A Matrix Approach to Point Group Symmetries G. D. Nigam Department of Physics, Indian Institute of Technology, Kharagpur 721302, India
An understanding of crystallographic point groups is essential in solving a variety of problems in quantum chemistry, solid state physics, and many other branches of physics. They have been studied from a geometrical standpoint by many authors (1-2). Recent approaches make use of specialized topics in group theory in the derivation of crystallographic point groups (3-8). In this paper we present a derivation of the 32 point groups that use elementary knowledge of matrices and group theory. Since the crystallographic point groups are intimately connected with crystals, we start our discussion with a brief introduction of crystals and their associated symmetries. A crystal is a three-dimensional regular array of constituents: neutral atoms, charged ions, and molecular complexes. A symmetry transformation or operation of a crystal is an arrangement interchange of the constituents which leaves unchanged or invariant the positions of the constituents. The basic symmetry of a crystal is its translation symmetry which implies that by parallel displacements with constant lengths in certain directions, the crystal structure is brought into self-coincidence. The set of the translations of a crystal structure is called the translation lattice. Each latti . - arc h i t visualized by their stere&rams. E'igure 2 shows a few point groups as they operate on a triangular motif. It is possible to show that all space lattices will he characterized by one of the seven point groups marked by boxes in Table 3. These seven groups are called holohedral groups and the 32 crystallographic groups are the subgroups of these seven holohedral groups. They are the triclinic (I), monoclinic (2/m), ofthorhombic (mmm), tetragonal(4/mmm), rhombohedra1 (3 m), hexagonal (6/mmm), and cubic (m3m) systems. Acknowledgment
The author takes this opportunity to thank Professor H. Wondratobek who initiated him to .erouu. theoretical methods in c r y s r i r l l o g m p h g w h r n h r WI.,,I H u ~ ~ ~ h oi l ~d fl l i~~t~ht eu111s t i t u t t , fur t i r i s t n l l o ~ r a i ) h i , . ( l e r ( ' ~ t i t e r * i r a t (Tlii tiarlsruhe. West Germany. ~ e - a l i othanks Dr. B. Deppiscb for many useful discussions. Thanks are due to Professor G. B. Mitra, Department of Physics, I. I. T., Kbaragpor for encouragement.
Table 3.
The 32 C r y s t a l l o g r a p h i c Point Groups and Their Orders
The 11 proper crystallographic ~ o i nC~~OUDS t
.
- .
Order
The 21 Improper Crystallographic Point Groups
Laue Groups
Not Containing
Containinq
i
Order
1
1
2
2
m
4
3
3
3
6
4 6
4 6
4im 6im
8 12
2
i
Order
-
2
m
4
-
6
6
4
Literature Cited ( I ) Buerger, M.J., IIElomentary Crystallography-An Introductiont o the Fundamental Geometry Features of Crystals." 1st Ed., John Wiley and Sons, Inc, NmYork, 1956, p. 46. (2) Philips. F. C.. "An Introduction to Crystallography; Longmans, 3rd 1963, p. 103. (19631. (31 A1tman.S. L , R m Mod. Phy~~35.641 (4) Boisen. Monte 8.. Jr., and Gibbs, G. V., American Mineroluiist, 61,145 (1976) ( 5 ) senechai.M . . Z ~ ~ i ~ t ~ i i 142, o g rI .(19751. , (61 Bradley.C. J.. and Ciacknell, A. P., "The MsthematiealTheowof Symrnetiy; 1st Ed.. Clarendon Press. Oxford. 1972. p. 24. (7)Burckhsrdl, J. J.. "Die Bewegungsgruppender Kristellographie: 1st Ed.. vertag Birkhauner, Bssel. 1947.p. 34. (81 Henry. N. F. M., and Lonsdeie, K., "Internstional Tables for X-Ray Crystallography. I . Svmmetn Grou~s."Kvnoch Preas. Birrnineham, 1952.
Ed.,
Volume 60 Number 11 November 1983
921
