〉 unshared 3
where ot is the observed octahedral thickness and Σ 〈O–O〉unshared is the mean value of the M(1)and M(2) unshared edges [i.e, the mean octahedral triad value: Σ 〈O–O〉unshared = (〈O–O〉unshared M(1) + 2 × 〈O–O〉unshared M(2))/3]. The resulting equation of regression is 〈ψ〉 = -34.352⋅otcor / Σ〈O–O〉unshared + 54.779 (r = 0.982). The trend in Figure 23 indicates that the ψ mean value depends nearly entirely on the displacement of the O(4) atom along the [001] direction. The first-order constant in the regression equation (i.e., -34.352) is greater than the calculated value from the
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Figure 23. Variation of the mean octahedral flattening angle vs. the otcor/〈O–O〉unshared. otcor represents the difference between the octahedral thickness (actual) value and the thickness of an ideal octahedron whose edge is equal to 〈O–O〉unshared. 〈O–O〉unshared are the octahedral unshared edges (i.e., the octahedral triads). Symbols used: filled circles = annite; filled circles, x-hair = magnesian annite; open circle = phlogopite; open circle, x-hair = ferroan phlogopite; filled circle, dotted = tetra-ferriannite; open circle, dotted = tetra-ferriphlogopite; filled hexagon, x-hair =norrishite; crosses = preiswerkite; open triangle up = clintonite; filled triangle up, x-hair =ferrokinoshitalite; filled trangle up = kinoshitalite.
geometrical model which assumes that only O(4) is displaced parallel to the [001] direction. Thus, we assume for the O(3) position (i.e., the apical oxygen atoms) a similar displacement along the [001] direction. This displacement suggests the existence of a very small corrugation in the O(3) and O(4) oxygen-atom plane. Similar results can be obtained using octahedral mean bond distances 〈M–O〉. For C2/m micas, the geometrical model also predicts the effect of the O(4) atom displacement along [100] (i.e, the difference between ψ values of M(1) and M(2) increases concomitant with the displacement of O(4) along [100]). Toraya (1981) used energy-based arguments to explain the difference between the M(1) and M(2) sites and, in particular, to explain why M(1) is usually larger than M(2). The results obtained here seem to confirm the interpretation of Toraya. In particular, it is shown that a displacement of the O(4) oxygen atom in the (001) plane should not change the 〈M-O〉 mean value, i.e., ⎡ M(1) - O + 2 M(2) - O ⎤ 1 . It can be easily shown that the value of + ⎢ ⎥ 3 〈M(1) − O〉 ⎣ ⎦ 1 1 + increases. Therefore, the energy of the M–O(4) bond decreases. 〈M(2) − O〉 〈M(3) − O〉
Assuming equal charges in each of the three octahedra, the following relationship (see Appendix I) can be derived:
36
Brigatti & Guggenheim ⎛ ⎞ 1 1 1 E M −O(4) ∝ QO(4) ⋅ QM ⋅ ⎜ + + ⎟ ⎝ 〈M(1) − O〉 〈M(2) − O〉 〈M(3) − O〉 ⎠
where EM-O(4) is the M-O(4) bond energy. This equation indicates that cation ordering resulting in an increase in the charge of M(2) and M(3) and a decrease in charge in M(1) is even more energetically favorable. In contrast, as O(4) is displaced away from the center of the ditrigonal silicate ring, the energy related to the O(4)-O(3) bonds is expected to increase. This mechanism may account for the difference concerning the two octahedral sites, but further studies are needed to confirm this. Bailey (1984b) suggested that the large dimensions of the vacant M(1) octahedral site in dioctahedral micas cause an “overshift”, (i.e., the intralayer shift parameter), where the upper tetrahedral sheet is shifted relative to the lower sheet by a value greater than 0.33a1 of the pseudo- hexagonal cell. Apical oxygen atoms of the 2:1 layer are linked to the diagonal edges of the trans M(1) octahedron, thus causing the M(1) diagonal edge to increase. This produces an intralayer shift. The intralayer shift results from a displacement of the apical oxygen atoms from their ideal positions midway among the three octahedral sites. Trioctahedral micas with all three sites equal in size have the smallest intralayer shifts whereas the dioctahedral micas show the greatest. The anomalous behavior of norrishite is related to octahedral distortions induced by the Jahn-Teller effect (Fig. 24).
Figure 24. Differences between the 〈M(1)–O〉 and 〈M(2)–O〉 octahedral mean bond distances vs. the intralayer shift. Symbols and samples as in Figure 3.
Crystal chemistry of micas in plutonic rocks Most rock-forming silicates have solid solution involving the substitution of different cations in one or several symmetrically different sites. Their compositions, cation ordering and topology are sensitive to many environmental conditions occurring during crystallization (e.g., Ganguly 1982; Hirschmann et al. 1994). In particular, micas
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are rock-forming silicates of great interest for their petrological, crystal-chemical and thermodynamic significance. In igneous rocks, they participate in mineral-mineral, mineral-melt, and mineral-fluid equilibria. Thus, their structures may reflect the different reactions involved (Icenhower and London 1995, 1997; references therein). However, rock-forming micas have received less attention by crystallographers as petrogenetic indicators owing to the difficulty in obtaining quality data from a structure determination. We review below the relatively few systematic studies on the crystal chemistry of naturally occurring micas crystallized from a melt. Phlogopite-annite appears to be the most widespread species in plutonic parageneses, whereas dioctahedral mica occurrences are limited, typically, to peraluminous granitoids. Many authors have suggested the dependence of the composition of igneous micas on factors such as bulk host-rock chemistry, oxygen fugacity, H2O fugacity, and (other) fluid activity (e.g. Arima and Edgar 1981; Barton 1979; Edgar and Arima 1983; Speer 1984). The partition coefficients (D) between micas and coexisting phases were also recognized as important indicators for melt evolution (e.g., De Albuquerque 1975; Monier and Robert 1986; Icenhower and London 1995; Wolf and London 1997). The increase of Mg/(Mg+Fe) in phlogopite with increase in MgO/(MgO+FeO) of the rock as well as with increase temperature is well-documented (Speer 1984; Puziewicz and Johannes 1990). Figure 25a suggests that the MgO/(MgO+FeO) ratio of the rock is related approximately to the content of the M(1) site. In particular, as the MgO/(MgO+FeO) ratio of the rock increases, the mica composition becomes more phlogopitic and the 〈M(1)-O〉 mean bond distance decreases. Different crystallization conditions may account for the poor fit of the data in Figure 25a. For example, the Tapira carbonatite complex crystallized at a high crustal level (e.g., lower pressure, faster cooling rates) and is characterized by oxygen fugacity above the NNO buffer (Brigatti et al. 1996a), whereas peraluminous granites of Sardinia and Antarctica are the products of ultrametamorphism leading to anatexis of aluminous metasedimentary rocks in the continental crust (Brigatti et al. 2000a). Figure 25b shows the interlayer separation vs. bulk rock Al content. Interlayer separation increases as the Al content of the rock decreases. This relationship is the structural parameter which better fits the Al rock content. The micas, for which data are shown in Figure 25b, come from different rock-types, but share a relationship of increasing Ti and Al with increasing bulk-rock Al. The decrease in the interlayer separation (Fig. 25b) is therefore related to the exchange vector [6](Mg,Fe)2+-1 OH--2 [6]Ti4+ O-22 required to achieve layer-charge neutrality. Bigi et al. (1993) described ferroan phlogopite and magnesian annite crystals from mafic rocks occurring at different stratigraphic levels of the Ivrea-Verbano Mafic Complex (Western Alps, Northern Italy). They noted that some properties, such as polytypism with different amounts of stacking disorder coexist in all samples and do not have a direct or simple petrological cause, whereas octahedral cation disorder in a crystal close to the contact with metasediments may suggest the incorporation of a restitic assemblage in the melt. The hotter melt may have preserved the mica (and other refractory minerals), but may have induced cation disorder of the octahedral sites. Similar results were found in the study on Fe3+-rich phlogopite from the Tapira alkaline carbonatitic complex (Brigatti et al. 1996a). The crystal-chemical features of these micas (mostly phlogopite1M and tetra-ferriphlogopite-1M) are related to the variation of f(O2), a(H2O) and a(CO2), in addition to the magma composition during fractional crystallization and cumulus processes responsible for the generation of the rock sequence. In addition, the disorder in octahedral cation distribution agrees with field relationships and textural features which suggest crystallization at high crustal levels and a high cooling rate.
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Figure 25. (a) Compositional dependence of 〈M(1)–O〉 bond length in trioctahedral mica from Mg/(Mg+Fe) of the host rock; (b) dependence of interlayer separation from Al content in the host rock. Symbols: open squares = ferroan phlogopite and magnesian annite crystals from “Diorites” of Ivrea-Verbano Zone, Italy (Bigi et al. 1993); open circles = ferroan phlogopite crystals from synitic complex of Valle del Cervo, Northwestern Italy (Brigatti and Davoli 1990; Bigi and Brigatti 1994); crosses = magnesian annite from “UZ gabbros” of Ivrea Verbano Zone, Italy (Bigi et al. 1993); filled triangles up = ferroan phlogopite from monzonitic complex of Valle del Cervo, Nortwestern Italy (Brigatti and Davoli 1990); open diamonds = magnesian annite crystals from peraluminous granites of Sardinia Island (Italy) and Antarctica (Brigatti et al. 2000a); open triangles down = phlogopite, tetraferriphlogopite and ferroan phlogopite crystals from Tapira carbonatite complex, Brazil (Brigatti et al. 1996a).
The partition coefficients of major and trace elements between trioctahedral Mg-, Fe-rich micas and dioctahedral muscovite are a key to understanding thermodynamic relationships. In addition, the coefficients can lead to an understanding of detailed kinetics of anatexis of aluminous metasediments and the evolution of peraluminous granitic suites (De Albuquerque 1975; Tracy 1978; Speer 1984; Dymeck 1983; Monier and Robert 1986; Patiño Douce and Johnston 1991,1993; Icenhover and London 1995). The octahedral site intercrystalline partitioning between coexisting magnesian annite and muscovite was recognized as a good indicator of chemical processes during crystallization (Brigatti et al. 2000a). In peraluminous granites, the compositions of
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trioctahedral micas (mostly magnesian annite) are characterized by significant [6]Al contents, whereas dioctahedral mica (muscovite) has noticeable celadonite-like substitutions. Structure refinements of magnesian annite suggest that the M(2) site topology is largely controlled by [6]Al, whereas M(1) topology is related to both the Fe/(Fe+Mg) ratio and to [6]Al content. In addition, crystals have significant cation ordering which is attributed to either crystal-chemical constraints (i.e., the preference of small high-charge cations for octahedra with OH in cis-orientation), or to intensive variables acting during crystallization. Corresponding features found in magnesian annite and in muscovite indicate equilibrium during subsolidus crystallization. For example, the unit-cell volumes of coexisting micas increase in similar ways and the variation of the cell volume of magnesian annite depends on the Al intercrystalline partition coefficient between the M(2) sites of trioctahedral and dioctahedral mica [(D(Al)M(2)Ma/Ms (Figs. 26a and 26b)]. The volumes of the M sites of both micas decrease with decrease of the Al saturation index of the rock, thus reflecting the influence of melt composition (Fig. 26c). In contrast, the behavior of Ti is opposite to that of Al. An increase in Ti content produces an increase in volume of the M(2) sites and an increase in the volume of the unit cell in coexisting micas (Fig. 26d) and this, according to Patiño Douce et al. (1993), is related to the temperature during mica growth. ATOMISTIC MODELS INVOLVING HIGH-TEMPERATURE STUDIES OF THE MICAS Studies of samples having undergone heat treatment Where a transformation occurs in the solid state and the rearrangement of the atoms in the product is limited relative to the reactant, the use of Pauling’s electrostatic valency principle may delineate the transformation process in detail. Dehydroxylation reactions are topotactic in dioctahedral micas, complete recrystallization does not occur, and thus there is strong crystallographic control, thereby allowing the use of the electrostatic valency principle (e.g., Guggenheim et al. 1987 for muscovite). This procedure follows a transformation step-by-step and describes how bond lengths and bond strengths are affected at each transitional step. In contrast, decomposition and recrystallization occur nearly simultaneously with dehydroxylation in trioctahedral micas, and transitional forms are not known. Thus, the electrostatic valency principle cannot readily be applied to these materials. Takeda and Ross (1975) compared two polytypes (1M, 2M1) of Fe-rich phlogopite (previously referred to as “biotite”) after passing hydrogen gas at 700°C over the sample to produce reduced and hydrogenated products, and Ohta et al. (1982) used hot argon gas to produce oxidized (“oxy-mica”) and hydrogen-depleted versions of each polytype for comparison. Thus, the effect of iron oxidation/reduction and hydrogenated/deprotonization was examined for each polytype. A similar study was made by Russell and Guggenheim (1999) for Fe-rich phlogopite-1M for comparison with Mössbauer data for the same material. Another approach is to examine how thermal behavior differs for apparently similar materials to deduce how the materials may differ (as in the case of smectite and illite, Tsipursky and Drits 1984; Drits et al. 1993). For these materials, single-crystal structural studies have not been attempted because the structures have considerable (stacking) disorder. Thus, thermal behavior was used initially to determine that some aspect of the structure, in this case cation or vacancy ordering, may differ for different samples of the same species. Powder X-ray diffraction was then used to confirm that these differences existed.
Figure 26. Relationships between coexisting magnesian annite-1M and muscovite-2M1 crystals in peraluminous granites from Sardinia Island and Antarctica (Brigatti et al 2000a). (a) unit cell volume of magnesian annite vs. unit cell volume of muscovite; (b) magnesian annite unit cell volume vs. D(AlM(2))Ma/Ms partition coefficients (AlM(2) = Al in M(2) sites; Ma = magnesian annite; Ms muscovite); (c) ratio between M(1) and M(2) site volumes in magnesian annite vs. the Al saturation index (A.S.I.). of the host rock; (d) Muscovite cell volume vs. the M(2) site titanium content of magnesian annite and muscovite (TiMa / (TiMa + TiMs) (from Brigatti et al. 2000a).
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Dehydroxylation process for dioctahedral phyllosilicates Bailey (1984b) described the general differences between the orientation of the O–H vector between trioctahedral and dioctahedral micas. In trioctahedral micas where each of the three octahedral sites is occupied by the same cation, the O–H vector points nearly vertical to the (001) plane. In this way, the proton is nearly equidistant from the three octahedral cations, and thus, equidistant from the sources of the positive charges originating from the octahedral sheet. The interlayer cation sits directly above the OH group so that the proton is between the oxygen atom of the OH and the interlayer cation. Because they are of like charge, the interlayer cation and the proton interact (repulsion), but the repulsions are directed along the vertical to the (001) plane, thereby positioning the interlayer cation away from the proton and creating an increased separation between adjacent 2:1 layers. Hence, fluorine substitution for OH in trioctahedral micas produces a smaller interlayer separation and smaller c-axis dimension because the negatively charged F anion attracts, rather than repulses, the interlayer cation. This is also why Frich trioctahedral micas generally have a greater thermal stability than a mica of equal composition, but F-poor.
Figure 27. Projection near the (001) plane to illustrate the relation between the interlayer cation (K), the (underlying) O-H vector, and the O(2) atom. Note that the hydrogen is closely associated with the K-O(2) bond. The K-O(2) bond is weaker than the other K-O bonds and weakens further at high temperature. Note also that the silicate ring is ditrigonal and not hexagonal.
In contrast to trioctahedral forms, dioctahedral micas with the vacant site located in the trans position (“trans-vacant” or “tv” 2:1 layers) have a relatively non-symmetric distribution of positive charges around the OH (Fig. 27). The O–H vector is directed away from the two occupied sites containing trivalent cations and points, in plan view, toward the vacant site. Thus, in muscovite, the O–H vector does not point directly toward the interlayer cation, and instead the proton is near the interlayer cation, K, and a bonded oxygen [K–O(2)] of the basal oxygen-atom plane. In muscovite (Rothbauer 1971), the O– H vector is inclined by 12° from the basal oxygen plane and, in plan, a straight line may be drawn from the oxygen of the OH group, through the proton, and to O(2). Of the three symmetry-unique nearest-neighbor K–O bond distances, Guggenheim et al. (1987) found that the K–O(2) bond distance is the longest at room temperature, and therefore, the weakest. At all elevated temperatures studied (to 650°C), this bond remains longer and increases in length at a faster rate than the other K–O bonds. The proximity of the proton to the K–O(2) bond and the rate at which this bond weakens at high temperatures suggest a preferred path for dehydroxylation, although the actual mechanism is not known. Dehydroxylation is not a result of destabilization of the structure owing to an increase in misfit between the tetrahedral and octahedral sheets at higher temperatures as was suggested by Hazen (1977), because the measured rotation angle, α, indicates that the tetrahedral and octahedral sheets mesh comfortably at temperatures to dehydroxylation.
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The initial process of dehydroxylation in dioctahedral micas remains problematic. In muscovite, Guggenheim et al. (1987) noted that the weakening of the K–O(2) bond with temperature is consistent with (i) the angular relationship of the O–H vector to K and O(2), and (ii) the O(2) atom deviating from the mean basal plane by an out-of-plane tetrahedral corrugation as measured by Δz (see above). Results from a high-temperature study of paragonite (see below) suggest that the weakening of the K–O(2) bond in muscovite is probably related to an increase in tetrahedral corrugation at high temperatures. Guggenheim et al. (1987) noted also that mean interatomic distances of the Al octahedral site do not vary greatly with temperature and that the Al–OH bond expands in a limited way (from 1.906 Å at 20°C to ∼1.918 Å at 650°C). However, the latter expansion is greater than the other Al–O bonds of the octahedra. These results clearly indicate that the O–H bond does not weaken as a simple function of temperature below the dehydroxylation temperature. If the length of the O–H bond did expand, the charge on the oxygen atom would be affected (i.e., the proton, as it moves away from the oxygen atom, would cause undersaturation of the oxygen atom with respect to positive charge) and the Al–OH bond distance would be expected to either decrease or increase less rapidly than those of the other Al–O bonds. Drits (pers. comm.) suggested that the tetrahedral bridging and non-bridging bond lengths would be affected also, in accord with Bookin and Smoliar (1985), if the O–H bond weakens. It is possible that the O–H bond length is affected only at conditions immediately below the dehydroxylation event such that, at a critical temperature, the O–H bond of muscovite destabilizes with the formation of H2O molecules. This is consistent with the hightemperature study of Guggenheim et al. (1987). Other processes are possible and careful optical (infrared, etc.) studies at high temperature would be useful to obtain a better understanding. A potential problem of such studies, however, is that the dehydroxylation process does not appear to occur uniformly within a crystal, and the proton interactions are complex. Some of these interactions are discussed below. Dehydroxylation, for example in muscovite, involves H2O loss, and not (OH)- or H2 loss alone. Therefore, it is likely that the OH group that destabilizes initially must attract the H+ from the adjacent OH group following the reaction: 2(OH) → H2O(↑) + Or. The remaining oxygen, Or, is referred to as the “residual” oxygen and remains in a muscovitelike dehydroxylate structure where the Al cations are in five-coordination (Fig. 28). This rearrangement, to maintain a “dioctahedral” configuration after H2O loss, distinguishes the dioctahedral micas from trioctahedral varieties, because there is insufficient room in trioctahedral micas for such an adjustment and recrystallization occurs upon dehydroxylation. The muscovite dehydroxylate structure and the corresponding dehydroxylate of pyrophyllite were described by Udagawa et al. (1974) and Wardle and Brindley (1972), respectively. In addition to comparing bond lengths as a measure of bond strength, the strength of an electrostatic bond (Pauling bond strength, PBS) is defined as the ratio of the valence, v, of the cation and the (first) coordination number, n. Guggenheim et al. (1987) used Pauling bond strengths and the second rule to examine the muscovite structure, the transitional forms by simulating structures by considering the loss of H2O groups one at time, and the dehydroxylate structure. More sophisticated models involving bond strengths and bond lengths are not suitable because these structures have not been refined and bond lengths are unavailable. Although OH groups are initially equal with respect to energy in muscovite and would be expected to respond to temperature in identical ways, the implication here is that dehydroxylation is temporal in that as H2O evolves, the structure changes to compensate for changes in charge distribution. Evidence for a nonhomogeneous loss of H2O in dioctahedral 10-Å phyllosilicates was found by Heller-
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Figure 28. A fragment of the crystal structure of an Al-rich dioctahedral mica with the trans octahedral sites vacant. Part (A) shows the octahedral sheet of muscovite, part (B) shows the corresponding portion of the muscovite dehydroxylate, and part (C) is muscovite in transition between a hydroxylate to a dehydroxylate structure. Values refer to the summations of the contributing positive charge of neighboring cations to the anion. For simplicity, tetrahedral sites (not shown) have an occupancy that is considered to be Si only (from Guggenheim et al. 1987).
Kallai and Rozenson (1980) by Mössbauer analysis in Fe-containing muscovite and by Guggenheim et al. (1987) in the thermal analysis of muscovite and by MacKenzie et al. (1985), as reinterpreted by Guggenheim et al. (1987), in the thermal analysis of pyrophyllite. Guggenheim et al. (1987) and Evans and Guggenheim (1988) showed that analogous processes are found in both muscovite and pyrophyllite, suggesting that the process may be described by considering Si tetrahedra rather than partially Al-substituted tetrahedra. Thus, for simplicity in illustration, we assume that the tetrahedral sites contain only Si to explain the dehydroxylation process atomistically in dioctahedral mica-like phyllosilicates. Otherwise, we must consider each oxygen atom as having electrostatic bond strengths with values dependent on the probability of Al0.25Si0.75 occupancy, as was done by Guggenheim et al. (1987). Dehydroxylation models for trans-vacant 2:1 layers Figure 28A shows a dioctahedral 2:1 layer. Each oxygen atom has a formal charge of -2.0 electrostatic valence units (evu). In the case of a dioctahedral sheet that is not undergoing dehydroxylation, each oxygen atom is fully charge-balanced by neighboring cations, and selected atoms in Figure 28A have associated summations of positive charge contributions shown. The sums for all the oxygen atoms in the structure shown in Figure 28A indicate that each is balanced also. In contrast, however, the forms undergoing either complete dehydroxylation (Fig. 28B) or partial dehydroxylation (Fig. 28C) do not have fully charge-balanced oxygen atoms. For the dehydroxylate form, the residual oxygen atom is greatly undersaturated with respect to positive charge (e.g., Or has a coordination of two Al3+ cations only, each in five-coordination, thus ΣPBSOr = 2 × PBSAl = 2 × 3/5 = 1.2 evu). The other oxygen
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atoms are each bonded to five-coordinated Al cations and one four-coordinated (Si4+) cation and thus are oversaturated at 2.2 evu (ΣPBSO = 2 × PBSAl + 1 × PBSSi = 1.2 + 4/4 = 2.2 evu). The arrow in Figure 28B emanating from the Al cations shows the resulting direction of movement of the Al cation away from the oversaturated oxygen atoms. This cation has a high positive charge (3+), which helps saturate the (undersaturated) Or atom. The PBS involves only approximate electrostatic relationships and it is not an accurate measure of bond strength in comparison to bond length. However, the anticipated movement of the Al toward the Or atom will produce a stronger Al–Or bond based on the bond length and the simultaneous movement away from the oversaturated oxygen atoms produce weaker Al–O bonds. Figure 28C shows a partially dehydroxylated form where some aluminum cations are in five coordination and others are in six coordination. In the latter case, the distribution of oversaturated oxygen atoms (labeled 2.1 evu) are such that the direction of anticipated Al movement (note arrows) is toward the OH group and away from the oversaturated oxygen atoms. Moving a charged atom closer to an OH group that is balanced (at 2.0 evu for the oxygen atom of the OH) must result in a readjustment which, in this case, requires the H+ ion to move further away from its oxygen-atom neighbor, thereby weakening the O–H bond. Aines and Rossman (1985) and Gaines and Vedder (1964) showed that heating of muscovite produces a shift to lower wave numbers for the O–H stretching frequency, indicating an overall weakening of the O–H bond during dehydroxylation over the average structure of the crystal. Thermal analysis (TGA, DTA, DTG) curves are consistent with the model in that dehydroxylation does not occur homogeneously throughout the sample. Thus, dehydroxylation initially occurs relatively rapidly. However, as some OH groups are lost and sections of the structure contain fivecoordinated Al cations, the remaining OH groups become more tightly bound as Al–OH bonds strengthen, thereby slowing further dehydroxylation. As temperature increases, the number of five-coordinated Al atoms increases to a point where the Al–OH bonds have all been affected and dehydroxylation proceeds rapidly again. The model predicts that there is a bimodal loss of H2O involving dehydroxylation and the temperature interval will be large, as is observed (Fig. 29). Because the thermal energy required to produce dehydroxylation changes with variations in the bond strength during the process, the area under the curve of the DTA (or DTG) cannot be used as an estimate of the number of OH groups involved in either of the two distributions. Mazzucato et al. (1999) examined the kinetics of muscovite dehydroxylation by in situ powder X-ray diffraction. They found that the results were compatible with the above structural model and followed a multi-step process where (i) two adjacent OH groups form a H2O molecule within the octahedral sheet, followed by (ii) diffusion along the c* axis through the six-fold silicate ring (Rouxhet 1970), and then (iii) diffusion of H2O in the interlayer to the crystal surface. The rate limiting step is step (ii). However, this study is consistent with Heller-Kallai and Rozenson (1980), who advocated nonhomogenous loss of H2O (see above), and Kalinichenco et al. (1997), who suggested a continuous nucleation process. Dehydroxylation models for cis-vacant 2:1 layers Guggenheim and co-workers considered only trans-vacant dioctahedral micas in the dehydroxylation process. The possibility of dioctahedral 2:1 layers with the vacant site located in the cis position (“cis-vacant” or “cv” 2:1 layers) had been suggested for montmorillonite and other dioctahedral smectites (e.g., Méring and Glaeser 1954; Méring and Oberlin 1971; Besson 1980; Besson et al. 1982; Drits et al. 1984; Tsipursky and Drits 1984). Although most illite consist of trans-vacant layers, Zvyagin et al. (1985), Drits et al. (1993), and Reynolds and Thompson (1993) described illites with cis-vacant sites.
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Figure 29. Thermal analysis curves [thermal gravimetric analysis (DTA) and derivative thermal gravimetric (DTG)] of muscovite (A) and pyrophyllite (B) showing two thermal analysis events in each phase. See text for discussion (from Guggenheim et al. 1987).
The postulated configuration (Drits et al. 1995) of the cis-vacant 2:1 layer is shown in Figure 30. In this configuration, the shared edge between any two Al-containing octahedra is either two oxygen atoms or an oxygen atom and a hydroxyl group. This differs significantly from the configuration involving the trans-vacant site by both the position of the vacancy and the shared edge between Al octahedra which consists of either two oxygen atoms or two hydroxyl groups. Dehydroxylation is still expected to involve adjacent OH groups, so that a residual oxygen atom is retained in the structure and one oxygen atom and two hydrogen atoms are liberated (= H2O). For a cis-vacant 2:1 layer, dehydroxylation produces a structure where there is a five-coordinated Al(2) site, with one Or atom and four additional oxygen atoms, and an octahedral site [Al(2)] in trans-orientation with respect to Or (Fig. 31). Pauling’s electrostatic valency rule may be used to predict how the structure responds to dehydroxylation. As before, the tetrahedral sites may be assumed to contain only Si. Figure 31 shows the postulated structure with each oxygen atom labeled with the sum of the bond strengths that reach the atom from the neighboring cations (including the Si cations, which are not shown). Al(2) cations are surrounded by oversaturated oxygen atoms, except for one very undersaturated Or. Thus, it is expected that the position of this cation will be further away from the oversaturated oxygen atoms and closer to the residual oxygen atom. In contrast, the Al(1) cation is located in a very distorted and large site with two opposing under-saturated residual oxygen atoms, and this arrangement is inherently unstable. To compensate, the Al cation positions itself closer to one of the residual oxygen atoms (illustrated in Fig. 31 as the upper Or). Thus, each Or is partly
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Figure 30 (left). The octahedral sheet of an Al-rich dioctahedral mica with the cis octahedral sites vacant (after Drits et al. 1995). Figure 31 (right). The structural model of a cis-vacant Al-rich octahedral sheet after initial heating. The OH groups have been lost and a residual O atom remains. Compare this figure to Figure 30. Al(1) atom is located in a very distorted octahedral site, and this atom will readjust its position to a structure as depicted in Figure 32 (after Drits et al. 1995).
compensated by an adjacent Al(1) and the resultant nearest neighbor arrangement around Al(1) becomes five-coordinated (Fig. 32). Drits et al. (1995) noted that the movement of Al(1) toward Or also requires a closer approach to O(5), which is destabilizing. They suggested that this structure (Fig. 32) transforms to the trans-vacant dehydroxylated structure (Fig. 28) with increasing temperature by the migration of Al(1) to the vacant site depicted in Figure 32. The models as presented in Figures 28 and 32 are significantly different and thus they have unique diffraction patterns and different a and b cell parameters. Drits et al. (1995) considered several dioctahedral cis-vacant phyllosilicates, including illite and montmorillonite. In montmorillonite, the b value initially increases with increasing temperature, but decreases from 500 to ~650°C, after which it again increases with increasing temperature. The decrease in the b value is related to the change in structure illustrated from Figure 31 to Figure 32. Drits et al. (1995) suggested that the interlayer cation prevents tetrahedral rotation in illite, but because montmorillonite does not have an interlayer cation that resides within the silicate ring, tetrahedral rotation is not inhibited. Thus, unlike cis-vacant illite, the lateral dimensions in cis-vacant montmorillonite can decrease, and cis-vacant montmorillonite can more easily adjust to dehydroxylation than
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Figure 32. The Al(1) atom has move closer to the residual oxygen atom and the octahedral coordination of Al(1) (Fig. 31) becomes five-coordinated. As temperature increases, Al(1) moves to the vacant site (note arrow) and the structure becomes trans vacant.
cis-vacant illite. At temperatures greater than ~ 700°C, both cis-vacant illite and cisvacant montmorillonite transform to the model shown in Figure 28 (derived from the dehydroxylation of the trans-vacant dioctahedral phyllosilicate) by the migration of Al(1) of Figure 32 into the vacant site. Drits et al. (1995) argued that although crystallinity and particle size effects are important considerations for the dehydroxylation temperatures for illite and montmorillonite, these effects do not explain why cis-vacant forms dehydroxylate at higher temperatures (by 150-200°C) than trans-vacant forms. Drits et al. (1995) suggested that one possible reason involves “the probability for hydrogen to jump to the nearest OH group to form a water molecule strongly depends on the distance between the adjacent OH groups. The shorter the distance, the lesser the thermal energy required for the dehydroxylation of OH pairs”. Studies involving electrostatic modeling would be useful to relate how the proximity between adjacent OH pairs affects the path the proton must travel to form an H2O molecule. A thermodynamic assessment for aluminum-rich cis-vacant and trans-vacant dehydroxylation reactions is useful. Figure 33 shows a schematic reaction path for both reactions with respect to temperature. The low-temperature, cis-vacant hydroxyl-rich form (phase a) appears to have a greater thermal stability than the corresponding transvacant form (phase b). The trans-vacant dehydroxylate structure (c) is the end product in Figure 33 for both reactions, and this phase appears also to have a different thermal stability for the two reaction series.
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Temperature increasing →
Figure 33. A schematic showing two reactions, A and B, with apparent stability fields with respect to temperature. Reaction A involves phase (a), which is cis-vacant, and Reaction B involves a trans-vacant reactant, (b). Reaction A produces a transition phase, a’, that is OH poor before transforming to a trans-vacant dehydroxylate, phase (c). Reaction B transforms directly to phase (c). Phase (c) appears to have two different thermal “stability” ranges depending on the reaction involved, which is a clear indication that kinetic effects are important in determining where the transformation to phase (c) occurs. Likewise, the upper temperature limits for phase (a) and phase (b) are not related to stability, but must be related to kinetic effects. Note that Pauling bond strength calculations for the residual oxygen atom (note the values, in evu, at the top right corner above each box) are consistent with decreasing PBS at elevated temperatures.
Muller et al. (2000c) showed that rehydroxylated illite consisted of only trans-vacant layers, regardless of the nature of the starting material. Thus, reaction B involves a (metastable) equilibrium reaction phase b → phase c, which is reversible. In contrast, reaction A has not been reversed, suggesting that if this is an equilibrium reaction, a significant kinetic barrier prevents a reversal. Thus, the different temperatures for phase a → phase a’ and phase b → phase c are a consequence of resultant processes where the two forms (a and b) follow separate paths to phase c. This may be the reason why there is a significant difference in thermal stability for phase c for reaction A vs. reaction B. The reaction series for reaction A and reaction B must follow a structural pathway consistent with Pauling’s second rule. Note that increasing temperature produces structures where the charge on the residual oxygen atom (Fig. 33) deviates from 2.0 by greater amounts with increasing temperature for each structure in a given series. The situation differs, however, for dioctahedral micas containing significant amounts of divalent cations, and this would be expected, considering the differences in composition. Tsipursky et al. (1985) inferred that dehydroxylation of celadonite and glauconite is accompanied by cation migration from cis to trans sites, with cation migration occurring if Fe is greater than Al. For celadonite, Muller et al. (2000a) found that trans-vacant layers are transformed to cis-vacant layers upon dehydroxylation and both types of layers exist upon rehydration (at 80% cis-vacant, 20% trans-vacant). The transformation involves cation migration and several intermediate structures (see below). For the glauconite sample studied, which had a significant amount of octahedral aluminum, the trans-vacant layers were transformed to cis-vacant layers by dehydroxylation and then to trans-vacant layers upon rehydroxylation. Muller et al. (2000b) suggested that the dehydroxylated structure of Fe- and Mgbearing dioctahedral micas contains structural fragments with different cation occupancy within the same layer, with each fragment being dioctahedral. In a study using selectedarea electron diffraction, Muller et al. (2000c) found that at near 650°C, Mg preferentially migrates unequally to trans sites, after which it remains fixed. After Mg
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migration, Fe3+ ions redistribute unequally over all cis sites and one of the two trans sites (a reduction in symmetry from a C cell to a P cell, which occurs here, requires two trans sites per cell on average). Because the mica structure has partial site occupancy over both cis and trans sites, the residual oxygen atoms (Muller et al. 2000b) partially occupy former OH-group sites (instead of the average position between two former OH-group sites). Perhaps this is related in part to iron oxidation where an oxy-component forms by loss of hydrogen rather than by a dehydroxylation process only, although some of the studied materials have relatively low Fe2+-content. Muller et al. (2000b) suggested that misfit between the octahedral sheet and the tetrahedral sheet may play a role in the cation migration at high temperatures; the octahedral cations (Fe, Mg) are considerably larger in size than Al. Thus, if the tetrahedral sheet limits thermal expansion of an Fe, Mg-rich octahedral sheet, then rearrangement of the octahedral cations is required to produce a more efficient packing of cations and anions. Muller et al. (2000b) noted that cation migration is dependent on Al content of the octahedra, with cation migration absent where octahedral Al-content is greater than that of the other octahedral cations. At temperatures near 750°C, cation migration of a different type was inferred to occur. Superlattice reflections appear in hk0 electron diffraction patterns, with different crystals showing that two superstructures may form, some with superperiodicity along a and some with superperiodicity along b. For the latter it was suggested that the scattering efficiencies of alternating domains of Mg-rich and Fe-rich regions may produce the structural modulation. In a study of interstratified illite-smectite minerals to determine cis-vacant vs. transvacant content of natural and untreated samples, McCarty and Reynolds (1995) determined that there is a linear trend with a decrease in cis-vacant layers with an increase in Mg and Fe substitution of octahedral Al, although such trends are not universally observed (e.g., Ylagan et al. 2000). Characterizing the octahedral ordering pattern is of potential value in determining if the smectite to illite transformation occurs dominantly by a solid-state transformation or by dissolution and crystallization. If the latter mechanism dominates, then cis-vacant illite-smectite may form crystallites containing greater proportions of trans-vacant sites with increasing illitization, for example, as a result of temperature changes. Comparison of Na-rich vs. K-rich dioctahedral forms High-temperature studies of paragonite-2M1 (Comodi and Zanazzi 2000) showed that the dehydroxylation process in this Na-rich mica is analogous to that of muscovite2M1 and trans-vacant micas. Like muscovite, the O–H vector points toward a basal oxygen atom; in paragonite, this oxygen atom is defined as O(4) (Comodi and Zanazzi 2000), rather than O(2) as in muscovite. Thus, the Na-O(4) bond is the longest nearestneighbor bond and remains such to 600°C, the highest temperature studied for paragonite. The increase in the Na-O(4) bond appears related to tetrahedral corrugation (with Δz increasing from ∼0.232 to 0.243). A similar result was suggested by Guggenheim et al. (1987) for muscovite, although the trend in muscovite was equivocal. The weakening of Na-O(4) suggests a possible path for H2O migration during dehydroxylation. Upon dehydroxylation, paragonite transforms to a dehydroxylate structure much like the muscovite-dehydroxylate and the pyrophyllite-dehydroxylate structures (see above). With increasing temperature, the interlayer separation increases. Also, the tetrahedral rotation angle, α, for both muscovite and paragonite decreases (muscovite, 11.8 to 9.2° at 650°C, paragonite, 16.2 to 12.9° at 650°C) indicating that the tetrahedral sheet must extend laterally in both structures to compensate for an expanding octahedral sheet. Differences between the two structures are more apparent by comparing layer offset.
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Brigatti & Guggenheim
Lin and Bailey (1984), in a study of the room-temperature structure of paragonite-2M1, noted that the large layer offset in paragonite in comparison to that in muscovite is related to both the smaller size of Na relative to K and the corrugation of the basal oxygen-atom surfaces owing to tetrahedral tilting around the vacant M(1) site. The tetrahedral tilting places the O(3) basal oxygen atoms of adjacent layers in paragonite close to each other, thereby producing repulsion that causes a layer offset. The high-temperature study of Comodi and Zanazzi (2000) appears to confirm this explanation because the corrugation of the basal oxygen atoms, Δz, increases at higher temperatures and produces a greater layer offset. In contrast, the K in muscovite is much larger than Na in paragonite and the adjacent 2:1 layers are sufficiently separated so that O(3)-O(3) repulsions are minimized. Heat-treated trioctahedral samples: the O,OH,F site and in situ high-temperature studies Takeda and Morosin (1975) obtained cell dimensions to 802°C and refined the crystal structure of a synthetic F-rich phlogopite-1M at 700°C. They developed a model involving misfit between the octahedral and tetrahedral sheets to explain thermal decomposition. In this model, the octahedral sheet expands at high temperature and the tetrahedral sheet responds to this expansion by becoming more hexagonal in symmetry (i.e., the tetrahedral rotation angle, α, approaches zero) because the apparent size of the individual tetrahedra does not expand commensurately. Takeda and Morosin (1975) stated that thermal decomposition resulted from loss of contact of the anions around the interlayer cations as α approaches zero. Hazen (1977) suggested that misfit and the lack of congruency between the tetrahedral and octahedral sheets results in decomposition as a general mechanism. However, this general mechanism has since been discredited by Guggenheim et al. (1987), although it may apply for certain unusual compositions. Toraya (1981) noted that a fully extended tetrahedral sheet may not be the only condition for establishing an upper-temperature limit before decomposition, and he also discussed how octahedra may change in size and shape in response to thermal effects. Russell and Guggenheim (1999) studied a near-OH end-member, natural phlogopite1M crystal to 600°C and found that the K–O bond distances increased similarly with increasing temperature. In trioctahedral micas where each octahedral site is similarly occupied and the O–H vector points along the [001] toward the interlayer cation, all the K–O bond distances would be expected to lengthen as the proton moves away from the oxygen atom of the OH group (owing to H+ to K+ repulsions). Thus, it is not apparent from a structural analysis how dehydroxylation is initiated. Dehydroxylation in phlogopite, however, is consistent with generally higher thermal stabilities for F-rich trioctahedral micas compared to OH-rich trioctahedral micas of otherwise similar compositions. Comparison of the F vs. OH end-member phlogopite structures at high temperatures showed that the effect of this substitution produced different modes of octahedral expansion with increasing temperature. For F-rich octahedra, Takeda and Morosin (1975) found that the octahedra become elongate parallel to the c axis above 400°C and there is a change in thermal expansion for each cell parameter (a, b, or c) at this temperature. In contrast, although the OH-rich octahedra are larger at elevated temperatures, these octahedra do not change shape, and there is no change in the rate of linear expansion of the cell parameters (Russell and Guggenheim 1999). The attractive forces between K+ and F- and the repulsive forces between K+ and H+ affect the interlayer K octahedron for these two phases: in accordance, the K–O octahedron in phlogopite is elongate parallel to c*, but in fluorophlogopite the octahedron is compressed along c*. The octahedral distortions for both the Mg-rich octahedra and the K-rich octahedron attributed by Russell and Guggenheim (1999) to the difference in composition between the OH-rich
Mica Crystal Chemistry and Influence of P-T-X on Atomistic Models
51
vs. F-rich site assumes that the substitutions of Na, Al and Fe in the natural sample (K0.82Na0.115)Σ= 0.935(Mg2.28Al0.495Fe0.12)Σ=2.895 do not greatly affect the results. Heat-treated trioctahedral samples: polytype comparisons The location and influence of the proton in micas is important in understanding how mica structures alter (e.g., Bassett 1960; Norrish 1973), the formation of mica polytypes (e.g., Takéuchi 1965), and thermal decomposition (e.g., Guggenheim et al. 1987). Micapolytype derivations and descriptions are included in detail elsewhere in this volume. However, heat-treatment studies have been useful in understanding the influence of the OH site on mica polytypes. These studies have involved structure determinations of ironbearing phlogopite (“biotite”) that have been reduced (“hydroxy” component with Fe2+ production) or oxidized (“oxy” component with Fe3+ production), so that the “OH” site may or may not contain H+, respectively. Using samples heated under hydrogen for an unreported period at 700°C, Takeda and Ross (1975) studied 1M and 2M1 polytypes of similar compositions to determine the effect of stacking on atom positions. They found that adjacent layers exert an influence on the 2:1 layer and that two oxygen atoms associated with the octahedra are displaced in the 2M1 polytype relative to the 1M polytype within the (001) plane. This causes the octahedra to have a deformation that results in variations in the octahedral bond distances (some longer and some shorter than most octahedra). In a follow-up study using similar material from the same locality but with treatment by hot argon gas (oxy-micas), Ohta et al. (1982) found that the interlayer separation increased significantly for the hydrogenated material relative to the oxy-mica; this is caused by H+ to K+ repulsions. Ohta et al. (1982) found also that for both polytypes, Fe was incorporated in the tetrahedral sites upon oxidation. In contrast, in a study involving heat-treatment and oxidation, Russell and Guggenheim (1999) did not find a change in tetrahedral-site occupancy by oxidation and heat-treatment, although they did find results similar to those of Ohta et al. (1982). Russell and Guggenheim suggested that the apparent change in the tetrahedral site occupancy as found by Ohta et al. (1982) was a result of using crystals of different starting composition. ACKNOWLEDGMENTS We thank V.A. Drits for comments on an early version of this manuscript and D.R. Peacor and G. Ferraris for comments on the final version. Special thanks to Marco Poppi for helping in deriving models of mica crystal chemistry. Partial support for this work was made possible by the donors of The Petroleum Research Fund, administered by The American Chemical Society, under grant PRF-32858-AC5 and by the U.S. National Science Foundation, under grant #EAR-0001122. We also thank the Italian MURST (project “Layer silicates: Crystal chemical, structural and petrological aspects”) and the CNR for financial support.
52
Brigatti & Guggenheim APPENDIX I: DERIVATIONS This section shows the derivation of some formulae reported in the text.
Derivation of “tetrahedral cation displacement”, Tdisp. T d isp . ( A ) =
( T − O basal )2 −
(O−O
basal
)2
3
−
(T − O apical ) 3
Figure A1. Geometrical considerations to derive Tdisp. T represents the tetrahedral cation, the three Obasal atoms define the basal oxygen-atom plane and Oapical repre-sents the oxygen atom of the tetrahedral apex. Oapical–H is the height of the tetrahedron.
The variable Tdisp. is the displacement of the tetrahedral cation from its ideal position (i.e., from the center of mass of the tetrahedron). In a tetrahedron (Fig. A1), the center of mass divides the tetrahedral height (Oapical–H) into two parts. The part containing the vertex (T–Oapical) is three times larger than the distance of the tetrahedral (T) cation from the basal oxygen-atom plane (T - H). In the above formula the part under square root is T-H. Using The Pythagorean theorem and assuming that all Obasal–Obasal edges are equal, we obtain:
(O
basal
− H) =
⎞ 2 ⎛ 3 ⋅ (O basal − O basal )⎟ ⎜ 3 ⎝ 2 ⎠
By equating the two latter relations:
(T − H) = (T − O basal ) − 2
2 1 ⋅ (O basal − O basal ) 3
For a regular tetrahedron:
(T − H) = (T − Oapical ) 3 Thus, for a regular tetrahedron Tdisp. is zero. If the tetrahedral cation shifts from the center of mass of the tetrahedron Tdisp. differs from zero and relates to the modulus of the shift. Derivation of ΔE1, ΔE2, ΔE3 ΔE1 =
−3 ⋅ q T ⎛ −9⋅ q T ⎞ −⎜ ⎟ d TπOb ⎝ T − O apical ⎠
⎛ ΔE 2 = ⎜⎜ ⎜ ⎝
ΔE3 =
q T ⋅ (q A / 4 )
(IS / 2 + d TπOb )2 +
O apical − Oapical
2
⎞ ⎛ ⎟ −⎜ ⎟ ⎜ ⎟ ⎜ ⎠ ⎝
⎛ q 2T q 2T −⎜ IS + 2 ⋅ d TπOb ⎜⎝ IS + 2 / 3 ⋅ T − Oapical
(
⎞ ⎟ ⎟ ⎠
)
q T ⋅(q A / 4)
(IS / 2 + (T − O )/ 3) + O 2
apical
apical
− Oapical
2
⎞ ⎟ ⎟ ⎟ ⎠
Mica Crystal Chemistry and Influence of P-T-X on Atomistic Models
53
The “asymmetric unit” and charge distribution produces a pattern of charge as shown in Figure 9 (see text). The electrostatic energy (E) associated with two charged atoms is directly proportiol to the product of charges (Q) and inversely proportional to the distance (d12) between atoms (1,2). The energy E can be estimated as E = ηQ1 × Q2 d12, where η is a constant set equal to 1 (Coulomb’s law). Energy E1 is the electrostatic interaction between the tetrahedral cation and the basal oxygen atoms. Each oxygen atom contributes −1 × q t d pob where -1 is the formal oxygen-atom charge (i.e., the charge of the basal oxygen atom divided by the number of nearest neighbor tetrahedral cations, qt is the tetrahedral-cation charge, and dpob is the distance between the tetrahedral cation and the basal oxygen atom. ΔE1 represents the difference between the E1 calculated for the configuration vs. an ideal configuaration. The ideal configuration occurs where the position of the tetrahedral cation is located in the center of mass of the tetrahedron [ i.e., dTπOb is substituted by (T-Oapical)/3 ]. Thus, for a regular tetrahedron, ΔE1 is equal to zero. If the tetrahedron is distorted, ΔE1 differs from zero and increases (in absolute value) as the tetrahedral cation shifts from the center of mass of the tetrahedron. ΔE2, ΔE3 can be calculated similarly. E2 is the electrostatic interaction between the tetrahedral cation and the interlayer cation. Each tetrahedral cation interacts with three neighboring interlayer cations (contributions from more distant interlayer cations are omitted). Each interlayer cation contributes q T ⋅ (q A / 12)
(IS / 2 + d Tπ Ob )2 + ( Oapical − Oapical
)
2
where each interlayer cation is surrounded by 12 tetrahedral cations and the term involving the square root is the distance between the interlayer cation and the tetrahedral cation). An expression for ΔE2 is obtained similarly to ΔE1 (i.e., substi-tuting in the second term (T-Oapical)/3 for dTπOb). E3 is the electrostatic interaction between two opposing tetrahedral cations (the influence of additional surrounding tetrahedral cations is omitted). The atoms involved here have the same charge (qT). The product of the charges ( q T2 ) is thus divided by the distance between atoms (as for E1 and E2). ΔE3 is obtained the same way as ΔE1 and ΔE2. Derivation of α ⎛ 1 / 3 + k ⋅ 4 / 3 − k2 α = tan −1 ⎜ k2 − 1 ⎝
⎞ ⎟ − 60 ⎠
or
⎛ 3 ⎞ α = cos −1 ⎜ ⋅ k⎟ ⎝ 2 ⎠
The following assumptions are made:(1) The hexagonal ring distorts by varying the internal-angle value from 120° by an amount of either +2α or -2α alternating around the ring. (2) Symmetry plane is present. (3) Each tetrahedron is rigid (i.e., with constant tetrahedral edges and with each tetrahedral oxygen atom and with each tetrahedral cation occupying ideal positions). Thus tetrahedral and octahedral coordinates for the basal and apical oxygen atoms in a (001) plane (Fig. A2) may be written. From this, as expression for k as a function of α, can be obtained k=
3 ⋅ 3
2 α − 60 ⎞ 60 + α ⎞ α α − 60 ⎞ 2 60 + α ⎞ α ⋅ π ⎟ ⋅ sin⎛⎜ ⋅ π⎟ + cos ⎛⎜ ⋅ π⎟ ⋅ cos⎛⎜ ⋅ π⎞⎟ + 2 − cos ⎛⎜ ⋅ π⎟ + cos ⎛⎜ ⋅ π⎞⎟ + 3 ⋅ cos⎛⎜ ⎝ 180 ⎠ ⎝ 180 ⎠ ⎝ 180 ⎠ ⎝ 180 ⎠ ⎝ 180 ⎠ ⎝ 180 ⎠
60 + α ⎞ α α − 60 ⎞ 60 + α ⎞ 60 + α ⎞ 60 + α ⎞ ⋅ π⎟ ⋅ cos ⎛⎜ ⋅ π⎟ + 3 ⋅ sin⎛⎜ ⋅ π ⎟ ⋅ cos ⎛⎜ ⋅ π⎞⎟ + sin⎛⎜ ⋅ π⎟ ⋅ sin⎛⎜ ⋅ π⎟ + 3 ⋅ sin⎛⎜ ⎝ 180 ⎠ ⎝ 180 ⎠ ⎝ 180 ⎠ ⎝ 180 ⎠ ⎝ 180 ⎠ ⎝ 180 ⎠
and simplifying:
54
Brigatti & Guggenheim k=
2 ⎛ α ⎞ ⋅ cos ⎜ ⋅ π⎟ . ⎝ ⎠ 3 180
Figure A2. The distribution of basal and apical oxygen atoms for a six-fold silicon tetrahedral ring with α = 0. Black circles represent tetrahedral basal-oxygen atoms whereas gray circles represent tetrahedral apical-oxygen atoms.
Explanation of otcor ot cor = ot −
6 ⋅ 〈O − O〉 unshared 3
In a regular octahedron, the thickness is equal to 6 3 l , where l is the length of an octahedral edge. The parameter otcor is the difference between the observed value for the octahedral thickness of an octahedron with a given mean basal edge and the octahedral thickness of an ideal octahedron with each edge being the same length as the mean basal edge. Explanation of ΕΜ−Ο(4) ⎛ ⎞ 1 1 1 E M −O (4 ) ∝ QO ( 4) ⋅ QM ⋅ ⎜ + + ⎟ ⎝ 〈M(1) − O〉 〈M(2) − O〉 〈M(3) − O〉 ⎠
The energy related to two charged points is proportional to the product of the magnitude of the charges divided by the distance between the charged particles (Coulomb’s law). The energy related to the interaction between the octahedral cation (M) and the O(4) atom can be estimated from the above formula.
Mica Crystal Chemistry and Influence of P-T-X on Atomistic Models
APPENDIX II: TABLES 1-4 TABLE 1. STRUCTURAL DETAILS OF TRIOCTAHEDRAL TRUE MICAS
55
56
Brigatti & Guggenheim
Table 1a. Structural details of trioctahedral true Micas-1M, space group C2/m
Cell parameters Reference (sample number)
Species, locality
Rock type
Composition
3+
2+ 0.12
R
A
b
c
E
(Å)
(Å)
(Å)
(°)
(%)
1. Alietti et al. 1995 (n # 1a)
Phlogopite, Mt. Monzoni (Italy)
Skarn
(K0.93Na0.04) (Al0.24Fe 0.09 Fe Mg2.48Mn0.01Ti0.02) (Si2.74Al1.26) O9.99 F0.06 (OH)1.95
5.306(1)
9.195(3)
10.272(3)
100.01(2)
2.9
2. Alietti et al. 1995 (n # 1b)
Phlogopite, Mt. Monzoni (Italy)
Skarn
(K0.93Na0.04) (Al0.24Fe3+0.07 Fe2+0.11 Mg2.55Mn0.01Ti0.02) (Si2.65Al1.35) O9.96 F0.09 (OH)1.95
5.309(2)
9.180(5)
10.291(4)
100.00(4)
2.8
3. Alietti et al. 1995 (n # 2a)
Phlogopite, Mt. Monzoni (Italy)
Skarn
(K0.95Na0.02Ba0.01) (Al0.18Fe3+0.15 Fe2+0.03Mg2.63Ti0.01) (Si2.60Al1.40) O9.93 F0.11 (OH)1.96
5.305(2)
9.189(3)
10.286(3)
99.96(2)
2.9
4. Alietti et al. 1995 (n # 3a)
Aluminian phlogopite Mt. Monzoni (Italy)
Skarn
(K0.95Na0.02Ba0.01) (Al0.47Fe3+0.15 Fe2+0.07Mg2.23Mn0.04 Ti0.01) (Si2.50 Al1.50) O10.02 F0.04 (OH)1.94
5.299(1)
9.179(2)
10.279(3)
99.90(2)
3.0
5. Alietti et al. 1995 (n # 4a)
Phlogopite, Mt. Monzoni (Italy)
Skarn
(K0.90Na0.02Ba0.02Ca0.02) (Al0.20 Fe3+0.11Fe2+0.04 Mg2.64Mn0.01) (Si2.60Al1.40) O9.92 F0.06 (OH)2.02
5.307(2)
9.199(2)
10.291(2)
99.89(2)
2.5
6. Bigi and Brigatti 1994 (n # M7)
Ferroan phlogopite, Valle Cervo (Italy)
Syenite
(Na0.02K0.95) (Al0.05Fe3+0.50 Fe2+0.70 Mg1.54Mn0.02Ti0.20) (Si2.81Al1.19) O10.73 (OH)1.27
5.335(2)
9.244(2)
10.206(3)
100.08(2)
3.3
7. Bigi et al. 1993 (n Magnesian annite, # MP9) Ivrea (Italy)
Gabbro
(Na0.02K0.81Ba0.10) (Fe2+1.05 Mg0.92 Mn0.01Ti0.67) (Si2.50Al1.37 Fe0.13) O9.99 F0.06 (OH)1.95
5.349(2)
9.244(6)
10.132(7)
100.38(4)
3.1
8. Brigatti and Davoli 1990 (n # M14)
Ferroan phlogopite, Valle Cervo (Italy)
Monzonite
(K0.90Na0.03) (Fe3+0.45Fe2+0.79 5.343(3) Mg1.43Mn0.01Ti0.23 Li0.01) (Si2.78 Al1.19 Fe3+0.03) O10.44 Cl0.04 (OH)1.52
9.258(1)
10.227(2)
100.26(2)
3.3
9. Brigatti and Davoli 1990 (n # M32)
Ferroan phlogopite, Valle Cervo (Italy)
Syenite
(K0.92Na0.01Ca0.01) (Al0.01Fe3+0.46 Fe2+0.71Mg1.50Mn0.03 Ti0.15Li0.01) (Si2.80Al1.20) O10.25 Cl0.02 (OH)1.73
5.346(2)
9.252(2)
10.238(4)
100.02(3)
2.4
10. Brigatti and Davoli 1990 (n # M62)
Ferroan phlogopite, Valle Cervo (Italy)
Granite-monzonite transition
(K0.94Na0.02) (Al0.05 Fe3+0.39 Fe2+0.95 5.337(1) Mg1.35Mn0.03Ti0.20 Li0.01) (Si2.79 Al1.21) O10.55 Cl0.01 (OH)1.44
9.242(2)
10.211(2)
100.15(2)
3.5
11. Brigatti and Davoli 1990 (n # M73)
Ferroan phlogopite, Valle Cervo (Italy)
Monzonite
(K0.91Na0.02) (Al0.02 Fe3+0.36 Fe2+0.86 5.345(1) Mg1.39Mn0.02Ti0.25 Li0.01) (Si2.74 Al1.26) O10.32 Cl0.05 (OH)1.63
9.258(2)
10.222(2)
100.23(2)
2.1
12. Brigatti and Davoli 1990 (n # M13)
Ferroan phlogopite, Valle Cervo (Italy)
Granite
(K0.99Na0.01) (Al0.05Fe3+0.34Fe2+0.91 Mg1.35Mn0.03 Ti0.23Li0.02) (Si2.85 Al1.15) O10.54 Cl0.01 (OH)1.45
5.355(1)
9.251(4)
10.246(4)
100.15(3)
6.2
Lamproite
(K0.93Na0.06Ba0.01) (Al0.01Fe3+0.18 Fe2+0.06Mg2.33Mn0.01 Ti0.41) (Si2.94 Al1.06) O10.96 F0.79 (OH)0.25
5.320(2)
9.207(3)
10.100(2)
100.24(2)
2.0
14. Brigatti and Poppi Aluminian phlogopite, Leucitic 1993 (n # 20) Grotta dei Cervi basanite (Italy)
(K0.88Na0.07Ca0.03Ba0.03) (Al0.93 Fe3+0.41Fe2+0.39 Mg1.10Mn0.03Ti0.14) (Si2.68 Al1.32) O11.36 F0.14 (OH)0.50
5.323(1)
9.219(1)
10.219(4)
100.03(2)
2.7
15. Brigatti and Poppi Ferrian phlogopite, 1993 (n # 21) Grotta dei Cervi (Italy)
(K0.92Na0.05Ba0.03) (Al0.14Fe3+0.38 Fe2+0.31Mg2.00Mn0.01Ti0.17) (Si2.68 Al1.32) O10.57 F0.16 (OH)1.27
5.326(1)
9.222(1)
10.223(2)
100.04(1)
2.3
13. Brigatti and Poppi Titanian phlogopite, 1993 (n # 18) Jumilla (Spain)
Leucitic basanite
57
Mica Crystal Chemisrty and Influence of P-T-X on Atomistic Models
Tetrahedral D (°)
W (u 2) (°)
Octahedral <M1 (°)
<M2 (u 2) (°)
Sheet thickness Interlayer Basal Mean bond lengths Tetra- Octa- Separa- oxygen ¢T-O² ¢M1-O² ¢M2-O² hedral hedral tion 'z (u 2) (u 2) (Å) (Å) (Å) (Å) (Å) (Å) (Å)
Intralayer ¢A - O² Inner Outer shift
(A)
a1
Layer offset
Overall shift
a1
a1
10.2
110.1 59.0
59.0
2.252
2.129
3.482
0.008
1.660
2.066
2.065
2.947
3.408
-0.336
-0.001
-0.337
1.
10.7
110.3 58.9
58.9
2.259
2.134
3.482
0.005
1.662
2.066
2.065
2.935
3.417
-0.335
-0.002
-0.337
2.
11.1
110.1 59.0
58.9
2.258
2.130
3.485
0.003
1.663
2.067
2.064
2.927
3.429
-0.335
0.000
-0.335
3.
12.5
109.6 59.3
59.1
2.251
2.112
3.512
0.006
1.666
2.067
2.054
2.903
3.466
-0.334
0.000
-0.334
4.
10.7
110.3 58.8
58.7
2.253
2.149
3.483
0.002
1.661
2.072
2.070
2.937
3.420
-0.334
0.001
-0.333
5.
6.4
110.4 58.8
58.5
2.247
2.169
3.386
0.005
1.654
2.095
2.078
3.016
3.306
-.0.334
-0.001
-0.335
6.
7.1
110.1 58.7
59.0
2.248
2.151
3.319
0.021
1.667
2.067
2.088
2.983
3.311
-0.336
-0.005
-0.341
7.
5.6
110.4 59.0
58.7
2.257
2.160
3.391
0.008
1.657
2.095
2.077
3.037
3.294
-0.337
-0.004
-0.341
8.
6.4
110.4 58.9
58.7
2.255
2.164
3.408
0.014
1.657
2.092
2.081
3.026
3.316
-0.333
0.000
-0.333
9.
6.3
110.3 58.8
58.6
2.248
2.164
3.391
0.017
1.655
2.088
2.079
3.020
3.306
-0.334
-0.003
-0.337
10.
5.6
110.4 58.9
58.6
2.252
2.169
3.386
0.006
1.656
2.097
2.081
3.037
3.292
-0.336
-0.004
-0.340
11.
6.0
110.6 58.9
58.7
2.270
2.164
3.382
0.026
1.663
2.096
2.080
3.028
3.302
-0.335
-0-002
-0.337
12.
5.9
110.6 59.6
59.2
2.275
2.104
3.286
0.018
1.652
2.077
2.054
2.993
3.261
-0.335
-0.002
-0.337
13.
9.1
110.2 59.1
58.9
2.258
2.136
3.411
0.015
1.659
2.077
2.068
2.956
3.371
-0.333
-0.001
-0.334
14.
9.3
110.2 59.1
58.9
2.261
2.135
3.410
0.008
1.660
2.076
2.070
2.952
3.377
-0.334
-0.001
-0.335
15.
58
Brigatti & Guggenheim
16. Brigatti and Poppi Ferroan phlogopite 1993 (n # 22) (Antartica)
Leucitite
(K0.85Na0.11Ba0.04) (Fe2+0.74Mg1.70 Mn0.01Ti0.49) (Si3.25Al0.75) O11.14 F0.31 (OH)0.55
5.330(3)
9.245(2)
10.192(9)
100.35(6)
3.4
17. Brigatti and Poppi Ferroan phlogopite, 1993 (n # 19) Colli Euganei (Italy)
Trachyte
(K0.90Na0.07Ba0.03) (Al0.02Fe3+0.39 Fe2+0.60Mg1.61Mn0.01 Ti0.37) (Si2.75 Al1.25) O11.93 F0.23 (OH)0.84
5.331(1)
9.230(2)
10.160(2)
100.19(1)
3.2
18. Brigatti and Poppi Ferrian phlogopite, 1993 (n # 23) Alto Adige (Italy)
Lamprophire
(K0.88Na0.08Ba0.04) (Al0.12Fe3+0.47 Fe2+0.42Mg1.85Mn0.01 Ti0.14) (Si2.65 Al1.35) O10.56 F0.01 (OH)1.43
5.328(3)
9.219(2)
10.233(3)
99.88(3)
3.4
19. Brigatti and Poppi Ferrian phlogopite, 1993 (n # 24) Alto Adige (Italy)
Lamprophire
(K0.91Na0.06Ba0.04) (Al0.13Fe3+0.72 Fe2+0.30Mg1.67Mn0.01 Ti0.18) (Si2.62 Al1.38) O10.87 F0.04 (OH)1.09
5.328(1)
9.224(2)
10.247(3)
100.01(2)
2.7
20. Brigatti and Poppi Ferroan phlogo-pite, 1993 (n # 25) Grotta dei Cervi (Italy)
Leucitic basanite
(K0.89Ba0.12) (Al0.24Fe3+0.23Fe2+0.76 Mg1.58Ti0.17) (Si2.59Al1.41) O10.52 F0.26 (OH)1.22
5.333(1)
9.241(1)
10.180(1)
100.10(1)
2.2
21. Brigatti et al. 1991 (n # 8)
Ferroan phlogopite, Puebla de Mula (Spain)
Lamproite
(K0.96Na0.02Ca0.03) (Al0.22Cr0.05 Fe2+0.39Mg2.17Mn0.02 Ti0.14) (Si2.86 Al1.14) O10.43 F0.20 (OH)1.37
5.317(1)
9.207(1)
10.232(2)
99.98(2)
2.5
22. Brigatti et al. 1991 (n # 9)
Phlogopite, Cancarix (Spain)
Lamproite
(K0.95Na0.02Ca0.01) (Cr0.03Fe2+0.28 Mg2.42Mn0.01Ti0.18) (Si2.91Al1.09) O10.12 F0.72 (OH)1.16
5.306(1)
9.190(1)
10.163(1)
100.11(1)
2.2
23. Brigatti et al. 1991 (n # 10)
Ferroan phlogopite, Fortuna (Spain)
Lamproite
(K0.96Na0.02) (Al0.09 Cr0.05Fe2+0.59 Mg1.60Mn0.03Ti0.52) (Si2.93Al1.07) O10.88 F0.57 (OH)0.55
5.322(1)
9.228(3)
10.102(1)
100.25(1)
2.2
24. Brigatti et al. 1991 (n # 11)
Ferroan phlogopite, Jumilla (Spain)
Lamproite
(K0.96Na0.03Ca0.01) (Al0.15Cr0.07 Fe2+0.50Mg1.90Mn0.03 Ti0.33) (Si2.87 Al1.13) O10.71 F0.30 (OH)0.99
5.315(1)
9.204(1)
10.168(1)
100.13(2)
1.9
25. Brigatti et al. 1991 (n # 12)
Ferroan phlogopite, Jumilla (Spain)
Lamproite
(K0.95Na0.03) (Al0.04Cr0.05Fe2+0.50 Mg2.09Mn0.02 Ti0.27) (Si2.90Al1.10) O10.43 F0.44 (OH)1.13
5.314(1)
9.190(1)
10.160(3)
100.18(2)
2.1
26. Brigatti et al. 1991 (n # 15)
Ferroan phlogopite, St. Alkaline Hilaire (Canada) gabbroperalkaline syenite
(K0.92Na0.01Ca0.01) (Al0.01Cr0.01 Fe2+0.94Mg1.48Mn0.02 Ti0.39) (Si2.73 Al1.27) O10.15 F0.07 (OH)1.78
5.329(1)
9.235(2)
10.190(3)
100.20(2)
2.3
27. Brigatti et al. 1991 (n # 16)
Ferroan phlogopite, Sande (Norway)
Monzonitealkali syenite
(K0.97Na0.02Ca0.01) (Al0.08Cr0.01 Fe2+1.24Mg1.40Mn0.02 Ti0.23) (Si2.81 Al1.19) O10.32 F0.31 (OH)1.37
5.333(1)
9.256(6)
10.186(4)
100.17(3)
3.0
28. Brigatti et al. 1991 (n # 17)
Magnesian annite, Capo Vaticano (Italy)
Quartz diorite
(K0.91Na0.02) (Al0.19 Cr0.01Fe2+1.30 Mg1.24Mn0.01 Ti0.20) (Si2.76Al1.24) O10.18 F0.02 (OH)1.80
5.323(1)
9.215(2)
10.210(2)
100.14(2)
2.6
29. Brigatti et al. 1996a (n # Tae 23-1a)
Phlogopite, Tapira (Brazil)
Alkaline carbonatitic complex: bebedourite
(K0.93Na0.05Ba0.02) (Fe3+0.16Fe2+0.09 Mg2.65 Ti0.08) (Si2.84Al1.04Fe3+0.12) O10.17 F0.01 (OH)1.82
5.321(1)
9.211(2)
10.287(1)
99.93(1)
2.7
30. Brigatti et al. 1996a (n # Tae 23-1b)
Phlogopite, Tapira (Brazil)
Alkaline carbo natitic complex: bebedourite
(K0.88Na0.05Ba0.01) (Fe3+0.22Fe2+0.09 Mg2.60Ti0.09) (Si2.82Al1.13 Fe3+0.05) O10.18 F0.01 (OH)1.81
5.330(2)
9.230(3)
10.256(4)
99.92(3)
2.7
31. Brigatti et al. 1996a (n # Tae 23-1c)
Phlogopite, Tapira (Brazil)
Alkaline carbonatitic complex: bebedourite
(K0.87Na0.05Ba0.02) (Fe3+0.23Fe2+0.09 Mg2.57Ti0.10) (Si2.81Al1.14Fe3+0.05) O10.18 F0.01 (OH)1.81
5.318(1)
9.219(3)
10.274(4)
99.88(3)
3.0
32. Brigatti et al. 1996a (n # Tpg 63-2B)
Ferroan phlogopite, Tapira (Brazil)
Alkaline carbo natitic complex: bebedourite
(K0.98Ba0.02) (Fe3+0.24Fe2+0.62Mg1.90 5.341(1) Mn0.02 Ti0.18) (Si2.71Al1.20Fe3+0.09) O10.25 F0.02 (OH)1.73
9.244(2)
10.253(3)
100.09(2)
2.3
59
Mica Crystal Chemisrty and Influence of P-T-X on Atomistic Models
4.4
110.3
59.0
58.7
2.270
2.154
3.332
0.020
1.655
2.093
2.072
3.045
3.245
-0.336
-0.007
-0.344.
16.
7.6
110.0
59.3
59.0
2.261
2.125
3.353
0.008
1.657
2.083
2.064
2.976
3.325
-0.335
-0.002
-0.337
17.
9.3
110.2
58.9
58.7
2.250
2.154
3.426
0.010
1.659
2.083
2.074
2.957
3.380
-0.331
0.001
-0.330
18.
9.1
110.3
58.9
58.7
2.262
2.153
3.413
0.001
1.662
2.084
2.073
2.959
3.373
-0.334
0.000
-0.334
19.
8.6
110.4
59.1
59.0
2.271
2.135
3.344
0.007
1.663
2.080
2.071
2.953
3.347
-0.334
-0.001
-0.335
20.
7.6
110.5
58.9
58.8
2.259
2.145
3.413
0.005
1.654
2.079
2.068
2.989
3.333
-0.333
-0.001
-0.334
21.
6.6
110.7
59.2
59.1
2.269
2.116
3.352
0.010
1.650
2.067
2.058
2.989
3.289
-0.335
-0.001
-0.336
22.
5.3
110.0
59.6
59.2
2.261
2.107
3.313
0.010
1.648
2.081
2.056
3.017
3.256
-0.336
-0.002
-0.338
23.
6.8
110.5
59.2
59.0
2.265
2.122
3.357
0.009
1.652
2.074
2.061
2.989
3.300
-0.334
-0.002
-0.336
24.
6.7
110.5
59.2
59.0
2.258
2.123
3.360
0.009
1.649
2.071
2.061
2.990
3.297
-0.335
-0.003
-0.338
25.
7.5
110.2
59.0
58.7
2.252
2.150
3.374
0.013
1.656
2.089
2.071
2.985
3.327
-0.335
-0.004
-0.339
26.
5.3
110.5
58.9
58.6
2.248
2.164
3.365
0.000
1.653
2.094
2.077
3.036
3.276
-0.335
-0.002
-0.337
27.
8.0
110.1
58.7
58.5
2.230
2.170
3.421
0.015
1.650
2.085
2.077
2.984
3.346
-0.336
-0.002
-0.338
28.
8.9
110.6
58.7
58.7
2.259
2.161
3.453
0.000
1.659
2.077
2.077
2.972
3.376
-0.333
0.000
-0.333
29.
8.9
110.3
58.7
58.7
2.248
2.158
3.449
0.000
1.658
2.079
2.079
2.974
3.380
-0.332
0.001
-0.331
30.
8.9
110.3
58.8
58.8
2.255
2.149
3.462
0.006
1.659
2.076
2.073
2.975
3.378
-0.333
0.002
-0.331
31.
8.1
110.2
58.9
58.7
2.254
2.159
3.428
0.009
1.659
2.087
2.079
2.990
3.359
-0.335
-0.001
-0.336
32.
60
Brigatti & Guggenheim
33. Brigatti et al. 1996a (n # Tas 22-1a)
Tetra-ferriphlogopite, Tapira (Brazil)
Alkaline carbo natitic complex: glimmerite
(K0.99Na0.01) (Fe3+0.05Fe2+0.17Mg2.70 5.357(2) Ti0.01) (Si3.11Fe3+0.89) O10.08 F0.14 (OH)1.78
9.270(4)
10.319(4)
99.96(3)
3.2
34. Brigatti et al. 1996a (n # Tas 22-1b)
Tetra-ferriphlogopite, Tapira (Brazil)
Alkaline carbo natitic complex: glimmerite
(K0.98Na0.02) (Fe3+0.06Fe2+0.17Mg2.75 5.358(2) Mn0.01Ti0.01) (Si3.07Fe3+0.93) O10.17 F0.05 (OH)1.78
9.277(3)
10.308(2)
99.99(4)
3.3
35. Brigatti et al. 1996a (n # Tpt 17-1)
Phlogopite, Tapira (Brazil)
Alkaline carbonatitic complex: perovskitemagnetitite
(K0.98Na0.01Ba0.02) (Fe3+0.15Fe2+0.08 Mg2.68Mn0.01 Ti0.08) (Si2.82Al1.11 Fe3+0.07) O10.16 F0.11 (OH)1.73
5.332(1)
9.239(2)
10.291(2)
99.94(2)
2.8
36. Brigatti et al. 1996a (n # Tas 27-2Ba)
Phlogopite, Tapira (Brazil)
Alkaline carbo natitic complex: dunite
(K0.96Na0.03Ba0.01) (Fe3+0.19Fe2+0.07 Mg2.68Ti0.05) (Si2.85Al1.07Fe3+0.08) O10.16 F0.03 (OH)1.81
5.318(2)
9.214(1)
10.279(2)
100.01(2)
2.8
37. Brigatti et al. 1996a (n # Tas 27-2Bb)
Phlogopite, Tapira, (Brazil)
Alkaline carbo natitic complex: dunite
(K0.96Na0.03Ba0.01) (Fe3+0.21Fe2+0.07 Mg2.64Mn0.01Ti0.06) (Si2.85Al1.10 Fe3+0.05) O10.13 F0.06 (OH)1.81
5.330(1)
9.235(1)
10.301(1)
99.92(1)
2.5
38. Brigatti et al. 1996a (n # Tag 15-4)
Ferroan phlogopite, Tapira (Brazil)
Alkaline carbo natitic complex: bebedourite
(K0.92Ba0.04) (Fe3+0.30Fe2+0.38Mg2.17 5.333(1) Mn0.01Ti0.13) (Si2.76Al1.19Fe3+0.05) O10.26 F0.06 (OH)1.68
9.238(2)
10.267(2)
99.96(2)
2.8
39. Brigatti et al. 1996a (n # Tag 15-3)
Ferroan phlogopite, Tapira (Brazil)
Alkaline carbo natitic complex: bebedourite
(K0.92Ba0.02) (Fe3+0.25Fe2+0.34Mg2.19 5.329(2) Mn0.01 Ti0.13) (Si2.74Al1.15Fe3+0.11) O10.04 F0.05 (OH)1.91
9.228(2)
10.258(3)
100.03(3)
2.8
40. Brigatti et al. 1996a (n # Tpq 16-4A)
Tetra-ferriphlogopite, Tapira (Brazil)
Alkaline carbonatitic complex: perovskitemagnetitite
K0.99 (Fe3+0.10Fe2+0.22Mg2.64Mn0.01 Ti0.03) (Si2.91Al0.71Fe3+0.38) O10.06 F0.08 (OH)1.86
5.338(2)
9.247(1)
10.300(2)
99.96(2)
2.8
41. Brigatti et al. 1996a (n # Tpq 16-6B)
Tetra-ferriphlogopite, Tapira (Brazil)
Alkaline carbo natitic complex: glimmerite
(K0.95Na0.02) (Fe3+0.23Fe2+0.20 Mg2.54Ti0.02) (Si3.15Al0.04Fe3+0.81) O10.34 F0.10 (OH)1.56
5.356(1)
9.284(2)
10.309(3)
100.03(2)
3.1
42. Brigatti et al. 1996b (n # Tas 22-1c)
Tetra-ferriphlogopite, Tapira (Brazil)
Alkaline carbo natitic complex: glimmerite
(K0.99Na0.01) (Fe3+0.08Fe2+0.17Mg2.73 5.362(1) Ti0.01) (Si3.05Fe3+0.95) O10.17 F0.04 (OH)1.79
9.288(1)
10.321(2)
99.99(1)
3.1
43. Brigatti et al. 1996b (n # Tpq 16-6B)
Tetra-ferriphlogopite, Tapira (Brazil)
Alkaline carbo natitic complex: glimmerite
K1.02 (Fe3+0.11Fe2+0.20Mg2.68Mn0.01) (Si3.05Fe3+0.95) O10.18 F0.07 (OH)1.75
5.365(1)
9.292(1)
10.326(1)
99.99(1)
2.5
44. Brigatti et al. 1998 (n # wa3H)
Ferroan phlogopite, Warburton (Australia)
Granodiorite
(K0.92Na0.03Ca0.02Ba0.04) (Al0.18 Fe3+0.18Fe2+1.01Mg1.26Mn0.02 Ti0.28) (Si2.77Al1.23) O10.58 F0.08 Cl0.02) (OH)1.32
5.341(1)
9.252(1)
10.229(2)
100.17(2)
2.9
45. Brigatti et al. 1998 (n # wa8E)
Magnesian annite, Warburton (Australia)
Microgranitoid enclave in granodiorite
(K0.93Na0.03Ca0.02Ba0.01) (Al0.21 Fe2+1.37Mg1.15Mn0.03 Ti0.25) (Si2.85 Al1.15) O10.61 F0.16 Cl0.06 (OH)1.17
5.345(1)
9.263(4)
10.234(6)
100.11(2)
3.9
46. Brigatti et al. 1998 (n # wa8H)
Magnesian annite, Granodiorite Warburton, (Australia)
(K0.89Na0.03Ca0.03Ba0.02) (Al0.18 Fe3+0.13Fe2+1.20Mg1.19Mn0.02 Ti0.29) (Si2.82Al1.18) O10.76 F0.14 Cl0.05 (OH1.05)
5.344(1)
9.258(1)
10.232(1)
100.15(1)
3.3
47. Brigatti et al. 1998 (n # wa23e)
Ferroan phlogopite, Warburton (Australia)
Microgranitoid enclave in granodiorite
(K0.92Na0.03Ca0.04Ba0.01) (Al0.31 Fe3+0.16Fe2+1.10Mg1.23Mn0.01 Ti0.19) (Si2.77Al1.23) O10.67 F0.12 Cl0.02 (OH)1.19
5.347(1)
9.260(2)
10.229(3)
100.07(3)
2.8
48. Brigatti et al. 1999 (n # TAG15-4b)
Ferroan phlogopite, Tapira (Brazil)
Alkaline carbo natitic complex: bebedourite
(K0.95Na0.02Ba0.03) (Fe3+0.23Fe2+0.38 Mg2.25Mn0.01 Ti0.13) (Si2.76Al1.17 Fe0.07) O10.28 F0.05 (OH)1.68
5.332(1)
9.230(2)
10.267(1)
99.99(1)
2.8
Alkaline carbonatitic complex: perovskitemagnetitite
(K0.97Na0.01Ba0.02) (Fe3+0.20Fe2+0.11 Mg2.59Mn0.01 Ti0.05) (Si2.90Al1.06 Fe0.04) O10.12 F0.06 (OH)1.82
5.323(1)
9.219(1)
10.282(1)
99.93(1)
2.4
Phlogopite, Tapira 49. Brigatti et al. 1999 (n # TpQ16- (Brazil) 4Ab)
61
Mica Crystal Chemisrty and Influence of P-T-X on Atomistic Models
10.8
109.6
58.7
58.7
2.254
2.168
3.487
0.002
1.670
2.088
2.089
2.952
3.446
-0.334
0.001
-0.333
33.
10.9
110.1
58.9
58.9
2.271
2.156
3.454
0.000
1.677
2.087
2.085
2.941
3.440
-0.333
-0.001
-0.334
34.
8.8
110.7
58.8
58.7
2.265
2.159
3.447
0.001
1.663
2.081
2.080
2.977
3.379
-0.333
0.000
-0.333
35.
8.5
110.7
58.6
58.6
2.256
2.164
3.448
0.000
1.656
2.078
2.077
2.980
3.364
-0.335
-0.001
-0.336
36.
8.5
110.7
58.6
58.6
2.261
2.167
3.457
0.001
1.660
2.081
2.082
2.987
3.372
-0.333
0.000
-0.333
37.
8.6
110.2
58.8
58.7
2.257
2.157
3.442
0.005
1.660
2.083
2.078
2.981
3.372
-0.333
0.000
-0.333
38.
8.5
110.4
58.7
58.6
2.252
2.167
3.431
0.001
1.658
2.085
2.079
2.977
3.365
-0.335
0.000
-0.335
39.
9.1
110.2
58.7
58.7
2.257
2.164
3.467
0.003
1.662
2.084
2.083
2.978
3.394
-0.334
0.000
-0.334
40.
10.2
110.2
58.8
58.7
2.268
2.168
3.447
0.004
1.673
2.091
2.089
2.955
3.422
-0.334
-0.001
-0.335
41.
11.5
109.9
59.0
59.0
2.274
2.151
3.465
0.001
1.680
2.087
2.087
2.931
3.462
-0.334
0.000
-0.334
42.
11.5
119.9
58.9
58.9
2.271
2.159
3.469
0.001
1.679
2.091
2.089
2.934
3.464
-0.333
-0.001
-0.334
43.
7.0
110.3
59.1
58.8
2.257
2.148
3.407
0.009
1.658
2.091
2.073
3.010
3.330
-0.336
-0.002
-0.338
44.
6.6
109.7
58.5
58.5
2.229
2.186
3.431
0.026
1.654
2.092
2.090
3.027
3.329
-0.336
0.000
-0.336
45.
7.1
110.2
59.0
58.8
2.256
2.152
3.408
0.012
1.659
2.091
2.076
3.010
3.332
-0.335
-0.002
-0.337
46.
6.8
110.3
59.0
58.8
2.262
2.153
3.395
0.010
1.657
2.090
2.075
3.010
3.319
-0.333
-0.001
-0.334
47.
9.1
110.5
58.7
58.7
2.260
2.163
3.428
0.000
1.662
2.082
2.079
2.965
3.378
-0.334
0.000
-0.334
48.
8.8
110.7
58.6
58.6
2.259
2.164
3.446
0.002
1.659
2.078
2.078
2.974
3.372
-0.333
0.000
-0.333
49.
62
Brigatti & Guggenheim
50. Brigatti et al. 1999 (n #TPQ164Ac)
Alkaline carboFerroan tetraferriphlogopite, Tapira natitic complex: perovskite(Brazil) magnetitite
(K0.99Na0.01) (Fe3+0.30Fe2+0.54Mg1.99 5.370(1) Mn0.02 Ti0.01) (Si3.01Al0.13Fe0.86) O10.04 (OH)1.96
9.306(1)
10.319(1)
100.00(1)
3.0
51. Brigatti et al. 2000a (n # a4)
Magnesian annite, Sos Peraluminous granite Canales pluton, Sardinia (Italy)
(K0.95Na0.04) (Al0.35Fe3+0.01Fe2+1.45 Mg0.77Mn0.04 Ti0.21) (Si2.71Al1.29) O10.15 F0.05 (OH)1.80
5.352(1)
9.268(3)
10.255(3)
100.27(2)
3.2
52. Brigatti et al. 2000a (n # b1)
Magnesian annite, Tinker Glacier (Antarctica)
Peraluminous granite
(K0.93Na0.03Ca0.01) (Al0.54Fe3+0.01 Fe2+1.41Mg0.83Mn0.03 Ti0.17) (Si2.62 Al1.38) O10.46 (OH)1.54
5.336(1)
9.239(2)
10.200(2)
100.29(2)
2.7
53. Brigatti et al. 2000a (n # c3-31)
Magnesian annite, Tinker Glacier (Antarctica)
Peraluminous granite
(K0.96Na0.03Ca0.01Ba0.01) (Al0.48 Fe2+1.48Mg0.70Mn0.06 Ti0.20) (Si2.63 Al1.37) O10.38 F0.01 (OH)1.61
5.347(2)
9.257(1)
10.211(1)
100.27(2)
3.1
54. Brigatti et al. 2000a (n # cc1)
Magnesian annite, Tinker Glacier (Antarctica)
Peraluminous granite
(K0.96Na0.01) (Al0.64Fe2+1.33Mg0.73 Mn0.04Ti0.17) (Si2.68Al1.32) O10.44 (OH)1.32
5.328(1)
9.222(2)
10.197(2)
100.26(1)
3.2
55. Brigatti et al. 2000a (n # Gfs15a)
Magnesian annite, Sos Peraluminous granite Canales pluton, Sardinia (Italy)
(K0.96Na0.02Ca0.03Ba0.01) (Al0.60 Fe2+1.36Mg0.73Mn0.02Ti0.14) (Si2.69 Al1.31) O10.31 F0.12 (OH)1.57
5.339(1)
9.232(2)
10.208(2)
100.30(2)
3.6
56. Brigatti et al. 2000a (n # H87)
Magnesian annite, Riu Peraluminous granite Morunzu, Sardinia (Italy)
(K0.98Na0.02) (Al0.50Fe2+1.46Mg0.70 Mn0.03Ti0.16) (Si2.72Al1.28) O10.25 F0.15 Cl0.03 (OH)1.57
5.344(2)
9.256(3)
10.237(2)
100.27(2)
3.2
57. Brigatti et al. 2000b (n # 120)
Annite, Pikes Peak, Colorado
Granitic pegmatite
(K0.99Na0.01) (Al0.13Fe3+0.21Fe2+2.29 Mg0.10Mn0.01Ti0.25) (Si3.14Al0.86) F0.26 O10.95 (OH)0.79
5.384(1)
9.324(1)
10.254(1)
100.86(1)
2.6
58. Brigatti et al. 2000b (n # 26 )
Siderophyllite, Pikes Peak, Colorado
Granitic pegmatite
(K0.95Rb0.02Na0.05) (Al0.84Fe3+0.24 Fe2+1.63Mg0.10 Zn0.01 Li0.17Ti0.02) (Si2.94Al1.06) O10.93 F0.90 (OH)0.17
5.358(2)
9.280(3)
10.151(2)
100.10(1)
3.3
59. Brigatti et al. 2000b (n # 33 )
Aluminian annite, Pikes Peak, Colorado
Granitic pegmatite
(K1.00Na0.01) (Al0.35Fe3+0.16Fe2+2.22 Mn0.08Ti0.11 Li0.08) (Si3.09Al0.91) O10.95 F0.26 (OH)0.79
5.372(1)
9.313(1)
10.204(1)
100.52(1)
3.6
60. Comodi et al. 1999
Cesian, tetra-ferriannite
Synthetic
Cs0.89 (Fe3+0.03Fe2+2.97) (Si3.07 Fe3+0.90Al0.03) O10 OH2
5.486(1)
9.506(1)
10.818(1)
99.67(6)
3.7
61. Donnay et al. 1964
Tetra-ferri-annite
Synthetic
K1.00 Fe2+3.00 (Si3.00Fe3+1.00) O10 (OH)2
5.430(2)
9.404(5)
10.341(3)
100.1(1)
9.3
62. Guggenheim 1981 Trilithionite, Radkovice, Jihlava, Moravia (Czech Republic)
Pegmatite
(K0.79Rb0.07Cs0.03Na0.03Ca0.01) (Li1.48Fe2+0.02Fe3+0.008 Mg0.05 Mn0.03Al1.30) (Si3.49Al0.51) O10 (OH,F)2
5.209(2)
9.011(5)
10.149(5)
100.77(4)
3.5
63. Guggenheim and Kato 1984 (n # 1)
Manganoan phlogopite, Nodatamagawa, Iwate Prefecture (Japan)
Metamorphosed (K0.85Na0.19Ba0.06) (Fe3+0.06Mg1.74 manganese Mn2+0.95Mn3+0.18) (Si2.75Al1.15 deposit Ti0.03Fe3+0.07) O10.01 F0.09 (OH)1.90
5.380(2)
9.295(2)
10.318(4)
99.96(2)
5.4
64. Guggenheim and Kato 1984 (n # 5)
Barian, manganoan, phlogopite, Nodatamagawa, Iwate Prefecture (Japan)
Metamorphosed (K0.58Na0.09Ba0.35) (Fe3+0.04Al0.35 manganese Mg2.10 Mn2+0.52 Mn3+0.22) (Si2.33 deposit Al1.65Ti0.01) O10.75 F0.07 (OH)1.18
5.330(2)
9.245(3)
10.240(3)
99.92(2)
3.8
65. Hawthorne et al. 1999
Rubidian, cesian, phlogopite, Red Cross Lake, Manitoba (Canada)
Granitic pegmatite
K0.46Cs0.23Rb0.28 (Al0.38Fe2+1.00 Mn0.04Ti0.04Mg1.20 Li0.34) (Si2.91 Al1.09) O10 F0.45 (OH)1.55
5.343(1)
9.247(2)
10.397(3)
100.04(2)
4.5
66. Hazen and Burnham 1973
Annite, Pikes Peak, Colorado
Granite
(K0.88Na0.07Ca0.03) (Al0.09Fe3+0.19 Fe2+2.22Mg0.12Mn0.05 Ti0.22) (Si2.81 Al1.19) O10.35 F0.22 Cl0.05 (OH)1.38
5.3860(9)
9.3241(7)
10.2683(9) 100.63(1)
4.5
63
Mica Crystal Chemisrty and Influence of P-T-X on Atomistic Models
10.8
110.0
58.9
58.8
2.269
2.170
3.455
0.006
1.678
2.097
2.092
2.948
3.446
-0.334
0.000
-0.334
50.
8.5
110.1
59.3
58.8
2.259
2.148
3.424
0.011
1.665
2.101
2.073
2.985
3.373
-0.338
-0.004
-0.342
51.
7.6
110.1
59.4
58.9
2.255
2.130
3.395
0.019
1.658
2.094
2.063
2.991
3.337
-0.338
-0.003
-0.341
52.
7.6
110.2
59.3
58.9
2.258
2.140
3.391
0.019
1.661
2.097
2.070
2.993
3.341
-0.338
-0.002
-0.340
53.
8.1
110.2
59.5
58.9
2.251
2.128
3.404
0.021
1.657
2.093
2.058
2.978
3.348
-0.338
-0.003
-0.341
54.
7.7
110.2
59.3
58.9
2.251
2.137
3.404
0.008
1.658
2.091
2.067
2.992
3.340
-0.339
-0.003
-0.342
55.
8.1
110.1
59.4
58.8
2.260
2.138
3.415
0.015
1.663
2.101
2.066
2.989
3.357
-0.338
-0.004
-0.342
56.
1.5
110.2
58.8
58.4
2.250
2.196
3.374
0.000
1.658
2.117
2.098
3.143
3.212
-0.336
-0.023
-0.359
57.
5.0
110.1
59.0
59.4
2.252
2.123
3.367
0.009
1.668
2.059
2.087
3.051
3.279
-0.333
0.001
-0.332
58.
2.0
110.2
58.5
58.4
2.241
2.201
3.349
0.002
1.656
2.109
2.100
3.122
3.211
-0.338
-0.009
-0.347
59.
0.3
110.5
59.3
59.2
2.293
2.180
3.899
0.013
1.688
2.132
2.128
3.359
3.372
-0.333
+0.002
-0.331
60.
6.4
110.3
59.3
59.3
2.318
2.152
3.394
0.010
1.687
2.107
2.106
3.055
3.351
-0.334
0.001
-0.333
61.
7.3
112.1
61.0
58.6
2.259
2.056
3.397
0.070
1.632
2.118
1.970
2.950
3.270
-0.357
-0.007
-0.364
62.
6.8
110.8
58.1
57.9
2.254
2.245
3.409
0.008
1.663
2.122
2.110
3.027
3.337
-0.331
-0.001
-0.332
63.
11.0
110.6
58.8
58.7
2.273
2.161
3.380
0.002
1.672
2.086
2.078
2.910
3.413
-0.332
0.001
-0.331
64.
3.5
110.8
59.5
59.0
2.261
2.125
3.591
0.017
1.652
2.093
2.064
3.141
3.296
-0.338
-0.001
-0.339
65.
1.6
110.4
58.6
58.3
2.252
2.207
3.380
0.014
1.660
2.121
2.101
3.143
3.215
-0.334
-0.018
-0.352
66.
64
Brigatti & Guggenheim
67. Hazen and Burnham 1973
Phlogopite, Franklin, New Jersey
Marble
(K0.77Na0.16Ba0.05) Mg3.00 (Si2.95Al1.05) O10 F1.30 (OH)0.70
5.3078(4)
9.1901(5)
10.1547(8) 100.08(1)
4.1
68. Hazen and Finger 1978 (high pressure)
Phlogopite, Franklin, New Jersey
Marble
(K0.77Na0.16Ba0.05) Mg3.00 (Si2.95Al1.05) O10 F1.30 (OH)0.70
5.260(1)
9.100(1)
9.791(9)
100.68(4)
14.7
69. Hazen et al. 1981 (n # Y253)
Tetra-ferriphlogopite, Cupaello (Italy)
Volcanic melilite
(K0.97Na0.01Ba0.02) (Fe2+0.03Mg2.46 Ti3+0.09 Li0.23Na0.11) (Si3.31Al0.04 Fe3+0.65) O10 F2.00
5.329(1)
9.230(2)
10.219(1)
99.98(1)
3.0
70. Joswig 1972
Phlogopite (Madagascar)
(K0.90Na0.02) (Al0.07Fe2+0.16Mg2.70 Ti0.03) (Si2.91Al1.09) O9.90 F1.13 (OH)0.97
5.314(1)
9.2024(5)
10.1645(7) 100.05(1)
2.0
71. Kato et al. 1979 (n # 2)
Manganoan phlogopite, Nodatamagawa, Iwate Prefecture (Japan)
Metamorphosed (K0.73Na0.32Ba0.11) (Fe3+0.03Mg2.27 5.349(2) manganese Mn2+0.49) (Si2.86Al1.07Ti0.02 Fe3+0.05) deposit O10.34 F0.09 (OH)1.57
9.241(2)
10.282(4)
99.96(2)
10.6
72. McCauley et al. 1973
Fluoro phlogopite
Synthetic
5.308(2)
9.183(3)
10.139(1)
100.07(2)
6.1
73. Medici 1996 (n # TPP16-6a)
Phlogopite, Tapira (Brazil)
(K0.98Na0.01) (Fe3+0.08Fe2+0.13Mg2.73 5.330(1) Alkaline carbo natitic complex: Ti0.06) (Si2.82Al1.04Fe3+0.14) O10.01 garnet magnetite F0.11 (OH)1.88
9.239(1)
10.305(1)
99.89(1)
3.3
74. Medici 1996
(K1.00Ba0.01) (Fe3+0.01Fe2+0.60Mg2.36 5.360(1) Alkaline carbo Octa-ferroan tetraferriphlogopite, Tapira natitic complex: Mn0.01 Ti0.01) (Si3.03Al0.07Fe3+0.90) garnet magnetite O10.08 F0.01 (OH)1.91 (Brazil)
9.293(1)
10.314(2)
100.01(1)
2.8
75. Medici 1996 (n # TPP16-6c)
(K1.97Ca0.03 Ba0.01) (Fe2+0.60Mg2.38 Alkaline carbo Octa-ferroan tetraferriphlogopite, Tapira natitic complex: Mn0.01Ti0.01) (Si3.02Al0.06Fe3+0.92) garnet magnetite O10.05 F0.04 (OH)1.91 (Brazil)
9.2908(8)
10.321(1)
99.995(9)
2.5
76. Medici 1996 (n # TAX27-1)
Ferroan phlogopite, Tapira, Brazil
Alkaline carbo natitic complex: clinopyroxenite
(K0.95Na0.03) (Fe3+0.04Fe2+0.43Mg2.39 5.351(1) Mn0.01Ti0.08) (Si2.94Al0.78Fe3+0.28) O10.00 F0.05 (OH)1.95
9.267(2)
10.311(1)
99.99(1)
2.6
77. Medici 1996 (n # TAI17-1)
Ferroan phlogopite, Tapira (Brazil)
Alkaline carbo natitic complex: bebedourite
(K0.98Na0.02Ba0.01) (Fe3+0.10Fe2+0.44 Mg2.36Mn0.01 Ti0.09) (Si2.82Al1.10 Fe3+0.08) O10.12 (OH)1.88
5.3355(8)
9.2457(7)
10.294(2)
99.94(1)
2.5
78. Medici 1996 (n # TAA11-1a)
Ferroan phlogopite, Tapira (Brazil)
Alkaline carbo natitic complex: bebedourite
(K0.98Na0.02) (Fe3+0.06Fe2+0.60 Mg2.23Mn0.01 Ti0.10) (Si2.84Al1.14 Fe3+0.02) O10.11 F0.05 (OH)1.84
5.329(2)
9.244(2)
10.271(3)
99.97(2)
3.6
79. Medici 1996 (n # TA9)
Ferroan phlogopite, Tapira (Brazil)
(K0.98Ba0.02) (Fe2+1.14Mg1.73Mn0.04 Alkaline carbo naitic complex: Ti0.09) (Si3.00Al0.90Fe3+0.10) O10.17 garnet magnetite F0.01 (OH)1.82
5.344(1)
9.259(2)
10.280(2)
100.01(1)
2.8
80. Medici 1996 (n # LI12a)
Ferroan phlogopite, Limeira, Brazil
Kamafugite
(K0.95Na0.04) (Fe2+0.44Mg2.51Ti0.05) (Si3.01Al0.92Fe3+0.07) O10.11 F0.18 (OH)1.71
5.331(1)
9.227(1)
10.275(2)
99.96(2)
3.9
81. Medici 1996 (n # MA-1)
Phlogopite, Malaquias Kamafugite (Brazil)
(K0.97Na0.02 Ba0.02) (Fe3+0.03Fe2+0.35 Mg2.07Ti0.33) (Si2.94Al1.06) O10.21 F0.93 (OH)0.86
5.317(1)
9.208(2)
10.118(2)
100.15(1)
2.9
(n # TPP16-6b)
(K0.98Na0.04) Mg2.97 (Si2.98Al1.02) O9.90 F1.94 (OH)0.16
5.3637(5)
82. Mellini et al. 1996 Cesian tetra-ferriannite
Synthetic
Cs0.89 (Fe3+0.03Fe2+2.97) (Si3.07 Fe3+0.90Al0.03) O10 (OH)2
5.487(1)
9.506(2)
10.826(6)
99.83(3)
5.5
83. Oberti et al. 1993 (n # KP9)
Ultramafic complex
(K0.02Na0.83) (Al0.93 Fe0.17Mg1.90 5.225(4) Cr0.01) (Si2.12Al1.88) O9.99 (OH)2.01 mean composition
9.050(8)
9.791(9)
100.27(6)
3.8
Preiswerkite, Geisspfad (Switzerland)
65
Mica Crystal Chemisrty and Influence of P-T-X on Atomistic Models
7.5
110.5
59.0
59.0
2.261
2.125
3.352
0.002
1.650
2.063
2.065
2.969
3.312
-0.334
-0.001
-0.335
67.
9.7
109.3
60.1
59.3
2.280
2.059
3.002
0.000
1.66
2.07
2.02
2.81
3.26
-0.345
0.000
-0.345
68.
5.7
110.8
58.8
58.8
2.275
2.151
3.364
0.003
1.655
2.077
2.077
3.021
3.282
-0.332
0.000
-0.332
69.
7.7
110.6
59.2
59.1
2.268
2.116
3.356
0.001
1.654
2.066
2.063
2.970
3.319
-0.334
0.000
-0.334
70.
6.2
111.3
58.4
58.2
2.265
2.202
3.395
0.018
1.657
2.101
2.090
3.025
3.307
-0.330
-0.002
-0.332
71.
5.9
110.1
59.0
59.0
2.252
2.124
3.356
0.004
1.642
2.062
2.064
3.006
3.273
-0.334
0.000
-0.334
72.
8.6
110.7
58.7
58.6
2.263
2.168
3.458
0.008
1.661
2.084
2.081
2.986
3.374
-0.333
0.001
-0.332
73.
10.3
110.0
58.8
58.8
2.265
2.170
3.457
0.001
1.674
2.092
2.091
2.953
3.435
-0.334
0.000
-0.334
74.
10.4
110.1
58.8
58.8
2.270
2.165
3.460
0.003
1.675
2.091
2.090
2.958
3.433
-0.334
0.000
-0.334
75.
8.0
110.5
58.7
58.6
2.265
2.174
3.451
0.006
1.664
2.091
2.089
3.004
3.367
-0.334
0.000
-0.334
76.
9.0
110.6
58.7
58.7
2.269
2.162
3.440
0.002
1.665
2.084
2.081
2.973
3.382
-0.333
0.000
-0.333
77.
8.5
110.4
58.5
58.5
2.245
2.178
3.447
0.007
1.658
2.087
2.084
2.983
3.372
-0.333
-0.001
-0.334
78.
6.7
110.7
58.6
58.5
2.259
2.178
3.428
0.005
1.659
2.092
2.087
3.024
3.329
-0.334
0.000
-0.334
79.
6.7
110.8
58.6
58.6
2.258
2.169
3.435
0.018
1.655
2.082
2.081
3.019
3.324
-0.333
0.000
-0.333
80.
4.5
110.8
59.3
59.0
2.271
2.123
3.294
0.011
1.648
2.077
2.061
3.026
3.230
-0.334
-0.001
-0.335
81.
0.2
110.0
59.4
59.3
2.284
2.168
3.930
0.015
1.688
2.130
2.125
3.370
3.380
-0.335
-0.002
-0.337
82.
20.0
107.7
59.6
59.5
2.255
2.051
3.073
0.005
1.695
2.025
2.020
2.573
3.514
-0.334
0.000
-0.334
83.
66
Brigatti & Guggenheim
84. Oberti et al. 1993 (n # KP17)
Preiswerkite, Geisspfad (Switzerland)
Ultramafic complex
(K0.02Na0.83) (Al0.93 Fe0.17Mg1.90 Cr0.01) (Si2.12Al1.88) O9.99 (OH)2.01
85. Otha et al. 1982
Ferrian phlogopite, Ruiz Peak, Valles Mountains, New Mexico
Rhyodacite
86. Rayner 1974
Phlogopite
5.228(7)
9.049(10)
9.819(12)
100.41(13) 4.6
(K0.77Na0.16Ba0.02) (Al0.16Fe3+0.86 Fe2+0.01Mg1.67Mn0.01Ti0.34) (Si2.84 Al1.16) O11.62 F0.17 (OH)0.21
5.320(4)
9.210(1)
10.104(1)
100.10(1)
5.0
(K0.93Na0.04Ca0.03) (Fe2+0.10Mg2.77 Ti0.11) (Si2.88Al1.12) O10 F0.51 (OH)1.49
5.322
9.206
10.240
100.03
6.6
Aluminian phlogopite, Metamorphosed (K Na ) (Al Fe Mg ) 87. Russell and 0.82 0.12 0.50 0.12 2.28 Guggenheim 1999 White Well (Australia) ultrabasic schist (Si Al ) O F (OH) 2.79 1.21 10 0.07 1.93 (T°C = 20)
5.3030(4)
9.1805(6)
10.2483(7) 100.05(6)
Aluminian phlogopite, Metamorphosed (K Na ) (Al Fe Mg ) 88. Russell and 0.82 0.12 0.50 0.12 2.28 Guggenheim 1999 White Well (Australia) ultrabasic schist (Si Al ) O F (OH) 2.79 1.21 10 0.07 1.93 (T°C = 300)
5.3193(7)
9.207(1)
10.286(1)
Aluminian phlogopite, Metamorphosed (K Na ) (Al Fe Mg ) 89. Russell and 0.82 0.12 0.50 0.12 2.28 Guggenheim 1999 White Well (Australia) ultrabasic schist (Si Al ) O F (OH) 2.79 1.21 10 0.07 1.93 (T°C = 450)
5.3331(7)
9.2316(9)
10.3159(8) 100.004(8) 11.9
Aluminian phlogopite, Metamorphosed (K Na ) (Al Fe Mg ) 90. Russell and 0.82 0.12 0.50 0.12 2.28 Guggenheim 1999 White Well (Australia) ultrabasic schist (Si Al ) O F (OH) 2.79 1.21 10 0.07 1.93 (T°C = 600)
5.342(3)
9.238(4)
10.357(5)
99.99(1)
6.9
(K0.93Na0.08) (Mg1.57Fe2+1.07Fe3+0.10 5.3346(7) Ti0.10 Mn0.06) (Si2.97Al1.00Ti0.03) O10 F0.94 Cl0.01 (OH)1.05
9.2417(8)
10.182(2)
100.26(1)
3.9
(K0.93Na0.08) (Mg1.57Fe2+1.07Fe3+0.10 5.3099(5) Ti0.10 Mn0.06) (Si2.97Al1.00Ti0.03) O10 F0.94 Cl0.01 (OH)1.05
9.185(1)
10.093(2)
100.07(1)
3.9
Ferroan phlogopite 91. Russell and Guggenheim 1999 Silver Crater Mine, Bancroft (Ontario) (room temperature) Ferroan phlogopite 92. Russell and Guggenheim 1999 Silver Crater Mine, Bancroft (Ontario) (heated)
mean composition
Calcite veins hosted within nepheline syenites Calcite veins hosted within nepheline syenites
9.7
100.042(9) 12.8
93. Sartori 1976
Trilithionite, Elba Island (Italy)
Granitic pegmatite
(K0.88Na0.06Rb0.05Ca0.01) (Al1.13 Li1.31) (Si3.36Al0.64) O10 F1.53 (OH)0.47
5.20(2)
9.01(1)
10.09(1)
99.3(3)
6.7
94. Semenova et al. 1977
Tetra-ferriphlogopite, Kovdor massif
Ultrabasic and alkaline rocks
(K1.03Na0.09Ca0.04) (Mg2.89Fe2+0.16 Mn0.01) (Al0.08Fe3+0.85Ti0.03Si2.98) O10 (OH)2
5.358(3)
9.297(3)
10.318(2)
100.02(5)
4.2
95. Steinfink 1962
Tetra-ferriphlogopite, Langhan (Sweden)
(K0.90Mn0.10) Mg3.00 [Si3.00 (Fe3+, Mn)1.00] O10 (OH)2.0
5.36(1)
9.29(2)
10.41(2)
100.0(2)
13.1
96. Takeda and Burnham 1969
Polylithionite
Synthetic
K1.00 (Li2.00 Al1.00) Si4.00 O10.00 F2.00 5.188(4)
8.968(3)
10.029(5)
100.45(1)
5.1
97. Takeda and Donnay 1966
Lithium-containing phlogopite
Synthetic
(K0.95) (Mg2.80Li0.20) (Si3.25Al0.75) O10 F2
5.31
9.21
10.13
100.02
7.5
98. Takeda and Morosin 1975
Fluoro phlogopite (room temperature)
Synthetic
(K0.98 Na0.04) Mg2.97 (Si2.98Al1.02) O9.90 (OH0.16,F1.94)
5.3074(6)
9.195(2)
10.134(1)
100.08(1)
4.3
99. Takeda and Ross 1975
Ferroan phlogopite, Ruiz Peak, Valles Mountains, New Mexico
Rhyodacite
(K0.78 Na0.16Ba0.02) (Al0.19Fe3+0.19 Fe2+0.71Mg1.68Mn0.01 Ti0.34) (Si2.86 Al1.14) O11.12 F0.17 (OH)0.71
5.331(2)
9.231(4)
10.173(4)
100.16(3)
4.4
Tetra-magnesian phlogopite
Synthetic
(K0.96Na0.03) Mg2.84 (Si3.63Mg0.31 Fe0.03Al0.03) O10 (OH)2
5.321(2)
9.238(1)
10.287(1)
100.06(1)
10.4
100. Tateyama et al. 1974
67
Mica Crystal Chemisrty and Influence of P-T-X on Atomistic Models
19.6
107.8
60.0
59.1
2.247
2.061
3.103
0.065
1.688
2.059
2.007
2.592
3.512
-0.342
0.003
-0.339
84.
7.3
110.3
59.5
59.2
2.275
2.112
3.287
0.021
1.655
2.077
2.059
2.962
3.294
-0.331
-0.002
-0.333
85.
8.7
109.0
59.2
59.0
2.270
2.126
3.418
0.005
1.659
2.076
2.064
2.967
3.360
-0.332
-0.003
-0.335
86.
11.2
110.2
59.1
58.9
2.252
2.192
3.459
0.001
1.662
2.072
2.059
2.915
3.424
-0.328
-0.009
-0.337
87.
9.6
110.2
59.1
58.9
2.252
2.130
3.495
0.014
1.660
2.074
2.065
2.968
3.402
-0.336
-0.001
-0.337
88.
8.7
110.2
59.1
59.0
2.252
2.138
3.517
0.006
1.660
2.076
2.072
3.000
3.393
-0.335
-0.001
-0.336
89.
6.7
110.5
58.9
58.8
2.257
2.154
3.532
0.001
1.658
2.085
2.078
3.051
3.352
-0.324
-0.012
-0.336
90.
4.3
110.7
58.7
58.5
2.250
2.175
3.345
0.002
1.650
2.090
2.082
3.051
3.247
-0.336
-0.004
-0.340
91.
4.7
110.6
59.2
59.0
2.269
2.119
3.280
0.016
1.645
2.070
2.059
3.013
3.227
-0.331
-0.001
-0.332
92.
7.4
112.2
60.8
58.5
2.255
2.060
3.387
0.062
1.631
2.113
1.972
2.942
3.269
-0.330
0.017
-0.313
93.
11.5
109.9
59.0
59.0
2.277
2.146
3.460
0.008
1.680
2.086
2.085
2.933
3.458
-0.334
-0.001
-0.335
94.
11.1
110.2
58.1
58.2
2.280
2.218
3.475
0.008
1.681
2.101
2.105
2.945
3.452
-0.338
0.001
-0.337
95.
3.0
113.8
60.2
58.1
2.247
2.095
3.274
0.036
1.619
2.106
1.981
3.000
3.132
-0.351
0.000
-0.351
96.
6.2
110.6
59.4
59.3
2.273
2.102
3.328
0.006
1.651
2.061
2.060
2.995
3.278
-0.333
0.001
-0.332
97.
6.5
110.7
59.4
59.4
2.277
2.095
3.329
0.008
1.650
2.056
2.058
2.987
3.282
-0.335
0.001
-0.334
98.
7.6
110.4
59.2
58.9
2.271
2.138
3.334
0.014
1.659
2.086
2.068
2.972
3.318
-0.335
-0.002
-0.337
99.
7.4
110.8
58.0
58.2
2.230
2.204
3.465
0.016
1.65
2.08
2.09
3.01
3.34
-0.335
-0.003
-0.338
100.
68
Brigatti & Guggenheim
(K0.88Na0.02Ca0.02) (Fe2+2.31Mg0.28 Mn0.02Al0.18Fe3+0.01 Ti0.10Li0.04) (Si2.71Al1.29) (H3O)0.04 O10 F0.14 (OH)1.86
5.366(5)
9.311(5)
10.16(1)
100.2(2)
13.5
Synthetic
K1.0 (Mg1.04Mn1.96) (Ge3.00Al1.00) O10 F2
5.489(1)
9.509(1)
10.462(3)
100.12(2)
5.0
Tetra-germanatian, 103. Toraya and Marumo1981 (n # manganoan fluoro phlogopite X0.68)
Synthetic
K1.00 (Mg2.36Mn0.64) (Ge3.00Al1.00) O10 F2
5.435(1)
9.413(2)
10.458(3)
100.03(3)
4.0
104. Toraya et al. 1976 Tetra-silicic fluorophlogopite
Synthtetic
K Mg2.5 Si4O10 F2
5.253(1)
9.086(2)
10.159(1)
99.89(3)
3.8
105. Toraya et al. 1977 Tainiolite
Synthetic
K (Mg2Li) Si4O10 F2
5.231(1)
9.065(2)
10.140(1)
99.86(2)
2.4
106. Toraya et al. 1978a (n # 1)
Tetra-germanatian fluoro phlogopite
Synthetic
K1.00 Mg2.50 Ge4.00 O10 F2
5.421(2)
9.353(4)
10.533(2)
100.14(4)
5.5
107. Toraya et al. 1978a (n # 2)
Tetra-germanatian tainiolite
Synthetic
K1.00 (Mg2.00Li1.00) Ge4.00 O10 F2
5..395(1)
9.341(2)
10.547(1)
99.87(2)
3.8
108. Toraya et al. 1978b (n # c)
Fluoro phlogopite
Synthetic
K Mg2.75 (Si3.5Al0.5) O10 F2
5.292(1)
9.164(5)
10.143(1)
100.07(2)
2.9
109. Toraya et al. (1978c)
Tetra-germanatian fluoro phlogopite
Synthetic
K Mg3 (Ge3Al) O10 F2
5.417(6)
9.345(5)
10.468(1)
100.03(3)
3.7
110. Toraya et al. 1983 Fluoro phlogopite
Synthetic
K (Mg2.44Mn0.24) (Si3.82Mn0.18) O10 F2
5.285(1)
9.157(1)
10.190(2)
99.97(2)
4.3
111. Tyrna and Norrishite, Grenfell Guggenheim 1991 New South Wales (Australia)
Metamorphosed K (Li Mn3+2) Si4 O12 stratiform unit
5.289(3)
8.914(3)
10.062(7)
98.22(5)
7.8
5.3655(6)
9.293(1)
10.198(2)
100.47(1)
3.8
101. Tepikin et al. 1969
Annite
102. Toraya and Marumo 1981 (n # X1.96)
Tetra-germanatian, manganoan fluoro phlogopite
112. Weiss et al. 1993
Aluminian fluoro annite, Brooks Mountain, Seward (Alaska)
(K0.92Na0.09Ca0.01Rb0.01) (Fe2+2.02 Al0.47Li0.33Mn0.07Mg0.03) (Si2.98 Al1.02) O10 F0.99 Cl0.03 (OH)0.98
69
Mica Crystal Chemisrty and Influence of P-T-X on Atomistic Models
1.3
110.5 59.6
59.6
2.277
2.107
3.340
0.060
1.668
2.080
2.078
3.134
3.191
-0.330
-0.004
-0.334
101.
13.2
111.4 59.5
59.5
2.419
2.158
3.304
0.006
1.749
2.128
2.123
2.892
3.527
-0.335
0.000
-0.335
102.
15.0
111.5 59.9
59.9
2.421
2.102
3.354
0.004
1.746
2.094
2.095
2.846
3.557
-0.335
0.000
-0.335
103.
1.4
111.8 58.0
58.0
2.243
2.186
3.337
0.003
1.625
2.062
2.064
3.079
3.142
-0.333
+0.001
-0.332
104.
1.1
112.7 57.8
57.9
2.251
2.192
3.297
0.000
1.625
2.058
2.061
3.068
3.116
-0.332
0.000
-0.332
105.
12.9
114.3 60.1
58.4
2.446
2.170
3.306
0.051
1.744
2.178
2.070
2.872
3.480
-0.339
-0.003
-0.342
106.
13.5
114.3 59.3
59.3
2.458
2.138
3.338
0.009
1.744
2.093
2.092
2.861
3.494
-0.333
-0.002
-0.335
107.
3.6
111.1 58.8
58.8
2.258
2.137
3.334
0.000
1.638
2.062
2.063
3.045
3.209
-0.335
0.000
-0.335
108.
15.9
111.5 60.2
60.2
2.425
2.063
3.395
0.005
1.744
2.076
2.078
2.824
3.577
-0.335
-0.002
-0.337
109.
1.7
111.2 58.4
58.4
2.246
2.172
3.372
0.000
1.632
2.071
2.070
3.097
3.174
-0.333
-0.001
-0.334
110.
0.6
112.9 58.6
57.2
2.246
2.213
3.253
0.055
1.621
2.123
2.040
3.063
3.086
-0.274
0.002
-0.272
111.
1.3
110.6 58.9
58.5
2.247
2.180
3.355
0.005
1.656
2.108
2.088
3.133
3.195
-0.341
-0.004
-0.345
112.
70
Brigatti & Guggenheim
Table 1b. Structural details of trioctahedral true Micas-1M, space group C2
Cell parameters Reference
Species, locality
Rock type
(sample number)
Composition
R
a
b
c
E
(Å)
(Å)
(Å)
(°)
(%)
113. Backhaus 1983
Polylithionite, Wakefield (Canada)
5.216(3)
9.005(4)
10.084(3)
100.72(5)
7.3
114. Brigatti et al. 2000b (n # 114)
5.262(1) Ferroan polylithionite, Miarolitic cavity (K0.91Na0.01Rb0.05) (Al1.06Li1.41 in granitic Sentinel Rock, Pikes Fe3+0.05Fe2+0.40Mn0.04 Mg0.002Zn0.002) pegmatite Peak (Colorado) (Si3.54Al0.46) O10.11 (OH)0.14 F1.75
9.085(2)
10.099(2)
100.72(1)
3.4
115. Brigatti et al. 2000b (n # 55a)
Ferroan polylithionite, Miarolitic cavity (K0.94Na0.002Rb0.003) (Al1.11Li1.11 5.270(1) Wigwam Creek, Pikes in granitic Fe3+0.05Fe2+0.53Mg0.01Mn0.04Ti0.01 pegmatite Peak (Colorado) Zn0.003) (Si3.41Al0.59) O10.17 (OH)0.20 F1.63
9.092(1)
10.080(2)
100.70(1)
3.7
116. Brigatti et al. 2000b (n # 55b)
Ferroan polylithionite, Miarolitic cavity (K0.96Na0.01Rb0.02) (Al1.06Li1.22 Wigwam Creek, Pikes in granitic Fe3+0.06Fe2+0.55 Mg0.005Mn0.05 pegmatite Peak (Colorado) Zn0.01Ti0.005) (Si3.41Al0.59) O10.23 (OH)0.24 F1.53
5.263(1)
9.085(1)
10.078(1)
100.75(1)
3.2
117. Brigatti et al. Ferroan polylithionite, Miarolitic cavity (K0.97Na0.02Ca0.01) (Al1.01Li1.08 5.290(1) 2000b (n # 130-1) Devils Head area, Pikes in granitic Fe3+0.09Fe2+0.70Mg0.02 Mn0.06Zn0.005 pegmatite Peak (Colorado) Ti0.002) (Si3.30Al0.70) O10.26 (OH)0.19 F1.55
9.128(1)
10.093(1)
100.80(1)
3.0
118. Brigatti et al. Ferroan polylithionite, Miarolitic cavity (K0.96Na0.02Ca0.006) (Al1.02Li1.09 5.275(2) 2000b (n # 130-2) Devils Head area, Pikes in granitic Fe3+0.06Fe2+0.71Mg0.02Mn0.06Zn0.005 pegmatite Peak (Colorado) Ti0.005) (Si3.33Al0.67) O10.27 (OH)0.19 F1.54
9.105(2)
10.084(1)
100.70(1)
3.9
5.279(1)
9.114(2)
10.077(2)
100.79(1)
3.6
5.285(1)
9.122(2)
10.101(2)
100.85(1)
3.3
(K0.90Na0.08Rb0.04 Cs0.003) (Al1.10 Li1.51Fe3+0.03Fe2+0.15 Mn0.16 Ti0.01) (Si3.48 Al0.53) O10.38 (OH)0.41 F1.67
119. Brigatti et al. 2000b (n # 137)
Polylithionite-sidero- Miarolitic cavity (K0.94Na0.02) (Al1.05Li0.97Fe3+0.07 phyllite intermediate, in granitic Fe2+0.67 Mg0.01Mn0.07Zn0.006Ti0.005) pegmatite Lake George Ring, (Si3.21Al0.79) O10.02 (OH)0.24 F1.74 Pikes Peak (Colorado)
120. Brigatti et al. 2000b (n # 104)
Polylithionite-siderophyllite intermediate, Crystal Park, Pikes Peak (Colorado)
121. Brigatti et al. 2000b (n # 54b)
Polylithionite-sidero- Miarolitic cavity phyllite intermediate, in granitic Harris Park, Pikes Peak pegmatite (Colorado)
(K0.94Na0.02Rb0.004) (Al1.05Li0.94 5.283(1) Fe3+0.12Fe2+0.61Mg0.01Mn0.06Ti0.006 Zn0.002) (Si3.31Al0.69) O10.10 (OH)0.25 F1.65
9.123(2)
10.072(2)
100.76(1)
3.8
122. Brigatti et al. 2000b (n # 177)
Polylithionite-sidero- Miarolitic cavity phyllite intermediate, in granitic Wigwam Creek, Pikes pegmatite Peak (Colorado)
(K0.94Na0.04Rb0.003) (Al0.88Li0.86 Fe3+0.24Fe2+0.65Mg0.01 Mn0.06 Zn0.01Ti0.005) (Si3.23Al0.77) O9.93 (OH)0.15 F1.92
5.288(1)
9.133(1)
10.088(1)
100.81(1)
3.4
123. Brigatti et al. Polylithionite-sidero- Miarolitic cavity (K0.96Na0.01) (Al1.02Li0.86Fe3+0.04 2000b (n # 140-1) phyllite intermediate, in granitic Fe2+0.81Mg0.05Mn0.06 Zn0.01Ti0.03) pegmatite Lake George Ring , (Si3.17Al0.83) O10.13 (OH)0.25 F1.62 Pikes Peak (Colorado)
5.283(1)
9.118(1)
10.092(1)
100.78(1)
2.9
124. Brigatti et al. Polylithionite-sidero- Miarolitic cavity (K0.96Na0.01) (Al0.98Li0.85Fe3+0.05 2000b (n # 140-2) phyllite intermediate, in granitic Fe2+0.80Mg0.05 Mn0.06 Zn0.01Ti0.025) pegmatite Lake George Ring, (Si3.24Al0.76) O10.11 (OH)0.25 F1.64 Pikes Peak (Colorado)
5.297(1)
9.146(1)
10.102(1)
100.81(1)
2.7
(K0.82Na0.03Rb0.09) (Al1.11Li0.77 5.295(1) Fe3+0.05Fe2+0.78Mn0.08 Ti0.006Mg0.004 Zn0.002) O10.24 (Si3.31Al0.69) (OH)0.17 F1.59
9.139(2)
10.077(2)
100.83(2)
3.7
Miarolitic cavity (K0.96Na0.02Ca0.001) (Al1.03Li0.97 in granitic Fe3+0.14Fe2+0.64 Mg0.01 Mn0.01 pegmatite Zn0.005Ti0.01) (Si3.30Al0.70) O10.15 (OH)0.24 F1.61
125. Brigatti et al. 2000b (n # 24)
Polylithionite-sidero- Miarolitic cavity phyllite intermediate, in granitic Wigwam Creek, Pikes pegmatite Peak (Colorado)
126. Brigatti et al. 2000b (n # 47)
Lithian siderophyllite, Lake George Ring complex, Pikes Peak (Colorado)
Quartz core, granitic pegmatite
(K0.99Na0.01) (Al0.81 Li0.41Fe3+0.09 Fe2+1.40Mg0.04 Mn0.08Zn0.02Ti0.10) (Si3.06Al0.94) O10.64 (OH)0.28 F1.08
5.339(1)
9.233(1)
10.135(2)
100.73(1)
3.3
127. Brigatti et al. 2000b (n # 103)
Lithian siderophyllite, Crystal Park, Pikes Peak (Colorado)
Quartz core, granitic pegmatite
(K0.99Na0.04Rb0.002) (Al0.90Li0.62 Fe3+0.09Fe2+1.19 Mg0.02 Mn0.05Zn0.01 Ti0.03) (Si3.23Al0.77) O10.55 (OH)0.15 F1.29
5.300(1)
9.144(1)
10.089(2)
100.74(1)
3.6
71
Mica Crystal Chemisrty and Influence of P-T-X on Atomistic Models
Tetrahedral D (°) 6.0
WT1 WT2 (°) 112.3
Octahedral <M1 (°) 60.5
112.3 3.7
111.9
111.8
60.6
112.0
60.6
111.2
60.6
111.7
60.8
111.6
60.7
111.4
60.7
111.5
60.8
111.3
60.8
111.5
60.7
111.3
60.6
111.1
60.7
110.5
60.9
111.3 111.3
2.087
3.319
0.134
56.2
56.7
56.6
56.4
56.6
56.3
56.6
56.4
56.6
56.6
59.8
58.0
2.246
2.092
3.316
0.135
56.6 60.3
2.124
1.641
1.634
2.255
2.075
3.331
0.129
1.649
2.125
2.082
3.323
0.144
1.641
2.129
2.084
3.314
0.109
1.638
2.128
2.081
3.336
0.130
1.644
2.125
2.087
3.308
0.133
1.642
2.129
2.083
3.326
0.128
1.643
2.133
2.093
3.323
0.137
1.643
2.136
2.088
3.326
0.127
1.646
2.130
2.077
3.320
0.141
1.645
2.133
2.138
3.339
0.058
1.658
2.135
2.107
3.310
0.125
1.643 1.641
3.195
-0.355
-0.002
-0.357
114.
1.888
3.022
3.199
-0.354
-0.001
-0.355
115.
1.882
3.022
3.193
-0.355
-0.002
-0.357
116.
1.890
3.009
3.236
-0.354
-0.004
-0.358
117.
1.890
3.023
3.206
-0.353
-0.002
-0.355
118.
1.885
3.018
3.209
-0.355
-0.002
-0.357
119.
1.890
3.018
3.226
-0.356
-0.004
-0.360
120.
1.883
3.033
3.195
-0.354
-0.002
-0.356
121.
1.891
3.021
3.222
-0.355
-0.003
-0.358
122.
1.891
3.028
3.208
-0.354
-0.003
-0.357
123.
1.896
3.026
3.224
-0.354
-0.004
-0.358
124.
3.015
3.230
-0.355
-0.003
-0.358
125.
3.070
3.224
-0.348
-0.005
-0.353
126.
3.043
3.201
-0.352
-0.003
-0.355
127.
2.131 2.134
1.885 2.133
2.126
1.637 2.247
3.029
2.129
1.643 2.241
2.122
2.131
1.643 2.250
113.
2.132
1.637 2.254
-0.360
2.123
1.641 2.250
-0.004
2.131
1.639 2.250
-0.356
2.125
1.640 2.250
3.226
2.127
1.641 2.252
2.958
2.127
1.637 2.250
Overall shift a1
2.124
1.640 2.252
Layer offset a1
1.885
1.636
59.2 60.4
1.632
2.058
Intralayer ¢A - O² Inner Outer shift a1
1.914
1.635
60.9
110.5 3.5
2.249
60.7
111.1 3.4
56.5
2.098
1.640
60.6
111.3 4.8
0.131
60.7
111.6 4.4
3.333
60.7
111.2 4.0
2.086
60.7
111.5 4.5
2.252
60.7
111.4 3.6
60.6
1.631 1.633
60.7
111.5 4.6
0.109
60.8
111.6 4.3
3.328
60.5
111.3 4.1
2.066
60.6
111.8 5.0
2.257
56.4
111.7 3.8
59.9
Sheet thickness Interlayer Basal Mean bond lengths Tetra- Octa- Separa- oxygen ¢T1-O² ¢M1-O² ¢M2-O² hedral hedral tion 'z ¢M3-O² ¢T2-O² (Å) (Å) (Å) (Å) (Å) (Å) (Å)
57.3
111.9 4.0
<M2 <M3 (°)
2.017 2.085
2.131
1.916 2.129
72
Brigatti & Guggenheim
128. Guggenheim 1981 Polylithionite, Tanakamiyama, Ohtsu, Japan
(K1.01Na0.01 Rb0.03) (Si3.87Al0.13) (Al1.13 Li1.41Fe2+0.07Mn0.05) O10 (OH, F)2
5.242(3)
9.055(6)
10.097(7)
100.77(5)
6.2
129. Guggenheim and Bailey 1977
(K0.90Na0.05) (Al1.05Li0.67Fe3+0.16 Fe2+0.77Mg0.01Mn0.05 Ti0.01) (Si3.09Al0.91) O10 (OH)0.79 F1.21
5.296(1)
9.140(2)
10.096(3)
100.83(2)
5.7
(K0.90Na0.08 Rb0.07) (Si3.33Al0.67) (Al0.98Li1.27Mn0.50Fe3+0.03Fe2+0.09 Ti0.005) O9.82 (OH0.60F1.58)
5.262(2)
9.102(3)
10.094(3)
100.83(2)
4.6
Polylithionite-siderophyllite intermediate, Sadisdorf Mine, Germany
130. Mizota et al. 1986 Masutomilite,Tanakamiyama, Ohtsu, Japan
Table 1c. Structural details of trioctahedral true Micas-2M1, space group C2/c
Cell parameters Reference (sample number) 131. Bigi and Brigatti 1994 (n # M7)
Species, locality
Rock type
Ferroan phlogopite Syenite Valle del Cervo (Italy)
Composition
3+
R
a
b
c
E
(Å)
(Å)
(Å)
(°)
2+
(%)
(K0.95Na0.03) (Mg1.55Fe 0.52Fe 0.70 5.339(1) Mn0.02Ti0.22) (Si2.78Al1.22) O10.73 (OH)1.27
9.249(1)
20.196(1)
95.06(1)
2.7
132. Bigi e al. 1993 (n # Magnesian annite, MP16) Ivrea-Verbano Zone (Italy)
Gabbroic calcalkaline rock
(K0.92Na0.01Ca0.001Ba0.04) (Fe2+1.36 Mg0.80 Al0.40Fe3+0.17Mn0.01Ti0.26) (Si2.84 Al1.16) O10.95 (OH)1.05
5.335(2)
9.242(3)
20.106(7)
95.07(3)
3.7
133. Bigi et al. 1993 (n Ferroan phlogopite # MP17a) Ivrea-Verbano Zone (Italy)
Gabbroic calcalkaline rock
(K0.98 Ca0.001 Ba0.02) (Mg1.63Al0.23 5.328(4) Fe2+0.81 Mn0.002Ti00.33) (Si2.79 Al1.21) O10.70 F0.31 (OH)0.99
9.220(3)
20.118(3)
95.11(3)
2.7
134. Bigi et al. 1993 (n Ferroan phlogopite # MP17b) Ivrea-Verbano Zone (Italy)
Gabbroic calcalkaline rock
(K0.95Na0.02Ca0.003Ba0.02) (Mg1.57 Al0.34Fe2+0.79Ti0.30Mn0.002) (Si2.87 Al1.13) O10.83 F0.26 (OH)0.91
5.323(1)
9.222(3)
20.130(5)
95.06(2)
3.4
135. Bohlen et al. 1980 Magnesian annite, Au Orthogneiss Sable Forks, New York (Northeast Adirondacks)
(K0.99 Ca0.003Na0.02) (Al0.12Mg1.16 5.357(6) Fe2+1.39 Mn0.007 Ti0.32) (Si2.79 Al1.21) O10.56 F0.08Cl0.14 (OH)1.22
9.245(5)
20.234(5)
94.98(4)
4.2
136. Brigatti et al. 2000a (n # C6c)
Magnesian annite, Tinker Glacier, Antarctica
Peraluminous granite
(K0.98Na0.02Ca0.01) (Fe2+1.36Al0.60 Mg.0.71Mn0.04Ti0.16) (Si2.71Al1.29) O10.36 (OH)1.64
5.335(1)
9.242(2)
20.181(4)
95.20(2)
2.8
137. Otha et al. 1982
Ferrian phlogopite, Ruiz Peak, Valles Mountains, New Mexico
Rhyodacite
(K0.77Na0.16Ba0.02) (Mg1.67Fe3+0.86 Fe2+0.01Mn0.01Ti0.34 Al0.16) (Si2.84 Al1.16) O11.62 F0.17 (OH)0.21
5.3175(7)
9.212(2)
19.976(3)
95.09(1)
3.9
138. Swanson and Bailey 1981
Polylithionite, Biskupice, Czech Republic
(K0.80Na0.004Cs0.02Rb0.06) (Li1.65 5.199(1) Al1.24Mg0.002Fe2+0.002Ti0.001 Mn0.04) (Si3.62Al0.38) O10 F1.52 (OH)0.48
9.026(2)
19.969(5)
95.41(2)
9.1
139. Takeda and Ross 1975
Hydrogenated, ferroan Rhyodacite phlogopite, Ruiz Peak, Valles Mountains, New Mexico
(K0.78Na0.16Ba0.02) (Mg1.68Fe3+0.19 Fe2+0.71Mn0.01Ti0.34 Al0.19) (Si2.86 Al1.14) O11.12 F0.17 (OH)0.71
9.234(3)
20.098(7)
95.09(3)
5.6
5.329(2)
73
Mica Crystal Chemisrty and Influence of P-T-X on Atomistic Models
3.5
112.6
60.4
112.5 5.8
111.0
108.8
60.8
56.5
60.5
56.4
Octahedral
WT1,T2
<M1
<M2 (u2)
(°)
(°)
(°)
(°)
58.9
58.6
110.6
3.318
0.087
2.252
2.078
3.333
0.127
109.8
2.238
2.098
3.340
0.115
110.2
110.1
2.257
2.170
3.376
0.018
110.2
59.3
58.8
2.255
2.144
3.359
0.026
110.2
59.3
59.0
2.258
2.128
3.374
0.020
110.2
58.9
58.9
2.244
2.146
3.393
0.006
112.3
58.7
58.4
2.250
2.185
3.393
0.008
110.2 110.3
-0.360
128.
1.882
2.990
3.251
-0.354
-0.004
-0.358
129.
3.017
3.215
-0.356
-0.004
-0.360
130.
Intralayer shift
Layer
Overall
offset
shift
a2, 3
a1, 1
a1
2.131 2.128
1.893 2.123
¢A- O² Inner
Outer
(Å)
(Å)
2.097
2.082
3.023
3.297
0.333
-0.001
-0.334
131.
1.639
2.100
2.071
3.006
3.317
0.330
-0.002
-0.333
132.
1.656
2.085
2.064
2.977
3.331
0.336
-0.001
-0.336
133.
1.662
2.074
2.076
2.980
3.337
0.332
0.000
-0.334
134.
1.661
2.106
2.086
3.046
3.289
0.334
0.001
-0.328
135.
2.099
2.063
2.988
3.346
0.338
-0.002
-0.343
136.
2.076
2.060
2.960
3.300
0.331
-0.001
-0.333
137.
2.107
1.977
2.964
3.237
0.357
-0.003
-0.362
138.
2.087
2.068
2.970
3.323
0.334
-0.001
-0.335
139.
1.656
59.4
58.8
2.254
2.135
3.406
0.024
1.658 1.660
59.4
59.1
2.270
2.113
3.295
0.026
1.653 1.656
60.7
58.6
2.256
2.061
3.366
0.072
112.2
7.7
-0.004
1.661
110.2
6.2
-0.356
1.658
110.1
7.4
1.657
3.18
1.677
110.4
7.8
1.626
3.02
1.654
110.0
5.3
2.132
Sheet InterBasal Mean bond lengths Thickness layer Tetra- Octa- separa- Oxygen ¢T1-O² ¢M1-O² ¢M2-O² tion hedral hedral 'z ¢T2-O² (u2) (Å) (Å) (Å) (Å) (Å) (Å) (Å)
110.2
7.8
1.646
1.88 2.12
1.641
109.9
7.8
2.12
1.639
110.7
6.8
1.64 1.63
60.4
D
6.0
2.092
60.8
111.3
Tetrahedral
2.254
60.5
111.1 4.4
56.2
1.628 1.631
59.2
58.9
2.269
2.135
3.337
0.020
1.662 1.657
74
Brigatti & Guggenheim
Table 1d. Structural details of trioctahedral true Micas-2M1, space group Cc, C1
Cell parameters Reference (sample number)
Species, locality
Rock type
R
a
b
c
D, E, J
(Å)
(Å)
(Å)
(°)
Composition
(%)
140. Rieder et al. 1996 Lithian siderophyllite, Space group Cc Barbora mine, Krupka, Czech Republic.
(K0.80Na0.04Rb0.05Ca0.02) (Fe2+1.07 5.292(1) Al0.97Li0.50Fe3+0.14 Mn2+0.03Mg0.02) (Si3.00 Al1.00) O10.00 F0.91 (OH)1.09
9.187(2)
19.935(3)
90 5.8 95.40(1) 90
141. Slade et al. 1987 Space group C1
5.123(2) (Na0.94K0.001Ca0.03) (Al2.01Li0.85 Fe3+0.01Mn0.005Mg0.03) (Si2.01 Al1.99) O10.00 (OH)2
8.872(3)
19.307(3)
89.97(2) 4.7 95.15(2) 89.96(2)
Ephesite, Postmasburg district, South Africa.
Table 1e. Structural details of trioctahedral true Micas-2M2, space group C2/c Cell parameters Reference (sample number)
Species, locality
Rock type
142. Guggenheim 1981 Trilithionite, Radkovice, Jihlava, Moravia (Czech Republic) 143. Sartori et al. 1973 Polylithionite, Elba Island, Italy
Pegmatite
144. Takeda et al. 1971 Trilithionite, Rozna, Moravia, (Czech Republic)
Composition
R
a
b
c
E
(Å)
(Å)
(Å)
(°)
(%)
(K0.79Rb0.07Cs0.03Na0.03Ca0.01) 9.023(2) (Li1.48Al1.30Mg0.05Fe3+0.008Fe2+0.002 Mn0.03) (Si3.49Al0.51) O10 (F, OH)2
5.197(2)
20.171(3)
99.48(2)
4.8
(K0.92Rb0.06Na0.06Cs0.004Ca0.01) 9.04(2) (Li1.76Al1.26Fe3+0.003Mn0.003 Mg0.007) (Si3.36Al0.64) O10 F1.53, (OH)0.47
5.22(2)
20.210(1)
99.6(3)
9.6
(K0.87Rb0.06Na0.12Cs0.005Ca0.02) (Li1.05Al1.40Fe2+0.07Mn0.03 Mg0.05) (Si3.39Al0.61) O10 F1.2 (OH)0.8
5.200(3)
20.15(4)
99.8(2)
7.2
9.032(2)
Table 1f. Structural details of trioctahedral true Micas-3T, space group P3112 Cell parameters Reference
Species, locality
Rock type
(sample number) 145. Brown 1978
Polylithionite, Coolgardie (Australia)
146. Weiss et al. 1993
Lithian siderophyllite, Kymi stock, Finland
Composition
(K0.85Na0.11 Rb0.05) (Al1.25Li1.62 Mg0.01Fe0.015Mn0.09) (Si3.48Al0.52) O10 (OH)0.44 F1.54 Granitic rock
R
a
c
(Å)
(Å)
(%)
29.76(1)
4.7
29.818(6)
3.0
5.200(5)
(K0.92Na0.03Rb0.04Ca0.01) (Al0.68 5.309(2) Li0.37Fe2+1.25Fe3+0.34 Zn0.02Mn2+0.04) (Si2.97Al1.03) O10 (OH)0.94 F1.06
75
Mica Crystal Chemisrty and Influence of P-T-X on Atomistic Models
Tetrahedral
Octahedral
D
WT1
WT2
<M1 <M2 <M3
(°)
(°)
(°)
(°)
(°)
Sheet thickness Tetra- Octahedral hedral (Å)
(°)
InterMean bond lengths ¢A- O² layer Basal sepa- oxygen ¢T1-O² ¢T2-O² ¢M1-O² ¢M2-O² ¢M3-O² Inner Outer ration 'z (Å) (Å) (Å) (Å) (Å) (Å) (Å) (Å) (Å) (Å)
5.9 109.0 109.7 60.8 56.9 60.8 2.241 2.077 3.401 0.123 1.642 1.645 2.131
Intralayer Layer Overall shift offset shift
a2, 3
a1, 1
a1
1.901
2.131
3.002 3.265 0.334 -0.002 -0.355
140.
22.1 108.8 108.1 61.6 58.0 58.4 2.277 2.024 3.077 0.158 1.653 1.734 2.128
1.910
1.933
2.491 3.522 0.348 -0.036 -0.338
141.
21.8 108.1 108.5 61.8 58.4 58.2 2.235 2.018 3.081 0.165 1.750 1.625 2.132
1.925
1.914
2.494 3.520
5.5 109.1 108.0
2.204
0.075 1.651 1.650
22.6 107.6 108.2
2.237
0.138 1.732 1.641
22.5 108.0 107.7
2.280
0.148 1.633 1.748
Tetrahedral
Octahedral
D
WT1,T2
<M1
<M2 (u2)
(°)
(°)
(°)
(°)
112.1 61.1
58.6
6.6
Sheet InterMean bond lengths Thickness layer Basal Tetra- Octa- separa- Oxygen ¢T1-O² ¢M1-O² ¢M2-O² hedral hedral tion 'z (u2) ¢T2-O² (Å) (Å) (Å) (Å) (Å) (Å) (Å) 2.254
2.052
3.387
0.083
112.1 6.5
111.9 60.8
58.4
2.241
2.074
3.409
0.094
58.2
2.246
2.076
3.360
0.095
112.3
D (°) 7.6
4.1
WT1 WT2 (°)
1.967
shift
(Å)
a1 a1
a1
(Å)
b3 b2
2.961
3.251
1.630
2.123
1.980
2.976
3.262
1.627
112.2 61.1
Tetrahedral
2.121
offset
Inner
1.629
112.0 5.4
1.629
Outer
Intralayer Shift
¢A- O²
1.620
2.144
1.967
2.980
3.220
1.633
Octahedral
Sheet thickness Interlayer Basal
<M1
Tetra- Octahedral hedral (Å) (Å)
(°)
<M2 <M3 (°)
112.6 59.6
60.8
111.8
57.6
110.7 60.5
56.8
111.1
60.2
2.257
2.059
Mean bond lengths
Separa- oxygen ¢T1-O² ¢M1-O² ¢M2-O² tion 'z ¢T2-O² ¢M3-O² (Å) (Å) (Å) (Å) (Å) 3.347
0.131
1.652
2.036
1.617 2.251
2.109
3.328
0.110
1.651 1.644
2.113
¢A - O² Inner
Outer
Layer Overall
-0.358
-0.005
+0.358
-0.005
-0.361
-0.005
+0.361
-0.005
-0.364
-0.007
+0.364
-0.007
Intralayer Layer shift a2,3,1
-0.368
142.
-0.372
143.
-0.380
144.
Overall
Offset a1,2,3
Shift
2.925
3.265
-0.355
-0.006
0
145.
3.041
3.225
-0.347
-0.003
0
146.
1.920 2.143
1.926 2.123
76
Brigatti & Guggenheim
TABLE 2. STRUCTURAL DETAILS OF DIOCTAHEDRAL TRUE MICAS
Table 2a . Structural details of trioctahedral true Micas-1M, space groups C2/m and C2 Cell parameters Reference (sample number)
Species, locality
Rock type
a
b
c
E
(Å)
(Å)
(Å)
(°)
Composition
(K0.65Na0.03) (Al1.83Fe3+0.03Fe2+0.04 Mg0.10 Mn0.04) (Si3.51Al0.49) O10.13 F0.07 (OH)1.80
Muscovite, 1. Sidorenko et al. 1975 (Space group Transbaikal, Siberia, Russia C2)
R
(%)
5.186
8.952
10.12
101.8
10.9
Paragonite 2. Soboleva et al. 1977 (Space group C2/m)
Synthetic
Na0.91 Al1.88 (Si3.45Al0.55) O10 (OH)2 5.135
8.890
9.74
99.7
12.1
3. Zhukhlistov et al. Celadonite, Krivoj 1977 (Space group Rog, Ukraine C2/m)
Iron-ore basin
(K0.83Na0.01Ca0.04) (Al0.05Fe3+1.15Fe2+0.36 Mg0.41Ti0.01) (Si3.94Al0.06) O10 F0.01 (OH)1.99
9.05
10.15
100.5
10.8
5.23
77
Mica Crystal Chemisrty and Influence of P-T-X on Atomistic Models
Tetrahedral D
W
Octahedral <M1 <M2,M3
vacancy
(°) 9.3
(°)
(°)
(°)
110.1
61.6
56.6
Basal Mean bond lengths Intralayer Layer Sheet thickness Interlayer oxygen ¢A - O² Tetra- Octa- separaoffset 'z ¢T-O² ¢M1-O² ¢M2-O² Inner Outer shift a1 hedral hedral tion a1 vacancy ¢M3-O² (Å) (Å) (Å) (Å) (Å) (Å) (Å) 2.196
2.113
3.399
0.220
1.614
2.221
1.920
Overall shift a1
2.897
3.306
-0.376
-0.024
-0.400
1.
1.633
1.957
111.1
57.3
19.1
110.4u2 59.9
57.8u2 2.222
2.099
3.059
0.096
1.659u2 2.091
1.971u2 2.561
3.441
-0.338
0.020
-0.319
2.
1.3
112.6u2 58.3
56.6u2 2.248
2.249
3.233
0.000
1.636u2 2.141
2.043u2 3.044
3.103
-0.354
-0.002
-0.356
3.
78
Brigatti & Guggenheim
Table 2b. Structural details of trioctahedral true Micas-1M, space group C2/c Cell parameters Reference (sample number)
4. Birle and Tettenhorst 1968
Species, locality
Rock type
Composition
3+ 0.12
Muscovite, Hartz Range, Australia
(K0.94Na0.06) (Al1.83Mg.0.06 Fe (Si3.11Al0.89) O10 (OH)2
R
a
b
c
E
(Å)
(Å)
(Å)
(°)
(%)
) 5.194(6)
8.996(6)
20.10(2)
95.2(1)
12
5. Brigatti et al. 1998 Muscovite, Maddalena Pegmatite (n # GA1) Island, Italy
(K0.99Na0.01Ba0.01) (Al1.65Fe2+0.29 Mn0.07Ti0.01) (Si3.30Al0.70) O10.01 F0.22 (OH)1.77
5.226(1)
9.074(2)
20.039(2)
95.74(1)
2.5
6. Brigatti et al. 1998 Muscovite, Antarctica (n # RA1)
(K0.92Na0.09) (Al1.78Mg.0.06 Fe2+0.12 Ti0.04) (Si3.18Al0.82) O10.08 F0.07 (OH)1.85
5.182(3)
8.982(5)
20.002(5)
95.72(2)
3.0
7. Brigatti et al. 1998 Muscovite, Sos Peraluminous (n # A4b) Canales pluton, Central granite Sardinia, Italy
(K0.92Na0.09) (Al1.88Mg.0.05 Fe3+0.09 Ti0.02) (Si2.92Al1.08) O10.01 F0.11 (OH)1.88
5.186(1)
8.991(3)
20.029(7)
95.77(3)
3.6
8. Brigatti et al. 1998 Muscovite, Sos Peraluminous (n # GFS15Ab) Canales pluton, Central granite Sardinia, Italy
(K0.92Na0.08) (Al1.86Fe3+0.01Mg.0.07 5.192(2) Fe2+0.06Ti0.02) (Si3.03Al0.97) O10 F0.09 (OH)1.91
9.013(5)
20.056(7)
95.83(3)
2.9
9. Brigatti et al. 1998 Muscovite, Riu (n # H87b) Morunzu, Sardinia, Italy
Two-mica leucogranite
(K0.96Na0.05) (Al1.71Fe3+0.16 Fe2+0.13 Mn0.01) (Si3.09Al0.91) O10.01 F0.22 (OH)1.77
5.209(3)
9.035(6)
20.066(9)
95.68(3)
3.9
10. Brigatti et al. 1998 Muscovite, Frontier (n # CC1b) Mountains Area, Antarctica
Peraluminous granite
(K0.93Na0.08) (Al1.83Mg.0.07 Fe2+0.07 Ti0.06) (Si3.18Al0.82) O10.21 (OH)1.79
5.186(1)
9.005(1)
20.031(3)
95.78(1)
2.9
11. Brigatti et al. 1998 Muscovite, Tinker (n # C3-29b) Glacier, Antarctica
Peraluminous granite
(K0.88Ca0.06Na0.06) (Al1.88Mg.0.06 Fe2+0.07Ti0.03) (Si3.07Al0.93) O10.17 F0.19 (OH)1.64
5.188(1)
8.996(3)
20.082(2)
95.78(1)
2.8
12. Brigatti et al. 1998 Muscovite, Tinker (n # B1b) Glacier, Antarctica
Peraluminous granite
(K0.94Na0.07) (Al1.83Mg.0.07 Fe2+0.07 Ti0.06) (Si3.09Al0.91) O10.12 F0.23 (OH)1.65
5.187(2)
9.004(2)
20.036(2)
95.73(2)
2.1
13. Brigatti et al. 1998 Muscovite, Tinker (n # C6Cb) Glacier, Antarctica
Peraluminous granite
(K0.92Na0.09) (Al1.78Mg.0.15 Fe2+0.13 Ti0.04) (Si3.17Al0.83) O10.25 F0.19 (OH)1.56
5.186(1)
9.003(1)
20.030(4)
95.84(2)
3.9
14. Brigatti et al. 1998 Muscovite, Tinker (n # C6Bb) Glacier, Antarctica
Peraluminous granite
(K0.93Na0.05) (Al1.80Mg.0.15 Fe2+0.07 Ti0.05) (Si2.87Al1.13) O9.91 F0.41 (OH)1.68
5.196(2)
8.997(3)
20.034(4)
95.80(2)
3.1
15. Brigatti et al. 1998 Muscovite, Tinker (n # C3-31b) Glacier, Antarctica
Peraluminous granite
(K0.93Na0.05Ca0.01) (Al1.64Fe3+0.08 Fe2+0.08Mg0.16Ti0.02) (Si3.18Al0.82) O9.93 (OH)2.07
5.197(1)
9.022(2)
20.076(4)
95.79(2)
2.8
16. Brigatti et al. 2001 Chromium-containing muscovite, Westland, (Westland) New Zealand.
In glacial moraines
(K0.86Na0.10Ba0.04) (Al1.86Mg0.08 Fe2+0.04Cr0.06) (Si3.11Al0.89) O10.17 (OH)1.83
5.192(1)
9.011(1)
20.028(2)
95.74(1)
2.5
17. Brigatti et al. 2001 Chromium-containing (Campbell Creek) muscovite, Northwest Nelson, Campbell Creek, New Zealand
Biotite schist
(K0.73Na0.27) (Al1.84Mg0.02 Fe2+0.02 5.175(1) Cr0.10Ti0.02) (Si3.07Al0.93) O10.05 F0.03 (OH)1.92
8.979(2)
19.915(2)
95.66(1)
3.1
(K0.96Na0.03Ba0.01) (Al1.83Mg0.11 Fe2+0.10Cr0.11Ti0.03) (Si3.14Al0.86) O10.50 F0.04 (OH)1.46
9.040(3)
20.058(9)
95.79(4)
3.3
Pegmatite
18. Brigatti et al. 2001 Chromium-containing Quartz schist Northwest (Anatoki River) muscovite, Nelson, Anatoki River, New Zealand
5.206(1)
79
Mica Crystal Chemisrty and Influence of P-T-X on Atomistic Models Table 1a . Structural details of trioctahedral true micas-2M1, C12/c(1) layer symmetry (May 20, 2000)
Tetrahedral D (°) 12.0
WT1, WT2 (°)
Octahedral vacancy
<M2 (u2)
Tetrahedral
(°)
(°)
(Å)
110.6 62.4
57.0
2.243
<M1
Inter- Basal Mean bond lengths layer oxygen Octa- separa¢T1-O² ¢M1-O² ¢M2-O² 'z tion hedral ¢T2-O² vacancy (u2) (Å) (Å) (Å) (Å) (Å) (Å)
2.097
3.427
0.236
111.2 7.7
111.5 61.8
111.0 62.3
57.4
2.247
2.107
3.368
0.179
111.1 62.1
57.1
2.242
2.088
3.378
0.223
110.9 62.3
57.1
2.242
2.095
3.385
0.223
111.2 62.0
57.2
2.242
2.087
3.405
0.225
111.0 62.2
57.3
2.251
2.099
3.383
0.214
111.0 62.3
57.2
2.242
2.090
3.391
0.221
111.1 62.3
57.2
2.246
2.089
3.409
0.230
111.0 62.3
57.1
2.245
2.091
3.388
0.219
111.1 62.2
57.2
2.241
2.088
3.393
0.225
111.2 62.2
57.2
2.242
2.090
3.391
0.219
111.1 62.2
57.1
2.248
2.097
3.393
0.224
111.0 62.2
57.1
2.250
2.095
3.369
0.223
111.5 62.1 111.2
0.002
-0.349
4.
1.640
2.230
1.953
2.943
3.287
0.374
-0.005
-0.384
5.
1.643
2.243
1.923
2.849
3.351
0.378
-0.002
-0.385
6.
1.642
2.242
1.928
2.854
3.354
0.378
-0.005
-0.388
7.
1.644
2.244
1.928
2.864
3.362
0.379
-0.006
-0.392
8.
1.641
2.237
1.941
2.887
3.336
0.374
-0.004
-0.381
9.
1.642
2.243
1.928
2.860
3.354
0.377
-0.006
-0.389
10.
1.645
2.247
1.925
2.858
3.364
0.377
-0.006
-0.390
11.
1.642
2.245
1.927
2.865
3.347
0.376
-0.005
-0.386
12.
1.644
2.245
1.926
2.857
3.358
0.378
-0.007
-0.393
13.
1.641
2.244
1.928
2.863
3.353
0.377
-0.006
-0.390
14.
1.647
2.251
1.931
2.868
3.356
0.377
-0.006
-0.390
15.
1.646
2.244
1.931
2.848
3.358
0.377
-0.005
-0.386
16.
2.244
1.925
2.810
3.361
0.376
-0.002
-0.380
17.
2.246
1.937
2.875
3.344
0.379
-0.006
-0.389
18.
1.646 57.0
2.242
2.095
3.329
0.228
110.9 10.5
0.368
1.645
111.1 12.3
3.385
1.644
111.1 11.4
2.852
1.642
111.0 10.9
1.925
1.641
110.9 11.0
2.259
1.644
111.1 11.3
a1
1.642
111.0 10.8
a1, 1
1.647
111.0 11.4
shift
a2, 3
1.643
111.1 11.1
(Å)
Overall
1.640
110.8 10.1
(Å)
Layer offset
1.639
111.1 11.2
Outer
1.640
111.1 11.2
1.660
Inner
Intralayer shift
1.636
111.5 11.3
¢A- O²
Sheet thickness
1.645 1.644
57.2
2.252
2.099
3.374
0.196
1.642 1.650
80
Brigatti & Guggenheim
19. Catti et al. 1989 (T Muscovite, Monte Pegmatite = 25°C) Botte Donato, Calabria, Italy
(K0.86Na0.11) (Al1.93 Fe0.07Mg.0.02) (Si3.08Al0.92) O10 (OH)2
5.191(1)
9.006(3)
20.068(6)
95.77(2)
4.8
20. Catti et al. 1989, (T = 700°C)
Muscovite, Monte Pegmatite Botte Donato, Calabria, Italy
(K0.86Na0.11) (Al1.93 Fe0.07Mg.0.02) (Si3.08Al0.92) O10 (OH)2
5.229(1)
9.076(3)
20.322(8)
95.74(3)
6.0
21. Catti et al. 1994 (Room pressure)
Muscovite, Effingham Township, Ontario
(K0.90Na0.07) (Al1.63Fe0.23Mg0.16 Ti0.03) (Si3.20Al0.80) O10 (OH)2
5.2108(4)
9.0399(8)
20.021(2)
95.76(1)
4.0
Paragonite, Western 22. Comodi and Zanazzi 1997 (n # Alps, Italy AL433, 0.001 Kbar)
(Na0.88K0.10Ca0.01Ba0.01) (Al1.97 Fe0.01Mn0.002Mg0.006Ti0.007) (Si3.01 Al0.99) O10 (OH)2
5.135(1)
8.906(1)
19.384(4)
94.6(1)
2.1
Paragonite, Western 23. Comodi and Zanazzi 1997 (n # Alps, Italy AL433, 0.5 Kbar)
(Na0.88K0.10Ca0.01Ba0.01) (Al1.97 Fe0.01Mn0.002Mg0.006Ti0.007) (Si3.01 Al0.99) O10 (OH)2
5.134(3)
8.906(5)
19.32(1)
94.5(2)
6.1
Paragonite, Western 24. Comodi and Zanazzi 1997 (n # Alps, Italy AL433, 25.4 Kbar)
(Na0.88K0.10Ca0.01Ba0.01) (Al1.97 Fe0.01Mn0.002Mg0.006Ti0.007) (Si3.01 Al0.99) O10 (OH)2
5.082(2)
8.813(5)
18.91(1)
94.7(2)
7.0
25. Comodi and Paragonite, Western Zanazzi 1997 (n # Alps, Italy AL433, 40.5 Kbar)
(Na0.88K0.10Ca0.01Ba0.01) (Al1.97 Fe0.01Mn0.002Mg0.006Ti0.007) (Si3.01 Al0.99) O10 (OH)2
5.062(2)
8.769(3)
18.64(2)
95.2(2)
6.5
Paragonite, Guatemala 26. Comodi and Zanazzi 2000 (n # AMNH104213) (T = 25°C)
(Na0.91K0.07Ca0.01Ba0.01) (Al1.99 Fe0.01Mn0.001Mg0.02Ti0.005) (Si2.92 Al1.08) O10 (OH)2
5.140(2)
8.911(5)
19.38(1)
94.62(1)
3.7
Paragonite, Guatemala 27. Comodi and Zanazzi 2000 (n # AMNH104213) (T = 210°C)
(Na0.91K0.07Ca0.01Ba0.01) (Al1.99 Fe0.01Mn0.001Mg0.02Ti0.005) (Si2.92 Al1.08) O10 (OH)2
5.152(2)
8.941(5)
19.46(1)
94.26(1)
2.5
Paragonite, Guatemala 28. Comodi and Zanazzi 2000 (n # AMNH104213) (T = 450°C)
(Na0.91K0.07Ca0.01Ba0.01) (Al1.99 Fe0.01Mn0.001Mg0.02Ti0.005) (Si2.92 Al1.08) O10 (OH)2
5.173(2)
8.985(5)
19.55(1)
93.58(1)
2.9
Paragonite, Guatemala 29. Comodi and Zanazzi 2000 (n # AMNH104213) (T = 600°C)
(Na0.91K0.07Ca0.01Ba0.01) (Al1.99 Fe0.01Mn0.001Mg0.02Ti0.005) (Si2.92 Al1.08) O10 (OH)2
5.190(3)
9.011(6)
19.60(2)
92.96(1)
4.4
30. Evsyunin et al. 1997
(K0.82Ba0.14Na0.04) (Cr3+1.42Al0.27 V3+0.13Mg0.18Fe2+0.01) (Si3.02 Al0.98) O10 F0.30 (OH)1.66
5.240(3)
9.103(2)
19.93(4)
95.59(3)
4.8
31. Guggenheim et al. Muscovite, Diamond Pegmatite 1987 (T = 20°C) mine, Keystone, South Dakota
(K0.93Na0.08Ca0.01) (Al1.83Fe0.16 5.200(4) Mg0.01Mn0.01) (Si3.10Al0.90) O10 F0.17 (OH)1.83
9.021(7)
20.07(2)
95.71(7)
4.0
32. Guggenheim et al. Muscovite, Diamond Pegmatite 1987 (T = 300°C) mine, Keystone, South Dakota
(K0.93Na0.08Ca0.01) (Al1.83Fe0.16 5.215(2) Mg0.01Mn0.01) (Si3.10Al0.90) O10 F0.17 (OH)1.83
9.053(4)
20.15(1)
95.72(3)
5.3
33. Guggenheim et al. Muscovite, 1987 (T = 20°C) Panasqueira, Portugal
(K1.00Na0.03Ca0.01) (Al1.93Fe2+0.01 5.1579(9) Mg0.01Mn0.01) (Si3.09Al0.91) O10 F0.12 (OH)1.88
8.9505(8)
20.071(5)
95.75(2)
5.2
34. Guggenheim et al. Muscovite, 1987 (T = 525°C) Panasqueira, Portugal
5.182(1) (K1.00Na0.03Ca0.01) (Al1.93Fe2+0.01 Mg0.01Mn0.01) (Si3.09Al0.91) O10 F0.12 (OH)1.88
8.993(1)
20.232(5)
95.75(2)
6.9
35. Guggenheim et al. Muscovite, 1987 (T = 650°C) Panasqueira, Portugal
5.189(1) (K1.00Na0.03Ca0.01) (Al1.93Fe2+0.01 Mg0.01Mn0.01) (Si3.09Al0.91) O10 F0.12 (OH)1.88
9.004(1)
20.256(6)
95.74(2)
7.3
Granite pegmatite
Chromphyllite, Slyudyanka, Irkutsk region
Mica Crystal Chemisrty and Influence of P-T-X on Atomistic Models
11.8
111.0
62.4
57.2
2.248
2.089
3.398
0.232
111.0 8.2
111.2
109.8
62.5
57.2
2.257
2.100
3.495
0.212
110.4
62.0
57.2
2.241
2.108
3.370
0.229
110.3
62.0
56.9
2.243
2.085
3.089
0.234
111.5
62.1
57.0
2.247
2.080
3.056
0.193
111.5
62.6
57.9
2.306
1.998
2.814
0.188
110.4
62.5
57.8
2.252
1.980
2.797
0.167
110.4
62.0
56.9
2.246
2.092
3.074
0.232
110.4
62.1
57.0
2.247
2.091
3.118
0.225
110.3
62.2
57.0
2.248
2.096
3.167
0.238
111.6
62.2
57.1
2.244
2.101
3.200
0.243
111.0
61.2
57.3
2.252
2.136
3.278
0.155
111.0
62.1
57.2
2.249
2.098
3.388
0.216
110.9
62.2
57.1
2.246
2.105
3.427
0.229
111.2
62.2
57.1
2.234
2.081
3.436
0.218
111.1 111.2
-0.005
-0.389
20.
1.645
2.241
1.943
2.897
3.322
0.372
-0.005
-0.386
21.
1.654
2.225
1.912
2.642
3.375
0.373
0.036
-0.303
22.
1.65
2.22
1.91
2.63
3.37
0.372
0.036
-0.295
23.
1.68
2.17
1.88
2.50
3.30
0.364
0.024
-0.305
24.
1.68
2.14
1.86
2.48
3.33
0.366
0.015
-0.334
25.
1.656
2.225
1.915
2.634
3.378
0.373
0.037
-0.304
26.
1.655
2.233
1.918
2.672
3.375
0.374
0.049
-0.281
27.
1.655
2.243
1.925
2.722
3.372
0.374
0.070
-0.236
28.
1.652
2.256
1.932
2.768
3.359
0.374
0.090
-0.195
29.
1.644
2.213
1.976
2.933
3.260
0.368
-0.002
-0.370
30.
1.646
2.243
1.935
2.858
3.364
0.376
-0.004
-0.384
31.
1.649
2.256
1.939
2.898
3.358
0.377
-0.004
-0.385
32.
1.635
2.234
1.916
2.848
3.368
0.377
-0.006
-0.390
33.
2.246
1.925
2.916
3.347
0.379
-0.006
-0.391
34.
2.249
1.926
2.935
3.340
0.379
-0.006
-0.390
35.
1.637 62.3
57.1
2.241
2.091
3.492
0.213
111.1 9.2
0.380
1.645
110.8 9.8
3.332
1.647
111.0 11.8
2.970
1.643
111.2 10.3
1.936
1.648
111.6 11.3
2.274
1.651
110.3 7.3
1.642
1.652
110.5 12.9
19.
1.654
110.4 14.3
-0.389
1.65
110.4 15.3
-0.005
1.67
112.1 16.2
0.377
1.65
112.7 18.4
3.373
1.651
110.0 17.9
2.848
1.640
110.4 16.3
1.925
1.646
110.7 16.0
2.251
1.647
111.1 9.5
1.647
81
1.637 1.635
62.3
57.1
2.241
2.091
3.506
0.220
1.636 1.634
82
Brigatti & Guggenheim
[K0.86Na0.10 (H+3°)0.01] (Al1.90 Mg.0.06Fe3+0.02Fe2+0.05 Ti0.01) (Si3.02Al0.98) O10 F0.01 (OH)1.99
5.1906(2)
9.0080(3)
20.0470(6) 95.757(2)
3.5
(K0.87Na0.07Ba 0.01Ca 0.02) (Al1.43 Mg.0.50Fe3+0.05Fe2+ 0.09 Ti0.01) (Si3.39Al0.61) O10.08 (OH)1.92
5.2112(3)
9.0383(4)
19.9473(6) 95.769(5)
4.5
Metamorphosed (K Na Ba 5.2044(8) 0.93 0.05 0.007) (Al1.72Mg.0.10 sedimentary Fe3+0.15Mn3+0.02Ti0.02) (Si3.06 Al0.94) manganese O10 (OH)2 deposits
9.018(2)
20.073(5)
95.82(2)
2.7
Glaucophane(Na0.92K0.04Ca 0.02) (Al1.99Mg.0.01 bearing Fe0.03Ti0.003) (Si2.94Al1.06) O10 metamorphosed (OH) 2 eclogite
5.128(2)
8.898(3)
19.287(9)
94.35(3)
4.5
40. Brigatti et al. 2001 Muscovite, Fregeneda, Granitic (n # 39) Portugal pegmatite
(K0.94Na0.05Rb0.01) (Al1.94Fe2+0.08 Mg0.02Li0.03) (Si3.07Al0.93) O10.12 F0.09 (OH)1.79
5.193(1)
9.016(3)
20.114(5)
95.77(2)
3.5
Granitic 41. Brigatti et al. 2001 Lithian, ferroan (n # 147) muscovite, Pikes Peak, pegmatite Colorado
(K0.96Na0.01Rb0.03) (Al1.49Fe3+0.07 Fe2+0.39Mn0.01Li0.28) (Si3.24Al0.76) O10.0 F0.43 (OH)1.57
5.209(2)
9.038(3)
19.997(5)
95.70(3)
4.1
42. Brigatti et al. 2001 Lithian, ferroan Granitic (n # 129) muscovite, Pikes Peak, pegmatite Colorado
(K0.98Na0.02Rb0.02) (Al1.45Fe3+0.08 Fe2+0.33Mn0.03Ti0.01 Li0.37) (Si3.28 Al0.72) O10.0 F0.57 (OH)1.43
5.224(1)
9.081(3)
19.952(4)
95.63(2)
3.5
43. Brigatti et al. 2001 Muscovite, Argemela, (n # 2b) Portugal
Granite
(K0.86Na0.15Rb0.02) (Al1.79Fe2+0.13 Mg0.01Mn0.01Li0.12) (Si3.19Al0.81) O10.0 F0.29 (OH)1.71
5.190(2)
9.022(3)
20.057(4)
95.60(7)
3.3
44. Brigatti et al. 2001 Muscovite, Argemela, (n # 2a) Portugal
Granite
(K0.86Na0.15Rb0.02) (Al1.81Fe2+0.13 Mg0.01Mn0.01Li0.12) (Si3.13Al0.87) O10.0 F0.29 (OH)1.71
5.197(1)
9.019(2)
20.068(3)
95.71(1)
4.2
(K0.88Na0.03Ca0.01) (Al1.87Ti0.03 Mg0.06 Mn3+0.03) (Si3.01Al0.85 Fe3+0.14) O10 (OH)2
5.199(2)
9.027(2)
20.106(4)
95.78(4)
9.9
(K0.85Na0.09) (Al1.81Fe2+0.14 Mg0.12) (Si3.09Al0.91) O9.81 F0.19 (OH)2
5.1918(2)
9.0155(5)
20.0457(7) 95.735(3)
2.7
(K0.95Na0.05Ba0.03) (Al1.51Mg0.27 Fe0.14Cr0.10Ti0.01Mn0.003) (Si3.25 Al0.75) O10 (OH)2
5.2153(5)
9.043(2)
19.974(9)
95.789(9)
3.3
8.894
19.365
94.10
11.1
36. Güven 1971b
Muscovite, Georgia
37. Güven 1971b
Magnesian muscovite, Schist Tiburon Peninsula, California
38. Knurr and Bailey 1986
Muscovite, Minas Gerais, Brazil
39. Lin and Bailey 1984 (n # PWB 1705)
Paragonite, ZermattSaas Fee, Swiss Alps
45. Richardson and Richardson 1982
Muscovite, Archer’s Post, Kenya
46. Rothbauer 1971
Muscovite, Diamond Mine, Black Hills, South Dakota
47. Rule and Bailey 1985
Magnesium-containing muscovite, Rio de Oro, Sahara
48. Sidorenko et al. 1977a
Paragonite, Southern Urals
Pegmatite
Pegmatite
Pyroclastic rock (K0.10Na0.60Ca0.03) (Al1.93 Mg0.10 5.135 Fe2+0.02) (Si2.98 Al1.02) O8.78 (OH)2.22
83
Mica Crystal Chemisrty and Influence of P-T-X on Atomistic Models
11.4
111.0 62.1
57.0
2.239
2.104
3.392
0.227
111.1 6.0
111.5 61.4
111.1 62.1
57.1
2.219
2.126
3.359
0.161
110.3 62.1
57.0
2.243
2.106
3.394
0.224
110.9 62.4
57.0
2.243
2.077
3.053
0.231
111.9 61.7
57.2
2.249
2.087
3.422
0.232
111.7 61.5
57.2
2.241
2.110
3.356
0.179
111.5 61.8
57.4
2.238
2.114
3.337
0.147
111.3 62.1
57.8
2.255
2.072
3.399
0.172
111.1 61.9
57.5
2.252
2.081
3.399
0.198
111.0 62.2
56.8
2.236
2.123
3.406
0.218
111.6 61.7
57.2
2.245
2.089
3.393
0.217
111.0 60.6 112.3 57.0
1.956
2.971
3.236
0.376
-0.004
-0.385
37.
1.644
2.252
1.935
2.873
3.353
0.378
-0.006
-0.391
38.
1.653
2.221
1.908
2.457
3.370
0.376
0.045
-0.285
39.
1.647
2.249
1.927
2.866
3.372
0.378
-0.006
-0.389
40.
1.634
2.226
1.949
2.911
3.297
0.377
-0.006
-0.381
41.
1.629
2.214
1.963
2.973
3.237
0.374
-0.004
-0.375
42.
1.639
2.190
1.944
2.896
3.324
0.369
-0.005
-0.377
43.
1.643
2.219
1.937
2.877
3.347
0.373
-0.005
-0.384
44.
1.639
2.253
1.941
2.871
3.361
0.378
-0.005
-0.389
45.
1.645
2.241
1.930
2.857
3.362
0.376
-0.005
-0.386
46.
2.233
1.952
2.924
3.278
0.376
-0.005
-0.386
47.
2.161
1.950
2.586
3.386
0.355
0.053
-0.274
48.
1.644 57.1
2.237
2.121
3.341
0.185
111.6 17.4
2.222
1.646
110.9 7.9
1.621
1.643
110.9 11.3
36.
1.639
111.2 11.0
-0.387
1.637
111.6 10.6
-0.005
1.638
111.6 9.7
0.377
1.646
111.6 5.9
3.362
1.652
110.9 8.7
2.855
1.644
110.3 11.4
1.933
1.634
111.0 16.2
2.245
1.642
111.4 10.8
1.643
1.636 1.637
2.264
2.125
3.005
0.13
1.671 1.661
84
Brigatti & Guggenheim
Table 2c. Structural details of trioctahedral true Micas-2M2, space group C2/c
Cell parameters Reference (sample number)
49. Ni and Hughes 1996
Species, locality
Nanpingite, Nanping, Fujian, China
50. Zhoukhlistov et al. Muscovite, North 1973 Armenia.
Rock type
Granitic pegmatite
Composition
(Cs0.88K0.06Rb0.01) (Al1.64 Fe2+0.17Mg0.22Li0.15) (Si3.16Al0.84) O10 (OH)1.79 F0.21
R
a
b
c
E
(Å)
(Å)
(Å)
(°)
9.076(3)
Metasomatized (K0.68Na0.09) Al1.93 (Si3.5Al0.5) O10.06 8.965 pyrite deposits (OH)1.94
5.226(2)
21.41(5)
99.48(6)
5.8
5.175
20.31
100.66
11.7
Table 2d. Structural details of trioctahedral true Micas-3T, space group P3112
Cell parameters Reference
Species, locality
Rock type
(sample number) 51. Amisano Canesi et Muscovite, (North al. 1994 (n # KZ) Kazakhstan)
Shist, conditions: 6 ΘD, where ΘD is the Debye temperature of the compound under study, given that PTH can be
approximated as Pt = αV ,0 K 0 (T − T0 ) (7) where αV,0 stands for the bulk thermal expansion coefficient at a reference temperature (T0) and ambient pressure. Such an approximation, which was empirically proven (Anderson and Isaak 1995) to hold below ΘD for many materials, allows one to linearize the dependence of the EoS on T, and leads to a significant numerical simplification. Other EoS have not been considered in this brief outline for the sake of brevity: Poirier and Tarantola (1998), Jackson and Rigden (1996), Kumar and Bedi (1996), Holzapfel (1996), Kumar (1995). Also the use of the Rankine-Hugoniot equations related to shock wave methods (Ahrens 1987) is not discussed here. Dioctahedral micas Compressibility of white mica . Several methods were used to study the elastic behavior of dioctahedral micas. After the first compression data of Bridgman (1949), Aleksandrov and Ryzhova (1961) measured an incomplete set of elastic constants of muscovite by ultrasonic methods, assuming a hexagonal symmetry for mica. Vaughan and Guggenheim (1986) measured muscovite elasticity by Brillouin spectrometry in its proper monoclinic symmetry. Sekine et al. (1991) obtained the EoS of muscovite by a shock wave method. Faust and Knittle (1994) studied muscovite under pressure by powder X-ray methods to 270 kbar, where complete amorphization of mica occurs. Catti et al. (1994) determined the compressibility of muscovite 2M1 by powder neutron diffraction up to 20 kbar. Comodi and Zanazzi (1995, 1997) measured the compressibility of two samples of muscovite with different Na content, and that of paragonite by singlecrystal diffraction methods in the range of 0-40 kbar. The EoS of a phengite 3T was modeled by Pavese et al. (1999b) on the basis of synchrotron powder diffraction measurements, performed at high P and T (P in the range of 0-50 kbar, T to 1000K). Finally, Smyth et al. (2000) determined the bulk elastic properties of synthetic 2M1 and 3T Si-rich phengites. No data are available for brittle micas. Figure 1 plots the equations of state of muscovite on the basis of the literature data. Differences in samples and methodologies produce scattered values of the compressibility (Table 1). The main observed features, however, show the strong control the atomic arrangement has on the anisotropy of the elastic behavior. The compressibility along [0 0 1] direction is between three and five times (depending on chemical composition) greater than that along the a or b axes. The main deformation mechanisms based on the single-crystal structural refinements under pressure for muscovite and paragonite-2M1 (Comodi and Zanazzi 1995, 1997) are: 1. The high compressibility along the [001] direction, largely related to the reduction in the interlayer thickness. This behavior is expected because of the weak bonding of the interlayer cation. The effect is more evident if the cation is potassium, and decreases if potassium is substituted by sodium (Fig. 2). 2. The smaller compressibility observed as Na increases relative to K is explained by stronger repulsion between the basal oxygen atom planes on both sides of the Na interlayer cation, owing to shorter csinβ and greater α tetrahedral rotation than for samples containing K. 3. (Si,Al)-O bond lengths do not vary significantly in the P range of 0-40 kbar. On the whole, tetrahedral volume increases slightly. Tetrahedral thickness also apparently increases, probably owing to tetrahedral tilting out of the (001) plane. A reduction of
104
Zanazzi & Pavese
the corrugation parameter, Δz, as defined by Güven (1971), is observed as P increases. 4. In the octahedral sheet there is a small decrease in the volume of the octahedra. A larger decrease is observed in the nominally vacant M1 octahedron, whereas in M2, which is Al rich, a smaller decrease occurs. On the whole, the octahedral sheet is thinned along the [0 0 1] direction. 5. The compressibility of the octahedra is greater than that of the tetrahedra. This results in an increased dimensional misfit between tetrahedral and octahedral sheets, so that there is an increase in the tetrahedral rotation angle, α, with P (from 16.0 to 18.4° at 41 kbar for paragonite, and from 11.5 to 12.7° at 28 kbar for muscovite). 6. The partial occupancy of the M1 site by divalent cations, and a greater Si content in the tetrahedral sheet, should decrease the compressibility. Therefore the presence of a phengitic component probably increases the mica stiffness, as shown by the increased bulk modulus of phengite-3T in comparison with that of end-member muscovite (Pavese et al. 1999b). The difference found, however, could be partially ascribed to the different mica polytype. A statistical study on the occurrence of metamorphic phengites by Sassi et al. (1994) seems to show that the crystallization of the 3T polytype is mainly favored by high P/T ratio conditions.
Figure 1. Equation of state of muscovite. Elastic parameters from Table 1. Full line: Vaughan and Guggenheim (1986); dashed line: Sekine et al. (1991); dot-dashed line: Faust and Knittle (1994); dotted line: Catti et al. (1994) and Comodi and Zanazzi (1994).
Presently, the lack of reliable compressibility data precludes any affirmation about the relative stiffness of the various polytypes of mica, and hence any speculation on their relative stability under high-pressure conditions. The progressive Si-Al ordering in tetrahedral sites of muscovites with increasing pressure, as claimed by Velde (1980), was not confirmed by Flux et al. (1984) on the basis of infrared (IR) spectra of synthetic 2M1 samples crystallized under different pressures. Above 180 kbar, at ambient temperatures, muscovite begins to amorphize (Faust and Knittle 1994). The amorphization is complete near 270 kbar. In a diamond anvil cell heated at about 800°C by laser, muscovite transforms between 18 and 37 kbar into an assemblage containing sanidine, corundum, H2O, and a cymrite phase KAlSi3O8.H2O. Between 37 and 40 kbar, muscovite breaks down to wadeite, kyanite, corundum and H2O. Over 109, and to ~210 kbar, an assemblage of hollandite, corundum and H2O is stable (Faust and Knittle 1994).
Micas at High Temperature and High Pressure
105
Table 1. Bulk modulus at P = 0 (K0 in kbar)* and its first derivative versus pressure for dioctahedral micas. * K0 values obtained from static compression measurements are the isothermal moduli, values obtained from Brillouin spectroscopy and shock wave measurements are the adiabatic moduli. Sample
Muscovite-2M1
K0
K’
0
Technique
582
-
Brillouin spectroscopy
520
3.2
shock wave
614
6.9
static compression powder diffraction
560
4
static compression neutron powder diffraction
560
-
static compression single crystal diffraction (β -1)
600
-
static compression single crystal diffraction (β -1)
655
-
static compression single crystal diffraction (β -1)
583
6.6
620
9
static compression single crystal diffraction
570
9.2
static compression single crystal diffraction
(Vaughan & Guggenheim 1986)
Muscovite-2M1 (Sekine et al. 1991)
Muscovite-2M1 (Faust and Knittle 1994)
Muscovite-2M1 (Catti et al. 1994)
Muscovite-2M1 (Comodi and Zanazzi 1995)
Na-Muscovite-2M1 (Comodi and Zanazzi 1995)
Paragonite-2M1 (Comodi and Zanazzi 1997)
Phengite-3T (Pavese et al. 1999)
Phengite-3T (Smyth et al. 2000)
Phengite-2M1 (Smyth et al. 2000)
static compression synchrotron X-ray diffraction
Figure 2. β-1 (kbar) as a function of Na/(Na+K) in dioctahedral micas. Data from Comodi and Zanazzi (1997).
106
Zanazzi & Pavese -1
Table 2. Axial thermal expansion coefficients (αx in °C ) for the cell edges and the volume, and the first derivative of tetrahedral rotation angle α versus T in dioctahedral micas. Sample Muscovite-2M1 (Guggenheim et al. 1987) Muscovite-2M1 (Catti et al. 1989) Muscovite-2M1 (Symmes 1986) Phengite-2M1 (Pavese et al. 1999) Phengite-2M1 (Mookherjee et al. 2000) Phengite-3T (Amisano Canesi 1995) Phengite-3T (Pavese et al. 1997) Paragonite-2M1 (Comodi and Zanazzi 2000) Paragonite-2M1 (Symmes 1986) Margarite (Symmes 1986)
αa
αb
αc
αV
0.99×10-5
1.11×10-5
1.38×10-5
1.12×10-5
1.18×10-5
1.89×10-5
4.2×10-5
1.4×10-5
1.4×10-5
1.9×10-5
4.7×10-5
-
0.89×10-5
0.98×10-5
1.66×10-5
3.43×10-5
-
0.86×10-5
0.99×10-5
2.15×10-5
4.05×10-5
-
0.57×10-5
-
2.53×10-5
4.26×10-5
0.57×10-5
-
2.14×10-5
3.31×10-5
-
1.51×10-5
1.94×10-5
2.15×10-5
5.9×10-5
-5.74×10-3
1.5×10-5
1.4×10-5
1.7×10-5
4.9×10-5
-
0.86×10-5
0.65×10-5
1.2×10-5
2.8×10-5
-
dα/dT
3.54×10-5 - 4.12×10-3 -5.3×10-3
-4.5×10-3
Thermal expansion of dioctahedral micas. The thermal behavior of dioctahedral micas has been studied with single-crystal methods by Guggenheim et al. (1987) and Catti et al. (1989) for muscovite, by Comodi and Zanazzi (2000) for paragonite, and by Amisano Canesi et al. (1994) for phengite. Symmes (1986) reported isobaric expansion data from the powder diffraction method for muscovite, paragonite, margarite, and other phyllosilicates to 500°C. The structural response of phengite to temperature by in situ powder neutron diffraction studies was reported by Pavese et al. (1999a) and Mookherjee et al. (2000) for the 2M1 polytype, by Amisano Canesi et al.(1994), Amisano Canesi et al. (1995) and Pavese et al. (1997) for the 3T polytype. Results on the thermal expansion of micas are summarized in Table 2. The study of thermoelastic properties is more difficult for micas than for other rockforming silicates, because of the partial hydroxyl loss at high temperature. The mechanism of this reaction is 2OH- → H2O↑ + O2-, with H2O diffusing through the tetrahedral sheets and the interlayer region. A kinetic analysis of the process was performed by Mazzucato et al. (1999) (and references therein) by in situ powder X-ray diffraction. Dehydroxylation of (Fe3+, Mg)-rich dioctahedral micas [celadonite and glauconite samples] was studied by Muller et al. (2000a,b) both in terms of structural transformations and of cation migration, combining powder X-ray diffraction, selected area electron diffraction and modeling. Such an investigation was performed by in situ and ex-situ measurements, and revealed migration of cations, mostly Mg, from the cissites to the empty trans-sites, which acquire 5-fold coordination. This reaction causes a transformation of the C-centered cell into primitive. The dehydroxylation path of muscovite was modeled by Abbott (1994) on the basis of theoretical energy calculations. Results suggest that the release of OH depends on the [4]Al/Si ratio of the environment. Since this ratio is variable, owing to the many possible patterns of Al-Si tetrahedral order, it is not surprising that muscovite dehydroxylation takes place over a wide temperature range. This conclusion is however to be assessed with due care, as it relies upon the
Micas at High Temperature and High Pressure
107
assumption of a precise knowledge of the hydrogen position, which thing is hard to be determined at an adequate level of precision. The formation of a metastable dehydroxylated phase is shown by a marked change in slope of the lattice constants and cell volume upon increasing T. This was found in muscovite above 800°C (Guggenheim et al. 1987) and in paragonite above 600°C (Comodi and Zanazzi 2000). This is in agreement with the greater thermal stability of muscovite than paragonite’s obtained from petrographic evidence (Guidotti 1984) and experimental phase relations (Hewitt and Wones 1984). However, the temperature range of dehydroxylation is variable, depending on the rate of the change in T and the possible non-equilibrium conditions of the sample, e.g. those occurring during thermal analysis, or on a hypothetic locally different Al-Si order in tetrahedral sites (Abbott 1994). For paragonite, the dehydroxylation process is completed below 700°, as shown by Raman spectra. Structure refinements by single-crystal method (Comodi and Zanazzi 2000) show that the dehydroxylated phase is similar to the dehydroxylation product of muscovite (Udagawa et al. 1974). In this phase, the “octahedral” sheets undergo the greatest changes, because the Al ions become five-fold coordinated to form distorted trigonal bipyramids. The sheet of these polyhedra is sandwiched between two tetrahedral sheets, with an atomic arrangement similar to that of the precursor mica. This reaction is topotactic. Because of the enlarged dimensions of the Al sheet, tetrahedral rotation angle decreases, going from 16.2° in paragonite to 13.3° in the anhydrous phase. The effects of temperature on the mica structure, based on the single-crystal refinements of muscovite at 650°C (Guggenheimet al. 1987) and of paragonite at 600°C (Comodi and Zanazzi 2000), can be summarized as follows: 1.
the expansion of dioctahedral micas, like that of other phyllosilicates, is strongly anisotropic, with maximum value along the [001] direction (Table 2). The anisotropy is slightly smaller in paragonite than in muscovite. The effect on the c-cell parameter is mainly related to the expansion of the interlayer thickness. The variation in the interlayer thickness is 0.07 Å in muscovite and 0.13 Å in paragonite. The β angle does not change with T in muscovite, but decreases in paragonite.
2.
the tetrahedral (Si,Al) volume does not change significantly in the investigated interval (25-650°C for muscovite; 25-600° C for paragonite). The Al octahedra expand slightly: the volume of M2 increases by 1.5% in muscovite, and by 2.5% in paragonite. The volume of the M1 vacant site increases by 1.8% in muscovite and by 4% in paragonite. The resulting total increase of the [2:1] layer thickness is 0.02 Å in muscovite and 0.013 Å in paragonite.
3.
the effects of intensive variables P and T on the structure of mica are roughly similar, but opposite in sign. This applies to variations in cell edges and to their anisotropy, as well as to polyhedral deformations. The greatest effects concern the interlayer region, where the difference of the six "inner" and six "outer" distances in the interlayer cation polyhedron decreases with T and increases with P. The same trend is observed for the tetrahedral rotation angle α, going from 11.8 to 9.2° in muscovite and from 16.2 to 12.9° in paragonite. Assuming that
1. 2. 3.
the variations induced by T and P are cumulative, the derivative of the thermal expansion coefficient versus P at constant T, (∂α/∂P)T , is negligible, the derivative of the compressibility coefficient versus T at constant P, (∂β∂T)P , is negligible,
108
Zanazzi & Pavese
the conditions which do not change the structure with respect to ambient conditions can be determined, at least volumetrically. At first approximation, the resulting EoS in the PTV space for paragonite and muscovite are: V/V0 = 1 + 5.9×10-5 T – 0.00153 P and V/V0 = 1 + 4.3×10-5 T – 0.0017 P (T is in °C andP is in kilobars), showing that paragonite is more expandible and less compressible than muscovite. This is a significant datum concerning the shape of the Pg-Ms solvus at high P and T. Order-disorder of Si and Al in tetrahedral sites as a function of temperature was the object of several studies, with controversial results. The behavior of phengite-3T from Dora Maira at high T was studied by in situ single-crystal X-ray diffraction (AmisanoCanesi et al. 1994; Amisano-Canesi 1995) and powder neutron diffraction (Pavese et al. 1997, 2000). These authors found that this sample seems to exhibit cation ordering both on the tetrahedral and on the octahedral sites, supporting the hypothesis of Sassi et al. (1994) that a high P/T ratio induces cation order. Powder neutron diffraction on a Fe-rich phengite-2M1 (Pavese et al. 1999a) showed that cation partitioning is disordered at room temperature, but Al fully orders into the T1 site at 600°C. This resultis in contrast with the findings of Mookherjee et al. (2000) on phengite-2M1 from Greece, who find no evidence for changes in tetrahedral cation order on heating (in situ powder neutron diffraction to 650°C). Low-temperature powder neutron diffraction measurements were used by Pavese et al. (2001) to investigate the fractional occupancy of the M1 site of phengite-3T and -2M1, commonly assumed to be empty. Trioctahedral micas High pressure studies. High pressure investigations on trioctahedral micas are hitherto very scarce. Hazen and Finger (1978) first pioneered the crystal structures and compressibilities of layer minerals at high pressure, investigating natural phlogopite and chlorite. There was a twenty-year long lack of high pressure studies on trioctahedral micas until the work of Comodi et al. (1999), who dealt with high-pressure and hightemperature behavior of Cs-tetra-ferri-annite-1M (composition: Cs1.78(Fe2+5.93Fe3+0.07) (Si6.15Fe3+1.80Al0.05)O20(OH)4), and Comodi et al. (2001), who studied the compressibility of Rb-tetra-ferri-annite-1M; hereafter, referred to “C s-annite” and “Rb-annite,” respectively. Note that (1) the investigated samples are synthetic, and that (2) the results therein reported can only be extrapolated to natural micas (i.e., K- and Na-rich micas) with care. The above studies provide indications about the microscopic mechanisms governing the behavior of trioctahedral micas under pressure and explore the baric range to 50 kbar, which is sufficient for applications of geological interest. In Table 3, the compressibility of the lattice parameters and of the polyhedral building units are reported, along with other structural parameters useful to understand the high-pressure behavior of Cs-annite, Rb-annite and phlogopite. The points outlined below show the effect of how different structural units of trioctahedral micas relate with one another, and then respond as a whole to pressure: 1.
Most of the volume reduction under pressure occurs along the [001] direction. This is readily understood by observing, in Table 3, the remarkable anisotropy of the axial compressibilities. The shortening of the c axis is responsible of about 81% (Cs-annite) and 70% (phlogopite and Rb-annite) of the reduction in volume. An analysis of the change of the tetrahedral, octahedral and interlayer thicknesses as a function of pressure reveals that the interlayer thickness undergoes a shortening at least ten times as large as the others, which, in turn, are comparable with one another.
2.
The interlayer site exhibits a significantly smaller bulk modulus than the other sites,
Micas at High Temperature and High Pressure
109
Table 3. Polyhedral bulk modulus for T1, T2, M1, M2 and interlayer sites, in kbar. Axial and -1 volume compressibilities (βa,b,c,V), in kbar-1, and first derivative versus P of β/β0, in ºkbar . Bulk modulus at P = 0 (K0, in kbar) calculated by the Birch-Murnaghan EoS, constraining its first derivative versus P (K’0) to be equal to 4. Tetrahedral rotation angle (α) and tetrahedral -1 -1 tilting (Δz) first derivative versus P, in °kbar and Å kbar , respectively.
T1 T2 M1 M2 K/Cs
3.
4.
5.
phlogopite
Cs-annite
Rb-annite
(Hazen & Finger 1978)
(Comodi et al. 1999)
(Comodi et al. 2001)
negligible negligible 1200(2000) 1200(2000) 200(30)
1370(400) 1370(400) 1040(300) 1190(300) 260(70)
ßa ßb ßc d(β/β0)/dP
2.5×10-4 2.8×10-4 11.7×10-4 1.7(5) ×10-4
1.6(2) ×10-4 1.7(1) ×10-4 14.0(5) ×10-4 1.5(1) ×10-4
K0 K’ 0
510(14) 4
419(6) 4
dα/dP dΔz/dP
51.4×10-3 2.1×10-4
1.6(1) ×10-4 2.13(7) ×10-4 10.1(4) ×10-4 negligible 500(3) 4
4.1×10-3 6.9×10-4
in keeping with the fact that the contraction along the [001] direction occurs mainly at the expense of the interlayer polyhedron, which yields under load. The discrepancies of the βc values are likely related to differences of the interlayer composition. Note that Cs- and Rb-cations are expected to be softer than K [Cs, Rb and K have the same oxidation state, but decreasing ionic radius, leading to decreasing cation-oxygen bond lengths], which is consistent with the compressibilities of the c-axis. This conclusion is consistent also with the polyhedral bulk-modulus values of the interlayer sites, although care is required because of the large uncertainties. The polyhedral bulk moduli of the octahedral sheet do not allow a reliable comparison between the Fe-bearing sites and the Mg-bearing ones because of the large uncertainties affecting the issues of Hazen and Finger (1978). The βa and βb values, which are reflective of the compliance of the T-O-T layer across the (001) plane, indicate Cs-annite behaves more rigidly than phlogopite. This observation is in keeping with Zhang et al. (1997) who maintain, for clinopyroxenes, that sites containing Mg are softer than those occupied by Fe. Rb-annite exhibits a value of βb in disagreement with the one of Cs-annite, presumably owing to the structural transition from A type to B type (Franzini 1969; see next point 5), which mainly affects the compressibility of a and b lattice parameters. The tetrahedra are nearly rigid units, and tend to relax under pressure by rotations described by the α angle. These rotations are sensitive to composition, as Table 3 shows. In particular, the tetrahedral sheets adjacent to the Fe-bearing octahedral sheet, in Cs-annite, exhibit a significantly smaller α rotation than in phlogopite. This
110
Zanazzi & Pavese is consistent with point (4): the rigidity of the Fe-rich octahedral sheet seriously limits the rotational degree of freedom of the tetrahedral sheet to comply with pressure. The α value and the tetrahedral sheet corrugation (Δz) increase upon increasing pressure, regardless of the composition; such an observation suggests that the tetrahedral rotation and the corrugation of the basal oxygen atom plane are a very general way of relaxation of the tetrahedral sheet in micas. The structural evolution of Rb-annite with pressure (Comodi et al. 2001) shows a peculiar behavior of the tetrahedral rotation angle α: although it increases with pressure as well as in other micas, it assumes positive values at pressures lower than about 45 kbar and negative values at higher pressure, indicating a phase transition from a type A to a type B structure (Franzini 1969).
6.
The differences on the bulk-modulus values must be assessed with care, because of the small number of points obtained by Hazen and Finger (1978) in deriving K0 for phlogopite. Taking into account the large uncertainties on the data, the difference in K0 reflects the greater softness of Cs/Rb-annites than phlogopite because of the replacement of K with Cs/Rb in the interlayer. In particular, note that the larger K0 of Rb-annite than that of Cs-annite is consistent with predictions based on the ionic radii, as discussed in point (3) above.
7.
The obliquity of the monoclinic cells of trioctahedral micas increases upon pressure.
High-temperature studies. In this section we discuss in situ high-temperature investigations, and studies with heat-treated samples to induce oxidation and dehydration/ dehydroxylation processes. Takeda and Morosin (1975) compared the behavior at high temperature of a synthetic fluorphlogopite with predictions relying upon the geometrical approach of Donnay et al. (1964). Hogg and Meads (1975) investigated by Mössb auer spectroscopy the thermal decomposition of biotites, owing to Fe2+ → Fe3+ oxidation accompanied by dehydroxylation on specimens previously heated. Tripathi et al. (1978) studied the effects of high-temperature reactions in biotite and phlogopite by Mössbauer spectroscopy. Rancourt et al. (1993) reported the first kinetic study of iron oxidation in biotite by Mössbauer spectroscopy. They also suggested a simple activation model accounting for iron oxidation and ordinary dehydroxylation, based on the following reactions: OH- → O2- + H+, i.e., local dissociation of OH-groups either by Fe2+ + H+ → Fe3+ + H, i.e., Fe-oxidation, or by O2- + 2 H+ → H2O, i.e., dehydroxylation. Twenty-four years after the Takeda and Morosin’s investigation, Russell and Guggenheim (1999), and Comodi et al. (1999) discussed the high-temperature behavior of a near end-member phlogopite and of a “Cs-annite,” respectively. These investigations consider structural aspects observed in situ and Fe-oxidation mechanisms in heat-treated specimens. The latter topic is discussed in the light of bond-length changes and cell volume versus heating-time curves. Tutti et al. (2000) studied the thermal expansion of a natural phlogopite from Pargas (Finland) and monitored by TGA and DTG the Fe-oxidation and dehydroxylation processes triggered by temperature. They observed that the curves of a, b and c as a function of temperature are split by a discontinuity slightly above 400°C into two regions exhibiting quite different thermal expansion coefficients. Such a behavior upon heating is presumably related to the different mechanisms governing the structural rearrangements of phlogopite below and above the oxidation temperature of Fe, which takes place about 500-600°C. Note that the discontinuity observed by Tutti et al. (2000) is consistent with that reported by Takeda and Morosin (1975) for F-phlogopite. It is worthy of interest to analyse how the structure of trioctahedral micas responds to heating, to achieve a full understanding of the mechanisms driving the high-temperature processes in these
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-1
Table 4. Axial thermal expansion coefficients (αa,b,c,V), in ºC , and first derivative of β/β0 angle -1 of the monoclinic cell versus T, in ºC . Tetrahedral rotation angle (α) and octahedral flatness -1 angle (Ψ) first derivatives versus T, in degºC . The results from Tutti et al. (2000) are reported on two rows: the upper and the lower refer to thermal expansion coefficients below and above 412°C, respectively. F-phlogopite (Takeda and Morosin 1975)
phlogopite (Russell and Guggenheim 1999)
αa
0.89×10-5
1.40×10-5
αb
0.77×10-5
1.34×10-5
αc
1.8×10-5
1.81×10-5
-1.3
-1.5
-5.6×10-3 -0.7×10-3
-7.9×10-3 -0.3×10-3
d(β/β0)/dT dα/dT dΨ/dT
phlogopite (Tutti et al. 2000)
Cs-annite (Comodi et al. 1999)
3.74×10-5 0.86×10-5 1.09×10-5 0.80×10-5 1.19×10-5 1.93×10-5
negligible negligible 3.12×10-5 -0.36 -1.0×10-3 -0.7×10-3
minerals. In Table 4, we report some of the parameters required to understand structural adjustments occurring as a function of temperature: 1. All trioctahedral micas (Table 4) expand mainly along the [001] direction, although at quite different rates according to the compositions of the octahedral sheet and, principally, of the interlayer sites. For Cs-annite, the Fe(O,OH)6 octahedra are moderately sensitive to heating, so that the thermal expansion coefficients along the [100] and [010] directions are negligible and the volume expansion occurs entirely along the [001] direction. The “soft” Cs-cation in the interlayer site causes βc in Csannite to be nearly twice as large as the corresponding value in phlogopite. This conclusion is consistent with the linear thermal expansion coefficients of the interlayer at approximately 6.3 × 10-5 and 3.7 × 10-5 per °C for Cs-annite and phlogopite, respectively. In Figure 3, the relative change of the c cell dimension for Cs-annite and phogopite is reported as a function of T. 2.
The octahedral flatness angle, ψ, decreases upon heating for both Mg- and Fe-bearing trioctahedral micas, which reduces the diagonal elongation versus the sheet thickness. Note that the dψ/dT values are not simply related to the cation occupancy of the Msites. For example Mg(O,OH)6 shows remarkable differences in phlogopite or in Frich phlogopite.
3.
The tetrahedral rotation angle as a function of T exhibits a negative slope, and changes at different rates in Mg- or Fe-bearing trioctahedral micas. In particular, the rotation angle changes slightly in the latter, and this result is related to the inertness of the Fe(O,OH)6 or Fe(O,F)6 octahedra and implies a modest rearrangement in the T-sheet.
4.
The obliquity of the cell decreases on heating, indicating a tendency to approach a 90° β angle.
In brief, from the discussion above, trioctahedral micas, under thermal or baric conditions, respond as follows: 1. structural changes occur mainly along the [001] direction, at the expense of the interlayer site, owing to the weaker interactions. The axial thermal expansion or compressibility coefficients reflect this aspect.
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Figure 3. (c-c0)/c0 ratio as a function of temperature, for Cs-annite (Comodi et al. 1999) and phlogopite (Russell and Guggenheim 1999). c0 is the value of c at ambient conditions.
Figure 4. (a) (filled squares) and <M-O> (M1 and M2, filled and empty diamonds respectively) bond lengths versus pressure, normalized to their values at room conditions. pg: paragonite. Data from Comodi and Zanazzi (1997). Subscripts HP and RP stand for high pressure and room pressure, respectively. (b) (T1 and T2, filled and empty circles, respectively) and <M-O> (M1 and M2, filled and empty diamonds, respectively) bond lengths versus temperature, normalized to their values at room conditions. pg: paragonite. Data from Comodi and Zanazzi (2000). Subscripts RT and HT stand for room temperature and high temperature, respectively.
2. the adjustment of the T-O-T layer is accomplished by volumetric changes of the octahedra by either expansion or compression, and by rotations and corrugations of the tetrahedral sheets. The tetrahedra behave as quasi-rigid bodies, to achieve matching lateral dimensions between tetrahedral and octahedral sheets. 3. the obliquity of the unit cell increases under pressure, and decreases upon heating. The behavior of micas as a function of P and/or T is therefore strictly dependent on the stacking structural features of these minerals. Thus, trioctahedral and dioctahedral micas show similar responses to thermobaric stress. Most of the compression/expansion occurs along c*, at the expense of the interlayer polyhedra. The changes occurring across the (0 0 1) plane are out-of-plane tilting and in-plane rotation of the tetrahedra of the tetrahedral sheets, to minimize the misfit between the tetrahedral and octahedral sheets,
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the latter accounting for the most significant bond-length variations in the 2:1 layer. As an example, in Figures 4a and 4b the average T-O and M-O bond lengths in paragonite are plotted as a function of pressure and temperature, respectively, to illustrate the sensitivity of the tetrahedra and octahedra to thermobaric stress. The results are consistent with the general rule that temperature promotes regularity whereas pressure favors distortion in structures: Figures 5a and 5b show the difference between the average of the interlayer cation-oxygeninner and cation-oxygenouter bond lengths in paragonite and muscovite, normalized to ambient conditions, versus P and T, respectively. Note that pressure tends to increase the twelve-fold site strain, whereas temperature shows the reverse trend, where atoms produce a more regular arrangement. Note also the remarkable dependence of the thermoelastic parameters on the chemical composition, apparent in the case of the interlayer cation replacement. Figure 6 shows the bulk modulus plotted as a function of the interlayer cation size. The figure is computed as an average of the interlayer chemical composition, assuming a twelve-fold coordination. A shift of approximately 0.5 Å in ionic radius corresponds to more than 100-kbar decrease in K0.
Figure 5. (a). |inner-outer| versus pressure, normalized to their values at room conditions. pg (filled circles): paragonite (Comodi and Zanazzi 1997); ms (empty circles): muscovite (Comodi and Zanazzi 1995). Subscripts i and o stand for inner and outer; HP and RP for high pressure and room pressure, respectively. (b)|inner-outer| versus temperature, normalized to their values at room conditions. pg (open circles): paragonite (Comodi and Zanazzi 2000); ms (filled circles): muscovite (Guggenheim et al. 1987). Subscripts i and o stand for inner and outer; HT and RT for high temperature and room temperature, respectively.
The modest number of experimental studies on the behavior of micas as a function of pressure and/or temperature and the difficulty in the experiments that leads to lack of precision, make a statistically reliable comparison between trioctahedral and dioctahedral micas difficult. The main differences between the two originate from the octahedral sheet behavior. Differences between the two forms would be expected between the rearrangements of the 2:1 layers upon P and/or T. Based on the current data, and assuming a K-bearing interlayer sheet, the following conclusions can be drawn: 1. Dioctahedral micas are less thermally stable, and have larger bulk thermal expansion than trioctahedral micas, presumably as a consequence of the vacant M1 site.
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2. Dioctahedral micas have a slightly larger bulk modulus than trioctahedral micas (i.e. are slightly stiffer), because of the presence of trivalent cations instead of divalent cations in octahedral coordination. This produces a larger polyhedral bulk modulus (Hazen and Finger 1982).
Figure 6. Bulk modulus as a function of interlayer cation size, calculated using the interlayer chemical composition. Na(Pg): paragonite (Comodi and Zanazzi 1997); Na(mu): Na-rich muscovite (Comodi and Zanazzi 1995); K(phl): phlogopite (Hazen and Finger 1978); K(mu): K-muscovite (Comodi and Zanazzi 1995); Cs(Cs-tfa): Cstetra-ferri-annite (Comodi et al. 1999); Rb(Rb-tfa): Rb-tetra-ferri-annite (Comodi et al. 2001).
ACKNOWLEDGMENTS We are grateful to Steve Guggenheim for reviewing this chapter and for his comments on the manuscript. REFERENCES Abbott RN (1994) Energy calculations bearing on the dehydroxylation of muscovite. Can Mineral 32:87-92 Ahrens TJ (1987) Shock wave techniques for geophysics and planetary physics. In: CG Sammis, TL Henyey (eds) Methods of experimental physics. p 185-235. Academic Press, San Diego, CA Ahsbahs H (1987) X-ray diffraction on single crystals at high pressure. Prog Crystal Growth and Charact 14:263-302 Aleksandrov KS, Ryzhova TV (1961) Elastic properties of rock-forming minerals, II. Layered silicates. Izv Acad Sci USSR, Phys Solid Earth, Engl Transl 1165-1168 Amisano Canesi A. (1995) Studio cristallografico di minerali di altissima pressione del complesso Brossasco-Isasca (Dora Maira Meridionale): PhD dissertation, University of Torino Amisano Canesi A, Ivaldi G, Chiari G, Ferraris G (1994) Crystal structure of phengite-3T: thermal dependence and stability at high P/T. Abstracts 16th General Meeting IMA, Abstr, 10 Anderson OL (1995) Equations of state for geophysics and ceramic sciences. Oxford University Press, Oxford, UK Anderson OL and Isaak DG (1995) Elastic constants of mantle minerals at high temperature. In Mineral Physics and Crystallography: A Handbook of Physical Constants. Ahrens TJ (ed) AGU Reference Shelf 2 Angel RJ (2001) Equations of state. Rev Mineral Geochem 41:35-59 Angel RJ, Downs RT, Finger LW (2001) Diffractometry. Rev Mineral Geochem 41:556-559 Barnett JD, Block S and Piermarini GJ (1973) An optical fluorescence system for quantitative pressure measurement in the diamond-anvil cell. Rev Sci Instrum 44:1-9 Birch F (1986) Equation of state and thermodynamic parameters of NaCl to 300 kbar in the hightemperature domain. J Geophys Res 83:1257-1268 Bridgman PW (1949) Linear compressions to 30,000 kg/cm2, including relatively incompressible substances. Proc Am Acad Arts Sci 77:189-234
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Catti M, Ferraris G, Ivaldi G (1989) Thermal strain analysis in the crystal structure of muscovite at 700°C. Eur J Mineral 1:625-632 Catti M, Ferraris G, Hull S, Pavese A (1994): Powder neutron diffraction study of 2M1 muscovite at room pressure and at 2 GPa. Eur J Mineral 6:171-178 Chung DDL, De Haven PW, Arnold H, Ghosh D (1993). X-ray diffraction at elevated temperatures, VCH Ed., New York Comodi P, Zanazzi PF (1995) High-pressure structural study of muscovite. Phys Chem Minerals 22:170177 Comodi P, Zanazzi PF (1997) Pressure dependence of structural parameters of paragonite. Phys Chem Minerals 24:274-280 Comodi P, Zanazzi PF (2000) Structural thermal behavior of paragonite and its dehydroxylate: A hightemperature single-crystal study. Phys Chem Minerals 27:377-385 Comodi P, Zanazzi PF, Weiss Z, Rieder M, Drábek M (1999) Cs-tetra-ferri-annite: High-pressure and hightemperature behavior of a potential nuclear waste disposal phase. Am Mineral 84:325-332 Comodi P, Drábek M, Montagnoli M, Rieder M, Weiss Z, Zanazzi PF (2001) Pressure-induced phase transition in a new synthetic Rb-mica. FIST-Geoitalia 2001, Chieti, Sept. 5-8 2001 Donnay G, Donnay JDH, Takeda H (1964) Trioctahedral one-layer micas. II. Prediction of the structure from composition and cell dimensions. Acta Crystallogr 17:1374-1381 Duffy TS, Wang Y (1998) Pressure-volume-temperature equations of state. Rev Mineral xx 425-457 Faust J, Knittle E (1994) The equation of state, amorphization, and high-pressure phase diagram of muscovite. J Geophys Res 99:19785-19792 Flux S, Chatterjee ND, Langer K (1984) Pressure induced [4](Al,Si)-ordering in dioctahedral micas? Contrib Mineral Petrol 85:294-297 Franzini M (1969) The A and B layers and the crystal structure of sheet silicates. Contrib Mineral Petrol 21:203-224 Guidotti CV (1984) Micas in metamorphic rocks. Rev Mineral 13:357-467 Guggenheim S, Chang YH, Koster van Groos AF (1987) Muscovite dehydroxylation: High-temperature studies. Am Mineral 72:537-550 Güven N (1971) The crystal structures of 2M 1 phengite and 2M1 muscovite. Z Kristallogr 134:196-212 Hazen RM and Finger LW (1978) The crystal structures and compressibilities of layer minerals at high pressure. II. Phlogopite and chlorite. Am Mineral 63:293-296 Hazen RM and Finger LW (1982) Comparative Crystal Chemistry. John Wiley and Sons, New York. Hewitt DA, Wones DR (1984) Experimental phase relations of the micas. Rev Mineral 13:357-467 Hogg CS, Meads RE (1975) A Mössbauer study of th ermal decomposition of biotites. Mineral Mag 40:7988 Holzapfel WB (1996) Physics of solids under strong compression. Reports Progress Phys 59:29-90 Jackson I and Rigden SM (1996) Analysis of P-V-T data—Constraints on the thermoelastic properties of high pressure minerals. Phys Earth Planet Int 96:85-112 Jeanloz R (1988) Universal equation of state. Phys Rev B 38:805-807 Kumar M (1995) High pressure equation of state for solids. Physica B 212:391-394 Kumar M and Bedi SS (1996) A comparative study of Birch and Kumar equations of state under high pressure. Phys Stat Sol B 196:303-307 Mazzucato E, Artioli G, Gualtieri A (1999) High temperature dehydroxylation of muscovite-2M1: a kinetic study by in situ XRPD. Phys Chem Minerals 26:375-381 Mookherjee M, Redfern SAT, Hewat A (2000) Structural response of phengite 2M1 to temperature: an in situ neutron diffraction study. EMPG VIII, Bergamo, April 16-19, Abstracts, p 75 Moriarty JA (1995) First-principles equations of state for Al, Cu, Mo and Pb to ultrahigh pressures. High Press Res 13:343-365 Muller F, Drits VA, Plançon A, Besson G (2000a) Dehydroxylation of Fe3+, Mg-rich dioctahedral micas: (I) structural transformation. Clay Minerals 35:491-504 Muller F, Drits VA, Tsipursky SI, Plançon A (2000b) Dehydroxylation of Fe3+, Mg-rich dioctahedral micas: (II) cation migration. Clay Mineral 35:505-514 Pavese A, Ferraris G, Prencipe M, Ibberson R (1997) Cation site ordering in phengite-3T from the DoraMaira massif (western Alps): a variable-temperature neutron powder diffraction study. Eur J Mineral 9:1183-1190 Pavese A, Ferraris G, Pischedda V, Ibberson R (1999a) Tetrahedral order in phengite-2M1 upon heating, from powder neutron diffraction, and thermodynamic consequences. Eur J Mineral 11:309-320 Pavese A, Ferraris G, Pischedda V, Mezouar M (1999b) Synchrotron powder diffraction study of phengite 3T from the Dora-Maira massif: P-V-T equation of state and petrological consequences. Phys Chem Minerals 26:460-467
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Pavese A, Ferraris G, Pischedda V, Radaelli P (2000) Further powder neutron diffraction on phengite-3T: cation ordering and methodological thoughts. Mineral Mag 64:11-18 Pavese A, Ferraris G, Pischedda V, Fauth F (2001) M1-site occupancy in 3T and 2M1 phengites by low temperature neutron powder diffraction: Reality or artefact? Eur J Mineral (in press) Poirier JP, Tarantola A (1998) A logarithmic equation of state. Phys Earth Planet Int 109:1-8 Rancourt DG, Tume P, Lalonde AE (1993) Kinetics of the (Fe2++ OH-)mica→ (Fe3++O2-)mica + H oxidation reaction in bulk single-crystal biotite studied by Mössbauer spectroscopy. Phys Chem Minerals 20:276-284 Russell RL, Guggenheim S (1999) Crystal structures of near-end-member phlogopite at high temperature and heat treated Fe-rich phlogopite: the influence of the O,OH,F site. Can Mineral 37:711-720 Sassi F P, Guidotti C, Rieder M, De Pieri R (1994) On the occurrence of metamorphic 2M1 phengites: some thoughts on polytypism and crystallization conditions of 3T phengites. Eur J Mineral 6:151-160 Saxena SK, Zhang J (1990) Thermochemical and pressure-volume-temperature systematics of data on solids, examples: tungsten and MgO. Phys Chem Minerals 17:45-51 Sekine T, Rubin AM, Ahrens TJ (1991) Shock wave equation of state of muscovite. J Geophys Res 96:19675-19680 Shinmei T, Tomioka N, Fujino K, Kuroda K, Irifune T (1999) In situ X-ray diffraction study of enstatite up to 12 GPa and 1473 K and equation of state. Am Mineral 84:1588-1594 Smyth JR, Jacobsen SD, Swope RJ, Angel RJ, Arlt T, Domanik K, Holloway JR (2000) Crystal structures and compressibilities of synthetic 2M1 and 3T phengite micas. Eur J Mineral 12:955-963 Symmes GH (1986) The thermal expansion of natural muscovite, paragonite, margarite, pyrophyllite, phlogopite, and two chlorites: The significance of high T/P volume studies on calculated phase equilibria. B.A. Thesis, Amherst College, Amherst, Massachusetts Takeda H and Morosin B (1975) Comparison of observed and predicted structural parameters of mica at high temperature. Acta Crystallogr B31:2444-2452 Tripathi RP, Chandra U, Chandra R, Lokanathan S (1978) A Mössbauer study of the effects of heating biotite, phlogopite and vermiculite. J Inorg Nucl Chem 40:1293-1298 Tutti F, Dubrovinsky LS, Nygren M (2000) High-temperature study and thermal expansion of phlogopite. Phys Chem Minerals 27:599-603 Udagawa S, Urabe K, Hasu H (1974) The crystal structure of muscovite dehydroxylate. Japan Assoc Mineral Petrol Econ Geol 69:381-389 Utsumi W, Weidner DJ, Liebermann RC (1998) Volume measurements of MgO at high pressure and temperature. In MH Manghnani, Y Yagi (eds) Properties of Earth and Planetary Materials at High Pressure and Temperature, p 327-334, Am Geophys Union, Washington, DC Vaughan MT, Guggenheim S (1986) Elasticity of muscovite and its relationship to crystal structure. J Geophys Res 91:4657-4664 Velde B (1980) Cell dimension, polymorph type, and infrared spectra of synthetic white micas: the importance of ordering. Am Mineral 65:1277-1282 Vinet P, Ferrante J, Smith JR, Rose JH (1986) A universal equation of state for solids. J Phys C 19:L467L473 Vinet P, Smyth JR, Ferrante J, Rose JH (1987) Temperature effects on the universal equation of state of solids. Phys Rev B 35:1945-1953 Vinet P, Rose JH, Ferrante J, Smyth JR (1989) Universal features of the equation of state of solids. J Phys Cond Mater 1:941-1963 Wallace DC (1972) Thermodynamics of Crystals. John Wiley and Sons, New York. Zhang L, Ahsbahs H, Hafner S, Kutoglu A (1997) Single-crystal compression study and crystal structure of clinopyroxenes up to 10 GPa. Am Mineral 82:245-258 Zanazzi PF (1996) X-ray diffraction experiments in (moderate) high-P / high-T conditions; high-P and high-T crystal chemistry. High Pressure and High Temperature Research on Lithosphere and Mantle Materials. Proc Int’l School Earth Planetary Sci, Siena, December 3-9, 1995, p 107-120
3
Structural Features of Micas Giovanni Ferraris1 and Gabriella Ivaldi2 1,2
Dipartimento di Scienze Mineralogiche e Petrologiche Università di Torino 10125 Torino, Italy and 1 Istituto di Geoscienze e Georisorse Consiglio Nazionale delle Ricerche 10125 Torino, Italy
[email protected] [email protected] INTRODUCTION The large number of mica species (end-members) and varieties is based on chemical variability and peculiar structural features like polytypism, local and global symmetry. In addition, mainly because of an inherent misfit between the constituent tetrahedral and octahedral sheets, in the specific mica structures several structural parameters undergo adjustments relative to their ideal values. Consequently, the mechanisms ruling distortions from ideal models must be considered when investigating a mica behavior under geological conditions. Micas are important rock-forming minerals and petrographers consider them mainly for their chemical aspects. The importance of the chemical composition is well known to all researchers dealing with minerals. To give emphasis to the chemical composition, the official classification of the micas (Rieder et al. 1998; Rieder 2001) allows exceptions (note the introduction of ‘species that are not end member’ in Rieder et al. 1998) to the rules which are normally used to define mineral species (Nickel and Grice 1998). However, as shown throughout this book, the role of structure features (including some aspects of polytypism) in determining fields of stability of micas and, therefore, in providing geological insights, is increasingly recognized as crucial. Thus, it seems justified that a chapter dedicated to the description of the general structural background of micas should be presented independently from specific cases, which are discussed in other chapters. This chapter is an introduction to the symmetry and geometric aspects of micas. Nevertheless, some less conventional topics not covered in other chapters are reported in appendices. Appendix I concerns the wide presence of mica-like modules in the growing group of natural, layer titanosilicates (Khomyakov 1995; Ferraris et al. 2001b,d) and other more or less exotic structures belonging to the expanding field of modular crystallography (Merlino 1997). Important results have been obtained by the obliquetexture electron diffraction method (OTED; cf. Zvyagin et al. 1996, and references therein). Only a few, limited treatments are in English. Thus, this method is discussed in Appendix II. NOMENCLATURE AND NOTATION Bailey (1984c) recommended a notation system for structural sites in micas. However, there is no agreement to the labeling of these sites; e.g., either an italic or roman font is used with or without parentheses to separate the alphanumeric parts. Following recent papers (Nespolo et al. 1999c: Nespolo 2001) and in agreement with the chapter of Nespolo and Durovic (this volume), the nomenclature of the OD theory of polytypes (Dornberger-Schiff et al. 1982; Durovic 1994) is here adopted to label sites, 1529-6466/02/0046-0003$05.00
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planes, sheets and layers. Emphasis is given to symmetry aspects more than to structural and chemical features. Thus, rather than to a distinction between dioctahedral and trioctahedral micas, preference is given to a classification in homo-octahedral, mesooctahedral and hetero-octahedral families according to the layer symmetry [H`( 3 )1m, P`( 3 )1m and P(3)12; see Table 1 below] of the octahedral sheet in the TOT or 2:1 layer, here referred to as the M layer. Two types of M layers are introduced according to the position of the origin assigned to the reference system in the octahedral sheet: layer M1 if the origin is in the octahedral trans site M1, and layer M2 if the origin is in the octahedral cis sites M2 or M3. Trans and cis refer to the position of the OH groups (cf. Fig. 5 below). The distinction between M1 and M2 is necessary also because of the different role played by these two types of layer in generating polytypes (Nespolo 2001). The introduction of the notation M1 and M2 for the M layer follows directly from the letter “M” which is previously used to indicate the TOT layer (e.g., Takéuchi and Haga 1971), before the existence of two types of layer was recognized. Because the roman font is reserved for the layer, we adopt the italic font to indicate the octahedral sites, namely M1, M2 and M3. For the ordinal label 1, 2 and 3 see the below “Structural symmetry” below. Summarizing: roman font is used for planes (cf. Fig. 2 below), sheets (O octahedral, T tetrahedral) and layers (M, TOT); italics font is used for structural sites (M octahedral, T tetrahedral) and cations occupying them (Y octahedral, Z tetrahedral, I interlayer). MODULARITY OF MICA STRUCTURE Thanks to the pioneering paper on the biopyribole polysomatic series by Thompson (1978), the structure of micas, together with those of amphiboles and pyroxenes, lead to the development of the modern modular description of the crystal structures. According to the modular crystallography principles (Merlino 1997), the same structural modules (fragments) larger than single coordination polyhedra may occur in different structures. The emphasis on modules is not only important in describing series, it is also useful in describing aspects ranging from a single structure to classification, genesis, solid state reactions (e.g., Baronnet 1997; Ferraris et al. 2000), structural modeling, and defect structures (cf. chapter by Kogure, this volume). Micas are layer silicates (phyllosilicates) whose structure is based either on a brucite-like trioctahedral sheet [Mg(OH)2 which in micas becomes Mg3O4(OH)2] or a gibbsite-like dioctahedral sheet [Al(OH)3 which in micas becomes Al2O4(OH)2]. This module is sandwiched between a pair of oppositely oriented tetrahedral sheets. The latter sheet consists of Si(Al)-tetrahedra which share three of their four oxygen apices to form a two-dimensional hexagonal net (Fig. 1). In micas, the association of these two types of sheet produces an M layer, which is often referred as the 2:1 or TOT layer. As mentioned in the Introduction, the wide variety of micas (Rieder et al. 1998) derives not only from chemical composition but also from structural features such as the many (infinite, in principle) possibilities of stacking the M layer, particularly the special type of polymorphism known as polytypism, discussed by Nespolo and Durovic (this volume). The mica module The mica module, consisting of an M (TOT or 2:1) layer plus an interlayer cation, is conveniently considered to be built by eight atomic planes in the following sequence, starting from the bottom in Figure 2. • Obl Lower (l) plane of the basal (b) oxygen atoms (O) belonging to the tetrahedra; these oxygen atoms also participate in the coordination of the interlayer cation I. • Zl Lower plane of the four-coordinated tetrahedral cations Z (these cations are
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often indicated by T, but here this letter is used to indicate a tetrahedral site T and a tetrahedral sheet T). Oal Plane of the lower apical (a) oxygen atoms of the tetrahedra; these oxygen atoms are shared between one tetrahedral and one octahedral sheet. The Oal plane contains also hydroxyl (OH)- groups (and their substitutions) which belong only to the octahedral sheet. Y Plane of the octahedral cations Y which are often indicated by M (here this letter is used to indicate an octahedral site M and the M layer; the symbol O is used to indicate the sheet containing the M sites). Oau Plane of the upper (u) apical oxygen atoms (Oa) of the tetrahedra (cf. Oal). Zu Upper (u) plane of the tetrahedral cations (cf. Zl). Obu Upper (u) plane of the basal (b) oxygen atoms (Ob) belonging to the tetrahedra (cf. Obl). I Plane of the interlayer cations I (interlayer sites).
Figure 1. Ideal trioctahedral brucite-like (a) and dioctahedral gibbsite-like (b) sheets. Two ideal tetrahedral sheets (c) share their apical oxygen atoms with an octahedral sheet to form an M layer (d) which is also known as 2:1 or TOT layer. Hydroxyl (OH)- groups are represented by black circles.
Planes are combined to form three types of sheets: Ob + Z + Oa form two tetrahedral sheets (Tl and Tu); Oa + Y + Oa form one octahedral (O) sheet. The whole M mica layer corresponds to the Tl-O-Tu (also termed 2:1) sequence; this layer is also called the conventional mica layer and is often designated as the TOT layer. The interlayer cations are located between two successive M layers in the I plane and their coordination is discussed below. The separation between two I planes is about 10 Å.
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The apical oxygen atoms and the hydroxyl (OH)- groups forming the Oa plane (Fig. 2) are arranged according to a two-dimensional closest-packing of spheres; Oa is also called the hydroxyl plane. In this plane, the packing is however not tight; in fact, the spacing between the oxygen anions is about 3.1 Å compared to 2.6 Å in a typical closestpacking of oxygen atoms (e.g., in olivine or spinel). Between two adjacent hydroxyl planes (Oal and Oau), the octahedral and tetrahedral sites that are typical of a threedimensional closest-packing of spheres occur (Fig. 3). This type of tetrahedral sites is vacant in micas; the octahedral sites M instead are fully (trioctahedral micas) or partially.
Figure 2. Cross-section perpendicular to the M layer of the mica structure seen along [110]. Sequence and labeling of eight distinct building atomic planes are shown. Hydroxyl (OH)- groups are represented by black circles (see text for explanation of labeling).
Figure 3. Projection of a closest-packing AB sequence along the planes formed by apical oxygen atoms Oa. Positions of the octahedral (M) and tetrahedral (small circles) sites are shown. These tetrahedral sites are not occupied in the octahedral sheets of micas.
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Figure 4. Stacking of two closest-packing planes of spheres (plane A dark gray, plane B light gray) consisting of oxygen atoms and (OH)- groups. The primitive hexagonal cell (A1 = A2) and the conventional C-centred orthohexagonal cell [(a, b) in the C1 orientation according to Arnold (1996)] normally used for phyllosilicates are shown. Cells are shown also for a closest-packing of equal spheres [smaller (ah, bh) cells]. Each n-th plane of spheres (e.g. B) is ±a/3 staggered relative to the (n-1)-th plane (A). The interstitial sites between the spheres appear either as open holes or as gray ‘triangles’. Each interstice is surrounded by three packing spheres in its plane; between the two planes A and B tetrahedral and octahedral sites occur (Fig. 3).
(dioctahedral micas) occupied by Y cations. Note (Fig. 4) that in the Oa plane each (OH)group is surrounded by six oxygen ions which, in turn, are surrounded by three (OH)groups, and three oxygen atoms. The distances (∼2.7 Å) between the anio ns within the basal Ob plane are closer to the expected value for a closest-packing of oxygen ions (∼2.6 Å). However, relative to real closest-packing, the Ob plane shows vacancies. In fact, this plane can be formally derived from a closest-packing of spheres by removing one third of the spheres which, otherwise, in a (001) projection would occupy the center of the hexagonal rings (Fig. 1). The same configuration of the Ob plane is obtained by removing the (OH)- groups in Figure 4. The Ob plane shows ideal closest-packing without vacancies if the maximum value (30°) of the ditrigonal rotation occurs (cf. the paragraph “Ditrigonal rotation” and Fig. 8 below). Closest-packing and polytypism The pseudo-closest-packing feature of the Oa planes is key to the understanding of widespread polytypism of micas (Bailey 1984a). A plane closest-packing of equal spheres (Fig. 4) is based upon a plane hexagonal Bravais lattice with cell parameter ah which is equal to the diameter of the packed sphere. In the plane, each sphere is in contact with six translationally equivalent spheres and two translationally independent sets of small vacant sites; each of these two sets (open circles and gray ‘triangles’ in Fig. 4) contains three translationally equivalent vacant sites. To maintain a closest-packing arrangement in three dimensions, the stacking of two successive planes of spheres (A and B) implies that the upper plane (e.g., B) is staggered (shifted) in such a way that its spheres overlie one set of vacant sites belonging to the lower plane (A). Owing to the hexagonal symmetry in the plane, six equivalent staggers are possible along six directions separated by 60° (cf. Ferraris 2002).
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The possibility of multiple staggering is the basis for different periodicities along c, a structural aspect known as polytypism (Verma and Krishna 1966). In the case of micas (and, generally, of phyllosilicates) there are two types of packing spheres: oxygen ions and (OH)- groups. A larger, orthohexagonal C-centered cell (a,b) must be chosen, as shown in Figure 4, and the typical closest-packing stagger between Oal and Oau corresponds to an ±a/3 shift (intralayer stagger). Note that, in module, the parameter a in micas corresponds to the parameter bh of the orthohexagonal cell in a standard closestpacking plane of equal spheres. The ±a/3 stagger between Oal and Oau reflects in the mutual postion of the Tl and Tu sheets as shown in Figure 5. Particularly in K-micas (Radoslovich 1960) and in dioctahedral micas (Bailey 1975) the intralayer stagger may slightly differ from ±a/3. This effect is related either to the size of the I cation or to the distortion of the vacant M1 site, as defined below.
Figure 5. Reference axes in the M layer plane. Hydroxyl (OH)- groups are represented by black circles. The OH groups are in trans position in M1 and in cis position in M2 and M3. The stagger (offset) ±a/3 between lower and upper T sheets is shown. The T1u and T2u tetrahedra are translationally independent; the same for the M1, M2 and M3 octahedra.
To build the crystal structures of the mica polytypes, the M layer is stacked along c in steps of about 10 Å. Commonly, at least in the homo-octahedral (i.e., all octahedra are equal in content and size; cf. below) approximation, the derivation of the mica polytypes is achieved by considering rotations between adjacent M layers (Smith and Yoder 1956) rather than stacking directions. These rotations are performed around the normal to the layer and leave the layer unchanged if multiples are of 60°. The insertion between two Ob planes of interlayer cations I is possible only if each (ideally) hexagonal ring of the atomic plane Obu, belonging to the nth layer, faces an (ideally) hexagonal ring of the plane Obl belonging to the (n+1)th layer. Because this structural requirement can be achieved by different relative rotations between two adjacent layers, different mica polytypes are possible. COMPOSITIONAL ASPECTS Ideally, the crystal-chemical formula of micas can be written as I(Y3-xx)[Z4O10]A2
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(Rieder et al. 1998). Labels represent the following chemical elements and groups [the commonest elements and groups are shown in bold face and their ionic radii (Å) according to Shannon (1976) are given in parentheses]. • I = Cs, K (1.38), Na (1.02), NH4, Rb, Ba, Ca (1.00), . • Y = Li (0.76), Fe2+ (0.78), Fe3+ (0.645), Mg (0.72), Mn2+, Mn3+, Zn, Al (0.535), Cr, V, Ti (0.605), Na (unpublished results on the occurrence of an analogue of tainiolite with octahedral Na instead of Li). In the crystal-chemical formula, the coefficient 3-x together with the symbol of vacancy () means that in principle the occupancy of the octahedral sheet (O) can span from 2/3 (x = 1, dioctahedral micas) to all the available sites (x = 0, trioctahedral micas). Actually, not many examples of intermediate di/trioctahedral micas are known. Some of the examples might leave doubts on their ‘intermediate’ nature because of unsatisfactory chemical (cf. below) and/or structural data. A 2M2 lepidolite with (Li0.35Al0.100.55) in M1 (Takeda et al. 1971) and a Li-Berich mica bityite with (Li0.550.45) in M1 (Lin and Guggenheim 1983) should be true octahedrally intermediate micas. Cases as the M1-deficient Li-rich micas refined by Brigatti et al. (2000), where the maximum vacancy in M1 is 0.23, look more like octahedrally-deficient trioctahedral micas than intermediate di-/trioctahedral micas. On chemical basis only, a Si-rich mica with slightly less than two Y cations has recently been reported (Burchard 2000). • Z = Be, Al (0.39), B, Fe3+ (0.49), Si (0.26), Ti (?) (no vacancies have been reported). • A = Cl, F, OH, O, S (no vacancies have been reported). It should be noted that: 1. The same site may be occupied (either in an ordered or a disordered way) by different ions. 2. At least two elements (Al and Fe3+) may occupy both octahedral and tetrahedral sites; as mentioned above, Na is reported also in octahedral coordination. 3. The same element (e.g., Fe) may be present in different oxidation states. 4. Most of the recent chemical data are obtained by electron microprobe analysis; consequently, they are often incomplete because oxidation state, light elements and water (hydrogen) are not analyzed [cf. Dyar (this volume) and Pavese et al. (2002) for a recent case of synergic use of neutron-diffraction data and Mössabauer spectroscopy]. Features 1-4 imply that the crystal-chemical formula of a mica cannot be established on the basis of a chemical analysis only (even if it is complete); detailed structural knowledge is necessary. Structurally, the occupancy of a site can be obtained by combining chemical constraints (chemical analysis) with other information like the following. Scattering power of a site (Sp ). If a site is fully occupied by two elements with scattering power S1 and S2 and occupancy x and 1-x, respectively, the distribution of the elements can be obtained by solving the equation Sp = xS1 + (1-x)S2. This procedure cannot be applied without further information when (1) vacancy and/or more than two elements occur in the same site; (2) the difference in the scattering power is small as, with X-ray diffraction, in the common cases of substituting elements differing by only one electron (Na-Mg, Mg-Al, Al-Si, Mn-Fe). If suitable wavelengths are available (e.g., synchrotron radiation) anomalous scattering may be used to recognize different atoms that randomly occupy the same site. For case (2), neutron-diffraction data would represent the best solution; but, for powder-diffraction data, cf. a discussion in Pavese et al. (2000). Note that the occurrence of stacking faults in the structure may create peculiar problem in the refinement procedure (Nespolo and Ferraris 2001).
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Distributions of the bond lengths . Good quality structural data allow the use of the average bond length of a coordination polyhedron to determine quantitatively the fraction of occupying atoms. In micas, a reasonable determination of the tetrahedral Si (xSi) and Al (xAl) fractional contents as a function of the average tetrahedral bond lengths (Z-O)av can be obtained by the equation (Z-O)av = 0.163[xAl /( xSi + xAl )] + 1.608 (Hazen and Burnham 1973; cf. Brigatti and Guggenheim (this volume) for an equation which also takes into account tetrahedral Fe). SYMMETRY ASPECTS Metric (lattice) symmetry Because of the pseudo-closest-packing nature of the atomic planes mentioned above, the two-dimensional Bravais lattice of both the T and the O sheets (Fig. 1), idealized and undeformed according to the Pauling (1930) structural model of micas, is hexagonal 6mm. Both sheets can be described in terms of a primitive hexagonal lattice, defined by two hexagonal axes A1 and A2, or of a C-centered orthohexagonal lattice defined by the two shortest perpendicular translation vectors, a and b, between which the orthohexagonal relation b = a√3 ideally holds (Fig. 5; Nespolo et al. 1997a, 1998). The two-dimensional lattice of the real sheets, as well as of the whole M layer they form, is no longer hexagonal. The A1 and A2 axes are no longer exactly identical in length and their interaxial angle is no longer exactly 120°: they define a lattice that is only pseudo-hexagonal and corresponds to a centered rectangular lattice whose a and b axes only approximately obey the orthohexagonal relation b = a√3. Structural symmetry The T sheet. In an ideal T sheet (Fig. 1), the tetrahedra are regular polyhedra and their centers (Z cations) coincide with the nodes of a hexagonal plane lattice; the corresponding point group symmetry is 6mm. The layer symmetry (λ-symmetry) of this sheet is P(6)mm; the symmetry of the direction without periodicity, which is perpendicular to the layer, is shown in parentheses according to the layer group notation (Dornbenger-Schiff 1959). In each T sheet there are two translationally independent tetrahedral sites (Fig. 5). On the whole there are four T sites in the M layer: T1u, T2u, T1l and T2l (u = upper; l = lower). Following Bailey (1984), tetrahedral sites in the upper sheet that, in the (001) projection, are at -1/3[010], +1/3[310] and –1/3[⎯310] from the upper OH group are labeled T1, whereas those at +1/3[010], -1/3[310] and +1/3[⎯310] are labeled T2. The same definition applies to the lower T sites with respect to the lower OH group. The O sheet . In the O sheet (Fig. 5) the number of translationally independent M sites is three: one site (M1) has two (OH)- groups in trans configuration, whereas the other two sites (M2 and M3) have two (OH)- groups in cis configuration. The definition of M2 and M3 is however not straightforward. Bailey (1984c) suggested labeling M3 the site on the left of the (pseudo)-mirror plane, but most authors have labeled that site as M2. Here we retain the definition prevailing in the literature, calling M2 (M3) the site with negative (positive) y coordinate in the layer-fixed reference, namely on the left (right) of the (pseudo)mirror plane looking down the positive direction of the c axis. Families of micas accordin g to the symmetry of the O sheet . The type of occupancy (number of electrons in the site, if the exact cation composition of the site is unknown) of the three M octahedral sites defines the three following families of micas as introduced by OD theory (Dornberger-Schiff et al. 1982; Durovic 1994). The λ-symmetry of the O sheet is different in the three families (Table 1): homo-octahedral family (the three M sites have the same cation occupancy), meso-octahedral family (two M sites are identical,
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one is different), and hetero-octahedral family (the three M sites are differently occupied). The distinction in three families is based on structural features and reflects the symmetry of the O sheet; it is operatively determined on the basis of the number of electrons filling each of the three M sites, and it is used by the OD theory to fix unequivocally the origin in the layer. Table 1 contains a comparison of the common division of the mica families into tri- and dioctahedral classifications. Table 1. Families of micas based on the symmetry of the octahedral sheet. Comparison with dioctahedral and trioctahedral classification is given. (Modified after Durovic 1994). Family
λ-symmetyr
Homo-octahedral
H`( 3 )1m
•••
---
Meso- octaeh dral
P`( 3 )1m
•♦♦
••
Heteor - octaeh dral
P(3)12
•♦♣
♦♣
Trioctaeh dral
Dioctaeh dral
•♦ ♣ = different electron occupancy of the M sites; = vacancy
Dioctahedral/tir octahedral distinction. As mentioned above, the O sheet of micas is traditionally described with reference to the minerals brucite (brucite-like sheet or trioctahedral sheet, namely homo-octahedral sheet) and gibbsite (gibbsite-like sheet or dioctahedral sheet, namely meso-octahedral sheet). This description is helpful to emphasize the modular nature of the mica layer; however, whereas the brucite-like sheet corresponds to the highest symmetry (homo-octahedral), the gibbsite- sheet does not correspond to the lower symmetry, being only meso-octahedral. Symmetry of the O sheet . The plane point group symmetry of the ideal O sheet (Fig. 1) is 3m (a subgroup of 6mm) and its layer-symmetry is either H`( 3 )1m (brucite-like sheet) or P( 3 )1m (gibbsite-like sheet). In fact, the symmetry of the two types of octahedral sheets differs at least for the following reasons. 1. In the ideal brucite-like sheet (Fig. 1) all the octahedral sites are metrically equivalent; each oxygen atom has coordination number three and the O-H bonds of the hydroxyl (OH)- groups are perpendicular to the sheet. 2. In the ideal gibbsite-like sheet (Fig. 1) only 2/3 of the octahedral sites are occupied by the same cation and the other 1/3 is vacant; each oxygen atom has coordination number two and the O-H bond is parallel to the sheet and directed towards the vacant site. Symmetry of the M layer . Because Oal and Oau correspond to two successive planes of a (pseudo)-closest-packing of spheres, Tl and Tu of an M layer are ±a/3 staggered (Fig. 5); consequently, both in the brucite-like and in the gibbsite-like case, the λ-symmetry of the entire M layer is lowered to C12/m(1). Symmetry and cation sites Mainly because of a dimensional misfit between the T and O sheets (cf. below), in real mica structures the Pauling model (in which there are no structural distortions) is too abstract and must be replaced at least by a model which takes into account a rotation of the tetrahedra within the (001) plane. This ditrigonal rotation is discussed below; the resulting model has been called the trigonal model by Nespolo et al. (1999c).
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In the homo-octahedral family, the three M sites are by definition identical in content and size. Any difference in one of the M sites violates the H centering, lowering the symmetry of the O sheet to that of the meso-octahedral family. A difference between the other two M sites destroys also the inversion center and lowers the symmetry of the O sheet to that of the hetero-octahedral family. From the practical viewpoint, differences among the M sites are often small and must be evaluated on statistical grounds. As discussed by Bailey (1984c) for the specific case of micas [cf. an application in AmisanoCanesi et al. (1994)], if σl is taken as the estimated standard deviation (esd) of an individual quantity and σn = σl/n1/2 is the esd of the mean of n values, the esd of a difference (Δ) between two mean values is given by σΔ = 21/2σn. Two quantities are considered different at the 0.1% level of significance if Δ > 3.1σΔ (usually known as the 3σ rule). In both the Pauling and the trigonal models, the stagger of the two T sheets reduces the λ−symmetry of the M layer to monoclinic (Fig. 5). Within the highest layer-group C12/m(1), M2 = M3 and only one symmetrically independent tetrahedral site exists (most of 1M polytypes have this symmetry). Depending on the cation distribution and the presence of structural distortions, the M layer may however have a lower λ-symmetry corresponding to a subgroup of C12/m(1). In principle the following lower symmetries can occur. Layer group symmetry C1m(1). The m mirror plane coincides with the ac plane of the layer, M2 = M3 and two symmetrically independent tetrahedral sites occur according to the following scheme: T1u = T2u and T1l = T2l. No structures are known with this symmetry. Layer group symmetry C12(1). The two-fold axis is along the b axis of the layer and the M2 and M3 sites are no longer equivalent. Two symmetrically independent tetrahedral sites are present in each sheet, but the two T sheets of a layer are symmetrically equivalent: T1u = T1l and T2u = T2l. This symmetry occurs in some meso- and hetero-octahedral 1M polytypes and in the hetero-octahedral 3T polytypes. Layer group symmetry C 1 . The M2 and M3 sites are equivalent. There are two symmetrically independent tetrahedral sites in the M layer according to the following scheme: T1u = T2l and T2u = T1l. Most of the known 2M1 polytypes show this symmetry. Layer group symmetry C1. The three M sites and the four T sites are all symmetrically independent. This symmetry occurs, e.g., in ephesite-2M1 (Slade et al. 1987). In real structures, the λ-symmetry of the M layer with a given pattern of cation ordering may be lower than the ideal one described by the trigonal model as a function of the concrete stacking mode in a polytype: this phenomenon is known as desymmetrization (Durovic 1979). A primitive P lattice for the layer occurs in the unique example of anandite-2O (space group Pnmn; Giuseppetti and Tadini 1972, Filut et al. 1985). However, anandite-2O cannot be considered a real mica polytype (Ferraris et al. 2001c) for the following reasons: its P cell is not compatible with the C-centered cell common to all mica polytypes; its space group is not that expected (Ccmm) for the 2O mica polytype according to the OD theory; S substituting OH is coordinated by the interlayer cation which thus has coordination number 13. Summarizing, in an M layer: 1. The maximum of symmetrically independent M sites is three. 2. At least two symmetrically independent M sites are always present; in dioctahedral micas an independent site is always represented by the vacant M1 site.
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3. In real polytypes the maximum number of independent T sites is two. 4. If the lattice of the layer is P, the maximum number of independent T sites can reach four, but the resulting structure is no longer strictly polytypic (cf. anandite-2O). Table 2. λ-symmetry (S) and type of layer (L) in the three families of mica polytypes within the Trigonal model. δ indicates the electron density of the octahedral site (site occupancy). Origin S of in S of δ L Family M layer O sheet O sheet Homo-octahedral (M1 = M2 = M3) Meso-octahedral
H`( 3 )1m
δ(M1) = δ(M2) = δ(M3)
C12/m(1)
---
M1
M1 ≠ M2 = M3
P`( 3 )1m
δ(M1) ≠ δ(M2) = δ(M3)
C12/m(1)
M1
M1
M1 = M2 ≠ M3
P`( 3 )1m
δ(M1) = δ(M3) ≠ δ(M2)
C12(1)
M2
M2
M1 = M3 ≠ M2 Hetero-octahedral
P`( 3 )1m
δ(M1) = δ(M2) ≠ δ(M3)
C12(1)
M3
M2
(M1 ≠ M2 ≠ M3)
P(3)12
δ(M1) ≤ δ(M2), δ(M1) ≤ δ(M3)
C12(1)
M1
M1
P(3)12
δ(M2) < δ(M1), δ(M2) < δ(M3)
C12(1)
M2
M2
P(3)12
δ(M3) < δ(M1), δ(M3) < δ(M2)
C12(1)
M3
M2
Two kinds of mica layer: M1 and M2 layers
The λ-symmetry of the M layer depends on the number of identical M sites in the O sheet. The origin of this sheet is fixed by the OD theory and taken at the site with the point symmetry corresponding to the λ-symmetry of the sheet (Dornberger-Schiff et al. 1982). The complete scheme is given in Table 2 and is summarized as follows. Homo-octaeh dral family . In this family any of the three M sites has 3 1m point group symmetry. Meso-octahedral family . In this family only one of the M sites has symmetry 3 1m: it is the site with different occupancy/size. Depending on whether this site is M1 (trans) or M2/M3 (cis), the layer itself is termed M1 or M2 respectively; the highest layergroup for these two layers is C12/m(1) (M1) and C12(1) (M2). There is thus a basic difference in the nature of these two types of layer. Hetero-octaeh dral family . In this family, because of the chemical/size difference of the three M sites, the highest layer-group is C12(1) for both kinds of layer and any of the three M sites has point symmetry 312. In the hetero-octahedral family the origin can in principle be chosen in any of the M sites. In the case of dioctahedral micas, the origin of the O sheet is taken in the vacant octahedral site. By extending this criterion, in the case of hetero-trioctahedral micas the origin of the O sheet is taken as the site showing the lowest electron density δ (lowest X-ray scattering power; Durovic et al. 1984), which often corresponds to the largest M site (in most cases that site is M1). However, some examples are known (cf. Nespolo and Durovic, this volume) in which the site containing the lowest electron density is either M2 or M3, and the origin of the O sheet is thus in one of the two cis sites. Thus, as in the mesooctahedral family, two kinds of layer, M1 and M2, are distinguished.
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Examples . Although the M1 layer is much more common, several examples of micas constructed of M2 layers are known. So far, in dioctahedral micas constructed of M2 only examples of 1M polytype have been reported; in them the vacancy is ordered into one of the two cis sites (Zvyagin et al. 1985; Bloch et al. 1990; Zhukhlistov and Zvyagin 1991; Zhukhlistov et al. 1996). On the contrary, trioctahedral micas constructed of M2 layers are known for all the three most common polytypes (1M, 2M1 and 3T) (Guggenheim and Bailey 1977; Brown 1978; Guggenheim 1981; Mizota et al. 1986; Rieder et al. 1996; Brigatti et al. 2000). Both the M1 and the M2 layers can undergo the mentioned desymmetrization (Durovic 1979); a corresponding reduction to a space subgroup of the whole polytype may or may not occur. To distinguish the two kinds of layer, the occupancy and the size of the three octahedral sites must be known as a result of the refinement of the crystal structure. This knowledge is available only in a few cases and most of the mica structural studies are just based on the assumption that the crystal under investigation is built by M1 layers. For this assumption, Nespolo et al. (1999c), following a suggestion by S. Durovic (pers. comm.), introduced the term homooctahedral approximation. The interlayer configuration
In the Pauling model, the λ-symmetry of the interlayer is P(6/m)mm: the interlayer cations have twelve nearest neighboring oxygen atoms at the corners of a hexagonal prism, which is not modified by an n × 60° rotation between adjacent layers. On the other hand, in the trigonal model the ditrigonal rotation modifies the symmetry of the interlayer depending on the parity of n in the n × 60° rotation. This symmetry becomesP( 3 )1m for even n and P`( 6 )2m for odd n. In both cases the I cations have six oxygen atoms as nearest neighbors and other six oxygen atoms as next-nearest neighbors (Fig. 6).
Figure 6. The interlayer cation I (large circles) displays antiprismatic coordination and prismatic coordination in the subfamily A and B polytypes, respectively. First and second neighbor bonds are indicated by full and dashed lines respectively. For maximum ditrigonal rotation (30°), the first six neighbors form either a trigonal antiprism (subfamily A) or a trigonal prism (subfamily B).
Subfamilies A and B of polytyep s . Even rotations between adjacent layers lead to a trigonal (considering only the nearest neighbors oxygen atoms) or ditrigonal (considering also the next-nearest neighbors oxygen atoms) antiprismatic coordination for the interlayer cations I, whereas odd rotations lead to trigonal or ditrigonal prismatic coordination. The antiprismatic coordination of the nearest neighbors is often quoted as ‘octahedral’ coordination and, as discussed below, is presumed to be a stabilizing factor of the subfamily A polytypes, as are called those based on 2 n × 60° rotations. Instead, the subfamily B polytypes, which are based on (2n +1) × 60° rotations, are rarer and are
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considered less stable. MDO polytypes . The subfamily A polytypes 1M (n = 0), 2M1 (n = 1 and 2), and 3T (n = 1 or 2) and the subfamily B polytypes 2O (n = 1), 2M2 (n = 0 and 2), and 6H (n = 0 or 2; never found) are called homogeneous (Zvyagin 1988), MDO (Durovic et al. 1984), simple (Smith and Yoder 1956) or standard (Bailey 1980) polytypes. The ideal space groups of the six MDO polytypes are: C2/m (1M), C2/c (2M1 and 2M2), P31,212 (3T), Ccmm (2O) and P61,522 (6H) (Fig. 7).
Figure 7. Crystal structures of the five known MDO (homogeneous) polytypes of mica: 1M (a), 2M1 (b), 2M2 (c), 3T (d), and 2O (e).
Possible ordering schemes in the MDO polytypes
The actual λ-symmetry (maximum λ−symmetry) of the M layer in a polytype can be any subgroup of C12/m(1) which is not lower than the λ-symmetry required (minimum λ−symmetry) to the M layer by the space-group of the structure. For the five known MDO (homogeneous) polytypes the minimum λ−symmetry is as follows. Space groups C2/m and Ccmm. The minimum λ−symmetry required by these two space– groups is C12/m(1). The O sheet contains the global twofold axis in both space groups. One tetrahedral and two octahedral independent sites are allowed and no ordering of the tetrahedral cations is possible. The 1M and 2O polytypes show this symmetry.
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Space group C 2/c. The minimum λ−symmetry required by this space group is C 1 ; the global twofold axis passes through the interlayer cation. Two tetrahedral and two octahedral independent sites are allowed; consequently, both tetrahedral and, for trioctahedral micas, octahedral cation ordering are possible. The 2M1 and 2M2 polytypes show this symmetry. Space group P 31,212. The minimum λ−symmetry required by this space group is C12(1) with the O sheet containing a set of global twofold axes; the interlayer cation lies on a second set of global twofold axes. The 3T polytypes show this symmetry. Most of the 3T micas are dioctahedral and both tetrahedral (two independent T sites) and octahedral (three independent M sites, including the vacant one) cation ordering is possible.
Summarizing: among the five known MDO polytypes with ideal space-group, tetrahedral ordering is not possible in 1M and 2O polytypes. Octahedral ordering is instead possible in all five trioctahedral polytypes; for dioctahedral polytypes cf. phengite below. Note that there are some hints of a limited occupancy of M1 in ‘strictly’ dioctahedral micas (Brigatti et al. 1998, 2001; Pavese et al. 2001). The phengite case
The possibility of chemical order/disorder allowed by crystallographically independent M and T sites is a feature influencing the polytype stability (Pavese et al. 1997, 1999a,b; 2000), together with the interlayer coordination and the T/O dimensional mismatch as discussed later. The dioctahedral phengite micas are typical of high-pressure environment (Sassi et al. 1994) and represent a good example for this type of discussion. Owing to an optimum octahedral Mg/Al and tetrahedral Si/Al substitution, a good T/O match with very small ditrigonal rotation (see below) is possible in both 2M1 and 3T polytypes. Because this rotation increases with pressure P (cf. Zanazzi and Pavese's chapter), a small starting value at room-condition allows to maintain a still ‘reasonable’ rotation at higher P. Whereas this feature is present also in trioctahedral micas, only dioctahedral micas (and phengites in particular) show the following characteristics which favor stability at high P (Ferraris et al. 1995; Ferraris and Ivaldi 1993, 1994a,b). 1. The O-H bond tends to be parallel to (001) and in the direction of the vacant M site and does not hinder the compressibility of the interlayer. The O-H pointing towards the interlayer cation is instead an obstacle to this compressibility in the trioctahedral micas. 2. A higher structural flexibility In 3T following from the presence of more than one crystallographically independent octahedral site. 3. A spiral disposition in 3T of the O-H directions across the three M layers of a cell compared to the antiparallel disposition in the 2M1 polytype; consequently, a minor interlayer repulsion can be expected in 3T. In particular, the stability of 3T phengite at high P relative to the 2M1 polytype occurs by 3T phengite possessing the characteristics 2. and 3. DISTORTIONS The misfit
As already mentioned, in real mica structures the Pauling model is too abstract and must be replaced at least by the trigonal model, which considers a rotation of the tetrahedra around the perpendicular to (001). In fact, b being about 9.4, 8.6 and 9.3 Å in brucite, gibbsite and T sheet (with Si:Al = 3:1), respectively, the dimensions of the T and of the O sheets do not match. Consequently, as discussed below, some structural distortions are needed to overcome the misfit and to form these two sheets into a layer.
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Geometric parameters describing distortions
Practically, in all crystal structures, strictly regular coordination polyhedra occur only if regularity is constrained by the symmetry, i.e., in micas, if the coordinated cation occupies a crystallographic position with point symmetry 4 3m for a tetrahedron and m 3 m for an octahedron. That does not happen because the tetrahedral and octahedral sites occurring in micas, even in the ideal Pauling model, have maximum symmetry 1 and 2/m, respectively. Consequently, the polyhedra on which the M layer is comprised have some degrees of freedom to differentiate their bond lengths and angles. This freedom is exploited to compensate, at least in part, both internal strains, connected with the chemical composition and the misfit between T and O sheets, and external strains, like pressure and temperature variations. In dioctahedral micas, the vacant octahedron is by far larger (〈M-O〉 ~ 2.2 Å) than the occupied ones ( 〈M-O〉 ~ 1.9 Å) because of the repulsion between the unshielded Oa apical oxygen atoms forming the vacant octahedron. Because two types of quite different Oa-Oa octahedral edges occur in the O sheet of the dioctahedral micas, two different types of Oa-Oa distances are necessary also in the T sheet to fit with the O sheet. Larger distortions must therefore be expected in dioctahedral micas than in trioctahedral micas.
Figure 8. From left to right, undistorted, moderately and fully distorted tetrahedral sheets are shown. The distortion (ditrigonal rotation α) is obtained by rotation of the tetrahedra around the perpendicular to the sheet. As shown, the angle 2α is defined by the directions of two tetrahedral edges sharing a corner. In a fully distorted tetrahedral sheet, the basal oxygens form an ideal closest packing without vacancies.
Ditrigonal rotation
The trigonal model is the most important modification of the simple Pauling’s model; it was introduced in 1949 by Belov, although a later paper by Radoslovich (1961) is usually quoted. To match the T and O sheets into a TOT layer, structural distortions must be introduced. The most important of these distortions is the ditrigonal (or in plane) rotation, α, of the tetrahedra around the perpendicular to (001) (Fig. 8). This distortion was theoretically related to the misfit between T and O by several authors (cf. Bailey 1984c) and was first experimentally confirmed in the structure of clintonite (Takéuchi and Sadanaga 1959; these authors used the variety name ‘xanthophyllite’). The rotation α reaches its maximum value (30°) when thehexagonal ring becomes a perfect ditrigonal ring; in this case the Ob oxygen atoms form an ideal closest-packing. In both Pauling and trigonal models, the tetrahedra have a triangular base exactly parallel to the (001) plane and the Ob oxygen atoms form a flat (001) surface. The angle between the prolongation of one edge of a triangular base and the corresponding edge of the triangular base sharing the same oxygen atom corresponds to 2α (Fig. 8). Calling φi the internal angles of the basal ‘hexagon,’ the angle 2α is obtained by the equation 2α = Σi = 1,6 (|120 −φi|)/6 (e.g., Weiss et al. 1992). The λ-symmetry of the tetrahedral sheet is reduced to trigonal by the ditrigonal rotation [layer-group P( 3 )1m] but the entire M layer maintains the C12/m(1) symmetry. The physical limits for α are 0° (the fit between T and O is perfect) and 30°
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(maximum ditrigonalization). The effects of the two possible directions of the ditrigonal rotation are as follows (Franzini 1969): Type-A layer. The triangular bases of the tetrahedra are oriented in the opposite way relative to the underlying, parallel triangular faces of the octahedral sheet (Fig. 9A). Type-B layer. The triangular faces of the tetrahedral and octahedral sheets have the same orientation (Fig. 9B).
Figure 9. Type-A and type-B mica layers according to Franzini (1969) seen down the positive direction of the c axis. In type-A the triangular bases of the tetrahedra are oriented in the opposite way relative to the underlying, parallel triangular faces of the octahedral sheet. In type-B the triangular faces of the tetrahedral and octahedral sheets have the same orientation. The trans octahedra are shadowed.
In the type-A layer, the oxygen atoms of the Ob plane approach the perpendiculars passing through the octahedral cations because, according to Bailey (1984a), they are attracted by these cations. Thus a shielding effect between the Y octahedral cations and the I interlayer cations occurs. In the type-B layer the opposite situation arises and the shielding effect is reduced with respect to the undistorted Pauling’s model. In both cases, the b parameter of the distorted T sheet shortens with respect to that of an ideal T sheet and the T/O match is improved. As noted by Zvyagin (1967), and shown by the orientation of the triangles in Figure 9, the packing sequence of three apical sheets Oal-Oau-Obl approaches cubic-closest-packing (ccp) in type-A and hexagonal-closestpacking (hcp) in type-B. Contrary to Griffen’s (1992) statement that, although rarely, the type-B has been observed in dioctahedral micas, we have found no cases of this layer in micas. It occurs instead in other phyllosilicates, e.g., in 1:1 layer silicates such as lizardite (Mellini 1982) and in cronstedtite (Hybler et al. 2000) [actually, in these silicates the occurrence of unshielded octahedral cations is not a problem because there are no isolated interlayer cations]. Other distortions
Because of the sheet dimensions, which are determined by the Y-O and Z-O distances, the misfit between the T (b ~ 9.3 Å for tetrahedral occupancy Al:Si = 1:3) and O sheets is minimum and maximum when the octahedral sheet corresponds to a brucitelike (b ~ 9.4 Å) or a gibbsite-like ( b ~ 8.6 Å) layer, respectively. These two cases occur in the pure end-members micas phlogopite (trioctahedral), KMg3[AlSi3O10](OH)2, and muscovite (dioctahedral), KAl2[AlSi3O10](OH)2. Therefore, besides ditrigonal rotation, the chemical composition can contribute to match the dimensions between the T and O sheets. As seen in the paragraph “C ompositional aspects,” whereas in the T sites practically only Al and Si (sometimes Fe3+) can occur, a larger variety of cations, with octahedral ionic radii ranging from 0.535 Å (Al) to 0.76 Å (Li), can occur in the M sites. An appropriate distribution of cations can therefore favor the fitting between T and O sheets. The introduction of chemical substitutions at least in part contributes to various types of polyhedral distortions. These, besides the discussed ditrigonal rotation, are classified here below (Fig. 10). Some specific structural reasons for the appearance of
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Figure 10. Definition of the octahedral flattening (ψ), octahedral thickness (t0), tetrahedral elongation (τ), tetrahedral tilting (Δz) and octahedral counter-rotation (ω) which is related to εi as shown in the text.
these distortions are given here but more details can be found in Brigatti and Guggenheim (this volume). Tetrahedral elongation . This distortion is also known as tetrahedral thickening and implies an expansion for the tetrahedra in the direction perpendicular to the T sheet and a lateral compression (Radoslovich and Norrish 1962). The effect is measured by the angle τ = Σi=1,3(Ob-T-Oa)i /3 (τideal = 109.47º). It is particularly active in dioctahedral micas where it is related to the presence of the vacant octahedral site (Lee and Guggenheim 1981). Tetar eh dral it lting . Practically, this distortion is only found in dioctahedral micas because is caused by a great difference between the sizes of the octahedral sites. The tetrahedra rotate around a direction parallel to the (001) plane determining a departure from coplanarity of the Ob oxygen atoms (out-of-plane tilting). This tilting produces a corrugation of the basal plane which is measured by the parameter Δz = [zOb(max) – zOb(min)]csinβ. Octaeh dral flatet ning (or iht ckening) . This distortion is measured by the angle ψ between the body diagonal and the base of the octahedron (Donnay et al. 1964). Given the thickness t0 of the O sheet and the average octahedral distance 〈Y-O〉, ψ is calculated as ψ = cos-1[t0/(2 〈Y-O〉)]. Because ψideal = 54.73°, a flattening results in a larger value of ψ; vice versa for a thickening. Counter-rotation ω. This distortion (Newnham 1961) is measured as the angle of rotation between the two triangular octahedral faces parallel to (001) belonging to the same octahedron; it is calculated by ω = |(ε 1 + ε 3 + ε 5)/3 - 60°| = |(ε 2 + ε 4 + ε 6)/3 – 60°| (ωideal = 60° and 0°). The anglesεi correspond to the O-YO angles measured in the projection of the octahedron onto (001); in a regular octahedron εi = 60°. Generally, for all micas this effect is related to the difference in size of neighboring octahedra (Lin and Guggenheim 1983).
These distortions are not independent variables. In fact, besides specific aspects in part mentioned above, all of them are to some extent correlated with chemical substitutions and T/O misfit. Several correlations have been proposed, particularly for the ditrigonal rotation (e.g., Lin and Guggenheim 1983; McCauley and Newnham 1971; Toraya 1981; Weiss et al. 1985). Effects of the distortions on the stacking mode
All the distortions decrease in the order hetero-octahedral > meso-octahedral > homo-octahedral, because of the corresponding reduction in the size difference of
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different octahedra. All octahedra in micas are more or less distorted (they should thus more rigorously be termed trigonal antiprisms) and the distortions can be fully described by the flattening ψ and the counter-rotation ω (Weiss and Rieder 1997). It has been shown (Weiss and Wiewióra 1986; Weiss and Rieder 1997) that the ditrigonal rotation is most effective in influencing the diffraction intensities, in particular 20l and 13l reflections (i.e., the second ellipse in the OTED described in Appendix II); instead the counter-rotation affects mainly the basal diffractions.
Figure 11. Adjacent basal Ob oxygen atoms in the case of 1M and 2O polytypes, that show 0° and 180° rotations betweenadjacent M layers (left), and (right) of 2M1, 3T, 2M2 and 6H polytypes where instead the rotation between adjacent M layers is ±120° (2M1, 3T) or ±60° (2M2 and 6H). The tetrahedral tilting Δz (Fig. 10) is exaggerated.
Δz and stability of polytypes . The tetrahedral tilting Δz seems to have the most marked influence on the relative stability of the 1M and 2M1 polytypes, the latter becoming energetically favored when Δz increases (Appelo 1978 and 1979, Abbott and Burnham 1988). A general influence of Δz ≠ 0 on the relative stability of different polytypes can also be expected on geometric grounds by considering the interlayer configuration (Fig. 11) for different values of the n × 60° rotation between adjacent M layers (Güven 1971, Soboleva 1987). For polyt ypes based on 0° or 180° rotations (the MDO polytypes are1M and 2O), both the Ob planes delimiting the interlayer region have negative Δz in correspondence of the I cation. For polytypes based on ±120° (2M1 and 3T) or ±60° (2M2 and 6H) rotations, the two Ob sheets delimiting the interlayer region have opposite signs of Δz in correspondence of the I cation. In presence of a large Δz, the polytypes based on 0° and 180° rotations offer too a large cavity for theI cation and, e.g., 2M1 (but also 3T) is favored relative to 1M. At high Δz, (2n+1) × 60° rotations become favored and the relative stability of 2M2 and 2O seems then to depend on the ditrigonal rotation of the tetrahedra (Bailey 1984c; Abbott and Burnham 1988).
Figure 12. Stacking of the M octahedral sites for subfamily A and subfamily B polytypes. These sites lie on the same perpendicular to (001) in subfamily B but not in subfamily A polytypes.
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Relatiev rotation of two adjacent M layers . This plays a role in stabilizing polytypes also in connection with the relative positions of the M sites. In fact (Fig. 12), whereas for a 2n × 60° rotation (subfamily A) the octahedral sites in two adjacent M layers are staggered by ±a/3 and thus not overlapped in the (001) projection, in the case of a (2n+1) × 60° rotation (subfamily B) they are staggered ±b/3 and thus lie on the same perpendicular to (001) [Soboleva 1987; cf. also the polytypic stacking discussed in terms of configurations I and II of the octahedral cations in Bailey (1984a)]. Although a direct influence of the relative positions of octahedral cations belonging to adjacent layers is hardly conceivable, because of the large separation (~10 Å), the stacking of octahedra along the perpendicular becomes an indirect destabilizing factor through its effect on tetrahedral tilting Δz. This would be clear in the case of a hypothetical 2O dioctahedral mica, where the large and vacant octahedral sites would stack on the same perpendicular. As a consequence, tetrahedra on the opposite sides of the I cations are tilted in the same direction, increasing the repulsion between approaching Ob atoms. In the other two MDO subfamily B dioctahedral polytypes (2M2 and 6H), the stacking along a perpendicular alternates vacant and filled octahedral sites, reducing the Ob-Ob repulsion with respect to the 2O polytype. Dioctahedral 2M2, although rare, has been found (Zhukhlistov et al. 1973), whereas neither 2O nor 6H have been discovered so far in dioctahedral micas. The complete absence of the 6H polytype in any family of mica likely derives both from energetic factors (e.g., odd rotations) and kinetic reasons (low probability of formation and inheritance of a 6-layer period with hexagonal symmetry; Nespolo 2001). FURTHER STRUCTURAL MODIFICATION Pressure, temperature and chemical influence
Generally pressure (P) and temperature (T) have a major influence on the distortions of coordination polyhedra (cf. Zanazzi and Pavese, this volume). The tetrahedral dimensions are the ones least sensitive to P and T, whereas the compressibility and expansion of the octahedra are large, so the fit between tetrahedral and octahedral sheets improves with increasing T (larger O sheet) and worsen with increasing P (smaller O sheet). Therefore, in a first approximation, P and T shows an opposite behavior (Hazen and Finger 1982). The knowledge of the P-V-T equation of state would allow the calculation of the isochor, i.e., the P-T path which maintains constant the cell volume (cf. Pavese et al. 1999b) and, reasonably, also the distortions. For a rough estimate of the isochor, the values of the expansion at constant P (isobar) and of the compression at constant T (isotherm) can be combined (e.g., Comodi and Zanazzi 1995; Mellini and Zanazzi 1989). The main effect of P and T are on the interlayer because the I-Ob bonds are weak. Both the effect of modifying the length of the c parameter and the ditrigonal rotation are discussed below. Note that any change in the T/O match because of P and T variations modifies the ditrigonal rotation and consequently the interlayer coordination. Other sources of change for the I-Ob distances are tetrahedral substitutions (which modify the length l of the tetrahedral edge). Summarizing, for a given I, the parameter c modifies mainly under the following factors: 1. temperature T (expansion → longer c); 2. pressure P (compression → shorter c); 3. ditrigonal rotation α (I-Ob distances are modified). Thickness of the mica module
As mentioned at the beginning of this chapter, a mica module is intended to consist of an M layer plus the interlayer cation. The use of the module thickness tm = csinβ/n (n is the number of M layers in a unit cell) allows the comparison of data from different polytypes.
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Figure 13. Decreasing trend of the mica module thickness (top) and ditrigonal rotation α vs the increasing content of Si in 2M1 (rhombi and crosses) and 3T (open triangles and circles) natural phengitic micas. To obtain α the knowledge of the crystal structure is necessary. The values corresponding to samples with known structure are indicated by rhombi (2M1) and circles (3T). The values at 3.81Si represent the only two synthetic phengites which are included because their crystal structures are known (Smyth et al. 2000). R represents the correlation coefficients of the shown regression lines.
Sinking effect of the I cation . The smaller the ditrigonal rotation α, the larger is the more or less hexagonal cavity where the interlayer cation I can sink; consequently a shorter c parameter is expected. This effect has been observed by several authors. Guidotti et al. (2000) noted that a shorter c parameter is observed in the low pressure Fmrich muscovites [Fm = (Fe + Mg)/(Fe + Mg + Al)]. In fact, as expected from the values of the cation ionic radii, the Fm substitution for Al in muscovites (and the parallel Si/Al tetrahedral substitution) improves the T/O fit and, consequently, the ditrigonal rotation α decreases. Μassonne and Schreyer (1986, 1989) and, recently, Schmidt et al. (2001) showed, in synthetic phengites, a contraction of the c parameter with the increase of the Si content. Ivaldi et al. (2001a) have found the same result on natural samples (Fig. 13; the thickness of the mica module tm instead of c is used). The ditrigonal rotation α behaves as tm. The decrease of the ditrigonal rotation α with the increase of the Si content is well explained by the improvement of the fit between the tetrahedral and octahedral sheets promoted by the aluminoceladonitic substitution (Mg for VIAl and Si for IVAl). By decreasing α, more-hexagonal rings occur in the Ob plane where the interlayer cation can sink.
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Figure 14. A shift σ (exaggerated) of the basal plane Ob from the full line to the dashed line position reduces more (CD) the I-Ob(outer) distances than (AB) the I-Ob(inner) ones.
Interlayer separation and coordination. Ferraris and Ivaldi (1994b) showed that a variation of the interlayer separation influences the outer I-Ob distances more greatly than the inner I-Ob distances. This geometric effect appears clear in Figure 14. Therefore, an ‘at first sight unexpected’ behavior occurs under variation of P and/or T: the I-Ob(inner) distances, which represent shorter and thus stronger bonds, change by far more than the IOb(outer) ones, which instead represent longer and thus weaker bonds.
Figure 15. Undistorted (a) hexagonal ring of a T sheet showing that all basal Ob atoms have the same distance DA = l from the ring center D; D represents the intersection of the drawing plane with the perpendicular to the ring. In a ring (b) with maximum ditrigonal rotation α, the inner Ob atoms are closer (DC = l/31/2) to D than the outer Ob atoms (DB = 2l/31/2). The values of the internal ring angles are related to α as follows: BCB' = 120° + 2α, CBC' = 120° - 2α.
Ditrigonal rotation and interlayer coordination
In presence of the ditrigonal rotation, six Ob basal oxygen atoms are closer to (inner Ob oxygens) and six are farther from (outer Ob oxygens) the interlayer cation I. By increasing α from its minimum value (0°, absence of distortion) to its maximum value (30°), the distance d of the Ob oxygen atoms from the perpendicular to the layer containing I, expressed as a function of the tetrahedral edge l, changes (Fig. 15) from d = l to dinner = l/31/2 and douter = 2l/31/2. In other words, while the Ob(inner) atoms decrease their undistorted distance from the perpendicular by (dinner – d)/d = -42.3%, the Ob(outer) atoms move very little and increase the same distance by (douter – d)/d = 15.5%. Therefore, the consequence of the ditrigonal rotation on the interlayer coordination is dramatic. Precisely:
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Increase of the ditir gonal or at tion . An increase of α (e.g., under compression) causes the difference between I-Ob(outer) and I-Ob(inner) distances to become larger. Thus both the trigonal antiprismatic (ideally octahedral) coordination (subfamily A polytypes such as 1M, 2M1 and 3T ) and the trigonal prismatic coordination (subfamily B polytypes such as 2O, 2M2 and 6H) become more dominant. As far as the antiprismatic coordination is a stabilizing factor, the increase of the ditrigonal rotation at high P should not weaken a structure. Decrease of eht ditir gonal or at it on . A decrease of α (e.g., under expansion) causes the difference between I-Ob(outer) and I-Ob(inner) distances to become smaller: the interlayer coordination approaches the hexagonal prismatic coordination for both polytype subfamilies A and B. As far as the antiprismatic coordination is a stabilizing factor, the decrease of the ditrigonal rotation at high T should weaken a structure.
Note that, under the combined causes which influence the I-Ob distances and discussed above, overall decreases of the I-Ob(inner) has been reported in high-temperature refinements of micas (Catti et al. 1989, Guggenheim et al. 1987; Ivaldi et al. 1998; Russel and Guggenheim 1999; Takeda and Morosin 1975). Effective coordination number (ECoN)
ECoN is a useful generalization of the classical definition of coordination number (number of anions in contact with a cation); it considers the lengths of the bonds (Hoppe 1979). For a cation X establishing R(X)i bonds with equal anions, ECoN is defined as [Nespolo et al. (1999a) on the basis of Hoppe et al. (1989)]: ECoN(X) = Σiexp{1 – [R(X)i /R(X)av]6};
(1)
R(X)av represents a weighted average bond distance for the coordination polyhedron around the cation X and is defined as Rav(X) =Σi R(X)iexp{1 – [R(X)i /R(X)min]6}/Σiexp{1 – [R(X)i /R(X)min]6}
(2)
R(X)min being the shortest R(X)i distance (the exponent 6 is valid when the anion is O2-). The sum over i is in principle extended to all the oxygen atoms; practically, however, note that the contribution falls to zero as R(X)i exceeds R(X)av [Eqn. (1)] or R(X)min [Eqn. (2)] by more than about 20%. ECoN defined by Equation (1) is a non-integer number approaching the Pauling's coordination number (i.e., the number of first neighbor anions) and equal to it for regular coordination polyhedra where R(X)i = R(X)av = R(X)min. Recently the method has been extended to distorted and hetero-ligand polyhedra (Nespolo et al. 2001).
A correlation between ECoN calculated for the interlayer cation I and the ditrigonal rotation α can intuitively be expected from the discussion of this section. Such correlation has been investigated by Weiss et al. (1992) and found to be quite regular (Fig. 16). In fact, ECoN for I smoothly decreases from about 12 (null ditrigonal rotation) to 6 (for a ditrigonal rotation higher than about 16°). CONCLUSIONS
Even if the discussion is still open, the basic features of the mica structure reasonably explain a wide range of the micas properties, from polytypism and twinning (Nespolo et al. 1997b; Nespolo et al. 1999b; details in Nespolo and Durovic', this volume) to chemical variability and stability in a range of geological conditions. The matter of stability fields is of paramount interest in Earth sciences and concerns both the occurrence of a polytype more than others and the capability for a mica of existing at high P and/or T values. For a list of references to occurrences of associated polytypes of
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Figure 16. Correlation between ECoN (effective coordination number) of the interlayer cation I and the ditrigonal rotation α in micas. [Modified after Weiss et al. (1992)].
micas; cf. Ivaldi et al. (2001b) and Ferraris et al. (2001c). The following basic structural features play a role in structure stabilization. 1. The type of interlayer coordination, which is connected with the parity n of rotation (n × 60°) between adjacent M layers, justifies a wider occurrence of family A polytypes (not limited to the MDO polytypes 1M, 2M1, 3T) which show even rotations only. The amount of rotation between two adjacent M layers influences the relative stability of polytypes independently of the parity n. 2. A phengitic composition favors stability at high P/T values because the aluminoceladonitic substitution provides a good T/O fit and consequent small ditrigonal rotation α at high P/T also. 3. The argument of a small ditrigonal rotation α cannot stand alone (cf. the trioctahedral micas). It becomes effective as stabilizing factor at high P for dioctahedral micas because other aspects concur as: (1) the O-H bond points towards the vacant octahedral site and assumes a direction (almost) parallel to (001) thus minimizing its interaction with the interlayer cation; (2) the presence both of a pair of independent tetrahedral sites and of independent M2 and M3 octahedral sites in the 3T polytype which, with phengitic composition, is the most stable form of mica at high P/T. 4. In trioctahedral micas, the O-H bond is pushed away from the (001) Ob plane and tends to lie along the perpendicular to this plane. Thus, some repulsive interaction with the interlayer cation occurs that weaken the stability of the structure. However, in oxidized (e.g., Ohta et al. 1982) or fluorinated micas (e.g., Takeda et al. 1971) this repulsion is reduced proportionally to the O2- → OH- or F- → OH- substitution. This is particularly evident in synthetic fluoro-micas (e.g., Takeda and Burnham 1969). 5. Geometric effects connected with the variation of the ditrigonal rotation α and of the interlayer separation are as important as energetic factors in determining the variation of the interlayer bonds under compression, dilatation and effects in α that produce changes in the coordination (i.e., sinking effect) of the interlayer cation. 6. The presence of two types of M layer, M1 and M2, may play a role in the growth of long period polytypes (Nespolo 2001).
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Ferraris & Ivaldi APPENDIX I:
MICA STRUCTURE AND POLYSOMATIC SERIES Layer silicates as members of modular series?
The T and O sheets occurring in the mica structure are present in all layer silicates (phyllosilicates); the entire M (TOT) mica layer is present in 2:1 layer silicates only. The description of layer silicates is often given by emphasizing different stacking of T and O sheets, even if an explicit discussion in terms of modular series is absent from the literature. Several types of modular series have been defined (Makovicky 1997): Polysomatic (homologous accretional) series. The crystal structures of the members of these series are based on the same modules. Biopyriboles are a well known example (Thompson 1978). Merotyip c series . Both common and peculiar modules are present in the crystal structures of the members. The case of bafertisite-derivative structures, belonging to the heterophyllosilicate group of titanosilicates, is described below. Plesiotypic series. The crystal structures of the members of these series are based on modules which have common features but may contain additional peculiar details. The family of serpentine-like structures (lizardite, chrysotile, antigorite, carlosturanite) is an example reported by Makovicky (1997). The members of this plesiotypic series are based on variously curled, reversed and/or interrupted TO (serpentine) layers. From a topologic viewpoint, namely without considering the actual composition of the M layer but only that of the interlayer, the following modules are necessary to obtain all the structures of the 2:1 layer silicates: 1. three types of M layer: homo- meso- and hetero-octahedral layers; 2. interlayer modules of different chemical and structural nature ranging from single cations (micas), to octahedral sheets (chlorites) and a mixture of layers and various chemical groups (interstratified clay minerals). The entire group of layer silicates could therefore be classified as a mero-plesiotypic series in the sense that both structural details and nature of the building modules varies. Mica modules in polysomatic series The M mica module occurs not only in biopyriboles, chlorites and interstratified clay minerals as mentioned above, but also in some other polysomatic series. Because these series represent different possibilities for the presence of mica-like structures in minerals, it seems useful to shortly describe some of them. The heterophyllosicate polysomatic series
By analogy with phyllosilicates, a group of titanium silicates whose structures are based on TOT-like layers have been called heterophyllosilicates (Ferraris et al. 1997). In these structures, rows of Ti(Nb)-octahedra (hereafter, Ti-octahedra) are introduced in a T sheet along the direction which is parallel to a pyroxene tetrahedral chain (Fig. 17). HOH layers are thus obtained where H stands for hetero to indicate the presence of the Tioctahedra in a sheet corresponding to the T sheet of the layer silicates. Because the edges of the Ti-octahedra and Si-tetrahedra have close lengths dimensions, the insertion of the octahedra in a T sheet does not produce strain. As summarized by Ferraris (1997), three types of HOH layers (Fig. 18) are known so far. Bafertisite-like ( HOH)B layer . A bafertisite module B, I2Y4[Ti2(O)4Si4O14](O,OH)2, is one-to-one intercalated with a one-chain-wide mica-like module
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Figure 17. Different types of H sheets which are obtained by periodically introducing Ti-octahedra (light gray) in a tetrahedral T sheet: bafertisite-like (B), astrophyllite-like (A) and nafertisite-like (N) H sheets.
M,IY3[Si4O10](O,OH)2 (I and Y represent interlayer cations and octahedral cations, respectively). Astrophyllite-like ( HOH)A layer . With respect to the bafertisite-like layer, a second one-chain-wide mica-like module M is present between two bafertisite-like modules. Nafertisite-like ( HOH)N layer. With respect to the bafertisite-like layer, a second and a third one-chain-wide mica-like module M are present between two bafertisite-like modules [or, a second M module is added to (HOH)A]. The series. Bafertisite (Guan et al. 1963, Pen and Shen 1963, Rastsvetaeva et al. 1991), astrophyllite (Woodrow 1967) and nafertisite (Ferraris et al. 1996) are members of a polysomatic series BmMn which is based on B (bafertisite-like) and M (mica-like) modules and has a general formula I2+nY4+3n[Ti2(O)4Si4+4nO14+10n](O,OH)2+2n. In the formula, atoms belonging, even in part, to the H sheet are shown in square brackets; for n = 0, I = Ba and Y = (Fe,Mn) the formula of bafertisite is obtained. The heterophyllosilicates have also been described by using differently defined B and M modules (Christiansen et al. 1999), a possibility which is not rare in modular crystallography (Merlino 1997).
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Figure 18. Bafertisite-like (B), astrophyllite-like (A) and nafertisite-like (N) HOH layers.
Seidozerite derivatives . The bafertisite-like module (HOH)B is the building fragment of several layer titanosilicates (seidozerite or bafertisite derivatives; Ferraris et al. 1997) where only the interlayer content varies. These titanosilicates are B1M0 members of the heterophyllosilicate series with a peculiar interlayer content. They are represented by the formula XY4[Ti2(O)4Si4O14](O,OH)2, where X indicates the interlayer content which may consist of H2O, tetrahedral anions and cations. All the seidozerite derivatives are based on a common two-dimensional (sub)cell with a ~ 5.4 Å and b ~ 7 Å, whereas the value of the stacking c parameter depends on the nature of X. The set of seidozerite derivatives forms a merotype [or mero-plesiotype (Ferraris 2001d) series]. In the seidozerite derivatives, (HOH)B represents the common building layer and the X inter-layer content is variable. The palysepiole polysomatic series The palysepiole polysomatic series PpSs (Ferraris et al. 1998) includes minerals whose structures contain one or both the types of TOT ribbons (modules) which are present in palygorskite and sepiolite; these ribbons are reminiscent of the TOT modules occurring in amphiboles (Fig. 19). The two modules are: • P = Ax(Y2+,Y3+,)5[Si8O20(OH)2]·nH2O (palygorskite module; in palygorskite Y is mainly Mg and n ∼ 8); • S = Ax(Y2+,Y3+,)8[Si12O30(OH)4]·mH2O (sepiolite module; in sepiolite Y is mainly Mg and m ∼ 12).
The structures of sepiolite and palygorskite are based on chess-board arranged [001] TOT ribbons and intercalated channels. Each ribbon occurring in sepiolite, (TOT)S, is six pyroxene-chain wide and 50% wider than that occurring in palygorskite, (TOT)P, which in turn is four pyroxene-chain wide. A variable amount of alkali A cations and water molecules occurs in the channels. The third known mineral of the group, kalifersite, corresponds to the member P1S1 of the series and has formula K5(Fe73+,2)[Si20O50(OH)6]·12H2O. Its structure is based on an alternation, in the [010] direction, of (TOT)S and (TOT)P ribbons. Each of two types of [001] channels, which occur within the mixed palygorskite/sepiolite framework, is filled with a different strip of alkali-octahedra (not shown in Fig. 19).
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Figure 19. View along [001] of the crystal structures of palygorskite (P), kalifersite (K) and sepiolite (S). Kalifersite is based on a chessboard arrangement of ribbons (TOT)S (sepiolite) and (TOT)P (palygorskite). Cations and water molecules occurring in the channels are not shown.
Other modular structures
Guggenheim and Eggleton (1987, 1988) described some modular 2:1 layer silicates in terms of fragments of the M (TOT) mica module, with or without interlayer cations. The modularity of these silicates originates by the inversion of part of the tetrahedral linkage. On the basis of the inversion fragments, the basic TOT layers may form either islands (e.g., stilpnomelane and zussmanite) or strips (where the octahedral sheets remains continuous as in ganophyllite and minnesotaite, or discontinuous as in the above mentioned palysepioles). Conclusions
TOT modules occur in different mineral structures and the following examples have been discussed in this Appendix. • Infinite two-dimensional layers occur in micas, talc, pyrophyllite, chlorites and interstratified clay minerals. • Slices of the mica (talc) structure cut perpendicularly to the layer are present in amphiboles and palysepioles; they are inclined on the layer in heterophyllosilicates, according to the description of Ferraris et al. (1996). It seems reasonable to connect variety and frequency of occurrence with structural stability. The wide range of conditions under which the TOT layer is stable on its own occurs in talc and pyrophyllite, built up by this layer only. The TOT layers are even able to survive through reactions generating other minerals [cf. Baronnet (1997) and Buseck (1992) and references therein] including other micas (cf. Ferraris et al. 2001a). Probably features other than crystal chemistry concur to explain the wide distribution and persistence of variously sliced mica modules. The high symmetry of the mica modules could be a key feature, in the sense that it favors different stacking and connections with other modules both of the same kind and different nature. Mica polytypes and twins are clear examples of symmetry-assisted structures. The flexibility of
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the T and O sheets and of the TOT layer as a whole has been widely discussed in this chapter. This flexibility is usefully exploited to match mica modules with other modules, a role which can be played also by closeness of polyhedral dimensions (cf. the heterophyllosilicates case). APPENDIX II: OBLIQUE TEXTURE ELECTRON DIFFRACTION (OTED)
In addition to powder X-ray (Bailey 1988) and single-crystal X-ray diffraction methods, electron diffraction is widely used to characterize micas. In particular, the oblique-texture electron diffraction (OTED) method here described has been used to obtain important results from micro grained samples of layer silicates (cf. references below). A part some earlier sporadic papers, the OTED method was developed by Vainshtein (1956, 1964), following Pinsker (1953), and further improved during following years (Zvyagin 1967, Zvyagin et al. 1979, Vainshtein et al. 1992, Zvyagin et al. 1996). OTED has been used to obtain diffracted intensities for solving crystal structures (cf. quoted papers and Zhukhlistov et al. 1997), spite of the dynamic effects affecting the electron diffraction intensities. In the present context, however, we are interested in the application of OTED for polytype identification and follow Zvyagin (1967). The method is suitable for materials that show a very good cleavage where thin mounts can be prepared so that the cleavage planes are more or less perfectly parallel to the plane of the mount.
Figure 20. Cylindrical reciprocal lattice generated by rotation of reciprocal lattice rows around c* (left) and its elliptical intersection with the Ewald sphere (right) which, in the case of electron diffraction (only small Bragg angles are possible), can be approximated by a plane. At right, the effect of a non perfect planarity of the sample is shown by substituting the circles of the left figure with tori; the intersection of a torus with the Ewald sphere is an ‘arc’ (Fig. 22). Modified after Zvyagin (1967).
Let us suppose that the exploited cleavage is {001} and the cleavage lamellae are textured in a planar mount so that their orientations have a common perpendicular to (001), i.e., around c*. Under these conditions, each reciprocal row hkl (h and k are fixed) parallel to c* describes a so called cylindrical reciprocal lattice (Fig. 20). The nodes with the same hk indices are at a distance [(h/a)2 + (k/b)2]1/2 from c*. All the circles have their center on the rotation axis but not at the corresponding 00l node, except when the lattice is orthogonal. The projection of these circles on the ab plane is shown in Figure 21 together with the hk
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Figure 21. Distribution in the ab plane of the hkl reflections on concentric circles with radius [(h/a)2 + (k/b)2]1/2 in the case of a lattice with b = a31/2, as occurs in layer silicates. Modified after Zvyagin (1967).
indices for the case of those layer silicates where the relations b = a√3 holds. These circles correspond to the orbits which are defined in Figure 16 in the chapter by Nespolo and Durovic (this volume) in connection with the S, D and X classification of the rows (cf. below). However, whereas the orbits are the loci containing S, D or X rows with their individual nodes, in the cylindrical lattice one circle carries intensity contributed by all hkl nodes falling on that circle. Because of the small electron wavelength (∼10-2 Å) and a substantial diffraction intensity limited to quite small Bragg angles around the incident beam, the Ewald sphere can locally be approximated by a plane. For a given inclination angle φ between the incident beam and the plane of the mount, the intersection of the Ewald plane with the cylindrical lattice results in a series of ellipses which represent the loci of the diffracting nodes (Figs. 20, 22). The length of the minor axis bhk of each ellipse is independent of the inclination φ angle and corresponds to the intersection of the ellipse with the ab plane; the major axis ahk is given by bhk/cosφ. Α plane detector behind the sample, set at a distance L from the sample and parallel to it, is parallel to the diffracting ellipses and registers an undistorted image of the position of the diffracting nodes together with their diffraction intensities. The scale factor between the reciprocal lattice ellipse and the image ellipse is Lλ, where λ is the electron wavelength. For each hk image ellipse, the distance of a diffraction spot from the minor axis is Dhkl = (Lλ/sinφ)(ha*cosβ* + kb*cosα* + lc*) = hp + ks + lq
and the length of the minor axis is
(3)
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Figure 22. Theoretical OTED diffraction pattern [for a 1M mica polytype for which b = a31/2] showing the spots (arcs) arranged on ellipses. According to its order, each ellipse contains reflections with the type of indices shown in this Figure and in Table 3. The ‘spots’ are represented as arcs because in practice the lamellae of the sample can be slightly inclined relative to the average plane of the sample (Fig. 20). The distance of each ‘spot’ from the trace O of the incident beam (origin of the lattice) corresponds to d*(hkl); the case of d*(112) is shown in the figure. Modified after Zvyagin (1967).
bhk = Lλ(h2/a2 + k2/b2)1/2.
(4) 1/2
Because in the layer silicates b = a3 , Equation (4) becomes bhk = Lλ(3h2 + k2)1/2/b.
(5)
Note that in the OTED method the overlap of reflections is limited to those reflections with the same value of Dhkl (Eqn. 3) and belonging to the same ellipse; thus a dramatic improvement is obtained relative to the classic powder diffraction methods. Note also that, in the most general case, the reflections with l = 0 do not lie on the minor axis because the a*b* and ab planes are not coincident. These two planes coincide in orthogonal crystal systems where Equations (3) and (4) become, in the order, Dhkl = (Lλ/sinφ)lc* = lq
(6)
and
bhk = Lλ[h2(a*)2 + k2(b*)2]1/2. Some simplification is also obtained in the monoclinic case (α* = 90°) where Dhkl = (Lλ/sinφ)(ha*cosβ* + lc*) = hp + lq.
(7) (8)
Each ellipse is characterized by 3h2 + k2 = constant. Table 3 reports the absolute values of hk which occur when b = a31/2. According to their h and k values, the rows parallel to c* have been classified in the following way (Nespolo et al. 2000; cf. below and Nespolo and Durovic, this volume, for details). S or sw . These are the rows with h = 0(mod3) and k =0(mod3); they are family rows in the Pauling model and are common to all polytypes of the same family.
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Table 3. Absolute values allowed for the hk indices of the spots appearing, in an OTED pattern, on the ellipses with the shown order. Ellipse order
|h||k|
3h2 + k2
1
02, 11
4
Type of ellipse
Ellipse order
|h||k|
3h2 + k2
7
17, 35, 42
52
Type of ellipse
X
X 2
13, 20
12
D
8
08, 44
64
3
04, 22
16
X
9
28, 37, 51
76
X
4
15, 24, 31
28
X
10
19, 46, 53
84
D
5
06, 33
36
S
11
0.10, 55
100
X
6
26, 40
48
D
12
39, 60
108
S
X
D or sw . These are the rows with h ≠ 0(mod3) and k = 0(mod3). They may identify the general symmetry principle on which a polytype is comprised, distinguishing subfamily A polytypes (for which D rows are family rows in the trigonal model), subfamily B polytypes (for which D rows are again family rows in the trigonal model ) and mixed-rotations polytypes (for which D rows are non-family rows in both the Pauling and the trigonal model). X or sw . These are the rows with h ≠ 0(mod3); they are non-family rows in both the Pauling and trigonal model and are characteristic of each polytype.
Because of the condition 3h2 + k2 = constant, which constrains the values of h and k, each ellipse of OTED bears only reflections corresponding to one type of row (S, D or X). Consequently, the ellipses themselves can be usefully classified into S, D and X ellipses (Table 3). By using the equations shown above, indexing and cell parameters of the reflections registered on an OTED pattern can be obtained. Whereas in two-dimensional X-ray diffraction (XRD) studies (e.g., precession method), reciprocal central planes containing two types of rows (SD planes and SX planes) are recorded, in OTED each ellipse contains only one type of the S, D or X rows, as said above. Although the recording technique is different, the general principle of polytype identification is the same. Both techniques can identify polytypic and structural features upon inspection of specific rows (XRD) or ellipses (OTED), at least in the homo-octahedral approximation. This identification can be obtained not only for MDO polytypes of micas, but also for non MDO (inhomogeneous) polytypes [cf. examples in Borutsky et al. (1987) and Zhukhlistov et al. (1990, 1993)]. For polytypes with longer period (longer stacking sequence), a simplified procedure to analyze the intensity distribution is necessary, but the general principle of polytype (and twin) identification is the same, and consists in the inspection of D and X rows. The OTED patterns can be analyzed as follows in terms of S, D, and X rows (cf. Nespolo and Durovic, this volume, for details). Ellipses of 2+2n order . The D character of the rows belonging to the 2nd (as well as 6th, 10th) ellipse can discriminate among subfamily A polytypes, subfamily B polytypes and mixed-rotation polytypes by observing the number N of reflections in the c*1 repeat (about 0.1 Å -1): one reflection for subfamily A polytypes, two equally spaced reflections for subfamily B polytypes, N > 2 reflections for mixed-rotation
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polytypes. When indexing the reflections in the unit cell of a polytype, they correspond to the following presence criteria for the index l: 1M (no systematic absences of l), 2M1 (l = 2n), 3T (l = 3n), 2M2 (no absences of l), 2O (no absences of l), 6H (l = 3n), mixed-rotation (no absences of l). X-type ellipses . The intensity distribution in the 1st ellipse (as well as other X-type ellipses) is typical of each mica polytype. Knowing the symmetry principle (subfamily A or B, or mixed-rotation, revealed by the 2nd ellipse) helps to obtain the stacking sequence from the intensity distribution in the 1st ellipse. Because the X rows are non-family rows in both the Pauling and trigonal model, the computation of the intensities in the X-type ellipses, to be compared with those experimentally measured, can be performed even in the simplest Pauling model. Other ellipses . The intensity distribution on the 6th and 7th ellipses (and on the 2nd ellipse in some cases) can distinguish between di- and trioctahedral phyllosilicates (Zvyagin 1993). The ditrigonal rotation α is most effective in influencing the diffraction intensities of the 2nd ellipse (Weiss and Wiewióra 1986; Weiss and Rieder 1997). Rieder and Weiss (1991) have extended the method to XRD. In this case, however, because of the large curvature of the Ewald sphere (wavelength ∼1 Å) the reflections are no longer on ellipses and concentrated at low diffraction angles. Presumably, a synchrotron source could provide sufficiently short wavelengths to make XRD closer to OTED. ACKNOWLEDGMENTS
Discussions with M. Nespolo (University of Nancy) greatly influenced this chapter. Useful suggestions came from S. V. Soboleva (IGEM, Moscow). Constructive comments have been provided by the referees, S. Guggenheim and M. Rieder. Research was financially supported by MURST (‘Layer silicates: Crystal chemical, structural and petrologic aspects’ project) and CNR (‘Igneous and metamorphic micas’ project). REFERENCES Abbott RN Jr, Burnham CW (1988) Polytypism in micas: A polyhedral approach to energy calculations. Am Mineral 73:105-118 Amisano Canesi A, Chiari G, Ferraris G, Ivaldi G, Soboleva SV (1994) Muscovite- and phengite-3T: Crystal structure and conditions of formation. Eur J Mineral 6:489-496. Appelo CA (1978) Layer deformation and crystal energy of micas and related minerals. I. Structural model for 1M and 2M1 polytypes. Am Mineral 63:782-792 Appelo CA (1979) Layer deformation and crystal energy of micas and related minerals. II. Deformation of the coordination units. Am Mineral 64:424-431 Arnold H (1996) Transformations in crystallography. In Th Hahn (ed) International Tables for Crystallography, Vol A. Kluwer Academic Publishers, Dordrecht, The Netherlands, 69-80 p Bailey SW (1975) Cation ordering and pseudosymmetry in layer silicates. Am Mineral 60:175-187 Bailey SW (1980) Structures of layer silicates. In GW Brindley, G Brown (eds) Crystal Structures of Clay Minerals and Their X-ray Identification. Mineralogical Society, London, 1-123 p Bailey SW (1984a) Classification and structures of the micas. Rev Mineral 13:1-12 Bailey SW (1984b) Crystal chemistry of the true micas. Rev Mineral 13:13-61 Bailey SW (1984c) Review of cation ordering in micas. Clays Clay Minerals 32:81-92 Bailey SW (1988) X-ray diffraction identification of the polytypes of mica, serepentine, and chlorite. Clays Clay Minerals 36:193-213 Baronnet A (1997) Equilibrium and kinetic processes for polytype and polysome generation. In S Merlino (ed) Modular Aspects of Minerals. Eur Mineral Union Notes in Mineralogy 1:119-152 Belov NV (1949) The twin laws of micas and micaceous minerals. Mineral sb L’vovsk geol obva pri univ 3:2940 (in Russian)
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Ferraris G, Khomyakov AP, Belluso E, Soboleva SV (1997) Polysomatic relationships in some titanosilicates occurring in the hyperagpaitic alkaline rocks of the Kola Peninsula, Russia. Proc 30th Int’l Geol Congr (Mineralogy vol) 16:17-27 Ferraris G, Khomyakov AP, Belluso E, Soboleva SV (1998) Kalifersite, a new alkaline silicate from Kola Peninsula (Russia) based on a palygorskite-sepiolite polysomatic series. Eur J Mineral 10:865-874 Filut MA, Rule AC, Bailey SW (1985) Crystal structure refinement of anandite-2Or, a barium- and sulfurbearing trioctahedral mica. Am Mineral 70:1298-1308 Franzini M (1969) The A and B mica layers and the crystal structure of sheet silicates. Contrib Mineral Petrol 21:203-224 Giuseppetti G, Tadini C (1972) The crystal structure of 2O brittle mica: Anandite. Tschermaks mineral petrogr Mitt 18:169-184 Griffen DT (1992) Silicate Crystal Chemistry. Oxford University Press, Oxford, UK Guan Ya S, Simonov VI, Belov NV (1963) Crystal structure of bafertisite, BaFe2TiO[Si2O7](OH)2. Dokl Acad Nauk SSSR 149:1416-1419 (in Russian) Guggenheim S (1981) Cation ordering in lepidolite. Am Mineral 66:1221-1232 Guggenheim S, Bailey SW (1977) The refinement of zinnwaldite-1M in subgroup symmetry. Am Mineral 62:1158-1167 Guggenheim S, Chang YH, Koster Van Groos AF (1987) Muscovite dehydroxylation: high-temperature studies. Am Mineral 72:537-550 Guggenheim S, Eggleton RA (1987) Modulated 2:1 layer silicates: Review, systematics, and predictions. Am Mineral 72:724-738 Guggenheim S, Eggleton RA (1988) Crystal chemistry, classification, and identification of modulated layer silicates: Review, systematics, and predictions. Rev Mineral 19:675-725Guidotti CV, Sassi FP, Comodi P, Zanazzi PF, Blencoe JG (2000) The contrasting responses of muscovite and paragonite to increasing pressure: Petrological implications. Can Mineral 38:707-712 Güven N (1971) Structural factors controlling stacking se quences in dioctahderal micas. Clays Clay Minerals 19:159-165 Hazen RM, Burnham CW (1973) The crystal structure of one layer phlogopite and annite. Am Mineral 58:889-900 Hazen RM, Finger LW (1982) Comparative Crystal Chemistry. John Wiley and Sons, New York Hoppe R (1979) Effective coordination numbers (ECoN) and mean fictive ionic radii (MEFIR). Z Kristallogr 150:23-52 Hoppe R, Voigt S, Glaum H, Kissel J, Müller HP, Bernet K (1989) A new route to charge distribution in ionic solids. J Less-Common Metals 156:105-122 Hybler J, Petrícek V, Durovic S, and Smrcok L (2000) Refinement of the crystal structure of cronstedtite1T. Clays Clay Minerals 48:331-338 Ivaldi G, Ferraris G, Curetti N (2001a) Crystal structure paths to a phengite geobarometer? EUG XI, J Conf Abstr 6:540 Ivaldi G, Ferraris G, Curetti N, Compagnoni R (2001b) Coexisting 3T and 2M1 polytypes in a phengite from Cima Pal (Val Savenca, western Alps): Chemical and polytypic zoning and structural characterisation. Eur J Mineral 13:1025-1034 Ivaldi G, Pischedda V, Ferraris G (1998) Evoluzione termica di una flogopite 1M contenente bario. XXVIII Meeting Italian Crystallogr Assoc Abstr M3-15 Khomyakov AP (1995) Mineralogy of Hyperagpaitic Alkaline Rocks. Clarendon Press, Oxford, UK Lee JH, Guggenheim S (1981) Single crystal X-ray refinement of pyrophyllite-1Tc. Am Mineral 66: 350-357 Lin IC, Guggenheim S (1983) The crystal structure of a Li,Be-rich brittle mica: A dioctahedraltrioctahedral intermediate. Am Mineral 68:130-142 Makovicky E (1997) Modularity—different approaches. In S Merlino (ed) Modular Aspects of Minerals. Eur Mineral Union Notes in Mineralogy 1:315-343 Massonne HJ, Schreyer W (1986) High-pressure synthesis and X-ray properties of white micas in the system K2O-Mg-Al2O3-SiO2-H2O. N Jahrb Mineral Abh 153:177-215 Massonne HJ, Schreyer W (1989) Stability field of the high-pressure assemblage talc + phengite and two new phengite barometers. Eur J Mineral 1:391-410 McCauley JW, Newnham RE (1971) Origin and prediction of ditrigonal distortions in micas. Am Mineral 56:1626-1638 Mellini M (1982) The crystal structure of lizardite 1T: Hydrogen bonds and polytypism. Am Mineral 67: 587-598 Mellini M, Zanazzi PF (1989) Effects of pressure on the structure of lizardite-1T. Eur J Mineral 1:13-19 Merlino S (ed) (1997) Modular Aspects of Minerals. Eötvös University Press / European Mineralogical Union, Budapest, 448 p
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Mizota T, Kato T, Harada K (1986) The crystal structure of masutomilite, Mn-analogue of zinnwaldite. Mineral J (Japan) 13:13-21 Nespolo M (2001) Perturbative theory of mica polytypism. Role of the M2 layer in the formation of inhomogeneous polytypes. Clays Clay Minerals 49:1-23 Nespolo M, Ferraris G (2001) Effects of the stacking faults on the calculated electron density of mica polytypes —The Durovic effect. Eur J Mineral 13:1035-1045 Nespolo M, Ferraris G, Ivaldi G, Hoppe R (2001) Charge Distribution as a tool to investigate structural details. II. Extension to hydrogen bonds, distorted and hetero-ligand polyhedra. Acta Crystallogr B57:652-664 Nespolo M, Ferraris G, Ohashi H (1999a) Charge Distribution as a tool to investigate structural details: Meaning and application to pyroxenes. Acta Crystallogr B55:902-916 Nespolo M, Ferraris G, Takeda H (2000) Twins and allotwins of basic mica polytypes: Theoretical derivation and identification in the reciprocal space. Acta Crystallogr A56:132-148 Nespolo M, Ferraris G, Takeda H, Takéuchi Y (1999b) Plesiotwinning: oriented crystal associations based on a large coincidence-site lattice. Z Crystallogr 214:378-382 Nespolo M, Takeda H, Ferraris, G (1997a) Crystallography of mica polytypes. In S Merlino (ed) Modular Aspects of Minerals. Eur Mineral Union Notes in Mineralogy 1:81-118 Nespolo M, Takeda H, Ferraris, G (1998) Representation of the axial settings of mica polytypes. Acta Crystallogr A54:348-356 Nespolo M, Takeda H, Ferraris G, Kogure T (1997b) Composite twins of 1M mica: Derivation and identification. Mineral J (Japan) 19:173-186 Nespolo M, Takeda H, Kogure T, Ferraris G (1999c) Periodic intensity distribution (PID) of mica polytypes: Symbolism, structural model orientation and axial settings. Acta Crystallogr A55:659-676 Newnham RE (1961) A refinement of the dickite structure and some remarks on polymorphism of the kaolin minerals. Mineral Mag 32:683-704 Nickel HN, Grice JD (1998) The IMA commission on new minerals and mineral names: Procedures and guidelines on mineral nomenclature, 1998. Can Mineral 36:913-926 Ohta T, Takeda H, Takéuchi Y (1982) Mica polytypism: Similarities in the crystal structures of coexisting 1M and 2M1 oxybiotite. Am Mineral 67:298-310 Pauling L (1930) The structure of micas and related minerals. Proc Nat Acad Sci (USA) 16:123-129 Pavese A, Curetti N, Ferraris G, Ivaldi G, Russo U, Ibberson R (2002) Deprotonation and order-disorder reactions as a function of temperature in a phengite 3T (Cima Pal, western Alps) by neutron diffraction and Mössbauer spectroscopy. Eur J Mineral 13, submitted Pavese A, Ferraris G, Pischedda V, Fauth F (2001) M1-site occupancy in 3T and 2M1 phengites by low temperature neutron powder diffraction: Reality or artefact? Eur J Mineral 13:1071-1078 Pavese A, Ferraris G, Pischedda V, Ibberson R (1999a) Tetrahedral order in phengite 2M1 upon heating, from powder neutron diffraction, and thermodynamic consequences. Eur J Mineral 11:309-320 Pavese A, Ferraris G, Pischedda V, Mezouar M (1999b) Synchrotron powder diffraction study of phengite-3T from the Dora Maira massif: P-V-T equation of state and petrological consequences. Phys Chem Minerals 26:460-467 Pavese A, Ferraris G, Pischedda V, Radaelli P (2000) Further study of the cation ordering in phengite-3T by neutron powder diffraction. Mineral Mag 64:11-18 Pavese A, Ferraris G, Prencipe M, Ibberson R (1997) Cation site ordering in phengite-3T from the Dora Maira massif (western Alps): A variable-temperature neutron powder diffraction study. Eur J Mineral 9:1183-1190 Pen ZZ, Shen TC (1963) Crystal structure of bafertisite, a new mineral from China. Scientia Sinica 12:278-280 (in Russian) Pinsker ZG (1953) Electron Diffraction. Butterworth, London Radoslovich EW (1960) The structure of muscovite, KAl2(Si3Al)O10(OH)2. Acta Crystallogr 13:919-932 Radoslovich EW (1961) Surface symmetry and cell dimensions of layer-lattice silicates. Nature 191:67-68 Radoslovich EW, Norrish K (1962) The cell dimensions and symmetry of layer-lattice silicates. I. Some structural considerations. Am Mineral 47:599-616 Rastsvetaeva RK, Tamazyan RA, Sokolova EV, Belakovskii DI (1991) Crystal structures of two modifications of natural Ba, Mn-titanosilicate. Sov Phys Crystallogr 36:186-189 Rieder M (2001) Mineral nomenclature in the mica group: the promise and the reality. Eur J Mineral 13:1009-1012 Rieder M, Cavazzini G, D’yakonov YuS, Frank-Kamenetskii VA, Gottardi G, Guggenheim S, Koval’ PV, Müller G, Neiva AMR, Radoslowich EW, Robert JL, Sassi FP, Takeda H, Weiss Z, Wones DR (1998) Nomenclature of the micas. Can Mineral 36:905-912 Rieder M, Hybler J, Smrcok L, Weiss Z (1996) Refinement of the crystal structure of zinnwaldite 2M1. Eur J Mineral 8:1241-1248
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Rieder M, Weiss Z (1991) Oblique-texture photographs: More information from powder diffraction. Z Kristallogr 197:107-114 Russel RL, Guggenheim S. (1999) Crystal structures of near-end-member phlogopite at high temperature and heat-treated Fe-rich phlogopite: The influence of the O, OH, F site. Can Mineral 37:711-720 Sassi PF, Guidotti C, Rieder M, De Pieri R (1994) On the occurrence of metamorphic 2M1 phengites: Some thoughts on polytypism and crystallization conditions of 3T phengites. Eur J Mineral 6:151-160 Schmidt MW, Dugnani M, Artioli G (2001) Synthesis and characterization of white micas in the join muscovite-aluminoceladonite. Am Mineral 86:555-565 Shannon RD (1976) Revised ionic radii and systematic studies of interatomic distances in halides and oxides Acta Crystallogr A32:751-767 Slade PG, Schultz PK, Dean C (1987) Refinement of the ephesite structure in C1 symmetry. N Jahrb Mineral Monatsh, p 275-287 Smith JV, Yoder HS (1956) Experimental and theoretical studies of the mica polymorphs. Mineral Mag 31:209-235 Smyth JR, Jacobsen SD, Swope RJ, Angel RJ, Arlt T, Domanik K, Holloway JR (2000) Crystal structures and compressibility of synthetic 2M1 and 3T phengite micas. Eur J Mineral 12:955-963 Soboleva SV (1987) Mica polytypes: Theoretical and applied aspects. Mineral J (Ukraine) 9:26-41 Takeda H, Burnham CW (1969) Fluor-polylithionite: A lithium mica with nearly hexagonal (Si2O5)2- ring. Mineral J (Japan) 6:102-109 Takeda H, Haga N, Sadanaga R (1971) Structural investigation of polymorphic transition between 2M2-, 1M-lepidolite and 2M1 muscovite. Mineral J (Japan) 6:203-215 Takeda H, Morosin B (1975) Comparison of observed and predicted parameters of mica at high temperature. Acta Crystallogr B31:2444-2452 Takéuchi Y, Haga N (1971) Structural transformation of trioctahedral sheet silicates. Slip mechanism of octahedral sheets and polytypic changes of micas. Mineral Soc Japan Spec Paper 1:74-87 (Proc IMAIAGOD Meetings '70, IMA Vol) Takéuchi Y, Sadanaga R (1959) The crystal structure of xanthophyllite. Acta Crystallogr 12:945-946 Thompson JB Jr (1978) Biopyriboles and polysomatic series. Am Mineral 63:239 Toraya H (1981) Distorsion of octahedra and octahedral sheets in 1M micas and the relation to their stability. Z Kristallogr 157:173-190 Vainshtein BK (1956) Structure Analysis by Electron Diffraction. Akad Nauk SSSR, Moscow (in Russian) Vainshtein BK (1964) Structure Analysis by Electron Diffraction. Pergamon, Oxford, UK Vainshtein BK, Zvyagin BB, Avilov AS (1992) Electron Diffraction Techniques. Vol 1. Oxford University Press, Oxford, UK Verma AJ, Krishna P (1966) Polymorphism and Polytypism in Crystals. Wiley, New York Weiss Z, Rieder M (1997) Distortions of coordination polyhedra in phyllosilicates and their diffraction pattern. 11th Int’l Clay Conf Abstr, p A82 Weiss A, Rieder M, Chmielová M (1992) Deformation of coordination polyhedra and their sheets in phyllosilicates. Eur J Mineral 4:665-682 Weiss Z, Rieder M, Chmielová M, Krajícek J (1985) Geometry of the octahedral coordination in micas: A review of refined structures. Am Mineral 70:747-757 Weiss Z, Wiewióra A (1986) Polytypism of micas. III: X-ray diffraction identification. Clays Clay Minerals 34:53-68 Woodrow PJ (1967) The crystal structure of astrophyllite. Acta Crystallogr 22:673-678 Zhukhlistov AP, Avilov AS, Ferraris G, Zvyagin BB, Plotnikov VP (1997) Statistical distribution of hydrogen over three positions in the brucite Mg(OH)2 structure from electron diffractometry data. Crystallogr Rep 42:774-777 Zhukhlistov AP, Dragulescu EM, Rusinov VL, Kovalenker VA, Zvyagin BB, Kuz’mina OV (1996) Sericite with a non-centrosymmetric structure from the gold-silver base metal deposit Banská Stiavnika (Slovakia). Zap Vseross Min Obshch (Proc Russian Mineral Soc) 125:47-54 (in Russian) Zhukhlistov AP, Litsarev MA, Fin’ko VI (1993) First find of a six-layered triclinic 6Tc polytype of a Tioxybiotite. Dokl Acad Sci SSSR 329:188-194 (in Russian) Zhukhlistov AP, Zyagin BB (1991) The efficiency of electron diffraction in revealing 2:1 layer differing in structure and symmetry, found in dioctahedral micas and smectites. Proc 7th Euroclay Conf, Dresden, p 1211-1212 Zhukhlistov AP, Zvyagin BB, Pavlishin VI (1990) Polytypic 4M modification of Ti-biotite with non uniform alternation of layers, and its appearance in electron-diffraction patterns from textures. Sov Phys Crystallogr 35:232-236 Zhukhlistov AP, Zvyagin BB, Soboleva SV, Fedotov AF (1973) The crystal structure of the dioctahedral mica 2M2 determined by high voltage electron diffraction. Clays Clay Minerals 21:465-470.
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Zvyagin BB (1967) Electron-diffraction Analysis of Clay Mineral Structures. Plenum Press, New York Zvyagin BB (1988) Polytypism of crystal structures. Comp Math Applic 16:569-591 Zvyagin BB (1993) Electron diffraction analysis of minerals. MSA 23:66-79 Zvyagin BB, Rabotnov VT, Sidorenko OV, Kotel’nikov, DD (1985) Unique mica built of noncentrosymmetrical layers. Izvestia Akad Nauk SSSR (ser Geol) 5:121-124 (in Russian) Zvyagin BB, Vrublevskaya ZV, Zhukhlistov AP, Sidorenko OV, Soboleva SV, Fedorov AF (1979) Highvoltage electron diffraction in the study of layered minerals. Nauka, Moscow (in Russian) Zvyagin BB, Zhukhlistov AP, Plotnikov VP (1996) Advances in electron diffraction of minerals. In VI Simonov Structural studies of minerals. Nauka, Moscow, 225-234 p (in Russian)
4 Crystallographic basis of Polytypism and Twinning in Micas Massimo Nespolo LCM3B, UMR, CNRS 7036 Université Henri Poincaré Nancy 1, BP 239 F54506 Vandoeuvre-les-Nancy cedex, France
[email protected] Slavomil Ďurovič Slovak Academy of Sciences Institute of Inorganic Chemistry; Department of Theoretical Chemistry Dúbravská cesta, 9; SK-842 36 Bratislava, Slovakia
[email protected] INTRODUCTION Although the investigation of micas dates back to the pre-scientific era (see Cipriani, this volume), the idea of polytypism (originally not distinguished from “polymorphism”) in the micas did not ensue until 1934, when Pauling proposed it in a private conversation quoted by Hendricks and Jefferson (1939). The existence of several structural types was however known from goniometric measurements and morphological analysis performed in the 19th century (e.g., Marignac 1847; Baumhauer 1900) and collected in the 4th volume of the Atlas der Krystallformen (Goldschmidt 1918; for a comparative review and later measurements see Peacock and Ferguson 1943) and appears also in the different axial settings introduced to describe the unit cell of micas (e.g., Brooke and Miller 1852; Des Cloizeaux 1862; Koksharov 1875; Tschermak 1878). The systematic investigation by X-ray diffraction (XRD) started with Mauguin (1927, 1928), who pointed out that the c axis of phlogopite was half that of muscovite. Pauling (1930) was the first to solve the structure of a mica, a fuchsite (now termed “chromian muscovite”, according to Rieder et al. 1998), by visual comparison of a subset of intensities from photographs, and introduced the first model of the structure of phyllosilicates on the basis of the coordination theory. Jackson and West (1931) were the first to perform a complete structure determination, investigating a muscovite-2M1. Hendricks and Jefferson (1939) investigated one hundred samples of micas and discovered several “polymorphs”, many of which were however twins of simpler structural types (shorter-period polytypes). The symmetry of the 2:1 mica layer was not fully recognized until Pabst (1955) showed that the correct space-group type of 1M polytype was C2/m instead than Cm, as previously assumed by Hendricks and Jefferson (1939) and reported also by Peacock and Ferguson (1943). Since the accomplishment of such an apparently easy task as the determination of the structure of the single-layer polytype took so long time and so much effort, it is not surprising that the whole phenomenon of polytypism in micas occupied several researchers from different countries for a long run of time, and still keeps undisclosed some of its most interesting and challenging points. Although the causes of the complexity of the phenomenon of polytypism in micas are multifaceted, they can be simplified to “magic words”, local (partial) symmetry, and a “magic number”, 3. As shown hereafter, each atomic plane in mica has an ideal symmetry of at least trigonal, which is preserved in each of the two kinds of sheets (tetrahedral and octahedral), but it is reduced to monoclinic when considering the layer as 1529-6466/02/0046-0004$15.00
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a unit. The two T sheets of a layer are staggered along c and the amount of the stagger in the (001) projection is ideally |a|/3. For each non-orthogonal polytype an ideally orthogonal multiple cell can always be chosen, with 3-times the periodicity of the polytype in the stacking direction. In the real structure, some of the atoms move slightly from the positions corresponding to the ideal symmetry, but each atomic plane still preserves a trigonal pseudo-symmetry. Then, the (001) projection of the layer stagger deviates more or less from |a|/3, and the multiple cell is close to, but not exactly orthogonal. The magic words and magic number can be traced also in reciprocal space, where the reflections with k = 0(mod 3) reveal the symmetry principle on which a polytype is built, and the reflections with k ≠ 0(mod 3) permit the identification of the stacking sequence. The existence of a multiple cell with a metric pseudo-symmetry higher than the structural symmetry, together with the trigonal pseudo-symmetry of the planes of the basal oxygen atoms, is also the geometrical reason of the extensive occurrence of twinning in micas. Although polytypism and twinning can be reduced to relatively simple common geometrical bases, the development of general criteria to recognize the presence of twinning from the diffraction pattern took a long time, and still many questions remain open. The purpose of this chapter is to give a general overview of the factors, in terms of lattice geometry and of symmetry, which are responsible for polytypism and twinning in micas, and to provide general and simple criteria to be applied in the experimental practice of polytype and twin identification. For this reason, micas are hereafter regarded as built by layer archetypes, i.e. idealized layers where most of the structural distortions are not taken into account. The true atomic structure of the mica layer influences mainly the intensities but not the geometry of the diffraction pattern, and is discussed in detail in Ferraris and Ivaldi (this volume) and in Brigatti and Guggenheim (this volume). Rigorous mathematical demonstrations are not given here: readers wishing to acquire a deeper knowledge are invited to consult the original publications, quoted hereinafter, where those demonstrations are given in detail. The crystallographic terminology follows Wondratschek (2002). NOTATION AND DEFINITIONS The geometrical description of mica polytypes is given in terms of the OD theory developed by Dornberger-Schiff (e.g., 1964) and her successors. OD stands for “OrderDisorder” and indicates that the stacking of layers may produce both periodic (“ordered”) and non-periodic (“disordered”) structures. It has no relation with the chemical orderdisorder phenomena. The OD theory emphasizes particularly the role of polytypes which involve pairs, triples, quadruples etc. of geometrically equivalent layers, or, when this is not possible, the smallest number of kinds of triples, quadruples etc. of layers. These polytypes are termed Maximum Degree of Order (MDO) polytypes. The layer-group notation adopted here is the one developed by Dornberger-Schiff (1959), in which the direction of missing periodicity is indicated by parentheses. For example, C12/m(1) indicates a monoclinic holohedral C-centered layer, having (a,b) as the layer plane (for details see Merlino 1990). The indicative symbols for polytypes were introduced by Ramsdell (1947) and are written as NSn, where N is the number of layers, S indicates the symmetry and n is a sequence number, often (but not always) indicating the order in which polytypes have been discovered. Ramsdell’s symbolism is actually a mixed symbolism, since S (nowadays given with a single uppercase letter according to the IUCr Ad-Hoc committee
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recommendations: Guinier et al. 1984) is used to indicate the six crystal families, the trigonal syngony (syngony = crystal system) and the rhombohedral Bravais system: A = anorthic (triclinic), M = monoclinic, O = orthorhombic, Q = quadratic (tetragonal), T = trigonal, R = rhombohedral, H = hexagonal, C = cubic. This mixed symbolism is nowadays preserved for historical reasons and its use is accepted only for indicating polytypes. Q, R and C cannot appear in micas (Takeda 1971). To classify, but also to identify experimentally, mica polytypes, the relations between a lattice and its derivative lattices (superlattices, sublattices) are of fundamental importance. Different authors have given contrasting definitions. Here, we adopt the definition in terms of the group-subgroup relations, in agreement with the International Tables for Crystallography, Vol. A, 5th ed., in press (Th. Hahn, pers. comm..). Sublattice is termed a derivative lattice obtained from an original lattice by taking a subgroup of translations: its unit cell is larger than that of the original lattice. In contrast, superlattice is termed a lattice obtained from an original lattice by taking a supergroup of translations: its unit cell is smaller than that of the original lattice. Because the derivative lattice obtained from the original one by taking a subgroup (supergroup) of translations has a larger (smaller) unit cell, in some publications the terms superlattice and sublattice are defined in the opposite way. The superlattice-sublattice character of a derivative lattice is inverted when going from one space to its dual (i.e. from direct to reciprocal, or vice versa). The notations most often used in the following are summarized here for ease of consultation: The mica layer and its constituents T: O: I: Ob: Oa:
tetrahedral sheet = Ob-Z-Oa or Oa-Z-Ob octahedral sheet = Oa-Y-Oa plane of the interlayer cations (also: these cations as such). plane of the basal oxygen atoms of the tetrahedra plane of the apical oxygen atoms of the tetrahedra; this plane, contains approximately also OH groups and, depending on the kind of mica, F and, less frequently, Cl and S. T1,T2: the two translationally independent tetrahedral sites within a T sheet M1,M2,M3: the three translationally independent octahedral sites within an O sheet Ma,Me,Mi: average cations occupying the three translationally independent octahedral sites (Ma = Maximal; Me = Medium; Mi = Minimal, with reference to their scattering power) δ(Ma), δ(Me), δ(Mi): X-ray scattering power of the (average) cations Ma,Me,Mi Z: plane of the tetrahedral cations Y: plane of the octahedral cations (whose coordination polyhedra are however not regular octahedra but rather trigonal antiprisms). Tet: tetrahedral OD layer = Oau-Z-Ob-I-Ob-Z-Oal (l = lower, u = upper; see text) Oc: octahedral OD layer = Oal-Y-Oau p2j: OD packet pointing up = Tet2j/2 + Oc2j+1/2 q2j+1: OD packet pointing down = Oc2j+1/2 + Tet2j+2/2 M: the entire mica layer (T-O-T). There are two types: M1 and M2 depending on the location (M1 vs. M2/M3) of the origin of the O sheet. Standard character “M” indicates layer, italics “M” indicates cation sites.
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Axial settings, indices and lattice parameters a, b, c: monoclinic crystallographic axes in the space-fixed reference (in italics) a1~6, b1~6, c: monoclinic crystallographic axes in the crystal-fixed reference (in italics) A1, A2, A3, c: hexagonal crystallographic axes (in italics) AF1, AF2, AF3, CF: hexagonal crystallographic axes of the family structure (in italics) a, b, c / a1~6, b1~6, c / A1, A2, A3, c / AF1, AF2, AF3, CF: crystallographic basis vectors (in bold) C1, C2, C3: the three orthohexagonal cells (Fig. 1; cf. Arnold 2002) cn : the (001) projection of the of the c basis vector c0: vertical distance between two interlayer cations on the opposite sides of an M layer (c0 = c1cosβ*) c*1 : parameter along c* of the simplest polytype (1M): it corresponds to about 0.1Å-1 HK.L: diffraction indices expressed in hexagonal axes hkl: diffraction indices expressed in orthohexagonal axes lC1: l index in the C1 setting lT: l index in the twin setting ω: obliquity of the twin, divided into a component within the (001) plane (ω||) and a component normal to the (001) plane (ω⊥) ε: angular deviation from orthohexagonality of the (001) plane η: linear deviation from orthohexagonality of the (001) plane t(hkl): trace of the plane (hkl) onto the (001) plane n t(hkl): normal to t(hkl) in the (001) plane
Figure 1. Relation between the hexagonal cell P and the three orthohexagonal cells C1, C2, C3 (cf. Arnold 2002).
Symbols N: N′: Ti:
number of layers in the conventional cell number of layers in the unit cell of the (pseudo)-orthohexagonal setting: N′ = N for orthogonal polytypes, N′ = 3N for non-orthogonal polytypes character (“0”~“5”) indicating the mica OD packet orientation
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v2j,2j+1: character (“0”~“5”) indicating the displacement between two adjacent mica OD packets p2j and q2j+1 〈v〉: the vector assigned to the character v Σv: character (“0”~“5”, “*”, “+”, “–“) indicating vector sum of v2j,2j+1 over a complete polytype period and corresponding to the projection of the c axis onto the (001) plane RSiP: i-th Rotational Sequence of the polytype “P” Ri: i-th translationally independent reciprocal lattice row parallel to c* of the single individual (1 ≤ i ≤ 9). Ci: i-th “composite row”: translationally independent reciprocal lattice row parallel to c* of the twin (1 ≤ i ≤ 9). I j: symbol identifying the “node features” of a row of the reciprocal lattice parallel to c*. I is the number of reflections in the c*1 repeat, j a sequence number. Symmetry and symmetry operations λ-symmetry: the symmetry proper of an individual layer (λ-operation: a symmetry operation transforming a layer into itself; the set of λ-operations constitute a layer-group) σ-symmetry: the symmetry of a layer pair (σ-operation: a coincidence operation transforming a layer into the adjacent one) τ-operations: symmetry or coincidence operations which do not change the sign of the coordinate in the layer stacking direction. They are labeled λ-τ or σ-τ if they refer to λ- or σ-operations, respectively ρ-operations: symmetry or coincidence operations which change the sign of the coordinate in the layer stacking direction and thus turn a layer or a stack of layers upside down. They are labeled λ-ρ or σ-ρ if they refer to λ- or σoperations, respectively. Evidently, τ.τ = τ, τ.ρ = ρ, ρ.τ = ρ and ρ.ρ = τ. THE UNIT LAYERS OF MICA The conventional layer of mica is described in details in Ferraris and Ivaldi (this volume). Here we recall only those definitions that are referred to in the following. The conventional layer (also termed TOT layer or 2:1 layer) is constructed of seven atomic planes: Obl, Zl, Oal, Y, Oau, Zu, Obu, where “l” and “u” stand for “lower” and “upper” respectively. Interlayer cations occurr between two successive layers in the I plane. This layer is referred as the “M layer” and is subdivided into two kinds of sheets: T (Tl: Obl, Zl, Oal, and Tu: Oau, Zu, Obu) , and O (Oal, Y, Oau). On the basis of the occupation of the three octahedral sites, three families of micas exist: homo-octahedral (all three sites are occupied by one type of cation), meso-octahedral (one site is occupied differently from the other two), and hetero-octahedral (all three sites are occupied differently). In these three families the idealized λ-symmetry of the O sheet is H⎯(3)1m, P⎯(3)1m, and P312 respectively (Dornberger-Schiff et al. 1982). Two models were introduced to describe the λ-symmetry of the T sheet: the Pauling model (Pauling 1930), which neglects all the distortions and assumes λ-symmetry P(6)mm; and the Trigonal model, which considers only the ditrigonal rotation of tetrahedra and assumes λ-symmetry P(3)1m. Both these models neglect the distortions occurring in the O sheet. Although the Trigonal model may seem still rather abstract, it is sufficient to describe the diffraction features useful for polytype and twin identification, whereas the Pauling model is too abstract. In fact, the main influence on the conditions for a reflection comes from the ditrigonal rotation of the tetrahedra. The other distortions, not taken into account by the Trigonal model, are quantitatively less relevant; they influence mainly the diffraction
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intensities, and to a much lesser degree the geometry of the diffraction pattern; for this reason they can be neglected, in the first approximation. The stagger of the two T sheets reduces the λ-symmetry of the M layer to monoclinic. Depending on whether the origin of the O sheet (which must be taken at the site with the point symmetry corresponding to the λ-symmetry of the sheet) is in M1 (trans) or in M2/M3 (cis), the layer itself is termed M1 or M2 respectively, and the highest λ-symmetry for these two layers is C12/m(1) and C12(1) respectively (for details see Ferraris and Ivaldi, this volume). The preliminary stage of the experimental study of a mica sample, such as the identification of the polytypic stacking sequence, is normally performed by assuming that the structure is homo-octahedral, and thus in the hypothesis of all M1 layers. For this assumption Nespolo et al (1999d), following a suggestion by S. Ďurovič (pers. comm.), introduced the term homo-octahedral approximation. Alternative unit layers Besides the M layer, other unit layers were introduced, in most cases to simplify the description of some features, such as the diffraction pattern. Amelincks-Dekeyser’s unit layer . Amelinckx and Dekeyser (1953) pioneered the study of the spiral growth of micas. They also introduced the first vectorial and symbolic representation of the stacking sequence of layers in mica polytypes. These authors used a unit cell having the apical oxygen atoms and (OH/F) groups at the boundaries (Fig. 2). In this way, the unit cell is orthogonal and the monoclinic symmetry is achieved by stacking successive cells along three directions making 120º. Although this cell has nowadays no practical importance, it represents the first description alternative to Pauling’s (1930) model and the precursor of the TS unit layers described below. Franzini and Schiaffino’s A and B layers . Franzini and Schiaffino (1963a,b) assumed that the ditrigonal rotation of the tetrahedra was mainly not related to the misfit of the a and b parameters of the tetrahedral and octahedral sheets, but to an intrinsic tendency of the potassium to assume an octahedral (actually antiprismatic) coordination. Those authors concluded that, with a single type of layer, rotations of (2n+1)×60º were not possible for K-micas. To explain “polymorphs” and twins in which such rotations appear, Franzini and Schiaffino (1963a) introduced two kinds of monolayers, called A and B, in which the antiprismatic coordination for the interlayer cation is preserved for all the six rotations. These two layers differ for the orientation of the octahedral sheet with respect to the tetrahedral sheets: in practice, the slant of the octahedra is reversed in the two layers1. The ordered repetition of layers of the same kind (both A or both B) produces 1M, 2M1 and 3T “polymorphs”, while the alternate repetition of both A and B layers produces 2O, 2M2 and 6H “polymorphs”, however preserving the antiprismatic coordination for the interlayer cation. The co-existence of A and B layers was however regarded as highly improbable (Franzini 1966; 1969). The starting assumption of this theory, namely the impossibility of trigonal prismatic coordination for the interlayer cation, is not correct (Sartori et al. 1973), and the Franzini and Schiaffino theory lost its importance. Despite that, the terms Franzini-type A and B have found their way into the literature. As Franzini (1969) noted, owing to ditrigonalization, the basal-oxygen atoms in the type A approach the cations in the adjacent octahedral sheet, whereas they move 1
Griffen (1992) described the direction of the ditrigonalization of the T sheets with respect to the triangular bases of the octahedra in terms of the rotations “O” (opposite, corresponding to Franzini-type A of layer) and “S” (same, corresponding to Franzini-type B of layer). This terminology, borrowed from pyroxenes, is commonly not adopted for micas.
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Figure 2. Schematic representation of a slab b/4 thick, showing three layers of the 1M polytype. Four different unit cells are shown: the M layer (solid lines), the U layer (dashed lines), the TS D layer (dotted lines) and the cell used by Amelinckx and Dekeyser (1953) (dotted-dashed lines). The OD layers and packets are indicated directly in the figure.
apart in type B layers, if compared with the Pauling model. This holds also for phyllosilicates other than micas. Whereas in mica structures refined to date only the type A has been found, the type B has been encountered in some 1:1 phyllosilicates and in some chlorites, where the energetic handicap of the type B is balanced by a more favorable arrangement of hydrogen bonds elsewhere in the structure. The U layer. The origin of the entire M layer is in the I plane. By shifting the origin into the O sheet, the U-layer (Fig. 2) is obtained and, inside it, a smaller portion, called the u-layer, which does not represent a unit layer but consists of two tetrahedral sheets and the interlayer cations between. These layers were used as a tentative interpretation of the crystallographic transformation of biotites by means either of crystallographic slips (CS) of cation or oxygen planes corresponding to cation-to-cation or oxygen-to-coplaneoxygen distance (CS of the first sort) or of co-operative slip movements of two atomic planes (oxygen-oxygen or oxygen-cation) in a single octahedral sheet (CS of the second
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sort) (Takéuchi 1971; Takéuchi and Haga 1971). A CS of the first sort probably occurs during polytype formation when a strengthening of the interlayer bonding is accompanied by a destabilization of the O sheet (Nespolo 2001). The TS layers. Similarly to the choice of Amelinckx and Dekeyser (1953), Sadanaga and Takeda (1969) and Takeda and Sadanaga (1969) described the structure of micas by means of orthogonal unit layers. Whereas Amelinckx and Dekeyser (1953) had chosen the origin in the Oa / OH / F plane, Sadanaga and Takeda (1969) and Takeda and Sadanaga (1969) defined their TS unit layers between two octahedral sheets of successive M layers and preserved the origin in the plane of the I cations (Fig. 2). TS unit layers are defined within the Trigonal model and consist of four layers, labeled D, D*, T and T*, with λ-symmetry ⎯P (3)1m (D and D*) and P⎯(6)2m (T and T*). D is related to D* and T to T* by an 180º rotation about c* (see Fig. 2,3). Because of their trigonal λ-symmetry, which is higher than the monoclinic λ-symmetry of the M layer, four kinds of unit layers are necessary to describe all possible polytypes. These layers are related by only translations, without rotations, and next layers are staggered ±a/3 along one of the three hexagonal axes A1, A2, A3 in the plane of the layer. As shown by Nespolo et al (1999d), the TS unit layers represent the most suitable geometrical description for a simple computation of the PID function (see below). The letters D and T indicate a “ditrigonal” or “trigonal” coordination of the I cation respectively for the two kinds of layer. Actually, D/D* layers have the I cation in antiprismatic coordination, whereas in the T/T* layers the I cation is in prismatic coordination. In both cases, the coordination polyhedron is trigonal where only the nearest-neighbor oxygen atoms are considered, whereas it becomes ditrigonal by considering also the next-nearest-neighbor oxygen atoms. A symbolism like A/A* (for “antiprismatic”) and P/P* (for “prismatic”) instead of D/D* and T/T* respectively would perhaps had been more appropriate. The TS layers are constructed by half-pairs of M1 layers in the homo-octahedral approximation and, as shown hereafter, their use is in the calculation of the Periodic Intensity Distribution function to solve an unknown stacking sequence. The OD layers and the OD ap ckets . The OD interpretation presupposes that any polytype of a given polytypic substance may be considered as consisting of disjunct parts periodic in two dimensions, called OD layers, whose pairs remain geometrically equivalent in any polytype of the same family. The OD layers do not necessarily coincide with the layers commonly chosen on the basis of the chemical identity and/or cleavage properties. In other words, the layers by which a polytypic substance is most commonly described from the crystal-chemical point of view are not always the most suitable layers to describe the geometrical equivalence of layer pairs. Furthermore, the choice of the OD layers in general is not absolute (Grell 1984); their purpose is not to explain but to describe and/or predict polytypism of a substance based on symmetry. Micas are considered to consist of two kinds of OD layers. The octahedral OD layer (Oc) corresponds to the sequence Oal-Y-Oau, and the tetrahedral OD layer (Tet) to the sequence Oau-Z-Ob-I-Ob-Z-Oal, with the Oal and the Oau planes (au = apical upper; al = apical lower) half belonging to neighboring OD layers (Fig. 2). By denoting an OD layer with the general letter L, Tet and Oc OD layers are L2j and L2j+1 respectively, where j is a running integer. Another useful unit is the OD packet, which corresponds to half of the M layer plus half the plane of the I cations, and constitutes the smallest continuous part, periodic in two dimensions, representing fully the chemical composition of a polytype (Ďurovič 1974). OD packets are by definition polar and lie within one side or the other pointing alternatively along +c and –c: they are indicated with the letters p and q: p2j = Tet2j/2 + Oc2j+1/2; q2j+1 = Oc2j+1/2 + Tet2j+2/2 (Fig. 2). All mica packets within the same family are geometrically equivalent and their symmetry is P(3)1m (homo-octahedral
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Figure 3. The four TS unit layers. a, b: orthohexagonal axes. Black and open small circles represent M1 and M2 sites respectively. Double circles represent interlayer cations and OH/F groups, which are overlapped in the (001) projection. u and l indicate octahedral cations with z = +1/2 and z = -1/2 respectively, overlapped in (001) projection for T and T* layers (modified after Nespolo et al. 1999d).
family), C1m(1) (meso-octahedral family) or C1 (hetero-octahedral family) (DornbergerSchiff et al. 1982; Backhaus and Ďurovič 1984; Ďurovič, et al. 1984). This reduces the problem of handling two kinds of OD layers to that of one kind of OD packet and this facilitates, among others, also the systematic derivation of MDO polytypes (see below).
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Furthermore, both M1 and M2 mica layers within the same family consist of the same kind of OD packet. MICA POLYTYPES AND THEIR CHARACTERIZATION The crystal chemical reason for polytypism is that adjacent layers (twodimensionally-periodic units) can be linked to each other in many translationally nonequivalent ways. However, the nearest-neighbor relationships remain preserved. Translated into the language of symmetry, this means that the pairs of adjacent layers remain geometrically equivalent in all polytypes of the same family. The geometrical equivalence must be fulfilled not necessarily by the real layers, but by their archetypes, i.e. the (partially) idealized layers to which the real layers can be reduced by neglecting some distortions occurring in the true structure. The notion of polytypism becomes thus unequivocal only when it is used in an abstract sense to indicate a structural type with specific geometrical properties. In micas, these archetypes are the layers described by the Trigonal model. Of the several kinds of layers presented in the previous section, the OD layers, and the OD packets, are the most suitable ones to both show and exploit the geometrical equivalence. Micas as OD structures If the position of a layer is uniquely defined by the position of the adjacent layers and by the so-called vicinity condition (VC)2, which states the geometrical equivalence of layer pairs, the resulting structure is fully ordered. If, on the other hand, more than one position is possible that obeys the VC, the resulting structure is an OD structure and the layers are OD layers. VC structures may thus be either fully ordered structures or OD structures (Dornberger-Schiff 1964, 1966, 1979; see also Ďurovič 1999). All OD structures are polytypic; the reverse may or may not be true (see the arguments in Zvyagin 1993). Equivalency depends on the choice of OD layers and also on the definition of polytypism (see below). In each of the three mica families, the packet pairs p2jq2j+1 and q2j+1p2j+2 are geometrically equivalent through a ρ-operation of the Oc2j+1 OD layer and of the Tet2j+2 OD layer, respectively. These operations are denoted as 2j,2j+1[ρ(i)] and (j) 2j+1,2j+2[ρ ] respectively .The resulting polytype depends on the kind of these operations (they follow from the λ-symmetry of Oc or Tet) and on their sequence in the polytype. Since ρ.ρ = τ and, particularly for OD structures, a product like kl[ρ(i)]·mn[ρ(j)]is allowed only if l=m, each even number of such products, e.g., 01[ρ
(1)
]·12[ ρ(2)]· 23[ ρ(3) ]· 34[ρ (4) ]· …. ·2n-1,2n [ρ (2n)]
yields a 0,2n[τ]-operation. This operation can be either a translation, a glide operation or a screw rotation, whose translation component is the so-called repeat unit. The τ-operation can be continued, i.e. continuously repeated, and then it generates a periodic polytype. The operation is thus global (total) for the polytype obtained. Of special importance are the 02[τ]-operations which play a decisive role in the derivation of MDO polytypes, as shown below. If the distribution of subsequent ρ−operations is completely random so that no generating τ-operation can be found, the polytype is disordered. Disordered polytypes have been reported as 2n×60º rotations only (e.g., Ross et al. 1966; let us indicate them as 1Md-A, where “A” stands for “subfamily A”) and with both 2n×60º and (2n+1)×60º rotations (Kogure and Nespolo 1999a; let us indicate them as 1Md-M, where “M” stands 2
The vicinity condition (e.g., Dornberger-Schiff 1979) consists of three parts. VC α: VC layers are either geometrically equivalent or, if not, they are relatively few in kind; VC β: translation groups of all VC layers are either identical or they have a common subgroup; VC γ: equivalent sides of equivalent layers are faced by equivalent sides of adjacent layers so that the resulting pairs are equivalent.
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for “mixed-rotation”), whereas no examples with (2n+1)×60º rotations only (let us indicate them as 1Md-B, where “B” stands for “subfamily B”) have been reported to date. Note that in periodic polytypes some ρ-operations also become global, whereas the remaining ones are valid only in a subspace of the crystal space. Note also that, alone, a ρ-operation could not be used to construct a polytype, because its repeated application leads back to the area of the starting layer or packet. For more details concerning the OD interpretation of mica structures, see Dornberger-Schiff et al (1982); for the derivation of MDO mica polytypes see Backhaus and Ďurovič (1984); for the classification and abundance of MDO mica polytypes see Ďurovič et al (1984). The set of all the operations valid in the whole crystal space and in a subspace of the crystal space constitutes a space groupoid (Dornberger-Schiff 1964; Fichtner 1965, 1977, 1980). The theory of groupoids was introduced in mathematics by Brandt (1927) and applied in crystallography in Germany by the OD school (Dornberger-Schiff 1964, 1966), and in Japan by the school of Ito and Sadanaga, with special emphasis on those groupoids, termed twinned space groups, which are necessary to explain the existence of polysynthetic structures (e.g., Ito 1935, 1938, 1950; Ito and Sadanaga 1976; Sadanaga 1978; Sadanaga et al. 1980). The OD school used the terms total for a space-group operation, local or partial (as synonyms) for a symmetry operation valid in a subspace of the crystal space, and coincidence operation, represented by a single transformation matrix, for a non-symmetry operation that corresponds – approximately – to a one-way movement in the structure, i.e. an operation without the corresponding inverse operation. Sadanaga and Ohsumi (1979) and Sadanaga et al (1980) used instead global, local and partial in the same way the OD school used total, local/partial and coincidence respectively. To avoid any possible confusion, hereafter the word “partial” is not adopted, and the term “local” is used to indicate a symmetry operation valid in a subspace of the crystal space. Within the Pauling model, an isolated Tet layer has λ-symmetry P(6/m)mm, with 12 τ-operations and 12 ρ-operations. Within a group, the three axial and the three inter-axial directions are symmetry-related and thus one entry for each of these two sets in the conventional Hermann-Mauguin symbol suffices to characterize the corresponding symmetry operations. However, in the OD structures, any of these operations can play a specific role and this is why Dornberger-Schiff (1964 p. 44 ff; 1966 p. 54) introduced extended Hermann-Mauguin symbols consisting of seven entries: . . . (.) . . . where the unique direction is in parentheses, the three entries to the left refer to the three axial directions A1, A2, A3 (Fig. 1; cf. Fig. 4) and the three entries to the right refer to the three inter-axial directions B1,B2,B3 where Bi┴Ai. Such an extended Hermann-Mauguin symbol for the layer-group P(6/m)mm reads: P 2/m 2/m 2/m (6/m) 2/m 2/m 2/m Within the Trigonal model, this λ-symmetry reduces to trigonal. The extended Hermann-Mauguin symbols, depending on which of the two maximal non-isomorphic subgroups is preserved (either P (3)1m or P(6)2 m , become: P 1 1 1 (3) 2/m 2/m 2/m and P 2 2 2 (6) m m m. The individual operations in each of these two groups can be characterized either by the extended Hermann-Mauguin (H-M) symbols (as usual in the OD literature), or with reference to the orthogonal (ORT) axes. Although indexing in the orthohexagonal setting in unequivocal, the correspondence between H-M symbols and ORT symbols depends on which cell is adopted (C1 vs. C2: see Fig. 1). In Table 1, all correspondences are shown. The two λ-symmetries of the Tet layer correspond to 2n×60º and (2n+1)×60º rotations, respectively, of successive M layers about c*. Each family of polytypes is defined in terms of the λ-symmetry of the Oc layer, which is the same as that of the O
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Nespolo & Ďurovič Table 1. Extended Hermann-Mauguin (H-M) symbols and corresponding operations indexed in orthogonal (ORT) axes for the two λ-symmetries of the Tet layer within the Trigonal model. The extended H-M symbols consist of seven entries: . . . (.) . . . where the unique direction is in parentheses, the three entries to the left refer to the three axial directions A1, A2, A3 and the three entries to the right refer to the three inter-axial directions B1,B2,B3 (Bi┴Ai). The corresponding orthogonal indices are given with reference to both the C1 and the C2 cells. P⎯(3)1m
τ-operations H-M
ORT (C1)
1 (3)
ρ-operations ORT (C2)
H-M
ORT (C1)
ORT (C2)
1
⎯1
⎯1
⎯1
⎯3-1[001]
⎯3-1[001]
1 -1
3
-1
[001]
3
-1
⎯(3)
[001]
–1
(3)1
31[001]
31[001]
⎯(3)1
⎯31[001]
⎯31[001]
[. . . (.) m . .]
m (010)
m (110)
[. . . (.) 2 . .]
2[010]
2[310]
[. . . (.) . m .]
m (110)
m (⎯110)
[. . . (.) . 2 .]
2[310]
2[⎯310]
[. . . (.) . . m]
m (⎯110)
m (010)
[. . . (.) . . 2]
2[⎯310]
2[010]
P⎯(6)2m
τ-operations
ρ-operations
H-M
ORT (C1)
ORT (C2)
H-M
ORT (C1)
ORT (C2)
1
1
1
⎯(2)1
m(001)
m(001)
(3)-1
3-1[001]
3-1[001]
⎯(6)–1
⎯6-1[001]
⎯6-1[001]
(3)1
31[001]
31[001]
⎯(6)1
⎯61[001]
⎯61[001]
[m . . (.). . .]
m (100)
m (⎯130)
[2 . . (.) . . .]
2[100]
2[⎯110]
[.m . (.) . . .]
m (⎯130)
m (130)
[. 2 . (.) . . .]
2[⎯110]
2[110]
[. . m (.) . . .]
m (130)
m (100)
[. . 2 (.) . . .]
2[110]
2[100]
sheet (see Table 2 in Ferraris and Ivaldi, this volume), and is then divided into two subfamilies on the basis of the Tet λ-symmetry: subfamily A for P⎯(3)1m, and subfamily B for P(6)2 m . We suggest for the polytypes where Tet layers with both P⎯(3)1m and P⎯(6)2m λ-symmetries co-exist, the term mixed-rotation polytypes (see also Nespolo 1999). Both subfamily A and subfamily B polytypes are OD structures, because the layer stacking obeys the VC. However, the layer stacking in the mixed-rotation polytypes with the geometry of the Trigonal model violates the VC: these polytypes are OD structures only within the Pauling model, i.e. for a null ditrigonal rotation of the tetrahedra (Backhaus and Ďurovič 1984).
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Figure 4. The nine possible displacements in the structure of polytypes of phyllosilicates. Left: the OD symbols and corresponding vectors, within the primitive hexagonal unit cell (modified after Durovic 1999). The sum of any two vectors is indicated, and the result of the summation of any number of vectors should be taken modulo primitive hexagonal cell. The individual vectors are designated by their conventional numerical characters and signs “+” and “–”, whereas the zero displacement “*” is not indicated. The “+” and “–” vectors do not explicitly occur in micas. However, in Class b polytypes the total displacement, obtained as vector sum of the packet-topacket displacements (v2j,2j+1, second line of the full OD symbol) corresponds to “–”, namely cn = (0,⎯1/3). Right: the corresponding Z vectors (modified after Zvyagin et al.1979) (cf. Table 4). In the publications by Zvyagin and his school, the coordinate system is oriented so that the orthogonal a axis points up and the b axis to the left. Here we follow instead the conventions of the International Tables for Crystallography: the space-fixed references, and consequently the Z vectors, are rotated by 180º with respect to their orientation in the original publications.
Figure 5. An isolated hetero-octahedral Oc layer, with the three two-fold axes in the plane of the layer (ρ-operations).
In Figure 5 an isolated Oc layer is shown. Depending on whether Ma, Mi and Me are all equal, two different or all different the Oc layer is homo-, meso- or hetero-octahedral respectively, and the λ-symmetry is H⎯(3)1m, P⎯(3)1m and P(3)12 respectively. In the
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Figure 6. The hetero-octahedral Oc layer shown in Figure 5 after substitution of 2/3 of the OH/F groups with Oa from the upper and lower tetrahedra. Only one of the three ρ-operations remains, defining the relation between the type of site (M1, M2, M3) and its occupation (Ma, Me, Mi). (a) ρ-operation 2[010], Mi cation in the M1 site, layer type M1. (b) ρ-operation 2[310], Me cation in the M1 site, layer type M2. (c) ρ-operation 2[⎯310], Ma cation in the M1 site, layer type M2.
meso-octahedral family Oc includes six τ-operations (1, 3+[001], 3-[001], m(010), m(110), m(⎯110)) and six ρ-operations (2[010], 2[310], 2[⎯310],⎯1, ⎯3+[001],⎯3-[001]); these numbers in the homooctahedral family have, in fact, to be multiplied by three, owing to the H centering, whereas for the hetero-octahedral family Oc includes three τ and three ρ-operations (the first three of each set). In Figure 6 the same projection is given, but with the positions of the OH/F groups remaining after the substitution with Oa are indicated. This substitution destroys two-thirds of the λ-operations, leaving one (hetero-octahedral) or two (homoand meso-octahedral) τ-operations (the identity and one m reflection) and one or two ρoperations (one of the two-fold rotations in the plane of the layer, and the inversion). For meso- and hetero-octahedral Oc layer, for the sake of simplicity and without loss of generality, let us assume that δ(Mi) < [δ(Ma), δ(Me)]. The origin of the Oc layer is then, according to the convention described in Ferraris and Ivaldi (this volume), at the Mi average cation. In Figure 6a, one of the ρ-operations (the only one for hetero-octahedral Oc layer) is the two-fold rotation along [010], and the M1 (trans) site contains the Mi average cation. The M layer is thus of the type M1. Instead, in Figure 6b and 6c the ρoperation is the two-fold rotation along [310] and [⎯310] respectively: the M1 site contains the Me or the Mi cation respectively, and in both cases the M layer is of type M2. MDO op lytypes
. Polytypes in which not only the pairs of layers, but also the triples,
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quadruples etc. are geometrically equivalent, or, when this is not possible, contain the smallest number of kinds of triples, quadruples etc., are termed Maximum Degree of Order (MDO) polytypes. This definition originates in a simple philosophy: if a certain configuration (say a triple of layers) is energetically favorable, it will be repeated again and again and will not be intermixed with other, less favorable configurations.
Figure 7. The (001) projections of a Tet layer (the I cation, not shown, takes place in the hole between the two rings of tetrahedra). The two-fold axes in the plane of the layer (half of the ρoperations of the Tet layer) are indicated. (a) The configuration corresponding to the Pauling model, with zero ditrigonal rotation. The symmetry of the Tet layer is P(6/m)mm. (b) The configuration corresponding to subfamily A in the Trigonal model. The symmetry of the Tet layer is P⎯(3)1m. (c) The configuration corresponding to subfamily B in the Trigonal model. The symmetry of the Tet layer is P⎯(6)2m.
MDO polytypes of the subfamily A [P⎯(3)1m λ-symmetry of the Tet layer] are more favorable then MDO polytypes of the subfamily B [P⎯(6)2m λ-symmetry of the Tet layer], probably because of the different (staggered vs. eclipsed) configuration of the facing Ob atoms (Fig. 7). The most common polytypes are indeed MDO subfamily A. Of the MDO subfamily B, only 2M2 is relatively common in Li-rich trioctahedral micas, where an important structural role of the fluorine atoms has been proposed (Takeda et al. 1971). 2O has been found in its ideal space group in a fluor-phlogopite (Ferraris et al. 2001) and in the brittle mica anandite (Giuseppetti and Tadini 1972; Filut et al. 1985): the structure refinement of anandite indicates that it cannot be described in terms of an orthohexagonal C-centered cell, its space-group type being Pnmn. 2O was also obtained synthetically in fluor-phlogopite (Sunagawa et al. 1968; Endo 1968), and identified from direct
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observation of the growth spirals on the surface, but no diffraction study has been performed. Several non-MDO subfamily A polytypes have been reported, as well as some mixed-rotation polytypes, but their number is far smaller than MDO polytypes. The abundance of MDO subfamily A polytypes shows that the geometrical equivalence of OD layers is an important factor even when considering the real structures. The occurrence of non-MDO polytypes is easily understood when considering that the MDO concept specifically refers to a layer-by-layer growth. In all the environments where crystals grow in a fluid phase, the spiral-growth mechanism, to which the MDO criteria apply less strictly, becomes dominant as soon as the supersaturation decreases below a certain critical value (Sunagawa 1984). As a matter of fact, the appearance of long-period polytypes in micas has precise structural reasons. In polytypes based on the close-packed arrangement of atoms, such as SiC, CdI2 etc., the layer thickness is only a few Å and the long-range interactions are not negligible. In micas the layer is about 10Å thick and the long-range interactions are thus less relevant. The probability of the occurrence of nonMDO polytypes, as well as of non-periodic (disordered) polytypes, depends in general on how close are the layers to their archetypes (i.e. how close is the real symmetry to the ideal OD symmetry). The more a layer deviates from its archetype, the less valid are the equivalencies between adjacent layers. The consequence is that when the pairs of adjacent layers are not geometrically equivalent, they are also not energetically equivalent and the ambiguity in the stacking of layers is lost. The first derivation of the predecessors of the MDO polytypes dates back to Smith and Yoder (1956), who theoretically described the six non-equivalent polytypes (termed “simple polymorphs” by them) that can be obtained by stacking the M1 layer with the same rotation (in the two possible directions) between adjacent layers. All other polytypes were collectively termed complex polymorphs. The term polymorphism was also used by Zvyagin (1962) and by Franzini and Schiaffino (1963a,b), whereas the word polytypism when referring to micas was used for the first time probably by Amelinckx and Dekeyser (1953). The adjectives simple and complex used by Smith and Yoder (1956) represent a qualitative description, as well as the word standard used by Bailey (1980a). Zvyagin et al (1979) (see also Zvyagin 1988) introduced the notion of “condition of homogeneity3”, which identifies polytypes in which the position of any layer relative to the others and the transition from it to the adjacent ones, are the same or equivalent for all layers. These polytypes are called homogeneous polytypes; the remaining ones are called inhomogeneous polytypes. The condition of homogeneity is similar to the condition of the Maximum Degree of Order, but with less emphasis on chemical variations, and thus also on the symmetry distinguishing the three families. The main difference is that Zvyagin applies his condition to the entire crystal-chemical layer, whereas the algorithms for the derivation of MDO polytypes (Dornberger-Schiff et al. 1982, Dornberger-Schiff and Grell 1982) apply to OD layers or OD packets: the latter in micas roughly correspond to half-layers. Within the homo-octahedral approximation in micas, the procedures for the derivation of “simple”, “standard”, “homogeneous”, “MDO”, yield identical results (Table 2): this is however, in general, not true for other compounds, because the algorithms for the derivation of MDO polytypes are considerably different from those employed to derive “homogeneous” or “simple” polytypes. The difference becomes evident when considering that there are only 6 “simple” or “standard” polytypes (that become 8 when considering the non-congruent polytypes, i.e. counting separately each member of an enantiomorphous pair), but they simply correspond to homo-octahedral MDO polytypes. There are then 14 non-equivalent 3 In some texts, the Russian term “однородность” is translated as “uniformity” instead of “homogeneity”. Here we adopt the latter translation, closer to the original meaning.
Senaryf
Ternaryf
Complexe
Non-standard, groups I & II alternating
Senaryf
Non-standard, groups I & II mixed, non-alternating
>1 kind of triples of M layers within a given polytype
Inhomogeneous with mixed (parallel and antiparallel) and randomly alternating orientation of octahedra c
Non OD structures1
More than one type of triple of OD packets within a given polytype
Mixed-rotationb
Both 2nu60º and (2n+1)u60º
a
These are OD structures if the ditrigonal rotation of the tetrahedra is zero. Durovic et al (1984); bNespolo (1999); cZvyagin et al (1979) and Zvyagin (1988); dBailey (1980a); eSmith and Yoder (1956); f Ross et al (1966). “MDO” stands for “Maximum Degree of Order”.
1
Ternaryf
Simplee
Standard, groups I & II alternating
Non-standard, group Id
Inhomogeneous with octahedra parallel and antiparallel regularly alternatingc
Standard, group Id
Inhomogeneous with all octahedra parallelc
More than one kind of triples of M layers within a given polytype
Homogeneous with octa-hedra parallel and anti-parallel regularly alternatingc
More than one type of triple of OD packets within a given polytype
Subfamily B non-MDOa
(2n+1)u60º
All triples of M layers within a given polytype equivalent
Homogeneous with all octahedra parallelc
2nu60º Subfamily A non-MDOa
OD structures
Subfamily B MDOa
Subfamily A MDOa
All triples of OD packets within a given polytype equivalent
2O: 180º; 2M2: 60º and 300º; 6H: 60º or 300º
1M: 0º; 2M1: 120º and 240º; 3T: 120º or 240º
Relative rotations between successive M layers
Table 2. Comparative classification of mica polytypes in the homo-octahedral approximation.
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(22 non-congruent) polytypes in the meso-octahedral family, and 36 non-equivalent (60 non-congruent) polytypes in the hetero-octahedral family, which obey the condition of Maximum Degree of Order. Zvyagin’s “condition of homogeneity” applies to homo- and meso-octahedral polytypes, but not to the hetero-octahedral family. The reason for the derivation of the polytypes mentioned above is to single out, from the theoretically infinite number of periodic polytypes within a given family, those with relatively short periods in the stacking direction, which are most likely to be encountered in investigated specimens. To calculate theoretical single-crystal diffraction patterns is easy, provided that the structure of the single layer is known, and the distribution of intensities can then be used for their identification by simple visual comparison with patterns obtained experimentally (Weiss and Wiewióra 1986). Thus, it is irrelevant which set of polytypes as derived by different authors/schools is used, provided it fulfils its purpose, namely it allows the identification of the polytype. Identification of long-period (non-MDO) polytypes requires special algorithms exploiting the periodicity of the intensity distribution, and this is treated at the end of this chapter. SYMBOLIC DESCRIPTION OF MICA POLYTYPES The indicative symbolism developed by Ramsdell (1947) is not sufficiently informative for polytypes with more than 2-3 layers in the repeat unit. Because of the rapid increase of the number of possible polytypes with the number of layers in the repeat unit (Mogami et al. 1978; McLarnan 1981) the Ramsdell notation needs augmentation with another, descriptive symbolism, from which the structure, including its symmetry, can be reconstructed when the structure of the individual layer is known. Note that a symbol, which describes the stacking mode in an individual polytype, consists of a string of characters. Symbolism is a set of rules governing the construction of symbols. The symbols introduced to describe the stacking mode in mica polytypes can be broadly divided into two types, orientational (giving the absolute orientation of layers with respect to a space-fixed reference) and rotational (giving the relative rotations between pairs of layers). 1) Orientational symbols 1A) DA symbols. The first symbolic description is from Dekeyser and Amelinckx (1953), who used a set of vectors and numerical symbols to indicate the complete stagger of the layer, defined as the (001) projection of the vector connecting two (OH/F) sites on the two sides of the octahedral sheet. Six characters n = 1,2,3⎯,1⎯,2⎯,3 represent the stagger of the layer with respect to a space-fixed reference (Fig. 8). These symbols apply to the homo-octahedral approximation only and therefore cannot correctly describe polytypes containing M2 layers. Figure 8. Symbols used by Dekeyser and Amelinckx (1953) to indicate the orientation of a whole M layer. These symbols can be considered the predecessors of OD and Z symbols (cf. Fig. 4). With respect to the original figure in Dekeyser and Amelinckx (1953), the b and c axes have been taken in the opposite direction (b left instead of right, and c coming out from the plane instead of into) in accordance with the conventions of the International Tables for Crystallography.
1B) Z symbols. Zvyagin (1962) introduced a numerical/vectorial description giving the stacking of half-layers, as defined by the interlayer cations and the origin of the O sheet. This choice made Zvyagin’s symbols more general than the symbols introduced previously and also suitable for some other phyllosilicates. However,
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Zvyagin changed the notation three times. At first (Zvyagin 1962) he adopted the letters A, B, C, A,⎯B,⎯C to indicate the absolute orientation of the entire layer. He then adopted the characters σi and τi to indicate the intra- and inter-layers displacement of half-layers (Zvyagin 1967). Later (Zvyagin et al. 1979) the Greek letters were abandoned in favor of the corresponding Roman (si and ti) and with a sign inversion between τi and ti, to make homogeneous the definitions of si and ti. Finally, the “s” and “t” letters were dropped, leaving only their numerical subscripts as orientation characters (Zhukhlistov et al. 1990). These most recent symbols, and the vectors they represent, are here termed Z symbols and Z vectors. As for DA symbols, Z symbols are oriented symbols linked to a space-fixed, orthohexagonal reference with (a, b) axes in (001) plane (see also Zvyagin 1985). For nonorthogonal N-layer polytypes, the period along the c axis of this reference corresponds to 3N layers (Fig. 9). The vector connecting the origin of the octahedral sheet with the nearest interlayer site and vice versa, always looking at the sequence of layers in the same direction, is called intralayer displacement: its projection on the (001) plane has length |a|/3 and corresponds to the vector T2j-2 (or T2j-1, depending on which of the two half-layers is considered) in Figure 2. There are six possible orientations for each half layer, indicated by the six layer-fixed ai axes (i = 1~6). The projection of the intralayer displacement is indicated by the character i = 1,2,…6 when the ai axis is parallel to the space-fixed axis a (Fig. 4). The interlayer displacement is the vector giving the relative displacement between two adjacent layers: it can take any of the six orientations 1~6 described for the intralayer vector, and in some other phyllosilicates, also two independent orientations corresponding to ±b/3 (indicated as “+” and “–” respectively), but it can also be a zero vector (indicated as “0”). In micas, owing to the presence of interlayer cations, only the 0 interlayer displacement occurs. The (a, b) components (sx, sy) of Z vectors are given
Figure 9. The conventional monoclinic cell (dashed lines), the (pseudo)-orthohexagonal cell (solid lines), and the (pseudo)hexagonal cell [(001) base shaded] built overlapping three conventional cells. The scale along c is compressed (modified after Nespolo et al. 2000a).
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Nespolo & Ďurovič Table 3. OD symbols and Z symbols and the (a, b) components of the corresponding orientation vectors. OD symbol
Z symbol
(sx, sy)
3 2 1 0 5 4 + – 0
3 4 5 6 1 2 + – *
(1/3, 0) (1/6, 1/6) (-1/6, 1/6) (-1/3, 0) (-1/6, -1/6) (1/6, -1/6) (0, 1/3) (0, -1/3) (0, 0)
in Table 3. The complete symbolism, giving the stacking sequence of half layers, is ij0kl0mn0…. For micas containing only M1 layers, i=j, k=l, m=n etc.; the character 0 can be omitted and a shortened symbol IKM… is obtained (Zhukhlistov et al. 1990). M2 layers always correspond to intralayer displacement of the same parity; opposite parity would in fact produce a trigonal prismatic coordination for the Y cations. The Z vector for each layer corresponds to the (001) projection of a pair of intralayer displacement vectors and it is obtained by summing their (sx, sy) components. For micas built by M1 layers only, this is equivalent to twice the components, namely (2sx, 2sy) (Table 3). Z vectors are thus twice as long as DA stacking vectors (and also SY vectors, described below), and directed in the opposite way. Since ±2/3 is translationally equivalent to ∓1/3, in practice the (a, b) components of the Z vectors are the same as those of the intralayer displacements, but with the signs interchanged. The DA and SY stacking vectors are the (001) projections of vectors not passing through a cationic site in the O sheet. On the other hand, Z vectors are the (001) projections of vectors passing through that cationic site. As a consequence, Z vectors can distinguish between M1 and M2 layers, whereas the other two cannot. The latter simply correspond to the vector sum of Z vectors. The fundamental merit of Z symbols is that they can describe also mesooctahedral polytypes. Their shortcoming is that the symbols describing homooctahedral mica polytypes are identical with those describing meso-octahedral polytypes consisting of M1 layers, and additional information must be given also. Moreover, in their present form, they cannot handle hetero-octahedral polytypes. 1C) OD symbols. The OD school, inspired by Z symbols, derived the most general symbols to describe mica polytypes (Ďurovič and Dornberger-Schiff 1979; Dornberger-Schiff et al. 1982; Backhaus and Ďurovič 1984; Ďurovič et al. 1984; Weiss and Wiewióra 1986). These symbols consist of a sequence of characters referring to one period, placed between vertical bars; two lines of characters are used; the first line indicates the packet orientations, and the second line the packetto-packet displacements. A dot “.” separating the orientational characters for packets p2j and q2j+1 indicates the position of Oc layer. The OD symbols are thus expressed:
T0
⋅ T1 v0,1
T2 *
⋅ T3 v 2,3
*
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where Tj = 0~5, v2j,2j+1 = T2j+T2j+1 (v, T are the vectors corresponding to v and T characters, and the vector sum is taken modulo primitive hexagonal cell), and * indicates null vector (no displacement) (Fig. 4). Note that the parity of the orientational characters is necessarily opposite to that of the displacement characters. The vector sum of v2j,2j+1 over a complete polytype period (hereafter indicated as Σv, for shortness) corresponds to the cn projection of the c axis onto the (001) plane and gives the total displacement, which can correspond to the characters “0”~”5”, “*”, “+” and “–“ (see Tables 3 and 4). In the meso-octahedral family, the v2j,2j+1 characters in the second line are redundant because they follow unequivocally from the T2j·T2j+1··· characters in the first line: simplified symbols |T0 · T1 T2 · T3 …| can also be used. In the hetero-octahedral family the chirality of the packets is taken into account: right- and left-handed packets are indicated by a prime (′) or double prime (“), respectively, substituting the dot, where the chirality is conventionally determined by the direction connecting Ma to Mi (Fig. 10) (Ďurovič et al. 1984). Also in this case the v2j,2j+1 characters in the second line are redundant, and simplified symbols T2j ′ T2j+1 or T2j ″ T2j+1 for the individual packet pairs can be used. Although the v2j,2j+1 displacement characters are redundant in both these families, their vector sum Σv, as shown in the next section, allows the classification of mica polytypes in terms of their reticular features: the complete two-line symbols yield thus additional information. Finally, in the homo-octahedral family, there are only two distinguishable orientations of the packets. This follows from the fact that the Oc layer here is H centered and it can be attached to the Tet layer so that its three equivalent origins can be reached simultaneously by the three T vectors with evenor odd (uneven)- numbered characters, respectively. These two orientations of a packet, differing by a 1800 rotation, are indicated by orientational characters e and u, respectively. In this case, the first line of characters is redundant and simplified symbols consisting just of the line of displacement v2j,2j+1 characters may be sufficient (Dornberger-Schiff et al. 1982). The OD symbols for the packet orientations were defined with respect to hexagonal axes A1, A2, A3 as indicated in Figure 4. In Table 3 they are described in terms of the (a, b) orthohexagonal identity periods. The orthohexagonal cell used in the OD literature corresponds to the C2 setting (Fig. 1). In practice, for the mesooctahedral family, the OD symbols correspond to (6-Z)(mod 6), where Z are Zvyagin’s characters. For the hetero-octahedral family the same numerical relation holds, but the chirality of the packets is considered. For the homo-octahedral family, Zvyagin uses one of the three e- or u-vectors as representative: 6 or 3, respectively. Going from 0 to 5 instead than from 1 to 6, the OD symbols obey the closure property of the mod function. The corresponding OD vectors are disposed in a clockwise sequence, whereas Z vectors are defined counter-clockwise (Fig. 4), but their crystal chemical basis is the same. 1D) TS symbols (Sadanaga and Takeda 1969; Takeda and Sadanaga 1969) give the relative positions of the TS unit layers and are written as a sequence of N symbols Lj(ΔXj, ΔYj), j = 1-N, where Lj is the type of layer and N is the number of layers in the polytype period. Considering two successive repeats of N layers, (ΔXj, ΔYj) are the (a, b) components of the vector connecting the origin of the last (N-th) layer of a repeat and the origin of the j-th layer of the next repeat (Fig. 2). These symbols respect only the homo-octahedral approximation. 2)
Rotational symbols
2A) SY vectors. Smith and Yoder (1956) described the stacking sequence in a way similar to Dekeyser and Amelinck (1953). The stacking vectors are defined as the (001) projection of the vector connecting two nearest interlayer cations on the two
0 1 2 3 4 5 * + –
3 – 1 * 5 + 0 4 2
0 – 4 + 2 * 0 1 3 5
1 1 + 5 – 3 * 2 0 4
2 * 2 – 0 + 4 3 5 1
3
OD
5 * 3 + 1 – 4 2 0
4 + 0 * 4 – 2 5 1 3
5 0 1 2 3 4 5 * + –
* 2 3 0 5 2 1 + – *
+ 4 5 4 1 0 3 – * +
– 6 1 2 3 4 5 0 + –
3 – 5 0 1 + 6 2 4
6 – 2 + 4 0 6 5 3 1
5 5 + 1 – 3 0 4 6 2
4 0 4 – 6 + 2 3 1 5
3
Z
1 0 3 + 5 – 2 4 6
2
+ 6 0 2 – 4 1 5 3
5
6 5 4 3 2 1 0 + –
0
4 3 6 1 4 5 + – 0
+
2 1 2 5 6 3 – 0 +
–
Table 4. Table of vector sums for the nine possible displacement vectors appearing in the structure of most common phyllosilicates. The individual vectors ¢v² are represented by their respective characters “v” and the result of summation should be taken modulo primitive hexagonal cell (cf. Fig. 4). These nine vectors form a translation group with vector addition as the group operation.
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Figure 10. Construction of the stacking symbol for hetero-octahedral mica polytypes demonstrated on two one-layer (two-packet) polytypes, through the (001) projection of the Oc layer. Gray squares indicate the position of OH groups (coinciding in the projection to the interlayer cations). Shaded octahedra contain M1 (trans) sites. Thick hexagons are drawn through the lower and upper apical oxygen atoms, as in Durovic et al. (1984, Fig. 5). Thick solid arrows are orientational vectors T2j and T2j+1, thick dotted arrows are displacement vectors v2j,2j+1. The chirality (enantiomorphous hand) is determined by the curved arrow leading from Mi to Ma around the upper OH group: clockwise = right-handed, counter-clockwise = left-handed. Mi and Ma stand for octahedral sites with (Mi)nimal and (Ma)ximal average X-ray scattering power, respectively (modified after Durovic et al.1984).
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Nespolo & Ďurovič sides of a layer. Becuase interlayer cations and (OH, F) groups overlap in the (001) projection, in practice the methods of Dekeyser and Amelinckx (1953) and of Smith and Yoder (1956) are equivalent; however, Smith and Yoder (1956) did not adopt a symbolic notation. Also these vectors are correct only in the homo-octahedral approximation and cannot thus describe correctly polytypes containing M2 layers.
2B) RTW symbols. Ross et al (1966) introduced a numerical description (RTW symbols) giving the relative rotations between successive stacking vectors representing a sequence of M1 layers. This description is the most immediate, although not the most general (it applies to the homo-octahedral approximation only), to describe the mica-polytype stacking mode and to derive all possible mica polytypes with a given number of M1 layers (Takeda 1971; Mogami et al. 1978; McLarnan 1981). However, the method cannot distinguish between M1 and M2 layers. RTW symbols are orientation-free, rotational symbols written as a sequence on N characters Aj = 0,±1,±2,3, the j-th character giving the rotation angle between jth and (j+1)-th layers as integer multiple of 60º. A RTW symbol corresponds to the difference, with sign inverted, between pairs of displacement OD characters [Aj = – (v2j,2j+1- v2j-2,2j-1)] or to the difference between pairs of Z characters corresponding to successive M1 layers [Aj = +(Z2j+1-Z2j-1)]. The opposite sign between OD and Z symbols originates from the fact that Z and RTW symbols are defined counterclockwise, whereas OD symbols are defined clockwise. The closure of the periodicity after N layers is expressed by the condition (Takeda 1971):
∑
N j =1
Aj = 0 ( mod 6 )
(1)
2C) Thompson’s symbols. Thompson (1981) introduced an operatorial description of mica stacking, in which operators N1 and N-1 (N = 1,6) produce 2π/N counterclockwise (N1) or clockwise (N-1) rotation of the M layer. These operators are divided into dot [N = 1(mod 2)] and cross operators [N = 0(mod 2)]. Bailey (1980a,b) gave an alternative description of the polytypism of the micas, by classifying the six possible directions of the stagger of the tetrahedral sheets within a layer (positive and negative directions of the three hexagonal axes in the plane of the layer). The six possible positions of octahedral cations with respect to a space-fixed reference were divided into two groups, labeled I (negative stagger) and II (positive stagger). The first layer of each polytype was kept with tetrahedral stagger along –a1 (octahedral cation positions I): as a consequence, the axial setting used to derive the polytypes was not the most suitable to identify polytypes from their diffraction pattern, and a final axial transformation is necessary. Subfamily A, subfamily B and mixedrotation polytypes correspond to sequences of octahedral cations belonging to group I only, to groups I and II alternating, and to groups I and II mixed non-alternating. Bailey’s notation cannot distinguish between M1 and M2 layers and is not adopted here. We instead make reference to OD and Z (collectively termed “orientational symbols” when referring to both, for shortness) and to RTW symbols. RETICULAR CLASSIFICATION OF POLYTYPES: SPACE ORIENTATION AND SYMBOL DEFINITION Mica polytypes can belong to five symmetries: H, T, O, M and A (Takeda 1971). In both the Pauling and the Trigonal models, the lattice of triclinic polytypes is metrically monoclinic, and the (001) projection of the c axis, labeled cn, can take three values: 0, |a|/3, |b|/3, on the basis of which mica polytypes are classified into orthogonal, Class a and Class b respectively. The number N of layers building a polytype can be expressed as:
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N = 3n(3K+L) (K and n non-negative integers; L=1 or 2) (2) where n defines the Series and L the Subclass; K is a constant entering in the transformation matrices relating axial settings (Nespolo et al. 1998). The structural model of each polytype, as described by the stacking vectors, has six possible orientations with respect to the space-fixed (a, b) axes, each 60º apart. These orientations correspond to one sequence of characters in the RTW symbols, but to six different sequences of orientational symbols, and are in general non-equivalent. The cn projection may correspond to Σv = 〈*〉 (orthogonal polytypes), Σv = 〈0〉~〈5〉 (Class a polytypes), or to Σv = 〈+〉 or 〈–〉 (Class b polytypes), i.e. to cn = (0, 0), (±1/3, [0, ±1/3]) and (0, ±1/3) respectively. For non-orthogonal polytypes, cn can be reduced to ⎯(1/3, 0) (Class a) or (0,⎯1/3) (Class b) by means of the C-centering vectors and by rotating the structural model around c*. These six orientations can be grouped in the following way (Nespolo et al. 1999d). 1. Class a polytypes. Each orientation corresponds to a different cn projection, i.e. to a different character of Σv, from 〈0〉 to 〈5〉. Among these, there is only one that corresponds to the b-unique setting with an obtuse β angle: that with Σv = 〈0〉, i.e. cn = (1/3, 0). 2. Class b polytypes. Three orientations correspond to Σv = 〈+〉, i.e. cn = (0, 1/3) (acute α angle) and three others to Σv = 〈–〉, i.e. cn = (0,⎯1/3) (obtuse α angle). The three orientations with the same Σv (cn) are equivalent for triclinic polytypes, but not for monoclinic cases. The symmetry elements are oriented according to an a-unique setting with α obtuse. Only one of the three orientations leading to Σv = 〈–〉 agrees with such a requirement. 3. Orthogonal polytypes. The six orientations correspond to Σv = 〈*〉, i.e. cn = (0, 0), and they are equivalent for hexagonal, trigonal and triclinic polytypes, whereas for orthorhombic and monoclinic polytypes only two orientations, related by 180º rotation around c = c* axis, lead to the correct orientation of the symmetry elements. Because both the reticular features and the OD character are based on the geometry of the layer stacking in polytypes, some relations between the OD and the reticular classifications can be established (Backhaus and Ďurovič 1984; Nespolo 1999). 1. Subfamily A. These polytypes are described by orientational symbols with characters of the same parity, i.e. by all-even characters in the RTW symbol. They include the three most common MDO polytypes (1M, 2M1, 3T) and the great majority of nonMDO polytypes found so far. Successive layers are related by 2n×60º rotations; the x component of the stacking vector of each packet (half-layer) is either always +1/3 (odd orientational parity of characters in the orientational symbols) or always -1/3 (even orientational parity of characters in the orientational symbols). As a consequence, in Series 0 [n = 0 in Equation (2), i.e. polytypes with the number of layers not a multiple of 3] the x component of cn cannot be 0 and these polytypes belong to Class a. In Series higher than 0, the number of layers building the polytypes is a multiple of 3 and thus Σv is 〈*〉, 〈+〉 or 〈–〉 (the x component of cn is always 0). Therefore, these polytypes cannot belong to Class a. 2. Subfamily B. These polytypes are described by orientational symbols with characters of alternating parity, i.e. by all-odd characters in the RTW symbol. Successive layers are related by (2n+1)×60º rotations. Only polytypes with an even number of layers appear in this subfamily. In addition, because layers with different orientational parity have an opposite x component of the stacking vector, Σv is 〈*〉, 〈+〉 or 〈–〉 and it is not possible to have a Class a polytype. 3. Mixed-rotation polytypes. These polytypes correspond to orientational symbols with
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character of different, non-alternating parity and to mixed parity of the characters in the RTW symbol. Because there is no definite rule for the layer orientational parity sequence, the three kinds of polytypes (orthogonal, Class a, Class b) are possible. LOCAL AND GLOBAL SYMMETRY OF MICA POLYTYPES FROM THEIR STACKING SYMBOLS The main purposes of descriptive stacking symbols are: 1) to uniquely identify a polytype; 2) to enable the reconstruction of the structure of the polytype once the structure of the layer is known; 3) to enable a symmetry analysis of the polytype, not only for the systematic derivation of MDO polytypes but also to determine the symmetry (local and global) of a polytype from its symbol in a purely analytical way – without the need to draw auxiliary pictures (although these may be quite useful to visualize the stacking sequence); and 4) to calculate the Fourier transform of the polytype. It is thus necessary to know how the individual point operations influence the individual characters in the symbol. For mica polytypes, there are 24 point operations constituting the point group 6/mmm. The effect of each of them on the six vectors corresponding to the orientational symbols can be expressed in a general form, e.g., a 60º clockwise rotation converts an OD vector 〈j〉 into 〈j+1〉, or a Z vector 〈j〉 into 〈j-1〉, but as a “working tool” it is more convenient to compile a table of conversions to give the results explicitly. Note that the vectors given in Figure 2 and 4 are actually the (001) projections of the intralayer stacking vectors that give the absolute orientations of packets (half-layers in Zvyagin’s concept). The transformation of these projections is almost trivial for τ-operations, whereas it must be combined with an inversion for ρ-operations because the stacking vector must always to point in the same direction, namely along +c. For example, a 180º rotation around the b axis (H-M: [. . . (.) . . 2], ORT : 2[010] in Table 5a) converts a vector 〈0〉 into vector 〈3〉 but such a vector would point along -c. The corresponding vector directed along +c is 〈0〉. It follows that the 2[010] rotation applied to a packet p2j = 〈0〉 yields a packet q2j+1 = 〈0〉. Tables 5a and 5b give the conversion rules for OD symbols and Z symbols, respectively. Moreover, a τ-operation leaves unchanged the order of the sequence of characters in the orientational symbol, whereas a ρ-operation inverts it. The effect of the 24 point operations of the point group 6/mmm on the entire orientational symbol is given in Table 6. Derivation of MDO polytypes The derivation of MDO polytypes for homo-, meso-, and hetero-octahedral micas were described in detail by Backhaus and Ďurovič (1984). Therefore, only basic ideas are given here. The first step is to construct all packet triples compatible with the crystal chemistry. The use of meso-octahedral micas demonstrates the procedure. Let us take an M1 layer and begin with the packet p0 in the orientation 0 (any other initial orientation could be used). The packet q1 must then be also in the orientation 0 to preserve M1 layer. The packetpair is then 0 . 0 because the sum of the two orientational vectors 〈0〉 + 〈0〉 = 〈3〉, where 3 〈3〉 is the displacement vector (cf. Fig. 4). A single meso-octahedral packet has the symmetry C1m(1) but in the following we shall not consider the translations of the layer group. The point group m has the order 2 and consists of two operations: the identity (an operation of the first sort, whose transformation matrix has determinant +1) and the reflection (an operation of the second sort, whose transformation matrix has determinant −1). Therefore, any transformation of such a packet consists always of two operations with
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Table 5a. Conversion of characters appearing in the OD symbols of mica polytypes. The individual operations are characterized by their extended Hermann-Mauguin (H-M) symbols and by the corresponding operations indexed in orthogonal (ORT) C2 –setting axes. Cf. Table 1 and Backhaus and Durovic (1984).
τ-point operations H-M 1 (6)-1 (3)-1 (2)1 (3)1 (6)1
ORT 1 (6)-1 (3)-1 (2)1 (3)1 (6)1
Character conversion by point operation j: j: 1+j: 2+j: 3+j: 4+j: 5+j:
012345 012345 123450 234501 345012 450123 501234
eu* eu* ue* eu* ue* eu* ue*
ρ-point operations H-M ⎯1 ⎯(6)-1 ⎯(3)-1 ⎯(2)1 = m(001) ⎯(3)1 ⎯(6)1
′—′ ′′—′′
′—′′ ′′—′
τ-point operations H-M [m . . (.) . . .] [. . . (.) . m .] [. . m (.) . . .] [. . . (.) m . .] [. m . (.) . . .] [. . . (.) . . m] ′—′′ ′′—′
ORT ⎯1 ⎯(6)-1 ⎯(3)-1 ⎯(2)1 = m(001) ⎯(3)1 ⎯(6)1
ORT m(⎯130) m(⎯110) m(100) m(110) m(130) m(010)
Character conversion by point operation j: 5-j : 4-j : 3-j : 2-j : 1-j : -j :
012345 543210 432105 321054 210543 105432 054321
eu* ue* eu* ue* eu* ue* eu*
ρ-point operations H-M [2 . . (.) . . .] [. . . (.) . 2 .] [. . 2 (.) . . .] [. . . (.) 2 . .] [. 2 . (.) . . .] [. . . (.) . . 2]
ORT 2[⎯110] 2[⎯310] 2[100] 2[310] 2[110] 2[010]
′—′ ′′—′′
Table 5b. Conversion of characters appearing in the Zvyagin symbols of mica polytypes. The individual operations are characterized by their extended Hermann-Mauguin (H-M) symbols and by the corresponding operations indexed in orthogonal (ORT) C2–setting axes. Cf. Table 1 and Zvyagin (1997).
τ-point operations H-M 1 (6)-1 (3)-1 (2)1 (3)1 (6)1 [m . . (.) . . .] [. . . (.) . m .] [. . m (.) . . .] [. . . (.) m . .] [. m . (.) . . .] [. . . (.) . . m]
ORT 1 (6)-1 (3)-1 (2)1 (3)1 (6)1 m(⎯130) m(⎯110) m(100) m(110) m(130) m(010)
Character conversion by point operation j: j: 5+j : 4+j : 3+j : 2+j : 1+j : 1-j : 2-j : 3-j : 4-j : 5-j : -j :
654321 654321 543216 432165 321654 216543 165432 123456 234561 345612 456123 561234 612345
0 0 0 0 0 0 0 0 0 0 0 0 0
ρ-point operations H-M ⎯1 ⎯(6)-1 ⎯(3)-1 1 ⎯(2) = m(001) ⎯(3)1 ⎯(6)1 [2 . . (.) . . .] [. . . (.) . 2 .] [. . 2 (.) . . .] [. . . (.) 2 . .] [. 2 . (.) . . .] [. . . (.) . . 2]
ORT ⎯1 ⎯(6)-1 ⎯(3)-1 1 ⎯(2) = m(001) ⎯(3)1 ⎯(6)1 2[⎯110] 2[⎯310] 2[100] 2[310] 2[110] 2[010]
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Nespolo & Ďurovič Table 6. Transformation rules for OD and Z symbol under the effect of the λsymmetry operations of the hexagonal syngony. 〈i′〉,〈j′〉,…., 〈p′〉 (OD symbols) and 〈i〉,〈j〉,.…,〈p〉 (Z symbols) are the original symbols. The individual operations are characterized by their extended Hermann-Mauguin (H-M) symbols and by the corresponding operations indexed in orthogonal (ORT) C2 –setting axes. Cf. Table 1 (modified after Nespolo et al. 1999).
τ-point operation H-M
ORT
effect on OD symbol sequence
1 (6)-1 (3)-1 (2)1 (3)1 (6)1 [m . . (.) . . .] [. . . (.) . m .] [. . m (.) . . .] [. . . (.) m . .] [. m . (.) . . .] [. . . (.) . . m]
1 (6)-1 (3)-1 (2)1 (3)1 (6)1 m(⎯130) m(⎯110) m(100) m(110) m(130) m(010)
〈i′〉,〈j′〉,….,〈p′〉 〈1+i′〉,〈1+j′〉,….,〈1+p′〉 〈2+i′〉,〈2+j′〉,….,〈2+p′〉 〈3+i′〉,〈3+j′〉,….,〈3+p′〉 〈4+i′〉,〈4+j′〉,….,〈4+p′〉 〈5+i′〉,〈5+j′〉,….,〈5+p′〉 〈5-i′′〉,〈5-j′′〉,….,〈5-p′′〉 〈4-i′′〉,〈4-j′′〉,….,〈4-p′′〉 〈3-i′′〉,〈3-j′′〉,….,〈3-p′′〉 〈2-i′′〉,〈2-j′′〉,….,〈2-p′′〉 〈1-i′′〉,〈1-j′′〉,….,〈1-p′′〉 〈-i′′〉,〈-j′′〉,….,〈-p′′〉
〈i〉,〈j〉,…. 〈p〉 〈5+i〉,〈5+j〉,….,〈5+p〉 〈4+i〉,〈4+j〉,….,〈4+p〉 〈3+i〉,〈3+j〉,….,〈3+p〉 〈2+i〉,〈2+j〉,….,〈2+p〉 〈1+i〉,〈1+j〉,….,〈1+p〉 〈1-i〉,〈1-j〉,….,〈1-p〉 〈2-i〉,〈2-j〉,….,〈2-p〉 〈3-i〉,〈3-j〉,….,〈3-p〉 〈4-i〉,〈4-j〉,….,〈4-p〉 〈5-i〉,〈5-j〉,….,〈5-p〉 〈-i〉,〈-j〉,….,〈-p〉 Effect on Z-symbol sequence
ρ-point operation
effect on Z-symbol sequence
H-M
ORT
effect on OD symbol sequence
⎯1 ⎯(6)-1 ⎯(3)-1
⎯1 ⎯(6)-1 ⎯(3)-1
〈p′′〉,…., 〈j′′〉,〈i′′〉 〈1+p′′〉,…,〈1+j′′〉,〈1+i′′〉 〈2+p′′〉,…,〈2+j′′〉,〈2+i′′〉
〈p〉….,〈j〉,〈i〉 〈5+p〉,….,〈5+j〉,〈5+i〉 〈4+p〉,….,〈4+j〉,〈4+i〉
⎯(2)1 = m(001)
⎯(2)1= m(001)
〈3+p′′〉,…,〈3+j′′〉,〈3+i′′〉
〈3+p〉,….,〈3+j〉,〈3+i〉
⎯(3)1 ⎯(6)1 [2 . . (.) . . .] [. . . (.) . 2 .] [. . 2 (.) . . .] [. . . (.) 2 . .] [. 2 . (.) . . .] [. . . (.) . . 2]
⎯(3)1 ⎯(6)1 2[⎯110] 2[⎯310] 2[100] 2[310] 2[110] 2[010]
〈4+p′′〉,…,〈4+j′′〉,〈4+i′′〉 〈5+p′′〉,…,〈5+j′′〉,〈5+i′′〉 〈5-p′〉,….,〈5-j′〉,〈5-i′〉 〈4-p′〉,….,〈4-j′〉,〈4-i′〉 〈3-p′〉,...,〈3-j′〉,〈3-i′〉 〈2-p′〉,…,〈2-j′〉,〈2-i′〉 〈1-p′〉,….,〈1-j′〉,〈1-i′〉 〈-p′〉,….,〈-j′〉,〈-i′〉
〈2+p〉,….,〈2+j〉,〈2+i〉 〈1+p〉,….,〈1+j〉,〈1+i〉 〈1-p〉,….,〈1-j〉,〈1-i〉 〈2-p〉,….,〈2-j〉,〈2-i〉 〈3-p〉,….,〈3-j〉,〈3-i〉 〈4-p〉,….,〈4-j〉,〈4-i〉 〈5-p〉,….,〈5-j〉,〈5-i〉 〈-p〉,….,〈-j〉,〈-i〉
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the above properties. Accordingly, in the packet pair . 03. 0 the two packets are related . simultaneously with two ρ–operations, and a look at Table 5a shows that these, converting 0. into .0 and 3 into itself are [. . .(.) . . 2] ≡ 2[010] and an inversion. The packet pair has thus the point symmetry 2/m. The packet triples p0q1p2 compatible with the Trigonal model in the subfamily A are 0 . 0 0 , 0 . 0 2 plus its enantiomorphous 0 . 0 4 . The two 02τ-operations converting p0 into p2 3 * 3 * 3 * are in the first case (Table 5a) a translation (isogonal with the identity) and a glide operation (isogonal with […(.)..m(010)]). A continuation of any of these τ-operations leads to the same string …. 03. 0*03. 0*03. 0* … because any of them converts also displacement characters 3 in the same way. This string has modulus 03. 0 , the vector 〈3〉 is the interlayer vector of this one-layer monoclinic polytype. However, because 〈3〉 = +a/3 (acute β angle), it must be re-oriented by a rotation of 180o around c* to bring it into the standard, second setting (obtuse β angle). Evidently, this can be made (Table 5a) by adding 3 to all characters, thus 30. 3 is obtained. The basis vectors of this 1M polytype with symmetry C12/m1, are a, b, c0-a/3, where c0 is a vector perpendicular to the layer planes with length corresponding to the “layer width” (e.g., a distance between two closest planes of interlayer cations). The two 02τ-operations for the triple 03. 0*2 are (Table 5a) a clockwise three-fold screw rotation (first sort operation, isogonal with (3)-1 ≡ 3[001]) and a glide operation (second sort operation, isogonal with [. . . (.) m . .] ≡ m(110)). A continuation of the (3)-1, through a step-by step application onto the characters in the starting triple, converts 0→2, then 2→4 and 4→0 but also 3→5, 5→1 and 1→3, which closes the period. The resulting symbol 03. 0*25. 2*4.4 characterizes a three-layer, trigonal 3T polytype with symmetry P3212 1 * and basis vectors a1, a2, 3c0. A continuation of [. . . (.) m . .] ≡ m(110) converts 0→2 but then 2→0, and 3→5, 5→3 which closes the period. The symbol 03. 0*25. 2* describes a twolayer monoclinic polytype (glide operation is the global operation here) with an interlayer vector equal to the sum of the two displacement vectors 〈3〉 + 〈5〉 = 〈4〉 (cf. Fig. 4). Also this polytype must be clockwise rotated by 120o, by adding 2 to all characters, to bring it , the 2M1 polytype with symmetry into the standard setting. The final form is 25. 2*4.4 1 * C12/c1 and basis vectors a, b, 2c0-a/3. The packet triple 03. 0*4 , enantiomorphous to the previous example, yields analogous 2 . 2 , a 3T polytype with symmetry results. The continuation of the (3)1 gives string 03. 0*4.4 1 * 5 * P3112, the enantiomorphous counterpart to P3212. The continuation of [. . . (.) . m .] ≡ m 0 . 0 4.4 (⎯110)) gives a preliminary symbol 3 * 1 * and, after re-orientation by an anti-clockwise 2 . 2 , which is the same 2M polytype, just with another choice of rotation by 120º, 4.4 1 1 * 5 * origin on the glide plane. This example is instructive: from a pair of packet triples which are enantiomorphous to each other, we obtain, in general, three non-congruent MDO polytypes. Two of them, generated by first-sort operations, contain only packet triples of the one or of the other kind, and these two polytypes form an enantiomorphous pair. The third polytype, generated by second-sort operations, contains both kinds of packet triples, regularly alternating, and it is thus obtained twice in the process of the derivation of MDO
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polytypes. Thus, for the meso-octahedral micas of the subfamily A we have obtained the three known MDO polytypes constructed by M1 layers: 1M, 2M1 and 3T. The subfamily B is handled in the same way, yielding MDO polytypes 2O, 2M2 and 6H containing M1 layers. The derivation of MDO polytypes containing M2 layers is in principle the same; there is just a circumstance that not all the 02τ-operations constructed mechanically are suitable as MDO-generating operations. These details, however, are outside the scope of this paper and the reader should consult Backhaus and Ďurovič (1984). The list of all homo-, meso- and hetero-octahedral MDO polytypes can be found in Table 7, in context with relations of homomorphy described below. As shown above, whereas monoclinic and orthorhombic polytypes have to be oriented according to the crystallographic conventions, this is irrelevant for orthogonal polytypes of the triclinic, trigonal and hexagonal syngonies. Thus, e.g., the symbol for the 3T polytype is “equally good” in any of the six possible orientations. In general: any of the mutually congruent strings of characters describe the same polytype. The symmetry analysis from a polytype symbol The two meso-octahedral MDO polytypes derived in the previous section is now used to demonstrate a “reverse” procedure: to read-out the local and global symmetry from the descriptive symbol. The permanent use of Table 5a (or Table 5b, if Z symbols are to be analyzed) is not emphasized at every step. Before starting such a task, we must check the formal correctness of a symbol: the parity of any displacement character must be opposite to that of the two orientational characters above it which, in turn, must have the same parity. Also the rule T2j + T2j+1 = v2j,2j+1 must be observed. Otherwise, the symbol is wrong. Let us take an extended (more than one identity period) string of characters corresponding to the 3T polytype, which has six packets within the identity period: ...2 . 2 4.4 0 . 0 2 . 2 4.4 0 . 0 2 . 2... 5 * 1 * 3 * 5 * 1 * 3 * 5
τ-operations. Evidently, 02[3-1] is the only global non-trivial τ-operation because it converts any packet p or q into p+2 or q+2. In addition, there is a trivial 06τ-operation: a translation by the identity period (and its multiples, of course). The global 04[31] is a consequence. Other τ-operations are only local. The three packet pairs 03. 0 , 25. 2 and 4.4 1 have [. . . (.) . . m], [. . . (.) . m .] and [. . . (.) m . .] respectively as local operations converting each of the packet pair into itself, but these operations do not hold for the neighboring packets. There are also other local τ-operations. For example, the 02τ-glide reflection isogonal with [. . . (.) m . .], which is the MDO-generating (global) operation for the polytype 2M1 (see above), has only local character in the 3T polytype. ρ–operations. Within the packet pair 03. 0 there are two ρ–operations 01⎯[1] and 01[. . . (.) . . 2]. The latter converts not only 0→0, 2→4, and 4→2 (i.e. it mutually converts the two neighboring packet pairs), but also the entire string of characters into itself. Thus, this two-fold rotation is global. Similar statements hold for all two-fold rotations converting any p2j into q2j+1. And analogous results are obtained also for all twofold rotations converting q2j+1 into p2j+2, i.e. those, operating across the interlayer e.g., 12[. . . (.) 2 . .] converts 0→2, 2→0, 4→4, etc., for the entire string. All of the two-fold axes are in the inter-axial directions so that the space-group type is P3212.
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The inversions valid for each packet pair p2j q2j+1 are only local operations. If a string of characters corresponds to a centrosymmetric polytype, then this string, starting and ending with the same character(s), read forwards and backwards, must remain the same. This is not the case in polytype 3T. Let us now consider the meso-octahedral MDO polytype 2M1 derived above, already 2 . 2 . The only non-trivial τ-operation here is the glide in the standard orientation 4.4 1 * 5 * operation isogonal with [. . . (.) . . m], the local mirror reflections hold only for individual M1 layers as in the previous case, and also other τ-operations are local. On the other hand, this polytype is centrosymmetric. This becomes evident if we write down an extended string of characters so that it will contain an odd number of packet pairs pq. ... 4.4 2 . 2 4.4 2 . 2 4.4 2 . 2 4.4 ... 1 * 5 * 1 * 5 * 1 * 5 * 1
This symbolism remains the same when read forwards and backwards. In a way similar to the above, also all the two-fold rotations [. . . (.) . . 2] can convert any q2j+1 into p2j+2 and are global: 2→4, 4→2 5→1, 1→5. The other two-fold rotations, converting any p2j into q2j+1, remain local. The space-group type of this polytype is thus C12/c1, taking into account the C-centering with respect to the orthogonal axes a, b. As an example of a polytype containing also M2 layers, we consider the mesooctahedral polytype identified by the OD symbol |2.4 0.0| (Z symbol 420660). The extended string of characters containing an odd number of packet pairs is: ... 2 . 4 0 . 0 2 . 4 0 . 0 2 . 4 0 . 0 2 . 4 ... 3 * 3 * 3 * 3 * 3 * 3 * 3
from which it clearly appears that the polytype is non-centrosymmetric. The only global τ-operation is the trivial 04τ translation (and its multiples) corresponding to the identity 2.4 period. The packet pair 3 has no local τ-operations, but 01[…(.)..2] ≡ 2[010] as a local ρoperation. The λ-symmetry is C12(1) and the pair of packets corresponds to an M2 layer. 0.0 Instead, as seen in the example of 3T, the packet pair 3 has […(.)..m] ≡ m(010) as a local τ-operation, and 23⎯[1] and 23[. . . (.) . . 2] ≡ 2[010] as local ρ-operations. The λ-symmetry is C12/m(1) and the pair of packets corresponds to a M1 layer. The only global ρ-operation for the polytype is [. . .(.) . . 2] ≡ 2[010] located at both the Oc layers. The space-group type is C2. The complete analysis for the 8 meso-octahedral polytypes of Class a with period up to 2 layers is given in Table 8.
IV
3T
2M1
II
III
1M
Ramsdell symbol
I
Homomorphous MDO group
§ ¨ © e. e e. e e.e 3 * 5 * 1 *
e. e e.e e. e 3 * 1 * 5 *
e.e e.e 1 * 5 *
u.u 0 *
Homo
· ¸ ¹ § ¨ ©
§ ¨ ©
§ ¨ ©
1.5 0 *
Non-MDO
4. 2 0. 4 2.0 3 * 5 * 1 *
2. 4 0.2 4. 0 3 * 1 * 5 *
2. 4 4. 0 0.2 3 * 5 * 1 *
4. 2 2.0 0. 4 3 * 1 * 5 *
0. 0 2. 2 4.4 3 * 5 * 1 *
0. 0 4.4 2 . 2 3 * 1 * 5 *
Non-MDO
4.4 2.2 1 * 5 *
2.4 4.2 3 * 3 *
5.1 0 *
3.3 0 *
· ¸ ¹
· ¸ ¹
· ¸ ¹
subfamily A OD symbol Meso
|4’2 4”0 2’0 2”4 0’4 0”2| (|2”4 2’0 4”0 4’2 0”2 0’4|) |2’4 0”4 0’2 4”2 4’0 2”0| (|4”2 0’2 0”4 2’4 2”0 4’0|)
|2”4 0”2 4”0| (|4’2 0‘4 2’0|)
|2’4 0’2 4’0| (|4”2 0”4 2”0|)
|4”2 2”0 0”4| (|2’4 4’0 0’2|)
|4’2 2’0 0’4| (|2”4 4”0 0”2|)
|0’0 2”2 4’4 0”0 2”2 2’2 4”4| (|0”0 4’4 2”2 0’0 4”4 2’2|)
|0’0 4’4 2’2| (|0”0 2”2 4”4|) |0”0 4”4 2”2| (|0’0 2’2 4’4|)
|0’2 0”4| |0”2 0’4|
|4’4 2”2|
|2”4 4’2|
Hetero |3’3| (|3”3|) |0’0 0”0| |5’1| (|1”5|) |5”1| (|1’5|) |2’4 4”2|
2M2
2O
Ramsdell symbol
e.e u.u 5 * 4 *
u.u e.e 0 * 3 *
Homo
§ ¨ ©
1.5 4.2 0 * 3 *
Non-MDO
2.2 1.1 5 * 4 *
5.1 2.4 0 * 3 *
1.5 2.4 0 * 3 *
3.3 0.0 0 * 3 *
· ¸ ¹
subfamily B OD symbol Meso
|4’0 5”3| |4”0 4’3|
|2’2 1”1|
|5”1 2”4| (|1’5 4’2|)
Hetero |3’3 0”0| |3’3 0’0| (|3”3 0”0|) |1”5 2’4| |1’5 2”4| |5’1 2’4| (|1”5 4”2|)
Table 7. Relation of homomorphy between MDO polytypes of micas. Shortened (non-redundant) OD symbols are given for hetero-octahedral polytypes only: for enantiomorphous polytypes, they are shown in parentheses [the packet pair i’j is enantiomorphous of (-i)”(-j), where the orientational characters are expressed mod 6]. Modified after Durovic et al (1984).
186 Nespolo & Ďurovič
V
Homomorphic MDO group
Ramsdell symbol
Table 7 (continued). subfamily A OD symbol Homo Meso Hetero
6H
Ramsdell symbol
§ ¨ © e . e u . u e . e u . u e.e u . u 3 * 4 * 5 * 0 * 1 * 2 *
e . e u . u e.e u .u e . e u . u 3 * 2 * 1 * 0 * 5 * 4 *
Homo
· ¸ ¹ § ¨ ©
§ ¨ ©
§ ¨ ©
Non-MDO
4 . 2 5 . 3 0 . 4 1 . 5 2.0 3 . 1 3 * 4 * 5 * 0 * 1 * 2 *
2 . 4 1 . 3 0.2 5 . 1 4 . 0 3 . 5 3 * 2 * 1 * 0 * 5 * 4 *
2 . 4 3 . 5 4 . 0 5 . 1 0.2 1 . 3 3 * 4 * 5 * 0 * 1 * 2 *
4 . 2 3 . 1 2.0 1 . 5 0 . 4 5 . 3 3 * 2 * 1 * 0 * 5 * 4 *
0 . 0 1 . 1 2 . 2 3 . 3 4.4 5 . 5 3 * 4 * 5 * 0 * 1 * 2 *
0 . 0 5 . 5 4.4 3 . 3 2 . 2 1 . 1 3 * 2 * 1 * 0 * 5 * 4 *
subfamily B OD symbol Meso
· ¸ ¹
· ¸ ¹
· ¸ ¹
|4’2 1”3 2’0 5”1 0’4 3”5| (|2”4 5’3 4”0 1’5 0”2 3’1|) |2’4 3”1 0’2 1”5 4’0 5”3| (|4”2 3’5 0”4 5’1 2”0 1’3|)
|2”4 1”3 0”2 5”1 4”0 3”5| (|4’2 5’3 0’4 1’5 2’0 3’1|)
|2’4 1’3 0’2 5’1 4’0 3’5| (|4”2 5”3 0”4 1”5 2”0 3”1|)
|4”2 3”1 2”0 1”5 0”4 5”3| (|2’4 3’5 4’0 5’1 0’2 1’3|)
Hetero |0’0 5’5 4’4 3’3 2’2 1’1| (|0”0 1”1 2”2 3”3 4”4 5”5| |0”0 5”5 4”4 3”3 2”2 1”1| (|0’0 1’1 2’2 3’3 4’4 5’5|) |0’0 5”5 4’4 3”3 2’2 1”1| (|0”0 1’1 2”2 3’3 4”4 5’5| |4’2 3’1 2’0 1’5 0’4 5’3| (|2”4 3”5 4”0 5”1 0”2 1”3|)
Crystallographic Basis of Polytypism and Twinning in Micas 187
m(010) M1 C12/m(1)
q2n+1 W-operation
p2nq2n+1 type of layer
p2nq2n+1 O-symmetry
C2
02t
0
C2/c
c(010), 04t
C2
04t
C2/c
c(010), 04t
q2n+1p2n+2
q2n-1p2n
2[010],C1
C12/m(1)
M1
m(C110)
m(C110)
2[C310],C1
C12/m(1)
M1
m(110)
m(110)
2[310],C1
220440
4.4 2 . 2 (#) 1 * 5 *
C2
04t
22 C 2M1
q2n+1p2n+2
q2n-1p2n
2[010]
C12(1)
M2
m(010)
m(110)
2[C310],C3-1(a)
C12(1)
M2
m(C110)
m(010)
2[310],C3-1(a)
640260
0.2 4 . 0 1 * 5 *
Ramsdell symbol(b) 1M (a) These V-U operations are coincidence operations (one-way movement) and not local symmetry operations. (b) Symbols for the homo-octahedral polytypes homomorphic to the meso-octahedral polytypes listed in this table.
RTW symbol
C2/m
Space-group type
(b)
m(010), 02t
Global W-operations
q2n+1p2n+3 C(1)
q2n-1p2n C(1)
p2n+2q2n+3
q2n+1p2n+2
q2n+1p2n+2
p2n+2q2n+3(2[010])
p2nq2n+1
p2nq2n+1
Location
2[010] p2nq2n+1
2[010]C1
2[010],C1
Global U-operation
C12/m(1)
M1
m(010)
p2nq2n+1 (2[010])
M2 C12(1)
p2n+2q2n+3 type of layer
p2nq2n+1 O-symmetry
m(C110)
q2n+3 W-operation
m(010)
C12(1) 2[010],C1
C12(1)
M2
m(110)
m(C110)
2[010],C3-1 (a)
420660
2.4 0.0 3 * 3 *
2[010],C31 (a)
M2
m(110)
m(110)
2[010]
C12(1)
M2
m(110)
m(C110)
p2n+2 W-operation
p2n+2q2n+3 VU-operation
m(010)
m(C110)
420240 2[010], 3-1(a)
150 2[010],C3-1(a)
330 2[010],C1
2 . 4 4 . 2 (#) 3 * 3 *
5 . 1 (#) 0 *
3 . 3 (#) 0 *
p2n W-operation
p2nq2n+1 VU-operation
Z-Symbol
OD-Symbol (#) = MDO
Cc
c(010), 04t
-----
-----
C12(1)
M2
m(010)
m(C110)
2[310],C31(a)
C12(1)
M2
m(010)
m(110)
2[C310],C3-1(a)
260460
4 . 0 2.0 5 * 1 *
C1
04t
-----
-----
C12/m(1)
M1
m(C110)
m(C110)
2[C310],C1
C12(1)
M2
m(010)
m(C110)
2[310],C31(a)
460440
2.0 2 . 2 1 * 5 *
Table 8. The analysis of the local (see Tables 5a and 5b) and global (see Table 6) symmetry of the eight Class a meso-octahedral polytypes with period up to two layers. For the derivation of the independent polytypes see Backhaus and Durovic (1984) and Zvyagin (1997). The corresponding Ramsdell symbols apply to the homo-octahedral polytypes obtainable from the relation of homomorphy. The layer groups and space-group types are given in the Trigonal model: the possibility of symmetry reduction to a subgroup in the real structures should always be taken into account.
188 Nespolo & Ďurovič
Crystallographic Basis of Polytypism and Twinning in Micas
189
RELATIONS OF HOMOMORPHY AND CLASSIFICATION OF MDO POLYTYPES Polytypes are usefully classified not only within the same family, but also between different families. On the basis of the number of layers and of the parity of the corresponding characters in the orientational symbols, several meso-octahedral polytypes can be related to one homo-octahedral polytype; similarly, taking into account the chirality of the packets, several hetero-octahedral polytypes can be related to one mesooctahedral polytype. In mathematics, a n → 1 relation is a homomorphism, of which the 1 → 1 relation (isomorphism) is a special case: the n → 1 relation of polytypes of different families is hence termed relations of homomorphy. The recognition of such relations is also of practical importance. For instance, if during the refinement of a mica structure the homo-octahedral model fails, only the choice between the related meso- or hetero-octahedral models has to be made. All such polytypes have the same framework of all atoms except those octahedrally coordinated. Therefore, they have identical or very similar basis vectors, and the space-group type of the homo-octahedral polytype is their common supergroup. Also their diffraction patterns are closer to one another than to those of other polytypes: the geometry in reciprocal space is virtually the same and also the distribution of intensities is very similar owing to the fact that the framework of non-octahedral atoms in an “average” mica represents about 70 % of the total diffraction power. The relations of homomorphy can be easily revealed by analyzing the OD symbols (Ďurovič et al. 1984): 1)
by substituting the primes (′) or double primes (“) in the symbols of heterooctahedral polytypes with dots (.), the corresponding meso-octahedral polytypes are obtained;
2)
by substituting the Tj orientational characters in the symbols of meso-octahedral polytypes with “e” (for “even”) or “u” (for “uneven”), the corresponding homooctahedral polytypes are obtained;
3)
the relation of homomorphy between hetero- and homo-octahedral polytypes is obtained by combining steps 1) and 2);
4)
some of the hetero-octahedral MDO polytypes are in relation of homomorphy with non-MDO meso-octahedral polytypes, but the further homomorphy to the homooctahedral family yields again MDO polytypes (for details, see Ďurovič et al. 1984).
Note that the relations of homomorphy can, in some cases, make two or more subperiods identical although they are different in the original polytype: as a result, polytypes with a different periodicity can be in homomorphy. As an example, let us consider the meso-octahedral polytypes of Class a given in Table 8. Of the six 2-layer polytypes, the following four are homomorphous with the homo-octahedral 2M1 polytype. 4.4 2 . 2 0.2 4 . 0 4 . 0 2.0 2.0 2 . 2 1 * 5 *, 1 * 5 *, 5 * 1 *, 1 * 5 * e . e , for which the shortened In fact, the relation of homomorphy gives for each: e.e 1* 5 * symbol is |15|. The other two polytypes ( 23. 4*43. 2* and 23. 4*03. 0* ), however, are
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homomorphous with e3. e*e3. e* . In the homo-octahedral family, this polytype has 1-layer periodicity, with a shortened symbol |3|, and this corresponds to 1M rotated by 180º about c* (Fig. 11). This apparent reduction of periodicity occurs whenever: 1) the sequence of v2j,2j+1 displacement vectors of a meso-octahedral polytype contains two or more identical sub-periods, which are different for T2j.T2j+1 orientations of the packets; 2) the sequence of T2j.T2j+1 orientation vectors of a hetero-octahedral polytype contains two or more subperiods which differ only in the chirality of the packets. The relations of homomorphy in mica structures are summarized in Table 7. Full symbols are given for homo-and meso-octahedral polytypes, shortened symbols (the line of orientational characters) – for hetero-octahedral polytypes. The reason for the somewhat unusual layout of this table is related to the fact that two out of the six homooctahedral MDO polytypes, 1M and 2O, have the same projection normal to [010] (YZ projection). Thus, for the framework of the non-octahedral atoms in the homo-octahedral MDO polytypes (and also for the corresponding homo-octahedral approximations), there exist five different YZ projections labeled by Roman numbers I to V in the first column of Table 7. The significance of the YZ projections will be explained below in the section “Identification of MDO polytypes”. As an example for the relations of homomorphy, let us take the hetero-octahedral polytype 23' 4*01'2*45' 0* (subfamily A). This polytype is homomorphous to the meso4 . 0 and this, in turn, is homomorphous to the homo-octahedral octahedral polytype 23. 4*0.2 1 * 5 * polytype 3T e3. e*e1. e*e5. e* : all belong to the MDO group I. The two polytypes in the heteroand meso-octahedral families are constructed of M2 layers. However, in the homooctahedral family, the distinction between M1 and M2 layers becomes meaningless: the information about the type of layer is thus lost when applying the relation of homomorphy down to the homo-octahedral family. From the examples above it is evident that: 1) the homo-octahedral approximation corresponds to applying to a polytype the relation of homomorphy; 2) in micas, the classical Ramsdell notation rigorously applies to homo-octahedral polytypes only. BASIC STRUCTURES AND POLYTYPOIDS. SIZE LIMIT FOR THE DEFINITION OF “POLYTYPE” The term polytype implies that there is a family of structures to which the polytype belongs. The original idea of Baumhauer (1912, 1915), who introduced the term polytypism, was that the individual members of a family consist of identical layers and differ only in their stacking mode. Since that time, different views concerning the notion of polytypism were expressed, but the present official definition recommended by the Ad-hoc Committee on the Nomenclature of Disordered, Modulated and Polytype Structures (Guinier et al. 1984) is very close to the original concept of Baumhauer. According to this definition, “… an element or compound is polytypic if it occurs in several structural modifications, each of which can be regarded as built up by stacking layers of (nearly) identical structure and composition, and if the modifications differ only in their stacking sequence. Polytypism is a special case of polymorphism: the two-dimensional translations within the layers are essentially preserved”. The Ad-hoc Committee, however, admitted that this definition is
Crystallographic Basis of Polytypism and Twinning in Micas
191
Figure 11. Relation of homomorphy between the two-layer meso-octahedral 2 . 4 4 . 2 polytype (left) and the one-layer homo-octahedral e . e polytype (right), 3 * 3 * 3
illustrated by showing separately the two Oc layers. Solid vectors: packet orientation; dotted vectors: packet-to-packet displacements. Solid circles and open squares represent two different average cations. In the meso-octahedral polytype (left), the two Oc layers have the origin in either of the two cis-sites, where the different average cation is located: they correspond to M2 layers. The packet orientations, given by the vectors connecting the interlayer/OH sites (overlapped in projection) to the origin of the Oc layer, are 2 (packets p0 and q3) and 4 (packets q1 and p2). For both packet pairs, the vector sum (packet-to-packet displacement) is in orientation 3. By applying the relation of homomorphy, i.e., by making identical the content of the three octahedral cation sites, and obtaining the corresponding homo-octahedral polytype (right), both layers are transformed into the type M1, and the packet orientations change into e for both packets. The packet-to-packet displacements do not change. As a consequence, the two layers in the homooctahedral polytype have the same orientational vectors, but the periodicity is halved. The Σv, now coinciding with v0,1, corresponds to “3” (acute β angle), but can be transformed into “0” (obtuse β angle) by rotating the polytype by 180º about the normal to the layer.
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too wide because – except for the two-dimensional periodicity of layers – it imposes no restrictions on the sequence and stacking mode of layers. The fact that the definition is not sufficiently geometric prompted Ďurovič (1999) to suggest that the layers and their stacking must be limited by the vicinity condition (VC, see the section “Micas as OD structures”), and that a family can encompass only those polytypes which are built on the same structural and symmetry principle, i.e. only those which belong to the same OD groupoid family. This idea was in principle supported also by Makovicky (1997) who, at the same time, proposed to distinguish between proper polytypes, belonging to the same OD groupoid family, and improper polytypes, which cannot be interpreted as such. Recently, Christiansen et al. (1999) suggested a more detailed classification concept related to this subject. Makovicky also accepted the term polytypoids for polytypic substances in which more than 0.25 atoms per formula unit differ in at least one component as proposed by the IMA-IUCr Joint Committee on Nomenclature (Bailey et al. 1977). This term was applied also by Bailey (1980b) for the specific case of micas, and recommended also by the Ad-hoc Committee, as discussed above. Abstract polytypes The experience gathered over years with refined periodic structures of polytypic substances indicate that, sensu stricto, each such polytype is an individual polymorph with its own stability field, although the energy differences between polytypes of the same compounds are very small. This is caused by desymmetrization, i.e. by changes in the atomic coordinates of individual layers imposed by the influence of the neighboring layers and it is different for different stacking modes. Thus, even layers in different polytypes of the same substance are not identical. A prominent example in micas (1M and 2M1 polytypes of biotite with the same composition) was given by Takeda and Ross (1975), who not only found significant differences in the constituent layers of the polytypes but also postulated that these differences are "directly related to the atomic and geometric constraints imposed by the adjacent unit layers varying with the relative orientation of the adjacent layers". Desymmetrization occurs even in such less pliable structures as SiC, as convincingly reported by researchers at the former Leningrad Electrotechnical Institute (Sorokin et al. 1982a,b; Tsvetkov 1982; see also Tairov and Tsvetkov 1983) who showed that also the chemical composition (the ratio of Si/C) varies from polytype to polytype grown under (nearly) the same conditions. If these facts were taken absolutely at the face value, the notion of polytypism would loose its unifying significance. In order to overcome these difficulties, the concept of a polytype is often considered an abstract notion referring to a structural type with relevant geometric properties, belonging to an abstract family whose members consist of layers with identical structure and with identical bulk compositions. Such an abstract notion lies at the root of all systematization and classification schemes of polytypes. In micas (as well as in many other phyllosilicates) the Pauling model and also the homo-octahedral approximation are abstractions which are very useful, among others, for didactic purposes to gain first knowledge, but also for the calculation of identification diagrams of MDO polytypes, and for the calculation of PID functions, described in sections about experimental identification of mica polytypes below. A better approximation, but still an abstraction, is the Trigonal model, which is important for the explanation of subfamilies and for some features in the diffraction patterns. Also, when speaking of a specific polytype, a characteristic sequence of abstract mica layers is intended rather than deviations from stoichiometry, distribution of cations within octahedral sheets, distortion of coordination polyhedra, etc.
Crystallographic Basis of Polytypism and Twinning in Micas
193
Basic structures Owing to the fact that the energy difference between polytypes of the same substance is very small, the occurrence of different polytypes should be influenced mostly by the kinetics of crystal growth, and the frequency of occurrence of different polytypes is, in principle, directly related to the number of layers in the period. However, this statement is contradicted by the existence of a certain degree of structural control (Smith and Yoder 1956; Güven 1971) that governs the frequency of occurrence of polytypes as a function of the crystallization environment and of the crystal chemistry. As firstly noted by Ross et al (1966), a portion of the stacking sequence of the non-MDO mica polytypes coincides with the stacking sequence of one of the MDO subfamily A polytypes, similarly to what happens in SiC polytypes. The remaining portion represents a deviation from the sequence. For this reason, Baronnet and Kang (1989) introduced the term basic structures to indicate these three MDO polytypes, as well as 2M2 and 1Md-A. The non-MDO polytypes are thus said to belong to one structural series: the three structural series 1M, 2M1 and 3T were defined (Ross et al. 1966; Baronnet 1978; Takeda and Ross 1995). A structural series based on 2M2 has not been found, but its existence cannot be excluded in principle. The causes of the existence of a stacking memory in the basic structures are not well understood. Energy differences between two polytypes of the same family are small. However, the real structures are constructed not by layer archetypes, but by, more or less, desymmetrized layers: the corresponding energy differences may be sufficient to control the original stacking sequence. However, also when the crystal chemistry is practically identical, a certain degree of structural control exists, as shown by the fact that a few polytypes are clearly dominant, with the others appearing with much lower frequency. A general trend towards a relation between the formation environment, the crystal chemistry and the polytype frequency exists also (Nespolo 2001). The three basic structures may thus be not truly polytypic, even when the crystal chemistry is identical. HTREM observations and some implications The application of the High Resolution Transmission Electron Microscopy (HRTEM) (Iijima and Buseck 1978) has made possible the observation of several stacking sequences that would not be revealed by other techniques. At the same time, HRTEM has raised the question of the limits within which an observed stacking sequence should be considered a polytype. Kogure and Nespolo (1999b) stated that the stacking sequences revealed by HRTEM observation can be defined as a polytype only when they are repeated sufficiently to reveal the presence of a memory mechanism reproducing with regularity the stacking sequence; otherwise, they should rather be considered defects. It is questionable whether a sequence repeated only three times, like the 22-layer biotite reported by Konishi and Akai (1990), may be rigorously termed a “polytype”. In such cases it is recommended to speak of “a sequence corresponding to a certain polytype”. In such cases, we described the form as “a sequence corresponding to the polytype XY”. The problem is similar to that of nanocrystals where it is also questionable how many unit cells are necessary to determine a phase. IDEAL SPACE-GROUP TYPES OF MICA POLYTYPES AND DESYMMETRIZATION OF LAYERS IN POLYTYPES The ideal space-group type of a given polytype can be derived from the stacking sequence, as described above. However, three kinds of symmetries are required:
194
Nespolo & Ďurovič
1) the stacking symmetry, deduced from the sequence of packet orientations and displacements, which gives the space-group type in the Trigonal model; 2) the structural symmetry, which may be lower than the stacking symmetry because of structural distortions not taken into account by the Trigonal model; 3) the diffractional symmetry, which may be higher than the structural symmetry. This phenomenon is termed diffraction enhancement of symmetry (Ito 1950) and occurs when a crystal is constructed by substructures whose symmetry is higher than that of the crystal itself (e.g., Iwasaki 1972; Matsumoto et al. 1974). In micas, diffraction enhancement of symmetry was observed in the oxybiotite-10A1 from Ruiz Peak, which gave a monoclinic diffraction pattern, despite both the stacking symmetry and the structural symmetry were triclinic (Sadanaga and Takeda 1968). The validity of the local symmetry operations is often only approximate, and the atomic coordinates can deviate more or less from the values demanded by the corresponding space groupoid, depending on the stacking of the packets in the investigated crystal, and this is phenomenon known as desymmetrization (Ďurovič 1979). The λ-symmetry of the M layers can thus be lower than the λ-symmetry of the layer archetypes described by the Trigonal model (see Table 2 in Ferraris and Ivaldi, this volume). The space-group type corresponding to the stacking symmetry in general does not require the highest λ-symmetry compatible with the family (homo-, meso- or heterooctahedral) and the type of layer (M1 vs. M2). The layer is thus allowed, although not required, to attain a layer-subgroup. The general trend that results from the structure refinements performed on mica polytypes can be summarized as follows (see Table 9, and Tables 1-3 in Brigatti and Guggenheim, this volume): 1) 1M polytype has been refined only in the highest space-group types and layer-groups compatible with the type of layer: C2/m and C12/m(1) for the M1 layer; C2 and C12(1) for the M2 layer. 2) The highest space-group type for the 2M1 polytype is C2/c. All but one example of 2M1 polytypes refined so far belong to the meso-octahedral family and are constructed by M1 layers. Most of these polytypes have been refined in C2/c. This space-group type allows a desymmetrization of the layer-group to ⎯C1, which corresponds to the λ-symmetry normally obtained in 2M1 polytypes (Güven 1971; Zussman 1979; Takeda and Ross 1975). An important exception is oxybiotite-2M1 refined by Ohta et al (1982), where the highest λ-symmetry C12/m(1) was observed within experimental erro; this was also the λ-symmetry of coexisting oxybiotite-1M (Ohta et al. 1982). Three studies of meso-octahedral margarite-2M1 refined in the space-group type Cc have been reported (Guggenheim and Bailey 1975, 1978; Joswig et al 1983; Kassner et al. 1993), where the reduction of symmetry was related to the Si-Al ordering, that made the two T sheets no longer equivalent. The layer group is only C1, because of the destruction of the center of symmetry. A further reduction of symmetry was observed in the ephesite-2M1 reported by Slade et al. Table 9 (next nine ap ges ). Relevant properties of the MDO polytypes. Only polytypes for which the ccupancies of the octahedral sites were given in the original papers are reported. Following Durovic et al (1984), the effective scattering amplitude is taken directly from the original papers, when reported; otherwise it has been calculated assuming half-ionized atoms, even where the structure was refined using electron or neutron diffraction data. Polytypes built by M2 layers are in bold characters. References are given according to the sequence numbers in the tables of the Brigatti and Guggenheim chapter. For polytypes not reported there, the complete reference is given. (e) = electron diffraction data; (n) = neutron diffraction data; otherwise X-ray diffraction data.
195
Crystallographic Basis of Polytypism and Twinning in Micas
Reference
Type of mica
R factor
Spacegroup type
G(M1)
G(M2)
G(M3)
Full polytype symbol
Subfamily A – 1M polytype Homo-trioctahedral 1-95
Phlogopite
13.1
C2/m
11.0
11.0
11.0
u .u 0 *
1-61
Synthetic iron mica
9.3
C2/m
25.0
25.0
25.0
u .u 0 *
1-97
Synthetic lithian flourphlogopite
7.3
C2/m
10.4
10.4
10.4
u .u 0 *
3-15
Barium mica
7.1
C2/m
8.8
8.8
8.8
u .u 0 *
1-70
Phlogopite (n)
2.0
C2/m
11.7
11.7
11.7
u .u 0 *
1-97
Phlogopite
4.1
C2/m
11.0
11.0
11.0
u .u 0 *
1-72
Fluorophlogopite
6.1
C2/m
11.0
11.0
11.0
u .u 0 *
1-86
Phlogopite (n)
6.6
C2/m
11.9
11.9
11.9
u .u 0 *
1-98
Fluro phlogopite
4.3
C2/m
11.0
11.0
11.0
u .u 0 *
1-104
Synthethic fluormica
3.8
C2/m
9.4
9.4
9.4
u .u 0 *
1-94
Tetraferriphlogopite
4.2
C2/m
10.5
10.5
10.5
u .u 0 *
1-108
Fluoro phlogopite
2.9
C2/m
10.2
10.2
10.2
u .u 0 *
3.7
C2/m
11.0
11.0
11.0
u .u 0 *
3.0
C2/m
10.6
10.6
10.6
u .u 0 *
1.108 1-69
Tetra germanatian fluoro phlogopite Silica- and alkali-rich trioctahedral mica
1-103
Germanate mica
3.9
C2/m
14.0
14.0
14.0
u .u 0 *
1-102
Germanate mica
5.0
C2/m
19.5
19.5
19.5
u .u 0 *
1-110
Fluoro phlogopite
4.3
C2/m
10.8
10.8
10.8
u .u 0 *
Knurr and Bailey (1986)
Phlogopite
3.1
C2/m
12.1
12.1
12.1
u .u 0 *
3-7
Potassium Kinoshitalite (27)
2.5
C2/m
13.4
13.4
13.4
u .u 0 *
1-82
Cs-ferriannite
5.5
C2/m
25.0
25.0
25.0
u .u 0 *
1-45
Magnesian annite (WA8E)
3.9
C2/m
19.9
19.9
19.9
u .u 0 *
1-60
Cs-tetra-ferri-annite
3.9
C2/m
25.0
25.0
25.0
u .u 0 *
196
Nespolo & Ďurovič
1-87/92
Ferroan phologopite
3.9
C2/m
17.2
17.2
17.2
u .u 0 *
3-9
Ferrokinoshitalite
3.2
C2/m
20.0
20.0
20.0
u .u 0 *
3-8
Kinoshitalite
3.35
C2/m
12.0
12.0
12.0
u .u 0 *
Meso-trioctahedral Takéuchi and Sadanaga (1966)
Xantophyllite
10.8
C2/m
11.3
11.0
11.0
3.3 0 *
1-96
Synthethic fluor-polylithionite
5.1
C2/m
3.5
6.6
6.6
3.3 0 *
1-66
Annite
4.4
C2/m
22.6
22.7
22.7
3.3 0 *
1-100
Synthetic MgIV mica
9.2
C2/m
10.7
10.1
10.1
3.3 0 *
1-99
Biotite
4.4
C2/m
16.2
16.0
16.0
3.3 0 *
1-93
Lepidolite
6.7
C2/m
3.0
8.2
8.2
3.3 0 *
1-105
Taeniolite
2.4
C2/m
8.5
8.1
8.1
3.3 0 *
1-107
Germanate mica
3.8
C2/m
7.9
8.3
8.3
3.3 0 *
1-106
Germanate mica
5.5
C2/m
6.6
10.5
10.5
3.3 0 *
Sokolova et al (1979)
Ephesite
11.5
C2/m
3.2
11.4
11.4
3.3 0 *
1-62
Lepidolite
3.5
C2/m
3.6
8.2
8.2
3.3 0 *
1-128
Lepidolite
6.2
C2
4.7
10.1
4.7
5.1 0 *
1-85
Oxybiotite
4.4
C2/m
12.6
15.2
15.2
3.3 0 *
1-63
Manganoan phlogopite (1)
5.4
C2/m
15.2
16.1
16.1
3.3 0 *
1-64
Barian manganoan phlogopite (5)
3.8
C2/m
12.6
14.9
14.9
3.3 0 *
3-10
Clintonite (n)
2.0
C2/m
11.8
11.2
11.2
3.3 0 *
3-12
Clintonite (1782/5)
2.1
C2/m
12.1
11.2
11.2
3.3 0 *
3-13
Clintonite (94594)
3.9
C2/m
11.6+
11.6-
11.6-
3.3 0 *
3-14
Clintonite (105455)
2.1
C2/m
11.5
11.2
11.2
3.3 0 *
1-8
Ferroan phologopite (M14)
3.3
C2/m
18.6
17.8
17.8
3.3 0 *
1-9
Ferroan phologopite (M32)
2.4
C2/m
17.9
17.1
17.1
3.3 0 *
1-12
Ferroan phologopite (M13)
6.2
C2/m
20.4
19.8
19.8
3.3 0 *
197
Crystallographic Basis of Polytypism and Twinning in Micas
1-11
Ferroan phologopite (M73)
2.1
C2/m
19.0
18.2
18.2
3.3 0 *
1-10
Ferroan phologopite (M62)
3.5
C2/m
20.4
19.6
19.6
3.3 0 *
1-111
Norrishite
7.8
C2/m
2.5
23.3
23.3
3.3 0 *
1-21
Ferroan phlogopite (8)
2.5
C2/m
13.9
15.1
15.1
3.3 0 *
1-22
Phlogopite (9)
2.2
C2/m
13.7
14.0
14.0
3.3 0 *
1-23
Ferroan phlogopite (10)
2.2
C2/m
16.3
16.5
16.5
3.3 0 *
1-24
Ferroan phlogopite (11)
1.9
C2/m
14.7
16.8
16.8
3.3 0 *
1-25
Ferroan phlogopite (12)
2.1
C2/m
14.5
16.1
16.1
3.3 0 *
1-26
Ferroan phlogopite (15)
2.3
C2/m
17.5
17.0
17.0
3.3 0 *
1-27
Ferroan phlogopite (16)
3.0
C2/m
19.0
18.4
18.4
3.3 0 *
1-28
Magnesian annite (17)
2.6
C2/m
18.6
18.4
18.4
3.3 0 *
1-112
Protolithionite
3.8
C2/m
20.2
19.4
19.4
3.3 0 *
1-7
Magnesian annite (MP9)
3.1
C2/m
18.7
20.2
20.2
3.3 0 *
1-13
Titanian phlogopite (18)
2.0
C2/m
12.9
15.4
15.4
3.3 0 *
1-17
Ferroan phlogopite (19)
3.2
C2/m
17.6
18.1
18.1
3.3 0 *
1-14
Aluminian phlogopite (20)
2.7
C2/m
16.1
16.9
16.9
3.3 0 *
1-15
Ferrian phlogopite (21)
2.3
C2/m
15.3
16.1
16.1
3.3 0 *
1-16
Ferroan phlogopite (22)
3.3
C2/m
16.3
17.1
17.1
3.3 0 *
1-18
Ferrian phlogopite (23)
3.4
C2/m
16.2
16.8
16.8
3.3 0 *
1-19
Ferrian phlogopite (24)
2.7
C2/m
17.1
17.5
17.5
3.3 0 *
1-20
Ferroan phlogopite (25)
2.2
C2/m
16.6
17.7
17.7
3.3 0 *
Brigatti & Poppi (1993)
Potassium kinoshitalite (26)
2.6
C2/m
14.3
13.3
13.3
3.3 0 *
1-6
Biotite
3.33
C2/m
19.0
18.1
18.1
3.3 0 *
1-1
Phlogopite (1a)
2.9
C2/m
13.2
12.9
12.9
3.3 0 *
1-2
Phlogopite (1b)
2.8
C2/m
13.4
12.9
12.9
3.3 0 *
1-3
Phlogopite (2a)
2.9
C2/m
13.2
12.9
12.9
3.3 0 *
198
Nespolo & Ďurovič
1-4
Aluminian phlogopite (3a)
3.0
C2/m
13.3(1)
13.2(1)
13.2(1)
3.3 0 *
1-5
Phlogopite (4a)
2.5
C2/m
13.0
12.7
12.7
3.3 0 *
1-36
Phlogopite (Tas27-2Ba)
2.8
C2/m
14.0
13.1
13.0
3.3 0 *
1-37
Phlogopite (Tas27-2Bb)
2.5
C2/m
13.7
13.3
13.3
3.3 0 *
1-38
Ferroan phlogopite (Tag15-4)
2.8
C2/m
15.7
15.6
15.6
3.3 0 *
1-39
Phlogopite (Tag15-3)
2.8
C2/m
14.9
14.8
14.8
3.3 0 *
1-32
Ferroan phlogopite (Tpg63-2B)
2.3
C2/m
16.8
16.5
16.5
3.3 0 *
1-29
Phlogopite (Tae23-1a)
2.7
C2/m
13.4
13.3
13.3
3.3 0 *
1-30
Phlogopite (Tae23-1b)
2.7
C2/m
13.5
13.5
13.5
3.3 0 *
1-31
Phlogopite (Tae23-1c)
3.0
C2/m
14.0
13.7
13.7
3.3 0 *
1-40
Phlogopite (Tpq16-4A)
2.8
C2/m
13.8
13.6
13.6
3.3 0 *
2.8
C2/m
13.8
13.4
13.4
3.3 0 *
3.2
C2/m
12.9
12.8
12.8
3.3 0 *
3.3
C2/m
13.9
13.1
13.1
3.3 0 *
3.1
C2/m
14.6
13.8
13.8
3.3 0 *
3.1
C2/m
13.5
13.1
13.1
3.3 0 *
1-35 1-33 1-34 1-41 1-42
Phlogopite (Tpt17-1) Tetra-ferri phlogopite (Tas22-1a) Tetra-ferri phlogopite (Tas22-1b) Tetra-ferri phlogopite (Tpq16-6B) Tetra-ferri phlogopite (S1)
1-43
Tetra-ferri phlogopite (S2)
2.5
C2/m
13.8
13.5
13.5
3.3 0 *
Brigatti et al (1997)
Ferroan phlogopite (Tag15-4a)
2.8
C2/m
15.7
15.6
15.6
3.3 0 *
1-48
Ferroan phlogopite (Tag15-4b)
2.8
C2/m
15.2
15.4
15.4
3.3 0 *
Ferroan phlogopite (Tpq16-4Aa)
2.8
C2/m
13.8
13.6
13.6
3.3 0 *
1-50
Ferroan phlogopite (Tpq16-4Ab)
2.4
C2/m
13.7
13.4
13.4
3.3 0 *
Brigatti et al (1997)
Ferroan phlogopite (Tpq16-4Ac)
3.0
C2/m
15.9
15.3
15.3
3.3 0 *
Brigatti et al (1997)
Ferroan phlogopite (Tas22-1c)
3.1
C2/m
13.5
13.1
13.1
3.3 0 *
3-1
Clintonite (5a)
3.49
C2/m
13.0
12.6
12.6
3.3 0 *
3-2
Clintonite (7c)
3.73
C2/m
13.4
13.3
13.3
3.3 0 *
1-49
199
Crystallographic Basis of Polytypism and Twinning in Micas
3-3
Clintonite (8a)
3.11
C2/m
13.2
12.9
12.9
3.3 0 *
3-4
Clintonite (8d)
3.18
C2/m
12.7
13.0
13.0
3.3 0 *
3-5
Clintonite (9a)
3.29
C2/m
13.0
13.1
13.1
3.3 0 *
3-6
Clintonite (9b)
2.70
C2/m
12.6
13.0
13.0
3.3 0 *
1-65
rubidian cesian phlogopite
4.5
C2/m
16.0
15.8
15.8
3.3 0 *
1-44
Ferroan phlogopite (WA3H)
2.9
C2/m
18.3
18.2
18.2
3.3 0 *
1-46
Magnesian annite (WA8H)
3.3
C2/m
19.6
19.3
19.3
3.3 0 *
1-47
Ferroan phlogopite (WA23E)
2.8
C2/m
18.8
18.6
18.6
3.3 0 *
1-51
Magnesian annite
3.2
C2/m
19.6
18.9
18.9
3.3 0 *
1-55
Magnesian annite
3.6
C2/m
19.1
18.2
18.2
3.3 0 *
1-56
Magnesian annite
3.2
C2/m
19.6
18.8
18.8
3.3 0 *
1-54
Magnesian annite
3.2
C2/m
19.4
18.5
18.5
3.3 0 *
1-53
Magnesian annite
3.1
C2/m
19.9
19.6
19.6
3.3 0 *
1-52
Magnesian annite
3.7
C2/m
19.8
19.2
19.2
3.3 0 *
1-58
Fe-Li rich mica 26
3.3
C2/m
19.6
22.3
22.3
3.3 0 *
1-59
Fe-Li rich mica 33
3.6
C2/m
23.5
239
23.9
3.3 0 *
1-57
Fe-Li rich mica 120
2.6
C2/m
24.7
24.4
24.4
3.3 0 *
1-118
Fe-Li rich mica 130(2)
3.86
C2
12.7
13.0
12.7
5.1 0 *
Hetero-trioctahedral
1-129
Zinnwaldite
5.7
C2
15.0
11.5
13.5
5'1 0 *
1-113
Lepidolite
7.3
C2
3.7
11.4
11.5
3'3 0 *
Zhukhlistov et al (1983)
Li-Fe phengite (e)
10.2
C2
8.0
15.1
14.8
3"3 0 *
1-130
Masutomilite
4.6
C2
8.5
11.1
8.1
1'5 0 *
1-117
Fe-Li rich mica 130(1)
2.96
C2
13.5
12.6
12.8
5'1 0 *
1-123
Fe-Li rich mica 140(1)
2.89
C2
13.8
13.0
13.6
5'1 0 *
1-124
Fe-Li rich mica 140(2)
2.73
C2
13.3
12.4
13.7
5"1 0 *
200
Nespolo & Ďurovič
1-120
Fe-Li rich mica 104
3.34
C2
12.0
12.1
11.8
1'5 0 *
1-119
Fe-Li rich mica 137
3.63
C2
13.0
13.0
11.3
1'5 0 *
1-122
Fe-Li rich mica 177
3.39
C2
13.8
12.7
12.8
5'1 0 *
1-121
Fe-Li rich mica 54b 3.78
C2
11.9
11.6
13.0
5"1 0 *
1-125
Fe-Li rich mica 24
3.72
C2
14.2
12.3
13.0
5'1 0 *
1-115
Fe-Li rich mica 55a
3.74
C2
11.3
12.0
9.4
1'5 0 *
1-116
Fe-Li rich mica 55b 3.21
C2
11.3
13.0
9.9
1'5 0 *
1-126
Fe-Li rich mica 47
3.31
C2
19.2
15.8
19.4
5"1 0 *
1-127
Fe-Li rich mica 103
3.63
C2
16.0
14.3
17.6
5"1 0 *
1-114
Fe-Li rich mica 114
3.35
C2
10.2
8.5
12.2
5"1 0 *
Meso-dioctahedral 2-3
Ferrous celadonite (e)
10.8
C2/m
---
21.4
21.4
3.3 0 *
2-2
Paragonite (e)
12.1
C2/m
---
10.8
10.8
3.3 0 *
4-1
Boromuscovite
3.8
C2/m
---
12.5
12.5
3.3 0 *
---
12.8
11.5
3"3 0 *
12.8
12.8
12.8
u .u e.e 0 * 3 *
Hetero-dioctahedral 2-1
Dioctahedral mica (e)
10.9
C2
Subfamily B – 2O polytype Homo-trioctahedral Ferraris et al (2000)
Fluor-phlogopite
4.5
Ccmm
Meso-trioctahedral 3-17
Anandite*
6.1
Pnmn
3.3 0.0 0 * 3 *
3-18
Anandite*
6.4
Pnmn
3.3 0.0 0 * 3 *
Subfamily A – 2M1 polytype Meso-trioctahedral 1-139
Biotite
5.6
C2/c
15.8
16.3
16.3
4.4 2 . 2 1 * 5 *
Sartori (1977)
Lepidolite
11.3
C2/c
2.3
8.7
8.7
4.4 2 . 2 1 * 5 *
Sokolova et al (1979)
Bityite (e)
11.5
C2/c
2.3
8.7
8.7
4.4 2 . 2 1 * 5 *
1-135
Magnesian annite
4.2
C2/c
19.4
18.6
18.6
4.4 2 . 2 1 * 5 *
1-138
Lepidolite
9.1
C2/c
3.6
7.5
7.5
4.4 2 . 2 1 * 5 *
Crystallographic Basis of Polytypism and Twinning in Micas
201
1-137
Oxybiotite
3.9
C2/c
12.6
15.2
15.2
4.4 2 . 2 1 * 5 *
3-76
Li-Be rich mica
3.0
Cc
1.1
11.5
11.5
4.4 2 . 2 1 * 5 *
1-141
Ephesite
4.7
C1
2.9
11.5
11.5
4.4 2 . 2 1 * 5 *
1-132
Magnesian annite (MP16)
3.7
C2/c
20.8
20.1
20.1
4.4 2 . 2 1 * 5 *
1-133
Magnesian annite (MP17a)
2.7
C2/c
17.5
16.8
16.8
4.4 2 . 2 1 * 5 *
1-134
Magnesian annite (MP17b)
3.4
C2/c
17.2
16.6
16.6
4.4 2 . 2 1 * 5 *
1-131
Biotite
2.72
C2/c
18.8
18.3
18.3
4.4 2 . 2 1 * 5 *
1-136
Magnesian annite
2.8
C2/c
19.4
18.4
18.4
4.4 2 . 2 1 * 5 *
16.2
14.3
17.4
0"2 0 ' 4 1 * 5 *
Hetero-trioctahedral
1-140
Zinnwaldite
5.8
Cc
Meso-dioctahedral Radoslovich (1960)
Muscovite
17.0
C2/c
---
12.3
12.3
4.4 2 . 2 1 * 5 *
Takéuchi (1965)
Margarite
16.8
C2/c
---
11.5
11.5
4.4 2 . 2 1 * 5 *
2-4
Muscovite
12.8
C2/c
---
12.3
12.3
4.4 2 . 2 1 * 5 *
2-36
Muscovite
3.5
C2/c
---
11.8
11.8
4.4 2 . 2 1 * 5 *
2-37
Phengite
4.5
C2/c
---
12.3
12.3
4.4 2 . 2 1 * 5 *
2-46
Muscovite (n)
2.7
C2/c
---
12.3
12.3
4.4 2 . 2 1 * 5 *
Udagawa et al (1974)
Muscovite
14.2
C2/c
---
12.3
12.3
4.4 2 . 2 1 * 5 *
3-19
Margarite
4.0
Cc
---
11.5
11.5
4.4 2 . 2 1 * 5 *
Sidorenko et al (1977a)
Paragonite (e)
11.1
C2/c
---
11.6
11.6
4.4 2 . 2 1 * 5 *
2-39
Paragonite
4.5
C2/c
---
11.9
11.9
4.4 2 . 2 1 * 5 *
2-47
Phengite
3.3
C2/c
---
13.2
13.2
4.4 2 . 2 1 * 5 *
2-38
Muscovite
2.7
C2/c
---
12.7
12.7
4.4 2 . 2 1 * 5 *
2-19/20
Muscovite
4.8 (LT) 6.0 (HT)
C2/c
---
12.1
12.1
4.4 2 . 2 1 * 5 *
2-21
Muscovite (n)
4.0
C2/c
---
13.4
13.4
4.4 2 . 2 1 * 5 *
4-2
Boromuscovite
3.8
C2/c
---
12.5
12.5
4.4 2 . 2 1 * 5 *
2-30
Chromphyllite
4.8
C2/c
---
19.9
19.9
4.4 2 . 2 1 * 5 *
202
Nespolo & Ďurovič 2-5
Mg-, Fe-bearing muscovite
2.54
C2/c
0.64
15.5
15.5
4.4 2 . 2 1 * 5 *
2-6
Mg-, Fe-bearing muscovite
2.96
C2/c
0.97
13.9
13.9
4.4 2 . 2 1 * 5 *
2-7
Mg-, Fe-bearing muscovite
3.58
C2/c
0.46
13.5
13.5
4.4 2 . 2 1 * 5 *
2-8
Mg-, Fe-bearing muscovite
2.92
C2/c
0.44
13.7
13.7
4.4 2 . 2 1 * 5 *
2-9
Mg-, Fe-bearing muscovite
3.93
C2/c
0.84
15.0
15.0
4.4 2 . 2 1 * 5 *
2-10
Mg-, Fe-bearing muscovite
2.89
C2/c
0.32
13.8
13.8
4.4 2 . 2 1 * 5 *
2-11
Mg-, Fe-bearing muscovite
2.78
C2/c
0.49
13.7
13.7
4.4 2 . 2 1 * 5 *
2-12
Mg-, Fe-bearing muscovite
2.11
C2/c
0.38
13.7
13.7
4.4 2 . 2 1 * 5 *
2-13
Mg-, Fe-bearing muscovite
3.87
C2/c
1.73
14.0
14.0
4.4 2 . 2 1 * 5 *
2-14
Mg-, Fe-bearing muscovite
3.12
C2/c
0.88
13.6
13.6
4.4 2 . 2 1 * 5 *
2-15
Mg-, Fe-bearing muscovite
2.80
C2/c
0.39
13.8
13.8
4.4 2 . 2 1 * 5 *
Smyth et al (2000)
Phengite
1.3
C2/c
---
11.6
11.6
4.4 2 . 2 1 * 5 *
2-16
Cr-containing muscovite
2.5
C2/c
0.1
13.8
13.8
4.4 2 . 2 1 * 5 *
2-17
Cr-containing muscovite
3.1
C2/c
---
13.8
13.8
4.4 2 . 2 1 * 5 *
2-18
Cr-containing muscovite
3.3
C2/c
2.1
14.5
14.5
4.4 2 . 2 1 * 5 *
Subfamily B – 2M2 polytype Meso-trioctahedral 1-144
Lepidolite
7.2
C2/c
2.0
8.4
8.4
2.2 1.1 5 * 4 *
1-143
Lepidolite
9.6
C2/c
3.0
8.2
8.2
2.2 1.1 5 * 4 *
1-142
Lepidolite
4.8
C2/c
2.5
8.6
8.6
2.2 1.1 5 * 4 *
Meso-dioctahedral 2-50
Dioctahedral mica (e)
11.7
C2/c
---
11.2
11.2
2.2 1.1 5 * 4 *
2-49
Nanpingite
5.8
C2/c
---
12.9
12.9
2.2 1.1 5 * 4 *
Subfamily A – 3T polytype Hetero-trioctahedral
1-145
Lepidolite
4.7
P3112
5.2
3.4
10.3
4 ' 2 2 '0 0 ' 4 3 * 1 * 5 *
Pavlishin et al (1981)
Protolithionite
3.8
P3112
18.7
14.3
15.6
2 ' 4 0'2 4 ' 0 3 * 1 * 5 *
1-146
Protolithionite
3.0
P3112
16.1
14.4
17.6
2"4 0"2 4"0 3 * 1 * 5 *
Crystallographic Basis of Polytypism and Twinning in Micas
203
Hetero-dioctahedral 2-53
Muscovite
2.4
P3112
---
11.5
12.5
0 ' 0 4 '4 2 ' 2 3 * 1 * 5 *
2-54
Paragonite (e)
13.0
P3112
3.4
9.2
10.3
0 ' 0 4 '4 2 ' 2 3 * 1 * 5 *
2-51
Phengite (KZ)
3.6
P3112
---
13.4
13.7
0 ' 0 4 '4 2 ' 2 3 * 1 * 5 *
2-52
Phengite (DM)
4.5
P3112
---
12.5
13.0
0 ' 0 4 '4 2 ' 2 3 * 1 * 5 *
Pavese et al (1997)
Phengite (n)
7.0 (LT) 5.0 (HT)
P3112
---
11.5
11.1 (LT) 11.2 (HT)
0"0 4"4 2"2 3 * 1 * 5 *
Smyth et al (2000)
Phengite
0.9
P3112
---
12.7
13.0
0 ' 0 4 '4 2 ' 2 3 * 1 * 5 *
*The structure of anandite-2O cannot be described using an orthohexagonal C-centered cell and contains four independent octahedral positions. The symbol of this ‘polytype’ is therefore only an approximation.
(1987), where a different Si-Al ordering in the four tetrahedral sites reduced the space-group type to C1. Only one example of hetero-octahedral 2M1 polytype is known so far: the zinnwaldite refined by Rieder et al (1996). In the hetero-octahedral family, the highest layer-group for both M1 and M2 layers is C12(1): correspondingly, the highest space-group type for 2M1 is Cc, which is realized in this zinnwaldite-2M1. This mica is built up by M2 layers, with local V-U operations 2[310] and 2[C310] for the two layers respectively, as can be easily confirmed by analyzing the OD symbols (Table 9) on the basis of the conversion rules given in Table 5a. 3) The highest space-group type for the 3T polytype is P31,212, which is compatible with the highest layer groups in all the three families, namely C12/m(1) (homo- and meso-octahedral) and C12(1) (hetero-octahedral). We are aware of nine structure refinements of 3T polytypes in which the composition of the O sheet was given. All belong to the hetero-octahedral family, and three of them wereconstructed up by M2 layers. Refinement of meso-octahedral 3T polytypes is desirable to investigate (a) the desymmetrization of the layer group in this polytype; (b) the frequency of occurrence of M2 layers that, at least in Li-rich micas, seems higher than in other polytypes. 4) The highest space-group type for the 2M2 polytype is C2/c, the same as 2M1. All the polytypes refined so far have this symmetry. 5) The polytype 2O has ideal space-group type Ccmm, which was reported only recently in a fluor-phlogopite from the Khibiny massif (Kola Peninsula, Russia) (Ferraris et al 2000). Previously, two examples were reported in anandite (Giuseppetti and Tadini 1972; Filut et al. 1985), where however an unusual crystal chemistry, including tetrahedral Fe3+ and octahedral S2- and Cl-, reduced the space-group type to Pnmn, with some indications of further reduction to P21. The anandite-2O cannot be described with the orthohexagonal C-centered cell and contains four independent octahedral positions, two of which are on mirror planes. The symbols given in Table 9 for anandite-2O are thus only a rough approximation. In C2/c and P31,212 space-group types there are two independent T sites and the two independent M2/M3 sites. The possibility of cation ordering exists in these groups, and it is often verified in the O sheet, but more rarely in the T sheets (Bailey 1975; 1984; Amisano-Canesi et al. 1994; see also the examples of margarite and ephesite given above). If the O-symmetry C12/m(1) is maintained no ordering occurs, although it is not
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prevented by the space-group type. Thus, this is an example of local symmetry being higher than that required by the global symmetry. As shown by Güven (1971) and by Zussman (1979), the symmetry in the interlayer is different also, which is ⎯1 in C12/m(1) λ-symmetry and 2[010] in ⎯C1 and C12(1) λ-symmetries (for details see Ferraris and Ivaldi, this volume). CHOICE OF THE AXIAL SETTING A non-orthogonal mica polytype forms, besides the conventional (double) monoclinic C-centered cell, both a pseudo-orthorhombic C-centered sextuple cell and a pseudo-hexagonal P triple cell. For hexagonal and trigonal polytypes (ω|| = ω⊥ = 0) the triple cell is rigorously hexagonal. For all others, the orthohexagonal relation b = a31/2 is obeyed only approximately, the deviation being measured either by an angular parameter ε (Donnay et al. 1964) or by a linear parameter η (Zvyagin and Drits 1996), which is a function of ω|| (Fig. 12). For metrically monoclinic polytypes, β (Class a) or α (Class b) of the sextuple and triple cells are in general only close to 90º.
Figure 12. A small portion of the (001) two-dimensional hp lattice of micas. ε and η (exaggerated) are the angular and linear deviations from hexagonality. A1, A2: hexagonal axes (ε = 0. η = 0); aH, bH: orthohexagonal axes (ε = 0. η = 0) of the C1 cell (bH = aH⋅31/2); a, b: pseudo-orthohexagonal axes (ε ≠ 0. η ≠ 0). The figure is drawn for the case b > bH. Black circles: lattice nodes of the crystal lattice; dashed lines: H cell of the twin lattice; dotted lines: C1 cell built on the hexagonal and pseudo-hexagonal meshes (modified after Nespolo et al. 2000a).
The monoclinic setting in which, within the Trigonal model, cn is constant and the value of the monoclinic angle changes with the number of layers is labeled aS [Class a: cn = ⎯(1/3, 0); S stands for Standard] and bT [Class b: cn = (0,⎯1/3); T stands for Transitional]. The corresponding monoclinic l indices are labeled laS and lbT (Nespolo et al. 1997a). The metric equations in both direct and reciprocal space and the relations between l and h, k indices are given in Table 10. The bT setting is monoclinic a-unique
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and does not correspond to any of the settings commonly adopted to describe monoclinic crystals. Nevertheless, it facilitates the comparison of the atomic coordinates with other polytypes (Backhaus and Ďurovič 1984) and is thus the preferred setting to derive the family structure from a single polytype or vice versa. From bT a monoclinic b-unique setting is obtained through the exchange of axes by a → -b; b → -a; c → -c, so that a > b and β > 90º, as in the Smith and Yoder (1956) definition: this setting is labeled bS (Fig. 13). The exchange of axes is adopted when indexing the diffraction pattern (Nespolo et al. 1998; see also Takeda and Ross 1995).
Figure 13. Definition of the aS, bT and bS axial settings of mica polytypes. a S and bS settings have a < b, bT setting has b < a [used by permission of the editor of Mineralogical Journal, from Nespolo (1999) Fig. 2, p. 56].
For each Series and each Class, K = 0 of the Subclass 1, see Equation (2), determines the axial setting of the first polytype of the Series, which is termed the Basic axial setting. All the polytypes belonging to the same Series and the same Class can be indexed in a setting whose axes are parallel to the axes of the Basic axial setting but whose period along c is 3K+L [Eqn. (2)] times the corresponding period of the Basic axial setting. For each Series the angle is constant, within the Trigonal model, and the value of cn, nontranslationally reduced, changes with the number of layers: this nsetting is ntermed Fixedangle setting. For the two Classes this setting is symbolized by 3 ,aF and 3 ,bF, which for
206
laS. = (lC1 – h)/3 lC1 = h(mod 3)
lC1 = k(mod 3)
a*cosβ∗ = c*/3
b*cosα∗ = c*/3
Metric equations in direct space
ccosβ = -a/3
ccosα = -b/3
a
b
lbT = (lC1 – k)/3
relation between orthogonal and monoclinic l indices relation between lC1 and h, k indices Metric equations in reciprocal space
L ⎡( −1) L −1 0 ( −1) ⋅ ( K + L − 1)⎤ ⎢ ⎥ L −1 ⎥ = ( a b c ) 3n ,a ;3n ,b F 0 ( a b c ) 3n ,a ;3n ,b S ⎢ 0 ( −1) ⎢ ⎥ 0 1 ⎢ 0 ⎥ ⎣ ⎦
Class
Table 10. Metric equation in direct and reciprocal space and relation between Miller indices orthogonal and monoclinic settings for the two Classes (after Nespolo 1999).
Series 0 are shortened in aF and bF (Nespolo et al. 1997a; 1998) (Fig. 14). This setting is obtained from aS and bS by means of the transformation:
(3)
where L (Subclass) and K are defined in Equation (2). The choice of a common setting for polytypes belonging to different Series is instead geometrically not possible, because these polytypes are not based on the same Basic axial setting (Fig. 14). GEOMETRICAL CLASSIFICATION OF RECIPROCAL LATTICE ROWS By considering the lC1 (mod 3) index of reciprocal lattice nodes (Table 10) on rows related by n×60º rotations (0 ≤ n ≤ 5), Nespolo et al (1997b, 2000a) have shown that there are only nine translationally independent rows parallel to c* (Fig. 15) indicated as Ri, 1 ≤ i ≤ 9. In each Ri the same distribution of "present" and "absent" reflections is repeated along a* and b* with 3p and 3q translations (p and q are integers of the same parity). Ri are defined in terms of h and k as: [hi(mod 3), ki(mod 3), l] and are distributed along the edges and diagonals of a rhombusshaped unit, termed tessellation rhombus (Fig. 15, solid lines), which can tessellate the entire reciprocal space by (3p, 3q) translations. A smaller unit, termed minimal rhombus, can be drawn (Fig. 15, dotted lines), defined by the same Ri each taken only once. Opposite edges are different and, contrary to the tessellation rhombus, the minimal rhombus does not represent a translational unit. The two rhombi have six possible orientations, which represent equivalent descriptions of the same reciprocal lattice: they simply differ in the distribution of the Ri. Six equivalent rhombi are obtained by applying the five rotations (besides the identity) to the hi, ki indices of each of the nine Ri of the original rhombus and bringing the resulting Ri within the area spanned by the original rhombus through a (3p, 3q) translation between equivalent rows. The rows that can be obtained by rotating the original rhombus are within a star-polygon constructed by the six rhombi with the common origin (Fig. 15). The values of p and q to be considered are those connecting rows internal to the star-polygon but external to the original rhombus with rows internal to the original rhombus, i.e. (0, ±2), (1, ±1) and (2, ±2).
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Figure 14. Schematic view of the axial settings of mica polytypes. Black circles: direct lattice nodes. The number below each node indicates the number of layers of the polytype to which that node belongs. Horizontal axis is ±a or ±b depending on the Class and on the setting used. The c axes of S and F settings are shown as solid and dotted lines respectively. In all settings, the reference is right-handed. The superscript a or b in the S and F symbols is omitted, since the figure is drawn for both Classes (they differ in the label of the horizontal axis). The figure shows that in cases of polytypes with a number of layers multiple of 3, the c axis of the corresponding F setting does not pass on any lattice node: the F setting of the next Series has thus to be used.
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Figure 15. Minimal rhombus (dotted lines; in the foreground) and tessellation rhombus (solid lines) in the six orientations defining the star polygon. The nine translationally independent rows are distinguished by sequence numbers (R1 ∼ R9) (modified after Nespolo et al. 2000a).
The geometrical characteristics of the reciprocal lattice rows parallel to c*, each taken as a whole, are termed "row features". In the Trigonal model all mica polytypes have the same row features, described by the regular tessellation {3,6} (Takeda and Donnay 1965; see the section “Tessellation of the hp lattice”), and the nine Ri were classified into three types (Fig. 16): 1.
S (Single) rows [h = 0(mod 3) and k = 0(mod 3)].
2.
D (Double) rows [h ≠ 0(mod 3) and k = 0(mod 3)]. There are two translationally independent D rows, labeled Di: i = 1,2; h = i(mod 3); k = 0(mod 3).
3.
X (seXtuple) rows [k ≠ 0(mod 3)]. There are six translationally independent X rows, labeled Xi: 1 ≤ i ≤ 6; h = i(mod 3); k = 2×(-1)i(mod 3).
The nine Ri rows are thus classified as: R1 = S; R2-3 = D1-2; R4-9 = X1-6. This classification of Ri corresponds exactly to the classification in three types of rows introduced by Ďurovič (1982), who did not adopt specific names for each type of rows. Each of the three types lies on non-intersecting circular orbits centered on c*, of radius 3h2 + k2 (cf. Table 4 and Fig. 19 in Ferraris and Ivaldi, this volume). Each of these orbits contains only one type of rows (an n×60° rotation overlaps rows belonging to the same type only) and becomes an ellipsis when the incident beam is inclined by a general angle φ to the sample. This is the principle on which the oblique-texture electron diffraction method (OTED, see Zvyagin 1967) is based, and has been recently applied also to XRD (Rieder and Weiss 1991; for details, see Ferraris and Ivaldi, this volume). Figure 16 shows the orbits of S (solid lines), D (dashed lines), and X (dotted lines). For D and X
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Figure 16. Rotational relation between reciprocal lattice rows parallel to c*. Because of the pseudo-hexagonal symmetry of the 001 r.p., each type of row (S, D, X) lies on a circular orbit around c* with radius 3h + k. Solid, dashed and dotted orbits contain S, D and X rows respectively. D and X orbits are further subdivided into those containing only one set of six rows (DI and XI, thick orbits) and those containing two sets of six rows (DII and XII, thin orbits). The n × 60º rotations, which correspond to the relative orientation of twinned mica individuals, relate only rows of the same type and same set (S, DI, DII, XI, XII), whereas the noncrystallographic rotations typical of plesiotwins relate rows of the same type but of different sets (DI and DII; XI and XII) (modified after Nespolo et al. 2000a).
rows, two types of orbits exist: type I (DI and XI orbits, thick lines) connects one set of six D or X rows, whereas type II (DII and XII orbits, thin lines) connects two sets of six D or X rows. The n×60º rotations about c* lead to an alternate exchange of the two D-type Ri located on the long diagonal of the minimal rhombus, and they exchange the six Xtype Ri on the edges of the minimal rhombus in six different ways. SUPERPOSITION STRUCTURES, FAMILY STRUCTURE AND FAMILY REFLECTIONS By superposing two or more identical copies of the same polytype translated by a superposition vector (i.e. a vector corresponding to a submultiple of a translation period) a fictitious structure is obtained, which is termed a superposition structure. Among the infinitely possible superposition structures, that structure having all the possible positions of each OD layers is termed a family structure: it exists only if the shifts between
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adjacent layers are rational, i.e. if they correspond to a submultiple of lattice translations. The family structure is common to all polytypes of the same family (Dornberger-Schiff 1964; Ďurovič 1994). From a group-theoretical viewpoint, building the family structure corresponds to transforming (“completing”) all the local symmetry operations of a space groupoid into the global symmetry operations of a space-group (Fichtner 1977, 1980). Additional “virtual” atoms are created by the completed operations, and the resulting model may have physically unrealistic interatomic distances: they appear in the superposition structure, which is a purely mathematical construction, as a consequence of the group-theoretical process of completing the local symmetry operations. The group of translations of the polytype reciprocal lattice can be decomposed into a subgroup of translations, which corresponds to the Fourier transform of the family structure (family sublattice), and one or more cosets. The family sublattice is again common to all polytypes of the same family. This means that all polytypes of the same family, normalized to the same volume of scattering matter, have a weighted sublattice in common. The diffractions that correspond to the family sublattice are termed family diffractions (or, more commonly, family reflections). As discussed below, when indexed with respect to the basis vectors of any of the polytypes of the same family, the family sublattice shows several non-space-group absences, which indicate the existence of local symmetry operations. Clearly, the family reflections convey important information, because they reveal the symmetry of the family structure. The family reflections are always sharp, including the case of non-periodic (disordered) polytypes. In fact, the disorder of the stacking concerns the distribution of subsequent ρ-operations. If this distribution is periodic, after a finite even number of steps a period is closed and the product of those ρ-operations is the generating τ-operation (remember that the product of an even number of ρ-operations is a τ-operation). If instead the distribution of subsequent ρ-operations is not periodic, no generating τ-operation can be found, and the polytype is disordered. In the family structure the ρ-operations are completed to global operations: the family structure and its Fourier transform, which consists in the family reciprocal sublattice, are thus common to both periodic and non-periodic polytypes of the same family4 (Ďurovič and Weiss 1986; Ďurovič 1997, 1999). Because the family structure can be deduced from the symmetry principle of the polytype family, it is possible to illustrate its derivation by means of a very simple, hypothetical example, in which the actual atomic arrangement is not taken into account, and geometrical figures with the appropriate λ-symmetry are used instead. Let us consider the three hypothetical polytypes (Ďurovič 1999) and their geometric diffraction patterns in Figure 17. The polytypes are constructed by stacking equivalent layers perpendicular to the plane of the drawing, with λ-symmetry P(1)m1. The stacking direction is a, and the distance between adjacent layers is |a0|. The λ-symmetry is indicated by isosceles triangles with a mirror plane [.m.]. The three polytypes can be related to a common orthogonal four-layer cell with a = 4a0, inside which the cell of the polytype is shown by bold lines (Fig. 17). The first polytype (1A, MDO) has basis vectors a1 = a0 + b/4; b1= b; c1 = c and space-group P111. The only global τ-operation is the translation a0 + b/4. The second polytype (2M, MDO) has basis vectors a2 = 2a0; b2= b; c2 = c and space-group P1a1. The global τ-operations are the translation a = 2a0 and an aglide plane at y = 1/8 and 7/8. The third polytype (4M, non-MDO) has basis vectors a3 = 4a0; b3= b; c3 = c and space-group P1a1. The global τ-operations are the translation a = 4
The remaining diffractions, which correspond to the cosets of the weighted reciprocal lattice with respect to the family sublattice, are termed non-family reflections and are instead typical of each polytype: they can be sharp or diffuse, depending on whether the polytype is ordered or not, i.e. on whether the distribution of subsequent ρ-operations is ordered or random.
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Figure 17. Schematic representation of three hypothetical structures belonging to the same family. The layers are perpendicular to the plane of the drawing, and their constituent atomic configurations are represented by isosceles triangles with λ-symmetry [.m.]. All structures are related to a common, orthogonal four-layer cell with a = 4a0. The family structure is obtained by superposing two identical copies of the same polytype, translated by b/4, the superposition vector. The diffraction indices refer also to the common cell. Family diffractions correspond to ˆk = 2k (open circles), and the non-family diffractions, characteristic for individual polytypes, to ˆk = 2k+1 (close circles) (modified after Durovic and Weiss (1986).
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4a0 and an a-glide plane at y = 0 and y = 1/2. The geometric diffraction pattern of each of these polytypes can be divided into two parts: kˆ = 2k (open circles) and kˆ = 2k+1 (full circles). The kˆ = 2k are the family reflections, which define the family reciprocal sublattice, common to all the three polytypes. The Fourier transform of this subgroup of diffraction gives the family structure, with space-group C1m1, a = 2a0, b = b/2: the superposition vector is b/2. The non-family reflections are those for which kˆ = 2k+1: the number of reflections along each row in the four-layer reciprocal cell is the same as the number of layers in the period of the polytype. Family structure and family reflections of mica polytypes For micas, the family structure of the Pauling model is nine-fold (the supergroup of translation in direct space has the order nine) and the superposition vectors are ±a/3 and ±b/3; its symmetry is P6/mmm (Dornberger-Schiff et al. 1982). To any of the atoms in the layer, eight additional atoms are generated in the family structure, with coordinates (x±1/3, y); (x, y±1/3) and (x±1/3, y±1/3). The family reflections are those with h = 0(mod 3) and k = 0(mod 3), and correspond to S rows. The subgroup of translations in reciprocal space has the order nine. Because the layer stagger is |a|/3, the family vectors of the Pauling model complete the local symmetry operations of space groupoids to global symmetry operations of space groups after one single layer. Therefore, the period along the c axis of the family structure is c0 = 1/c*1 = c1Msinβ1M and thus corresponds to the vertical distance between two closest interlayer cations. The basis vectors of the family structure are AF1 = A1/3, AF2 = A2/3, CF = c0. (Backhaus and Ďurovič 1984; Ďurovič et al. 1984; Ďurovič 1994). In the Trigonal model each of the three families (homo-, meso- and heterooctahedral) splits into two subfamilies, A and B. For both subfamilies the family structure is three-fold and the superposition vectors are ±b/3. To any of the atoms in the layer, two additional atoms are generated in the family structure, with coordinates (x, y±1/3). The family reflections are those with k = 0(mod 3) and correspond to S and D rows. The subgroup of translation in reciprocal space has the order three. The family vectors complete the local symmetry operations of space groupoids to global symmetry operations of space groups after three layers for subfamily A, but after two layers for subfamily B. The basis vectors for the family structure are thus AF1 = (A1+2A2), AF2 = (2A1+A2), CF. For subfamily A, CF = 3c0; for subfamily B, CF = 2c0. The symmetry of the family structure is H⎯R31m (where the subscript R indicates that the smaller cell is rhombohedral) for subfamily A, and H63/mcm for subfamily B (Ďurovič 1994). The adoption of the H-centered cell allows the description of the family structures and the real structures in the same axes, but additional absences appear in the diffraction pattern (cf. Smrčok et al. 1994, Appendix, for cronstedtite-3T). Mixed-rotation polytypes are OD structures only when the ditrigonal rotation of the tetrahedra is zero. Their family structure and family reflections are those of the Pauling model (S rows). From the practical viewpoint, as noted by Ďurovič (1982), the family reflections of the nine-fold family structure (S rows) are common to all members of a family and are thus not useful for the purpose of distinguishing individual polytypes. D rows instead are characteristic of all members of a subfamily (A or B, in case of micas), permit to distinguish the kind of polytype (subfamily A, subfamily B or mixed-rotation). The real layers building micas deviate from their archetypes by several distortions, and the shifts between successive layers are in general not exactly rational. The intensities, but not the geometry, of the family reflections differ from polytype to polytype of the same family, and the divergence increases with the deviation of the real layers from their archetypes. Notwithstanding, the concepts of family structure and family reflections are useful in the identification of twins and polytypes, as shown below.
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REFLECTION CONDITIONS In the diffraction pattern of mica polytypes, systematic non-space-group absences extensively appear. The International Tables for Crystallography term this kind of absences additional reflection conditions (Hahn and Vos 2002). This definition does not provide anything about the kind of information one can get from these absences. As seen above, the absences along S and D rows derive from the existence of local symmetry operations that relate pairs of packets. These local symmetry operations are not accounted for in the space-group type. In the Trigonal model, any mica polytype of a given family is constructed from layer archetypes in which the atoms in each plane are distributed according to a hexagonal pattern. These atoms are either on special positions, or on positions that, without corresponding to any translation-free symmetry operation of the space-group type, have higher translational symmetry. These positions, under the symmetry operations of a space-group type, define sets of points (crystallographic orbits) the eigensymmetry group of which includes additional translations, and are known as extraordinary orbits of space-groups (Wondratschek 1976; Matsumoto and Wondratschek; 1979). The corresponding lattice of translation vectors is a proper superlattice of the polytype lattice. In reciprocal space, these vectors correspond to a sublattice, which shows systematic non-space-group absences when indexed with respect to the basis vectors of the polytype. The OD description is based on the existence of local symmetry operations, whereas the description in terms of crystallographic orbits is based on the points on which those local symmetry operations act. In spite of the different languages, the concepts are basically the same. The approach involving crystallographic orbits is not specifically related to VC structures but it is more general. The possible superlattices were however derived for all space-group types within the same syngony (Engel et al. 1984). There are no derivations yet for the cases in which the superlattice belongs to a Bravais system higher than that of the entire lattice. The superlattice common to all polytypes of a family (family superlattice, i.e. the lattice of the family structure) corresponds to this latter case (with the exception of trigonal-hexagonal polytypes, of which only 3T has been reported so far). A general symmetry analysis of mica polytypism in terms of crystallographic orbits is nowadays a completely open task, but the non-space-group absences along S and D rows are interpretable in terms of extraordinary orbits as well. The deviations of layers from their archetypes correspond to the movement of part of the atoms slightly away from the positions of higher translational symmetry, towards general positions. As a consequence, violations of the non-space-group absences appear as faint reflections between pairs of family reflections. These faint reflections can be recorded in dioctahedral micas (Rieder 1968) and, with longer exposure times, in Li-rich trioctahedral micas (Rieder 1970), but they are almost undetectable in Li-poor trioctahedral micas. This sequence is in accordance with the extent of the structural distortions, which decreases in the same order. The reflection conditions in the two subfamilies were derived by Nespolo (1999). The number and positions of reflections along the D rows reveal the symmetry of the family structure (H⎯R31m: subfamily A; H63/mcm: subfamily B; P6/mmm: mixedrotation). In addition, they are particularly useful in evaluating the possible presence of twins. Taking into account that for non-orthogonal polytypes only one out of three of the orthogonal l indices corresponds to integer monoclinic indices, and that subfamily B polytypes necessarily contain an even number of layers, the reflection conditions are (N and N′ are the number of layers in the conventional and orthogonal cell respectively): 1. S rows (family reflections of the nine-fold family structure): one reflection out of N always occurs, with presence criterion lC1= 0(mod N′). 2.
D rows: one reflection (family reflection) out of N occurs for subfamily A polytypes [presence criterion lC1 = (±N′h/3)(mod N′), “+” for the obverse setting of the family
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structure, “–” for the reverse setting], two (family reflections) for subfamily B [equally spaced, at lC1 = 0(mod N′/2)], and N′ (non-family reflections) for mixed-rotation polytypes. 3. X rows: N reflections appear in the c*1 repeat (non-family reflections for all polytypes). One or more of the N reflections along X rows (and for mixed-rotation polytypes also along D rows) may be very weak or absent. This non-space-group absence is related not to the symmetry of the family structure, as for family reflections, but to the stacking mode within the polytype. The family structure of subfamily A polytypes admits a primitive rhombohedral cell, and its lattice (family sublattice) can be overlapped for all polytypes belonging to subfamily A only if it is rotated by 180º around the normal to the layer when comparing polytypes built by layers of opposite orientational parity. This is because the rhombohedral primitive cell of the family structure for subfamily A polytypes is in the obverse setting for one orientational parity of the layers (odd orientational parity of the symbols), but in the reverse setting for the other (even orientational parity of the symbols). In Series 0, all polytypes belonging to subfamily A are Class a polytypes. Polytypes belonging to a different Subclass have opposite orientational parity. The aF setting alternates the directions of (a, b) and (a*, b*) axes with the Subclass (Fig. 14) and is exactly the axial setting leading to the overlap of the sublattice built on family reflections. In higher Series, polytypes belonging to subfamily A can be orthogonal or Class b polytypes and there is no longer a 1:1 correspondence. Subfamily B polytypes show two reflections along D rows. However, polytypes of this subfamily either are orthogonal or belong to Class b, for which the non-right angle is α (before the axes interchange) and the lC1 index of the superlattice nodes does not depend on h. The reciprocal sublattice in this case matches for all polytypes, which is consistent with the fact that the primitive cell of the family structure is hexagonal. In mixed-rotation polytypes, the family reflections are only those of the nine-fold family structure and appear along S rows. D rows convey important information, because the different number of reflections along the rows, or their diffuseness, unambiguously reveals the mixed-rotation character of the polytype. NON-FAMILY REFLECTIONS AND ORTHOGONAL PLANES Reciprocal central planes, which have c* in common, can be usefully classified, on the basis of the rows they contain, into SD and SX. Here we consider the six densest central planes, which are sufficient for a twin/polytype analysis. The three densest central reciprocal planes (r.p., hereafter) are of type SX: 0k*, hhl and⎯hhl. These planes have the shortest separation between pairs of reciprocal lattice rows parallel to c* (about 0.22Å-1), and are followed by the three densest SD central r.p. h0l, h.3h.l and ⎯h.3h.l (about 0.38Å1 ). These six central planes are shown in Figure 18, projected onto the (a*, b*) plane. The three SD central planes are 60º apart each, and the same holds for the three central SX planes. The two kinds of planes are each 30º apart. The SD central planes show the symmetry of the family structure. Then, from the intensities measured along one or more X rows, the stacking sequence can be determined. However, the presence of twinning must be excluded before analyzing the intensity distribution, and for this purpose the analysis of the geometry of the diffraction pattern, in particular the number and type of orthogonal planes, is of primary importance. A plane is orthogonal if the direction r* corresponding to the line perpendicular to c* and passing through the origin (a direction that belongs to the orthohexagonal cell) contains a node for each row parallel to c*.
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Figure 18. (001) projection of mica reciprocal lattice. Open circles: S rows; open triangles: D rows; close circles: X rows. The six central planes (three SD and three SX) that can commonly be recorded by a photographic technique such as precession camera are indicated (modified after Nespolo et al.1999d). Cf. Figure 4 in Sadanaga and Takeda (1969) and Figure 1 in Durovic (1982).,
In case of a non-orthogonal plane, no nodes are present on r* along the X rows, and the node closest to r* is at a height ±c*1/3N, where N is the number of layers in the conventional cell. If the node on r* or closest to it corresponds to an absent reflection, the orthogonality of the plane must be judged from the position of the two adjacent reflections, whose height is either ±c*1/N (orthogonal plane) or ∓2c*1/3N (nonorthogonal plane). For D rows the family character of the reflections should be considered. In subfamily A polytypes, reflections appear at ±c*1/3 (non-orthogonal SD plane); in subfamily B polytypes, reflections appear at 0 and c*1/2 (orthogonal SD plane); in mixed-rotation polytypes, the D rows correspond to non-family rows and the same criteria given for X rows hold. Finally, S rows always contain a node on r*. The number and features of the orthogonal planes (as defined above) depend both on the Class (lattice features) and on the subfamily (OD character). These are easily obtained by taking into account that polytypes in subfamily B and in subfamily A Series > 0 never belong to Class a, whereas polytypes in subfamily A Series 0 always belong to Class a. 1. 2. 3.
Orthogonal polytypes. In case of subfamily A polytypes, only the three SX central planes are orthogonal, according to the above definition. For subfamily B and mixed-rotation polytypes, all the six central planes are orthogonal. Class a polytypes. One SX central r.p. is orthogonal: 0kl. Class b polytypes. None of the three SX central planes are orthogonal. In subfamily A polytypes (Series > 0) the SD central planes are non-orthogonal and thus none of the six densest central planes is orthogonal. In subfamily B, the three densest SD
216
Nespolo & Ďurovič central planes are orthogonal. In mixed-rotation polytypes, D rows correspond to non-family reflections and on these rows, in general, N reflections occur. On the basis of the relation between l indices in bT and in C1 settings (lbT and lC1; Table 10), the three SD densest central planes are orthogonal also.
HIDDEN SYMMETRY OF THE MICAS: THE RHOMBOHEDRAL LATTICE Takeda (1971) analyzed the symmetry properties of the RTW symbols and showed that the stacking of the mica layers can produce polytypes belonging to five kinds of symmetries: A, M, O, T, H; it is thus not possible to obtain a polytype belonging to the rhombohedral Bravais system. Notwithstanding, the rhombohedral lattice appears in the geometry of the diffraction pattern and plays an important role in the twinning of the micas. Here the first aspect is briefly analyzed, whereas the effect on twinning is considered below. There are two categories of polytypes in which the rhombohedral lattice represents a kind of “hidden symmetry” for micas. 1)
2)
Subfamily A polytypes. As shown in the section dealing with the family structure, the family structure of subfamily A polytypes has symmetry HR⎯(3)1m, admitting a primitive rhombohedral cell. Within the Trigonal model the family reciprocal sublattice is rhombohedral both in its geometry and intensity distribution. In the real diffraction pattern the intensity distribution deviates from rhombohedral symmetry proportionally to the deviations of the layer from their archetypes described by the Trigonal model, but the geometry remains rhombohedral. Class b polytypes. Successive lattice planes parallel to (001) are shifted by 1/3 of the short (Class a) or the long (Class b) diagonal of the two-dimensional pseudohexagonal mesh built on (A1, A2) axes. For Class b polytypes a pseudo-rhombohedral primitive cell can be chosen, having (almost) the same volume of the reduced cell (Fig. 19). The primitive cell is closer to rhombohedral when the layers are closer to
Figure 19. Projection onto the (001) plane of the primitive, conventional (double, monoclinic), pseudo-hexagonal (triple), C1 (sextuple, pseudo-orthohexagonal) and pseudoto c axis of the orthogonal cell-rhombohedral (primitive) cells of Class b polytypes. Black, white and gray circles represent lattice nodes at z = 0, 1/3 and 2/3 (z is referred to c axis of the orthogonal cells). Thick lines: the C1 cell Dashed and borders 2/3 (z isofreferred to c and axisofofthe thepseudo-hexagonal orthogonal cells). cell. Thick lines: lines: borders of the upper plane of the conventional and primitive b d cellsf (the h lower C llplaned is fin hcommon d with h C1 cell l and ll pseudoD h d hexagonal cell respectively). The pseudo-rhombohedral cell (dotted lines) is best viewed by means of the pseudo-rhombohedral axes aR. a, b: (pseudo)-orthohexagonal axes. A1, A2: (pseudo)-hexagonal axes (modified after Nespolo 1999).
Crystallographic Basis of Polytypism and Twinning in Micas
217
their archetypes as described by the Trigonal model. The general reflection conditions for the rhombohedral lattice in hexagonal axes, -h+k+l = 3n, expressed in the C1 setting become: -3h+k+2l = 6n. Taking into account the C centering condition, the latter equation corresponds to l(mod 3) = k(mod 3), which is simply an alternative expression of the condition that monoclinic indices are integers, given in Table 10 for the bT setting (Nespolo 1999). Because non-orthogonal polytypes of subfamily A Series > 0 belong to Class b, in this case the “hidden” rhombohedral symmetry appears both in the family sublattice and in the entire polytype lattice. TWINNING OF MICAS: THEORY The definition and classification of twinning is given in Appendix A. The pseudosymmetries typical of micas made the recognition of the twin laws difficult, and Friedel initially classified mica twins among the “macles aberrantes” (Friedel 1904, p. 222), i.e. oriented crystal associations without either twin plane or twin axis stricto sensu. The derivation of the twin laws for mica polytypes must consider the point groups of the twin lattice and of the lattice of the individual, and the point group of the syngony of the individual. The twin operators are the point symmetry operators of the twin lattice not belonging to the point group of the individual and can be obtained by coset decomposition. The decomposition of the twin lattice point group (order m) yields one subgroup (the point group of the individual, order m′ < m) and n = m/m′-1 cosets corresponding to the twin laws. Hereafter the subgroup corresponding to the point group of the individual is always given first, and the twin laws follow as cosets No. 1 to n. All merohedral polytypes, in any syngony, may undergo twinning by syngonic merohedry: the twin laws depend on the point group of the polytype and should thus be derived case by case (see the example for 3T below). Instead, twins other than by syngonic merohedry can be derived with a general procedure. Hereafter, indexing is given in the (pseudo)orthohexagonal setting of the twin lattice. 1) Polytypes of the orthorhombic syngony with a hP lattice may undergo twinning by metric merohedry, the twin lattice coinciding with the lattice of the individual. The coset decomposition gives two twin laws:
{
}
6 / mmm = 1, 2[010] , 2[001] , 2[100] , 1, m( 010) , m( 001) , m(100) ∪
{ ∪ {3
} ) }.
∪ 6[+001] ,6[−001] , 2[110] , 2 ⎡1 10⎤ , 6[+001] , 6[−001] , m(130) , m(130) ∪ −
⎣
+
⎦
−
+
[001] ,3[001] , 2[310] , 2 ⎡⎣3 10⎤⎦ , 3[001] , 3[001] , m(110 ) , m(1 10
(4)
All the operators corresponding to the same twin law are equivalent under the action of the symmetry operators of the orthorhombic syngony. If the lattice is only oC, twinning is by pseudo-merohedry. The twin lattice (hP) does not coincide exactly with the lattice of the individual, because for the latter the orthohexagonal relation b = a31/2 is only approximated. However, the two lattices have the three orthohexagonal axes parallel. The coset decomposition is the same as given in Equation (4), but the non-zero obliquity (ω = ω|| ≠ 0, ω⊥ = 0) makes the operators in each of the two cosets not equivalent, as described in detail below.
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2) Polytypes of the monoclinic and triclinic syngony with an hP lattice may undergo twinning by metric merohedry. For the monoclinic syngony the coset decomposition gives five twin laws, each with four equivalent twin operators:
{ ∪ {3[
} {
}
6 / mmm = 1, 2[010] , 1, m( 010) ∪ 2[001] , 2[100] , m( 001) , m(100) ∪ − 001]
{
} {
}
, 2[310] , 3[−001] , m(110) ∪ 6[−001] , 2[110] , 6[−001] , m(130) ∪
} {
∪ 3[+001] , 2 ⎡3 10⎤ , 3[+001] , m(1 10) ∪ 6[+001] , 2 ⎡1 10⎤ , 6[+001] , m(130) ⎣
⎦
⎣
⎦
(5)
}
whereas for the triclinic syngony the coset decomposition gives eleven twin laws, each with two equivalent twin operators:
{
} {
} {
}
6 / mmm = {1, 1} ∪ 2[010] , m( 010) ∪ 2[001] , m( 001) ∪ 2[100] , m(100) ∪
{ ∪ {3[
} { ] } ∪ {2
} { ) } ∪ {6[
} { ] } ∪ {2
}
∪ 3[−001] , 3[−001] ∪ 2[310] , m(110) ∪ 6[−001] , 6[−001] ∪ 2[110] , m(130) ∪ + 001]
, 3[+001
⎣⎡3 10⎦⎤
, m(1 10
+ 001]
, 6[+001
⎣⎡1 10⎦⎤
(6)
}
, m(130) .
If the lattice of the individual is oC, the first two cosets in Equation (5) and the first four cosets [Eqn. (6)] correspond to metric merohedry, whereas the others correspond to pseudo-merohedry (ω = ω|| ≠ 0, ω⊥ = 0). If the lattice of the individual is mC Class a, the twin laws in Equations (5) and (6) correspond to reticular pseudo-merohedry. The hP twin lattice is a sublattice for the individual, with subgroup of translation 3: the twin index is thus 3. 3) Monoclinic and triclinic Class b polytypes with a two-dimensional hexagonal mesh in the (001) plane and a cn projection of exactly |b|/3 has a hR lattice. Twin elements belonging to the hR lattice but not to the monoclinic or triclinic syngony correspond to the twinning by metric merohedry, whereas twin elements belonging to the hP sublattice but not to the hR lattice correspond to twinning by reticular merohedry. The subgroup of translation defining the hP sublattice is 3, and thus the twin index is 3 also. The coset decomposition gives five (monoclinic syngony) or eleven (triclinic syngony) twin laws: monoclinic syngony:
{
} {
}
6 / mmm = 1, 2[100] , 1, m(100) ∪ 3[−001] , 2[110] , 3[−001] , m(130) ∪
{
} { } ∪ {6
}
∪ 3[+001] , 2 ⎡1 10⎤ , 3[+001] , m(130) ∪ 2[001] , 2[100] , m( 001) , m(100) ∪ ⎣
⎦
{
∪ 6[−001] , 2[310] , 6[−001] , m(110) triclinic syngony:
{
+
+
[001] , 2 ⎣⎡3 10⎦⎤ , 6[001] , m(1 10 )
} {
} {
(7)
}.
}
6 / mmm = {1, 1} ∪ 2[100] , m(100) ∪ 3[−001] , 3[−001] ∪ 2[110] , m(130) ∪
{
} {
} { ) } ∪ {2[
} { ) } ∪ {2
}
∪ 2 ⎡1 10⎤ , m(130) ∪ 3[+001] , 3[+001] ∪ 2[001] , m( 001) ∪ 6[−001] , 6[−001] ∪
{
⎣
⎦
} {
∪ 6[+001] , 6[+001] ∪ 2[310] , m(110
010]
, m( 010
⎡⎣3 10⎤⎦
}
, m(1 10) .
(8)
Crystallographic Basis of Polytypism and Twinning in Micas
219
The first two [Eqn. (7)] or four [Eqn. (8)] cosets give the twin laws by metric merohedry, the others give the twin laws by reticular merohedry. Twin operators in each coset are equivalent by the action of the symmetry elements of the syngony. If the two-dimensional mesh in the (001) plane is not rigorously hexagonal (ω|| ≠ 0), or if the cn projection is not exactly |b|/3 (ω⊥ ≠ 0), the hR lattice does not coincide exactly with the lattice of the individual; moreover, the hP sublattice is only an approximate sublattice for the individual. The twin laws derived in Equations (7) and (8) do not change, but they correspond to pseudo-merohedry and reticular pseudomerohedry instead of metric merohedry and reticular merohedry respectively. The operators in each coset are no longer equivalent. Choice of the twin elements
The twin element that relates a pair of individuals occurs in the morphology of the twin. Micas show two kinds of twin morphologies: rotation twins, with composition plane (001), and reflection twins, with composition plane (almost) normal to (001). As noted by Friedel (1904), the twin axis for rotation twins is within the composition plane, whereas the twin plane for reflection twins coincides with the composition plane. Whereas the morphological twin operation is unique, the geometrical operations bringing the twin lattice into self-coincidence are in general more numerous, as shown in the previous section. For zero obliquity, the operations within each coset corresponding to a twin law are equivalent, when considering only the lattice, by the action of the symmetry elements of the individual. The morphological twin operation is termed the representative operation of the coset (Nespolo and Ferraris 2000). For non-zero obliquity, however, they are no longer equivalent and the correct twin operations are those obeying the law of Mallard, which requires that the twin operations are crystallographic operations. As an example, let us consider the decomposition of the point group of the hP twin lattice with respect to the point group of the monoclinic syngony in Equation (5). If the monoclinic polytype has a hP lattice (twinning by metric merohedry) or sublattice (twinning by reticular merohedry) the six two-fold axes in the (001) plane are exactly 30º each apart and each of them is perpendicular to a plane (hk0): the four operations in each coset are truly equivalent, when considering only the lattice. Instead, if the lattice or sublattice of the individual is not exactly hexagonal (twinning by pseudo-merohedry and reticular pseudo-merohedry), either ω|| or ω⊥ (in general both) is non-zero. For ω|| ≠ 0 the 2[310], 2[⎯310], 2[110] and 2[⎯110] are (2n+1)×30±εº apart from 2[010] / 2[100] and they are no longer perpendicular to the (hk0) planes (Fig. 20). Twin axes and twin planes deviate thus from mutual perpendicularity: rotation twins and reflection twins are no longer equivalent, even for centrosymmetric crystals, and are called reciprocal twins (Mügge 1898) or corresponding twins (Friedel 1904, 1926). For ω⊥= 0 the equivalence relations become: 2[310] ⋅ 2[010] = 3[−001] ± 2ε
m(110) ⋅ m( 010) = 3[−001] ∓ 2ε
2 ⎡3 10⎤ ⋅ 2[010] = 3[+001] ± 2ε m(1 10) ⋅ m( 010) = 3[+001] ∓ 2ε ⎣
⎦
2[110] ⋅ 2[010] = 6[−001] ± 2ε
m(130) ⋅ m( 010) = 6[−001] ∓ 2ε
2 ⎡1 10⎤ ⋅ 2[010] = 6[+001] ± 2ε m(130) ⋅ m( 010) = 6[+001] ∓ 2ε ⎣
⎦
2[100] ⋅ 2[010] = 2[001]
m(100) ⋅ m( 010) = 2[001]
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Figure 20. Component of the obliquity within the (001) plane of the pseudo-hp lattice of micas. The six directions [hk0] (including the a and b axes) in the (001) plane (solid lines) would be equivalent in a hexagonal lattice. The dashed thick line is t(⎯130), i.e., the intersection of the (⎯130) plane with the (001) plane, which is almost but not exactly normal to [⎯110] direction (it would be normal to it in a truly hp lattice). The trace of the t(010) and t(100) coincide with a and b axes respectively (γ = 90º). To improve the clearness of the figure, the t(hkl) of the other three planes that would be equivalent in a truly hp lattice are not shown, but they can be easily traced (modified after Nespolo and Ferraris 2000).
Only the two-fold rotation about c of the twin lattice is a correct twin operation, in the sense that it restores the lattice, or a sublattice, of the individuals. If however ω⊥≠ 0, the c axis of the twin lattice is no longer exactly perpendicular to the (001) plane and the above rotations are defined only with respect to c* and not to c: none of them is thus a correct twin operation. The rotations about c* give simply the (approximate) relative rotations between pairs of twinned mica individuals, but are not true twin operations. Similar considerations apply also to the rotoinversion operations. ε depends upon the obliquity of the twin but, at least in Li-poor trioctahedral micas, is sufficiently small to be neglected for practical purposes (Donnay et al. 1964; Nespolo et al. 1997a,b, 2000a). In Table 11 the complete scheme developed above is summarized for ease of consultation. Effect of twinning by selective merohedry on the diffraction pattern
The above analysis does not consider the case of selective merohedry, which does not appear in the morphology of the twin but influences the diffraction pattern by relating lattice nodes corresponding to present reflections from one individual to nodes corresponding to non-space-group absences from another individual. Twinning by either syngonic or metric merohedry (for the definitions, see Appendix A) does not modify the geometry of the diffraction pattern. Instead, twinning by selective merohedry, i.e. when the twin operation belongs to the point group of the twin lattice but not to the point group of the family structure, produces an unusual diffraction pattern. The typical case is that of the 3T polytype orthogonal Series 1 subfamily A, space-group type P31,212, which has an hP lattice. As shown above, the family structure is rhombohedral and the family reflections (S and D rows) obey the presence criterion l = N′h/3(mod N′). With respect to
Table 11. Kind of twinning and twin laws for mica polytypes classified on the basis of the polytype syngony, polytype lattice and twin lattice. Syngony of Lattice of the Twin Kind of twinning Twin laws Twin Rotation between Polytypes the individual individual lattice index pairs of individuals‡ syngonic merohedry # 1 # merohedral polytypes H/T hP hP metric merohedry all polytypes O hP hP 1 ±(120)º [310] (110); [C310] (C110) pseudo-merohedry all polytypes oC hP 1 [310] (110); [C310] (C110) ±(120±2H)º syngonic merohedry # 1 # merohedral polytypes oC A/M metric merohedry all polytypes hP hP [310] (110); [C310] (C110) ±(120)º ±(60)º 1 [110] (130); [C110] (C130) (180)º [100] (100) pseudo-merohedry all polytypes oC hP ±(120±2H)º [310] (110); [C310] (C110) 1 [110] (130); [C110] (C130) ±(60±2H)º metric merohedry [100] (010) 1 (180)º all polytypes oC metric merohedry all polytypes hR hR 1 ±(60)º [110] (130); [C110] (C130) (Class b reticular merohedry all polytypes hP ±(60)º [310] (110); [C310] (C110) 3 polytypes) (180)º [100] (100) pseudo-merohedry all polytypes aC†/mC Class b hR 1 [110] (130); [C110] (C130) ±(120±2H)º hP reticular all polytypes [310] (110); [C310] (C110) ±(60±2H)º 3 pseudo-merohedry [010] (010) (180)º † reticular all polytypes aC /mC Class a hP [310] (110); [C310] (C110) ±(120±2H)º pseudo-merohedry 3 [110] (130); [C110] (C130) ±(60±2H)º [100] (100) (180)º syngonic merohedry [100](100) 1 (180)º merohedral polytypes M mC Class b mC syngonic merohedry [010](010) 1 (180)º merohedral polytypes mC Class a mC metric merohedry [100](100) 1 (180)º all polytypes A mC Class b mC metric merohedry [010](010) 1 (180)º all polytypes mC Class a mC mC pseudo-merohedry [100](100) all polytypes aC† Class b 1 (180±2H)º mC pseudo-merohedry [010](010) all polytypes aC† Class a 1 (180±2H)º † † aC aC syngonic merohedry merohedral polytypes 1 0º C1 † The unconventional C centring of triclinic polytypes is adopted to preserve the same pseudo-orthohexagonal axes (a, b) used for polytypes of the other Bravais systems. ‡Rotations about c*. #Symmetry elements and relative rotations depend on the point group of the individual
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the period of the family sublattice, 1/3c0, one reflection appears in the 1/c0 repeat, with presence criterion l = h(mod 3). The coset decomposition gives three twin laws:
{ ∪ {m(
} {
}
6 mmm = 1,3[+001] ,3[−001] ,2 [010 ] ,2 [310 ] ,2 [3 1 0 ] ∪ 2 [100 ] ,2 [1 1 0 ] ,2 [110] ,2 [001] ,6 [−001] ,6 [+001] ∪ 100 )
} {
, m(130 ) , m(1 3 0 ) , m(001) , 6[−001] , 6[+001] ∪ m(010 ) , m(110 ) , m(1 1 0 ) , iˆ, 3[−001] , 3[+001]
}
(9)
By expressing the twin laws through the Shubnikov’s two-color group notation (in which the twin elements are dashed: Curien and Le Corre 1958), the three twin laws are: 6′2′2; ⎯6′m′2; ⎯3′12/m′. The complete twin [i.e. twin by merohedry or reticular merohedry, in which the number of individuals generated from the original individual is equal to the number of possible twin laws (Curien and Donnay 1959)] contains four individuals and has symmetry 6′/m′′ 2′/m′′ 2/m′′′. The 6′2′2 and⎯6′m′2 twin laws correspond to syngonic selective merohedry class IIA, whereas the⎯3′12/m′ twin law corresponds to syngonic complete merohedry class I (Table A1). In the twins by syngonic selective merohedry, the twin operations do not belong to the point group of the family structure, and the two individuals in the twin are rotated by (2n+1)×60º, whereas layer rotations of subfamily A polytypes are 2n×60º. These twin operations produce the complete overlap of the reflections along X rows and S rows, but not of those along D rows. For example, the⎯h0l family row of one individual is overlapped to the symmetrically independent h0l family row of the other individual. Because of the presence criterion given above, the two reflections from the two individuals in the 1/c0 repeat along D rows are not overlapped, but are separated by 1/3c0 (Fig. 21). The 6′2′2′ and⎯6′m′2 twin laws, although being twin laws by merohedry according to the classical definition, produce the overlap of only one third of the family reflections (those along S rows), behaving thus as twin laws by reticular merohedry with respect to the family structure.
Figure 21. h0l r.p. (SD family plane) of the 3T polytype twinned by selective merohedry. Black circles: family reflections overlapped by the twin operation (common to both individuals). Gray and white circles: family reflections from two individuals rotated by (2n+1) × 60º, not overlapped by the twin operation (modified after Nespolo et al.1999a).
Crystallographic Basis of Polytypism and Twinning in Micas
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Diffraction patterns from twins
The twin reciprocal lattice results from the overlap of the reciprocal lattices of the individuals. From each individual, lattice rows of the same type (S, D or X) overlap into a single composite row. The reflections along a composite row are perfectly aligned for cn = |a|/3 or |b|/3, but slightly deviate from alignment where cn departs from those ideal values. Because of the physical (non-zero) dimension of the reflections, which for micas are commonly broad and oval-shaped, a zigzag disposition of reflections from different individuals can in practice be observed only for significant deviations of cn, typical of dioctahedral micas and, to a minor extent, for Li-rich trioctahedral micas (Rieder 1970). The zigzag disposition of the reflections along rows parallel to c* is indicative of twinning, but it is normally not noticeable in Li-poor trioctahedral micas. The presence of twinning has thus to be evaluated, in general, from the geometry of the SD and SX central planes. For non-orthogonal polytypes the metric relations lC1 = h (mod 3) (Class a) and lC1 = k (mod 3) (Class b) hold (see Table 10). Depending upon the twin law(s) (and thus the relative orientation of twinned individuals), non-family reflections from different individuals may either overlap or occur at positions separated by c*1/3N, where N is the number of layers in the repeat unit (Table 10). Where two of the three positions in a c*1/N repeat are occupied, the presence of twinning should be suspected. In contrast, where each of the three positions are occupied, the number of reflections in a c*1 repeat of a nonorthogonal twinned N-layer polytype is the same as that of an untwinned 3N-layer polytype. This phenomenon is known as “apparent polytypism” (Takano and Takano 1958). However, twinning in some cases modifies the appearance of the D rows, which, for subfamily A polytypes, may show two reflections at 1/3 and 2/3 of the c*1 repeat, as in case of selective merohedry. The number and the position of reflections along D rows, as well as the number of orthogonal planes, in most cases allows the presence of twinning to be distinguished. 1.
Twinning of subfamily A polytypes in which individuals are rotated by (2n+1)×60º corresponds to twinning by reticular pseudo-merohedry. This twinning produces a separation of the single reflection on D rows from each individual into two reflections, corresponding to l(c*1) = 1(mod 3) and l(c*1) = 2(mod 3); no reflection appears corresponding to l(c*1) = 0(mod 3); this pattern is clearly different from that of a subfamily B polytypes, where two equally spaced reflections appear. In addition, if rotation is by ±60º, for Series 0 polytypes (Class a) the orthogonal plane of one individual necessarily overlaps a non-orthogonal plane of another individual. The composite diffraction pattern has thus two or three SX orthogonal central planes.
2.
Twinning of subfamily A polytypes in which individuals are rotated by 2n×60º corresponds to twinning by reticular pseudo-merohedry for Class a (Series 0), but to pseudo-merohedry for Class b (Series > 0). Twinning produces overlap of the single reflection on D rows from each individual; no reflection appears corresponding to l(c*1) = 0(mod 3). However, for polytypes of Series 0 (Class a) two or three SX planes are orthogonal, depending on the number of individuals. When three such planes appear (three or more twinned individuals), the geometrical features of the diffraction pattern are the same as for orthogonal Series 1 polytypes. This situation corresponds to the 3T polytype vs. twinned 1M. For dioctahedral micas it is distinguished by careful examination of the appearance of weak reflections violating the reflection conditions (e.g., Nespolo and Kogure 1998), whereas for trioctahedral micas different techniques, such as microscopic observation of the crystal surface, may be necessary (e.g., Nespolo and Kuwahara 2001). If the twin involves only two
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4.
Nespolo & Ďurovič
individuals, successive reflections along X are unequally separated (1/3 and 2/3) and two SX planes are orthogonal: the presence of twinning is thus easily recognized. Subfamily B polytypes either are orthogonal or belong to Class b. In the latter case only three of the five pairs of twin laws correspond to twinning by reticular pseudomerohedry. However, the corresponding twin operations lead to the overlap of the two reflections on D rows from each individual; no SX plane is orthogonal, whereas the three SD planes are orthogonal. The presence of twinning is not evident. For mixed-rotation polytypes D rows are non-family rows. For Class a polytypes, two individuals rotated by 180º share one orthogonal r.p. 0kl, but reflections are unequally spaced. The presence of twinning is thus evident. In other cases, two or more SX planes are orthogonal, as for subfamily A polytypes of the same Class, but no SD plane is orthogonal. The presence of twinning is again evident. For Class b polytypes the three SD planes are orthogonal and the presence of twinning is not evident.
In Tables 12a-12c the complete scheme of the identification process is shown. The approximated relative rotations between twinned individuals are given: the corresponding twin laws are easily obtained from Table 11. For Class a polytypes (which represent most of the polytypes reported to date) the presence of twinning can be confirmed or excluded by simple inspection of the geometry of the diffraction pattern. Special attention is however needed to distinguish a 3N-layer orthogonal polytype from the spiral twinning of three non-orthogonal N-layer Class a polytypes in which the individuals are rotated by 2n×60º. For polytypes of Class b subfamily A Series 1 the presence of reticular pseudomerohedry twinning is also evident. In the other cases the presence of twinning cannot be confirmed or excluded by analyzing the geometry of the diffraction pattern. Allotwinning
The oriented association of two or more crystals differing only in their polytypic character is termed allotwinning, from the Greek αλλος, “different”, with reference to the individuals (Nespolo et al. 1999c). Allotwinning differs from twinning in that the individuals are not identical but have a different stacking sequence. Allotwinning differs also from oriented overgrowth (epitaxy: Royer 1928, 1954) and oriented intergrowth (syntaxy: Ungemach 1935) because the chemical composition is (ideally) identical and, because the building layer(s) are the same, at least two of the three parameters – those in the plane of the layer – are identical also. A cell common to the two individuals can always be found, which in general is a multiple cell for both crystals: the parameter not in the plane of the layer is the shortest one common to the cells of both individuals. As in case of triperiodic epitaxy, a three-dimensional common lattice exists (allotwin lattice): it may coincide with the lattice of one or more individuals or be a sublattice of it. Whereas a triperiodic epitaxy in general may or may not occur, depending on the degree of misfit of the lattice parameters of the individuals, there is no similar condition in allotwinning, because the individuals have a common mesh in the plane of the layer(s) even in polytypes with a different space-group type. The allotwin operation is a symmetry operation for the allotwin lattice, which may belong to the point group of one or more individuals also. The allotwin of N individuals is characterized by N allotwin indices: the allotwin index of the j-th individual is the order of the subgroup of translation in direct space defining the allotwin lattice with respect to the lattice of the j-th individual. Tessellation of the hp lattice
Assuming the mica two-dimensional lattice in the (001) plane is hp [ω|| = 0], the lattice can be described through a regular tessellation {3,6}, i.e. an assemblage of equal
-------------
Subfamily A Series 0 Class a untwinned polytype
Subfamily A Series 0 orthogonal polytype untwinned or ±120º-twinned -------------------------
1 (SX)
3 (SX)
3 (SD)
6 (SX and SD)
-------------
-------------
.
Subfamily A Series 0 orthogonal (±60º / 180º)-twinned polytype
lC1 = 1(mod 3) and 2(mod 3)
1 [lC1 z 0(mod 3)]
Number of planes with orthogonal appearance 2
------------Mixed-rotation Series 0 Class b polytype untwinned or ±120º -twinned Mixed-rotation Series 0 orthogonal polytype
Subfamily B Series 0 Class b polytype untwinned or ±120º-twinned Subfamily B Series 0 orthogonal polytype
Mixed-rotation Series 0 Class a untwinned polytype
N
-------------
-------------
[lC1 = 0(mod N) and N/2(mod N)]
Number of reflections in the c*1 repeat along reciprocal lattice rows corresponding to family reflections
Table 12a. Classification of diffraction patterns for N = 3K+L. For the correspondence between the relative rotations of twinned individuals and the twin laws see Table 11 (after Nespolo 1999).
Crystallographic Basis of Polytypism and Twinning in Micas 225
---------
---------
6 (SX and SD)
---------
---------
Mixed-rotation Series 0 Class a polytype (±60º / ±120º)-twinned (three individuals) Mixed-rotation Class b: Series 1 polytype untwinned or ±120º-twinned Series 0 (±60º / 180º)-twinned polytype Mixed-rotation Series 1 orthogonal polytype
Subfamily B Class b: Series 1 polytype untwinned or ±120º-twinned Series 0 polytype (±60º / 180º)-twinned Subfamily B Series 1 orthogonal polytype
Mixed-rotation Series 0 Class a polytype (±60º / ±120º)-twinned (two individuals)
Mixed-rotation Class a: Series 1 untwinned polytype Series 0 polytype 180º-twinned
---------
N
---------
---------
Subfamily A Series 0 Class a polytype ±60º-twinned (two individuals) Subfamily A Series 0 Class a polytype ±60º-twinned (three individuals) Subfamily A Series 1 orthogonal polytype (±60º / ±180º)-twinned
Subfamily A Series 0 Class a polytype ±120º-twinned (two individuals) Subfamily A Series 0 Class a polytype ±120º-twinned (three individuals) Subfamily A Series 1 orthogonal polytype untwinned or ±120º-twinned
3 (SD)
3 (SX)
2 (SX)
---------
Subfamily A Series 0 Class a polytype 180º-twinned
---------
1 (SX)
lC1 = 0(mod N) and N/2(mod N) ---------
Subfamily A Series 1 Class b polytype untwinned or ±120º-twinned
0
lC1 = 1(mod 3) and 2(mod 3)
2
---------
1 [lC1 z 0(mod 3)]
Number of planes with orthogonal appearance
Number of reflections in the c*1 repeat along reciprocal lattice rows corresponding to family reflections
Table 12b. Classification of diffraction patterns for N = 3(3K+L).
226 Nespolo & Ďurovič
Subfamily A Series n-1 Class b polytype (±60º / 180º)-twinned
--------Subfamily A Series n orthogonal polytype (±60º / 180º)-twinned
Subfamily A Series n Class b polytype untwinned or ±120º-twinned
---------
---------
Subfamily A Series n orthogonal polytype untwinned or ±120º-twinned
---------
---------
0
1 (SX)
2 (SX)
3 (SX)
3 (SD)
6 (SX and SD)
---------
---------
---------
lC1 = 1(mod 3) and 2(mod 3)
1 [lC1 z 0(mod 3)]
Number of planes with orthogonal appearance
Subfamily B Class b: Series n polytype untwinned or ±120º-twinned Series n-1 (±60º / 180º)-twinned polytype Subfamily B Series n orthogonal polytype
---------
---------
---------
---------
lC1 = 0(mod N) and N/2(mod N)
2
Mixed-rotation Series n orthogonal polytype
Mixed-rotation Class b: Series n polytype untwinned or ±120º-twinned Series n-1 (±60º / 180º)-twinned polytype
Mixed-rotation Series n-1 Class a polytype (±60º / ±120º)-twinned (three individuals)
Mixed-rotation Class a: Series n polytype untwinned Series n-1 polytype 180º-twinned Mixed-rotation Series n-1 Class a polytype (±60º / ±120º)-twinned (two individuals)
---------
N
Number of reflections in the c*1 repeat along reciprocal lattice rows corresponding to family reflections
Table 12c. Classification of diffraction patterns for N = 3n>1(3K+L). For the correspondence between the relative rotations of twinned individuals and the twin laws see Table 11 (after Nespolo 1999).
Crystallographic Basis of Polytypism and Twinning in Micas 227
228
Nespolo & Ďurovič
regular 3-gons (triangles), 6 surrounding each vertex, that covers the two-dimensional plane without overlap or interstices (Schläfli 1950). The tessellation {3, 6} defines the hexagonal mesh; its dual, {6, 3}, gives the H centering nodes (Coxeter 1973; 1989). If (u, v) are the coordinates of a node of {3, 6}, which define a vector: r = uA1 + vA2
(10)
(A1, A2, c; |A1| =|A2| = a ≅ 5.3Å; γ = 120º), the five other nodes produced by n×60° (0 ≤ n ≤ 5) rotations about the origin are: (u-v, u), (-v, u-v), (-u, -v) (v-u, -u), (v, v-u). If u = v = 1, these nodes together with the origin give the {3, 6} regular tessellation. If u ≠ 1 or v ≠ 1 the compound tessellation {3, 6}[n{3, 6}] is obtained, whose larger mesh has multiplicity n: n = u2+v2-uv
(11)
(Takeda and Donnay 1965). The length of the vector connecting the origin with a node of coordinates (u, v) is5: r = an1/2
(12)
and for the regular tessellation (u = v = 1) r/a = 1. A single set of six nodes with the same r exists when either u or v = 0, v = u or v = 2u: these nodes lie on the six directions corresponding to the reflection lines in the plane. In all other cases, there are two sets of six nodes with the same r, which lie outside the six reflection lines. The generating nodes of the two sets are defined as follows: set I: uI, v; set II: uII, v;
v > uII = (v – uI) > uI > 0
(13)
In reciprocal space (γ=60º), the relation corresponding to (11a) is given by: set I: H, K; set II: K, H; H = u; K = v – u
(14)
These are the conditions in reciprocal space given by Zvyagin and Gorshkov (1966) for the regularity of the secondary reflections in hexagonal nets being derivable from only geometrical considerations based on the superposition of the cells of both lattices. Reciprocal lattice nodes of set I correspond to the orbits S, DI, XI, and those of set II to the orbits DII and XII in Figure 16. Because the b axis of the C1 orthohexagonal cell is given by b = A1 +2A2, the generating node of set I is always between bC1 and A2 axes, whereas that generating set II is always between A1 and bC1 axes. Nodes belonging to the same set are still related by n×60º rotations, whereas those belonging to different sets are related by a noncrystallographic angle. These sets are symmetrically disposed with respect to the reflection lines in the plane, which thus bisect the rotation angle (Fig. 22, drawn for uI = 1, uII = 3, v = 4). Taking counter clockwise rotations as positive, the angle relating nodes belonging to sets I and II are6: ϕ ' : ( I → II ) = ( II → I ) +
ϕ : ( I → II ) = ( II → I ) −
−
+
⎛ 2v − u ⎞ ϕ ' = 2 cos −1 ⎜ 1/ 2 ⎟ mod 60 ⎝ 2n ⎠
(
)
(15)
ϕ = 60 − ϕ '
Takéuchi et al (1972) defined the vector r as r = ⎯uA1+vA2, i.e. with respect to a basis with interaxial angle 60º: correspondingly in the multiplicity of the mesh (Eq. 8) and in the length of the vector (Eq. 9) the term uv has opposite sign. Their definitions of (u, v) and n values correspond to reciprocal lattice values in our treatment. 6 The definition of the angles ϕ and ϕ′ is given according to Takéuchi et al (1972). 5
Crystallographic Basis of Polytypism and Twinning in Micas
229
Figure 22. Overlap of two hp lattices rotated about an axis normal to the plane and passing through the origin by the angle ϕ of the compound tessellation {3, 6}[13{3, 6}]. One node out of 13 is restored. Three hexagonal meshes containing each 13 nodes are also shown.
Figure 23. Definition of the tessellation angles ϕ, ϕ', δI, δII. The figure is drawn for the compound tessellation {3, 6}[13{3, 6}].
The relation between ϕ and ϕ′ is derived taking into account that a node belonging to one set is related to the two nearest nodes of the other set by two reflection lines that intersect at the origin. For the regular tessellation, only one set of six nodes with the same r exists, each node being 60º apart: in this case ϕ = ϕ′ = 0º (mod 60º). The space-fixed b orthohexagonal axis bisects the angle ϕ′ as defined in Equation (15). The angles between b and the directions (uI,v) (δI) and (uII,v) (δII) are simply given by (Fig. 23): δI = ϕ/2 = 30º - ϕ′/2
δII = -δI(mod 60º) = δI + ϕ′ = 30º + ϕ′/2
(16)
230
Nespolo & Ďurovič
In reciprocal space, n×60º rotations relate nodes on the same type of row and of the same set (S; DI, DII, XI, XII); instead, non-crystallographic rotations relate nodes on the same type of row but of different sets (DI and DII; XI and XII) and do not restore nodes of the same set (cf. Fig. 16). If u and v (and thus also h and k) are not co-prime integers (i.e. they have a common factor), or if u+v = 0(mod 3) [i.e. k-h = 0(mod 3)], the lattice constructed on the mesh defined by the compound tessellation is multiple. The same lattice is described by a primitive mesh with smaller multiplicity and corresponding to u and v co-prime integers and u+v ≠ 0(mod 3) Table 13 shows the features of compound tessellations {3, 6}[n{3, 6}] to r = 100Å [Eqn. (12) assuming a = 5.3Å], each of which describes a coincidence-site lattice (CSL) (Ranganathan 1961): the multiplicity n of its mesh is termed coincidence index or Σ factor and corresponds to the order of the subgroup of translation defining the twodimensional CSL with respect to the hp lattice. As shown in Table 13, the minimal value of the Σ factor for the hp lattice is 7 (see also Pleasants et al. 1996). Plesiotwinning
If the obliquity is neglected (ω|| = ω⊥ = 0), micas have a hexagonal lattice (orthogonal polytypes) or sublattice (non-orthogonal polytypes). The twin lattice coincides with the lattice of the individual (orthogonal polytypes) or with its (pseudo)hexagonal sublattice (non-orthogonal polytypes) and can be described through the regular tessellation {3,6}. A different kind of oriented crystal association occurs, although less frequently, whose lattice is based on one of the compound tessellations {3, 6}[n{3, 6}], and thus has been termed plesiotwinning, from the Greek πλεσιος, “close to” (Nespolo et al. 1999b). Plesiotwins are characterized by the following features: 1) 2) 3)
the lattice common to the individuals (plesiotwin lattice) is always a sublattice for any of the individuals; the order of the subgroup of translation (plesiotwin index) is usually higher than in twins; the operation relating the individuals corresponds to a symmetry or pseudosymmetry element of the plesiotwin lattice but not of the individuals, and that element has high indices in the setting of the individuals; pairs of individuals are rotated about the normal to the composition plane by a noncrystallographic angle, even neglecting the obliquity.
If Ξ is the hp lattice, two identical such lattices Ξ1 and Ξ2 with an origin in common can be brought into complete or partial coincidence by keeping Ξ1 fixed and rotating Ξ2 about c*, producing a two-dimensional CSL. The CSL corresponding to the {3, 6}[n{3, 6}] is produced through non-crystallographic rotations of Ξ2 about c*. For orthogonal polytypes the c axis is normal to Ξ and in each lattice plane parallel to Ξ the same twodimensional CSL is produced. Instead, for non-orthogonal polytypes the c axis is inclined, with a cn projection |a|/3 or |b|/3 (assuming ω⊥ = 0). The rotations normal to Ξ produce an identical CSL every third plane parallel to (001), namely the planes for which the normal to Ξ passes on a lattice point. The multiple cell containing three lattice planes is (ideally) orthogonal and defines either the twin lattice - {3, 6} tessellation - or the plesiotwin lattice - {3, 6}[n{3, 6}] tessellation. In micas, and more generally in layer compounds, plesiotwinning represents a generalization of the concept of twinning, at least from the lattice viewpoint. In twins the CSL produced in each plane (orthogonal polytypes) or in one plane out of three (nonorthogonal polytypes) has Σ factor 1, whereas in plesiotwins the CSL has Σ factor of n > 1 (n ≥ 7 for the hp lattice). The twin/plesiotwin index is thus 1 (twinning by merohedry)
231
Crystallographic Basis of Polytypism and Twinning in Micas
Table 13. Values of u, v (γ=120º), H, K (γ=60º) and corresponding angles (mod 60º) for the compound tessellation {3, 6}[n{3, 6}] up to r = 100Å (assuming a = 5.3Å). Set I, II I II I II I II I II I II I II I II I II I II I II I II I II I II I II I II I II
(u, v) (1,1) (1,3) (2,3) (1,4) (3,4) (2,5) (3,5) (1,6) (5,6) (3,7) (4,7) (1,7) (6,7) (3,8) (5,8) (4,9) (5,9) (2,9) (7,9) (1,9) (8,9) (3,10) (7,10) (1,10) (9,10) (5,11) (6,11) (3,11) (8,11) (2,11) (9,11) (5,12) (7,12)
(H, K) (1,1) (1,2) (2,1) (1,3) (3,1) (2,3) (3,2) (1,5) (5,1) (3,4) (4,3) (1,6) (6,1) (3,5) (5,3) (4,5) (5,4) (2,7) (7,2) (1,8) (8,1) (3,7) (7,3) (1,9) (9,1) (5,6) (6,5) (3,8) (8,3) (2,9) (9,2) (5,7) (7,5)
I II I II 133 61.1 I II I 139 62.5 II I 151 65.1 II # Regular tessellation {3,6}.
(6,13) (7,13) (1,12) (11,12) (4,13) (9,13) (3,13) (10,13) (5,14) (9,14)
(6,7) (7,6) (1,11) (11,1) (4,9) (9,4) (3,10) (10,3) (5,9) (9,5)
n 1#
r(Å) 5.3
7
14.0
13
19.1
19
23.1
31
29.5
37
32.2
43
34.8
49
37.1
61
41.4
67
43.4
73
45.3
79
47.1
91
50.6
97
52.2
103
53.8
109
55.3
127
59.7
ϕ 0º
ϕ′ 0º
δI 0º
δII 0º
21°47′
38°13′
10°54′
49°06′
32°12′
27°48′
16°06′
43°54′
13°10′
46°50′
6°35′
53°25′
42°06′
17°54′
21°03′
38°57′
9°26′
50°34′
4°43′
55°17′
44°49′
15°11′
22°25′
37°35′
16°26′
43°34′
8°13′
51°47′
7°20′
52°40′
3°40′
56°20′
35°34′
24°26′
17°47′
42°13′
48°22′
11°38′
24°11′
35°49′
26°00′
34°00′
13°00′
47°00′
49°35′
10°25′
24°47′
35°13′
6º01′
53º59′
27º00′
3º00′
29º25′
30º35′
45º18′
14º42′
40º21′
19º39′
39º50′
20º10′
11º00′
49º00′
54º30′
5º30′
5º05′
54º55′
57º27′
2º33′
51º23′
8º37′
34º18′
25º42′
25º02′
34º58′
47º29′
12º31′
34º32′
25º28′
42º44′
17º16′
18º44′
41º16′
50º38′
9º22′
or n (plesiotwinning) for orthogonal polytypes, and 3 (twinning by reticular merohedry) or 3n (plesiotwinning). For ω|| ≠ 0 or ω⊥ ≠ 0 this description is not modified, but the lattice overlap is only approximated and corresponds to pseudo-merohedry (n = 1) and reticular pseudo-merohedry (n > 1): the rotations normal to Ξ are ϕ±2ε, and do not obey
232
Nespolo & Ďurovič
the law of Mallard. These rotations are useful to describe the CSL and the corresponding twin/plesiotwin indices but, as shown dealing specifically with twins, they are not correct twin/plesiotwin operations: the latter correspond instead to two-fold axes in the (001) plane or reflection planes almost normal to (001). The plesiotwin axes and plesiotwin planes have higher indices than the twin axes (Table 14). Note that plesiotwin planes correspond to crystal faces usually not developed in micas: consequently, reflection plesiotwins have a probability of occurrence lower than rotation plesiotwins. Table 13, continued n
r(Å)
157
66.4
163
67.7
169
68.9
181
71.3
193
73.6
199
74.8
211
77.0
217
78.1
223
79.1
229
80.2
241
82.3
247
83.3
259
85.3
271
87.2
277
88.2
283
89.2
301
92.0
Set I II I II I II I II I II I II I II I II I II I II I II I II I II I II I II I II I II I II I II I II I II
(u, v) (1,13) (12,13) (3,14) (11,14) (7,15) (8,15) (4,15) (11,15) (7,16) (9,16) (2,15) (13,15) (1,15) (14,15) (3,16) (13,16) (8,17) (9,17) (6,17) (11,17) (5,17) (12,17) (1,16) (15,16) (3,17) (14,17) (7,18) (11,18) (2,17) (15,17) (5,18) (13,18) (9,19) (10,19) (7,19) (12,19) (6,19) (13,19) (4,19) (15,19) (9,20) (11,20)
(H, K) (1,12) (12,1) (3,11) (11,3) (7,8) (8,7) (4,11) (11,4) (7,9) (9,7) (2,13) (13,2) (1,14) (14,1) (3,13) (13,3) (8,9) (9.8) (6,11) (11,6) (5,12) (12,5) (1,15) (15,1) (3,14) (14,3) (7,11) (11,7) (2,15) (15,2) (5,13) (13,5) (9,10) (10,9) (7,12) (12,7) (6,13) (13,6) (4,15) (15,4) (9,11) (11,9)
ϕ
ϕ′
δI
δII
52º04′
7º56′
33º58′
26º02′
36º31′
23º29′
41º44′
18º16′
4º25′
55º35′
57º48′
2º12′
30º09′
29º51′
44º55′
15º05′
8º15′
51º45′
55º52′
4º08′
45º54′
14º06′
37º03′
22º57′
53º10′
6º50′
33º25′
26º35′
39º41′
20º19′
40º09′
19º51′
3º53′
56º07′
58º03′
1º57′
19º16′
40º44′
50º22′
9º38′
26º45′
33º15′
46º38′
13º22′
53º36′
6º24′
33º12′
26º48′
40º58′
19º02′
39º31′
20º29′
14º37′
45º23′
52º41′
7º19′
47º39′
12º21′
36º11′
23º49′
28º47′
31º13′
45º37′
14º23′
3º29′
56º31′
58º16′
1º44′
17º17′
42º43′
51º22′
8º38′
24º01′
35º59′
48º00′
12º00′
36º58′
23º02′
41º31′
18º29′
6º37′
53º23′
56º42′
3º18′
233
Crystallographic Basis of Polytypism and Twinning in Micas Table 13, concluded. n
r(Å)
307
92.9
313
93.8
325
95.5
331
96.4
337
97.3
343
98.2
349
99.0
Set I II I II I II I II I II I II I II
(u, v) (1,18) (17,18) (3,19) (16,19) (5,20) (15,20) (10,21) (11,21) (8,21) (13,21) (1,19) (18,19) (3,20) (17,20)
(H, K) (1,17) (17,1) (3,16) (16,3) (5,15) (15,5) (10,11) (11,10) (8,13) (13,8) (1,18) (18,1) (3,17) (17,3)
ϕ
ϕ′
δI
δII
54º20′
5º40′
32º50′
27º10′
43º07′
16º53′
38º27′
21º33′
32º12′
27º48′
43º54′
16º06′
3º09′
56º51′
58º26′
1º34′
15º39′
44º21′
52º10′
7º50′
54º38′
5º22′
32º41′
27º19′
44º01′
15º59′
38º22′
22º00′
Plesiotwinning is a macroscopic phenomenon that differs from twinning not only in a geometrical definition but also from a physical viewpoint. Whereas for twins the twin index and the twin obliquity directly influence the probability of twin occurrences, for plesiotwins a similar lattice control is not recognized. In fact, the lowest plesiotwin index for micas is 7, which becomes 21 for non-orthogonal polytypes. The degree of restoration of lattice nodes is too small for a lattice control to be active. The plesiotwin formation is thus structurally controlled. Twins are usually believed to form in the early stages of crystal growth (Buerger 1945), but the formation of twins from macroscopic crystals is also known (e.g., Gaubert 1898; Schaskolsky, and Schubnikow 1933). When two or more nanocrystals interact, they can adjust their relative orientation until they reach a minimum energy configuration, corresponding either to a parallel growth or to a twin. When two macrocrystals interact, the energy barrier to the mutual adjustment is higher, especially at low temperature. If two macrocrystals coalesce or exsolve taking at first a relative orientation corresponding to an unstable atomic configuration at the interface, they tend to rotate until they reach a lower energy configuration. Parallel growth and twinning correspond to minimal interface energy, whereas plesiotwinning corresponds to a lessdeep minimum. However, twin orientations are less numerous and are separated by larger angles, whereas plesiotwin orientations are more numerous and separated by smaller angles. In Figure 24 the plot Σ vs. ϕ for the hp lattice is given for Σ ≤ 100 and 0º ≤ ϕ ≤ 60º. Between the two extreme values of ϕ corresponding to crystallographic rotations and to Σ = 1, several discrete values appear, corresponding to Σ > 1 and to noncrystallographic rotations. Only limited adjustments may be necessary to reach plesiotwin orientations, which may thus represent a kind of compromise between the original unstable configuration and the too distant, although more stable, configuration of twins. This kind of origin is supported also by experiments of dispersion into a fluid and drying of flakes of crystals with layer structure: the result was simply a physical overlap of pairs of crystals, which however gave the same orientations of plesiotwins (Sueno et al. 1971; Takéuchi et al. 1972). TWINNING OF MICAS. ANALYSIS OF THE GEOMETRY OF THE DIFFRACTION PATTERN
A simple and straightforward method to derive the orientations of the individuals in a mica twin or allotwin is introduced. The following analysis is entirely based on the
non-orthogonal polytypes
Plesiotwin index
orthogonal polytypes
Plesiotwin index
Reflection plesiotwins
Rotation plesiotwins
Compound tessellation. {3, 6}[n{3, 6}]
21
39
13
C[120]S C[710]S C[C530]S [C120]S [C710]S [530]S C(160) C(730) C(C590) (C160) (C730) (590)
C[130]S C[510]S C[C210]S [C130]S [C510]S [210]S C(190) C(530) C(C230) (C190) (C530) (230) 7
n=13
n=7
57
19
C[150]S C[410]S C[C730]S [C150]S [C410]S [730]S C(1.15.0) C(430) C(C790) (1C.C15.0) (C430) (790)
n=19
93
31
C[230]S C[C11.10]S C[C750]S [C230]S [11C.10]S [750]S C(290) C(C11.30) C(7C.C15.0) (C290) (11C.30) (7.15.0)
n=31
111
37
C[170]S C[C11.30]S C[C520]S [C170]S [11C.30]S [520]S C(1.21.0) C(C11.90) C(C560) (1C.C21.0) (11C.90) (560)
n=37
126
43
C[570]S C[C13.10]S C[C430]S [C570]S [13C.10]S [430]S C(5.21.0) C(C13.30) C(C490) (5C.C21.0) (13C.30) (490)
n=43
147
49
C[140]S C[C13.30]S C[C11C.50]S [C140]S [13C.30]S [11.50]S C(1.12.0) C(C13.90) C(C11C.C15.0) (1C.C12.0) (13C.90) (11.15.0)
n=49
183
61
C[190]S C[720]S C[C13C.50]S [C190]S [C720]S [13.50]S C(1.27.0) C(760) C(C13C.C15.0) (1C.C27.0) (C760) (13.15.0)
n=61
2GI-2H 120º+2GI-2H 240º+2GI+2H 2GII-2H 120º+2GII+2H 240º+2GII+2H 2GI+2H 120º+2GI+2H 240º+2GI-2H 2GII+2H 120º+2GII-2H 240º+2GII-2H
Rotation about c*
Table 14. Plesiotwin laws for the hP lattice. Indices of plesiotwin axes and plesiotwin planes are given with respect to the orthohexagonal a, b axes, in counter clockwise orientation from b. GI and GII are given in Table 13 for the corresponding tessellation. Plesiotwin planes correspond to planes not developed as crystal faces: consequently, reflection plesiotwins have low probability of occurrence.
234 Nespolo & Ďurovič
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Figure 24. The coincidence index (Σ factor) vs. ϕ plot, in case of two-dimensional hp lattice. ϕ = 0 (parallel growth) and ϕ = 60º (twinning) correspond to Σ = 1. Between these two orientations, a large number of plesiotwin orientations exist, which are shown up to Σ = 100. The plot has been calculated by applying the compound tessellation theory and drawn for counter-clockwise rotations only. Clockwise rotations produce the same Σ in correspondence of 60º – ϕ rotations (modified after Nespolo et al.1999d).
geometry of the diffraction pattern, which is determined by the symmetry of the lattice of the individual, of the twin lattice and of the lattice of the family structure. The diffraction pattern is described within the Trigonal model and in terms of the weighted reciprocal lattice (w.r.l.), i.e. the reciprocal lattice (r.l.) in which each node has a weight corresponding to the resulting intensity. In particular, a node corresponding to a reflection with zero intensity in the Trigonal model is omitted from the w.r.l. The intensities that are actually obtained in a diffraction experiment are clearly influenced by structural deviations from the Trigonal model: two diffraction patterns with the same geometry, and thus considered equivalent hereafter, can thus be different when the actual structure (i.e., with distortions) is taken into account. Symbolic description of orientation of twinned mica individuals. Limiting symmetry
As seen in the previous section, rotations between pairs of individuals in a mica twin or allotwin are very close to n×60º about c*. The possible orientations of the individuals are thus almost identical to the possible orientations of the layers in a polytype. The absolute orientation of the individuals can be indicated by symbols similar to those used for polytypes. Nespolo et al (2000a) introduced the ZT symbols, where "T" indicates "twin", which are derived from the shortened Z symbols for polytypes. There are four main differences between Z and ZT symbols: 1. 2.
Because there cannot be two individuals in a twin oriented in the same way, the sequence of characters in a ZT symbol never contains the same character twice. The Z symbol of polytypes must take into account the space-group type, whereas ZT considers only the symmetry of the point group. The orthohexagonal setting of the first individual is taken to coincide with that of the twin lattice: the first individual is
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always fixed in orientation ZT = 3 (Fig. 4), and the orientations of the other individuals are determined by the twin laws. 3. Rotation by 180º of the entire twinned edifice around the a axis of the space-fixed reference changes the ZT symbol 3IJ…P into (6-P)…(6-J)(6-I)3; because the order of the individuals in the twin does not influence the diffraction pattern, this sequence of characters is equivalent to 3(6-I)(6-J)…(6-P), which corresponds to inverting the direction of rotation of the individuals in the twin about the cC1 axis. Considering the effect on the lattice, the 3IJ…P → 3(6-I)(6-J)…(6-P) transformation corresponds to reflecting the twin lattice across the (010) plane. 4. For polytypes in which layers are related only by proper motions7, like 3T, two twins operations with the same rotational part and differing only for the proper/improper character of the motion produce the same twin lattice. The corresponding two twin laws are however different, and thus an orientation produced by an improper motion is hereafter distinguished by a small black circle (•) after the ZT symbol. The number of independent orientations of the w.r.l. of an individual is determined by its limiting symmetry, i.e. the lower symmetry between the ideal crystal lattice (as described by the Trigonal model) and the family structure. The limiting symmetry is given in Table 15, which is easily understood remembering that: 1) for mixed-rotation polytypes the family structure is defined only within the Pauling model and the limiting symmetry always coincides with the symmetry of the polytype lattice; 2) for orthogonal polytypes, the lattice is (pseudo) hexagonal: for both subfamilies the limiting symmetry coincides with that of the family structure; 3) subfamily B polytypes cannot belong to Class a; 4) non-orthogonal subfamily A polytypes belong to Class a for Series 0 but to Class b for Series > 0. Table 15. Limiting symmetry defining the number of independent lattice orientations. The (idealized) symmetries of the lattice and of the family structure are given. The limiting symmetry corresponds to the lower of the two. For mixed-rotation polytypes the family structure is defined only within the Pauling model and the limiting symmetry by definition coincides with the symmetry of the lattice. Orthogonal polytypes (hP)
Class a polytypes (mC)
Class b polytypes (hR)
subfamily A (hR)†
hR
mC (Series 0)
hR (Series > 0)
†
hP
-----
hR
hP
mC
hR
subfamily B (hP)
mixed-rotation (hP)‡ †
Trigonal model. ‡Pauling model.
Class a polytypes . Each subfamily A Series 0 polytypes belong to Class a; mixedrotation polytypes may also belong to Class a. In both cases, the limiting symmetry is mC and the unique axis does not coincide with that of the family structure (b in the polytypes, c in the family structure). Each of the six possible orientations of the individuals correspond thus to independent orientations of the w.r.l. The possible composite twins are obtained by calculating the sequences of ZT symbols for sets of individuals from two to six. The orientation of the first individual is fixed (ZT = 3), and five possible orientations 7
A “motion” is an instruction assigning uniquely to each point of the point space an 'image' whereby all distances are left invariant. A motion is called proper (also: “first sort”) or improper (also: “second sort”) depending on whether the determinant of the matrix representing it is +1 or -1.
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remain where m individuals (1 ≤ m ≤ 5) must be distributed. The number of twins is then:
∑
5 m =1
5 ⎛ 5⎞ 5 5! N T ( m ) = ∑ m =1 ⎜ ⎟ = ∑ m =1 = 31 m !( 5 − m ) ! ⎝ m⎠
(17)
Table 16 gives the 12 sequences of independent ZT symbols; the other 19 simply correspond to a rotation of the entire twinned edifice followed by a shift of the origin along c, eventually coupled with the inversion of the direction of the rotation of the individuals in the twin [reflection of the lattice across (010)], as in ZT = 341. Class b op lytypes . Non-orthogonal polytypes belong to Class b in subfamily A Series > 0 and in subfamily B. The unique axis is a in the polytypes but c in the pseudorhombohedral lattice; the latter coincides with that of the family structure. The limiting symmetry is hR, which for subfamily A coincides both with the symmetry of the family structure and with the (pseudo) symmetry of the lattice. Only two orientations of the w.r.l. of the individual are independent, corresponding to the two parities of ZT symbols. A common symbol is thus used for the three equivalent orientations with the same parity, namely “U” (uneven) and “E“ (even). Twinning by pseudo-merohedry involves individuals with the same orientation parity of ZT symbols and produces complete overlap of the w.r.l. of the individual (neglecting the obliquity). The reciprocal lattice of the twin is thus geometrically indistinguishable from the reciprocal lattice of the individual. The three twins ZT = 35, ZT = 31 and ZT = 351 are equivalent to the single crystal, when considering the geometry of their lattice, and are thus represented as ZT = U. Instead, twinning by reticular pseudo-merohedry involves individuals with an opposite orientation parity of the ZT symbols and, considering the lattice only, they are represented as ZT = UE. Orthogonal polytypes . In the Trigonal model, the lattice is hP (ω = 0); in the true structure for orthorhombic polytypes the lattice is normally oC but pseudo-hP (ω ≠ 0). For subfamily B and mixed-rotation polytypes the limiting symmetry is hP and there is only one independent orientation of the w.r.l. Twinning is either by complete merohedry or by pseudo-merohedry and does not modify the geometry of the diffraction pattern.
Subfamily A polytypes have an orthogonal lattice only if they belong to Series > 0 and have a 1:1:1 ratio of layers with the three orientations of the same parity (odd or even). The only example reported to date is 3T, which is also the only possible orthogonal polytype in Series 1. Other subfamily A orthogonal polytypes may appear in Series > 1 but are at present unknown. The limiting symmetry is hR and the w.r.l. has two independent orientations, as for Class b polytypes, which correspond to the two settings (obverse/reverse) of the family structure. Twinning is by merohedry (ω = 0, either complete or selective, depending on the twin law) or pseudo-merohedry (ω ≠ 0). The 3T polytype has three twin laws, two of which correspond to selective merohedry and invert the parity of the ZT symbol, namely ZT = U → ZT = E (6′2′2) or ZT = E• ⎯(6′m′2); the third twin law ⎯(3′12/m′) corresponds instead to complete merohedry and preserves the parity of the ZT symbol (ZT = U → ZT = U•). Derivation of twin diffraction patterns
The number and disposition of nodes on the reciprocal lattice rows parallel to c* are termed node features and are identified by a symbol Ij, where I is the number of nodes within the c*1 repeat and j is a sequence number. Nespolo et al (2000a) introduced an orthogonal setting for the analysis of twins in terms of Ij, which is termed the twin setting. When dealing with a single polytype, the twin setting coincides with the C1 setting, which
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Table 16. Orientation of the individuals building a twin in Class a mica polytype. Angles in parenthesis express the counter clockwise rotations of the whole twinned edifice. “Shift” stands for the shift of the origin along c. (010) means reflection of the twin lattice across the (010).plane, which is equivalent to inverting the direction of rotation of the individuals in the twin, i.e. to the symbol transformation 3IJ…P → 3(6-I)(6-J)…(6-P). [After Nespolo et al. 2000a] ZT
Equivalent to
Equivalent to
Equivalent to
34 35 36 31 32 345 346 341 342 356 351 352 361 362 312
Unique Unique Unique 53(120º) 43(60º) Unique Unique 325(010) 453(60º) 134(240º) Unique 463(60º) 634(180º) 413(60º) 534(120º)
---------------------35(shift) 34(shift) --------------436(60º) 345(shift) 341(shift) -------346(shift) 346 341(shift) 345(shift)
-------------------------------------------------346(shift) -------346 ---------------------346 --------
ZT
Equivalent to
Equivalent to
3456 3451 3452 3461 3462 3412 3561 3562 3512 3612 34561 34562 34512 34612 35612 345612
Unique Unique 4563(60º) Unique Unique 5634(120º) 1345(240º) 4613(60º) 5134(120º) 6345(180º) Unique 45613(60º) 56134(120º) 61345(180º) 13456(240º) Unique
--------------3456(shift) --------------3456(shift) 3451(shift) 3461(shift) 3451(shift) 3456(shift) -------34561(shift) 34561(shift) 34561(shift) 34561(shift) --------
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is based on the cell of the twin lattice. To compare the geometry of the reciprocal lattice of polytypes with different periods, the twin setting is instead defined to have the shortest period along c* in the C1 setting among all the polytypes considered. The twin setting of the twin lattice is space-fixed and parallel to C1, whereas that of the crystal lattice is crystal-fixed for each of the individuals building a twin. Since the first individual of the twin is space-fixed (ZT = 3 for Class a, or ZT = U for Class b and orthogonal polytypes), its twin setting is parallel to C1. The l index in the twin setting is labeled lT. Rotations between pairs of individuals are taken counter-clockwise in direct space, and thus clockwise in reciprocal space. The n×60º rotations about c*, which give the approximate rotations between pairs of individuals, overlap only Ri belonging to the same type (S, D or X). Each of the Ri is rotationally related to five other Ri and along each of them a peculiar sequence of lT indices is obtained, which is termed a “Rotational Sequence”. Each Ri generates one rotational sequence, which is shortened to RSiP(n), where: the superscript P indicates the polytype; i is the same index defining Ri; n points to each of the six characters of the RS. RS1P corresponds to S rows and thus it is “000000” for all polytypes. The n-th values of RSiP correspond to the lT indices of the nodes on the row which is related to Ri by (n-1)×60º clockwise rotation. The two RSiP corresponding to D-type rows (R2-3) on the one hand, and the six RSiP corresponding to X-type rows (R4-9) on the other, can be transformed into each other by cyclic permutations. Since the orientations of the single-crystal lattices and of the twin lattice are fixed and determined by ZT, also the starting point of each RSiP is fixed, and the nine RSiP are independent. The node features of the composite rows are obtained from the corresponding RSiP by considering their relation with the ZT symbols. A twin of N individuals (2 ≤ N ≤ 6) is identified by N ZT symbols. The lT index of the q-th node coming on i-th row from the j-th individual is given by: [lT(i, j)]q = [RSiP(n) ]q , n = [(ZT)j+4](mod 6).
(18)
The node features of composite rows are completely defined by the nine RSiP and ZT symbols; therefore, there are only nine independent composite rows, for which the symbol Ci is adopted. Ri and Ci share the same row features and thus the description of the reciprocal lattice in terms of the tessellation rhombus and of the minimal rhombus is the same for both the single-crystal lattice and the twin lattice. Because of the metric relations (Table 10), the lT of both Ci and Ri of the same type and belonging to the same central plane are related by: ⎡⎣lT ( Di ) ⎤⎦ q = ⎡⎣6 − lT ( D3−i ) ⎤⎦ q
{
}
⎡⎣lT ( X i ) ⎤⎦ q = 6 − lT ⎡⎣ X (9−i )( mod 6) ⎤⎦
.
(19)
q
Knowing the lT of one D-type Ci / Ri and three X-type Ci / Ri, the lT of the remaining four Ci / Ri can be calculated. There are thus five truly geometrically independent Ci / Ri (one S-type, one D-type and three X-type), but nine translationally independent Ci / Ri. The distribution of Ij on the Ci of a minimal rhombus is the information necessary to derive and identify the diffraction patterns of mica twins. A short comparative analysis of the four periodic basic structures (1M, 2M1, 2M2 and 3T) is given below. For these four polytypes the twin setting has a period of c*1/6 along c*: lT (2M1, 2M2) = lC1(2M1, 2M2), but lT (1M, 3T) = 2lC1(1M, 3T). Table 17 gives the Ci and RSPi. The definition of Ij, is given in Table 18. The rules for combining Ij’s of the individuals into composite Ij’s of the twin are given in Nespolo et al (2000a).
1 2 1 2 0 1 2 0
D1 D2 X1 X2 X3 X4 X5 X6
C2
C3
C4
C5
C6
C7
C8
C9
2
1
2
1
2
1
0
0
0
k (mod 3)
044022
440220
220440
022044
402204
204402
424242
242424
000000
RS1M
022011 / 355344
220110 / 553443
110220 / 443553
011022 / 344355
201102 / 534435
102201 / 435544
242424
424242
000000
RS2M1
212121 / 545454
121212 / 454545
212121 / 545454
121212 / 454545
212121 / 545454
121212 / 454545
000000 / 333333
000000 / 333333
000000
RS2M2
000000 / 222222 / 444444
000000 / 222222 / 444444
000000 / 222222 / 444444
000000 / 222222 / 444444
000000 / 222222 / 444444
000000 / 222222 / 444444
424242
242424
000000
RS3T
(after Nespolo et al. 2000a).
41
0,1,2,4
l(mod6)
0
l(mod6)
Ij
11
Ij 2
12
0,1,3,4
42
4
13
43
0,4
22
0,2,3,5
0,2
21
0,2,3,4
44
0,3
23
45
2,5
25
0,2,4,5
1,4
24
31 0,2,4
1,2,4,5
46
2,4
26
0,1,2,3,4
51
0,1,4
32
0,1,2,4,5
52
0,2,3
33
0,2,5
34
0,2,3,4,5
53
0,3,4
35
61
2,4,5
37
0,1,2,3,4,5
1,2,4
36
Table 18. Definition of the Ij for the four basic polytypes and their twins. I indicates the number of nodes on the reciprocal lattice row. The subscript j is a sequential number (after Nespolo et al. 2000a).
0
S
C1
h (mod 3)
Type
Composite rows (Ci) and Rotational Sequences (RSiP) for the four basic polytypes
Composite row
Table 17.
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1M op lytype . The c*1 repeat coincides with the polytype period and along each Ri there is only one node, which obeys the relation lT = 2h(mod 6). D-type Ri are either 12 (D1) or 13 (D2) and the RS2-31M are "242424" and "424242". The n×60° rotations about c* produce the overlap of all the reciprocal lattice nodes belonging to D-type Ri when n is even, but to their separation when n is odd. X-type RS4-91M are the six cyclic permutations of "220440". On the basis of the relation between Ci and RSi1M (Table 17) seven different Ci appear in the twin lattice. One or two reflections can appear on D-type Ci (lT is never 0), whereas one, two or three reflections can appear on X-type Ci. Nine independent 1M twin patterns occur (Fig. 25).
Figure 25. The nine independent patterns of 1M twins as expressed through the corresponding minimal rhombi. For the ZT = 34 twin, the complete star polygon is given, with the minimal rhombus in it shaded. Inset: l (mod 6) indices of the nodes on reciprocal lattice composite rows [used by permission of the editor of Acta Crystallographica A, from Nespolo et al. (2000a) Fig. 8, p. 143].
When three equally spaced reflections in the c*1 repeat occur along non-family rows, in principle the diffraction pattern may correspond either to a 1M twin (apparent polytypism) or to a 3-layer polytype (real polytypism). The distinction is obtained by applying the geometrical criteria given in Tables 12a-12c. However, 1M twins with ZT = 351 cannot be distinguished geometrically from the 3T polytype (see also Nespolo and Kogure 1998). This ambiguity is removed when weak reflections appear along family rows, which can be expected for dioctahedral and Li-rich trioctahedral micas (Rieder 1968, 1970). The effect of these weak reflections on the twin diffraction pattern is analyzed in Nespolo et al (2000a). 2M1 polytype . Because the parity of layers is opposite for the 2M1 polytype (Z = 220440, T = |4.4 2.2|) with respect to the 1M polytype (Z = 330, T = |3.3|), the threefold family structure has an opposite setting (reverse / obverse) and the corresponding family rows have different reflection conditions, namely k = 0(mod 3), lT = 2h(mod 6) for 1M,
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but k = 0(mod 3), lT1 = 4h(mod 6) for 2M1 (Nespolo 1999). One reflection occurs in the c*1 repeat along family Ri, but two along non-family Ri. D-type Ri are the same as in 1M case, but, because of the opposite parity of the layers in the two polytypes, the two RS22M1 are inverted. X-type Ri have the three possible pairs of values of lT (mod 6): 0 and 3, 3 1 and 4, 2 and 5. For the X-type Ri the sequence of n×60° rotations corresponds to a double sequence of lT values: 011022 / 344355 or cyclic permutations, producing six independent double RSi2M1 (Table 6). As for the 1M polytype, the twelve composite twins produce nine different patterns, none of which can be mistaken for that of a 1M twin (Fig. 26).
Figure 26. The nine independent patterns of 2M1 twins as expressed through the corresponding minimal rhombi. Inset: l (mod 6) indices of the nodes on reciprocal lattice composite rows [used by permission of the editor of Acta Crystallographica A, from Nespolo et al. (2000a) Fig. 9, p. 144].
2M2 polytype. Being a Class b polytype, 2M2 has a markedly pseudo-rhombohedral lattice and two of the five pairs of twin laws, namely those corresponding to ±120º rotation about c*, correspond to pseudo-merohedry, whereas the remaining three correspond to reticular pseudo-merohedry. Each of the six n×60º rotations belong to the point group of the family structure (subfamily B), and thus the family sublattice of the individuals is always overlapped. RS22M2 and RS32M2 both correspond to the double sequence 000000/333333, whereas RS4-92M2 correspond to the cyclic permutations of the double sequence 121212/454545. There are only two kinds of patterns for 2M2 twins. Twinning by pseudo-merohedry gives a pattern geometrically indistinguishable from that of the single crystal (ZT = U). The other pattern corresponds to twinning by reticular pseudo-merohedry (ZT = UE) and differs from the single crystal pattern in the six X-type Ci, which show four reflections in the c*1 repeat (Fig. 27). Neither can be mistaken for any one of the 1M or 2M1 polytypes or twins. 3T polytype . The 3T polytype is an orthogonal subfamily A polytype, for which the six orientations of the structural model are equivalent. They can be grouped into two sets of odd or even parity, corresponding to obverse and reverse setting of the family structure
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respectively. Taking odd parities, as in Zvyagin (1967), D-type Ri and RSi3T are the same as those of 1M polytype. Taking the even orientation instead, as in Backhaus and Ďurovič (1984), D-type Ri and RSi3T are the same as those of 2M1 polytype. In both cases, there is only one triple sequence of X-type RS3T: 000000/222222/444444. The six orientations of the minimal rhombus are divided into two types, differing for the D-type Ri. The 2n×60° rotations belong to the symmetry of both the individual and the family structure and reproduce the same rhombus. On the other hand, (2n+1)×60° rotations do not belong to either symmetries and thus they exchange the two independent rhombi. Twinning by complete merohedry (ZT = UU•) by definition produces a diffraction pattern with the same geometrical appearance as the single crystal, which in its turn may be geometrically identical to the pattern of 1M twinned as ZT = 351. In contrast, for twinning by selective merohedry (ZT = UE, UE•, UU•E, UU•E•, UU•EE•), the two D-type Ci correspond to have two reflections at lT = 2(mod 6) and 4(mod 6). This is the same geometrical appearance of 1M twinned as ZT = 3451. The distinction between 1M twins and the 3T polytype (twinned or untwinned) requires by very careful examination of the violation of the additional reflection conditions (Nespolo et al. 2000a).
Figure 27. The two independent patterns of 2M2 twins as expressed through the corresponding minimal rhombi. Inset: l (mod 6) indices of the nodes on reciprocal lattice composite rows.
Derivation of allotwin diffraction patterns
The allotwin laws include the twin laws for each of the individuals, as well as the symmetry operations of the crystal(s) point group(s). The six rotations about c* now must be considered. By indicating the first individual with a superscript and the second one with a subscript, the allotwin ZT = 33 must be considered also, whereas the ZT = 33 twin simply corresponds to a parallel growth. Therefore, the number of possible laws increases and depends upon the number of different polytypes undergoing allotwinning. Because the geometrical appearance of the diffraction pattern of the 3T polytype and of its twins is ideally the same as 1M twinned as ZT = 351 or 3451, the contribution from 3T does not produce an independent pattern: it is not considered in the following systematic analysis. The three basic monoclinic polytypes can produce 3 binary (two-individual) allotwins (1M-2M1; 1M-2M2; 2M1-2M2) and 1 ternary (three-individual) allotwin (1M2M1-2M2). Binary (AB) and ternary (ABC) allotwins are indicated by AB and ABC respectively, where A, B and C represent the ZT symbols for each portion of the allotwin.
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These composite allotwins can be described on the following basis: 1.
2.
3. 4.
The allotwin is constructed by 2 (binary allotwin) or 3 (ternary allotwin) portions (A, B, C), each consisting only of individuals of the same polytype, which in turn can be twinned; 1M is taken as the first portion (A) of the allotwin; when 1M is not involved (binary allotwin 2M1-2M2), the portion A is 2M1. Because the individuals building the twin or allotwin are related by point group operations, the A-B-C sequence has no influence on the composite lattice and the two or three portions can be described as juxtaposed and non-mixed; for example, ZT = 3456 is equivalent to ZT = 3546. Within each single portion (A, B, C), the restrictions on the possible orientations derived for the twins are retained, but these restrictions are not applicable when comparing individuals belonging to different portions. The first individual of the first portion (A) is fixed in orientation ZT = 3, but this restriction is not applicable for the first individual of the other portions. Therefore the number of possible orientations for B and C portions must be multiplied by the number of independent orientations of the minimal rhombus, as determined by the limiting symmetry, namely six for 2M1, and two for 2M2.
The minimal rhombi of the allotwins are calculated as combinations of the minimal rhombi of each portion, but the number of minimal rhombi to be considered depends upon the limiting symmetry. Those minimal rhombi of two twins of 1M that are equivalent through an n×60º rotation about c* can produce two independent minimal rhombi when combined with a minimal rhombus of 2M1. Therefore, in the derivation of the reciprocal lattice of 1M-2M1 allotwins, the minimal rhombi of all the thirty-one twins for both polytypes in Table 16 must considered. To these, the minimal rhombus corresponding to the single crystal must be added. Moreover, keeping fixed the minimal rhombi of 1M (first individual in orientation ZT = 3), the six independent orientations of each of the thirty-two minimal rhombi of 2M1 must be considered. For the 2M2 polytype, there are only two independent orientations of the individual w.r.l. (ZT = U or ZT = E) and only one for the twin reciprocal lattice (ZT = UE). In deriving the reciprocal lattice of 1M2M2 or 2M1-2M2 allotwins, for Class a polytypes only the minimal rhombus of the single crystal and the minimal rhombi of the twenty-three twins related by (2n+1)×60º rotations must be combined with the three (U, E, UE) minimal rhombi of 2M2. The remaining eight minimal rhombi of Class a polytypes are related to some of the other twenty-three by 2n×60º rotations, which are symmetry operations for the minimal rhombi of 2M2 and cannot produce any further independent allotwin minimal rhombus. Finally, for the ternary allotwins 1M-2M1-2M2, the independent minimal rhombi of the binary allotwin 1M-2M1, and those related by (2n+1)×60º rotations, must be combined with the three minimal rhombi of 2M2. For each combination, the composite minimal rhombus obtained in this way, then rotated by n×60º (0 ≤ n ≤ 5), and finally – for each of these rotations – reflected across (010), is compared with those calculated for the previous combinations and, if equivalent, is discarded. The resulting minimal rhombi are given in Nespolo et al (2000a). IDENTIFICATION OF MDO POLYTYPES FROM THEIR DIFFRACTION PATTERNS Theoretical background
The identification of the stacking mode in an MDO polytype is based on two orthogonal projections, which are sufficient to characterize reliably any structure. For
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mica structures (but also for other phyllosilicates) the most suitable projections are the XZ and the YZ projections. A Fourier series calculated with coefficients derived from zonal diffractions only, yields a projection of the structure along the zone axis. It follows that the h0l and 0kl nets characterize unambiguously the projections XZ and YZ, respectively. The h0l net contains only the reciprocal rows with family diffractions (S and D rows) and, therefore, this set characterizes the family structure, i.e. the Subfamily. The set of diffractions 0kl contains both S and X rows (not D rows). Whereas the former are common (almost) for all polytypes in both subfamilies and thus useless for identification purposes, the latter are characteristic for any individual polytype and can be used for their identification, unless they are so diffuse that no discernible maxima can be obtained. Owing to the efficiency of atomic scattering factors as a function of sinϑ/λ, the diffractions close to the origin of the reciprocal lattice are best suited for identification purposes. Moreover, any family structure in micas is trigonal or hexagonal and from Friedel’s law it follows that the reciprocal lattice rows 20l, 13l,⎯13l, ⎯20l, ⎯⎯13l and ⎯13l carry the same information. Therefore, two reciprocal lattice rows, namely 20l and 02l, suffice to identify the subfamily and the MDO polytype, respectively. The positions of diffraction spots and the distribution of their intensities is so characteristic that a mere visual inspection of the diffraction patterns obtained experimentally with that calculated for a homo-octahedral structure with the expected chemical composition, leads to the solution, provided that the presence of twinning has been ruled out. This procedure was described first by Weiss and Ďurovič (1980) and explained in more details by Ďurovič (1981) (see also Ďurovič 1999, p.761). The recognition of the significance of the YZ projections (and thus also the five MDO groups given in Table 7), which can be derived also directly from the full polytype symbols (Ďurovič et al. 1984), is very important also for the interpretation of HRTEM images (Kogure, this volume). Identification procedure
The identification of the stacking mode of an MDO polytype in the homo-octahedral approximation is straightforward. It can be performed by visual inspection of the intensity distribution along two rows (one D and one X), and from visual inspection of the geometry of the diffraction pattern. 1. Intensity distribution. a) Calculate F2 values for each of the six homo-octahedral MDO polytypes given in Table 7 by using average atomic occupations in the octahedral sites, which correspond approximately to the chemical composition of the investigated polytype. Use the space-group type P1 and use a common orthogonal six-layer cell, which can "accommodate" each polytypes. Atomic coordinates from the ideal Pauling model may be used. The F2(0kl) values for the 1M and 2O polytypes must be the same (MDO group I, Table 7) and also the F2 values for the family diffractions must obey the trigonal/hexagonal Laue symmetry. Select the 20l and 02l rows, and construct identification diagrams for the determination of the subfamily (two rows for A and B only) and for the MDO polytype (four rows for the MDO groups I to IV) as indicated in Figure 28, where the size of each circle is proportional to the respective F2 values. In principle, the MDO V row should be given. However, this group contains only the 6H polytype, which has not been reported to date, and can be unambiguously identified by the geometry of its diffraction pattern, which has six orthogonal planes with two reflections in the c*1 repeat along D rows: this geometry cannot be obtained by the twinning of any other polytype. The program DIFK
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(Smrčok and Weiss 1993) is very convenient for the calculations of the F2 values. The program contains a subroutine to produce sequences of the F2 values along selected reciprocal rows. This program can be obtained free of charge from Smrčok8. Make a set of precession photographs, three from the SX planes and one from an SD plane. Select the 20l and 02l rows, and compare the intensities with the calculated values. Figure 29 and 30 show three examples.
Figure 28. Visual representation of calculated intensities of diffractions of MDO polytypes of phlogopite. The indexing refers to the six-layer orthogonal cell (C2 cell). Left: intensities along 20l (D row, containing family diffractions) reciprocal lattice row and intensity distribution within subfamilies A and B. Right: intensities along 02l (X row, containing non-family diffractions) reciprocal lattice row and intensity distribution within MDO groups I to IV. The strongest intensity of each subfamily (left) or MDO group (right) is drawn as the largest circle (modified after Weiss and Durovic 1989).
2. a) b) c)
Geometry of the diffraction pattern. Reciprocal lattice rows parallel to c* in the h0l r.p. have 1 (subfamily A) or 2 (subfamily B) reflections in the c*1 repeat; Reciprocal lattice rows parallel to c* in the 0kl r.p. have 1 (1M), 2 (2M1, 2M2 or 2O), 3 (3T) or 6 (6H) reflections in the c*1 repeat; 2M1 is the only 2-layer subfamily A polytype; 2M2 and 2O are distinguished because the 0kl r.p. is orthogonal for the latter but non-orthogonal for the former.
For the determination of the meso- and hetero-octahedral MDO polytypes, a complete structure refinement is necessary, because the occupancy factors of the three 8
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octahedral sites as well as the sizes of the corresponding octahedra must be determined. A complete structure refinement (e.g., using anomalous scattering) is necessary also to distinguish the two members of an enantiomorphous pair. Our experience shows that the ideal Pauling model is sufficient for identification purposes because the slight deviations from the actual atomic coordinates owing to desymmetrization are not important in these calculations.
Figure 29. Comparison of observed (obs.) and calculated (calc.) intensities along 02l (X row, containing non-family diffractions) and 20l (D row, containing family diffractions) reciprocal lattice rows of zinnwaldites 1M (MDO group I) and 2M1 (MDO group II), which are essential for the identification of MDO groups I, II and of subfamily A, respectively, Observed intensities are taken from 0kl and h0l precession photographs. The distribution of intensities of 20l diffractions is very similar for both zinnwaldite polytypes, and therefore only the distribution corresponding to the one-layer polytypes is given (modified after Weiss and Durovic 1989).
IDENTIFICATION OF NON-MDO POLYTYPES: THE PERIODIC INTENSITY DISTRIBUTION FUNCTION
The number of non-MDO polytypes in each family is infinite, and increases dramatically with the number of layers (Mogami et al. 1978; McLarnan 1981). The procedure for the identification of MDO polytypes described in the previous section becomes virtually impossible for non-MDO polytypes with longer periods, which require instead a simplified procedure. This simplified procedure was introduced by Takeda (1967) under of the name of Periodic Intensity Distribution (PID). The PID is an
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Figure 30. Comparison of observed (obs.) and calculated (calc.) intensities along 02l (X row, containing non-family diffractions) and 20l (D row, containing family diffractions) reciprocal lattice rows of lepidolite 2M2 (MDO group III), which are essential for the identification of MDO groups III and subfamily B, respectively. Observed intensities are taken from 0kl and h0l precession photographs.
approximation of the Fourier transform of the stacking sequence that can be obtained in a simple way from the diffraction intensities: it is defined within the Trigonal model and the homo-octahedral approximation, and gives thus the correct stacking mode for the case of all-M1 layers. If the polytype contains one or more M2 layers, the stacking mode obtained from PID analysis of. the diffraction pattern is simply an approximation: for each T2 j T2 j +1 e . e u . u v 2 i,2 j +1 M2 layer, the characters are replaced by the characters v2 j,2 j +1 or v2i,2 j +1 , depending on the parity of T2j and T2 j +1, and the displacement character obtained by the PID is simply v2j,2j+1. No indication can be obtained from the PID that the polytype may belong to the hetero-octahedral family. For the meso- and hetero-octahedral family, as well as for the distinction between the two members of an enantiomorphous pair, a complete structure refinement is required, similarly to the case of MDO polytypes. However, only the structural models corresponding to polytypes homomorphic to the homo-octahedral sequence obtained by PID analysis must be considered. The Fourier transform of a polytype (GN, where N is the number of layers) is given by the Fourier transform of the stacking sequence, which is a fringe function (Lipson and
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Taylor 1958), modulated by the Fourier transform of the layer (Gj): G N ( hkl ) = ∑ j =1 G j ( hkl ) exp 2π i ( t x , j h + t y , j k + t z , j l ) N
(20)
where tx,j, ty,j, tz,j are the (x, y, z) components of the stacking vector relating the j-th and the (j+1)-th layers (Takeda 1967). When the shifts between the building layers are rational and the rotations belong to the symmetry of the layer(s), their Fourier transform (Gj), which is a continuous function in the direction lacking periodicity, can be factorized from the expression of the structure factor GN. Thus, GN takes the simple form of the product of the layer transform and of the stacking sequence transform. The second term expresses the periodicity in reciprocal space appearing when a structure is constructed by a translation of subunits. This is the case of polytypes of binary compounds like SiC and ZnS (Tokonami and Hosoya 1965; Tokonami 1966; Farkas-Jahnke 1966; DornbergerSchiff and Farkas-Jahnke 1970; Farkas-Jahnke and Dornberger-Schiff 1970). In micas, the M layers are instead related by rotations belonging not to the layer symmetry but to the idealized symmetry of the Ob plane (with the obvious exception of the 1M polytype) and the same simplification is in principle not possible. However the Fourier transform of the M layer in the six possible orientations is almost unmodified in a subspace of the reciprocal space (Takeda 1967). By removing the modulating effect of the layer, the approximated Fourier transform of the stacking sequence is obtained. This is known as the Periodic Intensity Distribution (PID) function (Takeda 1967; Sadanaga and Takeda 1969; Takeda and Sadanaga 1969). Comparison of calculated and observed PID values along non-family reciprocal lattice rows parallel to c* is in principle sufficient to identify any mica polytype (Takeda and Ross 1995; Nespolo et al. 1999d). PID in terms of TS unit layers
A single type of non-polar unit layer (the M layer) is sufficient to describe polytypism of micas: the M layer is stacked with both translations along c and rotations about c* which do not belong to the layer symmetry. A different choice, employing more than one type of layers, is more suitable to describe the symmetry of the layer stacking and to simplify the process of identification of the stacking mode. As shown above, two kinds of non-polar OD layers (Tet and Oc) and one kind of polar OD packet (with two opposite orientations, p2j and q2j+1) are necessary to describe the OD character of mica polytypes. To compute the PID, Sadanaga and Takeda (1969) and Takeda and Sadanaga (1969) introduced the nonpolar TS unit layers, which are defined within the Trigonal model. The two layers D and T would be sufficient to describe any mica polytypes if two orientations, related by 180º rotation about c*, were permitted. To avoid the use of this rotation, which does not belong to the layer symmetry, four TS layers, including also the D* and T* layers, are employed. The relative positions of TS unit layers are given by the TS symbols, written as a sequence of N symbols Lj(ΔXj, ΔYj), 1 ≤ j ≤ N, where Lj is the kind of layer (D, D*, T, T*), and (ΔXj, ΔYj) are the (A1, A2) components in hexagonal axes of the total shift vector between the j-th TS layer and the N-th TS layer of the previous repeat (Fig. 2,3). The j-th TS unit layer is defined by the relation between the j-th and the (j+1)-th M layers and corresponds to the pair of packets q2j-1p2j. D and D* layers correspond to 2n×60º rotations between q2j-1 and p2j [i.e. the RTW symbol is Aj = 0(mod 2); q2j-1 and p2j have the same orientation parity], T and T* layers correspond to (2n+1)×60º rotations between q2j-1 and p2j [i.e. Aj = 1(mod 2); q2j-1 and p2j have an opposite orientation parity]. In the homo-octahedral approximation the two OD packets (p2j and q2j+1) describing each layer have the same OD symbol, and the two half-layers of an M layer have the
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same Z symbol. If “u” (uneven) and “e” (even) are the orientation parities of OD symbols of the OD packets, or of Z symbols of half M layers, the following equalities are obtained from Figure 3: D = u0u; D* = e0e; T = e0u; T* = u0e
(21)
The Fourier transform of an N-layer mica polytype [Eqn. (20)] in terms of TS unit layers in hexagonal axes becomes: N L( j − 1) ⎞ ⎛ G N (HK .L ) = ∑ j =1 G j (HK .LR ) exp 2 πi ⎜ HΔX j + KΔY j + ⎟. N ⎠ ⎝
(22)
The Fourier transform of the j-th TS unit layer, Gj(HK.LR), is two-dimensionally periodic and the reciprocal lattice coordinate in the direction lacking periodicity is not restricted to integral values but is a real variable, labeled LR. In Equation (22), Gj plays a role analogous to that of the atomic scattering factor in the expression of the structure factor. Because the j-th and the (j+1)-th TS layers must connect two packets p2j and q2j+1 with the same orientation parity (to preserve the octahedral coordination of the M cations), there are only eight possible pairs of TS unit layers (DD; D*D*; TT*; T*T; DT*; D*T; TD; T*D*). In addition, to match the cation positions, the layer stacked over a D or T layer must be shifted by –a/3, whereas the layer stacked over a D* or T* layer must be shifted by +a/3. Within the Pauling model only the octahedral cations have different coordinates in the four TS unit layers. However, their contribution to the layer Fourier transform becomes identical when the following conditions are satisfied: H = 0 ( mod 3) , all K ; H = 1( mod 3) , K ≠ 1( mod 3) ; H = 2 ( mod 3) , K ≠ 2 ( mod 3) h = 0 ( mod 3) , all k ; h ≠ 0 ( mod 3) , k ≠ 0 ( mod 3)
.(23)
Consequently, Gj is identical for all j (Gj = G0) and the contribution of the Fourier transform of the layer can be extracted from the summation in Equation (22), obtaining the PID function SN: S N (HK .L ) ≅
N L( j − 1) ⎞ G N (HK .L ) ⎛ = ∑ j =1 exp 2πi ⎜ HΔX j + KΔY j + ⎟. N ⎠ G0 (HK .LR ) ⎝
(24)
Within the Trigonal model also the Ob atoms have different coordinates, but again their contribution to Gj in all the four TS unit layers is the same when: H = 0, all K ; K = 0, all H ; H = − K h = 0, all k ; h = ± k
.
(25)
For these reflections, Gj = G0 and Equation (24) holds again. PID is thus defined in a subspace of the reciprocal space, which narrows from subfamily A polytypes to mixedrotation polytypes, but always includes at least the three r.p. 0kl, hhl,⎯hhl. The procedure for computing PID from the stacking mode is illustrated in Appendix B. A concrete example is hereafter analyzed in details for the 8A2 polytype. The PID is computed from the RTW symbols of the stacking sequence with the program PTST98
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(Nespolo et al. 1999d). This program can be obtained free of charge from its first author9. Derivation of PID from the diffraction pattern
The PID analysis of the diffraction pattern can be performed both in XRD and SAED (Selected Area Electron Diffraction) techniques. The experimental PID is easily obtained from the diffraction pattern once the data reduction has been applied. However, a complete data reduction is in general not necessary, because the stacking sequence is determined by the best match between the PID obtained from the diffraction pattern and the PID computed for all the homo-octahedral stacking candidates. For polytypes with a limited period, a direct visual comparison of the intensities with the computed PID can reveal the correct homo-octahedral stacking sequence (Ross et al. 1966). The experimental PID function is obtained from the intensities in a 0.1Å-1 repeat, within which the variation of the experimental factors is small, and the PIDs from several repeats are finally weighted, so that possible uncertainties are further reduced. For example, in general, the improvement in PID obtained by applying the absorption correction is smaller than the approximation of describing the mica structure with the TS layers, which are defined within the Trigonal model. Complete data reduction may improve the quality of the match of the experimental PID with that computed from the correct stacking sequence, but it does not change the sequence of stacking candidates. In other words, the homo-octahedral stacking sequence that best matches the experimental PID is not replaced by a different candidate when a more complete data reduction is applied. Some uncertainties can however be expected for a less complete data reduction in the hypothetical case of a long-period polytype (for which the number of possible stacking sequences is high) with a poor quality of the reflections, and consequently large uncertainties on the experimental PID, if two candidates show relatively close matches with the experimental PID. Such a hypothetical case has not appeared so far, but this is a possibility. A particularly intriguing case may occur when polytypes with different periodicities are in relation of homomorphy. As shown above, this may happen if a sub-periodicity exists in the sequence of v2j,2j+1 displacement vectors of meso-octahedral polytypes, or in the sequence of T2j,T2j+1 orientation vectors of hetero-octahedral polytypes when the chirality of the packets is neglected. In general, the number of reflections in the c*1 repeat corresponds to the number of layers in the full-period polytype. However, when the chemical difference between the family of the full-period polytype and the family of the shorter homomorphous polytype becomes smaller, some of the reflections weaken: if these weak reflections are overlooked, the homo-octahedral stacking sequence obtained from the PID analysis corresponds to an apparent periodicity shorter than the correct one. The visual comparison of the intensities, if performed, involves only the meso-octahedral polytypes homomorphous with the homo-octahedral polytype indicated by the PID, but with the same number of layers and the mistake may be overlooked. Special attention is necessary not to miss weak reflections along X rows. The general guidelines for the PID derivation from the diffraction pattern is summarized as follows: 1. For X-ray diffraction, the effect of the absorption on the PID is normally negligible for the purpose of polytype identification, if a sufficient number (e.g., four or more) of periods along the same row are considered and the corresponding PIDs are weighted. The LP factors are critical, however, if the diffraction pattern is taken with a precession camera, because the Lorentz-polarization effect in the precession motion is severe. 9
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For electron diffraction, the near-flatness of the Ewald sphere reduces greatly the effect of the experimental factors on the intensities. The pattern is, however, no longer kinematical, and the dynamical effects in general must be taken into account. However, the intensity ratio between adjacent reflections in a reciprocal lattice row can be treated as kinematical, and the PID analysis applies to electron diffraction as well (Kogure and Nespolo 1999b). Equation (24) is based on the approximation of the trigonal distribution of each kind of atom in the layer and G0 is thus an approximation of the Fourier transform of the layer. In the regions of reciprocal space where G0 passes through zero and changes sign, the relative error becomes large and Equation (24) is no longer applicable. In the practice of mica-polytype identification, the periods corresponding to l intervals including those regions should not be used to derive PID from the intensities. These intervals depend on the chemical composition: in the diffraction pattern they include very faint reflections and are easily recognized. The square root of the intensities, partially reduced when necessary, gives an approximant of the structure factors. By dividing these by the Fourier transform of the layer, an un-weighted, un-scaled PID is obtained. The mean value of PID along several period of the same reciprocal lattice row is computed, and the result is brought on the same scale [see Appendix B, Eqn. (B.4)]. EXPERIMENTAL INVESTIGATION OF MICA SINGLE CRYSTALS FOR TWIN / POLYTYPE IDENTIFICATION
Here we present the general guidelines for the experimental investigation of an unknown mica single crystal. The following represents an ideal outline and note that, depending on the availability and quality of the sample, and on the experimental equipment accessible to the investigator, not all the following steps may be possible. The local-scale investigation by TEM is described in detail by Kogure (this volume) and is thus not discussed here. Morphological study
The first step in the investigation of a mica single crystal consists in a morphological observation under the polarizing microscope. The sample should be observed immersed in a high refractive-index medium (an index oil if available; a natural fluid such as clove oil or glycerin may be used also) and not in air; otherwise the presence of twinning can be easily missed. In case of reflection twins [composition plane (quasi) normal to (001)] a twin results in different extinction positions under crossed polarizers and no complete extinction of light occurs for any orientation of the crystal. Instead, for rotations twins [composition plane parallel to (001)] the presence of twinning may be missed if the sample is observed only on one of the two surfaces, in case the uppermost crystal of a twin is larger than the others. A negative result from the morphological observation should thus be prudently taken as not conclusive about the absence of twinning. Surface microtopography
The second step should possibly involve a surface microtopography, which gives important information on both twinning and polytypism. The microtopography of a mica surface reveals spiral and parallel step patterns on the (001) crystal surfaces. Different techniques have been developed for this kind of investigation, such as phase-contrast microscopy, multiple-beam interferometry (e.g., Tolansky and Morris 1947a,b), surface decoration in TEM (Bassett 1958) and Atomic Force Microscopy. Three kinds of information, useful for the study of polytypism, are obtained by surface microtopography: 1) shape of the spirals; 2) height of the spiral step(s): 3)
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presence/absence of interlacing (Sunagawa 1964; Sunagawa and Koshino 1975). Micas of metamorphic origin are formed by alteration of the original rock and spiral growth is commonly not observed on the surface, which instead presents step systems as a consequence of Ostwald ripening typical of environments in which crystals grow or dissolve via a thin film of vapor or solution owing to an interstitial solvent (Sunagawa et al. 1975; Tomura et al. 1979). In contrast, micas formed in magmatic environments invariably show growth spirals on their surfaces, more or less polygonalyzed depending upon the strength of the solid-fluid interaction (Sunagawa 1977, 1978). A zigzag stacking sequence (i.e. a stacking sequence different from 1M) appears at the surface with an interlaced pattern: interlacing unambiguously indicates that the crystal under investigation is not 1M. In the case of metamorphic micas, multiple steps split into N unit layers with rhombic form, where N is the number of layers in the period of the polytype. In the case of magmatic micas, it is the spiral turn that decomposes into N unit layers. In both cases, the cause of interlacing is the anisotropy of the advancing rate, which is faster along the stagger direction and slower normal to it (Frank 1951; Verma 1953). No interlacing appears on the surface of the 1M polytype, because all the layers have the same stagger direction. The interlacing pattern of growth spirals is also observed in other phyllosilicates, and was exploited to identify kaolinite (single-layer, no interlacing), dickite (two-layer kaolinite with 60º or 120º rotations) and nacrite (two-layer kaolinite with 180º rotations) of hydrothermal metasomatic origin (Sunagawa and Koshino 1975). The shape of the growth spirals is controlled by the whole-layer symmetry, rather than by the symmetry of the sheet exposed on the growing surface. Typical growth spirals of trioctahedral micas are five-sided and show the monoclinic metric symmetry of the mica layer (Sunagawa 1964; Sunagawa and Tomura 1976) (Fig. 31). This shape of the growth spirals can be described as deriving from a regular hexagon through elongation and truncation. The growth is more rapid along [100] (the direction of the stagger) and results in longer sides parallel to the a axis (perpendicular to [010], the direction of slower growth), and the other four shorter edges and more largely spaced sides [±310, 3±10], corresponding to faster growth. The two sides ⎯[310] and ⎯[⎯310] are truncated to form a single line, eventually with a denticulated pattern, parallel to b. Truncation is not observed in 1:1 phyllosilicates, where there is no layer stagger. It is also not observed in dioctahedral micas: the reasons for this difference between trioctahedral and dioctahedral micas are not clear (Sunagawa and Koshino 1975; Sunagawa 1978). Because n×60º rotations are not equivalent when applied to a pentagonal spiral, the relative rotations of each component clearly appear at a surface observation and reveal the direction of stagger of each layer (Fig. 32). For short-period polytypes this information alone is sufficient to determine the stacking sequence. The height of the spiral step can also be directly measured by multiple-beam interferometry (step height as thin as 2.3Å were measured in hematite: Sunagawa 1960) and AFM (Kuwahara et al. 1998). In this way, Sunagawa et al (1968) identified polytypes 1M, 2M1, 2M2, 2O and 3T in synthetic fluor-phlogopite, and confirmed the presence of polytypes with longer period, whose stacking sequence was however too complex to be identified only on the basis of the surface morphology. Also the presence of twinning is clearly shown by surface microtopography. Sunagawa and Tomura (1976) reported beautiful examples of five-sided plateau-like patterns on the (001) face of phlogopite. These patterns derive from the agglutination of thin platy crystals, formed in the vapor phase and moving around as “flying magic carpets” while they are growing, onto the surface of a larger crystal, on which they settle with equal probability on any of the n×60º rotations, making thus either a parallel or a
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Figure 31. The five-fold growth spiral on the surface of the Mutsuré-jima phlogopite-1M, as revealed by multiple-beam interferometry (courtesy of I. Sunagawa) [from Sunagawa (1964) Fig. 1,2 p. 1429].
twin orientation (Fig. 33). Multiple platy crystals may come in contact when they agglutinate on the surface of the same larger crystal. In this case, a composite twin is formed: the platy crystals are twinned on (001) with respect to the substrate, forming a rotation twin, but they reciprocally contact on one of the (hk0) [orthohexagonal indexing] planes, thereby forming a reflection twin (Nespolo and Kuwahara 2001). Two-dimensional XRD study
The most common two-dimensional technique employs a precession camera, but any technique giving two-dimensional undistorted images of the reciprocal lattice is suitable as well. From these undistorted images, the geometry of the diffraction pattern can be analyzed by simple visual inspection. In the case of a precession camera study, the crystal must be mounted so as to have the (001) plane perpendicular to the goniometer rotation axis. In fact, the stacking of layers in micas is along c and the periodicity in reciprocal
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space appears along c*, thus it is necessary to have c* in all the images, i.e. to have c* aligned with the dial axis. A different mounting would show only one plane containing c*, which is insufficient for a twin/polytype analysis. The latter orientation shows diffraction from the (001) plane, with an almost hexagonal geometry. This plane is useless for twin/polytype identification, but is the richest in information for plesiotwins, because the Coincidence-Site Lattice (CSL) produced by the plesiotwin operations is parallel to (001). When the presence of a plesiotwin is suspected in a mica sample, the diffraction from (001) is necessary: it can be easily obtained by mounting the mica crystal so as to have the direction of elongation parallel to the glass fiber.
Figure 32. Schematic drawing of the interlaced pattern of the six homogeneous polytypes, resulting from the n×60º rotations of the five-fold growth spirals (modified after Endo 1968).
Figure 33. Tiny platy crystals agglutinated onto the (001) surface of a larger crystal. The tiny crystals are twinned on (001) with respect to the larger one, but on (hk0) with respect to each other. Notice the five-fold morphology (courtesy I. Sunagawa).
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The radiation to be employed initially can be either a short wavelength (e.g., Mo) or a longer wavelength (e.g., Cu of Fe). Mo is preferable for making easier the orientation of the crystal, but its wavelength is too short for the study of long-period polytypes, or even for twins of polytypes with period longer than three layers, resulting in an insufficient resolution between two successive reflections. Cu or Fe radiation is suitable for longer period polytypes (≤ 10-12 layers). The separation of the reflections can be improved by increasing the crystal-to-film distance, with slightly longer exposure times. This avoids the weakening of the reflections occurring when employing a longer wavelength. The choice of the radiation to employ initially is thus the result of a compromise between the ease of orienting the crystal (Mo) and the need of proper resolution. With some practice the orientation of a mica crystal on the precession camera becomes routine even with Cu radiation, which can thus be selected as the best compromise. For longer period polytypes, Fe or Cr radiation becomes necessary to obtain sufficient resolution, once the crystal is oriented. For investigating the possibility of apparent polytypism, one SD plane and three SX central planes must be recorded. From these planes, the geometry of the diffraction pattern is analyzed on the basis of the criteria given in Tables 12a-12c. If the crystal is twinned or allotwinned, the nine translationally independent rows forming a minimal rhombus, obtained from these four planes, allow the determination of the relative rotations between the individuals (see the example of ZT = 34 1M-2M1 allotwin below). If the crystal is not twinned, the stacking sequence in the homo-octahedral approximation can be obtained from the geometry of the diffraction pattern (MDO polytypes) or from the PID obtained along one or more X rows (non-MDO polytypes). This is the final stacking sequence if the polytype is composed of only M1 layers, otherwise it represents the homomorphic equivalent of the correct stacking sequence. In the meso-octahedral family, if the mesooctahedral character is pronounced (large difference between the average cations), the real stacking sequence can in principle be found by comparing the experimental intensities with the intensities computed for all the meso-octahedral polytypes homomorphic to the homooctahedral polytype obtained by the PID analysis. When the meso-octahedral character is not pronounced, the distinction is much more difficult. Moreover, as discussed in “Derivation of PID from the diffraction pattern”, when the sequence of displacement vectors contains one or more sub-periods, weak reflections occur along the X rows, and care must be taken to observe them. In both the meso- and the hetero-octahedral family, the true stacking sequence can be obtained only from a complete structure refinement, because the occupancies of the octahedral sites, and the sizes of the corresponding octahedra, must be refined. Unfortunately, the quality of the sample is often not sufficient to allow a complete data collection, and only the stacking sequence of the homomorphic polytype (PID stacking sequence) can be obtained. Diffractometer study
Once the stacking sequence in the homo-octahedral approximation is determined, if the quality of the crystal permits, the final stage consists of intensity measurements (usually by diffractometric measurement) and a structure refinement. The radiation to be employed is the same used in the preliminary (two-dimensional) study. The strongly anisotropic shape of mica crystals indicates applying an absorption correction through a ψ-scan procedure, rather than an analytical correction. Knowing the structure of the single layer and the homo-octahedral stacking sequence, the starting model is already very close to the final result, but the presence of one or more M2 layers must be determined. In other words, the meso- and hetero-octahedral stacking sequences, and not only the homomorphic sequence revealed by the PID, should be employed also as starting models, otherwise the presence of M2 layers may be overlooked. For instance, consider a hypothetical N-layer meso-octahedral polytype with biotitic composition, and suppose,
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for simplicity, that there are two Mg and one Fe2+ ions in the O sheet. Suppose also that n layers are of M2 type, and the remaining N - n layers are of M1 type. The occupation of the cation sites in the O sheet is described as: M1 layers: M1 = (1 - x)Fe + xMg; M2, M3: 0.5xFe + (1 - 0.5x)Mg; M2 layers: M2 = (1 - x)Fe + xMg; M1, M3: 0.5xFe + (1 - 0.5x)Mg. If the value of x is far from 2/3, the presence of the M2 layers should, in principle, be revealed even by a structure refinement employing only the homomorphic sequence as starting model. However, with the approach of x to 2/3 (where the difference between M1 and M2 disappears), the distinction between N layers of type M1 and (N-n) layers of type M1 plus n layers of type M2 becomes difficult. The presence of an M2 layer may be erroneously interpreted for disorder in the cation distribution. The site occupancies in the O sheet should be carefully checked; otherwise important information about the nature of the polytype under investigation can be easily overlooked (see also Nespolo 2001). APPLICATIONS AND EXAMPLES 24 layer Subfamily A Series 1 Class b biotite from Ambulawa, Ceylon
This polytype was found by Hendricks and Jefferson (1939) and is a typical example of how easily an incorrect stacking sequence may be accepted if the presence or absence of twinning is not properly evaluated. In most cases, the possibility of apparent polytypism may lead one to assume a longer stacking sequence, simulated by the twinning of a shorter polytype. In the present case, instead, a case real polytypism was incorrectly interpreted as apparent polytypism. Hendricks and Jefferson (1939) were the first to accomplish a systematic X-ray study of a large number (more than 100) of mica crystals, and the first to report the existence of non-MDO polytypes. At those times, the effect of twinning on the diffraction pattern was not understood yet and the authors implicitly assumed that the number of reflections in the c*1 repeat invariably corresponds to the number of layers in the polytype. They reported 1,2,3,6 and 24-layer polytypes, but later Smith and Yoder (1956) showed that the 3 and 6–layer polytypes were twins of 1 and 2-layer polytypes respectively. Smith and Yoder also re-analyzed the Weissenberg photographs of the 24layer polytype, concluding that it could be indexed on an 8-layer unit cell; the 3n-th, (3n+1)-th and (3n+2)-th reflections should thus come each from a different individual. Takeda (1969), adopting Smith and Yoder’s twin interpretation, performed a PID analysis based on the intensities of each third reflection. He derived a semi-quantitative intensity distribution from the sequence of w (weak), m (medium), or s (strong) given by Hendricks and Jefferson (1939). The best match with the PID values computed from the stacking sequences of all possible 8-layer polytypes corresponded to 8A2 polytype (for details about this polytype see below). Nespolo and Takeda (1999) re-analyzed the geometry of the diffraction pattern, as described in Hendricks and Jefferson’s paper, on the basis of the twin identification criteria given in Nespolo (1999) (see Table 12b) and showed that the pattern cannot correspond to a twin of an 8-layer polytype. They found: 1) the cell dimension given by Hendricks and Jefferson are: a = 5.3Å, b = 9.2Å, c = n×10Å, γ = 90º, β = 90º; it was thus a Class b polytype; 2) reflections hkl with k = 0(mod 3) were the same as the single-layer structure: it was thus a subfamily A polytype; 3) the heavy trace of continuous scattering from 060 on an over-exposed photograph did not pass through any 02l reflection but, rather it occurred at a distance of about one-third the periodicity from the closest reflection; the 0kl r.p. was thus not orthogonal and the diffraction pattern is typical of a Class b polytype.
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An 8-layer subfamily A polytype cannot belong to Class b; a twin of an orthogonal or Class a polytype cannot produce a diffraction pattern typical of Class b polytype. Therefore, the diffraction pattern reported by Hendricks and Jefferson was actually from a 24-layer polytype (Series 1), whose stacking sequence has not been resolved, and not a twin of the 8A2 polytype. On the basis of Takeda’s (1969) analysis of a subset of reflections, it can be inferred that Hendricks and Jefferson’s 24-layer polytype probably possesses a stacking sequence related to that of 8A2, belonging thus to the 2M1 structural series also. This example shows the danger of blindly applying a powerful method such as the PID. The direct determination of the polytype stacking sequence is easily obtained through comparison of the PID from the diffraction pattern with the PID computed for all the theoretical candidates, i.e. the polytypes with the same number of layers and the same OD character (subfamily A, subfamily B, or mixed-rotation). The correct stacking sequence corresponds to the best match between the experimental and the theoretical PID. If the presence of twinning is overlooked the experimental PID corresponds to the weighted mean of the PID from each individual, where the weight is the volume of the individuals. In contrast, as in the case of the 24-layer polytype shown here, if twinning is incorrectly assumed, the experimental PID is only a portion of the “true” PID. For a short-period polytype, with a limited number of candidates, the match with the computed PID is probably insufficient, and this should alert the investigator. However, for a longer period polytype a reasonable match may occur by chance, because the difference between the two closest PID decreases with the increase of the number of layers. Because the PID match is evaluated on a relative basis, taking the best match as the correct one, the possibility of a wrong interpretation exists. The presence/absence of twinning must therefore be correctly analyzed before PID analysis is applied to the diffraction pattern. 8A2 (subfamily A Series 0 Class a) oxybiotite from Ruiz Peak, New Mexico
This polytype was identified by Nespolo and Takeda (1999) in the oxybiotite from Ruiz Peak (New Mexico). Figure 34 is the diffraction pattern corresponding to the h0l (SD) r.p., with the geometry typical of a subfamily A polytype. Figure 35 shows the diffraction pattern corresponding to the⎯hhl (SX) r.p., which is non-orthogonal and with eight reflections in the c*1 repeat along X rows. The diffraction pattern is that of the subfamily A Series 0 Class a polytype and thus excludes the possibility of twinning: the crystal is an 8-layer polytype. Out of 9212 possible 8-layer homo-octahedral polytypes, only 94 belong to subfamily A (Ross et al 1966). Comparison of theoretically computed and experimentally recorded PID values was performed only for the 94 subfamily A homo-octahedral polytypes. In Table 19, the l indices in the three axial settings (C1, aS and aF) and the lˆ = l (mod 8) index are given, together with the corresponding observed structure factors corrected for the Lorentz and polarization effects, the Fourier transform of the single layer, the ratio of the latter two terms, and the scaled PID [Eqn. (B.4)]. The PID was not computed in the two periods in which the single-layer Fourier transform undergoes a sign change. The PID along the remaining five periods has been assigned weights (Table 20). The stacking sequences of all possible 8-layer homo-octahedral subfamily A polytypes were generated by the PTGR program (Takeda 1971). The PID of each polytype was computed by the PTST98 program (Nespolo et al. 1999d) and the closeness to the observed pattern was evaluated by means of an RPID index defined by analogy with the reliability index used structure refinements, namely: RPID
∑ =
N j =1
S Nj ( hkl )o − S jN ( hkl )c S jN ( hkl )o
(26)
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259
Figure 34. Precession diffraction pattern corresponding to the h0l SD r.p. of 8A2 polytype (Cu Kα). The a* axes of the three settings, C1, aS and aF, are shown [used by permission of the editor of Mineralogical Journal, from Nespolo and Takeda (1999) Fig. 2, p. 108].
Figure 35. Precession diffraction corresponding to the⎯hhl SX r.p. of 8A2 polytype (CuKα). The [⎯110]* directions of the three settings C1, aS and aF are shown. In aF setting the origin of PID is by definition in correspondence of l = 0(mod N). PID has been obtained from the intensities measured along the five periods indicated in the figure [used by permission of the editor of Mineralogical Journal, from Nespolo and Takeda (1999) Fig. 3, p. 109].
260
Nespolo & Ďurovič Table 19. Derivation of PID from measured intensities of 8A2 polytype. Observed structure factors (Fo) have been obtained from the intensities measured in five periods along the⎯11l reciprocal lattice row of aF setting (⎯11l of aS setting.). SLFT stands for Single Layer Fourier Transform (after Nespolo and Takeda 1999).
Period
1
2
3
4
5
l(C1)
85 82 79 76 73 70 67 64 61 58 55 52 49 46 43 40 13 10 7 4 1 -2 -5 -8 -35 -38 -41 -44 -47 -50 -53 -56 -59 -62 -65 -68 -71 -74 -77 -80
l(aS)
28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 4 3 2 1 0 -1 -2 -3 -12 -13 -14 -15 -16 -17 -18 -19 -20 -21 -22 -23 -24 -25 -26 -27
l(aF)
31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 7 6 5 4 3 2 1 0 -9 -10 -11 -12 -13 -14 -15 -16 -17 -18 -19 -20 -21 -22 -23 -24
lˆ
Fo 7 6 5 4 3 2 1 0 7 6 5 4 3 2 1 0 7 6 5 4 3 2 1 0 7 6 5 4 3 2 1 0 7 6 5 4 3 2 1 0
.25 .25 83.33 38.63 47.01 83.00 168.75 40.70 .22 30.06 183.05 76.75 91.39 169.81 246.62 32.92 24.77 33.28 111.86 23.28 41.45 60.84 134.28 30.39 18.80 20.98 172.90 89.73 99.68 188.67 288.53 28.74 .22 52.30 172.66 57.59 51.37 86.01 129.75 26.51
SLFT 12.99 16.77 20.65 24.53 28.28 31.77 34.86 37.43 39.34 40.51 40.85 40.31 38.88 36.57 33.46 29.65 12.46 15.41 17.64 19.11 19.78 19.64 18.71 16.98 22.08 26.78 30.99 34.58 37.43 39.46 40.59 40.83 40.21 38.78 36.64 33.88 30.65 27.06 23.25 19.35
Fo/SLFT .02 .01 4.04 1.574 1.66 2.61 4.84 1.09 .00 .74 4.48 1.90 2.35 4.64 7.37 1.11 1.99 2.16 6.33 1.21 2.10 3.10 7.18 1.79 .85 .78 5.58 2.59 2.66 4.78 7.10 .70 .00 1.34 4.71 1.70 1.68 3.18 5.58 1.37
PID .02 .02 4.44 1.73 1.83 2.87 5.32 1.20 .00 .57 3.47 1.47 1.82 3.59 5.70 .86 1.46 1.58 4.65 .89 1.54 2.27 5.26 1.31 .62 .57 4.07 1.89 1.94 3.49 5.19 .51 .01 1.26 4.42 1.59 1.57 2.98 5.23 1.28
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Crystallographic Basis of Polytypism and Twinning in Micas
Table 20. Comparison of measured and computed PID of 8A2 polytype (after Nespolo and Takeda 1999).
lˆ
Period 1
Period 2
Period 3
Period 4
Period 5
Mean
Calculated
7
.02
.00
1.46
.62
.01
.30
.23
6
.02
.57
1.58
.57
1.26
.89
.90
5
4.44
3.47
4.65
4.07
4.42
3.98
4.07
4
1.73
1.47
.89
1.89
1.59
1.52
1.73
3
1.83
1.82
1.54
1.94
1.57
1.74
1.78
2
2.87
3.59
2.27
3.49
2.98
2.98
3.35
1
5.32
5.70
5.26
5.19
5.23
5.13
5.31
0
1.20
.86
1.31
.51
1.28
1.04
1.00
Table 21. OD symbols (v2j-2,2j-1) and Z symbols (Z2j = Z2j-1) in the homo-octahedral approximation, RTW symbols (Aj) and TS symbols [Lj(Xj, Yj)] describing the stacking sequence of 8A2 polytype (after Nespolo and Takeda 1999).
j
v2j-2,2j-1
Z2j-1
Aj
Lj(Xj, Yj)
1
5
4
2
D(0,-1)
2
3
6
-2
D(0,-1)
3
5
4
2
D(0,1)
4
3
6
-2
D(0,1)
5
5
4
2
D(0,0)
6
3
6
-2
D(0,0)
7
5
4
-2
D(0,-1)
8
1
2
2
D(0,0)
Figure 36. The v2j,2j+1 displacement vectors of the 8A2 polytype in the homo-octahedral approximation, as revealed by PID analysis of the diffraction pattern in Figure 35 (modified after Nespolo and Takeda 1999).
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The best match corresponded to RPID = 0.04 (computed PID values for this sequence are in Table 20); the second best match to RPID = 0.33. This clearly shows that the homooctahedral stacking sequence has been uniquely identified. By employing the cell dimensions of the refined 1M polytype from the same sample (Ohta et al. 1982), the approximate cell parameters of this polytype was calculated through the axial transformations given in Equation (3) and the results are: a = 5.3Å, b = 9.2Å, c = 79.6Å, α = 90º. β = 91.3º, γ = 90º. The symbols for the homo-octahedral stacking sequence are given in Table 21, and the corresponding vector sequence is in Figure 36. The spacegroup type is ⎯C1, derived by applying the transformation rules given in Table 6. 1M-2M1 oxybiotite allotwin ZT = 34 from Ruiz Peak, New Mexico
This allotwin was also identified in the Ruiz-Peak oxybiotite and represents an example of apparent polytypism. Figures 37-40 present the diffraction patterns from three SX planes. The shortest separation between successive reflections along c* of X rows is c*1/6: the apparent period is six layers and thus the l index of all the reflections are expressed as (mod 6). Figure 40 shows the diffraction pattern from an SD plane of the same sample which, with one reflection for c*1 repeat, has the typical appearance of a subfamily A polytype. The presence of twinning is not evident from this plane. In principle, the investigated sample may be either a six-layer polytype, or a twin or allotwin involving the 2M1 polytype. However, two of the SX planes (Fig. 37 and 38) are orthogonal (i.e. reflections are present at l = 0 of the orthogonal six-layer cell, along each row parallel to c*). This geometry of the reciprocal lattice is impossible for a 6-layer subfamily A polytype, which would belong to Class b and should therefore have all the SX planes non-orthogonal (Table 12b). It follows that the sample is a twin or allotwin of the 2M1 polytype. Figure 41 shows the star polygon, comprised by the six possible orientations of the tessellation rhombus and the minimal rhombus, drawn by reporting the l (mod 6) indices of the reflections occurring in the four planes above, and applying the (3p, 3q) translations between translationally equivalent reciprocal lattice rows. None of the six orientations of the minimal rhombus matches any of the nine independent minimal rhombi which are possible for the 2M1 twins (Fig. 26). The sample is thus an allotwin. The pattern cannot involve a 3T crystal, otherwise three reflections corresponding to l = 0(mod 6), l = 2(mod 6) and l = 4(mod 6) would be present along all X rows. The sample is thus a 1M-2M1 allotwin. The shaded minimal rhombus matches the computed minimal rhombus of the 1M-2M1 allotwin with relative rotation of 60º between the two individuals and it corresponds to ZT = 34 in Nespolo et al (2000a). The cell of the allotwin lattice has a period of 6c0 along c and contains six lattice planes of the 1M polytype and three lattice planes of the 2M1 polytype. Of these, only the plane with z = 0(mod 6) has all the nodes from both polytypes overlapped by the allotwin operation, whereas in all the other lattice planes the nodes from the two polytypes are separated. Consequently, the allotwin index of 1M is 6, and that of 2M1 is 3. {3,6}[7{3,6}] biotite plesiotwin from Sambagawa, Japan
Sadanaga and Takéuchi (1961) performed a systematic study of micas of volcanic origin, and reported several examples of 1M twinning, and also one example of 2M1 twinning. Takéuchi et al (1972) foresaw that micas from a different environment, namely metamorphic, could reveal different kinds of “twinning” and investigated by electron diffraction a large number of small biotite crystals from the Sambagawa metamorphic belt in the Besshi area, Japan. They found several twins of the same type reported by
Crystallographic Basis of Polytypism and Twinning in Micas
Figure 37. Precession diffraction pattern from the first SX plane (SX1) of the allotwin ZT = 34. The l index of the reflections is expressed (mod 6) [used by permission of the editor of Acta Crystallographica B, from Nespolo et al. (2000b) Fig. 7, p. 644].
Figure 38. Precession diffraction pattern from the second SX plane (SX2) of the allotwin ZT = 34 (60º from SX1). l index as in Figure 37 [used by permission of the editor of Acta Crystallographica B, from Nespolo et al. (2000b) Fig. 8, p. 644].
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Figure 39. Precession diffraction pattern from the third SX plane (SX3) of the allotwin ZT = 34 (120º from SX1). l index as in Figure 37 [used by permission of the editor of Acta Crystallographica B, from Nespolo et al. (2000b) Fig. 9, p. 645].
Figure 40. Precession diffraction pattern from an SD plane of the allotwin ZT = 34 (30º from SX3). l index as in Figure 37 [used by permission of the editor of Acta Crystallographica B, from Nespolo et al. (2000b) Fig. 10, p. 645].
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265
Figure 41. Construction of the star polygon corresponding to the diffraction patterns in Figs. 3740. The SD plane in Figure 40 is taken coincident with the (a*c*) plane, and the three SX planes are reported counter clockwise according to the rotations indicated in Figures 37-40. The star polygon is then obtained by (3p, 3q) translations of the nine translationally independent rows in those four planes. The minimal rhombus and the tessellation rhombus are indicated in their six possible orientations. The shaded minimal rhombus corresponds to the ZT = 34 minimal rhombus tabulated in Nespolo et al. (2000a). Inset on the top-right: axes (a, b) of the space-fixed reference and of the individual-fixed references in the six possible orientations (a1 – a6), and corresponding ZT symbols (b1-b6 axes are not shown). Inset in the bottom-right: l (mod 6) indices of the reflections which are present on the composite rows of the lattice, and symbol of the rows. Ij is the symbol identifying the composite row, where I gives the number of reflections in the c*1 repeat and j is a sequence number [used by permission of the editor of Acta Crystallographica B, from Nespolo et al. (2000b) Fig. 11, p. 646].
Sadanaga and Takéuchi (1961), but they also found some flakes which gave a more complex diffraction pattern, and correspond to “plesiotwins” in the later definition introduced by Nespolo et al (1999b). One of these diffraction patterns is shown in Figure 42, where two (001) lattices rotated about the normal and with only one common node out of seven recognized. The angle between two corresponding reflections in the two rotated lattices is 21.8º, very close to the 21º47′ computed for the n = 7 plesiotwin. The slight difference is probably related to the deviation of the (001) plane from hexagonality. This kind of diffraction pattern is commonly obtained when flakes of layered crystals are suspended in water and dried in air (Sueno et al. 1971; Takéuchi et al. 1972). This process allows the flakes to settle over each other without alignment; the need for reducing the interface energy is not strong, because the flake-to-flake interaction is purely physical and there are no chemical bonds between them. In contrast, plesiotwins are formed by chemical interaction of crystals that have already reached a significant size, or by exsolution. The metamorphic environment, where crystals are less free of moving,
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favors the formation of plesiotwins. Plesiotwins, are less probable in a magmatic environment. In the presence of a fluid phase, crystals are more free to move and can overcome the kinetic barrier towards the more stable configuration of twins.
Figure 42. Composite diffraction pattern (right) produced by a single flake (left) of metamorphic biotite1M from the Sambagawa belt (courtesy Y. Takéuchi). Several pseudo-hexagonal lattices are overlapped with different orientation; two of these are rotated by 21.8º, close to the 21º47′ angle corresponding to the {3,6}[7{3,6}] composite tessellation. The two crystals to which these lattice belong form a plesiotwin with Σ factor 7 and plesiotwin index 21 [used by permission of the editor of Zeitschrift für Kristallographie, from Takéuchi et al (1972) Fig. 6, p. 219].
Crystallographic Basis of Polytypism and Twinning in Micas
267
APPENDIX A. TWINNING: DEFINITION AND CLASSIFICATION
Twinning is the oriented association of two or more individuals10 of the same crystalline compound, in which pairs of individuals are related by a geometrical operation termed twin operation. The twin operation is a symmetry operation that belongs to a crystallographic point group; it cannot belong to the symmetry of the crystal, otherwise it would produce a parallel growth instead of a twin. The lattice common to the twinned individuals is called twin lattice: it can either coincide (exactly or approximately) with the lattice of the individuals, or be a sublattice (exact or approximated) of them. A Twin element is a symmetry or pseudo-symmetry element for the twin lattice with respect to which the twin operation is defined. Twin index (n) is the order of the subgroup of translation in direct space defining the twin lattice, and coincides with the ratio of the number of lattice nodes of the individual to the number of nodes restored, exactly or approximately, by the twin operation. Twin obliquity (ω) is the angle, in the crystal setting of the individual, 1) between a twin axis and the normal to the lattice plane which is quasi-normal to the twin axis (rotation twins), or 2) between the normal to a twin plane and the rational direction closest to it (rotation twins).. The point group of the twin has the common symmetry of the individuals, as modified by the twin operation and may be lower, the same or higher than the point group of the single crystal (Friedel 1904, 1926; Buerger 1954). The French school (Bravais 1851; Mallard 1879; Friedel 1904, 1926) gave a classification of twinning based on the twin index and obliquity, introducing the four categories of merohedry (n = 1, ω = 0), reticular merohedry (n > 1, ω = 0), pseudomerohedry (n = 1, ω > 0), reticular pseudo-merohedry (n > 1, ω > 0). Twinning by merohedry has been further subdivided on the basis of the point groups of the Bravais class of the lattice, of the Bravais class of the space group, of the individual and, for OD structures, of the family structure (Table A1). The kind of merohedry the French school considered is that in which the Bravais class of lattice and the Bravais class of the space group coincide, and it has now been renamed syngonic merohedry. The case in which the Bravais class of the lattice is accidentally higher than the Bravais class of the space group includes two kinds of twinning: one is again a syngonic merohedry (the twin operations belong to the point group of the Bravais class of the space group), and the other is termed metric merohedry (the twin operations belong to the point group of the Bravais class of the lattice but not to the point group of the Bravais class of the space group) (Nespolo and Ferraris 2000). For each crystal family except the hexagonal, the “point group of the Bravais class of the space group” is tantamount to say “point group of the syngony”, because there is a 1:1 correspondence between crystal family, syngony and Bravais system, and for this reason the term “syngonic merohedry” was introduced. However, two syngonies (trigonal and hexagonal) and two lattices (hR and hP) correspond to the hexagonal crystal family. A trigonal crystal with lattice hR twinned within the same crystal family (h) may have two kinds of twinning: syngonic merohedry, with twin elements belonging to the hR lattice (only merohedral crystals) and reticular merohedry, with twin elements belonging to the hP sublattice of the hR lattice (twin index 3). Instead, a trigonal crystal with lattice hP twinned within the same crystal family (h) has only one kind of twinning and the twin elements belong to the hP lattice. This twinning corresponds to a syngonic merohedry. 10
The term “individual” is used to indicate one crystal of a twin, and the term “single crystal” to mean an untwinned crystal. Other authors (e.g., Hahn et al. 1999) use “component” instead of “individual”.
Class I
Syngonic Merohedry
Z = 0, n = 1
Syngonic Complete Merohedry
Syngonic Selective Merohedry
TPG > FSPG Metric Complete Merohedry
TPG d FSPG
Metric Selective Merohedry
TPG > FSPG
Metric Merohedry class IIB
Syngonic Merohedry class IIA TPG d FSPG
Z = 0, n = 1
p(TL) tp7 !p(BCSG)tp(La
Z = 0, n = 1
p(TL) = p(BCSG) = p(T) p(La) p(TL) =p(BCSG) tp(T) > p(La)
p(TL) = p(BCL)
Pseudomerohedry
Z > 0, n = 1
Reticular merohedry
Z = 0, n > 1
Reticular pseudomerohedry
Z > 0, n > 1
p(TL) > p(BCL) tp(BCSG)
Table A1. Classification of twinning. (p)TL = point group of the Twin Lattice; p(BCL) = point group of the Bravais Class of the Lattice of the individual; p(BCSG) = point group of the Bravais Class of the Spae Group of the individual;’ p(La) = Laue point group of the individual; p(T) = Twin point group; p(FS) = point group of the Family Structure. The point group of the twins has the common symmetry of the individuals, as modified by the twin operation (see Appendix B). Modified after Nespolo et al. (1999a) and Nespolo and Ferraris (2000).
268 Nespolo & Ďurovič
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Syngonic merohedry is subdivided, on the basis of the ratio between the order of the lattice point group and the order of the individual point group, into hemihedry (order 2), tetartohedry (order 4) and ogdohedry (order 8, possible only for the point group 3). Where the Laue symmetry of the individual is the same as the twin symmetry, the corresponding twins belong to class I. The diffraction pattern does not differ from that of a single crystal, unless anomalous scattering is substantial. The inversion center can always be chosen as a twin operation and the set of intensities collected from a twin is indistinguishable from that collected from a single crystal. Instead, when the Laue symmetry of the crystal is lower than the twin symmetry, the twins belong to class II and are then subdivided into class IIA (syngonic merohedry) and class IIB (metric merohedry). The twin operations relate non-equivalent reflections, and the presence of twinning may hinder a correct derivation of the symmetry from the diffraction pattern. In particular, when the number of individuals coincides with the order of the twin operation and the volumes of the individuals are identical, the symmetry of the diffraction pattern is higher than the Laue symmetry of the individual. An incorrectly chosen space-group type may thus be assumed in the initial stage of the structure refinement (Catti and Ferraris 1976; Nespolo and Ferraris 2000). In the case of OD structures, class II twins are further subdivided. The family structure may correspond to a Bravais system different from both the crystal lattice and the twin lattice. When the point group of the family structure is a subgroup of the point group of the twin lattice and twinning is by class II merohedry (both IIA and IIB), one or more of the twin laws do not belong to the point group of the family structure. This kind of twin law corresponds to merohedry for the polytype but to reticular merohedry for the family structure. These twin operations produce incomplete overlap of the family reciprocal sublattice; in particular, in terms of the polytype lattice, they overlap some of the nodes with zero weight of an individual to nodes with non-zero weight of another individual, and vice versa. Therefore, peculiar violations of the non-space-group absences along family rows appear in the diffraction pattern, where indexed in terms of the actual structure. This modifies the diffraction pattern, whose geometry no longer corresponds to that of the single crystal. This kind of merohedry, which restores only a part of the family sublattice of OD structures, is termed selective merohedry, whereas twinning by merohedry of OD structures in which the twin operation belongs to the point group of the family structure and restores the whole family reciprocal sublattice is termed complete merohedry (Nespolo et al. 1999a). In the case of layer compounds, it is useful to decompose the obliquity into two components, within and outside the plane of the layer, which for micas is (001). Labeling t(hkl) the “trace” of a plane (hkl) on the (001) plane, the component of the obliquity within the (001) plane (ω||) corresponds to the angle between the normal nt(hkl) to t(hkl) and the direction [hk0] quasi normal to t(hkl), i.e. ω||([hk0]^ nt(hkl)) (Fig. 20). The component normal to the (001) plane (ω⊥) corresponds to the angle between the normal to the (001) plane and the lattice row quasi-normal to (001). ω⊥ measures the deviation of the c axis of the triple and sextuple cells of non-orthogonal polytypes from the normal to (001); for Class b polytypes it measures also the deviation of the rhombohedral [111] direction, i.e. ω⊥([111]R^[001]*) (Fig. B1). ω|| measures the deviation from hexagonality of the (001) plane and is thus related to ε. In both the Pauling and the Trigonal models, nonorthogonal polytypes are metrically monoclinic, because γ = 90º, ω||([100]^nt(100)) = 0 and ω||([010]^nt(010)) = 0, but ω|| is non-zero for the other four directions that would be equivalent in a hexagonal lattice. Imposing ω|| = 0 for each of these four directions, the two-dimensional lattice in the (001) plane becomes hp, but the three-dimensional lattice
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is only pseudo-hP, because the c axis of the triple cell is not exactly perpendicular to (001). Imposing instead ω⊥ = 0, an oC lattice is obtained. Finally, imposing both ω|| = 0 and ω⊥ = 0, the lattice becomes hP, and for Class b polytypes the lattice is centered. Figure A1. Perspective view of the lattice of Class b mica polytypes. The monoclinic conventional cell (doubly primitive, thick dotted lines), the pseudorhombohedral cell (primitive, solid thin lines) and the pseudo-orthohexagonal cell C1 (sextuply primitive, thick solid lines) are shown. Thick dotted-dashed line: [111] row of the pseudo-rhombohedral cell. Thick dotted line: direction normal to (001). Thin dotted lines: directions normal to (001) passing through the Ccentering nodes on two successive lattice planes of the monoclinic conventional cell. ω⊥ is the component of the obliquity normal to the (01) plane. Black, gray and white circles represent lattice nodes at z = 0, 1/3 and 2/3 respectively (z is referred to c axis of the C1 cell). The stagger of the layer at z = 1/3 is -(b+δ)/3. For the ideal case δ = 0, ω⊥ = 0 (modified after Nespolo and Ferraris 2000).
APPENDIX B. COMPUTATION OF THE PID FROM A STACKING SEQUENCE CANDIDATE.
The calculation of the PID function requires the following steps. Step 1. Conversion from RTW symbols into “provisional” OD or Z symbols in the homo-octahedral approximation, by simply looking for Σv = “*”, “0” or “–” [i.e. cn = (0, 0), ⎯(1/3, 0) or (0,⎯1/3). This is straightforwardly obtained by means of a simple addition cycle: v 2 j ,2 j +1 = v 2 j −2,2 j −1 − A j (B.1) Z = Z + A ;Z = Z 2 j +1
2 j −1
j
2j
2 j −1
where j = 1~N. The initial value is fixed as v0,1 = 3 or Z1 = 3; if the resulting cn projection does not take one of the three expected values, v0,1 or Z1 is incremented and Equation (B.1) is recalculated. Step 2. Derivation of the correct homo-octahedral OD or Z symbol, by analyzing the symmetry properties. For orthogonal and Class b polytypes the symbols obtained from Equation (B.1) may correspond to an orientation of the symmetry elements not compatible with the space-group type. In such a case, the sequence of characters must be changed, by making v0,1 or Z1 taking one of the other values with the same parity. This is equivalent to rotating the structural model around c* by 2n×60º. The correct sequence is found when the characters in the OD or Z symbols are related by symmetry operators located along the lattice directions compatible with the space-group type requirements (Table 5a,b). Step 3. Expression of the stacking operators rj, which give the displacement between the (j-1)-th and the j-th TS layers, as a function of OD or Z symbols and calculation of TS symbols. The relation of the stacking operators rj with OD or Z symbols is straightforward for orthogonal polytypes, whereas for non-orthogonal polytypes the
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Subclass must be taken into account. OD and Z symbols for non-orthogonal polytypes always correspond to ⎯(1/3, 0)n (Class a) or (0,⎯1/3) (Class b). The PID is most n conveniently expressed in the (3 a, 3 b)F axial setting, which corresponds to cn = ⎯(1/3, 0) or (0,⎯1/3) for Subclass 1 and cn = (1/3, 0) or (0, 1/3) for Subclass 2. It follows that for orthogonal polytypes and Subclass 1 polytypes the stacking operators simply coincide with OD or Z symbols (rj = v2j-2,2j-1 or rj = Z2j-1), wherease for Subclass 2 polytypes they are related by a 180º rotation around c* (rj = v2j-2,2j-1 + 3 or rj = Z2j-1 + 3). Step 4. Computation of PID (SN) as a function of the (a, b) components of TS symbols. The components of the j-th TS layer referred to the (a, b) axes are indicated as (Xj, Yj), to distinguish from the components in (A1, A2) axes, which were labeled (ΔXj, ΔYj) (Eqn. (22) and (24)). (Xj, Yj) are equal to the sum of the (xrj, yrj) components of the stacking n noperators from the first to j-th stacking operators. However, because the c axis of the (3 a,3 b)F axial setting is displaced -1/3(n+1) (where n is the Series) along a or b (depending upon the Class), the additional displacement (–j/3(n+1), 0) (Class a) or (0, – to j/3(n+1)) (Class b) must be added to the (Xj, Yj) component of the j-th TSn symbol n express the layer stacking of non-orthogonal polytypes with respect to (3 a,3 b)F axial setting. In this way, TS symbols for Series 0 subfamily A polytypes always have Xj = 0 (c axis passing through the origin of each layer). ⎧ ⎪Orthogonal polytypes : ( X j ,Y j ) = ∑ j xr ,yr i i i =1 ⎪ ⎪ j ⎛ −j ⎞ (B.2) ⎨ Class a polytypes : ( X j ,Y j ) = ∑ i =1 xri ,yri + ⎜ n +1 , 0 ⎟ ⎝3 ⎠ ⎪ ⎪ j ⎛ −j ⎞ ⎪ Class b polytypes : ( X j ,Y j ) = ∑ i =1 xri ,yri + ⎜ 0, n +1 ⎟ ⎝ 3 ⎠ ⎩⎪ n Finally, in Class b the axes exchange a ↔ b expresses PID in the 3 bF axial setting. The complete TS symbols Lj(Xj, Yj) are obtained from Table 5 and Equation (B.2), and the PID function SN is: j −1 ⎞ ⎡ ˆ N N ⎛ S N hkl = ∑ j =1 S jN hklˆ = ∑ j =1 exp 2π i ⎜ hX j + kYj + lˆ ⎟ l = l ( mod N ) ⎤⎦ (B.3) N ⎠ ⎣ ⎝ with the normalizing condition:
)
(
)
(
)
( )
( )
∑
(
( )
2
⎡ S N hklˆ ⎤ = N 2 j =1 ⎣ j ⎦ . Symmetry of the PID N
(B.4)
Nespolo et al (1999d) have analyzed the symmetry of the PID in relation to the kind of polytype present. The results are briefly summarized here: for details, refer to the original paper. For Series 0 polytypes there is a well-determined relation between the PID sequences along rows related by 2n×60º:
(
)
(
S N 2h,2k , lˆ = S N h, k , N − lˆ
)
(B.5)
which, for the reciprocal lattice rows commonly used in the PID analysis, become:
( )
(
)
( )
(
)
( )
(
S N 04lˆ = S N 02, N − lˆ ; S N 22lˆ = S N 11, N − lˆ ; S N 22lˆ = S N 1 1, N − lˆ
)
(B.6)
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For all OD polytypes (both subfamilies A and B) of Series 0, the PID has also a translational symmetry reminiscent of that relation between pairs of translationally equivalent rows defining a minimal rhombus:
(
)
( )
S N h + 3 N , k + 3 N , lˆ = S N hk lˆ .
(B.7)
For subfamily A of Series 0 PID values have a trigonal symmetry: ± ± (B.8) S N 0,2k , lˆ = S N ⎛⎜ k k lˆ ⎞⎟ = S N ⎛⎜ k k , N − lˆ ⎞⎟ . ⎠ ⎝ ⎠ ⎝ which, for the reciprocal lattice rows commonly used in PID analysis, is expressed as:
(
( )
)
( )
( )
(
)
(
)
(
S N 02lˆ = S N 1 1 lˆ = S N 1 1 lˆ = S N 0 2, N − lˆ = S N 11, N − lˆ = S N 1 1, N − lˆ
)
(B.9)
For Series > 0, subfamily A polytypes either are orthogonal or belong to Class b; in the latter case the symmetry of the PID must take into account a shift of the origin. Class b polytypes have a pseudo-rhombohedral primitive lattice, which thus allows three equivalent orientations, related by 2n×60º rotations about c*. For monoclinic polytypes, only one of the three orientations leading to cn = (0,⎯1/3) corresponds to a correct disposition of the symmetry elements (a-unique setting for a < b). Instead, for triclinic polytypes these three orientations are truly equivalent. Z, OD and TS symbols are different for the three orientations, but they describe three equivalent orientations of the structural model. PID values expressed for a given reciprocal lattice row in a certain orientation of the structural model correspond to a different row in another orientation. (3na, 3nb) For Series > 0 the c axis of the F setting is displaced by 1/3n for each layer and the length of the axis displacement is a submultiple of the layer stagger: therefore, the origin of the PID is not the same in the three orientations of the structural model. An example is given for the 3A1 polytype in Table 14 of Nespolo et al (1999d). The existence of a similar ambiguity in chlorite was reported by Brindley et al (1950). ACKNOWLEDGMENTS
We wish to acknowledge Prof. Giovanni Ferraris (Torino University), Prof. Hiroshi Takeda (Chiba Institute of Technology), Prof. Yoshio Takéuchi (Nihon University, Tokyo), Prof. Takeo Matsumoto (Kanazawa University), Prof. Ichiro Sunagawa (Yamanashi Institute of Gemology and Jewelry Arts), Prof. Boris B. Zvyagin (IGEM – Russian Academy of Sciences, Moscow) and Prof. Theo Hahn (RWTH, Aachen) for several profitable discussions; Prof. Maria Franca Brigatti (Modena University) and Prof. S. Guggenheim (University of Illinois at Chicago) for letting us obtain the tables of their chapter while this manuscript was in preparation. The manuscript was reviewed by Prof. Stefano Merlino (University of Pisa) and Prof. Stephen. Guggenheim (University of Illinois at Chicago), to whom we express our gratitude. REFERENCES Amelinckx S, Dekeyser W (1953) Le Polytypisme des Minéraux Micacés et Argileux. Premiére partie: observations et leaurs interprétations. C R XIX Congr Geol Int’l, Comité International pour l'Étude des Argiles, Alger, fascicule XVIII, 1-22 Amisano-Canesi A, Chiari G, Ferraris G, Ivaldi G, Soboleva SV (1994) Muscovite- and phengite-3T: crystal structure and conditions of formation. Eur J Mineral 6:489-496 Arnold H (1996) Transformations in crystallography. Sect. 5 in International Tables for Crystallography Vol. A, 5th edition. Th Hahn (ed) Dordrecht / Boston / London: Kluwer Academic Publishers (in press) Backhaus K-O, Ďurovič S (1984) Polytypism of micas. I. MDO polytypes and their derivation. Clays Clay Minerals 32:453-463 Bailey, SW (1975) Cation Ordering and Pseudosymmetry in Layer Silicates. Am Mineral 60:175-187
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Polytype Structures. Report of the International Union of crystallography Ad-Hoc Committee on the Nomenclature of Disordered, Modulated and Polytype Structures Acta Crystallogr A40:399-404 Güven N (1971) Structural Factors Controlling Stacking Sequences in Dioctahedral Micas. Clays Clay Miner 19:159-165 Hahn Th, Vos A (2002) Reflection conditions. Sect. 2.13 in International Tables for Crystallography, Vol. A 5th edition. Th Hahn (ed) Dordrecht / Boston: London: Kluwer Academic Publishers (in press) Hahn T, Janovec V, Klapper H (1999) Bicrystals, twins and domain structures – A Comparison. Ferroelectrics 222:11-21 Hendricks SB, Jefferson ME (1939) Polymorphism of the micas with optical measurements. Am Mineral 24:729-771 Iijima S, Buseck PR (1978) Experimental study of disordered mica structure by high-resolution electron microscopy. Acta Crystallogr A34:709-719 International Tables for Crystallography Vol. A. (2002) 5th edition. Th Hahn (ed) Dordrecht / Boston / London: KLuwer Academic Publishers (in Press) Ito T (1935) On the symmetry of rhombic pyroxenes. Z Kristallogr 90:151-162 Ito T (1938) Theory of twinned space groups. J Japan Assoc Mineral Petr Econ Geol 20:201-210 (in Japanese) Ito T (1950) X-ray studies on polymorphism. Maruzen Co., Tokyo, 231 pp Ito T, Sadanaga R (1976) On the crystallographic space groupoids. Proc Jpn Acad 52:119-121 Iwasaki H (1972) On the diffraction enhancement of symmetry. Acta Crystallogr A28:253-261 Jackson WW, West J (1931) The crystal structure of muscovite KAl2(AlSi3)O10(OH)2. Z Kristallogr 76:211227 Joswig W, Takéuchi Y, Fuess H (1983) Neutron-diffraction study on the orientation of hydroxyl groups in margarite. Z. Kristallogr 165:295-303 Kassner D, Baur WH, Joswig W, Eichhorn K, Wendschuh-Josties M, Kupčik V (1993) A test of the importance of weak reflections in resolving a space-group ambiguity involving the presence or absence of an inversion centre. Acta Crystallogr B49:646-654 Kogure T, Nespolo M (1999a) First occurrence of a disordered stacking sequence including (±60º 180º) in Mg-rich annite. Clays Clay Miner 48:784-792 Kogure T, Nespolo M (1999b) A TEM study of long-period mica polytypes: determination of the stacking sequence of oxybiotite by means of atomic-resolution images and Periodic Intensity Distribution (PID) Acta Crystallogr B55:507-516 Kokscharow NV (1875) Materialen zur Mineralogie Russlands. Vol. 7. St. Petersburg Konishi H, Akai J (1990) HRTEM observation of new complex polytype of biotite from dacites in Higashiyama hills, Niigata, Central Japan. Clay Science 8:25-30 Knurr RA, Bailey SW (1986) Refinement of Mn-substituted muscovite and phlogopite. Am Mineral 34:7-16 Kuwahara Y, Uehara S, Aoki Y (1998) Surface microtopography of lath-shaped hydrothermal illite by tapping-mode™ and contact-mode AFM. Clays Clay Miner 46:574-582 Lipson H, Taylor CA (1958) Fourier Transforms and X-Ray Diffraction, Bell, London Mackovicky E (1997) Modularity – different types and approaches. In Modular aspects of minerals / EMU Notes in Mineralogy, vol. 1. S Merlino (ed) Eötvös University press, Budapest, p 315-343 Mallard E (1879) Traité de Cristallographie geometrique et physique, Vol. I. Paris: Dunod. 372pp Marignac C (1847) Notices minéralogiques. Suppl Bibl Universe Genève, arch Sci Phys Nat 6:293-304 Matsumoto T, Kihara K, Iwasaki H (1974) Conditions for the diffraction enhancement of symmetry of types 1 and 2. Acta Crystallogr A30:107-108 Matsumoto T, Wondratschek H (1979) Possible superlattices of extraordinary orbits in 3-dimensional space. Z Kristallogr 150:181-198 Mauguin MCh (1927) Êtude du mica muscovite au moyen des rayons X. CR Acad Sci Paris 185:288-291 Mauguin MCh (1928) Etude de Micas au moyen du rayons X. Bull Soc franç Minér Crist 51:285-332 McLarnan TJ (1981) The number of polytypes in sheet silicates. Z Kristallogr 155:247-268 Merlino S (1990) OD structures in mineralogy. Per Mineral 59:69-92 Mogami K, Nomura K, Miyamoto M, Takeda H, Sadanaga R (1978) On the number of distinct polytypes of mica and SiC with a prime layer-number. Can Mineral 16:427-435 Mügge O (1898) Über Translationen und verwandte Erscheinungen in Krystallen. N Jb Miner Geol Paläontol 1:71-158 Nespolo M (1999) Analysis of family reflections of OD-mica polytypes, and its application to twin identification. Mineral J 21:53-85 Nespolo M (2001) Perturbative theory of mica polytypism. Role of the M2 layer in the formation of inhomogeneous polytypes. Clays Clay Miner 49:1-23
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Nespolo M, Ferraris G (2000) Twinning by syngonic and metric merohedry. Analysis, classification and effects on the diffraction pattern. Z Kristallogr 215:77-81 Nespolo M, Kogure T (1998) On the indexing of 3T mica polytype. Z Kristallogr 213:4-12 Nespolo M, Kuwahara Y (2001) Apparent polytypism in the Ruiz Peak ferric phlogopite. Eur J Mineral 13 (in press) Nespolo M, Takeda H (1999) Inhomogeneous mica polytypes: 8-layer polytype of the 2M1 structural series determined by the Periodic Intensity Distribution (PID) analysis of the X-ray diffraction pattern. Mineral J 21:103-118 Nespolo M, Takeda H, Ferraris G (1997a) Crystallography of mica polytypes. In Modular aspects of minerals / EMU Notes in Mineralogy, vol. 1. S Merlino (ed) Eötvös University press, Budapest, p 81-118 Nespolo M, Takeda H, Ferraris G, Kogure T (1997b) Composite twins of 1M mica: derivation and Identification. Mineral J 19:173-186 Nespolo M, Takeda H, Ferraris G (1998) Representation of the axial settings of mica polytypes. Acta Crystallogr A54:348-356 Nespolo M, Ferraris G, Ďurovič S (1999a) OD character and twinning – Selective merohedry in class II merohedric twins of OD polytypes. Z Kristallogr 214:776-779 Nespolo M, Ferraris G, Takeda H, Takéuchi Y (1999b) Plesiotwinning: oriented crystal associations based on a large coincidence-site lattice. Z Kristallogr 214:378-382 Nespolo M, Kogure T, Ferraris G (1999c) Allotwinning: oriented crystal association of polytypes – Some warnings on consequences. Z Kristallogr 214:1-4 Nespolo M, Takeda H, Kogure T, Ferraris G (1999d) Periodic Intensity Distribution (PID) of mica polytypes: symbols, structural model orientation and axial settings. Acta Crystallogr A55:659-676 Nespolo M, Ferraris G, Takeda H (2000a) Twins and allotwins of basic mica polytypes: theoretical derivation and identification in the reciprocal space. Acta Crystallogr A56:132-148 Nespolo M, Ferraris G, Takeda H (2000b) Identification of two allotwins of mica polytypes in reciprocal space through the minimal rhombus unit. Acta Crystallogr B56:639-647 Ohta T, Takeda H, Takéuchi Y (1982) Mica polytypism: similarities in the crystal structures of coexisting 1M and 2M1 oxybiotite. Am Mineral 67:298-310 Pabst A (1955) Redescription of the single layer structure of the micas. Am Mineral 40:967-974 Pauling L (1930) The structure of micas and related minerals. Proc Nat Ac Sci 16:123-129 Pavese A, Ferraris G, Prencipe M, Ibberson R (1997) Cation site ordering in phengite 3T from the Dora-Maira massif (western Alps): a variable-temperature neutron powder diffraction study. Eur J Mineral 9:11831190 Pavlishin VI, Semenova TF, Rozhdesvenskaya IV (1981) Protolithionite-3T: structure, typomorphism and practical importance. Mineral Zh 3:47-60 (in Russian) Peacock MA, Ferguson RB (1943) The morphology of muscovite in relation to the crystal lattice. Univ Toronto, Studies in Mineral 48:65-82 Pleasants PA, Baake M, Roth J (1996) Planar coincidences for N-fold symmetry. J Math Phys 1029-1058 Radoslovich EW (1960) The structure of muscovite KAl2(Si3Al)O10(OH)2. Acta Crystallogr 13:919-932 Ramsdell LS (1947) Studies on silicon carbide. Am Mineral 32:64-82 Ranganathan S (1966) On the geometry of coincidence-site lattices. Acta Crystallogr 21:197-199 Rieder M (1968) Zinnwaldite: Octahedral ordering in lithium-iron micas. Science 160:338-1340 Rieder M (1970) Lithium-iron micas from the Krušné hory Mountains (Erzgebirge): Twins, epitactic overgrowths and polytypes, Z Kristallogr 132:161-184 Rieder M, Weiss Z (1991) Oblique-texture photographs: more information from powder diffraction. Z Kristallogr 197:107-114 Rieder M, Hybler J, Smrčok L, Weiss Z (1996) Refinement of the crystal structure of zinnwaldite 2M1. Eur J Mineral 8:1241-1248 Ross M, Takeda H, Wones DR (1966) Mica polytypes: systematic description and identification. Science 151:191-193 Rieder M, Cavazzini G, D’yakonov YuS, Frank-Kamenetskii VA, Gottardi G, Guggenheim S, Koval’ PV, Müller G, Neiva AMR, Radoslowich EW, Robert JL, Sassi FP, Takeda H, Weiss Z, Wones DR (1998) Nomenclature of the micas. Clays and Clay Miner 46:586-595 Royer L (1928) Recherches expérimentales sur l’épitaxie ou orientation mutuelle de cristaux d’espèces differentes. Bull Soc franç Minér Crist 51:7-159 Royer L (1954) De l’épitaxie; quelques remarques sur le problèmes qu’elle soulève. Bull Soc franç Minér Crist 77:1004-1028 Sadanaga R (1978) omplex structures and space groupoids. Rec Progr Nat Sci Jpn 3:143-151 Sadanaga R, Ohsumi K (1979) Basic theorems of vector symmetry in crystallography. Acta Crystallogr A35:115-122
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Sadanaga R, Sawada T, Ohsumi K, Kamiya K (1980) Classification of superstructures by symmetry. J Jpn Assoc Min Petr Econ Geol, Spec. Issue No. 2:23-29 Sadanaga R, Takeda H (1968) Monoclinic diffraction patterns produced by certain triclinic crystals and diffraction enhancement of symmetry. Acta Crystallogr B24:144-149 Sadanaga R, Takeda H (1969) Description of mica polytypes by new unit layers. J Mineral Soc Japan 9:177184 (in Japanese) Sadanaga R, Takéuchi Y (1961) Polysynthetic twinning of micas. Z Kristallogr 116:406-429 Sartori F (1977) The crystal structure of a 2M1 lepidolite. Tschermaks Min Petr Mitt 24:23-37 Sartori F, Franzini M, Merlino S (1973) Crystal Structure of a 2M2 Lepidolite. Acta Crystallogr B29:573-578 Schaskolsky M, Schubnikow A (1933) Über die künstliche herstellung gesetzmäβiger kristallverwachsungen des kalialauns. Z Kristallogr 85:1-16 Schläfli L (1950) Gesammelte mathematische Abhabndlungen (Vol. 1) Birkhäuser, Basel Sidorenko OV, Zvyagin BB, Soboleva SV (1975) Crystal structure refinement for 1M dioctahedral mica. Sov Phys Crystallogr 20:332-335 Sidorenko OV, Zvyagin BB, Soboleva SV (1977a) Refinement of the crystal structure of 2M1 paragonite by the high-voltage electron diffraction method. Sov Phys Crystallogr 22:554-556 Sidorenko OV, Zvyagin BB, Soboleva SV (1977b) The crystal structure of 3T paragonite. Sov Phys Crystallogr 22:557-560 Smith JV, Yoder HS (1956) Experimental and theoretical studies of the mica polymorphs, Mineral Mag 31:209-235 Slade PG, Schultz PK, Dean C (1987) Refinement of the Ephesite structure in C1 symmetry. N Jb Mineral Mh 1987:275-287 Smrčok Ĺ, Ďurovič S, Petříček V, Weiss Z (1994) Refinement of the crystal structure of cronstedtite-3T. Clays Clay Miner 42:544-551 Smrčok L, Weiss Z (1993) DIFK91: a program for the modelling of powder diffraction patterns on a PC. J Appl Cryst 26:140-141 Smyth JR, Jacobsen SD, Swope RJ, Angel RJ, Arlt T, Domanik K, Holloway JR (2000) Crystal structures and compressibilities of synthetic 2M1 and 3T phengite micas. Eur J Mineral 12:955-963 Sokolova GV, Aleksandrova VA, Drits VA, Bairakov VV (1979) Crystal structures of two brittle Lithia micas. In Kristallokhimiya i Struktura Mineralov. Frank-Kamenetskii VA (ed) Nauka, Moscow, p 55-66 (in Russian) Sorokin ND, Tairov YuM, Tsvetkov VF, Chernov MA (1982a) The laws governing the changes of some properties of different silicon carbide polytypes. Dokl Akad Nauk SSSR 262:1380-1383 (in Russian) Sorokin ND, Tairov YuM, Tsvetkov VF, Chernov MA (1982b) Crystal-chemical properties of the polytypes of silicon carbide. Sov Phys Crystallogr 28:539-542 Sueno S, Takeda H, Sadanaga R (1971) Two-dimensional regular aggregates of layered crystals. Mineral J 6:172-185 Sunagawa I (1960) Mechanism of crystal growth, etching and twin formation of hematite. Mineral J 3:59-89 Sunagawa I (1964) Growth spirals on phlogopite crystals. Am Mineral 49:1427-1434 Sunagawa I (1977) Natural crystallization. J Crystal Growth 42:214-223 Sunagawa I (1978) Vapour growth and epitaxy of minerals and synthetic crystals. J Crystal Growth 45:3-12 Sunagawa I (1984) Growth of crystal in nature. In Material Science of the Earth’s Interior. I Sunagawa (ed) Terra Publishing Company, Tokyo - D. Reidel Publishing Company, Dordrecht / Boston / Lancaster, p 63-105 Sunagawa I, Endo J, Daimon N, Tate I (1968) Nucleation, growth and polytypism of fluor-phlogopite from the vapour phase. J Crystal Growth 3,4:751 Sunagawa I, Koshino Y (1975) Growth Spirals on Kaolin Group Minerals. Am Mineral 60:407-412 Sunagawa I, Koshino Y, Asakura M, Yamamoto T (1975) Growth mechanism of some clay minerals. Fortschr Miner 52:217-224 Sunagawa I, Tomura S (1976) Twinning in phlogopite. Am Mineral 61:939-943 Tairov YuM, Tsvetkov VF (1983) Progress in controlling the growth of polytypic crystals. In Crystal Growth and Characterization of Polytype Structures. P Krishna (ed) Pergamon Press, Oxford / New York / Toronto / Sydney / Paris / Frankfurt, p 111-162 Takano Y, Takano K (1958) Apparent polytypism and Apparent Cleavage of the Micas. J Mineral Soc Jpn 3:674-692 (in Japanese) Takeda H (1967) Determination of the layer stacking sequence of a new complex mica polytype: A 4-layer Lithium Fluorophlogophite. Acta Crystallogr 22:845-853 Takeda H (1969) Existence of complex mica polytype series based on 2M1 sequence. Jpn Crystallogr Assoc Autumn Meeting, Iwate, 2-3 (in Japanese) Takeda H (1971) Distribution of mica polytypes among space gGroups. Am Mineral 56:1042-1056
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Takeda H, Donnay JDH (1965) Compound tessellations in crystal structures. Acta Crystallogr 19:474-476 Takeda H, Haga N, Sadanaga R (1971) Structural investigation of polymorphic transition between 2M2-, 1MLepidolite and 2M1 Muscovite. Mineral J 6:203-215 Takeda H, Ross M (1975) Mica polytypism: dissimilarities in the crystal structures of coexisting 1M and 2M1 biotite. Am Mineral 60:1030-1040 Takeda H, Ross M (1995) Mica polytypism: identification and origin. Am Mineral 80:715-724 Takeda H, Sadanaga R (1969) New unit layers for micas. Mineral J 5:434-449 Takéuchi Y (1965) Structures of brittle micas. Proc. 13th Natl. Conf. Madison, Wisconsin, 1964, Clays Clay minerals. Pergamon Press, 1-25 Takéuchi Y. (1971) Polymorphic or polytypic changes in biotites, pyroxenes, and wollastonites. J Mineral Soc Jpn 10:Spec. Issue No. 2:87-99 (in Japanese) Takéuchi Y, Haga N (1971) Structural Transformation of Trioctahedral Sheet Silicates. Slip mechanism of octahedral sheets and polytypic changes of micas. Mineral Soc Japan Spec Pap 1:74-87 (Proc. IMAIAGOD Meetings '70, IMA Vol.) Takéuchi Y, Sadanaga R (1966) Structural studies of brittle micas. I. The structure of xantophyllite refined. Mineral J 4:424-437 Takéuchi Y, Sadanaga R, Aikawa N (1972) Common lattices and image sets of hexagonal lattices, and their application to composite electron-diffraction patterns of biotite. Z Kristallogr 136:207-225 Thompson JB Jr (1981) Polytypism in complex crystals: contrast between mica and classical polytypes. In Structure and Bonding vol. II. M O'Keefe, A Navrotsky (ed), Academic Press, San Diego / London / Burlington, p 167-196 Tokonami M (1966) The structure determination of the 96R polytype of SiC by a direct method. Mineral J 4:401-423 Tokonami M, Hosoya S (1965) A systematic method for unravelling a periodic vector set. Acta Crystallogr 18:908-916 Tolansky S, Morris PG (1947a) An interferometric survey of the mica. Mineral Mag 28:137-145 Tolansky S, Morris PG (1947b) An interferometric examination of synthetic mica. Mineral Mag 28:146-150 Tomura S, Kitamura M, Sunagawa I (1979) Surface microtopography of metamorphic white micas. Phys Chem Miner 5:65-81 Tschermak G (1878) Die Glimmergruppe (I. Theil) Z Kristallogr 2:14-50 Tsvetkov V F (1982) Problems and prospects of growing large silicon carbide crystals. Izv Leningr Elektrotekh Inst 302:14-19 (in Russian) Udagawa S, Urabe K, Hasu H (1974) The crystal structure of muscvite dehydroxylate. J Japan Assoc Mineral Petr Econ Geol 58:381-389 (in Japanese, with English Abstract) Ungemach H (1935) Sur la Syntaxie et la Polytypie. Z Kristallogr 91:1-22 Verma AR (1953) Crystal Growth and Dislocations. London: Butterworths, 182p Weiss Z, Ďurovič S (1980) OD interpretation of Mg-vermiculite. Symbolism and X-ray identification of its polytypes. Acta Crystallogr A36:633-640 Weiss Z, Ďurovič S (1989) A united classification and X-ray identification of phyllosilicate polytypes. Collected abstracts, 9th International Clay Conference, Strasbourg (France), p. 430 Weiss Z, Wiewióra A (1986) Polytypism of micas. III. X-ray Diffraction Identification. Clays Clay Minerals 34:53-68 Wondratschek H (1976) Extraordinary orbits of space groups. Theoretical considerations. Z Kristallogr 143:460-470 Wondratschek H (2002) Introduction tp space-groups. Sect. 8 in international Tables for Crystallography, Vol. A,5th edition. The Hahn (ed) Dordrecht / Boston: London: Kluwer Academic Publishers (in press) Zhukhlistov AP, Zvyagin BB, Shuriga TN (1983) Electron-diffraction investigation of the crystal structure of di-trioctahedral Li,Fe-phengite 1M. Sov Phys Crystallogr 28:518-521 Zhukhlistov AP, Zvyagin BB, Pavlishin VI (1990) Polytypic 4M modification of Ti-biotite with nonuniform alternation of layers, and its appearance in electron-diffraction patterns from textures. Sov Phys Crystallogr 35:232-236 Zussman J (1979) The crystal chemistry of micas. Bull Mineral 102:5-13 Zvyagin BB (1962) A theory of polymorphism of micas. Sov Phys Crystallogr 6:571-580 Zvyagin BB (1967) Electron diffraction analysis of clay mineral structures. New York: Plenum Press, 364 p Zvyagin BB (1985) Polytypism in contemporary crystallography. Sov Phys Crystallogr 32:394-399 Zvyagin BB (1988) Polytypism of crystal structures. Comput Math Applic 16:569-591 Zvyagin BB (1993) A contribution to polytype systematics. Phase Trans 43:21-25 Zvyagin BB (1997) Modular analysis of crystal structures. In Modular aspects of minerals / EMU Notes in Mineralogy, vol. 1. S Merlino (ed) Eötvös University press, Budapest, p 345-372 Zvyagin BB, Drits VA (1996) Interrelated features of structure and stacking of kaolin mineral layers Clays Clay Miner 44:297-303
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Zvyagin BB, Gorshkov AI (1966) Effects of secondary diffraction in selected area patterns of mineral crystals superimposed with a relative rotation around the primary beam. Sixth Internat. Congress for Electron Microscopy, Kyoto. In Electron Microscopy 1966 Vol. I. R Uyeda (ed) Maruzen, Tokyo, p 603-604 Zvyagin BB, Vrublevskaya ZV, Zhukhlistov AP, Sidorenko OV, Soboleva SV, Fedotov AF (1979) Highvoltage electron diffraction in the study of layered minerals. Moscow: Nauka Press, 224 pp. (in Russian)
5
Investigations of Micas Using Advanced Transmission Electron Microscopy Toshihiro Kogure Department of Earth and Planetary Science Graduate School of Science, The University of Tokyo 7-3-1 Hongo, Bunkyo-ku Tokyo 113-0033 Japan
[email protected] INTRODUCTION After a long history of development and improvement, recent transmission electron microscopes (TEMs) with various analytical functions have become important in material science and engineering. These functions include not only obtaining magnified images of specimens, but also electron-diffraction patterns, chemical analyses, and chemical-state analyses with spacial resolution far greater than other methodologies. It is impossible to cover all of these functions considering page limitations and, more importantly, considering the author’s knowledge and ability even for topics limited to studies of mica. This chapter focuses on the investigations of mica using high-resolution transmission electron microscopy (HRTEM). HRTEM is generally defined as a technique to obtain information about atomic structures in crystals from TEM images formed by phase contrast at high magnifications. Although HRTEM is just one of many functions in TEMs, several examples in sections below demonstrate that HRTEM often plays a decisive role in determining the local atomic arrangements in mica. An early study of mica by HRTEM was reported by Buseck and Iijima (1974). They clearly observed three dark lines representing a mica layer (the lines correspond to the two tetrahedral sheets and one octahedral sheet) and that cleavage was formed at the interlayer. During a quarter century after this pioneering work, many HRTEM studies for mica and related phyllosilicates have been reported (for instance, see the references in Baronnet 1992). These included many studies of mica, e.g., polytypism, transformations, defects and interface research. In the following section, recent HRTEM and related techniques are briefly reviewed. Next, two topics of HRTEM investigation, polytype and defect analyses are presented based on studies, mainly by the author and his colleagues. TEMS AND RELATED TECHNIQUES FOR THE INVESTIGATION OF MICA Transmission electron microscopy After the invention of TEM by E.E. Ruska in 1932, this apparatus was improved rapidly in response to requests from many fields of science. In the 1950s, lattice fringes in crystals were recorded (Menter 1956), which indicated an exciting possibility that a tool was possible to observe atomic arrangements in a crystal directly on a screen. Imaging theory for HRTEM developed in the 1960s showed that contrast in magnified images can be observed, which corresponds to the projection of the electrostatic potential in specimens with a resolution (referred to as “point resolution”: δ) defined by the following equation: δ = 0.66 Cs1/4 λ3/4 1529-6466/02/0046-0005$05.00
DOI:10.2138/rmg.2002.46.05
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where Cs is the spherical-aberration coefficient of the objective lens and λ is the wave length of the electron beam determined by accelerating voltage (Spence 1981). Following this equation, TEMs with an objective lens of low aberration and a small wavelength (a high accelerating voltage) were developed to achieve high resolution. In the 1970s, TEMs with accelerating voltages of ∼1 MV, and the resultant point resolution of 2 χn) where the first element α0 is the atomic absorption coefficient and the second term α1 is always zero because H1m,1m= 0. For the K edge, in the plane-wave approximation, the expression for n = 2 is the usual backscattering amplitude, i.e., the EXAFS signal times the atomic part. Actually, the first multiple-scattering contribution is the α3 term, which can be written (Benfatto et al. 1989) as α3 = α0 Σi≠jIm { P1(cosφ) fi(ω) fj(θ) exp(2i(δ10 + kRtot))/kririjrj } where rij is the distance between atoms i and j, fi(ω) and fj(θ) are the relative scattering amplitudes, which now depend on the angles in the triangle that joins the absorbing atom to the neighboring atoms located at sites ri and rj, and Rtot= ri + rij + rj. In this expression, cosφ = - ri.rj, cosω = –ri.rij and cosθ = ri .r ij. As a consequence, the n = 3 term and all terms with n higher than 2 contain information about the higher order correlation function. It is possible to observe also that, in this framework, because of P1(cosφ) = cosφ, there is a selection rule in the pathways. As an example, consider the α3 term: in all the cases where ri is perpendicular to rj, the corresponding MS term does not contribute to the total cross section because cosφ = 0. Neglecting multi-electron contributions, this description makes clear the distinction between the FMS and IMS regions in a XANES spectrum, and assigns any differences to the local geometrical structure of the system.
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In a practical way, in the analysis of XANES spectra of condensed systems the first step is to identify the size of the relevant cluster of atoms (Garcia et al. 1986; Benfatto et al. 1986), i.e., the cluster of atoms around the central absorbing atom. The size of this cluster may range from the smallest one, including only the nearest neighbors, to clusters including several surrounding shells. Neither translation symmetry nor site symmetry of such a cluster is required, and the finite size of the cluster is determined only by the mean free-path for elastic scattering of the photoelectron and by the core-hole life time. In the energy range 1∼10 eV, where the mean free-path becomes longer than 0.1 nm, the size limitation due to the core-hole lifetime is the most important parameter. Actually, the contribution of further shells can be reduced or cancelled out by different degree of structural disorder. Experimental spectra recording Recent advances in X-ray spectroscopy of minerals are mainly related to the development of synchrotron radiation sources that overcame the limitations in energy range, intensity and stability of radiation that conventional X-ray tubes had. Currently, the availability of third generation electron storage rings and of special sources generated by insertion devices (wigglers and undulators) offers brilliant, tunable, and polarized sources in a wide range of energy, from IR to hard X-rays, and opens up new opportunities to all material sciences.
e-
Slit
Sample
Storage Ring
Incident flux (IO) Ionization Chamber Monochromator crystals
Transmitted flux (I1 ) Ionization Chamber
Figure 3. Schematic representation of a modern experimental setup for X-ray absorption spectroscopy in the transmission mode.
A schematic view of an experimental setup at one modern facility for X-ray absorption spectroscopy studies in the conventional transmission mode is shown in Figure 3. However, experimental setup and detection methods depend on several factors, the most important being the energy range of the X-rays to be used. In turn, this strictly depends upon the absorption edges to be analyzed. In the study of micas, it is opportune to investigate both low Z atoms (i.e., Na, Mg, Al, Si, K, etc.) and high Z atoms (e.g., Ti, V, Cr, Mn, Fe, etc). Such different energy ranges require different types of monochromators: (1) the soft X-ray energy range (4 KeV) requires double reflection Si or Ge crystals, the reflecting crystal plane being properly chosen to the purpose of achieving best resolution and high intensity. Moreover, the soft X-ray range requires special beam lines and experimental chambers and, because of the strong absorption of the radiation at these wavelengths in air, high (HV) or ultra-high vacuum (UHV) conditions are compulsory. The strong photon absorption of gases prevents the use of photo-ionization chambers; thus, in HV or UHV
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conditions, electron detection systems are usually employed. Metal grids can be used to monitor beam intensity, either by means of electron multipliers (channeltron) that collect all electrons extracted by the photon beam, or by direct measurement of the drained photo-electron current (Stöhr et al. 1980). The detection system depends on the concentration of absorbing atoms in the material and photon energy. For bulk experiments using hard X-rays (i.e., with hν > 4 KeV) on samples with concentrations above 10-3 (atomic ratio), standard X-ray transmission techniques are used. The incident and transmitted fluxes are typically measured by photo-ionization chambers. In the soft X-ray range (i.e., with hν < 1000 eV), absorption spectra may be efficiently measured by recording core-hole decay products. If we describe the inner-shell photo-ionization process as a two-step process, then in the first step the photon excites a core-hole electron pair, and in the second step the recombination process of the core-hole takes place. There are many channels suitable for core-hole recombination. These channels may produce the emission of photons, electrons, or ions, all of which are collected by special detectors. The recombination channel that is normally used to record bulk XAS spectra of dilute systems is the direct radiative core-hole decay that produces X-ray fluorescence lines. When fluorescence lines have high photon energies, this technique probes the bulk. In Figure 4 a beam line with an apparatus to record absorption spectra in the fluorescence mode is schematically represented.
Incident flux (IO) Ionization Chamber
Sample
Fluorescence Detector
Figure 4. Schematic representation of an apparatus designed to record X-ray absorption spectra in the fluorescence mode.
In the soft X-ray range, the Auger recombination has a higher probability than the radiative recombination (Stöhr et al. 1984). Because the energy of the Auger electrons is characteristic of a particular atom, the selective photoabsorption cross-section of an atomic species (in particular those chemisorbed on a surface) can be measured by monitoring the intensity of its Auger electrons as a function of photon energy. An intense Auger line is selected by an electron analyzer operating in constant final state (CFS) mode with an energy window of a few eV. A standard experimental setup for this type of XAS measurement is shown in Figure 5 (modified after Stöhr et al. 1984). Note, however, that Auger electrons arise from the uppermost impinged layers of atoms; consequently, this type of measurement is essentially probing the surface of the sample, i.e., it competes with surface EXAFS (i.e., SEXAFS), rather than with bulk XAS. For
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mmm Incident Flux (IO ) Metal Grid
Sample
ee-
Channeltrons
Figure 5. Schematic representation of a standard experimental setup for surface X-ray absorption measurements.
bulk measurements, the total electron yield (TEY), which has been found to be proportional to the absorption coefficient (Gudat and Kunz 1972), is used. This technique measures the integral yield over the entire energy range of the emitted electrons. The advantage of this method is that maximum counting rates are obtained, since all the emitted electrons over a large solid angle can be collected by applying a positive voltage to the detector. Another detection method used is the low-energy partial electron yield (PEY), where only the secondary electrons within a kinetic energy window around the maximum in the inelastic part of the electron energy distribution curve (EDC) are collected. Because of the long escape depth for low-energy electrons, the bulk absorption recording with this method makes use of an electron analyzer. High resolution, on the order of 0.15-0.2 eV (i.e., a resolving power in the range 104), is experimentally demanding in XANES spectroscopy because important physical information can be extracted from small variations in the intensity and/or energy shift of an absorption peak. For this reason, careful preparation of homogeneous pinhole-free samples and suppression of high harmonics in the incident photon beam are required. Using crystal monochromators, the energy band width ΔE of the photon beam monochromatized by Bragg diffraction is determined by the angular divergence ΔΘ and by the crystal rocking curve. In synchrotron radiation beam lines, the angular divergence depends upon the intrinsic vertical spread of the radiation, which is determined by both the energy of the electron beam circulating in the storage ring and the source size, i.e., the diameter of the electron beam and its divergence at the emission point as determined by the electron optics. Resolution can be improved by changing either the crystal or the reflection plane. In a double-crystal monochromator, two parallel reflections produce a monochromatised photon beam parallel to the incident one. These two reflections reduce the tails of the rocking curve, and consequently increase the resolution, but they leave the higher-order harmonic reflection content like that of a single reflection (Greaves et al. 1983). Less common are other types of high-resolution crystal monochromators with special geometries that make use of antiparallel reflections. Finally, high-resolution XANES spectra may be measured using higher-order reflections. Harmonic rejection may be achieved in devices with two crystals by detuning one crystal with respect to the other. In fact, when the two crystals are misaligned, the intensity of the harmonics drops off much more rapidly than the intensity of the fundamental, because bandwidth ωn(λ) is
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much narrower for n > 1 harmonics than for the fundamental one. The higher-order harmonic content in the synchrotron radiation beam is due to the intense continuum of the primary beam extending towards high energies, and it represents a significant contribution in all the high-energy third-generation synchrotron sources. Actually, rejection of higher-order harmonics may be obtained using either mirrors behaving as low-band pass filters, and/or by detuning crystals, or even by means of undulator sources. Optimization of spectra Orientation effect. Most experimental XANES spectra on micas were measured on powders, obtained by grinding hand-picked grains that had been gently settled on a flat sample-holder after dispersion in a liquid. The resulting mounts were considered to be randomly oriented, regardless of their grain-size homogeneity and distribution. However, experience gathered on other sheet-silicates (e.g., Manceau 1990; Manceau et al. 1988, 1990, 1998) has shown that, even in fine-grained powders, crystallite orientation strongly affects the shape of the final spectrum: primarily, it changes peak intensity, which is a significant component of the information and certainly reflects onto its quality (see above). If this is indeed the case, then among the mica XANES spectra performed so far (Table 2) only a few can be considered to be reliable. These include work by Osuka et al. (1988, 1990), Mottana et al. (1997) and Sakane et al. (1997), in which no special care was taken, but the investigated micas, being synthetic, were so homogeneously finegrained (1 μm) as to certainly lie on the sample-holder with their c axis more or less orthogonal to its surface and with their a and b axes oriented at random on it. A theoretical study of the orientation effect has been recently presented for selfsupporting clay-mineral thin films by Manceau et al. (1998), who also propose a tridimensional system of coordinates to record spectra in a standard setting. Their method, slightly modified by Cibin et al. (2001), has been adopted by Mottana et al. (in preparation) for single crystal mica blades (Fig. 6). Another approach used by Dyar et al. (in prep.) uses mica single crystals mounted on fibers in goniometer heads, which are then fitted onto a spindle stage mounted with the plane of rotation perpendicular to the path of the beam.
Sample surface
Figure 6. The coordinate system applicable to angular measurements on self-supporting phyllosilicate films as used for micas (Cibin et al. 2001; cf. Manceau et al. 1998, Fig. 2). Z-Y is the plane onto which the sample lies, with the X-ray beam impinging along X and linearly polarized on X-Y; α is the incidence (rotation) angle between the electric field vector ε and Y.
z
y hν
α x
ε
α
ε
For a perfectly random distribution of very small crystals (powder) there would be no angular variation effect on the experimental XAS spectra; however, for a fully oriented crystal structure such as that of a mica blade lying flat on the sample-holder, the amplitude of the scattered photoelectron wave depends on the angle α between the
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electric field vector ε of the impinging beam and the layers in the structure. This angle can be determined either by rotating the sample-holder on its vertical axis, or by preparing suitably oriented thin sections to be glued on the sample-holder in its routine setting orthogonal to the X-ray beam (α = 0°). Mottana et al. (in preparation) operated at SSRL at the 3-3 beamline (Hussain et al. 1982; Cerino et al. 1984), which is equipped with a double-crystal monochromator made Table 2. Published XAS data on mica species materials.
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of efficient crystals such as YB66 (Wong et al. 1990 1999). They scanned single-crystal mica blades lying flat on the vertical sample-holder and optically-oriented in such a way as to have a ≅ b // Z. Here Z is an axis lying parallel to the mica surface (Fig. 6). The synchrotron beam first impinges the mica at right angle (α = 0°); then the blade is rotated and α increased up to 60∼80°, this being the maximum angle allowed by the mechanics of the sample compartment and the geometry of detection, which uses channeltrons. Therefore, the electric vector ε always lies on the horizontal plane, but it impinges two almost perpendicular sections of the mica structure so as to scan its atoms under different angles, with their atomic bonds and angles geometrically modified. The orientation effects observed in this way are clearly visible in a natural muscovite compositionally close to the end member (Fig. 7). It is quite clear that orientation dramatically affects the intensity of all peaks, including the white-line, but also— although to a much lesser extent—the positions of some of them, by as much a 5 eV. A comparison between Figure 7 and the Al K-edge spectrum reported by Mottana et al. (1997; cf. Fig. 4) for synthetic muscovite, which is expected to be randomly oriented owing to its very fine-grained powdery nature, shows that best agreement is attained for a rotation angle α in between 45 and 70°.
E (eV)
Figure 7. Changes in an Antarctica muscovite Al K-edge spectrum due to changing the orientation of the same crystal blade against the impinging, horizontally-polarized synchrotron radiation beam. In the right panel a magnified view of the white-line intensity as a function of the α rotation angle.
A similar comparison between the Fe K-edge spectra of a phlogopite single crystal rotated in the same way (Fig. 8) and the several Fe XANES spectra of phlogopites in the literature (Table 1) confirms that best agreement is obtained when the crystal is rotated at α ca. 45°. Indeed, later work (unpublished) showed that best agreement for the same sample, when scanned as both single crystal (at various angles) and as a settled homogeneous powder having a grain size of ca. 5 μm, is obtained when α is equal or very close to the “magic angle” value 54.7° (Pettifer et al. 1990). Changes with orientation are also clearly evident in the XANES spectra of a number of di- and tri-octahedral micas and one brittle mica, respectively at the Mg (phlogopite: Fig. 9), Si (muscovite: Fig. 10, and tetra-ferriphlogopite: Fig. 11), K (muscovite: Fig. 12), and Fe (clintonite: Fig. 13, and tetra-ferriphlogopite: Fig. 14) K edges. Such changes
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imply
Figure 8. Changes in a Franklin phlogopite Fe K-edge spectrum due to changing the orientation of the same crystal blade against the impinging, horizontally-polarized synchrotron radiation beam. In the right panel a magnified view of the changes undergone as a function of the α rotation angle by the edge and FMS regions (top) and by the IMS region (bottom). Philgopite Mg K edge
Figure 9. Changes in the FMS region of a Franklin phlogopite Mg K-edge spectrum due to changing the orientation of the same crystal blade against the impinging, horizontally-polarized synchrotron radiation beam.
Absorption (Arb. Un.)
0º 30º 45º 70º
1300
1310
1320
1330
1340
Energy (eV) E (eV)
imply displacements in the peak positions from 0 up to 5 eV, and variation in the intensities by as much as 50%, with even reversals in the intensity of the edge top (Fig. 9) or appearance viz. disappearance (Fig. 11) of certain features. Most commonly, these changes occur gradually and trend always in the same direction, thus demonstrating their dependence upon the gradual rotation applied to the crystal. In turn, this rotation mostly reflects changes in the lengths of the bonds lying in the polarization plane, excited in the photoabsorption process, or in the lengths of multiple-scattering paths which are also probed in that geometry. Such spectral changes affect both the FMS and IMS regions, thus showing their dependence mostly upon the geometry of the section of the crystal that is being scanned by the synchrotron beam, as cosα. However, unexpected changes such as the one at the white-line in the phlogopite Mg K-edge spectrum (Fig. 9), or the sudden
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Figure 10. Changes in an Antarctica muscovite Si K-edge spectrum due to changing the orientation of the same crystal blade against the impinging, horizontally-polarized synchrotron radiation beam. In the right panel a magnified view of the changes undergone by the edge and FMS regions.
Figure 11. Changes in a Tapira tetra-ferriphlogopite Si K-edge spectrum due to changing the orientation of the same crystal blade against the impinging, horizontally-polarized synchrotron radiation beam. In the right panel a magnified view of the changes undergone by the edge and FMS regions.
appearance of a new low-energy peak, as in the muscovite Al and Si K-edge spectra (Figs. 7 and 10) and in the tetra-ferriphlogopite Si K-edge spectrum (Fig. 11), demonstrate the possibility that the electronic properties of the absorbing atom are also involved. We have to underline here that this interpretation of the near-edge structure is fully equivalent to the interpretation that is based on local geometrical distributions, such
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lk
Figure 12. Changes in an Antarctica muscovite K K-edge spectrum due to changing the orientation of the same crystal blade against the impinging, horizontally-polarized synchrotron radiation beam. In the right panel a magnified view of the changes undergone by the edge and FMS regions.
Figure 13. Changes in a Lago della Vacca clintonite Fe K-edge spectrum due to changing the orientation of the same crystal blade against the impinging, horizontally-polarized synchrotron radiation beam. In the right panel a magnified view of the changes undergone by the edge (top) and IMS regions (bottom).
as those expected when the different local atomic distributions in the micas are being compared. To summarize, in order to obtain XANES spectra that may be meaningfully compared, we recommend orienting the sample, when a single crystal, always at the same angle of rotation α = 54.7°. This is essentially the same conclusion reached by Manceau et al. (1998) for the self-supporting clay films they experimentally investigated by
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fffffffff
Figure 14. Changes in a Tapira tetra-ferriphlogopite Fe K-edge spectrum due to changing the orientation of the same crystal blade against the impinging, horizontally-polarized synchrotron radiation beam. In the right panel a magnified view of the changes undergone by the edge (top) and FMS and IMS regions (bottom).
polarized EXAFS and theoretically interpreted by performing full multiple-scattering calculations. Furthermore, we also recommend recording a full XAS spectrum of the same sample, after grinding it and settling in water for precisely determined times so as to obtain a well-classified powder possibly in the grain size range 1 to 2 μm. Dyar et al. (2000) used a different method of studying the orientation effects on the pre-edge region of Fe-bearing micas, with similar results. In that study, the microXANES probe at the National Synchrotron Light Source (NSLS), Brookhaven, NY, was used, allowing a beam size of 10 × 15 μm. Because the beam is smaller, samples on the order of 30 × 30 × 100 μm (orders of magnitude smaller than those used by other workers) could be studied, and concerns about sample homogeneity lessened. Each crystal was oriented with its cleavage perpendicular to a glass thin section, and then UVhardening epoxy was used to maintain it in that geometry. The mica+epoxy was removed from the thin section, and two mutually parallel faces were polished on each sample perpendicular to cleavage (though in an unknown orientation relative to the a and b axes: see Fig. 15). This preparation permitted acquisition of spectra in two important directions
Figure 15. Optical orientation of a model mica crystal showing the random position of the thin section cut across cleavage and used for microXANES measurements (Dyar et al. 2001).
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perpendicular and parallel to cleavage by simple rotation of the sample. A further advantage of this method is that parallel studies of the optical and IR absorption spectra of the identical crystals could be made. In more recent work (Dyar et al. in prep.) single crystals were analyzed while mounted on goniometer heads, so the beam could be polarized along the X, Y, and Z optical orientations. As with the work of Mottana et al. (in preparation), changes in peak intensity and, to a lesser extent, energy, were observed by Dyar et al. (2001) as a function of sample orientation. At the main edge, the difference in the intensity of the highest energy peak relative to the other prominent peak or peaks is generally greatest when the synchrotron beam is polarized in the direction of the cleavage plane, with a few exceptions. In the preedge region, intensity variations were also observed, but the maxima and minima were not necessarily parallel or perpendicular to cleavage, and the orientation at which maximum intensity occurred was different for various samples. This implies that there are variations in peak intensity not only perpendicular and parallel to the mica cleavages, but also within the sheets themselves as a function of orientation with respect to the unconstrained position in the XY plane. Such a conclusion is not surprising in a monoclinic mineral species: the XANES probe is sampling different bonds at different orientations relative to noncentrsymmetric Fe sites (Dyar et al. 2001; in prep.). Spectrum fitting. In standard XAS experiments, signal to noise (S/N) ratios in the range 103∼104 can be achieved. However, to fully enhance XANES potentials, these are not enough, especially in the soft-X-ray energy range where such ratios are only achieved after a perfect preparation of the sample. Consequently, with a lower S/N ratio, the best understanding of XANES critically depends upon a careful fitting of the experimental spectrum during which no fine details get lost. The standard procedure in XAS spectrum analysis follows two steps: the experimental spectrum is (1) corrected for background contributions from lower energy absorption edges by linear or polynomial fitting of the base line, then (2) normalized at high energy, i.e., close to the upper end of the XANES region at an energy position where no obvious features can be seen. In addition, for pre-edge analysis the contribution of the absorption jump is subtracted by an arctangent function. This procedure leaves a profile of the entire K-edge region that consists of a number of features, occasionally partially superimposed, that can be either evaluated visually or fitted by Gaussian or Lorentzian curves. The numerical values of the fitted curves (energy and intensity, with errors and significance bars) can then be used as solid data for interpretation. This standard procedure assures accuracy in energy position ±0.1 eV for the pre-edge, and ±0.03 eV for all other regions of the XANES spectrum. Both values are well within resolution, which increases with energy from ca. 0.3 to ca. 1.5 eV on going from the Na K-edge to the Fe one (Schaefers et al. 1992). Accuracy in the intensity measurements is estimated to be better than 10%. However, such intense structures as the "white line" are affected mainly by the harmonics content. At all synchrotron sources, a step preliminary to all this standard procedure consists of calibrating the energy positions of all peaks against standards (usually metal foils). An alternative way is to calibrate them against a “glitch”, i.e., a spurious absorption at constant energy in the spectra that is due to a planar defect present in the monochromator crystal (cf. Wong et al. 1999, for YB66). When high thermal loads heat the monochromator crystals, a further systematic correction is applied that takes into account the decrease of the ring current (and heat load) with time. As a matter of fact, in most mica studies a careful fitting procedure is seldom applied, and the “fingerprinting” method of evaluation is still predominant (Table 1). A
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recent improvement in the fitting procedure is based upon a novel software (Benfatto et al. 2001). However minor the error induced by such evaluation may be, any concomitant carelessness in taking into account orientation effects would, at the end, result in crowding the literature with spectra useless for interlaboratory comparisons. Systematics XAS studies on micas: a catalogue. Table 1 lists all XAS studies carried out on micas that could be retrieved in the relevant literature. They are presented in the alphabetical order of the di- and tri-octahedral mica species nomenclature approved recently (Rieder et al. 1998) and are further subdivided on the basis of the investigated atom. Almost all investigated samples are natural and are therefore intermediate in composition. However, some of them are close enough to end member compositions as to make it possible to classify them accordingly. Only seven true end members corresponding to natural mica species have been studied so far by XAS, i.e., the Tapira tetra-ferriphlogopite (Giuli et al. 2001) and the six synthetic micas investigated by Mottana et al. (1997). Even all other synthetic micas (Osuka et al. 1988 1990; Sakane et al. 1997) are intermediate, as they are doped crystals obtained for technological purposes. Furthermore, among the synthetic micas quite a few have no natural counterpart (Soma et al. 1990; Han et al. 2001). XAS studies on otherwise insufficiently characterized samples, or on samples with composition being complex solid solutions from the crystal-chemical viewpoint, are listed at the bottom of Table 1, in the section that accounts for the approved series names (cf. Rieder et al. 1998 Table 4). The first XAS spectra ever recorded on micas were those by Brytov et al. (1979) at the Si and Al K edges. However, as all these spectra were recorded in the late 1970s and early 1980s using a conventional X-ray tube as the source, they are practically useless for present-days studies because of the limited resolution: in practice, only the general shape is worth examining (e.g., Jain et al. 1980 Fig. 1). Nevertheless, these early attempts deserve to be remembered, for both the pioneering effort they record and their historical significance. The earliest synchrotron-activated experimental XAS spectrum for any mica was Calas et al.’s (1984) chromium muscovite at the Cr K edge. Although noisy, particularly in the pre-edge region, this spectrum satisfactorily compares with the recent spectrum of a similar mica at the same edge (Brigatti et al. 2001; see below), thus suggesting not only the high level of technical skill of the operators, but also that comparison of power spectra collected at very different times and on widely different synchrotron storage rings can be confidently made, provided the basic requirements of energy calibration and background subtraction were carefully applied (see above). Occasionally, mica has been used also to support epitaxially-grown layers that have been investigated by XAS (e.g., Blum et al. 1986; Drozdov et al. 1997). Although reported in Table 1, these XAS studies actually do not belong to mica studies. Finally, there has never been a spectrum published so far but those presented above to which the above-given precautions on orientation effects were applied (see also Dyar et al. 2001 and in prep.). Even the spectra that will be described in the following were obtained on ground powders, presumed to be homogeneous in their grain-size and randomly oriented, but never tested for those conditions. Determination of the oxidation state. Determining the effective charge on the absorbing atom from the chemical shift of the X-ray absorption threshold is a fundamental issue for XANES. However, a direct measure of the "ionization threshold" or "continuum threshold" (i.e., the energy at which the electron is excited in the
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continuum: e.g., the Fermi level in metals) is not possible because of the lack of any signature of it. Therefore, XANES is not a direct probe of core-level binding energy as other methods are (e.g., XPS or ESCA). However, there is evidence in both gas molecules and solid compounds that the energy shift of the first bound excited state at the absorption threshold follows the binding energy shift of the core level. Moreover, a linear dependence between core-level binding energy and atomic effective charge has been measured (Belli et al. 1980). By contrast, no linear relationship between the measured shift of the first strong multiple-scattering resonance and the effective atomic charge on the ion exists. The energy of multiple-scattering resonances is strongly dependent on interatomic distance, so their chemical shifts are much larger than that of the core excitation. Actually, the variation of the effective charge on an atom is often increased and a linear correlation with core-level binding energy indeed exists; however, this effect is always system-dependent. Moreover, within the same structure any correlation among the parameters of the potential is certainly confined only to small changes of the interatomic distances (e.g., less than 10%). Correct identification of the oxidation state of 3d transition metals is indeed important, but the quantification of the oxidation ratio is even more important in the case of potentially multivalent minerals such as the micas, a group where the number of elements occurring with more than one oxidation state is significant (Fe, Mn, Cr, V and possibly Ti: cf. Table 1) and their amounts may be so large as to even become essential and determine new end members. All transition element K-edge spectra display a preedge (Belli et al. 1980) and, mostly, all features of the pre-edge are strong enough to be easily recorded experimentally. Position and intensity of the peaks occurring in the preedge region can be reliably used to determine the oxidation state(s) of the absorbing atom (e.g., Waychunas 1987). However, as already seen (Fig. 2, above), coordination too plays a role, so that care must be made in discriminating the two effects, and to this purpose spectra need to be properly deconvoluted. As discussed above, the energy position of the peaks in the pre-edge region may be directly related to the increase in the oxidation state of the absorber atom: e.g., the preedge feature of Fe3+ is generally ca. 2∼3 eV higher in energy than the corresponding feature for Fe2+ (Waychunas et al. 1983; cf. Petit et al. 2001). The amount of such a “chemical shift” is different for the different transition elements, and depends on the final state reached by the electron. Implicitly, this weakens the possibility of reliably determining the oxidation state of a given atom when it occurs in different coordination sites of the same compound. However, when a significant part of the atom occurs in a tetrahedrally-coordinated site, the relevant pre-edge is strongly intensified owing to d-p mixing, and the determination of the oxidation state of the tetrahedral atom is made fairly easy to measure: e.g., amounts of Cr3+ in tetrahedral coordination as small as 0.5% could be detected even in the presence of a significant amount of Cr3+ in octahedral coordination (Brigatti et al. 2001; see below). Consequently, subtraction of the tetrahedrally-coordinated component can be made. The residual pre-edge spectrum of the octahedrally-coordinated atoms is then de-convoluted into its components to determine their oxidation state(s). Bajt et al. (1994 1995) and Sutton et al. (1995) have pushed the practice of pre-edge examination further to reach an effective quantification of the oxidation states for Fe, the atom which most frequently occurs in two oxidation states in the same site of minerals. They have developed, and Galoisy et al. (2001) and Petit et al. (2001) have recently improved upon, a procedure that makes use of the known positions of pre-edge peaks of Fe K-XANES spectra in mineral standards to fit a calibration line giving the Fe3+/ΣFe ratios of various minerals (Fig. 16).
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Figure 16. Plot of “pre-edge peak energy” vs. “Fe3+/ΣFe” for well characterized standards. The trend is linear with a correlation coefficient of 0.99 (after Sutton et al. 1995, p. 1465, Fig. 3).
However, extensive additional work by Dyar et al. (2001) on suites of Fe3+ and Fe2+ end members confirms that the energies of the end-member pre-edges vary considerably for several different mineral groups, and thus no single mineral species can be used to model all cases of any type of Fe (Fig. 17). Because different mineral groups have variably distorted coordination polyhedra, use of mineral group-specific standard end members will ultimately be necessary to interpret pre-edge positions assigned to different transitions. Examples of using this method to determine of the Fe3+/ΣFe ratios of a number of rock-forming micas are given elsewhere (Dyar et al. 2001). Determination of local coordination geometry. The position and intensity of the peaks in the pre-edge region do not solely depend upon the oxidation state of the absorber transition metal, but also upon the shape of the site (coordination polyhedron) where the absorber is located in the structure (Calas and Petiau 1983). An increase in coordination number provokes a positive energy shift, while the intensity of the peak is proportionally reduced (Waychunas et al. 1983). The first attempt at using the pre-edge features to determine quantitatively site geometry is Waychunas’ (1987) for the Ti K-edge of a suite of silicate and oxide minerals, including a biotite from Antarctica. He fitted Gaussian features to the entire edge region, and found that individual features are insensitive to changes in the Ti-O bond length, but sensitive to valence, with Ti3+ at ca. 2.0 eV lower energy than Ti4+. Moreover, the intensity of the second pre-edge feature at ca. 4969 eV turned out to be sensitive to both octahedral site distortion and to presence of tetrahedral Ti4+. A correlation was found for silicates between intensity and bond-angle variance σ2 in the octahedral Ti site, and for biotite σ2 could be quantified to be ca. 30 deg2, in fair agreement with the value computed from the X-ray diffraction crystal structure determination (Ohta et al. 1982). Cruciani et al. (1995) essentially followed the same
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gggg
Figure 17. Variation of the absolute pre-peak energy vs. Fe3+ content in the endmembers of several mineral groups; after Delaney et al. (in preparation).
SrCrO4
Absorption (arb.units)
Anatoki river
Westland Uvarovite
5990
6000
6010
6020
6030
6040
6050
Figure 18. Experimental Cr K-edge spectra for the Anatoki River and Westland E (eV) chromium muscovites, a synthetic SrCrO4 standard for tetrahedral Cr6+ (top) and an Outukumpu uvarovite standard for octahedral Cr3+ (bottom). See text for discussion (Brigatti et al. 2001, Fig. 6).
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Figure 19. Pre-edge fit of the Westland chromium muscovite Cr K-edge spectrum of Figure 18 and (inset) its de-convolution in two Gaussian components (Brigatti et al. 2001 Fig. 7).
procedure when trying to determine the [4]Fe3+ contents of a series of natural phlogopites, but came to a purely speculative result owing to the insufficient resolution of the monochromator crystal and the extremely low amount of sample available. As an example of successful evaluation, we report the case of two chromium muscovites worked out by Brigatti et al. (2001) at the Cr K pre-edge; the procedure they followed is the one developed by Peterson et al. (1997) for oxides. The Anatoki River and Westland chromium muscovites Cr K-edge spectra were compared with a synthetic SrCrO4 standard, for tetrahedral Cr6+, and a natural uvarovite, for octahedral Cr3+ (Fig. 18). The Anatoki River muscovite Cr K-edge spectrum proved to be too noisy for further evaluation, but the Westland one, after subtraction of the edge contribution by a pseudoVoigt function, had its pre-edge resolved in two Gaussian components: at 5991.3 eV and 5994.0 eV, respectively (Fig. 19). The second Gaussian component appears in the experimental spectrum only as a skew tail at the end of the pre-edge, owing to interference with the rapidly rising slope leading to the edge. However, after subtracting this interference, it can be reliably measured for both energy and intensity. The evaluation step that follows involves interpretation. If the second-component intensity is assumed to be the same as that of the single, symmetrical Gaussian pre-edge feature of a SrCrO4 standard in which the Cr6+ is entirely in tetrahedral coordination, then it can be appraised that amount of [4]Cr in muscovite, if any, cannot exceed 0.4-0.5% of total Cr (cf. Lee et al. 1995). By contrast, if both Gaussian components are considered to be due to [6]Cr3+, as in the uvarovite standard, and interpreted as a way to measure the distortions of the muscovite octahedral sites where Cr3+ is possibly hosted, then their relative
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intensities (1.3 and 1.2% nau [= normalized absorption units]) show that these two sites are very similar. Indeed, this is nothing more than an extension to Cr of the method for quantitatively determination of site distortion for octahedra centered by Ti4+ calibrated by Waychunas (1987). In the case of the already-mentioned Fe K pre-edge of tetra-ferriphlogopite, where Fe3+ is entirely in the tetrahedral site, the pre-edge is twice as strong and shifted to higher energy (ca. 2 eV) relative to annite, where Fe is mostly in the octahedral site (Fig. 2). This apparent irregularity can be explained by comparing the sharp single peak of tetraferriphlogopite, a synthetic endmember, and the broad, probably double peak of the Pikes Peak annite, the Fe of which is entirely octahedral, but partly Fe2+ and partly Fe3+. Clearly, the oxidation effect is more important than the coordination effect in determining the position of the Fe K pre-edge. However, the strong intensity of the tetraferriphlogopite peak also suggests that its Fe is constrained in a more tightly-bound coordination polyhedron than the annite one. Note, however, that there is an underlying problem in the pre-edge region that needs a more careful evaluation, and not only in these systems: this problem is the amount of quadrupolar effects present (see Giuli et al. 2001, for additional evaluation). A1 K edge
Absorption (Arb. Un.)
Grossularia
Polilithionite
Phlogopite
Albite
1560
1565
1570
1575 1580 E (eV)
1585
1590
1595
Figure 20. Shift of the white-line in the FMS region of the Al K-edge spectra of two synthetic micas as a result of two different coordination geometries: in phlogopite the Al atoms are entirely in a tetrahedral site geometry, and in polylithionite in an octahedral site geometry, as they are in the reference albite and grossular natural standards, respectively (Mottana et al. 1997, Fig. 3).
Coordination geometry also plays a role in shaping the FMS region of a XANES spectrum. This effect was clearly documented for the Al K edges of certain synthetic micas by Mottana et al. (1997), who showed that there is a shift of at least 2 eV between [4] Al as in phlogopite and albite, and [6]Al as in polylithionite and grossular (Fig. 20). Moreover, they found that it is possible, although difficult, to recognize the concomitant presence in the spectra of two white-line features arising from contributions of the same atom occurring in two different geometries ([4]Al and [6]Al in zinnwaldite and
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preiswerkite: Mottana et al. 1997 Fig. 4). Thus, the FMS region of the XANES spectrum of a mineral with Al in two coordinations can be seen as the weighted combination of the contributions arising from the two Al atoms, although the general appearance of the spectrum (and its ensuing evaluation) is somewhat blurred by next nearest neighbor effects due to the presence of other atoms in the same sites substituting for the absorber Al (cf. the muscovite vs. bityite spectra: Mottana et al. 1997 Fig. 4). In the following we will document visually and sparingly comment upon a series of XANES spectra obtained at different K edges for the powders of a number of natural micas close to the end members. The present state of our investigation, which is still under way, compels us to defer to a later moment for drawing conclusions (Mottana et al., in preparation): micas are no simple systems, and XAS literature is already cluttered by faulty reasoning and wrong conclusions reached when hastily evaluating even simpler systems!
Mg K edge
Absorption (Arb. Un.)
Phlogopite
Tetra-ferriphlogopite Biotite Clintonite
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Figure 21. Experimental Mg K-edge spectra for the powders of four natural tri-octahedral micas.
Figure 21 shows the experimental Mg K-edge spectra of three tri-octahedral micas (phlogopite, tetra-ferriphlogopite, and biotite) and one brittle mica (clintonite). All spectra are very similar and have no pre-edges, as magnesium is not a transition element. The FMS regions consist of three features, like the K edge of talc (Wong et al. 1995). However, the relative intensities of the three features differ significantly among the four spectra suggesting that there are substantial differences in the local order of their Mg that may be resolved via comparison with spectra taken for other absorbers. Note, moreover, that the three features in the clintonite spectrum are possibly doubled. Figure 22 shows the experimental Al K edge spectra of three tri-octahedral micas (phlogopite, annite, biotite) and one di-octahedral mica (muscovite). Again, Al is not a transition element, therefore the spectra have no distinct pre-edges. The FMS regions are apparently simpler than the ones occurring in the Mg K-edge spectra above, but in fact
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they Al K edge
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Figure 22. Experimental Al K-edge spectra for the powders of three natural tri-octahedral and one natural di-octahedral mica.
they contain the same three features, although with strongly different intensities and energies (cf. Mottana et al. 1999). Possibly, the fact that non-precisely oriented powders were used affects the recorded features (cp. this muscovite spectrum with that in Fig. 7). The IMS regions are poor in features, but they display shifts and relative differences that are enormous, considering the similarity of the local structures that originate such differences. The significant role of the outer shells around the Al absorber appears to be well depicted here, but it will create great problems when interpreting the spectra from a quantitative viewpoint. Figure 23 shows the experimental Si K-edge spectra of five micas: four tri- and one di-octahedral one. Nowhere is there a pre-edge, and the entire XANES spectrum is dominated by the strong white-line of Si in tetrahedral coordination (cf. Li et al. 1994; Li et al. 1995a). The regions in between FMS and IMS (inset) undergo subtle but significant variations as a result of changes in the local and medium-range ordering occurring in the relevant structures for the volumes that surround the Si tetrahedra. Such variations may also occur in the energies of certain peaks, but this variation is also certainly due to the tri- vs. di-octahedral structure of the investigated mica (inset: cf. muscovite with the other micas). The experimental K K-edge spectra of the same five micas are shown in Figure 24. These XANES spectra are rather complex, both to record experimentally and to reckon with. The FMS regions have no strong white-lines, and only small differences show up in the intensities of their IMS regions (inset). However, their analysis suggests that the K coordination number is less than the expected 12, possibly 8 or even 6. In a case like this, only XANES simulations by the multiple-scattering code may be able to reveal safely the actual site geometry around the potassium atom. Finally, Figure 25 shows the experimental Fe K-edge spectra of two trioctahedral
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Figure 23. Experimental Si K-edge spectra for the powders of five natural micas. The strong differences displayed by a portion of their FMS and IMS regions is shown as inset.
micas (biotite and tetra-ferriphlogopite) and one brittle mica (clintonite). Iron is a transition element, therefore all spectra exhibit significant pre-edges (inset), each one of them having properties of its own. In particular, the tetra-ferriphlogopite pre-edge is a singlet (cf. Fig. 2), as is the clintonite one, but at 1 eV lower energy. Fe is tetrahedrallycoordinated in both micas, but in the former one it is Fe3+ and in the latter one an additional contribution arising from Fe2+ is likely. The biotite pre-edge is weak, because it mostly arises from octahedral Fe2+. The three pre-edges require a deconvolution of the same sort as the one previously demonstrated for the Cr pre-edge of muscovite (Fig. 19) in order to reveal all the information they contain. The FMS regions of these spectra are dominated by the Fe white-line, which undergoes energy variations accounting for differences in both coordination and oxidation state. The presence of significant variations in the medium- to long-range ordering occurring in these mica structures is made evident by their greatly different IMS regions (and also by their EXAFS regions: cf. Giuli et al. 2001).
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Figure 24. Experimental K K-edge spectra for the powders of five natural micas. The strong differences displayed by a portion of their FMS and IMS regions are shown as an inset.
ACKNOWLEDGMENTS Our XAS work on minerals has enjoyed the support of numerous suggestions, discussions and contributions in many stages and levels over a number of years, the five more recent ones dedicated mostly to the micas. We thank all these colleagues, since it is by this form of synergy that we could carry out and develop our project over the years. A special thank goes to Maria Franca Brigatti, Jesús Chaboy, Paola De Cecco, Giancarlo Della Ventura, Gabriele Giuli, Antonio Grilli, Cristina Lugli, Jeff Moore, Takatoshi Murata, Eleonora Paris, Marco Poppi, Agostino Raco, Jean-Louis Robert, Claudia Romano, Michael Rowen, Francesca Tombolini, Hal Tompkins, Curtis Troxel, Joe Wong, Ziyu Wu and all others who allowed us to use for this review some of the data recorded together during painstaking sessions at the source. Most experimental XAS was carried out at SSRL, which is operated by Stanford University on behalf of D.O.E. Furthermore, M.D.D. acknowledges the insight and assistance of her collaborators at the N.S.L.S., Brookhaven National Laboratory: Jeremy Delaney, Tony Lanzirotti and Steve Sutton. Financial supports for our experimental work and for its evaluation and interpretation were granted by M.U.R.S.T. (Project COFIN 1999 “Phyllosilicates: crystalchemical, structural and petrologic aspects”), C.N.R. (Project 99.00688.CT05 “Igneous and metamorphic micas”), and I.N.F.N. (Project “DAΦNE-Light”) in Italy, and by N.S.F.
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EAR-9909587 and EAR-9806182, and D.O.E.-Geosciences DE-FG02.92ER14244 in U.S.A. Critical readings by C.R. Natoli and a unknown referee improved the quality of this paper in a substantial manner.
Figure 25. Experimental Fe K-edge spectra for the powders of two natural trioctahedral micas and one natural brittle mica. The pre-edge regions are shown as inset.
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