Symposium on Teaching CrystaIIography
The Teaching of Crystallography to Materials Scientists and Engineers -
Bernhardt J. Wuensch Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139 The concerns of materials science and engineering have their roots in structure, and classes that deal with crystallogranhv, diffraction, and crystal chemistry have long been an important part of the required undergraduate core in the materials science and engineering curriculum at MIT. This material is treated at the undergraduate level in two subjects. A maior materials laboratory deals with state-of-thea r t instruments that probe structure a t all of its levels and, with two lectures and six hours of laboratory work per week, renresents the lareest course in the curriculum in terms of crkdit units. A class in crystallographic symmetry, offered in the fall semester, serves as the introduction to the department curriculum for most students. Our conviction is that full understandina and the ability to use symmetry theory may be conveyed only if the results are derived systematically as opposed to being presented and rationalized after the fact. Crystallography is neither the easiest nor most popular topic to present to a class whose interests span a range from solid state science to engineering. Among the challenges are its seeming abstract and formal nature and the fact that it is founded upon geometry rather than converging to a paradigm in algorithmic form, a situation with which students are more familar. Our approach is to couple the treatment of symmetry with a topic in which crystallography has direct and taneible conseauences. examples heinn diffraction, tensor desiription of crystal &sotiopy or crystal chemistry, and defects. A second response to the challenge is to capitalize upon the considerable aesthetic and intuitive appeal of many of the manifestations of symmetry in the presentation of lectures and in selection of problems: Before discussing the philosophy, content, and approach to teaching of these courses, however, I would like to put them in the framework of the disciplines of materials science and eneineerine as thev have develoned over the nast 30 years, and furtTher, hriLfly indicate tke range of crystalloeranhic tooics contained in the materials science and ennineering curriculum a t M.I.T. ~~
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Development of Materials Sclence and Englneerlng Materials Science and Materials Engineering emerged as recognized disciplines relatively recently, no earlier than the 1950's, with the growing need for fundamental understanding of material behavior as opposed to empirical art. Such requirements were first encountered with metals when certain applications demanded the development of tailored, high-performance alloys. Similar demands were imposed on ceramic and, subsequently, polymeric materials a decade or two later. The shift from empiricism was, for all these materials. usuallv associated with a transition from low-cost, high-volume, predominantly structural uses of a given class of materials to high-technology, high-value-added applications. Appreciation gradually grew that these concerns all transcended separate classes of materials and the industries based upon them. An electron microscopist or diffractionist, for example, may he called upon to interpret disparate phe494
Journal of Chemical Education
Figure 1. Graduate degree programs within the Science and Engineering at MIT.
Department
of
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nomena in a metal, ceramic, or polymer, depending on the nature of the sample placed in an instrument. Engineers are increasingly called upon to design and fabricate systems or devices in which various classes of materials must be employed in concert.' This broader perspective has become known as materials science or engineering. Academic programs in these disciplines not uncommonly evolved from a traditional metallurgy department. Less frequently, established programs in metallurgy and ceramics were combined. At present, some 80 universities in the United States offer undergraduate programs or options in one or both of these areas ( I ) . Materials science, accordingly, is synonymous with the science of relations between the structure of matter and its properties. Materials engineering is concerned with the modification of properties and performance during and after nrocessinp, and with manufacture. "Structure" in the mateiials disciplines is understood in its broadest sense: not only crystal structure and ~lectronicand point defect structure, h& also structural features on more macrosconic scalesdislocations, grain boundaries and interfaces, polyphase assemhlaees.. and cornnosite materials. The concerns of materials science merge and overlap hroadly with those of solidstate physics and solid-stateihemisrry. 1ndeed.a distinction is igften impossihle. When one may he made, it is d e n on rhe basis that materials science, with its historic links tu the materials industries, is n~ncernedwith materials and prop. erties of rele\,ance to technology. Man). impurtnnt pruperties are dominated hy defects, impurities, or interfaces, and kinetic processes are commonly complicated by competing mechanisms. The challenge and interest of the field to many
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An electronic chip, with its ceramic substrate, semiconductor circuit elements, metallic connections, and polymeric encapsulation. provides an excellent example.
workers lies in this complexity and ambiguity. In other words, a system is often selected for study and analysis less because of its promise to provide a clean-cut experiment and unambiguous result than because of the relevance of the problem to technoloev or to real materials of eneineerine interest. This observiiion may serve to resolve theippare; contradiction that. at the author's institution and. indeed. a t most universities, department of materials science are housed within schools of engineering. Crystallography, in its broadest sense, is central to teaching and research in materials science and engineering, as these disciplines have their roots in structure. Classes and laboratories in X-ray diffraction and crystallography thus figure prominantly in the curriculum of nearly every academic program in these areas. Role of Crystallography and Dlffradlon In the Materials Curriculum The Department of Materials Science and Engineering at MIT
As content of a course mav be ~ r o n e r l vevaluated onlv within the context of the institution a&hGh it is presenteh and the student body it is designed to serve, a brief overview of the Department of Materials Science and Engineering a t MIT will be given. One of eight departments within the School of Engineering, it has a faculty of 40, approximately 250 graduate students, and 4 5 6 5 undergraduates per entering class. Although of moderate size relative to other engineering departments a t MIT, the department represents, nevertheless, one of the largest academic programs'in mate: rials science and engineering. The only undergraduate degree offered by the department is the SB in Materials Science and Engineering since our conviction is that a broad.. eeneric trainine in the fundamentals of materials science and engineering is desirable a t the undergraduate level. However, the eraduate curriculum is far too broad for this approach; more focussed degree programs (each of which mav lead to a masters. eneineers. or doctoral degree) are offerkd (Fig. 11, four of khiih are concerned with specific classes of materials. Two additional degree programs-materials science and materials engineering-slice through this range of interests in an orthogonal fashion, exploring the underlying science or engineering principles, respectively, of all types of materials. These latter programs correspond more ciosely to the present view of the field.
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Role of Crystaliography and Diffraction
Six subjects that are principally devoted to crystallography, diffraction, and crystal chemistry appear in our department curriculum. Four are intended for graduate students, with a pair devoted to electron optics and imaging, and a second pair to crystal chemistry and X-ray and neutron diffraction. One subject in each pair tends to emphasize theorv. the other instruments and annlications. The contents-of all four are loosely coordinated, but none carries completion of one of the other three as a nre-reauisite. A brief overview of their content is presented in the table. A laboratory and asuhject on crystallography are required subjects for undergraduates enrolled in the department. The content and philosophy of these two classes are examined in detail below. An Undergraduate Laboratory on Structure
The cornerstone of our undergraduate curriculum is a major laboratory taken predominantly by sophomores, but not infrequently by first-year students.The laboratory deals with the instruments and techniaues that are available to study structure at all levels, using light, X-rays, and electrons as probes. Primarv emphasis is placed unon the nrohes and i n ~ ~ u m e n tbut s , studedts are iniroducedto a full range of materials throueh careful selection of the specimens and experiments that i r e chosen to illustrate the Eapahilities of
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each instrument. The projects, performed by groups of six to eight students, are more or less eauallv distributed amone . tcese probes. Three to four experiments are devoted to each: A terse indication of the content of the experiments is included in the table. Undergraduate laboratories all too often stand the risk of being second-class appendages to a lecture component of a subject. Students are asked to confirm (perhaps to f50%) a . statement made by an instructor whose kord they would be perfectly willing to accept on the matter. I feel that it is necessary for tt department to decide to make a major commitment of student time and faculty resources in order to nresent an effective laboratorv .subiect. The nresent laboratory, with six hours of laboratory work and two lecture meetinas t former. rather than vice versa) - .(desimed - to s u n ~ o rthe per week, carriesm&e credit units than any other offering in the undergraduate or eraduate curriculum of our department. ~ n r o l l m e nhas, t i n one or two occasions, exceeded 100 students. As a result it has proved necessary to offer the ?
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Classes Oflered In Crystallography, Dlflracflon and Crystal Chemistry Under(lraduale Materials Labasfmy (2-6-7y Optical microscopy, preparation of polished sections (Fe-C phase diagram. decorated dislocations). inhared spectroscopy (polymer identification), Laue method (qua*. Si,'singie-=rystai turbine blade), Debye-Schener camera, pole flgure determination, X-ray topography, powder diffractomeby,and comPuterizd search-match procedures. quantitative diffractomeby(retainedaustinite; Zr0.I. fiumscence analysis with eneravdisoersive detector. electron -. . diffraction ievaDorated TiCi film. ouam: ~ h a s e Wansfmation and radioivsisl scanning elemon m!crMcopy (temper an0 glasp in ceramic anilacts: fracture S~rlaCeI,electron micropraoe analysis, e ectron-beam-inducedc ~ n e n(EBIC) t imaging of a Sem!Cana.CtOr device ~
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S n u c t m of Solids (4-0-8) Theory of crystallographic symmetry: derivation of point groups, space lattlces, and plane groups. Principles of space group derivation and Conventions for international notation. General and soeciai oasitions and use of lnternatiorr a1 Tables10 lnlerpret crystal strumres. Crysta cnemlsuy and suwey of basic strucbre ~YPS Pornt detects and d slocarians, panla ooslacat on*, stackcog tabits, and extendw detects. Graduate
Srrvctwe of Materiais (4-0-8) lnuoduction to X-ray and neuhon diffraction: Laue equations, Ewaid construction, and the reciprocal lattice. Lattice transformations. systematic absences. Fourier series and transforms. The Patterson function and direct methis. Crystal chemistty of ceramics and metals end survey of important structure types. Diffraction and SWuc1UTe (4-0-8) Review of symmetry theory. Systematic absences and diffraction symbols. instrumentation for diffraction analysis: Debye-Schener. Gandaifi, focusing, and Guinier cameras; rotating crystal. Weissenberg, de Jang-Baumann, and precession methods. Diffractometergeometries. Application of diffraction to precision lanice-constant determination, particie-sire measurement and taxlure analysis. ~ l s c t oM n I-w:
Image Interpretation (3-2-71
Relation between bansmission electron microscopy images and internal SWuCture on an atomic scale. Fourier theory of difhactlon. Lens action, aberrations, and transfer functions. Elastic and inelastic interactions: klnematic and dynamic lheories of electron diffraction. Phase ofject approximations and hlgh-resolution imaging. imaging of defects with displacement and replacement fields. Eleclmn optic^ and Microscopy(2-4-6) Electron optics and the interaction of eiechons with materiais. Application to chemistry and srmcture on a microscale. Electron dinraction and image formation in transmission electron microscopy. The electron microprobe. Scanning transmission elamon microscopy: energy dispersive detectors, Xray aMlysis. EELS. imaging, and microdiffraction.
l l w imeasra followinghe sublect tltle prwide he weekly number of lectures. h o w of IaborsmV work, and estimated number of Mlvro of work wtsde of class.
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June 1988
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laboratory in both the spring and fall semesters. This, in turn, requires the commitment of two faculty members, two technical instructors and 3-5 graduate student teaching assistants. The students have access to and employ state-of-the-art research equipment for many of the experiments. This is oossihle because of the dual function of the apparatus as Lotha teaching laboratory supported by the department and as a central research facility supported, in part, by the Center for Materials Science and ~ 3 i n e e r i nAmong ~. the modern instruments are a scanning transmission electron microscope, a diffraction facility including 12 kW and 18 kW rotating-anode generators, two computer-controlled powder diffraciometeri and automated searchlmatch procedures for the identification of unknowns. The laboratory thus has the salutory effect of introducing students to instruments and skills that they will later employ in upperclass lahoratories. thesis research. and their eventual careers. We remain sensitive, however, 'to the risk that the laboratory might become a button-pushing exercise involving black boxes and computer programs prepared by others, features that are undesirable in a laboratory that is part of an educational experience. Avoiding this is part of the function of the lecture comoonent of the subject. The great educational advantage of cbmputer-assistedinstrum&tation is the reduction of the tedious stages of data collection and processing. The more immediate &ailability of results not only permits a wider variety of experiments on a given topic that contributes to the student's experience, but also leads to a sense of excitement and involvement that encourages students to experiment beyond the level required for completion of a particular exercise. An Undergraduate Subject on Crystallography In addition to the laboratory, the undergraduate curriculum in materials science and engineering includes a required subject on crystallography. This course, which is the main focus of this paper, is required early in the program (at the beginning of the second year) and serves as an introduction to the departmental curriculum. The class, as currently presented, has the following goals: (1) Providing s sound foundation in symmetry and in lattice geometry for subsequent use in upper-class subjects. Diffraction. martensitic ohase transformations, slio for plas. svstems . ttcdefurn~tion,nnd quantum c h t m ~ : \ t r yprosidea few exan,pie, of later toprcs that d w q u e n t l y draw suhstantiallpupon this baekdrcund. t2, Eital,l~shmythe co,~vmtionsand uutation fur designation of cells and cumhit~ation* of symmetry elements, this as vocabulary f~ proper presentatim uf research rrsulrs as well as for
subsequent study. (3) Establishing familiarity with space groups at a level sufficient
to oermit readine of crvstalloeraohic literature and, with the aidof the intern&& ~ a b l i (i), s the capability of generating the cell contents of a structure from the notation and compact form in which this information is couched. Any study of the properties and behavior of a material in later work invariably starts with the question, "What is its structure?" Teachlng Symmetry Theory Most texts that deal with the chemistry and physics of the crystalline state invnriat~lycommence with an introductory chapter that lists theapace lattices and prwidei .wneexaniplea ,,I' syn,mrtry. 'l'he rumpilattm of crystallographic rrsults i n tables, followed bv im attempt to rarionalize their contents after the fact. has. to me. all of the intellectual stimulation of reading's telephone directory: the cast of characters is imoressive. but the nlot is difficult to discern. I hold the firm ;onvictioL that, k order to appreciate and understand svmmetrv theorv .fullv. and to be able to use this information, it is necessary to deriue the results-just as, indeed. one would present and develop most other bodies of scientific knowledge. M. J. ~uerger's-~lementary Crystal~
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lography (3)is one of the few hooks to adopt this approach and, moreover, to use little more than aeometrv tonroceed in a logical, thorough, and complete fashion. ~ i e r i e r ' streatment of the material, accordingly, serves as the basic a i d e for the ~~~-clans The lectures on crystallographic symmetry, although following Buerger's treatment cloiely, differ in several significant ways. The basic concepts of group theory are introduced early in the development. We stop far short of a full discussion from this perspective (e.g., representation theory and character tables). Groun theorv. -.however., nermits neat handling of concepts such as subsymmetry; the notion of "self-consistent" collections of svmmetrv elements that reproduce a finite, closed set of motifs; the addition of an extender to derive a point group of higher symmetry; and demystification of the basis of the Schonflies notation for point groups and space groups. Group theory is an elegant and concise way of handling symmetry once one has mastered the geometric relations involved. However, I feel strongly that extensive use of group theory in an introductory class contributes further toward making the material seem remote and abstract (one of the principal challenges in presenting the material effectively, as discussed later in this paper). Forms, important in classical development of crystallography, are given much less emphasis than in Buerger's text. An argument may clearly be made for the value of studying crystal morphology as a step in fully understanding point groups. Our reason for omitting appreciable development of the distribution of forms among the point groups is partly pragmatic: time constraints plus the logistical problems of working with traditional pearwood models with a class that ranges between 50 and 90 students. (Such models are, however, made available to interested students for individual studv.) are freauentlv encountered in .. Moreover., crvstals " materials science and engineering (although, arguably, not alwavs) bodv or as " . in the form of erains in a nolvcrvstalline . - . melt-grown single crystals whose morphology is determined by thermal gradients rather than being bounded by the rational planes that appear in an isotropic equilibrium environment. The principal departure from Buerger's treatment, however, is to develop two-dimensional crystallography fully after completion of the derivation of the three-dimensional point groups. With only five lattices and ten point groups to be considered, the derivation of the 17 two-dimensional space groups in a rigorous and thorough fashion is a manageable task. The steps in the derivation and identification of the properties of the general and special positions in the resultine". olane erouns~orovides a convenient and self-con. . tained microcosm of the analogous, but more involved, process for three dimensions. With the 17 olane erouns " . at hand. the constraints imposed upon the space lattices by the uonisometric noint eroups - . mav be more comoletelv and rieorously handled. (Buerger treats the combinHtions of lattrce nets with onlv rotation axes). The develonment concludes with derivation of the tricltntc and monoclinic space ynups, p l u ~ theconventions for, and uses of, the inftxn~arioncompiled in the symmetry tables of International Tables (2). Selection of Additional Tooics A treatment of symmetry theory that is both complete and presented a t a rate that can be fully absorbed by undergraduate students requires, in my experience, 50 to 60% of a semester for a class that has three to four meetings per week. Asizable block of time remains for the coverage of additional related topics. As X-ray diffraction is the primary means for unit cell and space group determination and, conversely, as a background in lattice geometry and point-group symmetry is necessary to development of the principles of diffraction, this is a logical and natural marriage of topics, and for many years ~~~~
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the two topics received equal coverage, using either Cullity ( 4 ) or Azboff (.5.~~~ ) as the text. The class also included a threehour laboratory in X-ray diffraction and flu~rescence.Very likelv. one susoects. . this is the traditional combination of topiES in courses elsewhere that deal with crystallographic svmmetrv. Paradoxicallv. however. I have never found this cimhinaiion entirely satisfactory, largely due to time constraints. It is difficult to discuss fullv space prouos and their applications within the confines of one-halfsem&er. SimiIarlv, the need to develop the production, spectral characteristiis, and scattering of ?(-rays did not permit the discussion to proceed much beyond simple diffraction methndsandusually not toalevel that included thesingle-crystal technique9 that are the most powerful tools for point group erouo and soace ,. . determination. A major reorganization of our undergraduate curriculum in 1970 resulted in the transfer and exoansion of the material dealing with diffraction and structure analysis into the maior materials lahoratorv, which is described above. What other topic of i m p r t a n c ~ t omaterials scientists and that draws strongly on a background in symmetry theory might be added to fill the time previously devoted to diffraction in the underrraduate subiect in crvstallopraphy? In responding to this-rhetorical q;estion, I i n t r o h c i d treatmknt of the tensor description of crystal properties and anisotropy. The level and breadth of coverigekas comparable to that contained in the first half of the hook by Nye ( 6 ) ,although this volume was not specifically employed as a text. After development of the formalism for transforming tensors and vectors from one coordinate svstem to another. the constraints imposed on second-order property tensors by the crvstallomaphic ooint grouos were derived, and the resulting aniso&obies were described in terms of the representation quadric and its properties. Available time usually permitted extension of the discussion to third- and fourth-order tensors and the examination of representation surfaces for the longitudinal piezoelectric modulus and Young's modu-
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This combination of topics proved rather successful. Reexamination of symmetry transformations in matrix form served to reinforce the earlier geometrical treatment and " developed formalism subsequently used in a class on quantum chemistrv. Most students had either comoleted. or were concurrently enrolled in, an introductory subject'dealing with the mechanical behavior of materials. This class introduced the concept of tensors in the context of stress and strain. Similarly, the students had encountered the fact that the elastic nmhtants and stiffnesses, for reasons that could not be examined in detail, displayed constraints and equalities that depended on the sy&m&ry of the material. ~ x a m i nation of tensors in a broader context and detailed examination of the basis for the constraints imposed by symmetry thus had a synergistic effect. In more terms, this coupling of topics had the advantage of connecting crystallography to a tangible collection of nhvsical ~rooerties.interest in which had undoubtedly con&uted t o students' initial attraction to materials science. The demonstration that point group symmetry precluded the existence of certain properties in a class of materials, or that the variation of certain properties (e.g., the longitudinal piezoelectric modulus) with crystallographic direction conformed to a universal surface for all materials possessing a certain point group, allowed one to conclude the semester's work with a clear-cut demonstration of the relevance of symmetry theory to the behavior of technologically relevant properties of materials. While this combination of tooics was verv successful. a re\,iewof our undergraduate curriculum five years ago led to a further rearrangement of material that madeit desirahle to include instead & this class a coverage of crystal structure plus a discussion of the defects that occur in real materials. The third and present incarnation of the introductory un-
dergraduate class (whose title was changed from "Crystallography" to "Structure of Solids" to reflect more accurately the content) accordingly combines symmetry theory with crystal chemistry and defects. The goals of the material treated in the second half of the course are: (1) Convey familiarity with the prototype atomic arrangements that are of importance in metallurgy, ceramics, and materials science, (2) To establish concepts for interpreting atomic arrangements at various scales (i.e., coordination polyhedra; slabs, chains, and rinm that appear in a varietvof relevant structure tvoes). relations betrue;" structure typis 1e.g.. polyfypism. ~ue.rger's concept of derivative structur? 17). and the strurturnl nature of phase transformations),and (3) To ornvide illustrations. on a case-bv-case basis.. of the link between strurture end physical hehsvior leg., predominant type of point defect, transport properties, ferrimngnetism in the spinel-typestructure).
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The treatment hegins with a discussion of the basis of svstems of atomic an: ionic radii and the role of radius ratidin determininn coordination numbers. I'aulinr's rules for ionic structures and the electronfatom ratios correlated with the occurrence of intermetallic phases are next established along with a discussion of sphere packing and the identification of the type and number of interstitial positions within these arrays. The latter treatment uses a short set of notes previously developed by the writer (8).This material serves as perspective for surveying a variety of inorganic structure types that proceeds to a level of complexity that encompasses the spinel, corundum, and garnet structure types, the clays and other layer silicates important in ceramics, and intermetallic structures such as the Laves phases and the sigma phases. Useful references for structure types and examples of phases that assume them are the hook on ternary phases by Muller and Roy for oxides (9) and that by Pearson (10)for metals. Apart from the specific structure types selected for discussion, one suspects that this represents a basic coverage of crystal chemistry common to many comparable class& in inorganic chemistry or mineralogy. The remainder of the coverage in the present class, however, is devoted to discussion of the point, line, and extended defects in realmaterials. The role of point defects in charge compensation and nonstoichiometry is examined, the equilibrium concentration of Schottkv and Frenkel defects is derived as a function of the enthalp; for their formation, and the widely used KriigerVink notation for defects and defect reactions is introduced. The discussion of line defects includes definition of the Burgers vector and atomic arrangements in the vicinity of disloEativns in edge, screw, andhtermediate mientarions and treats climb, glide, and the multiplication and intersection of dislocations. The treatment concludes with discussion of stacking faults, partial dislocations, and extended defects (e.g., shear in oxides). In keeping with the theme of the earlier portions of the class, i t is the structural aspects of defects that are emphasized; the energetics and interactions of defects and their role in controlling specific properties are left to uooer-class suhiects. The combination o i crystal chemistry and defects with symmetry in a single subject is also natural and logical and works successfully. The principal advantage of the arrangement is the opportunity to apply space groups in the generation and description of structures and, once again, to estahlish a tangible link to real materials and properties. As is the case with the combination of symmetry with diffraction, however, the topics described ahove represent a great deal of material to he covered within one semester if the treatment is to contain a depth sufficient to serve as a solid foundation for uooer-class subiects and not reoresent merelv a broad. ha6 introductory surve;. A recent soluAon to this been the introduction of comouters for the descri~tionand manipulation of structures. Volume 65 Number 6 June 1986
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Use of Computers in Teaching Crystallography Five years ago MIT, with the participation of Digital Equipment Corporation and IBM, initiated Project Athena, a iarge-scale experiment on the application of computers to undergraduate education. A capability for the display and maninhation of the atomic arraneement in a crvstal structure represented an interesting opportunity for the teaching of crvstalloeraohv. Over the oast several vears. therefore. ~ a u l " Bristowe ~. of our department, witi thesupport of Project Athena and assistance of its staff, has developed two user-friendly, menu-driven software packages for use in the introductory class on crystallography. A first package produces a skeletal, hall-and-pin representation of an atomic arrav in which the user specifies the rangeof distances (as a factor times the minimum separation present) for which neighboring atoms will be connected hv bonds. A second provides a polyhedral represent& tion of a structure in terms of a linkage of tetrahedra and octahedra. Both programs utilize the BLOX interface management system. All code is written in Fortran 77. The programs operate on VAX Station I1 hardware; clusters of workstations have been installed a t various locations throuehout camuus. includine some in livine erouns. The h a ~ l - & d - ~ i softwaie n has heen adapted for use from the three-dimensional molecular eraohics codes NAMOD 3 (11): . .. that for polyhedral representkon is based upon a program described by Fischer (12). Both modules may be used a t three different levels of sophistication. Data for most of the simple, close-packed structures that are initially treated in class are stored in a data hase. Students may display and manipulate these atomic arrays without the requirement of supplying crystal data or space group information. At the intermediate level of use, a student may retrieve information on the symmetry transformations of the space group from a data hase but must supply the atomic coordinates and lattice constants for a structure of interest. The transformations of all 230 space groups . . have been stored. but data for onlv a dozen or so commonly used, high-symmetry space groups are presently on-line. There is perhaps pedagogic value in requiring a student to supply the properties of the simpler space groups. A user a t the third level must supply all crystallographic data and space-group transformations, as well as the values for the parameters that control the . graphic aspects of the . display. If the sole function of the programs were to produce an illustration of the structure. one might iust as well distribute photocopies of a previouslipreparid f&e to the class. The educational value of the svstem lies in its interactive nature and its facilitation of iidependent study. In studying a structure with a traditional ball-and-pin model. one would turn the model over in one's hands andbeer at it from several different perspectives until at last (throueh mental steps that likel? differ from person to person) &e finally w o i d come to terms with the nature of the arrav. The interactive format and flexibility of the Athena mddules encourages similar experimentation and thought processes. The extent (i.e., number of cells) of the display may be independently selected along the three crystallographic axes. The size of the spheres that reuresent the individual soecies mav he varied. or some of them even eliminated, if, for example, one wished t o examine anion ~ a c k i n ealone. Circles that renresent the atom loeations may be iniuded or omitted in th;polyhedral representation, and the degree of shading on the faces of the polyhedra adjusted to convey an optimal illusion of three dimensions. Most importantly, the orientation in which the structure is viewed may he selected and varied: a projection along a specified direction may he selected or, alternatively, the structure may be rotated by selected amounts about any or all of the three reference axes. A specified increment of rotation may he repeated successively to convey a quasi-
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Flgure 2. A reproduction of a computer workstationscreen illustrating the ball&pin d u l e fw representation ot an atomic array.
dynamiceffect, but the code that currently generates thecell contents is sufficiently slow that a true kinematic impression is not possible. The contents of the display screen may, a t any stage, he converted almost immediately to extremely high quality hard cops for retention bv the student. An example o f a hall-andpin representatibn of the rocksalt structure is presented in Figure 2. The results of the polshedral representations are c$te comparable to those i n the illustrations presented by Fischer (12). These teaching modules were introduced in the class on an experimental basis in the fall of 1987. Their reception by the class was quite enthusiastic. The models not only convey the nature of an atomic array with great effectiveness hut also permit self-studv at an individual student's desired rate of s fnr a stuprogress. Also present is the d ~ v l o u opportunity dent to experiment with additional structures of interest in connection with research or other classes. Problems and Challenges In the Teaching of Crystallography Symmetry theory, quite apart from its elegance and heauty, serves as the language and foundation of crystal chemistry and diffraction. Much of my own research activity lies in these areas. In beginning to teach this material, I was thus somewhat surprised to discover that, despite my enthusiasm for the topic, crystallography was neither the most popular nor the easiest topic to present to undergraduate students. Part of the reason for this lies in the broad spectrum of backgrounds. interests. and eoals present amone students in a department of materials szenceand engineerGg. Students inclined toward the science of materials tend to auoreciate. as a matter of taste, both the rigor and completene~sof the theory as well as its relevance to the science of thecrsstalline state. Materials engineers, in contrast, are more concerned with the utility of material in solving.problems. They. per. . ceive no immediate relevance of a space group to, say, sintering of a ceramic or steel making. In short, more so than in departments of science, the mode of presentation must provide answer to the question: "What is it good for?" The position that (as with sulfur-and-molasses in the springtime) i t may he hell going down, hut is good for you in the long run, is not tenable. Thus, I feel that while, on the one hand, it is necessary to derive symmetry theory systemati-
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~ 3 ~ 1 . 3STRUCTURE FF SOLIDS J B. J , NUERSCH MiRKOe. Llu& ROOK 1.3-4C37
Figure 3. An opening step in sensitizing sMents to lhe presenceof symmetry upon intraducing the subject on the tint day of class.
cally in order to impart an understanding sufficient to permit its use. i t is nevertheless desirable to c o u ~ l eits treatment with; topic that demonstrates its conseq;ences or use in a phvsical context. w he nature of symmetry theory introduces a challenge in its teachine that is more basic than merelv the oroblem of inspiring student motivation. The material is distinct in several respects. Classes in most of the physical sciences converge toward some paradigm-E = hv, h = 2 d sin0, F = ma, or the like-a compact tool that may be tucked away for later application to a wide variety of problems. The development of symmetry theory proceeds in opposite fashion. starting with a few simple geometric concepts, one combines and builds to erect an elaborate, elegant, filigreed structure of great complexity. Moreover, the basis of the derivation is inherently geometric. Students a t a technically oriented universitv such as MIT have hiehlv develooed analvtic skills. hut hive little experience wythttreatments t h a t a r e based upon geometric relationships as opposed to algorithmic solutions. T o some, if it cannot be integrated, it need not be taken seriously. (A few, however, find this difference appealing, one student commenting that she enjoyed the "change of pace" and exposure to "a different way of understanding".) Symmetry theory, however, has one strong feature going for it: the intuitive appeal of spatial relationships and the not inconsiderable aesthetic pleasure provided by many of its manifestations. Some students (in spite of initial misgivings) find themselves intrigued. We capitalize on this intuitive aooeal . and. without sacrifice of rieor. - . enliven the oreceedings to as great an extent as possible. We attempt, from the opening moments of the first class, to sensitize students to presence of the examples of symmetry that surround them. Openina-dav protocol a t our institution requires that an instructor begin the proceedings by placing the name of the suhject plus the name, room numher, and telephone extension of the instructor upon the blackboard. In doing this, Figure 3, we are quick t o point out that the numher under which the suhject is listed in the catalog contains a horizontal line of mirror symmetry and that the instructor's telenhone extension is left invariant hv the onerationof a two-fild rotation axis. Little isaccompliHhedin'the first lecture apart from a review of procedural formalities for the semester and an overview of the topics to be discussed, yet the class is immediately issued an assignment. Figure 4 presents a problem2 whose solution requires recognition that the characters are grouped according to their symmetry. The mental change-of-gears that is required comes as a jolt to some students. (One team of individuals combined forces to attempt-without successful outcome-a sophisticated cryptographic analysis! Another student, ashen and somewhat shaken, appeared at the instructor's office the following day to ask if he possessed the necessary prerequisites for the class.) Figure 5 presents an analagous problem in which -
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Adapted from a problem which appears in ref. 13
all characters display the same point group, hut in which the asymmetric unit changes progressively in a recognizable fashion. Yet another problem asks the students to identify the symmetry elements that are present in a collection of familiar corporate logos and trademarks (which provide examples of a great variety of symmetries!) culled from magazines advertisements. Still another ~ r o h l e mserves as a srmple illustration of how symmetry may he used to simplify a ohvsical oroblem. A fieure that is orovided disolavs an arrav bf cubic-blocks that &e stacked in an arrahgeient that possesses "arms" that clearly conform to symmetry 2mm; portions of the array are obscured in the illustration. Students are asked to determine the number of blocks in the array. The homely point of the problem is that the simplest approach to the solution is to count the numher of blocks in one of the fully visible arms and multiply the result by four. (Just as, in determining the number of horseshoes necessary to shoe a herd, one does not count feet, but rather multiplies the numher of horses by four.) There is a more subtle motivation to the prohlem, however, that is discussed in the following class: the assignment of symmetry to a physical svstem is an experimental exercise with results that are dependent in pa& on the probes that are employed and that are suhiect to assumotions. In the specific nrohlem a t hand. forexa&ple,one munt assume that a.11interdtices in thearra; of blocks-in ~articular,those which are not \,isihle-are in fact occupied.-(Similarly, in recasting the problem in terms of a herd of horses, one must make the reasonable assumptions that five-legged horses are relatively rare, and that three-legged animals would likely enjoy but limited tenure a t the expense of the farmer.) A surprising number of students are sensitive to this aspect of solving the problem (beginning to suspect, perhaps, a devious streak in the instructor) andcorrectly list a range of numbersas thesolution along with the assumptions that were made.
Secret agent 007 crashes through the door of the headquarters of the infamous SMEHSH, but not quite in time to prevent the radio operator from attempting to eat ,he code sheet which our hero is s~eking.He is able to retrieve only the following scrap of paper:
What is the basis of this grouping of letters? Complete the arrangement for the rest of the alphabet Figve 4. An intraductwy pmbiem whose solution hinges upon recognizingthe symmetry oflhe charactem.
One of the archaeologicalfinds of the century is amysterious rune stone washed up upon the shores of Revere Beach. It is believed to
be a relic of an early settlement established by the intrepid Viking explorer Pierre-the-Lucky.
Write down the next five characters for this obscure Nordic alphabet. Figure 5. An inWdunory problem in which the characters POSMISS a m m o n point goup, but in which the asymmelric unit differs in a reccgnizabie fashion.
Volume 65 Number 6 June 1988
499
Figure 6. A collection of symmetry elements (not canfming to the group postulates)that is used to convey greetings of the season to students upon completion of the derivation of the crystallographicpaint groups in October. We attempt in the lectures to use an aphorism or bad pun to underscore a point or make a distinction in nomenclature. T o emphasize the distinction between form and habit: "Crystals undergoing plastic deformation never die; they just develop bad habits." (Others are presented for their sheer atrociousness: "If the Blarney Stone were to contract 'kissingdisease', i t would have to be transferred to the monodinir.") \'isuaI puns are also fair game. As the class meets during the fall semester, complerion of the derivatinn of the crystallographic point groups usually occurs in middle or late October. At the conclusion of this lecture we add a few additional threefold axes to point group 23 in a decidedly noncrystallographic fashion, Figure 6, extend greetings of the season, and exit the classroom to a stunned silence. A large number of patterns are analyzed in problem sets. Some are examined early on to apply the concept of translational periodicity,3 to identify symmetry elements and, ultimately, to assign plane groups. The patterns have been collected from a variety of sources. We examine many of the intriauing . ...periodic patterns of Escher (14,15) and examples drnwn frum a)mpilntions prepared for designers ( I f ? ) , as well as patterns d r a w from sume highly eclectic sources (171. A large number of fascinating are contained in the recent work by Griinhaum and Shepard (1.3, a marvelous book devoted to a serious mathematical analvsis of tilines. One hegins to sense some measure uf success in reaching a class when students are motivated to reroond in kind. Manv students experiment and try their hadd a t the design df patterns. An intricate (if somewhat chauvinistic) example in plane group p4gm is shown in Figure 7.4 students &en suggest an ingenious way of looking a t a problem. I t was proposed, for example, t h a t a convenient device for representation of the pattern generated by an n, screw axis is t o imagine a cylind;ical surface, concer&c with the axis, whose surface is divided into n angular increments. Let the surface further be divided, by planes normal to the axis, a t intervals equal to one nth of the lattice translation that is ~ a r a l l eto l the axis. This orovides an arraneement of referknce boxes to indicate theiocation of objectsrepeated by the symmetry element. Upon unrolling the cylinder, Figure 8, one has a convenient, two-dimensional means for representing the pattern that clearly displays the helical nature of the A surprising number of students, upon recognizing identical motifs that are related by symmetry, will altempt to assign lattice points to these locations.. desoite . the fact that the motifs are not in oaraiiei orientation. ' Designeo by graauate student Robert G. van oer Heiae. Created oy former gradLate srlrdenr William N. Schaffner.
500
Journal of Chemical Education
Figure 7. A studentdesignedpattern In plane group p4gm. repetition, and one in which the relation between the lattice translation and the translation component of the screw axes may be readily appreciated. This is a simple device, but one which I have not seen employed elsewhere. I t is especially useful in depicting patterns representative of screw axes of high symmetry and for examination of noncrystallographic screw axes in particular. Each year our graduate students conduct a contest for the design of a departmental T-shirt. I t is significant that the ereat maioritv of the winnine desiens - have featured a motif :hat is drawn"from crystallography or diffraction. One would like to think that this is an indication of student appreciation of the central role that crystallography plays in materials science and engineering. One of the writer's favorite designs is shown in Figure 9. On the front of the shirt5 one sees a forward face of a cell that disappears into the chest of the wearer; on the hack of the shirt it &emerges. This is clearly an example of a "body-centered" lattice! (Upon my later
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F b u m 8. me "unrolled cviinder"device suoaested bv a student tor convenient two-dmensiona represenlation of panerns generatea 3y a screw ax.%. Tne example shown is for S2 ~
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