Crystal Systems Verner Schomaker and E. C. Lingafelter University of Washington, Seattle, WA 98195 Dividing the 14 Bravais lattices into seven lattice systems accordine to their rotational svmmetries matches the classical division &crystals into crystal systems for the triclinic ( a , fur anorthic), monoclinic (m).orthorhotnhic (oJ,tetrnaonal ( 1 1 , and cubic (c) crystal systems hut differs for the ;ariously named rhomhohedral (r) and hexagonal (h) ( 1 4 , (or trigonal (Trg) and h (4-6), or Trglr and h(4), or, together, Trg-h (7), or just h (8))crystal systems. The difference is essential: the crvstal ooint swnmetw and the lattice ooint svmmetrv do not m k c h in the dtherwise simple way. M& of &e space groups with threefold orincinal rotational svmmetrv -reauire . the h Bravais lattice, which has sixfold principal rotational symmetrv and three tvDes of svmmetrv directions. while onlv a few require the r k t i c e with threefold principal rotational svmmetrv and onlv two tvoes of svmmetrv directions. (The spare g r o q i with sixt'old ;&tionil sylntnitry all require the h P 1atrice.1The lattice classification came first, but with the development of symmetry theory and of methods of determining the point symmetry of actual crystals and in the ahsence of methods (before the discovery of X-ray diffraction, see Dunitz (7)) for determining whether or not a lattice is primitive and in the absence of proper regard for morphology, the crystal point-symmetry notion of crystal system took over. So the troublesome case of r and h is indeed curious and of lone standine. What is convenientlv true of all the other svstems, namely that each includes all crystal symmetries (space groups) belonging to a certain set of point symmetries and that each of these crystal symmetries requires one or another lattice of that same symmetry, is not true of the point groups with a single threefold axis. Each of these is associated with one or two space ~. groups that require the lattice r P and with one or more that require the lat& h ~'That . r P canhr represented by a non-primitive h unit cell (or h P 11y a non-primitke r), does not help because the issue is lattice ~ y m m & ~and , the symmetries of the two lattices are different. The extra computational convenience of the use of h axes for an r is also inconsequential-the question is one of classification, and the h axes can he used for comoutation in anv case. The classical convention is by far the heit, as 1)onnay has consummately exolained ( I ) . with 'rwstal ivstwn" referred to lattice rather than to cr&.Ll symmetry. ~ l u s k e and r Trueblood (Z), and
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Sharma (3)also adopt it, although without explanation. Sharma (3)also emphasizes an old hut important point in this connection that is often left unclear in the textbooks: briefly, the symmetry of a crystal structnre-the full spacegroup symmetry-defines its Bravais lattice and so places characteristic restrictions on the axial lengths (a, 6 , c) and 0 , ~of) the unit cell used to describe the interaxial angles (a, lattice, whereas these restrictions may he satisfied accidentally by the unit cell of a structure of lower symmetry. There are many examples of this metrical pseudosymmetry (9). he point symmetry of the lattice is thus either a supergroup of, or the same as, that of the crystal (which is defined as the symmetry common to all its properties (10)) hut no more than an indication. The basic issue in principle is the symmetry of the crystal structure; in practice it is the symmetry of both the positions and the intensities of the X-ray or neutron diffraction spectra. The unit-cell restrictions may consist of relationships hetween edges or angles or of fixed values of angles. The tahle shows, for each lattice type, the number and nature of the restrictions in terms of the conventional unit cell as well as the seouence of lattices that can accidentallv show the same metrit:al symmetry (called dimensional symmetry hy Buerger t.l l ..) ~Note . that no case has iust one restriction. One miaht consider a unit cell with one a&e, say a,restricted to SOo, Lut there is no such crvstal svmmetrv. The several lattice tvDes of a particular system differ in that one of the symme&al unit cells conventionally used to describe them is primitive ("P," containing just one lattice point) and the others variouslv centered ("A." "B," or "C" in one face; "F" in all three; or "I" in the body center) and non-primitive, with two or four points per cell. Only the different types are listed in the tahle; C, A, I, and F are equivalent in m (as are P and B); A, B, and C are equivalent in o; and I and F are equivalent in t (as are P and C). tvnes m F and B. and t F and C. reouire in. (The . creases in multiplicity and are nkver used.) ~ o r ' e a g hof the centered cells.. a orimitive cell can he used instead. hut the . number of restrictions remains as shown in the tahle. The conventional centered cell has the advantage of exhibiting the full point symmetry of its lattice. In most cases, the corresponding primitive cell cannot he chosen to do so, and one cannot use a cell that is both primitive and fully sym-
".
Characteristicsof Crystal Systems
Crystal System
Lanice Types
Number of
Restrictions
a m
P P,C
0 2
o
P CAF
3 3 4 4 4 4 5 5
o
r h
P I P=R P
c
P
C
F.1
1
f
Nature of the ReSbictions none a = 7 = 90' a=p=r=9,y a = fj = r = 90' a=p=-f=90°,a=b a=fj=7=9O0.a=b a=p=y a=b== a = p = 90a. y = 120°, a = b a=P=r=900,a=b=c a=p=r=$o~,a=b=c
Volume 62
Othw Lattices Mat can Accidentally show the Same Metrical Symmetry*
a m(p) m (PC) 0 (P,C) o(1.F)
m (c) m ( P J .(C) ~ t(P).r f (p),,
Number 3 March 1985
219
metrical. I t may he noted, however, that the centered m cell (whether taken as A, C, or I) can he replaced by a fully symmetrical primitive cell, as can the side-centered o cell (A, B, or C); also that the h cell does not exhibit full symmetry either, but this situation cannot be helped, because all unit cells must be parallelepipeds and as such cannot have h symmetry. Note finally that the lattice symmetry that you find in practice is as likely too low as too high. This is hecause 1) the measurements may he less accurate than you think (e.g., you might call a truly m crystal a, having found a = 96.78(8)0,0 = 89.54(9)O, and y = 90.02(12)"; 2) the unit cell of appropriate metrical symmetry may he overlooked (e.g., you might find from perfectly precise measurements on a face-centered c crystal the primitive r cell, a = b = c and a = .13= . .I= 60". instead of the moresymmetrical non-primitivecell, a' = h' = d , n = 0 = = 90Dof the c system. Examdes of this kind are unfortuna&ly not rare (see Herbstein and Marsh (12)).
220
Journal of Chemical Education
The authors wish to acknowledge helpful suggestions from G. Donnay and J. D. H. Donnay. Literature Cited (1) Donnay, J. D. H., Aeto Crist., A33.979 (1977). and A34.638 (1978). (2) Gluliker,J. P.. and Tmeblmd, K. N., 'Cryeta1 Strunurehelysis: APrimq"0xfo.d University Press. New York, 1372, pp. 15,70.79,147.181. (31 Sbanna, B. D.. J. CHEM.EDUC., 59.742 (1982). (4) Stout,G.H.,andJensen.L.H.,'*X-rayStructureDetmination.AP~aetiealGuide," Maemilkn, New York, 1968, pp. 39.54. (51 "Fifty Yeamof X-ray Diffraction," (Editor:&aid, P. P.1,N.V. A Ooathak'aUi*eversmaatrbappij, UUhecht, 1962, p. 21. (6) Henry, Norman F. M.. and h s d s l e . Kathleen (Editan). "IntemtionalTablss for X-ray Crystailogrsphy," The Kynoch Press, Birmingham, 1952, p. 10. (7) Dunits. J. D., "X-ray Aoalyaiaand the Struelureof OrgsnieMoldes,"ComdlUniversity Preu, Ithaea, 1979. pp. 79-82. (8) Buelger, M. J.. "Elementary Clyatallography," Wiley. New York. 19.56, p. 100. Vol. . I"msmic," (9) Donnay. J. D. H., and Ondik H. M. (Editora),"ClyatalDafa," 3rd 4. 1972,Vol. I1 "lnorganie? 1913. NSRDS, Bmesvof Standards and Joint Committee on Powder Diffraction Standards, Wmhington. OC. (LO) See, lor example, Donnay. J.D. H., now. Amer Crysf. A.m., 3.74 (1967). 111) Ref. (81, p. 86. (12) Herbstein, F. H., and Mareh, R. E.,Acto Cryat., BPS,1061 (19821.
