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A Derivation of the 32 Crystallographic Point Groups Using Elementary

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A Derivation of the 32 Crystallographic Point Groups Using Elementary American Mineralogist, Volwne 61, pages 145-165, 1976 A derivationof the 32 crystallographicpoint groupsusing elementary group theory MoNrn B. Borsnu.Jn., nNn G. V. Gtsss Virginia PolytechnicInstitute and State Uniuersity Blacksburg,Virginia 2406I Abstract A rigorous derivation of the 32 crystallographic point groups is presented.The derivation is designed for scientistswho wish to gain an appreciation of the group theoretical derivation but who do not wish to master the large number of specializedtopics used in other rigorous group theoreticalderivations. Introduction have appealedto a number of interestingalgorithms basedon various geometricand heuristic arguments This paper has two objectives.The first is the very to obtain the 32 crystallographicpoint groups.In this specificgoal of giving a complete and rigorous deri- paper we present a rigorous derivation of the 32 vation of the 32 crystallographicpoint groups with crystallographicpoint groups that usesonly the most emphasisplaced on making this paper as self-con- elementarynotions of group theory while still taking tained as possible.The second is the more general advantageof the power of the theory of groups. objective of presenting several concepts of modern The derivation given here was inspired by the dis- mathematics and their applicability to the study of cussionsgiven in Klein (1884),Weber (1896),Zas- crystallography.Throughout the paper we have at- senhaus(1949), and Weyl (1952).The mathematical tempted to present as much of the mathematical the- approach usedin their discussionsis a blend of group ory as possible,without resortingto long digressions, theory and the theory of the equivalencerelation' so that the reader with only a modest amount of Each of these mathematical concepts is described mathematical knowledge will be able to appreciate herein and is used in the ways suggestedby these the derivation. Those facts that are used but not authors. However we depart from their approach in proven in this paper will be clearly identified and the determinationof the interaxialangles. While they referenceswill be given to sources where the appro- appeal to a study of Platonic solids,we continuewith priate discussionscan be found. the theory of groups and equivalencerelations in our Crystal symmetry was studied during the nine- determinationof theseangles. teenth century largely from a geometricalviewpoint. In the first part of the paper we discussthe defini- A comprehensiveoutline of the more important con- tion and propertiesof point isometriesand show that tributions to this subject by such men as Hessel, everypoint isometryis eithera proper or an improper Bravais, Miibius, Gadolin, Curie, Federov, Min- rotation. If a rotation leavesa lattice invariant, then nigerode, Schoenflies, and Miers is presented by it is shown that its turn anglemust be one of the eight Swartz (1909). It appears that Minnigerode (1884), followingpossibilities:0o, +60o, +90o, +120o, 180o. Schoenflies(1891), and Weber (1896)were the first to The method for composing two point isometriesis recast the problem of classifying crystal symmetry in discussed.The propertiesdiscovered about this conr- terms of group theory. Recent treatmentsmake ex- position motivate the definition of an abstractgroup, tensive use of specializedtopics in group theory in the which is then stated. We then show that the set of all derivation of the crystallographicpoint groups (see point isometriesthat leavesa given lattice invariant for example Seitz, 1934; Zachariasen, 1945; Burck- forms a finite group. These are the crystallographic hardt, 1947;Zassenhaus, 1949; Lomont, 1959;Weyl, point groups. In the secondpart, the proper crystal- 1952;McWeeny, 1963;Altmann, 1963;Yale, 1968; lographic groups are discussed.The cyclic monaxial Benson and Grove, l97l; Janssen, 1973; Coxeter, crystallographic groups are treated first. Then the 1973;andSenechal, 1976). In addition, severalwork- notion of an equivalencerelation and equivalence ers (for example,Donnay, 1942,1967; Buerger, 1963) classesare presentedin conjunction with a group t45 t46 M. B. BOISEN, JR,, AND G. V. GIBBS z Point isomefiies ln our study of point groups we are interestedin a special kind of transformation known as a point isometry. The mapping O from R3onto RBis a point isometry if and only if it satisfiesthe following four properties: (l) Given any element4e R3,its image O($ is a ' uniquely R3. (This - ->\y determined element of means that O is a mapping from Rato R3.)* Ftc l. Right-handed cartesian coordinate system with unit (2) Given any element4€ R3,there exists an ele- vectors ryr{ directed along the coordinate axesX, Y, and Z, respec- ment ,r E R3such that O(q) : r,. (This means tively, with the vector r expressedas a linear combination j of i, that O is a mapping from R3 R3.) and k onto (3) The magnitude of r is equal to the magni- tude of the image of r under O for all4€ R3, i.r., : (This meansthat o pre- theoretical development of the basic ideas leading lltll llotdll. servesmagnitudes and henceangles, sizes, and to the derivation of all the proper polyaxial shapes. crystallographicgroups togetherwith their interaxial ) (4) Givenany two elements4s g R3,O(r *-r) : angles.In the final part of the paper, the improper O(9,)+ O(s) and given any real numberx, crystallographic groups are constructed from the o(x4) : In particular,if4= yi-+ proper crystallographicgroups. xO(4). x!.+ z\,then o(d : x<D(i)+.yo(4) + zo(k). (This Discussion of basic concepts meansthat O is a lineartranformation and that O(4) is completelydetermined by O(r), O(7') Three dimensional space ando(&).) Our discussion of the 32 crystallographic point A consequenceof theseproperties is that ib is a one- groups will be conducted within the framework of to-onemapping. That is, if r andj arein Rssuch that real three-dimensionalspace, R3, where R designates : 414, thena(L) f o(s). Equivalently,if o(4) o({) the set of all real numbers.The elementsof RBcan be for somer,seR3, then4 mustequal s. A consequenceof describedgeometrically by placing three unit vectors, : : property(4) isthat o(q,) o(04 + E' + 04) 0og) i,1 and k along three mutually perpendicularaxes X, : + 0O ("1)+ 0O (Q 9.. Hencethe origin O of R3is Y, and Z which form a right-handed cartesiancoordi- : fixedunder Q, i.e.,O(g) 9.(fne word"point" in nate systemas in Figure 1. Accordingly, the vectors point isometryand point groupalludes to the prop- i j,k form a basisfor R' such that Rsmay be viewedas erty that somepoint, in this casethe origin, is left iiin-ptythe collectionof all linear combinationsof the fixed-that is, unchangedin position-by o.) : form r Jri + yj + zk where x, y, z areelements ol If €Dand O aretwo point isometries,then we say the sef Rlsfrrbolized i,y,, e R). Note that x, y. andz that theyare equal-that is O : O-if O(r) : O(r) are the X,Y and Z components of r. Stated more for all4 € R'. Hence in view of (4), tio poin't = concisely,n' {4t yj.+ t&lx,y,z e R} which is read isometries€) and O are equalif and only if O(r) : "R3is the set of all vect6rs x1+ yj f zk such that x,y,z o(i), o0r) : o(/) and o(ft) : o(&).A trivialaut are elements of the set n." ti is-impoiiunt to observe imp'ortan?exampG of a poinl isometiyis the identity that each vector in R3may be expressedin one and mapping1 on R3defined bV I(D: r for all r 6 Rs.In only one way as a linear combination of {i,7',&}.We the caseof the identity mapping,magnitudes are : call the vector O 0i + 0i + 0k the orisin"df'R'and preservedsince under / eachvector in Rsis mapped also useo to dJnotittre p'oint?t the oJgin. In gen- onto itself.Another example of a point isometryis a eral, unlessotherwise stated, we will not distinguish rotation. For our purposesa rotation will meana -l betweena vector and its end-point. lf r: xi yj.-f turningofspace about a line,called the rotation axis, z_(is someelement of R3,then its magnitudeor leng-th, that goesthrough Q and has a positivedirection llfll , it defined to be * fn general,amapping a of a setA into a set B is arulewhich assignsto each element in A a unique element in B. In this paper, R3 plays the role of both A and B, that is, the mapping we will be using will map R3into R3. GROUP THEORY DERIVATION OF THE 32 POINT GROUPS 147 defined on it. Let P denote such a rotation and let I unique point isometry O such that OO : O€D : /. denote its rotation axis. Then a turn anglep of P is In this caseO is called the inuerseo/O and is denoted an angle measuring the action of P on a plane by O : @-'. By uniquewe meanthat if @O : O€D perpendicularto /. The turn angleis consideredto be : 1 and ov = vo : /, then o : rlr. The existence positive if the action of the rotation is measuredin a of inverses implies the cancellation law which states counterclockwisedirection on the plane when viewed that if A, B and Q are any threepoint isometriessuch from the positive direction defined on / (seeFig.2a). that AB : Afl, then B : Q. If P' and P, are rotations It may be noted that the turn angle of P is not about the samerotation axisI with turn anglesp' and uniquely determined.In fact if p is a turn anglefor P, pr, respectively,then the composition PzP' is simply then so is p + k(360o)where k is any integer.Hence, the rotation about I with a turn angle of p1 * p2.ln any specific value for p is merely a representative particular, if pr: -pr, Ihen PrPr has a turn angle of of the turn angle of P. For example, the rotations zero and so PzPr: I (i.e.,Pr: Pr-t).Hence, by the about the line I with turn angles 240o and -120o, uniquenessof an inverse,the inverseof a rotation is a respectively,are really the same rotation P with two rotation.
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