Mechanics and Mathematics of Crystals: Selected Papers of J L Ericksen

Mechanics and Mathematics

CRYSTALS Selected Papers of J L Ericksen

Mechanics and Mathematics

(CRYSTALS Selected Papers of J L Ericksen

Editors

Millard F Beatty University of Nebraska-Lincoln, USA

Michael A Hayes University College Dublin, Ireland

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MECHANICS AND MATHEMATICS OF CRYSTALS Selected Papers of J L Erkksen Copyright © 2005 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-256-283-4

This book is printed on acid-free paper. Printed in Singapore by Mainland Press

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Jerald L. Ericksen P/iofo 6>> A Rochan Photography

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Contents Foreword M. F. Beatty and M. A. Hayes

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J. L. Ericksen's Autobiography

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Publications of J. L. Ericksen 1. Crystal Symmetry 1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7.

On the Symmetry of Crystals Nonlinear Elasticity of Diatomic Crystals On the Symmetry of Deformable Crystals Changes in Symmetry in Elastic Crystals Crystal Lattices and Sub-Lattices On Nonessential Descriptions of Crystal Multilattices On Groups Occurring in the Theory of Crystal Multi-Lattices

2. Constitutive Theory 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. 2.8.

Multi-Valued Strain Energy Functions for Crystals The Cauchy and Born Hypotheses for Crystals Constitutive Theory for Some Constrained Elastic Crystals Some Constrained Elastic Crystals Equilibrium Theory for X-ray Observations of Crystals A Minimization Problem in the X-ray Theory Notes on the X-ray Theory On Pitteri Neighborhoods Centered at Hexagonal Close-Packed Configurations

3. Defects 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7. 3.8. 3.9. 3.10.

Volterra Dislocations in Nonlinearly Elastic Bodies Twinning of Crystals Thermoelastic Considerations for Continuously Dislocated Crystals Some Surface Defects in Unstressed Thermoelastic Solids Stable Equilibrium Configurations of Elastic Crystals On Nonlinear Elasticity Theory for Crystal Defects On Correlating Two Theories of Twinning Twinning Analyses in the X-ray Theory Twinning Theory for Some Pitteri Neighborhoods On the X-ray Theory of Twinning

xxvii 1 3 7 14 27 38 47 77 111 113 117 134 148 165 185 191 209 239 241 249 266 272 281 295 311 340 369 383

viii 3.11. 3.12. 3.13. 3.14.

On the Theory of Rotation Twins in Crystal Multilattices On the Theory of Growth Twins in Quartz On the Theory of Cyclic Growth Twins Unusual Solutions of Twinning Equations in the X-ray Theory

4. Phase Transitions 4.1. 4.2. 4.3. 4.4. 4.5. 4.6.

Some Phase Transitions in Crystals Continuous Martensitic Transitions in Thermoelastic Solids Weak Martensitic Transformations in Bravais Lattices Bifurcation and Martensitic Transformations in Bravais Lattices Local Bifurcation Theory for Thermoelastic Bravais Lattices Thermal Expansion Involving Phase Transitions in Certain Thermoelastic Crystals 4.7. On the Possibility of Having Different Bravais Lattices Connected Thermodynamically 4.8. On the Theory of ot-(3 Phase Transition in Quartz

405 422 450 451 453 455 481 494 508 532 560 576 596

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Foreword This book celebrates the 80th birthday of Professor Jerald. L. Ericksen, our esteemed teacher and good friend for nearly 45 years now. The volume presents a selection of research papers by Ericksen on the mechanics and mathematics of crystals published over the past 35 years, just a fraction of his (currently) 145 journal and book articles published since 1948, all of which are listed below in the "Publications of J. L. Ericksen." With the exception of the first paper in this collection, an English version of the original Russian language publication unique to this volume, all others are reproduced entirely as originally published. Moreover, except for this brief Foreword, everything else in this book is written by Jerry Ericksen alone. The papers are arranged by Ericksen into four principal groups, or chapters, each of which is preceded by a summary introduction describing its contents. Unfortunately, we are unable to reprint Ericksen's two most recent papers which are expected to appear in 2005. Nevertheless, useful extended summaries of these works are provided in Chapter 3. This collection of selected works of Ericksen thus provides a splendid corpus of research papers which should prove valuable to a broad group of researchers in the mechanics, applied mathematics, and materials science communities. We are very pleased to have included in this volume a short autobiography of Professor Ericksen, which he very kindly prepared at our request. The narrative describes Jerry's family and boyhood years, his military service as a naval officer, meeting and eventually marrying his lovely wife Marion, his formal educational experience and development as an applied mathematician, and how he came ultimately to realize what he wanted to do in research. This volume contains, in all respects, the works of a gifted researcher who is noted for his originality and clarity of thought, alertness of mind, and soundness of judgment. Combined with these gifts he exhibits clear insights into basic problems in mechanics founded upon a continuing interest in experimental work of others, and its mathematical modeling. Jerry Ericksen is no follower of fashion, but rather a creator of it. He is comfortably at home in all areas of mathematics, and he is a world renowned master of mechanics. Jerry is most warmly admired by his former students, who are greatly appreciative of his generosity in freely sharing with them his ideas during the course of their research studies under his tutorship, never once assuming deserved credit for his contributions. And across the world, he is held in great esteem among colleagues in a wide variety of disciplines, including nonlinear elasticity, non-Newtonian and anisotropic fluids, rheological behavior of solids and fluids, liquid crystals, thermodynamics and stability of continua, and, of course, crystal physics. In recognition of his many accomplishments Jerry received the Bingham Medal of the Society of Rheology in 1968, the Timoshenko Medal of the American Society of Mechanical Engineers in 1979, and the Engineering Science Medal of the Society of Engineering Science in 1987. He was awarded the degree of Doctorate of Science honoris causa from the National University of Ireland in 1984 and from Heriot-Watt University in 1988; and in 1999 he was elected an Honorary Member of the Royal Irish Academy. Six years earlier, in 1993, Jerry Ericksen was elected to the U. S. National Academy of Engineering. In 2000, Jerry was named an Honored Member of the International Liquid Crystal Society. And most recently, in 2003 he was

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awarded the Panetti-Ferrari prize and gold medal of the Academia delle Scienze di Torino. We are truly and deeply proud to have the opportunity and to share in the effort to provide this unique volume of selected works of Ericksen on crystals.

Acknowledgment We are most grateful to Jerry Ericksen for his cooperation and tireless effort in bringing this publication project to fruition. Professor Lev Truskinovsky deserves special thanks for graciously providing the English translation of the original Russian version of Ericksen's first paper in this series. The cover artwork is based on a photograph, "Twin crossings in the orthorhombic phase of Cu-14.0% Al-3.5% Ni (mass percent), field of view: 0.4mm x 0.6mm," by courtesy of C. Chu and R. D. James. We thank Professor Richard D. James, University of Minnesota, for providing the electronic image and for his kind permission to use it here. Needless to say, the production of this book would not have been possible without the generous cooperation of all of the publishers of the original papers. To them we express our sincere appreciation for their kind permissions to reprint Jerry Ericksen's papers on crystals in this special volume celebrating his 80th birthday. We gratefully acknowledge and list below alphabetically by publisher the papers reprinted herein with permission granted by the named publisher or institution. Reprinted from Proceedings of an International Conference on the Mechanics of Dislocations, 1983, pp. 95-100, eds. E. Aifantis and J. Hirth, Thermoelastic considerations for continuously dislocated crystals by J. L. Ericksen, Copyright (1985), with kind permission by ASM International, the Materials Information Society. Reprinted from the Pergamon journal International Journal of Solids and Structures, with kind permission by Elsevier Science Ltd.: Vol. 6, J. L. Ericksen, Nonlinear elasticity of diatomic crystals, pp. 951-957, Copyright (1970). Vol. 18, J. L. Ericksen, Multi-valued strain energy functions for crystals, pp. 913916, Copyright (1982). Vol. 22, J. L. Ericksen, Constitutive theory for some constrained elastic crystals, pp. 951-964, Copyright (1986). Vol. 38, J. L. Ericksen, Twinning analyses in the X-ray theory, pp. 967-995, Copyright (2001). Reprinted from the Pergamon journal International Journal of Plasticity, Vol. 14, J. L. Ericksen, On nonlinear elasticity theory for crystal defects, pp. 9-24, Copyright (1998), with generous permission by Elsevier Science Ltd.. Reprinted from the Academic Press book Phase Transformations and Material Instabilities in Solids, 1984, ed. M. Gurtin, pp. 61-77, J. L. Ericksen, The Cauchy and Born hypotheses for crystals, Copyright (1984), with kind permission by Elsevier Science Ltd. and J. L. Ericksen. Reprinted from the Martinis Nijhoff Publishers book, Proceedings of the WTAM Symposium on Finite Elasticity, 1980, pp. 167-177, eds. D. E. Carlson and R. T. Shield, Changes in symmetry in elastic crystals by J. L. Ericksen, Copyright (1981), with kind permission of Kluwer Academic Publishers.

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Reprinted from the Kluwer Academic Publishers journal, Journal of Elasticity, with kind permission of Kluwer Academic Publishers: Vol. 28, 1992, pp. 55-78, Bifurcation and martensitic transformations in Bravais lattices by J. L. Ericksen, Copyright (1982). Vol. 55 (3), 1999, pp. 201-218, Notes on the X-ray theory by J. L. Ericksen, Copyright (1999). Vol. 63 (1), 2001, pp. 61-86, On the theory of the <x-p phase transition in quartz by J. L. Ericksen, Copyright (2001). Vol. 70 (1-3), 2003, pp. 267-283, On the theory of rotation twins in crystal multilattices by J. L. Ericksen, Copyright (2003). Reprinted from the Kluwer Academic Publishers journal, Meccanica, Vol. 31, 1996, pp. 473-488, Thermal expansion involving phase transitions in certain thermoelastic crystals by J. L. Ericksen, Copyright (1996), with kind permission of Kluwer Academic Publishers. Reprinted from Material Instabilities in Continuum Mechanics and Related Mathematical Problems, ed. J. Ball, pp. 119-135, Chapter 11, Some constrained elastic crystals by J. L. Ericksen, Copyright (1988), by kind permission of Oxford University Press. Reprinted from Analele stiinfifice ale Universitafii "Al. I. Cusa" din lasi, Vol. 23, 1977, pp. 423-430, Volterra dislocations in nonlinearly elastic bodies by J. L. Ericksen, Copyright (1977), by kind permission from T. Precupanu, Editor-in-Chief at the University "Al. I. Cusa," lasi. Reprinted from Rendiconti del Seminario Matematico dell'Universita di Padova, Vol. 68, 1982, pp. 1-9, Crystal lattices and sub-lattices by J. L. Ericksen, Copyright (1982), by kind permission of F. Baldassarri, Director of Rendiconti. Reprinted from Mathematics and Mechanics of Solids, with kind permission of Sage Publications Ltd.: Vol. 1, 1996, pp. 5-24, On the possibility of having different Bravais lattices connected thermodynamically by J. L. Ericksen, Copyright (1996). Vol. 4, 1998, pp. 363-392, On nonessential descriptions of crystal multilattices by J. L. Ericksen, Copyright (1998). Vol. 6, 2001, pp. 359-386, On the theory of growth twins in quartz by J. L. Ericksen, Copyright (2001). Vol. 7, 2002, pp. 331-352, On the X-ray theory of twinning by J. L. Ericksen, Copyright (2002). Reprinted from Archive for Rational Mechanics and Analysis, with generous permission by Springer-Verlag: Vol. 72, 1979, pp. 1-13, On the symmetry of deformable crystals by J. L. Ericksen, Copyright (1979). Vol. 73, 1980, pp. 99-124, Some phase transitions in crystals by J. L. Ericksen, Copyright (1980). Vol. 88, 1986, pp. 337-345, Some surface defects in unstressed thermoelastic solids by J. L. Ericksen, Copyright (1986).

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Vol. 94, 1986, pp. 1-14, Stable equilibrium configurations of elastic crystals by J. L. Ericksen, Copyright (1986). Vol. 107, 1989, pp. 23-36, Weak martensitic transformations in Bravais lattices by J. L. Ericksen, Copyright (1989). Vol. 139, 1997, pp. 181-200, Equilibrium theory for X-ray observations of crystals by J. L. Ericksen, Copyright (1997). Vol. 148, 1999, pp. 145-178, On groups occurring in the theory of crystal multilattices by J. L. Ericksen, Copyright (1999). Vol. 153, 2000, pp. 261-289, On correlating two theories of twinning by J. L. Ericksen, Copyright (2000). Vol. 164, 2002, pp. 103-131, On Pitteri neighborhoods centered at hexagonal close-packed configurations by J. L. Ericksen, Copyright (2002). Reprinted from Metastability and Incompletely Posed Problems, IMA Volumes in Mathematics and Its Applications, eds. S. Antman, J. L. Ericksen, D. Kinderlehrer, and I. Muller. Vol. 3, 1986, pp. 77-93, Twinning of crystals by J. L. Ericksen, Copyright (1987), with kind permission by Springer-Verlag. Reprinted from Microstructure and Phase Transitions, IMA Volumes in Mathematics and Its Applications, eds. D. Kinderlehrer, R. James, M. Luskin, and J. L. Ericksen. Vol. 54, 1993, pp. 57-84, Local bifurcation theory for thermoelastic Bravais lattices by J. L. Ericksen, Copyright (1993), with kind permission by Springer-Verlag. Reprinted from Continuum Mechanics and Thermodynamics, Vol. 14, 2002, pp. 249262, Twinning theory for some Pitteri neighborhoods by J. L. Ericksen, Copyright (2002), with kind permission by Springer-Verlag. Reprinted from the book Problems of Mechanics of Deformable Solid Bodies (in Russian), 1970, pp. 493-496, eds. Y. N. Rabotnov and L. I. Sedov, On the symmetry of crystals by J. L. Ericksen, Copyright (1970), with kind permission of 'Sudostroenie', St. Petersburg, to reproduce this article in English. Reprinted from the Hemisphere Publishing Corp. journal Journal of Thermal Stresses, Vol. 4, 1981, pp. 107-119, Continuous martensitic transitions in thermoelastic solids by J. L. Ericksen, Copyright (1981). Reproduced by permission of Taylor & Francis, Inc., http:/www.taylorandfrancis.com. Reprinted from Contributions to Continuum Theories — Anniversary Volume for Krzysztof Wilmanski, 2000, pp. 77-82, ed. B. Albers, A minimization problem in the X-ray theory by J. L. Ericksen, Copyright (2000). Reproduced by permission of B. Albers, Weierstrass Institute for Applied Analysis and Stochastics. Millard F. Beatty, Lexington, KY, USA Michael A. Hayes, Dublin, Ireland

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J. L. Ericksen's Autobiography

1. The Early Years From what my parents have told me, I was born December 20, 1924, in Portland, Oregon, during a bad winter storm. During the night, my mother, Ethel, went into labor, requiring a ride in a Model T Ford on icy streets. The delivery involved complications, but we survived. So, my arrival was traumatic for them. At the time, my father, Adolph, had a good job as foreman at a large creamery in Portland, and was winning prizes for his skills at making and judging butter. They bought a modest house and did much to improve it. When the Model A Ford replaced the various versions of the Model T, they bought a new one. From what little I remember and have been told, we lived the life of a rather typical middle class family for a time. Then, when I was about eight or nine, the Depression had hit. My father was suddenly fired, as happened to many others. I have lost track of the precise date. After looking at the dismal prospects for his getting a decent job, my parents decided to go into business for themselves. They found a rundown creamery at a low price in Vancouver, Washington, then a small town with a population of about 18,000. They managed to sell their house, taking a loss on this and, with the savings they had, scraped together enough money to buy the business, which was failing. They found a small apartment with low rent in Vancouver. My father was very strong physically and mentally, an efficient worker who was accustomed to working about sixty hours a week. By patching up decrepit equipment and producing good products, they managed to pay the rent and put food on the table. Close to this time, my brother A. Erwin (Erv) was born. My mother helped as much as she could, keeping the books, making sales contacts while taking care of us children, preparing meals and managing other household chores. Some brokers in Portland my father knew also helped him find markets for what he did not sell in town, this being sold under different brands. The quality of cream received was quite variable, butter made from the best being sold under his brand. When I became a teenager, I was put to work in the creamery after school, on Saturdays and during school vacations. By then, it had expanded to include making ice cream and operating a soda fountain. Most of my friends spent similar times at part-time jobs, to supplement meager money earned by their parents. Without minimum wage laws, benefits, many restrictions on hiring minors or the need for bookkeeping, it was fairly easy for youngsters to find low-paying jobs. Gradually, my parents' income increased enough to enable them to buy a small two-bedroom house, much like those most of my friends lived in, and we began to live somewhat more comfortably. I am impressed by how much the citizens of the town did to improve the lives of the children, during these hard times. As one example, there was a school bus, to bring in students living outside of town. Often, the operator would volunteer to take a busload to a skating rink in

xiv Portland at night, absorbing the transportation costs. As another example, I belonged to a Boy Scout troop, led by a reporter on the local paper, who managed to get local groups to supply minimal equipment for camping and skiing, along with whatever transportation was needed for the many outings he led. Often in the winter, we would gather in his home the night before a ski trip, to wax skis and enjoy wonderful pies his wife made for us. These were the old skis, with strap bindings that could be fit to any kind of boot, and most of us did not have any special kind of clothing. The next day, we were off to Mount Hood for a day of skiing at essentially no cost to us. One could use a rope tow at a cost but most of us climbed to the top. Occasionally, we stayed overnight at a lodge owned by the Scouts to enjoy skiing in moonlight. Essentially, this was one large room, devoid of plumbing since it would get frozen. Besides having such individuals go out of their way to help us, there were strong community efforts to make school-related activities better for us, for example in finding materials and equipment for extracurricular activities in the arts, music and sports.

2. World War II Then came the attack on Pearl Harbor. I was then a senior in high school, not quite seventeen. For a time, my life did not change very much, although there were many signs of mobilization, including formation of a shipyard in Vancouver to make Liberty ships. I graduated from high school and, in the fall of 1942, entered what was then called Oregon State College, in Corvallis, financed by my parents and the little I got from part-time jobs. Then, when I became eighteen, I persuaded my somewhat reluctant parents to let me enlist in the Navy. The Navy did not call me to active duty until I completed the year at Oregon State, when I was ordered to report to a new V12 unit at what was then the University of Idaho, Southern Branch, in Pocatello. To my surprise, I was to be trained as an officer. This involved taking some courses carrying academic credit, along with military training. Then, for reasons unknown to me, I was transferred to the NROTC unit at the University of Washington in Seattle. This involved a heavy dose of Navy courses on navigation, seamanship, rules of the road for ships, Navy regulations etc., along with more conventional academic courses. I was kept in officer training for what seemed to be an eternity. According to my Navy records, it was a long time, 85 1/2 weeks, including short stays in gunnery and firefighting schools. While I was in training, I met a Seattle girl, Marion Pook, and we fell in love, but marriage had to wait for some time. After completing the work to become an ensign, I was ordered to report to the LCI(R) 72, in the Philippines. After a train trip and a long ride on a troop ship, I complied. On the troop ship, we crossed the International Date Line and Equator, experiencing the traditional hazing associated with these events. I knew that LCI meant "Landing Craft Infantry," small ships used to land troops on beaches at invasions. However, I did not know what the (R) meant until I arrived. This represented "rocket". The ship no longer transported troops, being outfitted with numerous rocket launchers, used to clear beach heads prior to the landing of troops. These would demolish almost anything, including sizable trees, in a strip near the beach. It was manned by about 30 enlisted men and 4 to 6 officers, the latter being transferred from one ship to another rather frequently. This was in a group of LCI ships, modified in some different ways, often traveling together. For example, another kind carried a 5" gun, our largest being 40 mm anti-aircraft guns. While I did see some action of this kind, it was very mild compared to the fierce major battles that made headlines in newspapers. We also got involved in other

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kinds of work, for example following minesweepers and destroying mines that came to the surface, using rifles to set off their detonators. At different times, I served as commissary officer, engineering officer and executive officer. The only other possibility was to be captain. During my stay, we had two of these, both with the rank of lieutenant (j-g-)> standard for ships of this kind. One day, we got orders to come in for an overhaul, to prepare for an invasion of Japan, a frightening prospect, since very fierce opposition was expected. I must admit that we all cheered when we learned that dropping the bombs induced Japan to surrender. After a time, we were ordered to join a large convoy to sail for San Diego. Since the speed of a convoy is set by the slowest ship, this was a very slow trip, taking about 45 days, if my memory is correct. When we had drawn food supplies, we were obliged to take a quota of spam and much ingenuity had been exercised in finding better things to eat, so we and other ships had stored an oversupply of this. On the way back, much of this went to the bottom of the ocean at night. After arriving, we went about the boring business of decommissioning the ships, taking inventories of what was left, etc. However, it did involve the opportunity to get married, which Marion and I did in Seattle on Sunday, February 24, 1946.1 was scheduled to return to the Philippines, to help bring back another ship. However, a former shipmate then at the Bureau of Personnel in Washington, D. C. was nearly eligible to be separated from the service and he got my orders changed to be his replacement until I also became eligible. So, Marion and I had this time together. I needed a car to get to work, so we bought an old Chevrolet. After I was released from active duty, we drove to Seattle. I was required to stay in the Naval Reserve for a time and did take cruises on Navy warships from Seattle to San Francisco and from Seattle to Alaska and back. However, while this added a little to our income, I had enough of Navy life and resigned as soon as I could. Had I stayed in, it is likely that I would have become a participant in the Korean War, occurring after I had resigned. Including officer training, I was on active duty for about three years, ending in the early summer of 1946. The war effort expended large quantities of materials, requiring almost everything for civilian use to be rationed. That Model A that my parents had bought new had stopped running before the war started, was replaced by another car, but they still owned it. During the war, a man bought it for a good price. Gasoline was among the many things that were rationed, but some had found that some old cars could be made to run after a fashion on readily available stove oil and this is why he bought it. Such vehicles produced copious amounts of smoke and complaining engines made strange noises but they enabled some to drive more miles. With many in the military and many others put to work building ships, planes, etc., the previous shortage of jobs ended, replaced by numerous good-paying jobs. After the war, there was a pent up demand for all kinds of products and getting these into production certainly stimulated the economy. Until the shortages disappeared, a store having some item for sale, say a toaster, would have potential buyers standing in what was often a long line, until the supply was exhausted.

3. Back To School Marion and I had talked it over and decided that I should return to college, the GI Bill making this feasible, having traveled to Seattle to explore this. I took a job repairing tires to tide us over until classes started. The University of Washington was fairly generous in allowing

xvi credits for Navy courses and courses taken elsewhere. By taking a heavy course load and passing one or two courses by exam, I could complete requirements for a bachelor degree in one year. This I did, with a major in mathematics and a minor in naval science. About this time, the possibility of becoming a professor began to interest me. As a senior, I taught an elementary course in algebra, which did not dissuade me, and added a bit to our income. When time permitted, a friend and I took on various odd jobs. Also, Marion worked at an office that helped us find these. While I did well in mathematics, I wanted to do something useful with it, but was not sure what. I knew that the faculty members at Oregon State were more interested in applications, so we moved there for me to get a masters degree. To pad our income, Marion worked, while I did some teaching and odd jobs. Again, I did not see applications that really interested me. At the time, there was an opinion among many mathematicians that they should stick to pure mathematics, and some were disdainful of those who explored applications. My advisor, Howard Eves, was not an applied mathematician but was more tolerant of applications. He told me of a rather new program at Indiana University that seemed interesting. So, I applied and was accepted. We drove to Bloomington, stuffing all of our belongings into that old Chevrolet. The faculty included good mathematicians, some involved in continuum mechanics. I enjoyed the contacts with David Gilbarg, my advisor, Bill Gustin, Vaclav Hlavaty, Eberhard Hopf, Tracy Thomas, George Whaples and Max Zorn. From these, I learned a lot about mathematics and more conventional continuum theories of fluids and solids, but not about how such equations came into being. Then, during the latter part of my stay, Clifford Truesdell joined the faculty. I took a course from him based on a preliminary version of his paper that became well known, "The Mechanical Foundations of Elasticity and Fluid Dynamics," Journal of Rational Mechanics and Analysis 1, 125-300 (1952). Here, for the first time, I saw something of the reasoning that had led to successful continuum theories. I realized that this was what I was looking for and, since then, I have been trying to better understand the formulation of and techniques for exploring such theories. In another course he gave, I got a good introduction to the kinetic theory of gases, something I have done no research on but, on various occasions, I have found it important to know something about it. In Bloomington, another big event for us was the arrival of our first child, our daughter Lynn. I proceeded to get my doctorate, ending my formal education in 1951. Here again, I had done some teaching.

4. Comments On My Formal Education Looking back at my education, it had its ups and downs. In the lower grades, I did well enough to convince teachers that I should skip a grade. Neither of my parents finished high school. As a teenager, my father had to quit school to run the family's wheat farm because his father became bedridden with encephalitis: he was the oldest son. My parents believed that they could have done better with more education and were committed to helping my brother and me get a college education. No doubt, I disappointed them by being a mediocre student in high school. It would be wrong to blame this on the hours spent in the creamery. My parents were quite strict with me and, like some youngsters you might know, I was a bit rebellious. I responded by doing as little as possible to get C grades, as a way of rebelling. So, my judgment was bad. As soon as I entered college, I resumed getting very good grades although

xvii at that time I was very uncertain as to what I wanted to do afterwards. Without the war, I am sure that my parents would have found enough to help me get a bachelor's degree, probably for even more education, but my getting married would have overstrained their resources. Of course, most of this was made unnecessary by the investment of the Navy and the GI Bill. Certainly, the experience induced by naval service helped me to be a more mature person. On the negative side, most of the Navy courses I took were necessary for my work as an officer, but I did not find them useful after leaving the Service. Certainly, it would have been good to have more science courses. The quality of courses I had was quite variable. For example, I had taken an undergraduate course in thermodynamics, taught by a man who loved steam engines. We saw many impressive pictures of these, memorizing the cookbook formulae relating to them, but we learned nothing of other applications or more basic matters. An undergraduate course I had taken in physics was not much better. None of the mathematics courses I had was bad and some were quite good, which might explain why I drifted into mathematics. After the war, resources of universities were strained by the large influx of returning veterans but, in my limited experience, they did pretty well in coping with this by recruiting teachers and overloading professors. It was for this reason that I got a teaching assignment as a senior. The professors that I talked with willingly accepted being overloaded as their way of thanking returning veterans. In a perverse way, the Hitler regime helped us, by inducing refugee professors to come to the U.S. While I was serving in the South Pacific, we got almost no news about the happenings in Europe, in particular. It was after I returned to the U.S. that I learned about the atrocities, etc. I did chat with some survivors about their experiences. For example, one of these was Hlavaty. To avoid persecution in Czechoslovakia, he was befriended by a farm family, who hid him and provided food and other necessities. He stayed in the house during daylight, being careful not to be seen by others. When strangers approached, he crawled into a secret hiding place. They also gave him writing supplies that he used to write a book, although he had no access to libraries or personal papers. He escaped in the late 1930's and then published Differentialgeometrie der Kurven und Fldchen und Tensorechnung, P. Noordhoff N. V., Groningen-Batavia, 1939. So, he earned a little money from this labor. For understandable reasons, there are no specific references to the literature in this. I studied this as a graduate student, partly to help me improve my poor ability to read German. When I met him, he was a very nice old man, glad to have a secure position. After I left Indiana University, professors told me that Serrin and I were regarded as the best students while we were there. He was and still is the better mathematician. Certainly, the most valuable part of my formal education was obtained here. The most important tasks of a formal education are to teach students to learn by themselves and to improve their ability to think clearly. While my education had its ups and downs, I did learn these important things. Although there were weak spots in what I learned about science, I did get a good grounding in mathematics, particularly.

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5. U. S. Naval Research Laboratory (NRL) When I completed work on my doctorate, I needed a job and was hardly besieged with good job offers: continuum theory did not interest most mathematicians and only a few nonmathematicians were interested in the view being promoted by Truesdell. Truesdell offered to help me get a job in the group he had headed at NRL. From what he had told me, it seemed interesting, one attraction being that he visited this group occasionally, as a consultant. Another was that the group had much freedom. So, I accepted an offer to join the group. Serrin also joined us, for a temporary summer position. Others in the group were Paul Nemenyi, who headed the group, two physicists, Bill Saenz and Dick Toupin and a lovely young lady, Charlotte Brudno, perhaps best described as an expediter. Rather soon, she left to become Truesdell's wife. We missed her greatly, but she performed a much more important service, as a very able helpmate to him. The group was formed to do research, being quite free to pick topics and to serve as consultants to workers in other groups. At first, Nemenyi put me to work on some problems, and I managed to produce some results on water bells that he liked. However, I soon shifted to topics in continuum theory more like those I had learned from Truesdell. Toupin shared an interest in this and we had long discussions of it, as well as on electromagnetic theory and relativity theory. This helped us in doing our own researches and served to broaden our perspectives, since our educations had been rather different. Then, Nemenyi died and, after a time, Ronald Rivlin was hired as a replacement and he interacted strongly with us. Toupin and I had learned of his interesting work on rubber elasticity from Truesdell. Rivlin got me interested in viscoelastic effects in high polymers, leading to some collaboration. Also, he introduced me to the Society of Rheology. This was heavily populated by chemical engineers interested in high polymers. I made a serious effort to communicate with them, learning quite a bit in the process. Gradually, I became a respected member of this group. I started to serve as a consultant to the polymer group at what was then the National Bureau of Standards, an activity that continued for years. My other moonlighting included teaching mathematics for NRL employees after working hours, under the auspices of the University of Maryland. There were some restrictions on outside work I could do as a federal employee, but these activities were permissible. Largely because of Truesdell's efforts, our kind of continuum theory became quite popular. I believe that the experience I gained from this group was more valuable than that I would have acquired, had I gone directly to a university, but I never thought that I would stay there indefinitely. Although I did not suffer personally, I had some unpleasant experiences with the witch hunts conducted during the McCarthy era, being grilled about persons accused of being communist sympathizers. It was easy to be falsely accused, as I believe was the situation in cases I was involved in, much harder to get cleared. For this and other reasons, working conditions were getting worse. Eventually, Rivlin and I moved to universities, Toupin taking a research position at IBM. While I was at NRL, I was first classified as a mathematician. Later, I was reclassified as a mathematical physicist, largely to make it easier to promote me. On a happier note, it was during my stay at NRL that Marion gave birth to our son Randy. While I was here, I made a bad mistake. Truesdell invited me to help him write an article for the Handbuch der Physik. Of course, I was flattered and I accepted, without thinking much about the limits of my abilities. He had a remarkable ability to sift through a mountainous literature and find the few gems buried in it. I should have realized that I do not share this ability, which was very important for this project. When I look at a body of literature, I tend to

xix stop when I find something unclear, and stew about clarifying it. I much regret that I caused him much pain by missing deadlines, etc., particularly since he was quite patient with me. Eventually, I did produce something, my 1960 article "Tensor Fields," although I was not very happy with it. Since then I have known other young persons who got involved in writing books or long articles with senior workers and none of these found this to be a happy experience. So, I advise young persons to avoid making such agreements. This experience was an important factor in my decision to stop being a co-author of papers, a decision I have not regretted.

6. Johns Hopkins University (JHU) In 1957,1 got an offer to join the faculty in the Mechanical Engineering Department at JHU. Certainly, I had some qualms about being labeled as something I was not, an engineer. However, I felt better about this after exploring more about the people and activities there. So, after discussing it with Marion, I decided to gamble on it, although it meant a cut in salary. For a long time, it was a very good place for me. For one thing, professors had very much freedom, for example in deciding what and how to teach. The general philosophy could be described by a statement I have seen attributed to Booker T. Washington1, "Few things help an individual more than to place responsibility upon him, and to let him know that you trust him." Beyond the Department, there were just three administrators to deal with. If you wanted money for some unusual purpose, you would contact the Provost, who would give you a prompt and firm decision, depending on what he thought was in the best interests of the University. For academic matters, contact the Dean. For anything else, see the President. As a junior faculty member, I was free to go to his office and, if he was free, he would discuss the matter of concern. An amazing woman operated the main switchboard. If you called some professor who did not answer, she usually had a good idea where to find the person and would establish a connection when the person sought was on campus. The faculty club served as a good place to get acquainted with those in other departments at lunch. Once a week, the senior administrators had lunch at a large table, to be available to any faculty members that wanted to fill the other seats, to discuss any issues that were of concern to the latter. Certainly, some faculty members got research grants, but they were more individual contracts with grantors, not much involving the rest of the University, one of the things that made it feasible to keep the administration small. Well, it was like this at first, but changes occurred rather gradually and, in my opinion, they were not for the better, although I still had much freedom as long as I was there. The three Departments of Aerospace Engineering, Civil Engineering and Mechanical Engineering really acted more as one larger department, with Francis Clauser as a kind of overlord. Eventually, they merged to become the Mechanics Department. At first, it was headed by George Benton. He did his job very well, until he was moved to higher administrative positions. This was a diverse but close-knit group and, by interacting with them, I learned something about techniques and reasoning used in meteorology, metallurgy and turbulence, among other things. I enjoy learning independently of whether it seems useful to me. Though Clauser did not head the new department, he continued to exert leadership. He was a powerful man, influential with the upper administration and with granting agencies 1

Forbes, p. 32, March 29, 2004.

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involved in his area, aerodynamics. He did exert his influence in deciding who should be hired, promoted or fired, but he did not favor any particular interest in the department. He also had some success in raising standards in the various engineering departments. While I did not like all of his actions, my opinion was that, on the whole, he was a good influence. At one time, the possibility of making an offer to Truesdell was being considered. One day, Clauser came to my office, saying that he knew that Truesdell was a star, but that he could be difficult to get along with. He asked me whether I thought he would be impossibly difficult. My opinion was that he would not. This was enough for him to agree to the offer and, without his consent, I am sure the offer would not have been made. Soon, Truesdell accepted and joined us. While my relations with the mathematicians at JHU were quite friendly, my interests were more in mechanics, and the faculty members in my Department were congenial. James Bell had been particularly helpful to me when we first arrived and we and our students all became good friends. After Truesdell's arrival, the three of us discussed how to produce a stronger program for graduate students, postdoctoral associates and more senior visitors working with us, to supplement our individual interactions with these scholars. For one thing, we set up a weekly seminar, with the younger workers being obliged to attend and take turns speaking, we and other more senior workers speaking occasionally. Several other faculty members joined in to help this work better, particularly those interested in the behavior of solids. The audience was encouraged to speak up if they saw errors or lack of clarity. Generally, this was done in a friendly way. Privately, we discussed with those involved how they might improve their performances. Simply, we acknowledged the fact that the ability to speak well to a diverse audience and to field questions and criticisms is important to the career of younger workers, particularly. Of course, it is also important to write well, but this is better dealt with individually. This and our individual research conferences helped the group to understand one another's researches, hence to help each other. Also, we spent a lot of time individually in what is best described as mentoring. Conversely, the interactions with these diverse workers stimulated me to think more about theories relevant to their work, hence to broaden my perspectives. Truesdell and I got together to decide which applicants interested in theory were most deserving and how best to use funds available to us for their support. We did not much care what kinds of engineers or scientists they were, or whether they would neatly fit the restrictions of grants. We gave some preference to those who knew what research topics interested them, but did accept some others. We were looking for those we could help do better in research, long term, those who had a strong desire to do so and a willingness to work very hard. Senior visitors added variety to our small program and provided opportunities for younger workers to get acquainted with them. In fact, most of our former graduate students and postdoctoral workers have been successful in research, some now continuing to do so as retirees. I regard my part in this as my most important contribution to education. I cooperated with Truesdell in another venture, founding the Society for Natural Philosophy, to promote interactions between workers with diverse interests. Also, I served on two editorial boards of journals that were founded by him, the Journal for Rational Mechanics and Analysis and the Archive for Rational Mechanics and Analysis. The other editorial boards I served on were those for the Journal of Elasticity and the International Journal of Solids and Structures. For a time, my research interests did not change much. I was and still am fond of nonlinear

xxi elasticity theory. While I had become familiar with viscoelasticity theory and related experimentation, I did not think of ideas for developing it that really appealed to me, so did not do much with this. One term, I did commute to the University of Delaware to give a course on viscoelasticity theory for chemical engineers, the only side job I have had in this profession. While I got some pay for this, I did it more to promote a better understanding of continuum theory among such workers. Then, somewhat by accident, I got into liquid crystal research. In some works on fluid dynamics, I found some discussion of theories of anisotropic fluids that I considered to be unsound, and they were incompatible with the ideas about invariance of constitutive equations we had come to accept. So, I decided to formulate a simple, properly invariant theory of a fluid with a single preferred direction. I wondered whether there were any real fluids that might be roughly described by it. Chemists are likely to know about such things, so I asked Bernard Coleman, who has such a background. He suggested and told me a little about liquid crystals, materials I had never heard of. Soon, I made my first contact with a person working on these, James Ferguson, a physicist then working for Westinghouse in Baltimore. We exchanged ideas and became friends. Chemists had long known of liquid crystals, considered more as interesting scientific curiosities. Ferguson was working hard to find practical applications and to promote this idea, before there were important applications. In the years since, he has become very successful, holding many important patents. We learned of a symposium on these at a huge American Chemistry Society meeting, so we participated and made contact with the other participants. At the time, most of those interested in liquid crystals were chemists. However, it was not long before a number of physicists became involved, stimulated by the creation of the Orsay Group, headed by Pierre De Gennes. While I was not a chemist or a physicist, most of these welcomed me and each other to exchange ideas and information. A leading chemist in the area, Glenn Brown, promoted this in various ways, for example by urging us to start the International Liquid Crystal Society and to have contact with the Liquid Crystal Institute at Kent State that he had founded. Also, he invited Frank Leslie and me to write expository articles on related continuum theory. In the early days of my research in this area, some postdoctoral associates worked with me and we became a small but respected subgroup. Leslie was a leader in this group. Recently, we were saddened by his sudden and untimely death at the age of 65. I first got interested in crystals in a somewhat similar way. That is, I was uncomfortable with the theory of material symmetry, as it was applied to crystals in elasticity theory. Why should the invariance group for constitutive equations be the same as that describing the symmetry of a particular reference configuration? To me, this seemed reasonable enough for linear theory, but poorly motivated for nonlinear theory. In my first two papers on crystals, I proposed a different view of this. As I expected, I got the argument from workers in mechanics that this might be correct in principle, but that plasticity effects would dominate before the difference would become important. Being involved in other areas, I did not pursue this for quite a while. Another of my worries is loosely related. There had been quite a bit of discussion among workers in mechanics about what restrictions should be imposed on constitutive equations to exclude physically unreasonable predictions. For a time, some proposals seemed fairly reasonable to me, but they relied on what I saw as ad hoc arguments. Also, my early thoughts on crystals increased my doubts about these. My 1975 paper "Equilibrium of bars" was an attempt to illustrate what I considered to be some flaws in reasoning about thi s matter. Later analyses of twinning in crystals have provided better and realistic illustrations, some of which can be treated using thermoelasticity theory.

xxii On the personal side, Marion and I were saddened by the deaths of both of our fathers during this period. On the brighter side, we had moved into a large house near Baltimore, well built but rundown, so I spent what time I could spare remodeling it. It had a lovely yard bordered by a stream that fed a small pond. Here, our daughter Lynn married Spencer Cadmus in a wedding held outside, on a beautiful day. Later, they presented us with three grandsons, Adam, Brian and Luke, now residing in Rockford, Illinois. For a long time, I was happy enough with working conditions at JHU to turn down some better paying jobs elsewhere. Near the end of my stay, I decided that I should do better financially, to prepare for a comfortable retirement. At that time, some hostilities had developed in the Department, making it unrealistic to think that my salary would increase much. This led to the next chapter in my life.

7. University of Minnesota (UMN) In 1982, Pat Sethna, then head of the Aerospace Engineering and Mechanics Department (AEM) at UMN contacted me to ask whether I would consider an offer to join this Department. I knew a number of faculty members in different departments at UMN that I respected, and knew favorable things about some of their local activities. On the minus side, I was aware of drawbacks of large state universities, emphasized in advice from Truesdell. After some exploration, I believed that I could tolerate these until I retired. Of course, there were negotiations about financial matters, as well as whether I should have a joint appointment in the School of Mathematics. Sethna favored this, probably because it would save AEM some salary money. Not having any strong feelings about this, I did not object. This led to a satisfactory offer, involving the joint appointment, which I accepted. While Truesdell had hoped that I would stay at JHU, Sethna told me that he had received a very favorable recommendation for me from him. So we went through the traumatic business of selling and buying houses, moving and settling into my new duties. After my arrival, Sethna encouraged me to try to stimulate some activity on liquid crystals and I agreed. He arranged for me to give a lecture to a general audience, which was well received. I started a graduate course on these. Among those attending was a senior mathematician, David Kinderlehrer. He was very interested, soon becoming involved in research on these. Thus began our friendship and collaboration on several things. Actually, I had become more interested in phase transitions and twinning in crystals and covered related theory in other graduate courses. Again, Kinderlehrer attended and developed a serious interest. Primarily, these courses served AEM, although the audience included some mathematicians. Other faculty members and students were or became interested in one or both of these areas, strongly interacted with us and each other, producing some good new results. I thought it important to start something similar to the seminar that Bell, Truesdell and I originated at JHU, as did Roger Fosdick. For bureaucratic reasons, it was desirable to designate this as a course, which involved overcoming some opposition. We did this and launched the seminar, which was attended by interested faculty, graduate students and visitors, and it continued after I retired. AEM was having a problem with enrollment approaching zero in an intermediate

xxiii undergraduate-graduate level course dealing with solids. I volunteered to try something different with this, although I was not very optimistic about turning this around. After experimenting a bit, I made this a course on the thermodynamics of solids, covering elementary things that workers interested in solids should know, but rarely learn in school. It included a variety of simple but interesting demonstration experiments. There being no appropriate text, I wrote up and circulated copies of my notes as I went along. Eventually, this became Introduction to Thermodynamics of Solids, the only book I have written. The enrollment was not large, certainly not large enough to impress administrators, but the students attending were enthusiastic: they came from various departments.

The Institute for Mathematics and its Applications (IMA) had received initial funding and was operating, but there were serious doubts about whether funding would continue. The aim of IMA, suggested by the name, is to encourage stronger interactions between mathematicians and workers in other professions. After some discussion, Kinderlehrer and I decided to explore the possibility of producing a good year long program for IMA, dealing with continuum physics and partial differential equations, our areas of expertise. We wanted serious commitments from experts in these areas who were favorably disposed to interacting with those in other areas, as well as younger workers. So, we spent much time communicating with such people at meetings and elsewhere, getting the support and suggestions we needed to go ahead. We also discussed our ideas with representatives of funding agencies, whose past experiences had made them skeptical of such programs. We got their input and convinced some that ours looked promising. IMA accepted our proposal and agreed to have Millard Beatty come to help us for the year. We worked hard, doing whatever we could think of to make the program work better. While we were particularly fond of crystals and liquid crystals, we tried to encourage work in other areas as well. Generally, the staff and visitors were very cooperative, as were our wives. From what many participants told us afterwards, it was regarded as a very successful year, with some expert mathematicians finding and solving exciting problems, for example. Also, IMA got funds to continue. I believe that everyone should do some service for the community and that this is the best thing of this kind that I did atUMN. Generally, I was treated well at UMN and our finances improved enough for me to feel comfortable about retiring at age 65.1 never really enjoyed the things faculty members do to justify getting paid: teaching courses, responding to pressure to bring in money, serving on committees, etc. I was and am able to do research alone most of the time, with some occasional contacts with others, easily accomplished with modern communication systems. The kind of research I do is not costly enough to require outside funding. So, I retired at age 65.

8. Retirement When we began to think seriously about retiring, Marion was anxious to move to the West Coast, to be nearer to various relatives. I found the prospect of moving less than joyful. However, I like living on a lake and we both like being near the ocean. She found a place where we could have both, without paying a very high price for housing. So, I agreed to move there, near Florence, Oregon. Again, we bought and sold houses, buying one that

xxiv needed quite a bit of work, on a large lake. We arranged for a contractor to do some major remodeling and for a couple of years, I spent quite a bit of my time doing other improvements. When I retired, I made some decisions about what professional activities I would cease being involved in. I quit lecturing and editing, including serving on editorial boards of journals, thinking it better to have younger workers take over such activities. Also, I quit reviewing and writing letters of recommendation, except in rare cases where I could not suggest a competent substitute. As a personal matter, I never liked the idea of having retired workers influence the careers of active workers, although I grant that others can reasonably feel quite differently about this. I cannot stop thinking about research, but faced some hurdles. I was used to having office workers type papers, etc., being inept at using computers. So, I worked on this, developing enough skill to do what I want to do. Florence is a small town, with a library that tries hard to serve its users. However, there is too little demand for scientific works to make it reasonable for them to stock these. If I know a reference, they are good at borrowing it for me, if some Oregon library has it. With some help from friends and by using an internet service, I have done pretty well with library needs. Very occasionally, I attend a meeting, giving me a chance to chat with others about research, but I rely more on other kinds of communication to exchange ideas. Since I do pretty well working by myself, this, along with very occasional visits from some colleagues, gives me enough contacts with others. It would be feasible to form ties with closer universities, and this might have advantages, but I am leery of entangling alliances. Of course, being free of professorial duties and other professional activities have freed up some time for other activities. To some degree, this is countered by the fact that, like others, I am slowing up as I age but, all things considered, I do not feel hampered in doing what research interests me. I had been worrying about a difficulty brought up in discussions with my last doctoral student, Giovanni Zanzotto, now a professor at the Universita di Padova. He had been studying the literature on twinning, turning up numerous examples of cases of failure of the Cauchy-Born rule relating deformation to changes in lattice vectors. Further, he produced a reasonable argument that thermoelasticity theory is not likely to apply when such failures occur, and we did not see a viable alternative. This is covered in his paper On the material symmetry group of elastic crystals and the Born rule, Archive for Rational Mechanics and Analysis 121, 1-31 (1992). After stewing about this lack of theory, I worked out a partial theory, the X-ray theory presented in my 1997 article "Equilibrium theory for X-ray observations of crystals." With it, one can do something with analyses of X-ray observations of twins, as I have done in various later papers, and do some analyses of phase transitions. With this theory, it is to easy to use the Cauchy-Born rule, whenever one trusts it, to deal with deformation but, otherwise, it does not deal with deformation. Other workers have tried various ideas for covering both, but it seems very hard to find a reliable theory of this kind, and I have not found a new idea for dealing with this. More recently, another former student, Richard James, also a professor at UMN, brought to

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my attention a different worrisome matter concerning a theory of magnetism proposed by W. F. Brown. It involves a stress vector with a cubic dependence on the surface normal, instead of the usual linear dependence. Magnetism does involve long range interactions not considered in Cauchy's theory of stress. However, my prejudice is that the anomaly should not occur, if one uses a good field theory. In my 2002 paper "Electromagnetic effects in thermoelastic solids," I presented an internally consistent continuum theory of this kind, but it does involve giving up some entrenched assumptions. I am not content with this, but I have not yet thought much more about it, being preoccupied with other matters.

On a more personal note, our mothers both died after we moved, as did one of Marion's two brothers and some more distant relatives. Marion's other brother moved from Seattle to Southern California. So, one of our reasons for moving here no longer exists, but we cherish the time we had with the departed. My good friend Bell also died. Further, it was sad to see the decline in the health of another good friend, Truesdell, this becoming so bad that his death brought a sense of relief. On the brighter side, we attended the wedding of our son Randy to Mira Hyman, at a beautiful site in Haines, Alaska, where they live. This union has produced two more grandchildren for us, a boy, Correy, and a girl, Dana. Although neither of our children's families lives nearby, we are happy to see them occasionally. There seems to be a common belief that mechanics is not a profession, but something done by workers in various other professions. From this view, I do not fit neatly into any profession, but I have not felt handicapped by this. With some rare exceptions, I have been treated well by workers who share some of my interests, independent of their professional classification. I am touched by the efforts of good friends in different professions who have caused me to be awarded more honors than I deserve. While the honors are very nice in themselves, the approval of those that really understand some of my work is even more important to me. In writing this, I decided that it is not feasible to give due credit to more than a few of the many workers who have helped me do better in my career, and I regret this. Instead, I have tried to give some specific examples of how some people have been particularly helpful to me.

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Publications of J. L. Ericksen 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 1

Tangent lines and planes, (with F. H. Young), Am. Math. Monthly 55, 573-574 (1948). On the uniqueness of gas flows, J. Math. Phys. 31, 63-68 (1952). Thin liquid jets, J. Rational Mech. Anal. 1, 521-538 (1952). On the propagation of waves in isotropic incompressible perfectly elastic materials, J. Rational Mech. Anal. 2, 329-337(1953). Characteristic surfaces of the equations of motion for non-Newtonian fluids, TAMP 4, 260-267 (1953). On the uniqueness of ideal gas flows with given streamline patterns, Bull. Tech. Univ. Istanbul 6, 1-2(1953). Inequalities restricting the form of the stress-deformation relations for isotropic elastic solids and Reiner-Rivlin fluids, (with M. Baker), /. Wash. Acad. Sci. 44, 33-35 (1954). Large elastic deformations of homogeneous anisotropic materials, (with R. S. Rivlin), J. Rational. Mech. Anal. 4, 281-301 (1954). Deformations possible in every isotropic, incompressible, perfectly elastic body, ZAMP 5, 466-489 (1954). Stress-deformation relations for isotropic materials, (with R. S. Rivlin), J. Rational. Mech. Anal. 4, 323-425 (1955). Note concerning the number of directions which, in a given motion, suffer no instantaneous rotation, /. Math. Phys. 45, 65-66 (1955). Singular surfaces in plasticity, J. Math. Phys. 34, 74-79 (1955). Deformations possible in every compressible, isotropic, perfectly elastic material J. Math. Phys. 34, 126-128 (1955). A consequence of inequalities proposed by Baker and Ericksen, J. Wash. Acad. Sci. 45,268 (1955). Inversion of a perfectly elastic spherical shell, ZAMM 35, 382-385 (1955). Stress-deformation relations for solids, Can. J. Phys. 34, 226-227 (1956). Nonlinear elasticity, (with T. C. Doyle), Advances in Applied Mechanics 4, 53-115 (1956).*' Implications of Hadamard's condition for elastic stability with respect to uniqueness theorems, (with R. A. Toupin), Can. J. Math. 8, 432-436 (1956). Characteristic directions for certain equations of plasticity, Mathematika X, 56-62 (1956). Over determination of the speed in rectilinear motion of non-Newtonian fluids, Quart. Appl. Math. 14, 319-321 (1956). Characteristic directions for equations of motion of non-Newtonian fluids, Pacific J. Math. 4, 1557-1562(1957). On the Dirichlet problem for linear differential equations, Proc. Am. Math. Soc. 8, 521-522 (1957). Exact theory of stress and strain in rods and shells, (with C. Truesdell), Arch. Rational. Mech. Anal. 1, 295-323 (1958). Steady shear flow of non-Newtonian fluids, (with W. O. Criminale, Jr. and G. L. Filbey, Jr.), Arch. Rational Mech. Anal. 1,410-417 (1958). Work functions in hypoelasticity, (with B. Bernstein), Arch. Rational Mech. Anal. 1, 396-409 (1958). Hypo-elastic potentials, Quart. J. Mech. Appl. Math. 11, 67-72 (1958). Secondary flow phenomena in nonlinear fluids, TAPPI42, 773-775 (1959).* Anisotropic fluids, Arch. Rational Mech. Anal. 4, 231-237 (1960).

A * indicates an invited paper.

xxviii 29. Laminar shear flow of viscoelastic materials, pp. 77-91 in Viscoelasticity. Phenomenological Aspects, ed. J. T. Bergen, Academic Press, New York, I960.* 30. Theory of anisotropic fluids, Trans. Soc. Rheol. 4, 29-39 (1960). 31. Tensor fields, in S. Flugge's Handbuch derPhysik, 11 I/I, 794-858, Springer-Verlag, BerlinGottingen-Heidelberg, I960.* 32. Transversely isotropic fluids, Koll. Zeits. 173,117-122 (1960). 33. A vorticity effect in anisotropic fluids, J. Poly. Sci. 47, 327-331 (1960). 34. Poiseuille flow of certain anisotropic fluids, Arch. Rational Mech. Anal. 8, 1-8 (1961). 35. Conservation laws for liquid crystals, Trans. Soc. Rheol. 5, 23-34 (1961). 36. Kinematics of macromolecules, Arch. Rational Mech. Anal. 9, 1-8 (1962). 37. Hydrostatic theory of liquid crystals, Arch. Rational Mech. Anal. 9, 371-378 (1962). 38. Orientation induced by flow, Trans. Soc. Rheol. 6, 275-291 (1962). 39. Nilpotent energies in liquid crystal theory, Arch. Rational Mech. Anal. 10,189-196 (1962). 40. Singular surfaces in anisotropic fluids, Int. J. Eng. Sci. 1, 157-161 (1963). 41. Recoil of orientable fluids, Proc. 7th Congress on Theoretical and Applied Mechanics, Bombay, 1961,211-218(1963).* 42. Non-existence theorems in linear elasticity theory, Arch. Rational Mech. Anal. 14, 180-183 (1963). 43. Non-uniqueness and non-existence in linearized elasticity theory, Contr. Diff. Eqns. 3, 295-300 (1964). 44. Non-existence theorems in linearized elastostatics, J. Diff. Eqns. 1, 446-451 (1965). 45. Thermoelastic stability, Proc. 5 th National Cong. Appl. Mech., 187-193 (1966).* 46. Inequalities in liquid crystal theory, Phys. Fluids 9, 1205-1207 (1966). 47. A thermo-kinetic view of elastic stability theory, Int. J. Solids Structures 2, 573-580 (1966). 48. Some magnetohydrodynamic effects in liquid crystals, Arch. Rational Mech. Anal. 23, 266-275 (1966). 49. Instability of Couette flow of anisotropic fluids, Quart. J. Mech. Appl. Math. 19, 455-459 (1966). 50. Twisting of liquid crystals, J. Fluid Mech. 27, 59-64 (1967). 51. General solutions in the hydrostatic theory of liquid crystals, Trans. Soc. Rheol. 11, 5-14 (1967). 52. Continuum theory of liquid crystals, AMR 20, 1029-1032 (1967).* 53. Twisting of partially oriented liquid crystals, Quart. Appl. Math. 25, 474-479 (1968). 54. Propagation of weak waves in liquid crystals of nematic type, J. Acoust. Soc. Am. 44,444-446 (1968). 55. Twist waves in liquid crystals, Quart. J. Mech. Appl. Math. 21, 463-465 (1968). 56. A boundary layer effect in viscometry of liquid crystals, Trans. Soc. Rheol. 13, 9-15 (1969). 57. Twisting of liquid crystals by magnetic fields, TAMP 20, 383-388 (1969). 58. Continuum theory of liquid crystals of nematic type, Mol. Crystals and Liquid Crystals 7, 153-164(1969). 59. Simpler static problems in nonlinear theories of rods, Int. J. Solids Structures 6, 371-377 (1970). 60. Singular solutions in liquid crystal theory, pp. 181-192 in Liquid Crystals in Ordered Fluids, eds. R. S. Johnson and J. Porter, Plenum Press, New York, 1970.* 61. On the symmetry of crystals,2 pp. 493-496 in Problems of Mechanics of Deformable Solid Bodies, eds. Y. N. Robotnov and and L. I. Sedov, Acad. Sci. USSR (1970).* 62. Uniformity in shells, Arch. Rational Mech. Anal. 37, 73-84 (1970). 63. Nonlinear elasticity of diatomic crystals. Int. J. Solids Structures 6, 951-957 (1970). 64. Wave propagation in thin elastic shells, Arch. Rational. Mech. Anal. 43,167-178 (1971). 65. Symmetry transformations for thin elastic shells, Arch. Rational Mech. Anal. 47,1-14 (1972). 2

In Russian.

xxix 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90.

91. 92. 93. 94. 95. 96. 97. 3

The simplest problems for elastic Cosserat surfaces, J. Elasticity 2, 101-107 (1972). Simpler problems for elastic plates, Int. J. Solids Structures 9, 141-149 (1973). Apparent symmetry of certain thin elastic shells, J. Mechanique 12, 173-181 (1973). Apparent symmetry of plates of variable height and thickness, 1st. Lombardo (Rend. Sci.) A107, 71-82 (1973). Loading devices and stability of equilibrium, pp. 161-173 in Nonlinear Elasticity, ed. R. W. Dickey, Academic Press, New York, 1973.* Plane infinitesimal waves in homogeneous elastic plates, J. Elasticity 3,161-167 (1973). Complex exponential solutions of linear elasticity equations, J. Elasticity 4, 65-71 (1974). Plane waves and stability of elastic plates, Quart. Appl. Math. 32, 343-345 (1974). Liquid crystals and Cosserat surfaces, Quart. J. Mech. Appl. Math. 27, 213-219 (1974). Certain stability problems in nonlinear elasticity theory, pp. 559-565 in Advances in the Mechanics of Deformable Media, ed. A. Y. Ishlinski, Acad. Sci. USSR (1975). Equilibrium of bars, J. Elasticity 5,191-201 (1975). On equations of motion for liquid crystals, Quart. J. Mech. Appl. Math. 29, 203-208 (1976). Equilibrium theory of liquid crystals, pp. 233-298 in Advances in Liquid Crystals. Vol. 2, ed. G. H. Brown, Academic Press, New York, 1976.* Simple problems for elastic Cosserat surfaces, J. Elasticity 7, 1-11 (1977). The mechanics of nematic liquid crystals, pp.47-58 in The Mechanics ofViscoelastic Fluids, ed. R. S. Rivlin, ASME, New York, 1977.* Special topics in elastostatics, Adv. Appl. Mech. (ed. C.-S. Yih) 17, 189-244 (1977).* Semi-inverse methods in finite elasticity theory, in Finite Elasticity, ed. R. S. Rivlin, ASME, New York, 1977.* Volterra dislocations in nonlinearly elastic bodies, Analele Stiintifwe Univ. "AI. /. Cuza" Iasi 23, 423-430(1977).* On the formulation of St. Venant's problem, Nonlinear Analysis and Mechanics: Heriot-Watt Symposium (ed. R. J. Knops) 1,158-186, Pitman, London-San Francisco-Melbourne, 1977.* Transformation theory for some continuum theories, Iran. J. Sci. Tech. 6,113-124 (1977).* Research on the Mechanics of Continuous Media? ed. V. N. Nikolaevski, MIR, Moscow (1977). Bending a prism to helical form, Arch.Rational Mech. Anal. 66,1 -18 (1977). On the symmetry and stability of thermoelastic solids, J. Appl. Mech. 45, 740-744 (1978). Kirchoff and Gibbs revisited, J. Elasticity 8, 439-446 (1978). Theory of Cosserat surfaces and its applications to shells, interfaces and cell membranes, pp. 27-29 in Proceedings of the International Symposium on Recent Developments in Theory and Application of Generalized and Oriented Media, eds. P. G. Glockner, M. Epstein and D. J. Malcom, American Academy of Mechanics, Calgary, 1979.* On St. Venant's problem for thin-walled tubes, Arch. Rational Mech. Anal. 70, 7-12 (1979).* On the symmetry of deformable crystals, Arch. Rational Mech. Anal. 72, 1-13 (1979). On the status of St. Venant's solutions as minimizers of energy, Int. J. Solids Structures 16,195-198 (1980). Some phase transitions in crystals, Arch. Rational Mech. Anal. 73, 99-124 (1980). Periodic solutions for elastic prisms, Quart. Appl. Math. 7, 443-446 (1980). Variations on a bifurcation theorem by Poincare, Meccanica 13, 3-5 (1980). Changes in symmetry in elastic crystals, pp. 167-177 in Proc. IUTAM Symposium on Finite Elasticity,

This monograph is primarily a collection of papers by J. L. Ericksen translated into Russian.

XXX

98. 99. 100. 101. 102. 103. 104. 105.

106. 107. 108.

109. 110. 111. 112.

113. 114. 115. 116. 117. 118.

119.

120. 121. 122.

123.

Lehigh, 1980, eds. D. E. Carlson and R. T. Shield, Martinus Nijhoff Publishers, The Hague- BostonLondon (1981).* Some simpler cases of the Gibbs problem for thermoelastic solids, J. Thermal Stresses 4, 13-30 (1981).* Continuous martensitic transitions in thermoelastic solids, J. Thermal Stresses 4,107-119 (1981).* On generalized momentum, Int. J. Solids Structures 18, 315-317 (1982). Multi-valued strain energy functions for crystals, Int. J. Solids Structures 18 913-916 (1982). Crystal lattices and sub-lattices, Rend. Sem. Mat. Univ. Padova 68, 1-9 (1982).* Theory of stress-free joints, J. Elasticity 13, 3-15 (1983). Ill-posed problems in thermoelasticity theory, pp. 71-93 in Systems of Nonlinear Partial Differential Equations, ed. J. Ball, D. Reidel Publishing Company, Dordrecht, 1983.* The Cauchy and Born hypotheses for crystals, MRC Technical Summary Report No. 2591, University of Wisconsin, 1983; and pp. 61-77 in Phase Transformations and Material Instabilities in Solids, ed. M. Gurtin, Academic Press, Orlando, 1984.* A thermodynamic view of order parameters for liquid crystals, in Orienting Polymers, Springer-Verlag Notes in Mathematics 1063, 27-36 (1984). Thermodynamics and stability of equilibrium, in Appendix G3, Rational Thermodynamics, 2nd Edition, C. Truesdell, Springer-Verlag, New York- Berlin-Heidelberg-Tokyo, 1984.* Thermoelastic considerations for continuously dislocated crystals, pp. 95-100 in Proc. Int. Sym. on Mechanics of Dislocations, Houghton, 1983, eds. E. Aifantis and J. Hirth, Amer. Soc. Metals, Metals Park, 1985.* Some surface defects in unstressed thermoelastic solids, Arch. Rational Mech. Anal. 88, 337-345 (1986).* Stable equilibrium configurations of elastic crystals, Arch. Rational Mech. Anal. 94,1-14 (1986).* Constitutive theory for some constrained elastic crystals, Int. J. Solids Structures 22, 951-964 (1986). Twinning of crystals, pp. 77-93 in IMA Volumes in Mathematics and Its Applications, Vol. 3, eds. S. Antman, J. L. Ericksen, D. Kinderlehrer, and I. Miiller, Springer-Verlag, New York-Berlin-HeidelbergLondon-Tokyo, 1987.* Continuum theory of nematic liquid crystals, Res Mechanica 21, 381-392 (1987).* Liquid crystals ~ theory and some unsolved problems, pp. 4-5 Siam News, November 1987.* Some constrained elastic crystals, pp. 119-135 in Material Instabilities in Continuum Mechanics and Related Mathematical Problems, ed. J. Ball, Clarendon Press, Oxford, 1988.* Liquid crystals with variable degree of orientation, IMA Preprint Series #559, University of Minnesota, August 1989 and Arch. Rational Mech. Anal. 113, 97-120(1991). Weak martensitic transformations in Bravais lattices, Arch. Rational Mech. Anal. 107, 23-36 (1989).* Introduction to the thermodynamics of solids, mApplied Mathematics and Mathematical Computation, Vol. 1, eds. R. J. Knops and K. W. Morton, Chapman and Hall, London-New York-Tokyo-MelbourneMadras, 1991. Revised edition published as Applied Mathematical Sciences 131, Springer-Verlag, New York, 1998. Static theory of nematic liquid crystals, in Computer Analysis and Computer Graphics, eds. P. Concus, R. Finn and D. Hoffman, SpringerVerlag, New York-Berlin-Heidelberg-London-Paris-Yokyo-Hong Kong-Barcelona, 1991.* On kinematic conditions of compatibility, J. Elasticity 26, 65-74 (1991). Bifurcation and martensitic transformations in Bravais lattices, J. Elasticity 28, 55-78, (1992). Reversible and non-dissipative processes, Quart. J. Mech. Appl. Math. 45, 545-554 (1992); and pp. 29-38 in Nonlinear Elasticity and Theoretical Mechanics, eds. P. M. Naghdi, A. J. M. Spencer and A. H. England, Oxford University Press, Oxford-New York-Tokyo, 1994.* Local bifurcation theory for thermoelastic Bravais lattices, IMA preprint series #729, November 1990;

xxxi

124. 125. 126. 127. 128. 129. 130. 131. 132. 133.

134. 135. 136. 137. 138. 139. 140. 141. 142. 143. 144. 145.

and pp. 57-84 in IMA Volumes in Mathematics and Its Applications, Vol. 54, eds. D. Kinderlehrer, R. James, M. Luskin and J. L. Ericksen, Springer-Verlag, New York-Berlin-Heidelberg-London-ParisToyko-Hong Kong-Barcelona-Budapest, 1993.* Remarks concerning forces on line defects, ZAMP 46, Special Issue, S247-S271 (1995).* On the possibility of having different Bravais lattices connected thermodynamically, Math. Mech. Solids 1, 5-24 (1996).* Static theory of point defects in nematic liquid crystals, pp. 137-148 in Nonlinear Effects in Fluids and Solids, eds. M. M. Carroll and M. A. Hayes, Plenum Press, New York and London, 1996.* Thermal expansion involving phase transitions in certain thermoelastic crystals, Meccanica 31, 473-488 (1996).* Equilibrium theory for X-ray observations of crystals, Arch. Rational Mech. Anal. 139, 181-200 (1997). On nonlinear elasticity theory for crystal defects, Int. J. Plasticity 14, 9-24 (1998).* On nonessential descriptions of crystal multilattices, Math. Mech. Solids 4, 363-392 (1998). On groups occurring in the theory of crystal multilattices, Arch. Rational Mech. Anal. 148, 145-178 (1999). Notes on the X-ray theory, J. Elasticity 55, 201-218 (1999). A minimization problem in the X-ray theory, pp. 77-82 in Contributions to Continuum Theories Anniversary Volume for KrzysztofWilmanski, ed. B. Albers, Weierstrass Institut fur Angewandte Analysis und Stochastik, Report 18, 2000.* On correlating two theories of twinning, Arch. Rational Mech. Anal. 153, 261-289 (2000). On invariance groups for equilibrium theories, J. Elasticity 59, 9-22 (2000).* Twinning analyses in the X-ray theory, Int. J. Solids Structures 38, 967-995 (2001).* On the theory of growth twins in quartz.Mart. Mech. Solids 6, 359-386 (2001). On the theory of the a-p1 phase transition in quartz, J. Elasticity 63, 61-86 (2001). Electromagnetic effects in thermoelastic materials, Math. Mech. Solids 7, 165-189 (2002). Twinning theory for some Pitteri neighborhoods, Continuum Mech. Thermodyn. 14, 249-262 (2002).* On the X-ray theory of twinning, Math. Mech. Solids 7, 331-352 (2002).* On Pitteri neighborhoods centered at hexagonal close-packed configurations, Arch. Rational Mech. Anal. 164, 103-131 (2002). On the theory of rotation twins in crystal multilattices, J. Elasticity 70, 267-283 (2003). On the theory of cyclic growth twins, Math. Mech. Solids (in press). Unusual solutions of twinning equations in the X-ray theory, Math. Mech. Solids (in press).

1

1. Crystal Symmetry I use the classical molecular theory of crystal elasticity in the first paper, "On the symmetry of crystals," to motivate the idea that, for large deformations, the finite material symmetry groups commonly used should be replaced by an infinite group, that which can be represented by 3 x 3 unimodular matrices of integers. The subsequent article, "Nonlinear elasticity of diatomic crystals," essentially is an English version of the former paper originally published in Russian, although the details are rather different. For Bravais lattices, my work "On the symmetry of deformable crystals" analyzes the finite groups I called lattice groups and fixed sets, considered as connected sets of lattice vectors sharing the same lattice groups. Lattice groups distinguish nuances in symmetry that are not described by point groups. Although I did not realize it at the time, a simple example I presented is of some historical interest; see my 1996 paper "On the possibility of having different Bravais lattices connected thermodynamically" for a discussion and references. This work is reprinted below in Chapter 4 on Phase Transitions. "Changes in symmetry in elastic crystals" is an expository paper covering earlier thoughts on crystals. Often, lattice vectors are thought of as describing maximal translation groups of crystal configurations, used in what were later named essential descriptions. In practice, workers often use lattice vectors associated with what I called sublattice groups, describing less than maximal translation groups, used in what are now called nonessential descriptions. Some relations between lattice vectors for such descriptions are derived in "Crystal lattices and sublattices." It is noted that it can be advantageous to use nonessential descriptions to analyze some kinds of phase transitions. Briefly, a crystal n-lattice consists of n interpenetrating lattices, n-1 shift vectors describing how these are translated relative to each other. An nlattice can also be described as an n'-lattice with n' > n. All nonessential descriptions are characterized, along with procedures for extracting essential descriptions from them in "On nonessential descriptions of crystal multilattices." Some illustrative examples are presented. Results in "On groups occurring in the theory of crystal multi-lattices" deal with group theory associated with lattice groups. One aim is to make it easier to determine whether two descriptions describe the same symmetry. Another is to help classify the possible kinds of symmetry. Some special cases of the latter are treated.

3

On the Symmetry of Crystals* J. L. Ericksen Department of Mechanics Johns Hopkins University Baltimore, MD, USA Abstract We consider a group describing some symmetries of crystalline bodies which is not a subgroup of the orthogonal group, and draw some conclusions concerning elastic properties of simple cubic crystals.

1

The group

We associate a group with a crystalline material which allows one to obtain constraints on the form of constitutive relations for this material. It appears that this group has not been used in this area. In a less specific form, this group is being used in physical studies of crystal plasticity. Consider an infinite lattice of points with coordinates 3

N= 1,2,3...,

PN = ^m%aa,

(1)

where the three vectors aa (lattice vectors) are linearly independent and mfj are integers. In a simple cubic lattice, the lattice points can be identified with atoms. More complex crystals which can be represented as shifted lattices of the same type are sometimes called multi-lattices. They have the same lattice vectors and different sublattices are slightly displaced relative to each other. In this case it is common to consider the lattice points as the centers of mass of neighboring atoms belonging to different sublattices. The choice of lattice vectors is not unique. It is obvious that two sets aa and aa can be chosen in such a way that 3

^(m>

a

- m%aa) = 0 for all TV,

(2)

a=l

meaning that for any set of integers m ^ one can find another set of integers m ^ such that the equality (2) is satisfied and that the reverse statement is also true. By varying the values of m^, we obtain 3

«a = 5 3 / * ^ ,

(3)

,3=1 "This article first appeared in Problems of Mechanics of Deformable Solid Bodies (in Russian), eds. Y. N. Rabotnov and L. I. Sedov, Academy of Sciences USSR, Sudostroenie, Leningrad (St. Petersburg) 1970, pp. 493-496. English translation by Professor L. Truskinovsky, Departement de Mecanique, Ecole Polytechnique, with revisions by the author.

1

4 where the matrix

A* = 1 1 4 9 I I ,

(4)

is a nonsingular matrix with integer components. It is obvious that the inverse matrix fj,^1 has the same properties. Since the determinants of matrices /j, and / i - 1 are integers which are mutually inverse, we have det/x = ± l .

(5)

It is easy to see that any matrix with the above properties transforms one set of lattice vectors into another. The set of such matrices forms a group S and the group operation is defined in a standard way as matrix multiplication. If det/i = — 1, then obviously one can write At = (-l)/2,

det/i = l.

(6)

It is known [1, theorem 103.6] that any unimodular matrix whose elements are integers can be expressed as a product of a finite number of elementary matrices. Elementary matrices can be obtained from a unit matrix by one of the following operations: (a) permutation of two columns; (b) multiplication of a column by —1; (c) adding to the elements in one column the elements of another column, multiplied by an integer n. In view of relations of the type

(

l m 0 \ / l n 0 \ 0 1 0 0 1 0

o o i / y o o i /

=

/ 1 m+n 0 1

0 \ 0 ,

yo

\J

o

(7)

we can without loss of generality restrict our attention to the values of n equal to ± 1 . This gives a finite number of generating elements for our infinite group. In general, by prescribing lattice vectors, we do not fix the structure of the crystal. In multi-lattices we can change the relative shifts of the sublattices without changing the lattice vectors. In the classical theory of crystallographic groups one usually considers two groups. Describing the symmetry of particular configurations in terms of isometries leaving these invariant, these are the point and space groups. In elasticity theory, particular choices of the former are commonly used as material symmetry groups, but I will not do so. For simplicity, I will restrict considerations to (monoatomic) simple cubic lattices. In this case a subgroup of S can be considered with, say, det/i = 1 containing deformations which transform the lattice into an identical one. It is then natural to think that physical variables related to different configurations that are connected by a group transformation have the same value. In dealing with tensor quantities we, of course, must have in mind that their components can be different, if tensors are associated with different bases.

2

Elasticity

To relate the macroscopic deformation to changes in the lattice, we consider cubic crystals and base our analysis on the ideas used in classical molecular theories summarized, for instance, by Stackgold [2]. We begin with a lattice prescribed by three lattice vectors Aa.

5 Suppose that a macroscopically homogeneous deformation with deformation gradient F transforms the lattice into a new lattice with lattice vectors (8)

aa = FAa.

Further, suppose that the volumetric energy density W is a function of scalar products of the vectors aa, specifically W = W(aa • ap).

(9)

aa-ap = FAa • FAp = Aa • (CA0).

(10)

Then where

C = FTF = CT.

(11)

From the usual macroscopic point of view the argument of the function W is this Green's deformation tensor. It may happen that two different Fs generate lattice vectors related by a unimodular transformation from S 3

3

(12)

da = FAa = J2 »% = £ ^FAP0=1

/8=X

In accordance with our previous comments, we assume that if (12) is satisfied, then (13)

W = W(aa • ap) = W(aa • ap).

This is of course a much stronger assumption than the one which is usually made in elasticity theory. It states that the equality (13) is satisfied only for a subgroup of group S which is also a subgroup of the orthogonal group. Consider, for instance, vectors Aa that are mutually orthogonal and have equal lengths (the usual case) and shear a lattice in one of these direction by 7 a2 = 7^1 + A2;

ai=Ai;

03 = A3,

(14)

so that W becomes a function of 7 W = /(7).

(15)

By using transformations in the group S, &i = ai;

0,2 = nai±a,2 = (n±'y)Ai±A2;

a3 — ±a 3 ,

(16)

where n is an integer, we obtain

W = f{1) = f{n±1). Since (17) is true for any 7 and n, then, in particular,

f{l+\)

= /(l-(7+|))=/(^-7),

f(l + n) = /(2»-(7 + n))=/(n-7).

(17)

6 Therefore, / possesses maxima and minima at 7 = ^ , m = 0;±l;±2,...,

(18)

r=^=0.

(19)

and for these values

It is obvious that T is a tangential stress corresponding to 7. Prom (18) and (19) we find, starting at 7 — 0, that r must first reach its maximum or minimum at some 7 = 7*; moreover,

00,

(6)

F being identified with a macroscopic deformation gradient. It is envisaged that P will

9 Nonlinear elasticity of diatomic crystals

953

change into another constant vector p, though not necessarily that resulting from the same linear transformation. That is, in general, P -> p ± FP.

(7)

For given F, the notion is that p will take on some value permitting each atom to be subject to zero resultant force. Thus p is related to F in a complicated way, depending on the forms of the atomic force laws. The hope is to obtain p as a smooth, single valued function of F, reducing to P when F = 1. Cases normally considered involve small departures from a state which is, in a suitable sense, stable. When one or another of the implied properties of p(F) fails, it is almost a matter of definition that some instability will occur. With more general polyatomic crystals, the situation is much the same, except that p is replaced by a set of vectors. Of course, there is no analogous problem for the monatomic lattice. Elasticity theory does not provide a good vehicle for discussing this type of question. Said differently, there might be some merit in converting from atomic to continuum theory before facing this question. Without doing the molecular calculations, it is fairly easy to see what type of continuum theory should result. The conventional apparatus is designed to produce theories of materials whose response is determined once it is known for homogeneous deformation and constant "polarization". Our views on symmetry are geared to similar theories, so one might well have reservations about applying them to theories less local in character. We now attempt to make these ideas more coherent and more specific. 3. CONTINUUM THEORY As suggested above, we consider materials whose "state" is determined by four vector fields, suggestively labelled as P,

K-

(8)

We introduce a scalar function W, representing stored energy per unit mass W=W(p,»t).

(9)

W(Rp,Raa)= ^(P,aj,

(10)

It is assumed to be objective for every rigid rotation R" 1 = RT,

detR=l.

(11)

Here, classical molecular theory would imply that (10) also holds for improper orthogonal transformations, i.e. detR = - 1 .

(12)

Changes in aa are constrained by the requirement that they be derivable from fixed Aa and a smooth deformation X - x(X)

(13)

K) 954

J. L. ERICKSEN

as material vectors. That is K = FA.

(14)

where F denotes the usual deformation gradient F = Vx,

det F > 0.

(15)

Throughout, material coordinates X are taken as independent variables, the usual practice in elasticity. We can now write W = W(p, F) = W(p, FAJ.

(16)

To obtain equations of equilibrium, one possibility is to assume a principle of virtual work such as applies to nonlinear elastostatics, 8 fpWdV=&t.hxdS+

ff.dxdK

(17)

the integration extending over a fixed reference configuration, with mass density p. Here t and f have the usual interpretations as surface and body forces. Though we won't, we could generalize this to include a generalized body force doing work in changing p, possibly of some relevance in cases when electromagnetic fields are imposed. The only novelty in (17) involves the occurrence of p, which is to be varied independently. We then get the equilibrium equations

-w=°' V.T+f=0, dw T = P W,

(18) (19) (20)

plus natural boundary conditions of traction type, which we do not require. Here, (18) is analogous to the equilibrium equation arising in molecular theory, to be solved for p in terms of F. Formally, suppose (18) is satisfied by a certain smooth function P = P(F).

(21)

With (10), there is no loss in generality in assuming it is objective p(RF) = Rp(F).

(22)

W = W(F) = W[V(F),Y]

(23)

WfRF) = W(F).

(24)

With this, we can write with

Then, because (18) holds, SW_8W d¥ ~ d¥ '

(

'

n Nonlinear elasticity of diatomic crystals

955

We then arrive at elasticity theory. Of course, there is the possibility of nonuniqueness of solutions of (18) for p which could lead to different determinations of W, etc. We should now face the question of what invariance requirements should be imposed on W, other than (10), to account for material symmetries, presuming we are concerned with the diatomic crystals described before. Presumably, these should derive from the group obtained by combining (2), (5) and (10). To sum up, we should single out some subgroup, if not the entire group of transformations represented by aa = Rmfa^,

(26)

p = R(p + «X),

(27)

where the m's and M'S are rational integers and detm£=+l,

R-'^R7",

detR=±l.

(28)

In writing (10), we have already assumed one subgroup applies, that with m^ = 3%, n" = 0 and det R = 1. Some, but not all of the extended group of transformations can be accomplished by continuously varying p and aa, keeping the aa linearly independent. Those that can are characterized by the condition detRdetm£=l.

(29)

If R be restricted as in (11), (29) would then give detm£ = l.

(30)

For theories of such continuous variations, such as we consider, such restrictions do not seem entirely unnatural and there might well be differences of opinion as to which choice to make. Similarly, in the theory of crystallographic groups, there is occasional difference of opinion as to whether to take these to be subgroups of the orthogonal or the proper orthogonal group. If there is any other sensible reason to restrict the group, it escapes me. The remaining discussion applies to the full group or to restrictions deriving from (11) and (30). By itself, (29) is a bit awkward, for the possible subsets of m's and R's do not neatly divide into two separate groups. This induces some concern, for the group is much larger than that which has been used, with success, in the theory of infinitesimal deformations. However, the two are not as different as might first appear. In general, as represented by (26) and (27), the difference between aa, p and aa, p is not infinitesimal. Because of the discrete nature of the group, a nonzero difference is not easily converted to an infinitesimal difference. However, in special cases, the two sets of vectors need not differ at all. That is, for special choices of the vectors and certain transformations, K = RmCa,,

(31)

p = R(p + nX)-

(32)

For fixed aa and p, such transformations clearly form a subgroup. From (31), m% = R T a a ,

(33)

whence follows that the subgroup of m's form a group conjugate to a subgroup of the orthogonal group, the one which we would identify as the crystallographic group appropriate for this structure and configuration. With this hint, and what is discussed below, the reader might judge for himself whether our proposal is inconsistent with experience.

12 956

J. L. ERICKSEN

Reconsidering finite deformations, we use (13) to write Km* Up = Rm^FA^ = RFmfA^ = RFMA a ,

(34)

where M is the linear transformation such that MA a = n£A,.

(35)

The slightly ambiguous subgroup £fx represented by the m's can thus be thought of as applying to the reference lattice vectors. As the m's range over £/[, the M's range over a group £f2 conjugate to it. Said differently, they are but different representations of the same abstract group. Of course the form of the M's will be different for different choices of the reference configuration, as reflected in differences in the A a . Because of the invariance assumed, the solution of (18) indicated in (21) cannot be unique. For example, we can always add to p integral multiples of lattice vectors. However, again because of the assumed invariance of W, such different values of p, required by symmetry, yield the same value of W. Of course there remains the possibility of nonuniqueness of a less trivial character, possibly leading to a multi-valued W. To correlate with elasticity theory, we must somehow gloss over this problem. The assumed invariance then translates to W in the form W^RFM) = W(F),

Mey2.

(36)

That is, y2 is at least contained in the isotropy group. It is easily seen that it is not a compact group, hence cannot be a subgroup of the orthogonal group. This isotropy group is no different from that which we [1] previously proposed for monatomic crystals. Similar arguments suggest it should apply also to polyatomic crystals. It is only in an abstract sense that they exhibit this common symmetry, since the form of the matrices depends on the form of the vectors Aa and, of course, the form of W is expected to be different for different crystals. The situation is somewhat similar to that occurring in isotropic materials, which appear to be anisotropic when referred to most stressed configurations. The group 5^ has a finite set of generators, which can be obtained by applying a suitable similarity transformation to the generators of ^ described in [1], where some consequences of this symmetry are discussed in a simple situation. With the rather likely possibility that W is multi-valued, (36) would still hold, in the sense that the set of values of W would be transformed into itself by the indicated transformations. Of course, we are then somewhat outside the realm of elasticity theory. The system (18)-(20) seems preferable for studying such possibilities. Here, we have done little more than attempt to motivate and explain our proposed treatment of symmetry, leaving considerable room for work to be done in exploring its implications. I am firmly convinced that, by exploring these, we will gain a better understanding of the behavior of crystalline solids. Acknowledgment—This work was supported by a grant from the National Science Foundation.

REFERENCES [1] J. L. ERICKSEN, On the symmetry of crystals, pending publication. [2] W. NOLL, Arch, ration. Mech. Analysis!, 197 (1958). [3] I. STAKGOLD, Q. appl. Math. 8, 169 (1950). (Received 30 October 1969)

13 Nonlinear elasticity of diatomic crystals

957

AScTpaKT—Mcnojib3yH flJia HJiJiiocTpau,HH T e o p n n ynpyrocTH, nccneflyiOTCH coo6pa>KeHHa CHMMCTPHH, KOTopue MoryT oKa3aTbca yMecTHbiMH ana npocTbix MexaHHHecKHX Teopnft HfleajibHbix flHaTOMHbix KpHCTaJIJIOB. 3 T H COo6pa>KeHKH pa3HHTCH OT Tex, KOTOpblMH nOJlb3yK)TCH B MeXaHHKe CnJlOUJHOB Cpeflbl, HecMOTpa Ha T O , HTO 3 T H ABa noflxoflu He BbiflaioTCH 6 U T B HecoBMeCTHMbiMM. IlpM 6oJiee MMKpoCKonHHecKoK TOHKe 3peHHJi, noao6Hbie MACH BCTpenaioTca B o6cy>KAeHH5ix CKOJibJKeHmi HJIH flBoiiKOBoro CpaCTaHMH KpHCTaJIJIOB.

u Offprint from "Archive for Rational Mechanics and Analysis", Volume 72, Number 1, 1979, P. 1—13 © by Springer-Verlag 1979

On the Symmetry of Deformable Crystals J. L. ERICKSEN Communicated by R. A. TOUPIN 1. Introduction This study was motivated by my recognition of some difficulties in the theory of phase transitions. L A N D A U * [1] proposed a theory of phase transitions in crystals, covering cases where the crystal symmetry undergoes some change. Briefly and roughly, it excludes the possibility of various types of changes in symmetry. Unfortunately for the theory, various excluded transitions are observed. I do not know of a good reference describing the known exceptions. The discussion which follows refers to cubic-tetragonal transitions, which are observed in some superconductors. According to conversations I have had with experimentists, other types of exceptions are observed. It is not impossible to be fooled, for the distinction between a second order and weak first order transition is a fine one. However one assesses this, we have good reason to look at the theory with a critical eye, and to remove its imperfections. In discussing one such exception, ANDERSON & BLOUNT [3] claim to show that such transitions "... usually involve some change in internal symmetry other than mere strain ...". Said differently, thermoelasticity theory should be replaced by a theory involving a larger list of variables in the constitutive equations. If the suggested cure is to work, it should change the conclusions of LANDAU, most probably to decrease the number of changes excluded. Considering the nature of the calculations involved, it seems to me less than clear that merely enlarging the list of state variables will accomplish this, and they do not prove that it •will. Thus, I think that we should look at other weak spots in the LANDAU theory. ANDERSON & BLOUNT mention, but discount another, involving matters of continuity. I focus on another. It might well be that thermoelasticity theory is inadequate, but it should be given a fair hearing. As far as I know, we do not have a theory, accepted by knowledgeable workers, to replace LANDAU'S. Again considering the nature of LANDAU'S reasoning, it might help if we could attribute to thermodynamic potentials more invariance than is commonly done. Common molecular theories predict a rather unconventional theory of * The reference is to his collected works. As is noted there, the original papers were published in 1937. LANDAU & LIFSHITZ, [2, ch. XIV] cover much the same ground. There is a 1938 edition of this reference, which does not contain the same coverage. Archive for Rational Mechanics and Analysis, Volume 72, © by Springer-Verlag 1979

15 2

J. L. ERICKSEN

invariance for corresponding continuum theories, a fact which LANDAU and others seem not to have realized. The type of theory which results is discussed by ERICKSEN [4, 5] and PARRY [6, 7]. It does lead to greater invariance, although this can be obscured by common approximations. Perhaps the molecular theories rest on unsound assumptions but, if so, the error should be uncovered and satisfied. This is a rather global theory of invariance. To relate it to more conventional ideas of symmetry, one also needs a local theory, more like that discussed by ERICKSEN [8]. Also, one needs a theory of the symmetry of crystal configurations, which is not the same thing. My purpose is to develop some of this latter for crystals, in a manner which does not commit us to thermoelasticity theory. Also it is independent of, but could be used in conjunction with theories of the invariance of constitutive equations for crystals. Largely, observations of the transitions involve watching what happens to lattice vectors in crystals, in an environment in which the temperature, and sometimes the pressure are controlled and varied, on control paths which are reasonably considered to be differentiable. Normally, the lattice vectors will vary smoothly with the path parameter, except at isolated values corresponding to phase transitions. If, at such values, the lattice vectors remain continuous, but derivatives with respect to the parameter do not, one ascertains whether there is a latent heat associated with the discontinuity. Normally, we supplement this by requirements that elastic moduli etc. remain continuous. If there is no latent heat, etc., it is considered to be of second order. Of course, this is an operational definition, and one must exercise some judgment, in ascertaining whether data exhibits the requisite smoothness. If the thermodynamic state of the crystal can be considered to be determined by the lattice vectors and temperature, a classical theoretical definition of such transitions is essentially the same. Of course, this is what motivates the operational definition. Most expect that description of such states will involve lattice vectors, or the equivalent, and, perhaps, something else. It is rather well known that, by looking at lattice vectors alone, one cannot determine the crystal symmetry uniquely, and rather commonly, we take into account other information in judging it. Our analysis leaves open the possibility of using other data to determine it. This is a purely kinematical study, avoiding the thermodynamic reasoning which is involved in LANDAU'S theory. In these respects, it is like the classical theory of crystallographic groups. In kinematical terms, we are interested in continuous variations of lattice vectors, therefore in connected sets of these. We need to make sense of the notion that the symmetry can remain fixed, or change, as such vectors vary. The classical theory of crystallographic groups does a rather good job of treating this, if the crystal can be considered to move as a rigid body. With some care, one can relax this a bit, to cover the small deformations covered by classical linear theories of elasticity or thermoelasticity. Such things as second order transitions are outside the scope of such theories. Most nonlinear continuum theories further extrapolate, presuming that the invariance group is that used for linear theory. Thus, we have extrapolated ideas which work well enough for rigid bodies, to deformable media which might undergo large deformations, finite changes of temperature, etc. We all know that there is something wrong

16 Symmetry of Deformable Crystals

3

with this, since crystals can melt, etc., and various workers have been concerned with this. However, such theory is still in a tentative state. To some degree, I perpetuate the farce, by pretending that crystals remain crystals. For the moment, it is subtle enough to understand changes of symmetry left possible. For crystals, the theory of point and space groups is covered quite comprehensively by SEITZ [9, 10], for example, and I will presume some little familiarity with this part of the subject. In dealing with deformable crystals, I have found it advantageous to employ a third set of groups, the lattice groups. My purpose is to explain where they come from and how they illuminate the scene.

2. The Groups A set of lattice vectors is any set of three linearly independent vectors ea, a = 1, 2, 3, in E3. The collection of all of these form a set which divides naturally into two disjoint connected sets of opposite orientation, say

and

S1={ea:e1Ae2-e3>0} S2 =

(2.1)

{ea:elAe2-e3 Q2e1 = -eu

2ie2=-e2, Q2e2 = e2,

11

Qie3=-e3, Q2e3=-e3,

(5.2)

the basis being included in this set of lattice vectors. For C e / ^ J , we must have C12 = e1-e2 = {Q1el)-(Q1e2)=-C12

= O,

etc. By such calculations, we find that 1{LX)=

Cu 0 0

0 C22 0

0 0 , C33

(5.3)

where the remaining entries are abritrary, except for restrictions implied by C > 0 . Clearly, / ( L J can be viewed as the restriction to positive definite matrices of a three dimensional subspace of the six dimensional vector space of symmetric matrices. The same can be said of any other fixed set associated with V, because of the conjugacy of the groups involved. Similarly, there are other choices of lattice vectors, for example

-0-

"•(:)• •••(-!)•

which give rise to a different lattice group L 2 , with generators L\=

1 0 0 0-1 0 , 0 0-1

- 1 0 0 L 2= 0 0 1, 0 1 0 2

(5.4)

and an elementary calculation, similar to that given above, gives I(L2)=

Cn 0 0

0 0 C22 C23 , C23 C22

(5.5)

to be interpreted in the same way as (5.3). A third, L 3 , has generators of the form

L\=

i - i - i 0-1 0 , 0 0-1

L%=

- i i o 0 1 0 0 0-1

(5.6)

Cn 7(L 3 )= C 22 /2 C 33 /2

C 22 /2 C22 0

C 33 /2 0 . C33

(5.7)

with

Here, we have Clt 7(L1) + / ( L 1 ) n / ( L 2 ) = 0 0

0 C22 0

0 0 +7(L 2 ). C22

(5.8)

25 12

J.L. ERICKSEN

From theorem 5, (5.8) tells us that L1uL2 is a lattice group. Here, neither S(Lly L1^JL2) nor S(L2, L1KJL2) is the null set. It then follows from Theorem 6 that there are paths starting in the /(L^ and ending in I(L2), containing only one value of C in /(L 1 uL 2 ) = /(L 1 )n/(L 2 ). It is easy to construct some. Kinematically, it would then be feasible to have a second-order phase transition with the change of symmetry LloL2, although the point group remains V. I think that we should count it as a change of symmetry. At least I would find it noteworthy if I observed that e2 remained orthogonal to e3 on one side of the transition, but not on the other. Is this not a change of symmetry? Theoretically, we might wish to put such a transition in a different class from those in which the point group changes. I don't object to such classification, whenever it serves a useful purpose. Since C is positive definite, C 2 2 >0, whence follows that 7(L 1 )n/(L 3 )=0.

(5.9)

Thus, the change of symmetry Lx = 4>{ea)=

4>(Qea),

for any orthogonal transformation Q. Also, as is perhaps clear from the preceding discussion, it is invariant under G, (2.5)

(ea) = 4>{mbaeb\

m E G.

Actually, Cauchy did not calculate , but the Cauchy stress a, which can be T I am indebted to C. Truesdell for suggesting the name. * Forfinitedeformations of 2-lattices, such theory has been discussed by Ericksen [9] and Parry [10].

30

170 done, using his definition of it. Traditionally, this is done after introducing another assumption, but this is not necessary. For either the Cauchy theory of 1-lattices, or the Born theory, my calculations give it, in component form, as (2.6) where p is the mass density, in the present configuration. It follows from (2.4) that o- is a symmetric tensor, with the invariance property indicated by (2.7)

cr(Qea)=Q{F)=4>{Fea),

f As is discussed by Hartshorne and Stuart [12, Chapter 1], different workers have opted for different choices. Referring to choices suggested by Bravais, they remark that "These are not in all cases the ones now preferred by X-ray crystallographers."

31_

171 the ea being considered as fixed. The Cauchy stress a is related to this in the way which is commonly assumed in elasticity theory. Using the equation (2.11)

Mea=mhJb,

we can define a linear transformation M for each m £ G. It is easy to show that these form a group G', conjugate to G, a subgroup of the unimodular group. From (2.4) and (2.5) it follows that, for any of the indicated Q's and M's, (2.12)

4>(F) = 4>(QF) = 4>(FM).

In general terms, this seems consistent with the general theory of material symmetry of elastic materials, as presented by Truesdell [13, Chapter IV], for example. In his terminology, G' is the peer group, it seems. However, the peer groups commonly used for crystals are the (finite) point groups and G' is an infinite group. Use of the point groups represents extrapolation of successful practice for classical linear theories. Pitteri [8] and by private correspondence, C. M. Kwok, have made known to me different analyses, both implying that the use of G' for finite deformations is compatible with the use of a point group for infinitesimal deformations. This seems to eliminate a possible objection to the use of G' for finite deformations. There is a different issue which I now raise. We have made some rather tacit assumptions and accepted (2.9), in deducing (2.11). In terms of what is known about phase transitions in crystals, it is not too unreasonable to expect that such hypotheses will hold only for a limited range of deformations and that, for a given crystal, there might be more than one such range. Thus, we might have different peer groups applying locally, and molecular theory provides some guidelines for analyzing such possibilities. This idea remains to be explored. Later, we will say a bit more about it. Cauchy's theory and, in some cases, Born's theory leads to the well-known Cauchy relations, which are better ignored, being contradicted by experiment, for most crystals. To assess the soundness of predictions concerning invariance, we need to better understand what they are. What (2.12) does is to mathematize an old idea, that crystals must have a finite resistance to shear, because certain finite shears take the infinite crystal to an undistinguishable configuration. Rather intuitive discussions of this, such as are given in old works on applied mechanics, e.g. by Nadai [14, p. 34], make no explicit reference to molecular theories of elasticity. One can use more intuitive ideas of this kind to motivate (2.12), without introducing the assumptions of pair potentials etc. used by Cauchy and Born. Here, it is relevant that G contains elements of the simple shearing type, e.g.

32 172

(2.13)

1 0 0

r 1 0

0 0 1 ,

where r is any integer. It is one thing to characterize the invariance of the strain energy, a different matter to describe the symmetry of a particular configuration. The literature tends to confuse the two issues, presuming that the invariance of 1, it and /„ generate the same point sets, if they are related by G. I expect that crystallographers would object to calling all such vectors lattice vectors, and such terminology could well cause confusion. Thus, I will call them sub-lattice vectors. With this background, we restate these propositions: i) Lattice vectors ea, ea, etc. generate the equivalence class G of maximal translation groups. We can generate G by applying all transformations in G to one set of lattice vectors, in the manner indicated by (4). ii) Sub-lattice vectors fa can be obtained by applying a transformation, of the type (9) to any set of lattice vectors, with |detp\ > 1. Sub-lattice vectors /„ and /„ are regarded as equivalent, when one can be obtained from the other by applying a transformation in G. Actually, practice is somewhat variable, and sub-lattice vectors are, on occasion, used as lattice vectors. For example, a monatomic crystal might be described as being a body-centered cubic, suggesting lattice vectors which are orthogonal, identifiable with the edges of the cube. In our terminology, these are sub-lattice vectors. A set of lattice vectors can be obtained by using two of the edges issuing from one corner, plus the vector connecting the corner to the center of the cube. I t is easy to see, and known, that the maximal point group for a set of lattice or sub-lattice vectors is not changed, if we replace the vectors by an equivalent set. In the example just mentioned, the indicated lattice and sub-lattice vectors generate the same point group. However, in general, the point group for a set of lattice vectors differs from that for a set of sublattice vectors. For example, if we have orthogonal lattice vectors with || ex || = |e2f = ||e s |, P includes the 90° rotation (11)

Qet = et,

Qe1 = — e1,

Qe3 = e3.

42

Crystal lattices and sub-lattices

5

A possible set of sub-lattice "vectors is (12)

U = 2e x ,

U=e,,

f3=e3.

Applying Q to these, we get (13)

g/i=2/,,

QU = - l k ,

Qe3 = es,

and, because of the occurrence of the factor — | , this is not in the point group determined by /„. Similarly, the point group for a set of sublattice vectors can include orthogonal transformations not belonging too the point group for lattice vectors. I t is reasonable to expect that lattice vectors give a better estimate of the true symmetry of a crystal, and I know no reason to doubt this. Thus, some dangers are involved, in blurring the distinction between lattice and sublattice vectors. To restate remarks made in the introduction, there are cases where the kinematical assumption (1) fails to apply if we use lattice vectors, but. applies if we use selected subsets of sub-lattice vectors. Clearly, one then must use care, in properly accounting for crystal symmetries.

3. Simple observations. In the following, we consider any fixed configuration of a crystal, so the equivalence class G of lattice vectors is fixed. In describing the relations between different sets of sub-lattice and lattice vectors, we encounter another group, the group R of non-singular matrices which are rational numbers. Its significance is made clear by the following easy THEOEEM

(14)

1. If fa and fa are any two sets of sub-lattice vectors, we have

L = rlU

reE.

Conversely, if r e B, there exist two sets of sub-lattice vectors such that (14) holds. PROOF. If fa and /„ are sub-lattice vectors, and ea is any set of lattice vectors, we have matrices of integers p and p, with |det p~\ > |

43

6

J. L. Ericksen

and |det p | > | such that

(15)

l = vl>,

fa = P>»-

eliminating ea between these equations gives (14), with (16)

r=

pp-1.

Conversely, if r e E, we can write its entries in the form rl =

Pill,

where pi and q are integers, since a set of rationals has a common denominator. By multiplying numerator and denominator by an integer, if necessary, we can arrange that q > 1 and det \\p°\\ > 1. Then, if ea is any set of lattice vectors, and give two sets of sub-lattice vectors, with L =

with lattice vectors ea corresponding to the previous ea and so forth: the numbers n; and hi will be different but, of course, the two descriptions deal with the same species. One consideration is that for any species, the number of atoms per unit volume must not depend on how the configuration is described. A reasonable estimate of this is the number «, divided by the volume of a unit cell in any description. Using this, we get n,/|ei -e2Ae3| = w,-/|ei • e2 Ae 3 |, or that

(16)

52 368

J. L. ERICKSEN

hi/nt = | §i • e2 A e 3 /ei • e2 A e3 | = p/q < 1, (no sum),

(17)

where p and q are relatively prime integers. It then follows immediately that there are positive integers qi such that hi = qtp and «, = qtq.

(18)

A simple but useful result: if the numbers n, are relatively prime, the n-lattice description cannot be nonessential, for example. We will use (18) in various ways. We proceed on the assumption that the description is nonessential to get various conditions necessary for this. Since ea and efl both qualify as bases, we will have some numbers kba such that e a = kbatb,

(19)

and comparing this with (17), we get \detk\=

k = \\kb\\.

p/q,

(20)

Here, I request that the reader accept my claim that the kba must be rational numbers, which will be demonstrated later. Let d > 0 be the least common denominator (LCD) of these, so n=\\nb\\=dk

(21)

is a matrix of integers. We have said nothing about how to choose the two sets of lattice vectors, so we can transform them independently by elements of G. The effect is to change n by transformations of the form n -» mnm,

m & m e G.

(22)

So, I picked up my favorite reference on such matters (MacDuffee [5]) to see what can be done with this; all matters involving number theory used here are covered in it, almost all in his chapter 7. Other books do cover these rather elementary topics. First, there are three positive integers left invariant under these, labeled ha, called invariant factors. To calculate these, subject to the relevant assumption that det n ^ 0, (a) Find the greatest common divisor (GCD) of the numbers nba.

(23)

eabcnbdnce,

(24)

This is hi. (b) Find the GCD of the numbers

and divide this by hi. This is hi-

53 NONESSENTIAL DESCRIPTIONS OF CRYSTAL MULTILATTICES 369

(c) Set

h3=\detn\/hih2.

(25)

From (23), (24), and the formula for calculating the determinant, it follows that h3 is an integer. Besides the invariance noted above, there are two useful results: hi is a divisor of h2

and h2 is a divisor of h3,

(26)

and with transformations of the form (22), n can be put in the diagonal form n-> diag {huh2,h3} = s,

(27)

for example, this matrix being unique. This is called Smith's normal form. Said differently, n can always be represented in the form n = msm,

m&m e G.

(28)

These old results are used differently in the theory of coincidence site lattices, as discussed by Fortes [6], for example. Also, with hi, defined by (23), being invariant, it follows that the LCD denoted by d in (21) is also invariant. In addition, from the definitions of GCD and LCD, these integers are relatively prime. Let ha/d = ka/K,

(no sum)

(29)

where ka and ka are, for any fixed value of a, positive, relatively prime integers. It is immediate that k\ = h\ and k\ = d. Now, h2 and d may have a common divisor, which cancels from h2ld to reduce this to k2/k2, but this will leave h\ as a divisor of k2, and, of course, k2 is a divisor of d. Apply a very similar argument to h3ld to verify that k\ = h\ is a divisor of k2 and k2 is a divisor of k3 =>• k\ < k2 k2 > k3.

(31)

and

Given any set of numbers satisfying these conditions, it is routine to show that one can use (29) to determine admissible values of ha and d. One thing might be confusing. It is easy to construct diagonal matrices not of this form—for example, diag {2,4,6} does not qualify, 4 not being a divisor of 6. However, for any such diagonal matrix, it is a simple matter to calculate the ha. Here, the entries have 2 as a GCD so, from (23), we have h\ = 2. For (24), this amounts to looking at the products of two of these, 2 x 4, 2 x 6, and 4 x 6 , which gives h2 — 2. Then, using (25) gives

54 370

J. L. ERICKSEN

A3 = 12. The proof presented by MacDuffee [5] is constructive, a kind of blueprint for determining transformations, reducing this diagonal form to the Smith form, in particular. Of course, we want to apply these results to the X described by (19) and (20). Thus, we replace (22) by X = mam,

m & m e G,

(32)

where a=s/d = diag{k1/ki,k2/k2, h/h)

(33)

will be called the Smith matrix. With (20), this gives kik2k3/kik2k3 = det a = | det X. \= p/q.

(34)

Said differently, with a special choice of the lattice vectors §„ and e a , which I denote by fa and fa, respectively, we can put (19) in the form ia = (ka/ka)ta,

(no sum).

(35)

Rearranging the linear transformation in standard ways gives ea = Xbaeb = A e a ,

A = efl ea = Xbaeb ® e a .

(36)

4. REDUCIBLE n-LATTICES A strategy I have found useful is to understand as well as possible a particular kind of nonessential lattices, but I will do some analysis first to make this easier to understand. Here, we deal only with monatomic n-lattices, which could be a particular part of a polyatomic crystal. The idea is to determine conditions necessary for some part of it to reduce to an n-lattice, with n < n thought of as given, although this is unrealistic. So, let us pick any lattice vectors ea for the n-lattice, hereafter referred to as the original lattices, and any lattice vectors ea for the n-lattice, hereafter referred to as the reduced lattices. Consider one of the reduced lattices and all of the original lattices that share points with it, so there will be h < n of these. Of course, these must account for all points in the reduced lattice. At least one of these will share some set of noncoplanar points with the reduced lattice; otherwise, the reduced lattice would be contained in a finite number of planes, which is impossible. Pick one of the points common to this and the reduced lattice as the place from which to measure shifts. This is called the base point. Then, the assumption gives three independent equations of the form m"ea = maea, m" &ma e Z.

(37)

Obviously, this is enough to determine the numbers kba in (19) and, as promised, they will be rational numbers, so we can apply anything we have learned about them. So, we can

55 NONESSENTIAL DESCRIPTIONS OF CRYSTAL MULTILATTICES

371

choose the lattice vectors so that (35) holds, even though we really do not yet know how to accomplish this. Consider our reduced lattice. It will contain all integral combinations of the fa—in particular, fi = (£i/fci)fi,

(38)

and we know that k\/k\ is not an integer. So, this is not in the ^-lattice containing the base point. Thus, it must represent a shift to another one of the latter. Similarly, considering multiples of ix gives different shifts, but there is a limit because shifts that are integral linear combinations of fa and those whose differences are of this kind are not admissible. Working this out, we get k\ — 1 different shifts of the form mi(*i/*i)fi,

m, e S(*i),

(39)

where S(ki) is the set of nonnegative integers 0,1 • • • ifci — 1. Turning to the analog of (38) for f2, the only difference is that we can have jfc2 = 1, which gives no acceptable shifts and, from (31), this implies that k3 = 1. Then, since 5(1) contains only zero, the analog of (38) still holds, givingfc2— 1 shifts of the form m2(k2/k2)f2,

m2 € S(k2),

(40)

m3 e S(k3).

(41)

and, similarly, £3 — 1 of the form m3(k3/k3)h,

Then, considering general linear combinations of fa gives k\k2k3 — 1 shifts of the form 3

^ma(kalK)ta,

maeS{ka).

(42)

Of course, it is possible that not all points in these lattices occur in the one reduced lattice considered, so we turn it around, looking at fa = (ka/ka)fa, (no sum)

(43)

and the ^-lattice containing the base point. Similarly, this gives the k\k2k3 — 1 shifts of the form 3

Y,™a(.ka/ica)fa,

ma€S(ka).

(44)

At least formally, the shifts (42) and (44) describe an n-lattice and an n-lattice, with h = k\k2k3,

h = ^1^3,

(45)

56 372

J. L. ERICKSEN

and we obviously have h/h = fi • f2 A f 3 /f] • f2 A f3 = | ei • e 2 A e 3 /ei • e 2 A e 3 |,

(46)

checking that the two descriptions give the same number of points per unit volume. It is a bit tedious to check this in detail, but it can be done, and a brief sketch might make this clear. Consider any of the n-lattices—for example, one with points nata + (ki/kx)tu

naeZ,

(47)

Rewriting this in terms of the basis fa, we get {fi%/kx + l)fi + n2(k2/k2)i2 + n\k3 /JE3)f3.

(48)

Consider the coefficient of fi. If n 1 € S(k\), this part is covered by a shift. If not, divide nl by k\ to get a remainder in S (k\), plus an integer, and adding 1 to it also gives an integer. Similarly, the remaining terms are covered. Think about this and one can see that every n-lattice has some points in common with every n-lattice. Such nonessential descriptions are important enough to give them a special place. To this end, I introduce the following: Definition. A monatomic n-lattice is reducible if, for some h < n.itisalsodescribable as an n-lattice such that every n-lattice has some points in common with every n-lattice. What we have done establishes the following result. Theorem 1. For a reducible n-lattice, reducing to h, these numbers are related to the numbers ka and ka in the Smith matrix by n = kfak-} and n = k:\k2k3.

(49)

The above analysis only applies to that special choice of lattice vectors, and generally, one would begin by using some different set. However, these lattice vectors will also be associated with special kinds of shifts. If we begin with an n-lattice, referred to some set of lattice vectors, which might be reducible, we will not be given the numbers ka or the lattice vectors e a , and as will become clear, these need not be unique. So, we have more work to do. Until further notice, I will continue to use the special choice of lattice vectors. The next step is the following: Theorem 2. Suppose that a monatomic n-lattice is reducible, with (42) applying, for some particular choice of numbers ka and ka, along with the vectors fa. Then this collection of shifts can also be described by replacing ka by any other set of numbers ka e Z+ such that, for any fixed choice of a, ka and ka are relatively prime, making no other changes in (42). In particular, one can always take jfc! = j f c 2

=

fc3 = l .

(50)

57 NONESSENTIAL DESCRIPTIONS OF CRYSTAL MULTILATTICES

373

Said differently, a reducible n-lattice can always be reduced to a 1-lattice. Proof. First, note that such numbers ka will not be compatible with (30), in general. However, knowing the result, one can inspect the possibilities for any h < n, calculated using (49), with ica replaced by ka, rejecting integers violating (30). First, note that there is no real loss of generality in assuming that K e S(ka),

(51)

for if it is not, we can divide ka by ka, getting a remainder in S(ka) and a multiple of ka, which merely adds an integer times fa. This can be dropped since it gives equivalent shifts, a process that can be reversed. What happens if we multiply all the numbers in S(ka) by fcfl, now in this set, then reduce any of the products not in the set by dividing by ka to get a multiple that can be discarded, as before, and a remainder in S(ka)l Formally, one will get the required number of entries, so the only problem is that they might not all be different. Assuming that some two give the same, we get an equation, say, for a = 1, of the form (mi — m\)k\ +lk\ — 0,

rh\ > m\, m\ Scrh\ e S(k\), I € Z

(52)

or {mx -ml)icl+(l

+ l)Jfc, = * , .

(53)

With it] andfci relatively prime, ki must then be a divisor of mi—mi and/+1 by elementary numbertheory. However, it is easy to check that mi—mi e S(k\), making this impossible. This makes it clear that, generally, the values of ka are not unique for a given reducible n-lattice. However, the only choice I have found to be useful is that formalized by the following: Assumption:

k\ = k2 = h = 1-

(54)

What we have been doing is adjusting the diagonal entries in a, leaving the vectors fa unchanged. However, if we start with an /z-lattice, referred to some lattice vectors e a , the available shifts can generally be arranged in the required form for different choices of the vectors fa, leading to different vectors fa for the same constants ka. It might not be obvious how these are related, so I will analyze this. Let fa and fa denote two possible choices of fa. Considered only as shifts, we can add any integral combination of ea to h/h (no sum) and get equivalent shifts, for example, with ta=ta + kaPbeb,

(55)

where the pb are some integers. However, these must also be possible lattice vectors, so there must be some m & m e G such that fa = mbaeb, fa = mbaeb.

(56)

58

^ ^

374

J. L. ERICKSEN

Introduce integers qc by Pb = mbcqc,

(57)

and using (55), we get K = (K + kaqc)mbc =>fa = (Sba + kaqb)fb.

(58)

Then, taking determinants gives detpj + Kqc\\ = 1 + Kqa = ± 1 ,

(59)

kaqa = 0 or kaqa =-2.

(60)

so we must have

Calculating the corresponding barred vectors, we get ffl = fa/ka

and fa = (a/ka, (no sum).

(61)

Now, divide (58)2 by ka to get fa on the left side; on the right, replace fa by kata (no sum). This gives fa = q%,

(62)

where

\+q% q = 11^11=

q% q%

q2k2

\+q2h q2k2

q3k3 q% l+q3k3

.

(63)

What might not be obvious beforehand is that this turns out to be a matrix of integers. A calculation gives det q = 1 + kaqa = ±1 =» q € G.

(64)

So, this establishes that these two sets of lattice vectors describe the same 1-lattice; something would be very wrong with the theory if they did not.4 There is a useful generalization of the above result. We look for transformations of the form (22) that map a to itself or, equivalently, pairs m & m e G such that mar = am •&• m = crma" 1 1 such that n = kl. Then, there are numbers la(a — 1, 2, 3) such that la is a divisor of ka. In addition, Ij, is a divisor of l2, l2 is a divisor of l\, and / = hhh• Here, there is a possibility that h = 1 or l2 = h = 1» in which case, the corresponding divisor is unity. Proof. First, from (31), it follows that there are positive integers r and s such that k2 = rk^, ki = sk2 = rsk?, =$• n = k\k2k-$ — r2s(k3)3,

(69)

although these need not be prime numbers. Starting with £3, if £3 = 1, ignore it or consider it as a product of powers of prime numbers, any one being of the form p"1, where p is some prime number and n\ is some exponent. From (69), n will be divisible by p3ni. Some or all of these powers might be included in k. If all are, ignore this number. If only some are, we are left with a smaller exponent, say, p"2. Dividing n2 by 3 gives n2 = 3t + u,

ueS(3),

(70)

including the possibility t = 0. If u = 0, assign p' to each of the three sets of numbers la. If u = 1, give p' to l2 and l3, pt+i to l\. If u = 2, give pl+l to h and l2, p' to /3. Repeat the process, until all divisors of £3 are accounted for. Then turn to r and treat it similarly, the only difference being that we divide the analogous exponent by 2, giving an equal share to l\ and l2 when it is an even number and giving the extra power to l\ when it is odd. Finally,

60 376

J. L. ERICKSEN

whatever is in s and not used up by k is included in /]. Obviously, this accounts for all divisors of /, and the process gives la, satisfying all of the alleged conditions. Note that the conditions imply that l\ > I, although it is possible that I3 = 1 or that 12 — I3 = l. Theorem 3. For the situation described in the preceding lemma, the configuration can also be described as k identical I-lattices, which are simultaneously reducible merely by reinterpreting the shifts involved in the n-lattice and having the 1-lattice replaced by k identical 1-lattices. Proof. Consider the subset of shifts given by (m/*i)f,,

(71)

meS(kx).

If lx = k\, use the origin lattice and these to construct part of the /-lattice. If not, r\ — k\jl\ > 1. Then, S{kx) includes the numbers rx,2rx •••{l\ - l)rx. For these, (71) reduces to (mi//i)fi,

mx e S(h).

(72)

These shifts, applied to the origin, are to be used in constructing part of the /-lattice. This leaves the shifts with m not having r\ as a divisor, which I will reinterpret as shifting this part of the /-lattice to generate more lattices. To do so, divide such m by r\ to get m = mxrx + sx,

mi € S(lx),

sx > 0 € S(rx).

(73)

The first term on the right reduces to one of the shifts included in (72), and every value of sx allowed in (73) is attained for some m in the range mxrx + 1 < m < (mi + l ) n — 1. So, interpret the first term as shifting the base point to one of the lattices covered by (72), then shifting this by Csi//i)fi. Since this set depends only on r\, these shifts apply to all of the lattices generated by (72). I call this particular set of shifts a translation set, denoted by T\. That is, Tx = {(sx/lx)fx\sx>0eS(rx)}.

(74)

Clearly, this generates rx — 1 copies of the lattice generated by (72). If l2 — h = 1, / = lx, and we are done. If not, let r2 = ki/h and use the analog of the above argument for it to get another contribution to the /-lattice of the form (m2/l2)h,

m2 e S(/ 2 ),

(75)

\s2 > 0 € S(r2)}.

(76)

and another translation set T2 = {(s2/k2)f2

Similarly, if I3 = 1, / = lxl2, and we are done. If not, we get the last such contribution to the /-lattice as

61 NONESSENTIAL DESCRIPTIONS OF CRYSTAL MULTILATTICES 377

(m3/l3)f3,

m3 6 S(l3),

(77)

and translation set T3 = {(s3/k3)f3 | s3 > 0 e 5(r 3 )}.

(78)

Then, to get all of the original shifts covered, we need to add up these contributions. This gives the shifts 3

J2 ("WWfa,

maeS(la),

(79)

describing an / = /1/2/3-lattice, with shifts satisfying the conditions for a reducible lattice. Similarly, we get sums of elements in the three translation sets, forming a larger translation set,5 enabling one to shift the /-lattice as a unit. Note that Ta affects only the components of fa, and we have seen that this covers all of the possibilities. Clearly, these lattices all have the same Smith matrix—namely, o=diag{\lh,\/h,\/l3},

(80)

producing the same reduced lattice vectors fa for all. Of course, one needs to associate such a 1-lattice with each of the lattices in the /-lattice, and these need to be translated so that each pair has the necessary points in common. Obviously, there are infinitely many ways of describing this and, for our purposes, these details are not important. So, this completes the proof. With Theorem 3, one possibility is to arrange that / be a prime number; since k\ > 1, either it or a divisor is prime. Take such a number as / and set k = k\kik3ll. Then, with the conditions on divisors for the la, we have / prime =» h =l,h = h = 1-

(81)

Then, (67)3 is automatically satisfied, for example. Obviously, this also applies to any n-lattice, with n a prime number. Here, I have tried to elaborate the ambiguities in determining the Smith matrix for a reducible n -lattice as well as I can, and this flexibility is useful in treating more general kinds of nonessential descriptions. 5. NONESSENTIAL DESCRIPTIONS The reducible lattices are only a particular kind of nonessential lattices, albeit an important kind. Here, I will deal with the other kinds. First, we will restrict our attention to the monatomic crystals. It might be helpful to first consider two examples. Consider a configuration of a 4-lattice, involving three shifts that can be represented in the form

62 378

J. L. ERICKSEN

ei/2,

v and v + d / 2 ,

(82)

where v is any vector such that these are admissible shifts. If v = e 2 /2,

(83)

this is a reducible lattice, and we can take ta = e a . However, for most choices of v, it is not reducible. However, it contains two identical 2-lattices. One involves the lattice containing the base point and that obtained using the first shift, which has the form that makes this a reducible 2-lattice. Really, the other two shifts just translate this pair by v to give an identical 2-lattice, the simplest example of a translation set. We can reduce the two pairs to 1-lattices with lattice vectors given by f i = f 1 / 2 = e,/2,

f2 = f2 = e2,

f3 = f3 = e 3 ,

(84)

describing the 4-lattice as a 2-lattice. The shift for the latter is v, which can, of course, be expressed as a linear combination of the base vectors fa. So, all of these lattices are nonessential, although most are not reducible. If the 4-lattice is reducible, so is this 2lattice. Generally, it follows from the definition of reducible lattices and Theorem 2 that an n -lattice can be collapsed to a 1-lattice if and only if it is reducible. In this example, we have, in an implicit way, used Theorem 3 to consider the reducible 4-lattice as a pair of 2-lattices, but this is not helpful. However, it can be useful. Consider a 6-lattice, with three shifts given by (82), with v given by (83), to incorporate a reducible 4-lattice. Add two shifts of the form 7re 3 and7re 3 +ei/2,

(85)

say, to include a reducible 2-lattice. For the latter, the Smith matrix is o=diag{l/2,l,l),

(86)

and the only way to have this apply to the 4-lattice is to treat this as a pair of 2-lattices, as was done above. Do this and, using (84), one can represent the 6-lattice as a 3-lattice and nothing less: the last two shifts become equivalent to one for the 3-lattice, for example. In the general case of a monatomic nonessential ^-lattice, first recall the reasoning leading to the definition of reducible lattices. Certainly, some subset of the lattices must provide all of the points in a reduced lattice. With Theorem 2, we can arrange that just one reduced lattice suffices to replace these. If any other original lattices remain unused in this, repeat the process until all points are accounted for. There is the problem that the reduced lattice vectors are not given in advance, so the Smith matrix cannot be calculated, for example. So, it is good to consider what can be said about relations between the various numbers involved in attempting to decompose a monatomic nonessential rc-lattice, referred to some lattice vectors efl, into reducible lattices with the same lattice vectors. Of course, the latter will then be collapsed to reduced lattices, referred to some other vectors

63 NONESSENTIAL DESCRIPTIONS OF CRYSTAL MULTILATTICES

379

e a . For any particular choice of the two sets of lattice vectors, there will be a unique linear transformation relating them, determining the numbers ka. Then, from Theorem 1, the reducible lattices are all identical n-lattices, with n = kxk2h.

(87)

The total number of lattices must be unchanged by the decomposition, so h, the number of reducible n-lattices, must satisfy6 hn = nkik2k3 = n.

(88)

As is clear from the above examples, we can reduce these to identical 1-lattices because it is unimportant how these are translated relative to each other. Also, there will be one reduced lattice for each n-lattice, so we have h — number of reduced 1 - lattices.

(89)

Given n, its divisors then determine the possible values of n and ka, which might or might not be unique, as is clear from the above examples. Of course, one has to examine the shifts to determine whether an appropriate subset, suitably represented, fits one or more of the possible numbers. As a prelude to this, one might find useful the criterion presented by Pitteri [4] for a description of what is nonessential, which will be described later. If one finds that there is more than one possibility, which should one choose? The aim should be to have the n-lattice not be nonessential, which means using the smallest possible value of h. One could also use the aforementioned criterion to check whether further reduction is possible; in cases in which the calculations get complex, it can be useful to have cross-checks. Turning to the polyatomic crystals, an n-lattice is already decomposed into n,-lattices, representing the different species. We know that these and the corresponding numbers for reduced lattices n and n,- must satisfy (9), (15), (17), and (18) and that we cannot put atoms of different species in the same lattice. Thus, the n,-lattices must be reduced individually, albeit in a way that is compatible with all, meaning that the numbers ka and lattice vectors efl must be the same for all. For one thing, there is the implication that it does not matter how the different n;-lattices are translated relative to each other. As before, one concludes that the reducible lattices have the same dimension n = kxk2k3.

(90)

It follows that they are geometrically identical. What is different is that h gets replaced by a set of numbers n,-, one for each species, with hih = hik\k2k3 = n,-, and one gets a reduced 1-lattice for every n-lattice, giving the total as

(91)

64 380

J. L. ERICKSEN

m

number of reduced lattices = Y^ n,,

(92)

i=i

where m denotes the number of species. Otherwise, the analyses are essentially the same as before: one should make the number in (92) as small as possible, for example. I do not see much chance of doing much more with these matters in a general way, as they apply to a single configuration. There is one more feature that I will explain later in a discussion of point groups. However, for theory of the kind mentioned in Section 2 with fixed numbers n and, for polycrystals, n,, one should find and do some analyses on all nonessential configurations. I would not be surprised if clever persons found good tricks for making this easier, perhaps by using some different way of describing the translations involved. However, I do not have any good specific ideas about this. 6. SYMMETRY GROUPS By symmetry groups, I mean the groups used in describing the symmetry of configurations. This includes point groups, lattice groups, space groups, site symmetry groups, and what Pitteri [4] has called generalized lattice groups. Briefly, the two kinds of lattice groups can detect differences in symmetry too subtle to be detected by point, site symmetry, or space groups, as nicely illustrated by an example presented by Pitteri and Zanzotto [7, sect. 7]. First, we consider thefirsttwo listed, which are conjugate groups depending only on lattice vectors. As commonly defined, a point group P(ea) is described by P(fia) = {Q e 0(3) | Qefl = mbaeb, m = \\mbj e G},

(93)

the corresponding lattice group L(ea) consisting of these values of m, or L(efl) = {m e G \ mbeb = Qe a , Q 6 0(3)},

(94)

For a nonessential w-lattice, one could calculate these for lattice vectors originally used for these, or for those used for a corresponding reduced h -lattice that, to be definite, I assume to be essential. Since the choice of these is arbitrary, I will use the special choices described earlier, fa and fa, related by fa = ta/ka

(no sum).

(95)

So, for any Q e 0(3), we obviously have Qfa = ( Q U A «

(no sum).

(96)

Now, first suppose that Q e P(ta). Then, for some m e G, we have Qfa = mbfb =» Qfa = jL%,

(97)

65 NONESSENTIAL DESCRIPTIONS OF CRYSTAL MULTILATTICES

381

where jl is the matrix occurring on the right side of (66), that is,

m\ ji=

m\k2lk\

m\k3/k\

m\k\/k2 m\ m\k3/k2 m\k\/k-i m\k2/k3 m\

=*• det \L = detm = ±1.

(98)

As might be obvious from (65), ft is related to m by a similarity transformation, from which it follows that such jl form a group conjugate to L(ta). Generally, this is not a matrix of integers. However, it will be if (67) is satisfied, so we have

mfa/ku mfa/ki & m32k3/k2 e Z = ^ Q e P(ta).

(99)

Such Q form a subgroup of the point groups P(fa) and P(fa). Note that, generally, the two lattice group elements will then not be the same,7 although it is easy to determine conditions for any of them to coincide. Rather obviously, this subset also forms a group generally smaller than that just mentioned. This is one of various indications that lattice groups distinguish differences in symmetry better than point groups. Conversely, if Q e P(fa), corresponding to m, we can do the analogous argument to get Qffl = m% =» Qffl = ,i%

(100)

where m\ li=

m\k2/k\

m\k\/k2 rh\k\/ki m\

m\k2lh

m\k-i/k\ m\k3/k2

(101)

m\

is similar to in, the set of these forming a group conjugate to L(fa). From (68), we get the analog of (99) as m\k2/kum\k3/ki &mjk3/k2 e Z ^ Q e P(fa).

(102)

This generates the same subgroup of the point groups, and one can similarly get the same subgroup of the lattice groups. The two formulations are equivalent, except for one thing relating to the two point groups; when they are different, it is P(ta)mat correctly describes this symmetry associated with the lattice vectors only. In some cases, workers knowingly use nonessential descriptions. For example, a monatomic body-centered cubic crystal is commonly described as a 2-lattice, with mutually orthogonal lattice vectors ea of the same length and a shift given by p = ( e , + e 2 + e 3 )/2, this being of the form indicating that this is a reducible 2-lattice, with

(103)

66 382

J. L. ERICKSEN

f i = e i + e 2 + e3.

(104)

It is easy to verify that we must have *i = 2,

k2 = k3 = 1

(105)

and that a possible choice of lattice vectors is given by (104) and f2 = e2,

f3=e3,

(106)

giving fi=fi/2,

f2

=

f2 = e2,

f3 = f3 = e3

(107)

as the corresponding lattice vectors for the reduced 1-lattice. Here, it has long been known that these two descriptions give the same point group, but I think it worthwhile to consider one element as an example. For what is a random choice, I will pick a 90° rotation, with ei as axis Qe[ = ej,

Qe2 = e3,

Qe3 = - e 2 .

(108)

In terms of the lattice vectors ffl, this becomes Qf1=f,-2f2,

Qf2 = f3,

Qf3 = -f2.

(109)

Read off the lattice group element from this, and one finds that it conforms to (99), as it must, from the general theory. This gives Qt^h-h,

Qf2 = f3,

Qf3 = -f2,

(HO)

which conforms to (102), as it should. Even in this example, the 2-lattice description does give one result concerning symmetry that is at least misleading. It gives a lattice group describing a simple cubic crystal, whereas the 1-lattice version correctly gives that for a body-centered cubic. These two cubics are of different Bravais lattice types, which means that there is no way to choose the two sets of lattice vectors for essential descriptions so that their lattice groups coincide. For readers not familiar with this theory of symmetry, I recommend the paper by Pitteri and Zanzotto [8], which corrects common misconceptions about this classification. By common consent, the point and lattice groups adequately describe the symmetry of 1-lattices but not that of n-lattices for n > 1, where space groups are also commonly used. These involve translations as well as some of the transformations Q and m defined by (93) and (94). To introduce these groups, describe the translations involved in characterizing an n-lattice configuration by some set of shifts

67 NONESSENTIAL DESCRIPTIONS OF CRYSTAL MULTILATTICES

p,-,

i= l - n - \

383

(111)

in the manner mentioned in Section 2. As is discussed by Pitteri [4], to be included in an element of a space group, along with some translation, Q must act on these as indicated by QP,-=a/p;+/?ea,/?eZ,

(112)

where the as are elements of a group generated by permutation matrices and matrices obtained by replacing one column in the unit matrix by elements all equal to - 1 . As was explained in Section 2, Pitteri uses a different convention, making my columns correspond to his rows, a convention also used by Pitteri and Zanzotto [8, 9], for example. These cover the transformations given by (7) and (8). In various cases, requiring (112) to be satisfied eliminates some Q and m allowed by (93) and (94). An obvious way to fit these together for nonessential and essential descriptions is to use the lattice vectors fa and fa. An example of a case for which the allowed ms form a proper subgroup of the lattice group is included in the discussion of (145) in Section 7. Pitteri and Zanzotto [9, sect. 4.4.1] note that if (93) and (112) are satisfied for a nonessential description, there is a translation t such that (t, Q) is an element of the space group, properly calculated using some essential description. One implication is that when an element of L(fa) is consistent with (112), it also satisfies (99). However, the space group obtained from an essential description can include elements that cannot be obtained from solutions of (112) for the nonessential description, as might be obvious from my earlier discussions. For present purposes, we do not need any more of the theory of space groups.9 However, it is relevant to state a result by Pitteri [4], mentioned earlier, that does involve the same concepts. This characterizes nonessential lattices in the following way: n — lattice nonessential • 3 a\ ^ Sf and If 1 j" such that p, = ajpj + lfea.

(113)

Essentially, this is (112), with Q = 1, the point group associated with ea not being involved, making this consistent with the remarks leading to (14). Pitteri and Zanzotto [9, chap. 4] present (113) in a different but equivalent way. Both of these look very different from my characterizations, which might lead one to suspect that something is wrong. However, the two are consistent, and with my results, one can calculate possible values of a\ and I". Pitteri [4] has pointed out that certain types of as lead to solutions of (113), so I will merely fill in some details for these. I sketch an analysis of this, partly to explain this and partly to cover a point not explained before. First, consider a reducible n-lattice, referred to lattice vectors fa. In Section 4, we found that if we took the base point to be a point in one of these lattices, the shifts could be put in the form (42), later simplifying this, using (50), to become 3

J](i« fl /* a )t,,m fl €5(A o ). 0=1

(114)

68 384

J. L. ERICKSEN

There is an implication that this does not depend on which point is selected as the base point, but we did not analyze the process of shifting the base point to a point in a different lattice. This amounts to picking one of the shifts in (114), a particular choice of the numbers ma that I will label as na. Then, the obvious choice of the new shifts is 3

YlV~ma-na)/K]ia,

(115)

a=\

which need some adjusting to again be of the form (114). To do this, define integers rha by ma = ma — na if ma > na, ma =ma —na+ka

if ma < na.

(116)

It is easy to verify that, as ma takes on all values in S(ka), so does ma. Also, 3

J2(mfl/fca)ffl

(117)

is a set of shifts equivalent to those given by (115), differing from them by those integral multiples of fa. To get relations of the form (113), number the shifts given by (114) in some way, representing them as p,, and pick one corresponding to that associated with the numbers na above. This will generate the equivalent of (115) as Pi=ajvj,

(118)

where a is obtained from the unit matrix by replacing the Jfcth column by elements all equal to — 1. It has the property that « 2 = 1,

(119)

P< = a/py

(120)

so

With the juggling indicated by (116), one will get p,=fi/p;-+/jIffl,/?eZt

(121)

where a is a permutation matrix. Substitute this in (120) to get Vi=aji>j+l?ta,

(122)

« = «5,/f=a//J.

(123)

with

69 NONESSENTIAL DESCRIPTIONS OF CRYSTAL MULTILATTICES

385

For nonessential lattices that are not reducible, one needs to represent them as some number n of reducible lattices, as is explained above. To introduce shifts, locate the base point in any one of these, so the reducible lattice containing it is described as above. Then the rest can be represented in the form 3

vm + ]T(mfl/£fl)fa, ma e S(ka), m = 1 • • • n - 1,

(124)

a=l

where the vectors vm describe how the effective base points of the different reducible lattices are translated relative to the base point, forming a translation set. Alternatively, this set of vectors is a set of shifts for the corresponding essential description. They also serve as particular shifts for the nonessential. Of course, for fixed m, the set of shifts describes one of the geometrically identical reducible n-lattices, with n = kfak-}. Also, the atoms are all identical in any one of the reducible lattices. What I suggest is to shift the base point to another lattice in the same reducible n-lattice and to treat this as before. For this, label the shifts in this with the integers 1 • •• n — 1. For the rest, use the labels Pmn=V m ,

m = \--h-\,

Pmn+k = Pmn + Vk,

I

k = 1 • •• Tl - 1 J '

(125)

Some shift with an index in the set 1 • • • n — 1 will be subtracted from all, described by a matrix a of the same kind as before, giving acopy of (118) through (120). From the form of the shifts, the effect is to map each reducible lattice onto itself. To get the analogous permutations and so forth, consider each reducible n-lattice and treat it as before, with one difference. Except for that containing the two base points, there will be n shifts associated with a reducible lattice instead of n — 1, so the calculations must take this into account, but this is essentially the same for all such reducible lattices, with the labeling (125). Note that we are respecting the requirement that permutations can be applied only to lattices containing identical atoms. Put these pieces together to get the analogous permutation matrix a and integers I" to get a copy of (121), proceeding as before to get this version of (122). Of course, this covers only one of the two implications in (113), and it seems difficult to get the other from my analyses. Rather obviously, the relations between shifts and lattice vectors in (113) lead to ambiguities when one tries to use (112) for nonessential lattices, so it is tricky to base a comparison with corresponding essential descriptions on this. Since the nonessential description does cover positions of all atoms, one could, in principle, use it to calculate correct point and space groups, but it seems difficult to find a routine useful for this. However, with results obtained here, it is feasible to find a corresponding essential description and to determine the true space group, which is not always an easy task. 7. GROUPS AS LIMITS Considering the difficulties encountered in calculating correct space and lattice groups for nonessential descriptions, it seems sensible to look for alternative approaches, and one seems worth pursuing. Introduce the components of p, in the basis ea,

70 386

J. L. ERICKSEN

P, = P°*a,

pf = P,- • efl.

(126)

Then, using (93), we get the equivalent of (112) as pfmba=ajpb

+ lb,

(127)

which is trustworthy for essential descriptions. Suppose that we have some particular solution of this, giving a particular set of ms, as, and Is. These are the numbers defining an element of a generalized lattice group for multilattices. The aim is to fix these and let pf vary in a continuous manner, approaching a nonessential configuration. The ea could also vary in a manner consistent with fixing the ms. This should give estimates of symmetry of the nonessential descriptions that are relevant to continuum theory, even if they differ from those produced by analyzing reduced lattices. If it is possible to get more than one such limit, one should look at these as generators of some group. So, this is the plan. I have only explored it a bit, but I believe that it has some merit and that it is likely to be possible to develop some general theory of this kind. For the plan to work, it is obviously necessary that there exist numbers pf not all zero, such that9 pfmba = « / # .

(128)

which is a bit complicated to deal with in a general way. Also, one needs to have pf + pf take on values for some nonessential description for some values of pf. While we now understand this pretty well, it is somewhat cumbersome to deal with in a general way. So, for a first look, we will look at the simplest case, dealing with monatomic 2-lattices, the only case I will consider. Then, for the one shift p, (127) reduces to pamba=±pb

+ lb,

(129)

and (128) becomes pamba = ±pbr

o

- r ^ n ae a

when a 4= p.

(4)

Generally, one makes some particular choice of kinds of atoms and number of lattices n,- containing the ith species, considered to be fixed numbers such that m

(5) i=l

m being the number of different kinds of atoms. Also, I will make some use of the reciprocal lattice vectors (dual basis) e" such that ea-eb

= %,

ea®ea

= ea®ea

= 1.

(6)

Those interested in related continuum theories generalize this, regarding ea and pi as vector fields but, for the present study, this is not important.

79 On Groups Occurring in the Theory of Crystal Multi-Lattices

147

2.2. Some Relevant Transformations Certain transformations of ea and ra give values of these vectors describing exactly the same lattices, leave the atoms in each set the same, and change ra by simply replacing the point in the ath lattice by another. These are of the form ea -> ea = mbaeb, ra-+ra=ra+naaea,

m = \mba\ e G. < 6 Z.

(7)

(8)

Changing one point from which the position vectors are defined to another gives other transformations of the form fa - * ra + C,

C = Const.,

(9)

which could be interpreted as translating the whole configuration, something commonly regarded as inconsequential. One way to eliminate this ambiguity is to use what PITTERI [3] has named shifts: take the differences between each of the ra and any given one, for example, replace ra by ra — rv+\, reducing the number of these vectors by one. These describe how the different lattices are translated relative to some point in one of the lattices, this being called the base point. I denote a possible set of shifts by Pi,

i = l

v.

(10)

In matrices, for example those occurring in (7), I use the lower index to label rows, to get one common way of correlating composition of linear transformations with multiplication of corresponding matrices. That is, if ea —»• ea —> ea, I would describe this by ea = mbaeb, ea = mbaet = mbmcbec, the matrix for the latter being the product mm. PITTERI and some others use a different convention. Consider two linear transformations described by L(ea) = mhaeb and L(ea) = mbaef,. If one applies the first, then the second, the result is described by mbamcbec. This corresponds to the matrix product mm, if the upper index labels rows. Another thing is rather arbitrary: If two lattices consist of identical atoms and if we interchange the atoms in the two, we get a physically equivalent configuration. Thus, when the ath and /3th lattices consist of identical atoms, we get the permutation group generated by the transformations ra ->• rp,

rp - > ra.

(11)

As is noted by PITTERI [3,6], combining (9) and (11) gives transformations on shifts of the form Pi^pi=ajpj+l?ea,

Z?eZ,

(12)

where, for monatomic crystals, the v x v matrix a = \\uj \\ is an element of a group I denote by F(v), with matrix multiplication serving as group multiplication. With

80 148

J. L. ERICKSEN

my conventions, F(v) is generated by two kinds of matrices which are square roots of the unit matrix. One consists of representations of pair permutations, i.e., Matrices n{i, j) = n(j, i) obtained by interchanging the ith and jth rows (or columns) in the unit matrix => n2(i, j) = 1.

(13)

These generate the permutation group TJ(v), a subgroup of V(v). Throughout, it, along with embellishments such as n, it, etc., almost always denotes elements of 77(v). The one exception will be described later. The other set of generators consists of Matrices T ( / ) , obtained by replacing the ith column of the unit matrix by a column with all entries equal to —1. These do satisfy T (;') = 1.

(14)

For polyatomic multi-lattices, one cannot use permutations corresponding to interchanging different kinds of atoms, so F(v) gets replaced by a smaller group, a subgroup which depends on the numbers n,- referred to in (5). This requires that permutations applying to different kinds of atoms commute with each other. Of course, some permutations applying to like atoms also commute. Here, my main purpose is to achieve a better understanding of the group F(v) and some other matters relatingtolattice groups. For this, v will be considered to be a fixed integer, large enough to fit the requirements of particular analyses considered: it is not hard to determine which of the results apply to smaller values of v. Also, I allow all the permutations applying to monatomic crystals, to make this group as large as possible. It is not hard to restrict this to cover any of the various kinds of polyatomic crystals. This is a small step toward understanding lattice groups for multi-lattices, what PITTERI [6] has called generalized lattice groups. Some assumptions to be made are motivated by what little we know about these, so it is important to have some understanding of these. The above descriptions of multi-lattices can be either essential or non-essential. An n-lattice description of a configuration is of the latter kind when the configuration can also be described as an n'-lattice, for some n' < n, and essential if it cannot. Concerning this, PITTERI [6] deduced that an n-lattice description is non-essential O 3 a ^ l e r(v) B pi = pi in (12). (15)

Later I [7] worked out more explicit descriptions of non-essential configurations, along with algorithms for finding corresponding essential descriptions and examples of a's occurring in (15). The lattice groups, point groups and space groups involve additional transformations of the form ea^Qea,

ra-+Qra,

QQT = h

Q e 0(3).

(16)

All 1-lattice descriptions are essential, and, for these, the lattice group L(ea) is commonly defined by L(ea) = [m e G \ mbaeb = Qeb, Q e 0(3)},

(17)

81_ On Groups Occurring in the Theory of Crystal Multi-Lattices

149

these Q forming the corresponding point group P(ea)- Briefly, such lattice groups distinguish differences in symmetry, which point groups cannot 2 , and they form a part of the lattice group for a multi-lattice. For multi-lattices, one also uses (12), with pi replaced by Qpi, Qpi=ajPj+l?ea.

(18)

This forces Q to belong to a subgroup of P(ea), which can be proper3 denoted by P{ea, pi), the point group for the multi-lattice described by ea andp,.Like P(ea), it is invariant under the transformations replacing ea and pi by an equivalent set. Then, a lattice-group element is described by the ordered set (19)

(m,«,L),L=\\l?\\.

The lattice group L(ea, p{) for a multi-lattice then consists of such elements and, to within the conventional conjugacies, it determines a point group P(ea, /»,•) and a space group, elements of which are indicated by (t, Q), where t is a translation: if x denotes the position vector of any point, this acts as indicated by * - > Qx+t,Q€

P(ea,Pi).

(20)

While a lattice group determines a point and a space group, to within the usual conjugacies, the converse is not true, this being a reason why lattice groups better distinguish some differences in symmetry. A base point corresponds to x = 0, so its translation determines t. For a lattice group, group multiplication, denoted by a large dot, is described in terms of matrix products by (in, d, L) • (m, a, L) = (mm, aa, uL + Lm),

(21)

with the conventions used here. As is well known and easy to prove, any Q e P(ea) satisfies Qm = 1,

for m = 1, 2, 3, 4 or 6,

(22)

and combining this with (15) and (21), one gets 4 For essential descriptions, Qm = 1 => am = 1.

(23)

This provides some motivation for later discussion of elements of F(v) which satisfy am = l,

for m = 1,2, 3,4 or 6.

(24)

2 For readers unfamiliar with such theory for Bravais lattices (1-lattices), I recommend the paper by PITTERI & ZANZOTTO [8], which corrects common misconceptions concerning the logical basis for the classical classification of these. 3 This occurs in some examples presented later, for example. 4 Finitely generated groups with elements fi satisfying /3m = 1, called Burnside groups, have long been studied by group theorists, as is discussed by HALL [9, Ch. 18], for example. This is a difficult theory, and remains incomplete, as far as I know. Tofitconventions used for these, take m = 12 to cover all possibilities noted in (22).

82 150

J. L. ERICKSEN

Known examples make clear that the converse of (23) is not true: for example, one can have a = 1 when Q2 = 1. This occurs in some examples presented later. Various difficulties are encountered in trying to apply these ideas to non-essential descriptions. For one thing, (15) leads to ambiguities in the determination of lattice groups for these; (24) can be satisfied by one and not by another. Using any one, one can calculate a space group, which might be the same as that calculated using an essential description, or a subgroup of the latter5. For 2-lattices only, I [7] proved that, by a limiting process, one can always find a choice satisfying (24). I do not know whether this is true generally. This is not the place for a long discussion of such difficulties. Suffice it to say that the restriction (24) is of some interest.

3. The Group T(v) 3.1. Canonical Descriptions The pair permutation matrices defined by (13) have the property that, for any v x v matrix (i, interchanging its j'th and yth rows is described by (25)

x(i,j)it, while its ith and yth columns are interchanged by

(26)

tLK{i,j).

More generally, it is represented by a product of these generators, a word of this kind and fin applies this permutation to the columns of \i. The reverse product jr/i permutes the rows by the inverse permutation. The other generators of r(v) are the r(i), defined by (14). So any a is some word in these two kinds of matrices. This seems very complicated if v not a small integer. However, these matrices satisfy some nice commutation relations, enabling us to reduce the words to simpler forms, in a way which is more illuminating. For words containing T'S, there are two rules that can be verified by doing the matrix multiplications involved: n(i,j)r{k)

= r{k)n{i,j)

when i * k =# j,

(27)

(no sum).

(28)

jt(i, j)r(i) = r(j)n(i, j)

Among other things, (28) implies that the T'S are all similar matrices. Clearly, by using these repeatedly on any word, one can shift T'S to the left (or right) to get some permutation multiplied by a word in T'S only. For the permutations, one can also rearrange words, using the more familiar it(i, j)n(k, 5

1) = n(k, l)a(i, j)

if i, j , k, I are all unequal,

(29)

For more discussion of matters mentioned in this Section, cf. PITTERI & ZANZOTTO [2, Ch. 4].

83 On Groups Occurring in the Theory of Crystal Multi-Lattices

j r ( i , j ) n ( j , k) = j r ( i , k ) x ( i , j ) ,

(no sum).

i + j ^ k ^ i

151

(30)

The next step is to replace the latter word by a single r, using another relation, along with x2(i) = 1, i.e., r(i)r(j) = TU)yt(i,j) = n(iJ)r(i)

when i + j.

(31)

Of course, this can also be verified by matrix multiplication. By using this, one gets that any a is either a permutation or is of the form (32)

a = x(i)n. From the above discussion one gets a canonical description:

Theorem I. Any a e P(v) is either a permutation or, for some integer i, it is obtained by permuting the columns ofx(i) in some way. Alternatively, one can shift to the right, replacing (32) by a = jrT(i)

(33)

getting an equivalent to Theorem I as another canonical description: Theorem II. Any a e F(v) is either a permutation or, for some integer i, it is obtained by permuting the rows ofx(i) in some way. Those a's arise from accounting for two groups, the permutations and the translations described by (9). The x(i) can also be interpreted as permutations, so the F(v) can be interpreted as unorthodox representations of permutation groups acting on v + 1 points. To see this, go back to those ra, considered as describing v + 1 points. When I use the latter interpretation, I denote F(v) by Fv+\, with elements nv+\, to avoid confusion. Any permutation nv+\ of ra, generated by (11), induces a transformation on the translation vectors {#•! - / • „ + ! , r 2 - r

v + u

. . . ,rv - rv+\},

(34)

a possible choice of shifts. For this, one uses the obvious convention indicated by itv+\{ra-rv+\)

= nv+\(ra)~nv+\(rv+x),

a = \,...,v.

(35)

To get n{i, j), interchange r,- and ry, i, j = 1 , . . . , v, leaving the remaining ra fixed. To get T(/), use the permutation interchanging r, and rv+\ leaving the remaining ra fixed. Of course, one represents these by a's in the obvious way. However, this also covers all pair permutations needed to generate the permutation group on v + 1 points. So, what I have called F(v) can also be interpreted as a kind of representation of this permutation group, justifying the identification with Fv+\. Since this representation is unfamiliar, I will expend some ink to elaborate various properties of it. I think it important to learn how to use these representations, so I will establish some things using it directly. In T(I) the rows all contain the entries —1 and 1 except for the ith one, which has — 1 only. Permuting columns does not change this, so these permutations do not produce a different x. Also, one sees that different permutations give different

84 152

J.L. ERICKSEN

matrices. Count the total number of matrices obtained, or use familiar properties of Fv+i and you get Theorem III. The order of F(v) is (v + 1)! So, it does grow rapidly with v. Another thing worth noting is that, from (31), r(i, j) = T(i)T(y)T(i) = T(y)T(i)T(y)

(no sum),

(36)

implying Theorem IV. The matrices r(i) suffice to generate P(v). However, I do find it useful to continue to separate out those permutations denoted by n. Now, I note a few properties of this representation. Note that, in T(I), any permutation of rows leaves the ith column unaltered, since all entries are — 1. The ith row is distinguished as the only row having only one non-zero entry, - 1 in the ith place. Thus, any permutation of columns leaving the ith column in place also leaves the ith row in place. Remove the ith row and column, and what remains is a unit matrix. Recall that n is just the unit matrix with the columns permuted in some way. With such observations, it is easy to establish Theorem V. The following statements are all equivalent. (a)

The ith row of it coincides with that of the unit matrix.

(b)

The ith column of it coincides with that of the unit matrix.

(c)

jrT(i) = T(i)jr.

(d)

The ith rows ofnr{i)

(e)

The ith columns ofx(i)n

(37)

and r(i) coincide. and r(i) coincide.

It is worth noting that, if any of these statements holds and if /i is any v x v matrix, then fi and iin have the same ith row, while \LTC and //, have the same ith column. Another result is Theorem VI. T(i)jrr(i) € Tl{v) (no sum) •£> nx{i) — T{i)n.

(38)

Proof. Obviously, the equation on the right implies the statement on the left. If the statement on the left applies, use Theorem II to get an integer j and permutationt n such that JTT(i) = T{j)K =• T(i)7TT(i) = T(i)T(J)*-

(39)

From (31), this is impossible if i =(= y, implying that r(j) is a permutation in /7(v), which it is not. So, 7TT(i) = x{i)n.

(40)

85 On Groups Occurring in the Theory of Crystal Multi-Lattices

153

On the left, the product leaves the ith column fixed to agree with that of r(i). Then (37 (c), (e)) imply that the matrices on the right commute, hence that n = n, so T(Y) and n commute. Corollary. For any a e F(v), a £ Fl(v), there are integers i and j and it € TT(v) satisfying a = T(i)*=xT(j).

(41)

Proof. From Theorems I and II, we have relations almost the same as this, i.e., a = x(i)K = jrr(y),

(42)

so it is a matter of proving that n = n. If i = j , this follows from above. If i =j= j \ use (28) to put (42)2 in the form T(i)itjt(i, j) = *jr(i, y)T(i),

(43)

and then use the reasoning applied to (40). According to the theory of finite groups, matrices representing one are similar to orthogonal matrices. One easy proof involves taking any symmetric, positivedefinite tensor that is invariant under the group6 and a linear transformation transforming this to the unit matrix, then applying this to the group matrices as a similarity transformation, to convert them to orthogonal matrices. So, if all eigenvalues have the value 1, a = 1, and if all are — 1, a = —1. I think it obvious that the latter cannot be a permutation. Also, any of the remaining elements has that column of - l ' s , so7 Having all eigenvalues of a equal to — 1 O a = — 1,

(44)

and this is possible only for v = 1. 3.2. Some Examples

Here, I record examples satisfying (24) which will be useful later. One possibility is covered by x(i)n = nr(i)

& nm = 1 => (r(i)jc)m = 1,

m = 2, 4, 6,

(45)

the analog for m — 3 being impossible. Another simple possibility is that n is a pair permutation not commuting with T ( 0 , or a(i, j) = x{i)n(i, j),

i 4= j (no sum).

(46)

Then, using (28) and (31), we obtain « 2 (i, j) = r(i)T(y) = T(j)ji(i,

j) = x{i, y)T(i) = [T(i)jr(i, j)]-1

(no sum), (47)

6 7

One can find one by averaging a non-invariant tensor of this kind over the group. Essentially the same result is mentioned by PITTERI & ZANZOTTO [2, Sect. 4.3.1].

86 154

J.L.ERICKSEN

from which a?(i,j) = l,

(48)

a possibility for m = 3 . Complicating this a bit, we have « 4 (7, j , k) = 1,

(49)

a(i,j,k) = T(i)jt(i,j)jc{i,k), i =f= y 4= k 4= i (no sum).

(50)

where

The permutation occurring here is one way of describing the cycle i -*• j -+ k -+ i, which can be described by another word, using (30). To verify (49), one can do a calculation much like that above to get « 2 (i, j , k) = T(*)jr(i, y) = * ( i , y)T(fc),

(51)

using the fact, obvious from the above remark, that [ic(i, j)n(i, k)]3 = 1 (no sum)

(52)

and (45). The last example involves a cycle8 n of five different numbers, say i —> y —»&—>•/—» m —>•(,

(53)

which is representable in various ways as a product of four pair permutations. I do not belabor the calculations showing that a(i,j,k,l,m)

= r(i)ic

(54)

satisfies a6(i,j,k,l,m)

= l.

(55)

They are routine. Except for (45), the examples indicate how cycles in nv+\ are represented in F{v), when the points they permute include the base point. Later, we will need this. Now is the time to take stock, to characterize all other possibilities for satisfying (24). 4. Characterizations The aim is to characterize all elements of F(v) satisfying (24). It is to be understood that trivial possibilities are to be excluded, like having a = 1, or having a 4 = 1 be satisfied because a2 = 1.1 will make use of results concerning permuta8 Group theorists use a different notation for cycles, but different workers use different conventions. Some write (53) as (j, j , k, I, m), others as (m, I, k, j , i). Here, I use a different notation to avoid confusion.

87 On Groups Occurring in the Theory of Crystal Multi-Lattices

155

tion groups which are presented in various books, for example, those by GALLIAN [10, Ch. 5], JUDSON [11, Ch. 4] or M A C D U F F E E [12, Ch. II]. One is that any permu-

tation can be represented as a product of disjoint cycles, meaning cycles permuting disjoint point sets. Of course, matrices representing these commute. First, we have Characterization 1. There are three kinds of possibilities for having a2 = 1 : (a)

v=1

and a = - I ,

a=ii,jt

(b)

2

=l

9

(56)

or

(57)

it2 = 1.

(58)

(c) For any possible value ofi, one can have a = T(i)it = nr{i), Here, it = 1 is allowed in (c) only. Proof. To satisfy a2 = 1, the eigenvalues of a must be ± 1. If all are — 1, then (44) implies that (a) holds. If all are 1, then a = 1 which is excluded. If a is such that neither (a) nor (b) applies, Theorem I implies that (32) must hold. Then, it is easy to use Theorem VI and (45) to show that (c) must hold. So, we find nothing new. Note that for v = 1, the only possibilities are a = ± 1, so we have already covered all of the possibilities for this case. Next, we have Characterization 2. There are two kinds of possibilities for satisfying a? = 1 : It = Tl\Tt2 = Tt2lti,

(79)

where n] = 1,

n\ = 1.

(80)

One can take JTI=JT3

and

it2 = it~~.

(81)

So, as is implied above, one can construct such permutations for v ^ 5 or represent a cycle on six numbers in this way, for example. In the latter case, it\ and it 2 are not disjoint cycles. With the characterizations, we describe all elements of F(v) satisfying (24). However, in considering most lattice groups, one needs to deal with products of these and for this, results contained in Sect. 3 are useful.

90 158

J.L.ERICKSEN

5. Classifying Configurations 5.7. A Criterion For classifying configurations, what seems to me to be the most obvious criterion is to copy what is done to get the usual fourteen Bravais symmetry types for 1-lattices. This is a bit ambiguous, since descriptions for multi-lattices can be essential or non-essential, while those for 1-lattices are always essential. However, with the known examples of cases where non-essential descriptions give incorrect indications of symmetry, it seems clear that we should not rely on these. On these grounds, I support the proposal by PITTERI & ZANZOTTO [4]. In my words, this is Two multi-lattices are of the same symmetry type provided that it is possible to choose lattice vectors and shifts providing essential descriptions so that their lattice groups coincide.

(82)

This seems to me to be the obvious generalization of what is done for 1-lattices and I know of no alternative proposal, or of evidence that crystallographers have given thought to the matter.This does not distinguish between enantiomorphs. Here, I do not deal with the corresponding theory of enantiomorphism, although it deserves to be treated. To elaborate this a bit, suppose that (17) and (18) are satisfied by some values of the matrices involved, and some particular choice of ea and />,•. Using (7) and (12), we can introduce any different choice ea and pt by transformations of the form ,-

ea — mbneb, -

Pi=aJipj+lfea,

m € G, -

« e r ( v ) , If € Z.

(83)

Here, I see no clear reason for restricting a to satisfy (24). A routine calculation then gives Qea = mbaeb,

QPi = a{Pj + Ifea,

(84)

where the transformed lattice group is given by in = m~ mm,

a. = a

aa,

L=a

[aL + Lm — Lm)].

(85)

Note that (85)2 gives am — 1 implying that am = 1, even if a does not satisfy (24). Ideally, we would like to understand the structure of all lattice groups, with equivalencies eliminated, but it seems clear that this is a formidable task. So, I think it sensible to begin by attacking simpler problems. I will demonstrate that one general kind of problem is tractable for at least some kinds of multi-lattices. It is described as follows: 1. Fix a value of v and pick a conventional description of lattice vectors for one of the fourteen types of 1-lattices.

9)_ On Groups Occurring in the Theory of Crystal Multi-Lattices

159

2. Consider the corresponding groups L(ea) and F(v) restricted by (24), and all of their subgroups. 3. For each subgroup S c L(ea) and S c F(v), determine whether there are any corresponding lattice groups and, if so, find a maximal set of inequivalent ones. With the indicated restrictions on v and the type of lattice vectors, this determines all symmetry types. Later, I present solutions of this problem for some rather simple situations. Here, I discuss some general ideas for solving such problems. In the theory of 1-lattices, dealing with (17) is a familiar problem. For the rather standard choices of lattice vectors for the fourteen types, the lattice groups L(ea) are known, and it is not hard to pick out or look up their possible subgroups. Essentially, this takes care of all possible solutions of (17) and fixes corresponding m's. Set m = 1; there are still some remaining transformations applying to shifts. However, as will soon become clear, fixing the m's does not force m = 1, and it is important to consider the other possibilities. It is well known and easy to prove that, for m and Q related as in (17), mm = 1 Qm = 1,

(86)

so (23) can be replaced by mm = 1 => am = 1,

m = 1, 2, 3, 4, 6.

(87)

With this and the characterizations given in Sect. 4, one can, in principle, sort out the possible choices of the subgroups 5 and £ referred to above, for any given value of v, although this can be laborious unless v is quite small. Turning to (18), I [7] suggested introducing the components p°=Pi-ea,

Pi = P-ea,

(88)

using this to replace (18) by the equivalent p^mba=aJiPf

or pm = ap + L,

(89)

where p = \\p"\\ etc. The iih row of L is denoted by Lj. So, the problem is to characterize all equivalence classes of solutions of sets of equations of the form (89), where equivalence is defined by (82), with the m's and a's taken as the elements of the possible choices of S and Z, respectively. Of course, the p" in (88) must satisfy the admissibility conditions implied by (4): for fixed i and j , neither these numbers nor any of the differences pf — p", i =j= j , can all be integers. One general result that follows from (87) and (89) is m = 1 => a = 1 and L = 0,

(90)

so the identity is the only lattice group element with m — \. Now, with 5 fixed, we can still transform the lattice vectors without changing it, using the normalizer group of S, denoted by N(S), with N(S) = {m€G\meS^>

mmm~x

e S\.

(91)

92 160

J.L.ERICKSEN

Introduce the transformation ea = mbaeb,

in e N(S).

(92)

For the same shift vectors, this transforms the matrix p of components to p = pm\

(93)

Transforming (89) gives pm = ap + L,

(94)

with L = Lm~\

m = ihmih1

6 S.

(95)

So if (89) were satisfied before transforming it, (94) would be satisfied. In a similar way, one can treat any subgroup E of F(v), using its normalizer N[Z] = {a e r(v) | a e E => aaeT1 e Z1}

(96)

and, as an analog of (92)-(95), one has, assuming that the lattice vectors are unchanged and that shifts are transformed as vectors, P = «/>,

pm={

\uau.~lp

+ L,

a e S,ae

. [ap + L,

(97) N(S), (98)

a 6 T,

where L = aL.

(99)

Of course, one can compose these two kinds or generalize them by also adding integers to components of p as in (12). Some readers might find it easier to follow later discussions if they use these transformations to write down a formal definition of a normalizer 12 of a lattice group L(ea, Pi)- The set of equations of the form (89) covering all of S and Z is mapped onto itself by these transformations which preserve admissibility conditions and lattice vectors, but obviously parts thereof get changed. As is well-known, there is no way to choose lattice vectors for different types of 1 -lattice vectors to make their lattice groups coincide. Thus, lattice groups involving two such different lattice groups cannot be equivalent. However, two such do share subgroups in some cases, and use of these transformations enables one to find equivalencies associated with this, for example. There is another way to use essentially the same transformations. Consider replacing N(S) by L(ea) in (92)-(95). Often, this maps a subgroup S to one which is different, but conjugate, 12

The descriptions of normalizers used here are adequate for analyses to be done here, but some generalization might be useful for other purposes. Implicitly, the calculations allow for multiplying ea and p, by the same scalar, for example.

93 On Groups Occurring in the Theory of Crystal Multi-Lattices

161

and is also equivalent. Similarly, one can replace N(E) by F(v) in (97)-(99) to similarly relate groups which are conjugate to £ in F(v) and equivalent. Once one understands these kinds of equivalencies, it is not really necessary to use these modified equations. In using lattice groups to determine symmetry types of particular configurations, one needs to exercise some care. First, given values of ea and pi providing an essential description, these uniquely determine a maximal lattice group: equivalent choices determine equivalent lattice groups, and any of these can be used to characterize the symmetry type of the configuration. Eventually, workers may establish conventions for making particular choices but, here, I make some arbitrary choices. Second, one lattice group applies to various configurations, sometimes including some for which it is not maximal. For these, it does not properly describe their symmetry types. In the theory of 1-lattices, the theory of fixed sets helps in accounting for this, but we do not yet have an analog of this for multi-lattices. To represent a symmetry type, a putative lattice group should be maximal for at least one choice of ea and pt, so one should bear this in mind. With these facts, it might not be so obvious how best to define equivalence of lattice groups. What I think best is to regard two as equivalent if and only if every configuration represented in one is also represented in the other, by values of ea and Pi which are generally different but equivalent. In the discussions to follow, I use these ideas freely without belaboring them. 5.2. Remarks While the general problem just outlined seems formidable, something might be learned by studying simpler special cases. I think that it is feasible to work out all of the symmetry types for the smallest values of v, although even this seems to require quite a bit of calculation. For 2-lattices (v = 1), one has only a = ±1 to consider, but in combination with all subgroups of the 14 lattice groups for 1-lattices. For 3-lattices, one can get the six elements of F(v) from the listing of PITTERI & ZANZOTTO [2, Sect. 4.5.2] and it is also obvious from the routine implied by Theorem I. They are a\ = 1, three elements satisfying a2 = 1, which are 0 1 «2 = JT(1,2)=

i

-10 ,

«4 = T(2) =

«3 = T(1)=

1 o

^ ,

(100)

-1 _{ ,

and two satisfying a 3 = 1, «5 = T(1)JT(1,2)=

i

,

O6 = T(2)3T(1,2)=

,

(101)

94 162

J.L. ERICKSEN

these being inverses. Concerning (100), a.2, L = (0, 0, 0), pl = \,p2=0 1

=» L = (-1,0,0),

1

p =0, p = \ px =P2 = \

=> L = (0,-1,0),

(124)

=> L = ( - 1 , - 1 , 0 ) ,

these being selected so that adding integers to one set does not give another. The value of p3 not important as long as it gives an admissible shift vector. Working out the numbers for the other elements for each of the choices of shifts, one gets four lattice groups of order four, with the m = - 1 contribution given by (121), the remainder by (mp, 1,0, 0,0),

(-ffip,-1,0, 0, 0),

(m p , 1 , - 1 , 0 , 0 ) ,

(-mp,-1,1,0,0),

(mp, 1 , 0 , - 1 , 0 ) ,

( - n i p , - 1 , 0 , 1,0),

(wtp, 1 , - 1 , - 1 , 0 ) ,

( - / M P , - 1 , 1, 1,0).

(125)

Now consider (123) with (114) and (115) for the primitive monoclinics. Here the values of px and p2 do not matter, with the same proviso as before, and one gets two possibilities for p3, which can be taken as p3 = 0 or 1.

(126)

Calculations then give two lattice groups, with the m = — 1 contribution given by (121), the rest by (wip,-1,0,0,0),

(-win, 1,0, 0, 0),

(wip,-1,0,0, 1),

(-wip, 1 , 0 , 0 , - 1 ) .

(127)

So, altogether, we get down to six lattice groups of order four for the primitive monoclinics, by the method used for establishing some equivalencies. Proceeding

98 166

J. L.ERICKSEN

in the same way for the base-centered monoclinics, starting with (122), we take choices of pa from (117) as p1 =0,p2

= -±p3

=> L = (0, 0, 0),

Pl = \, P2 = -\p" =*• L = (-1,0,0), 1

7

2

1

( 1 2 8 )

%

3

p = 0, p = i ( l - p ) =» L = (0, - 1 , 0), p3 being arbitrary, with the usual proviso. This gives four lattice groups. Working these out, one finds that they are described by (121) and (mi, 1,0,0,0),

( - m f e , - 1 , 0 , 0,0),

(mh, 1 , - 1 , 0 , 0),

( - m * , - 1 , 1 , 0,0),

(mb, 1,0, - 1 , 0 ) ,

( - m t , - 1 , 0 , 1,0),

(mh, 1 , - 1 , - 1 , 0 ) ,

( - m i , - 1 , 1 , 1,0).

(129)

Proceeding in the same way with (123), we start with (118), and here it does not matter what values we take for p1 and p2, but there is just one choice for p3, to within the usual equivalencies. It can be taken as p3 = 0 = > L = (0,0,0).

(130)

This gives one lattice group, described by (121) and (mi,-1,0,0,0),

( - m i , 1,0,0,0).

(131)

There is the question of whether we have found all of the lattice groups. Since in m = ± 1 must be included in any lattice group, there are no others of order two. These, combined with any element involving ± m p or ±m(, give one of the groups of order four listed above. If we try to form a group containing elements of the form (nip, 1, —)and(m p , —1, — ),for primitive monoclinics, we need to make the values of pa given by (113) and (114) agree. This allows for non-essential descriptions only, a possibility I have excluded for determining lattice groups. Examination of other combinations indicates that they are to be rejected for the same reason. So there are no other possibilities. 6.2. Equivalencies We need to determine which of the groups of order four we found are equivalent. For either kind of monoclinics, these involve all elements of L(ea). From the theory of 1-lattices, we know that there is no way to choose lattice vectors to make these two kinds of lattice groups coincide, implying that multi-lattice lattice groups for them cannot be equivalent. As to lattice groups for the same kind, consider the possibility that one is of the kind (122), the other of the kind (123). For this, it is easy to see that some transformation included in (85) must either convert mp (nib)

99 167

On Groups Occurring in the Theory of Crystal Multi-Lattices

to — mp {—nib), or transform a = 1 to a = — 1, which is impossible. This leaves the possibility that some of the kind (122) and/or that some of the kind (123) are equivalent. Consider the primitive monoclinic kinds. One can put e2 — e\,

e\=ei,

£3 = C3,

(132)

an admissible change of lattice vectors which is not equivalent to an orthogonal transformation, not in L(ea). However, it is easy to check that L(ea) = L{ea), so it is in N[L(ea)]. Introduce a new orthonormal basis by i = -cosyi + sinyj,

j = sinyi + cosyj,

k = k,

(133)

which gives e\ = b(sin yi + cos yj),

e-i = aj,

£3 = ck.

(134)

Essentially this is (103) with a and b interchanged. For any shift P = Paea = Paea

(135)

P2 = p\

(136)

where P1=p2,

P3 = P3-

Thus for primitive monoclinics, just interchanging the values of /?' and p2 in one lattice group produces an equivalent lattice group. Checking through those listed above, one finds that The groups described by (125)2 and (125)3 are equivalent.

(137)

There is another equivalence. Introduce the transformation of lattice vectors ^2 = «2.

e\=e\+e2,