Structure of Solids

1 Elements of Symmetry in Periodic Lattices, Quasicrystals Walter Steurer

Institut fur Kristallographie und Mineralogie, Universitat Miinchen, Miinchen, Federal Republic of Germany

List of 1.1 1.2 1.2.1 1.2.2 1.2.3 1.2.4 1.2.5 1.2.6 1.3 1.3.1 1.3.2 1.3.3 1.3.4 1.3.5 1.3.6 1.4 1.4.1 1.4.2 1.4.3 1.5 1.5.1 1.5.2 1.5.3 1.5.4 1.5.5 1.5.6

Symbols and Abbreviations Introduction Symmetry of Crystals Morphology Crystallographic Axes Crystal Faces - Miller Indices Zones and Forms Symmetry Elements External Form and Internal Structure Crystal-Lattice Symmetry Crystal Patterns, Vector and Point Lattices The 7 Crystal Systems The 14 Bravais Lattices The Reciprocal Lattice Topological Properties of Lattices Lattice Transformations: Axes, Indices and Coordinates Crystallographic Point-Group Symmetry Group-Theoretical Terminology Symmetry Operations The 32 Crystallographic Point Groups Crystallographic Space-Group Symmetry Symmetry Operations The 230 Space Groups Wyckoff Positions and Site Symmetries Crystallographic Orbits and Lattice Complexes Subgroups and Supergroups of Space Groups Representation of Space Group Symmetry in the International Tables for Crystallography 1.6 Quasicrystals 1.6.1 Morphology 1.6.2 Quasiperiodic Tilings 1.6.2.1 Fibonacci Chain 1.6.2.2 Penrose Tilings 1.6.3 Decoration of Tilings 1.6.4 Non-Crystallographic Point-Group Symmetry Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. Allrightsreserved.

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1.6.5 1.6.5.1 1.6.5.2 1.7 1.8

1 Elements of Symmetry in Periodic Lattices, Quasicrystals

JV-Dimensional Crystallography Symmetry Operations Space Groups for Some Icosahedral Structures Acknowledgements References

55 55 58 59 59

List of Symbols and Abbreviations

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List of Symbols and Abbreviations a, b, c a, b, c, n, d a, m, o9 t, h9 c a, b, c or at a, b9 c or af an Z) z q> (N) X i/^x y z

interaxial angles, cell parameters, i = 1, 2, 3 matrix representation of the group element gk Kronecker symbol electron density distribution golden mean angles characterizing the unit face (111) Euler numbers character of a representation angles between the normal of a face (mno) and the axes of the crystal coordinate system

1, 2, 3 ... N 2l9...,Nm T 3, 4,... JV *

AT-fold rotation axes AT-fold screw axes center of symmetry, inversion center rotoinversion axes symbols referring to the reciprocal lattice, e.g., the basis vectors, are marked with the index *

GDM HRTEM k MI PBC SHG t

generalized dual-grid method high-resolution transmission electron microscope klassengleich (characterization of subgroups of a spacegroup) morphological importance periodic bond chain second-harmonic generation translationengleich (characterization of subgroups of a spacegroup)

1.1 Introduction

1.1 Introduction The regular polyhedral shape of crystals has long fascinated the observer by their beauty and brightness and by the perfect planarity of their faces (Fig. 1-1), which exceeds the proficiency of the work of artisans. The beliefs of Babylonians and Egyptians in the magical and healing powers of minerals and gemstones has been passed on to other civilizations, a revival taking place for instance in the Middle Ages. Indeed, the important learned man and bishop "doctor universalis" Albertus Magnus (1193-1280) dedicated a part of his book De Mineralibus et Rebus Metallicis Libri V, which appeared in 1276, to the curative properties of crystals. The more rational minds of antiquity dealt with the problems of the formation and composition of minerals. This is reflected in the application of the word %Q\)<j%(xX'koq,Figure 1-1. Rock-crystal (quartz, SiO2). The single crystals show plane faces and trigonal symmetry which means something like "solidification (from Hochleitner, 1981). by freezing" and had originally only been used for ice, to rock-crystal (quartz) during the time of Platon (428-348 B.C.). Following some of the ideas of antiquity with general validity, however, by JeanGeorgius Agricola (1494-1555) was one of Baptiste Rome de Vlsle (1736-1790) as late the learned men to overcome the mystical as 1783. He was able to verify this hypothesis by angular measurements with a conassumptions of medieval times. His books tact goniometer (Fig. 1-2), which was conare not only a collection of the empirical structed in 1780 by his assistant Maurice mining knowledge about minerals of his Carangeot, thus opening the way to quantime but they contain many hypotheses titative crystal morphology. In 1820 about crystal growth and properties. In the William Hyde Wollaston (1766-1829) following centuries the external shape of greatly increased the possible accuracy of crystals, their morphology, attracted more measurement with his optical goniometer and more interest. Thus Niels Stensen (from about 1° to 1'). (1636-1686) found from his crystallization studies that a correlation exists between With the work of the Abbe Rene Just crystal growth and form - and that the Haiiy (1743-1822) a new chapter of crysmorphology of crystals was not accidental. tallography began. A structural way of The law of constant angles between equiva- thinking received strong impetus from his work relating the internal structure of cryslent crystal faces was noticed by Stensen tals to their external shapes. From the ob1669 on the examples of quartz and heservation that the fragments obtained by matite. It was formulated explicitly and

1 Elements of Symmetry in Periodic Lattices, Quasicrystals

Figure 1-2. A historical contact goniometer for measuring the interfacial angles of crystals (from Haiiy, 1801).

repeatedly cleaving a crystal preserved the initial crystal form, Haiiy derived primitive forms ("molecules integrantes") for the basic building units of all crystals. His decrescency theory described the formation of different crystal shapes from basic parallelepipeds (Fig. 1-3). Haiiy's book Traite de Miner alogie (Haiiy, 1801) soon became the standard work on crystallographic mineralogy of the nineteenth century. The morphological school received new stimulus from Christian Samuel Weiss (1780-1856) who polemized against the atomistic basis of Haiiy in an appendix to his German translation of the Traite de Mineralogie. Weiss focused on the dynamic character of matter and the dominating influence of the external crystal form. He discovered the vectorial nature of some physical properties. From the symmetrical arrangement of sets of crystal faces he derived the existence of 2-, 3-, 4- and 6-fold zone axes. He then described the faces by their integer intercepts with these axes to form three-dimensional coordinate systems. As a consequence, Weiss formulated the law of ratio-

nal parameter coefficients which had implicitly already been found by Haiiy in 1784. The more convenient way of indexing used today, based on the reciprocal values of the intercepts, was introduced by William Hallowes Miller (1801-1880). Franz Ernst Neumann (1798 -1895) found a correlation between the morphology of a crystal and the anisotropy of its physical properties. His work on crystal physics was continued by his former student Woldemar Voigt (1850-1919). Interest in crystal symmetry was initiated by the symmetric arrangement of crystal faces. By way of analyzing the morphology of crystals, in 1830 Johann Friedrich Christian Hessel (1796-1876) ordered them into 32 possible crystal classes according to their symmetry. Using the concept of a mathematical point lattice, Auguste Bravais (1811-1863) deduced in 1848 the 14 possible 3-D space lattices in 7 groups which correspond to the 7 crystal systems detected by Weiss. The symmetry of these point lattices (holohedries) was too high in many cases to explain the symmetry properties of the respective elastic tensors determined experimentally. This contradiction could be overcome by occupying the lattice nodes with point complexes, so lowering the symmetry. After the preliminary work of Leonhard Sohncke (18421897), who in 1879 detected 65 space groups (the subset containing symmetry operations of the first kind only) using group theoretical tools, in 1891 all of the 230 possible space group symmetries were derived by Evgraf Stepanovic Fedorov (1853-1919) and Arthur Schonflies (18531929) independently. The atomistic structural theory based on the space lattice concept was confirmed by the first X-ray diffraction experiment (Fig. 1-4) which was suggested by Max von Laue (1879-1960) in 1912.

1.1 Introduction

(b) (d) Figure 1-3. Haiiy's decrescency theory of crystal growth: crystals of the same chemical composition but with a different habit are built from the same basic parallelepipeds ('molecules integrantes'). Schematic drawings of the crystal forms and of their construction using cubic unit cells are shown: (a) and (b) rhomb-dodecahedron, (c) and (d) pentagon-dodecahedron with inscribed cubes (from Hauy, 1801).

1 Elements of Symmetry in Periodic Lattices, Quasicrystals

#, ;

* igure 1-5. Monochromatic zero-layer X-ray precession photograph of the decagonal quasicrystal Al 70 Co 15 Ni 15 showing clearly non-crystallographic tenfold rotational symmetry.

Figure 1-4. One of the very first X-ray photographs taken by Laue, Friedrich and Knipping (1912). The diffraction pattern of zinc blende (ZnS) reflects the fourfold rotation symmetry along one of the main axes of this cubic crystal (from Laue, 1961).

1.2 Symmetry of Crystals 1.2.1 Morphology

The theory of crystal symmetry, i.e., of symmetric transformations in 3-D space under restrictions imposed by the existence of the periodic crystal lattice, appeared to be a rather closed part of crystallography until 1984, when the sensational discovery of quasicrystals by Shechtman, Blech, Gratias and Cahn occurred. The understanding of well-ordered crystals yielding diffraction patterns with non-crystallographic (icosahedral, decagonal, ...) symmetry, i.e., incompatible with a 3-D periodic translation lattice, has been a new challenge for crystallography (Fig. 1-5).

A very extensive and rich collection of crystal drawings was edited by Victor Mordechai Goldschmidt (1852-1933) in the years 1913-1923. He ordered the published information about the morphology of natural crystals, systematically in a nine volume atlas. Figure 1-6 shows one page of Vol. VIII illustrating different natural crystal forms of silver. Single crystals always have a convex polyhedral form, concave parts indicate that two single crystals are grown together. If these two individuals can be transformed into each other by a particular symmetry operation then we call the crystal twinned. A plane of intergrowth, for instance, may correspond to a reflection plane, and we say that this crystal is twinned according to that particular

1.2 Symmetry of Crystals

Figure 1-6. One page of the famous Atlas der Krystallformen in nine volumes edited by Goldschmidt during the years 1913-1923. Silver crystals with different habits grown under different conditions are shown in schematical drawings (from Goldschmidt, 1913 — 1923).

plane. Some of the drawings in Fig. 1-6 show twinned crystals, e.g., the last one of row one, and the second and third ones in row three. The polyhedra characterizing the shape of crystals grown under equilibrium conditions tend to show a particular symmetry, i.e., they are invariant under particular motions around symmetry elements centered in the crystal. These motions, which may be a rotation of the crystal around an axis or a reflection on a mirror plane or on a point (center of sym-

metry), transform a symmetrical object into itself. The transformed object is not distinguishable from the untransformed one, its position in space and its shape coincide with the original ones. Phenomenologically, crystals are defined as chemically homogenous materials with anisotropic physical properties. The most conspicuous manifestation of this anisotropy is the formation of plane faces reflecting the internal symmetry of the crystal. Whereas the crystal habit, defined by

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1 Elements of Symmetry in Periodic Lattices, Quasicrystals

the relative sizes of the faces, may vary widely between different crystal individuals of the same material due to different growth conditions, the interfacial angles remain constant (law of constant angles). Examples of crystals with the same crystal form of a cuboctahedron but with a different habit are shown in Fig. 1-6 (the second and third drawing in row two). The one polyhedron shows large hexahedron faces h and small octahedron faces o, for the other polyhedron the ratio is inverse. In both cases, however, the interfacial angles are the same as well as the directions of the face normals. The interfacial angles can be measured by means of a contact goniometer or, more accurately, by means of an optical two-circle goniometer, as was developed by Fedorov and Goldschmidt in 1892, where a crystal is mounted on a goniometer head so that the rotation axis of the goniometer is parallel to the edges formed by the intersecting faces to be measured. A collimated light beam falls on the crystal, and in each case, when a face is rotated to a reflecting position a signal is seen in a telescope. The light beam, the normal to the reflecting face and the telescope have to be in a plane perpendicular to the rotation axis of the goniometer. Hence, during one complete revolution of the crystal the angles between the faces of one zone can be measured. The angle between two face normals equals n minus the interfacial angle. The complete set of interfacial angles allows establishment of the eigensymmetry (corresponding to the group of point symmetry operations bringing the crystal into self-coincidence) of the crystal. It is even possible to derive a kind of morphological unit cell which is characteristic for a material with a given chemical composition. It is represented by a parallelepiped with edge lengths a\ br, d in relative units and

angles a, /?, y. Usually the ratio a':b':d is given normalized tofe'= l. The disturbing influence of the individual crystal habit may be eliminated by representing the faces by their normal vectors. The commonly used graphical method for the representation of a crystal form is the stereographic projection of its pole figure. Figure 1-7 (a) shows how the face poles result from the intersection of the face normals with a circumscribed sphere having a common center with the crystal. In the next step, the face poles of the northern hemisphere are connected with the south pole and those of the southern hemisphere with the north pole [Fig. l-7(b)]. The stereographic projection of the face poles is obtained by projecting the poles along the connecting lines upon the equatorial plane [Fig. l-7(c)]. The stereographic projection of the face poles is independent of individual variations of the relative dimensions of crystal faces, it shows the inherent symmetry of the crystal form in an unbiased way. For the practical application of the stereographic projection, it is useful to perform the construction on a Wulffs net, i.e., a stereographic projection of meridians and parallels with 2° divisions (Fig. 1-8). The construction and evaluation is facilitated by the fact that interfacial angles of a crystal form appear as true angles in the projection whilst circles on the sphere appear as circles in the projection. 1.2.2 Crystallographic Axes

The symmetrical arrangement of faces (zone-faces) bounding a crystal grown under equilibrium conditions led Weiss to the idea to refer all crystal faces to a 3-D coordinate system formed by three non-coplanar symmetry axes (zone-axes) (Fig. 1-9). Usually they are given by the basis vectors a, A, c with lengths a, b, c, coordinates x, y9

1.2 Symmetry of Crystals

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Figure 1-8. Wulff's net: a stereographic projection of meridians and parallels.

z, and interaxial angles a, /?, y. Another notation frequently used is: basis al9 a 2 , a 3 , lengths a1, a2, a 3 , and interaxial angles a x , a2 , a3 , respectively. The metric properties can also be represented by the metric tensor G, a (3 x 3) square matrix of elements gik = (at - ak) with z, k = 1, 2, 3, i.e., the scalar products of all pairs of basis vectors:

(

a- a ab a- c\ b a b b b c\ c-a

cb

(1-1)

c c)

Figure 1-7. (a) Representation of a crystal form by spherical projection. The normals on the crystal faces intersect a circumscribed sphere in so-called face poles (black dots if they are above, white dots if they are below the plane of drawing). All faces with poles on one meridian belong to one zone, (b) The principle of the stereographic projection: the face poles (e.g., PJ on each hemisphere are connected with the opposite pole (e.g., S) of the equatorial projection plane (dashdotted). The projection of the face poles along these connecting lines upon the equatorial plane (e.g., P/) is called a stereographic projection (c).

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1 Elements of Symmetry in Periodic Lattices, Quasicrystals

Table 1-1. The seven crystallographic coordinate systems : the names, metrical relationships of the lattice parameters and the orientation of the basis vectors with regard to unique symmetry elements are given.

Crystal system

Lattice parameters

Triclinic Monoclinic Monoclinic Orthorhombic Tetragonal Rhombohedral Hexagonal

a^b^c a^b^c a^b^c

Orientation

a ^/?^y^90° a = /? = 9 0 V y c\\2, 1st setting a: = y = 90 V P A || 2, 2nd setting

a^b^c a = P = y = 90° a, b, c||2's a = b^c a; = j3 = y = 9O° c||4 a — b — c; =a j5 = y^9O° a = b^c en: = j5 = 9O°,

(a + 6 + c)||3 c||6

y- = 120° a = b = c a, = P = y = 90° (a + b + c)W5

Cubic

1.2.3 Crystal Faces - Miller Indices Figure 1-9. (a) Schematical drawing of a crystal form and some of its symmetry elements: 4-, 3-, and 2-fold rotation axes. The basis vectors of a coordinate system for this cubic crystal form are oriented parallel to the 4-fold axes, (b) General (triclinic) right-handed coordinate system spanned by the basis vectors a, b, c.

For orthogonal bases the diagonal elements, only, do not equal zero. Table 1-1 lists the seven crystallographic coordinate systems, their metrics and the characteristic orientation of particular unique symmetry elements. In the case of one unique rotation axis N, it is chosen parallel to c, in the monoclinic case another setting with the axis parallel to b is in use too. Instead of rhombohedral axes, with the 3-fold rotation axis parallel to (a + b + c), the hexagonal coordinate system with rotation axis 3 parallel to c is often used. The 3-fold axes of the cubic system always have to be set parallel to the space diagonals (a + b -f c) of the cube.

On the basis of the seven crystallographic coordinate systems all crystal faces can be given in terms of their intercepts with the three axes (Fig. 1-10). The resulting numbers may be expressed as integral multiples m, n, o of the respective unit lengths on these axes yielding the so-called Weiss indices {mno). Face indices are always given in parentheses. The equation of the plane (m n 6) can be written (x, y, z in units of a,fe,c): x

y

z

(1-2)

-+-+-=1 m n o the largest common divisor for Finding 1/m, 1/n and \jo we obtain

nox 4- moy + mnz = mno

(1-3)

and with h = no, k = mo, I = mn and mno =j we end up with

hx + ky + Iz =j

(1-4)

/z, k, I are called the Miller indices and they define the face symbol (h k I) which is more

1.2 Symmetry of Crystals

Figure 1-10. A plane defined by its intercepts on a crystallographic coordinate system. The intercepts at x = 2, y = 3, z = 2 lead to the Weiss indices (232) and the Miller indices (323). The vector normal to the plane H is indicated.

commonly used than the Weiss symbol. j may be a positive or negative integer number. The derivation of the Miller indices may be illustrated by the example given in Fig. 1-10. The face intersects a at x = 2, b at y = 3 and c at z = 2 leading to the Weiss indices (232). Taking the reciprocal of the intercepts 2, 3, 2 we get 1/2, 1/3 and 1/2. Reducing to a common denominator we find 3/6, 2/6, 3/6, and writing the numerator only, the Miller indices (323) result. The indices, on the other hand, define a vector normal to the plane H given on a reciprocal basis (see Sec. 1.3.4). Another plane, parallel to the just mentioned one but two times the distance from the origin, is characterized by the Weiss indices (464) and the Miller indices 1/4, 1/6, l/4=>3/12, 2/12, 3/12 => (323). Both parallel planes have the same Miller indices and generally, the Miller indices (h k I) define an infinite set of parallel planes (lattice or net planes) (Fig. 1-11). Any two crystal faces with Weiss indices (mno) and (m'n'o') have a rational ratio m/mr: njri: ojo' of their intercepts on the crystallographic axes ("law of rational

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parameters") as a direct consequence of the fact that the numbers m, m', n, n\ o, d are integers. In the case of a trigonal or hexagonal coordinate system it is advisable to use the fourfold Bravais-Miller symbol (hkil) with h + k + i = 0 to facilitate the derivation of symmetrically equivalent indices. A threefold rotation, for instance, acting on a hexagonal coordinate system, transforms the axes a 1? a 2 , a3 into a 2 , — (a1+a2), a3 then into — {a1 + a2\ al9 a 3 and finally back into al9 a2, a3. Thereby, the face indices (h k i I) are changed to (k i h /), (i hkl) and (h k i /), respectively (Fig. 1-12). Generally, the Weiss or Miller indices transform like the axes. The angles i/fx, \jjy, \j/x between the normal of a face (m n o) and the axes of the crystallographic coordinate system obey the relations (see Fig. 1-10) OP ma OP

(1-5)

OP me

(110)

(010)

Figure 1-11. Point lattice on the basis a, b, c (c is perpendicular to the plane of drawing) with different sets of lattice (net) planes (hkO) parallel to c. The density of lattice points on (410), for instance, is smaller than on (010), it is proportional to the inverse of the respective interplanar spacings dhk0.

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1 Elements of Symmetry in Periodic Lattices, Quasicrystals

absolute scale they are referred to b = l9 generally. We get, consequently, a=

C O S (£>v

- and c =

C O S (7>v

cos