The Wigner-Seitz Unit Cell Sidney F. A. ~ e t t l e 'and Lars J. ~ o r r b ~ ' Department of Chemistry and Chemical Engineering, Royal Military College of Canada, Kingston, ON, Canada K7K 5L0 In a recent communication (1) in this Journal we argued for a change in the way that some of the basic concepts of solid-state chemistry are presented to chemistry undermaduates. A motivation for this was the recoenition ., that physicists and theoreticians have an approach to the suhiect that i n auite different from that currentlv adopted by chemists and crystallographers. The increased interest in the solid state being shown by chemists, in such phenomena as conductivity and superconductivity a s well a s surface properties, points to a need to better understand the approach of the physicist and the reasons for it. I n our earlier paper (1) we addressed this problem. We were specifically concerned that an emphasis on the existence of both primitive and centered lattices (there are seven of each in the traditional presentation of the Bravais lattices) may inhibit a ready understanding of important conce~tsof the solid state. such as the Brillouin zone and the enumeration of electronic or vibrational quantum states. For such aspects of the solid state it is necessary to work exclusively with primitive unit cells.3 Each and every one of the fourteen Bravais lattices is, ultimatelv, primitive and has a primitive unit cell. Not surprisingly, this emphasis on primitive unit cells is precisely the attitude adopted by solk-state physicists and-theoreticians. Seven of the Bravais lattices are usually presented to chemists a s being centered because centered unit cells show the full point-group symmetry of the lattice. They show the svmmetrv of the characteristic point mouo (the holohedry )"of the Erysta~system. So, both bod;-centered cubic (BCC) and face-centered cubic (FCC) lattices are associated with unit cells of cubic shape and symmetry (Oh). In contrast, when the primitive translation vectors (also called primitive generators) of solid-state physics, presented in our earlier communication (I), are used to define the primitive unit cells ofthe centered cubic lattices, those ) , cubic. A unit cells are of rhombohedra1 symmetry ( D J ~not similar situation holds for the bodv-centered tetraeonal and the hody-centered and face-centered orthorhomhic luttices. Their primitive unit cells are distorted ~.hombohedra of lower sy&metry than the corresponding lattices, that is, of lower svmmetw than the holohedw of their respective crystal system. Is this unavoidable, or is it possible to obtain primitive unit cells of the centered lattices that do display the full point-group symmetry of the lattice? The answer to the second part of the question is "yes". Primitive unit cells with full holohedral symmetry are known for all 14 Bravais lattices. These are usually called Wigner-Seitz unit cells becanse they first appeared (in 1933) in a now classic paper by Wigner and Seitz "On the Constitution of Metallic Sodium" (2). Such unit cells also appear under other names, such as Delaunay (or Delone) reduced cells (3)and Dirichlet domains (41,this multiplicity reflecting their versatility and importance. 'Adjunct Professor. 'Author to whom correspondence should be addressed. 3Stri~tlvsoeakina orimitive unit cells in the reciorocal lanice. but the relevant cokepts i;e best explored in the direci lanice before passing to the reciprocal.
Despite this, WignerSeitz unit cells (W-S cells) are little known to chemists. To our knowledge, they appear in no chemistry textbook. In contrast, KittelH classic textbook on solid-state physics ( 5 )introduces the W-S cell a s early as Chapter 1. The purpose of the present paper is to take a step towards making W-S cells as familiar to chemists as they are to physicists. We therefore now turn to the way that W-S cells are obtained. Wigner-Seitz Unit Cells Constructing the Cells A W-S cell of any Bravais lattice is constructed as follows. Draw tie lines from a lattice point chosen as the origin to all nearby lattice points. Then erect a plane normal to each tie line and placed halfway between the lattice points. These planes will intersect, forming a sometimes elaborate three-dimensional box. The smallest volume enclosed by this construction is the W-S cell. It has one lattice point a t its center and by construction contains no other lattice points, so it is indeed a primitive unit cell. I t has, as we shall see, the full point-group symmetry of its lattice. I t is therefore understandable that there is yet another name for W-S cells: symmetric unit cells (6). Prirnitiue Cubic Bravais Lattice In Figure 1 the construction of a W-S cell is demonstrated stepwise for the primitive cubic (PC) Bravais lattice, the simplest possible case and also a case of highest symmetry, Oh I n Figure l a a PC unit cell is shown. The lattice point in red is chosen to be the center of the W-S cell, and tie lines to the six nearest-neighbor lattice points are drawn. I n addition, tie lines are drawn across the face diagonals to two of the twelve next-nearest-neighbor lattice points. The midpoints of these eight tie lines are indicated by stars. The second step, shown in Figure lb, is the erection of miniplanes perpendicular to the tie lines a t their midpoints. The third step, shown in Figure lc, is making the miniplanes larger until they intersect. The volume enclosed by these planes, outlined in red, is the W-S cell that we are seekincr. The planes arising from the two tie lines to the next-nearhnefghbor latticepoints have been left as miniplanes in Figure l c because thev contain only edges of t h e L w 3cell an2 do not contribute-faces to it. As we see, the enclosed volume is a simple cube, of Oh symmetry, of the same size a s the conventional unit cell. The only-but crucial-difference from the conventional primitive unit cell is that the W-S cell has its lattice point a t the center, whereas the conventional unit cell has "fractional" lattice points a t each corner (which are really lattice points, each shared with eight adjacent unit cells). The two tie lines drawn across the face diagonals in step l a were deliberately included to show that only the truly nearest-neighbor lattice points in each direction contribute to the W-S cell. AW-S cell contains all the space around a lattice point that is closer to it than to any other lattice point. Other Unit Cells In Figure 2 we give similar constructions for the primitive tetragonal (Fig. 2a) and primitive orthorhomhic (Fig. Volume 71
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2b) unit cells, hut do not include any noncontributing planes. For these two lattices the W-S cells also look like their conventional counterparts and are of Ddh and DZhsymmetry. However, for the other 11Bravais lattices the W-S cells differ from the conventional u n i t cells found i n textbooks. These W-S cells a r e no longer parallelepipeds; they have more than six boundary faces." One of the simplest examples i s provided by the end-face-centered orthorhomhic lattice, shown i n Figure 2c. The lattice points that give rise to W-S cell faces here are the four adjacent centering lattice points together with the two nearest corner lattice points in the same plane plus one lattice point above the plane and one below. We thus ohtain a cell with eight faces, which i n cross section is a distorted hexagon of DZhsymmetry. The geuuinely hexagonal lattice, likewise, gives rise to a W-S cell with eight faces, but the cross section is a regular hexagon (Fig. 2d), so its symmetry is Dsh. There are two p o i n t s of i n t e r e s t h e r e . T h i s hexagon-shaped cell is primitive, like all W-S cells. In contrast, to obtain a truly hexagonal unit cell (i.e., ofDsh symmetry) in the conventional crystallographers' approach i t must be made triply primitive, containing three lattice points a n d with a threetimes larger volume. The above development makes it clear that there is a close relationship between the end-face-centered orthorhomhic and hexagonal lat- F gJre 1 a Tne prlm love CJDK -n t ce I The lance pomr n reo w oe tne cenler of the W~gner-SeIZ tices; one may be smoothly trans- ,W-SI Ln I ce o M nlplanes erecfed na Inray oetween !he cenlra anlce po n l an0 11ss x nearesl ne gnformed into the other by letting oors ano two of 1s we ve next-neareslne gnoors c S x m npanes enlarge0 mt tney nlersect,ocf nthe a and b axes of the orthor- ing the W-S unit cell (Onsymmetry), hombic cell eraduallv become (there are only seven possible symmetries, anyhow). So, for equal (a = b) a s in the hexagonal crystal system. example, for the hody-centered tetragonal lattice there are two quite different W-S cells, both of Dph symmetry, deFacetting pending on whether a > c or a < c, as shown in Figures 3a We have seen how a W-S cell may have more than six and 3b. As a result there are 24 distinct, differently facetfaces. Indeed, the more symmetrically a lattice point is ted W-S cells in three-dimensional space ( 6 8 ) .When the surrounded, the more lattice points there will he a t more symmetry of the lattice is high enough, i t removes any axor less the same distance from it, and the more facets the ial ratio freedom, and the W-S cell becomes both fixed in W-S cell will have. This is indeed so. W-S cells have in shape and highly facetted. The BCC and FCC lattices are some cases as many as fourteen faces and often look someexamples of this. The BCC W-S cell (Fig. 3c) may be rething like the idealized crystals pictured in many introducgarded as a development of the body-centered tetragonal torv textbooks. Conseoueutlv. the ~ r i m i t i v egenerators (Fig. 3a), with rectangular faces becoming square and sixmcntmncd in the introduction normally do not correspond sided faces becoming regular hexagons (but having only a t u ihe edges of the W-S n,ll.;. threefold axis due to the adjacent faces). Alternatively, i t However, the edges and faces of a W-S cell may be exmay be derived from the constructional procedure depressed a s functions of the primitive generators; explicit scribed above. expressions have been given by Koster (6) and Turrell(7). In Figure 3c six square faces (of which only three are seen) T6ere are fourteen ~ r a v a i lattices, s but there are more Wof the W S cell arise due to hody-centering lattice points at a S cells than that because the W-S facetting depends on the distance ofa (the cube cell edge) in the six adjacent body-cenrelative magnitudes of the conventional unit cell edges and 4Becauseevery lattice is centrosymmetric, the facets of any W S angles. Variations i n these quantities give rise to W-S cells cell always come in pairs of parallel faces. that look quite different, although of the same symmetry
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edges. I n other words, these W-S cells display the full point-group symmetry of their lattices. All W-S cells illustrated above display the full holohedry of their respective crystal systems. This is a general property and a key feature of W-S cells.
Figure 2. a. The W-S unit cell (Dm)of the primitive tetragonal lattice. b. The W-S unit cell (D2h)of the primitive orthorhombic lanice. c. The W S unit cell of the end-centered orthorhombic lattice, of distorted hexagonal shape (Dz~).2d. The W S unit cell of the hexagonal lanice, of regular hexagonal shape 1Dsl. , -.,
tered unit cells. Eight hexagonal faces (of which only four are seen) arise from cube-wmer lattice points a t a distance of (4312)~= 0.87a from the origin. As is general, these shorter distances lead to the larger facets. Thus there are seven pairs of faces, a total of 14. The W-S cell of the FCC Bravais lattice looks quite different. 'helve diamond-sha~edfaces (of which only six are seen i n Figure 3d) arise from the 12 nearest neighboring lattice mints. each a t a distance of (d212)a = 0.71a from the W-s cell has six pairs of parallel faces, a origin. 'Thus, total of 12. the crysta1 Vstem The three different W-S (Figs. lc, 3c, and 3d) all explicitly display full cubic symmetry. For instance, all the pure rotation operations of the Oh point group are manifest in faces, edges, or patterns of
this
plore t h e symmetry of cepk5
Symmetry and Space T h e theory of point-group symmetry has had a major impact on chemistry, although surprisingly little on first- and second-year undergraduate c h e m i s t r y c o u r s e s , notwithstanding the availability of a t l e a s t one textbook which attempts to make the subject accessible a t this level (9).In contrast, the theory of translational symmetry has had very little influence a t all on the subject, notwithstanding-the presentation in . first- and second-year chemistry courses of highly relevant material; t h e basics of solid-state structural chemistry. The reason is easy to see. Chemists like to t h i n k i n t e r m s of a t o m s a n d molecules. A tonic such a s the Bravals lattlccs ir thcrcfurc seen as a rather broad-brush class~fication of crystal structures, to he discussed before passing to the topic of real interest: the crystal structures themselves. The fact t h a t t h e Bravais lattices a r e empty is generally ignored. Translational symmetry is actually obscured hy the presence of atoms and molecules. If there i s a set of primitive lattice vectors connecting equivalent points in space, then it simply does not matter, from the point of view of translational s y m m e t q what atoms, if any, occupy these points or w h a t pattern of atoms surround them. The point is made as in this paper, One evident ~0ncentratesOn a n empty lattice. Actually, the present paperis one i n a series of t h r e e (10, I) i n which we have attempted to exempty lattices and related con-
51t is a common mistake among students to fail to distinguish properly between lattice points and atoms. The unit cells associated with three-dimensional lattices are always empty and have one of only seven symmetries. However, filled unit cells, as encountered in real crystal structures, have one of 230 symmetries. In most textbooks the unit cells of the BBC and FCC lattices are exemplified with crystal structures like a-Fe and Cu. In both cases the basis. the atoms associated with each lanice point, is a single atom. This means that a structure model (of the kind normally used in teaching) of an empty unit cell and one containing atoms look the same. However, this only happens when the basis is a single atom. Progressing to such a simple structure as CsCI, where all of the atoms cannot be placed at the corners of the unit cell, we can easily see the differencebetween its empty primitive cubic unit cell and the filled one. Volume 71
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I n t h i s series we have stopped short of a full diseussion of the irreducible repres e n t a t i o n s of t r a n s l a t i o n groups b u t have rather attempted to show that the concept of the empty lattice is neither trivial nor valueless. I n logical sequence, the concept of all Bravais lattices a s primitive ( I ) leads to primitive unit cells showing the full point-group symmetry of the lattice, to Wigner-Seitz unit cells. These, in turn, when applied to the reciprocal lattice, are identified with the first Brillouin zone. I t is with Brillouin zones that we find the connection with the irreducihle representations of transl a t i o n groups. Indeed, we have suggested (10)that the first Brillouin zone may be taken a s the translation group equivalent of the character tables well-known to chemists i n t h e context of point groups. However, in our opinion, a proper understanding of Brillouin zones requires a familiarity with the concept of Wigner-Seitz unit cells presented in this papers.
3a
Acknowledgment One of u s ( S F A K ) i s indebted to the Department of National Defence of Canada for financial support (ARPgrant FUHGC). Appendix I t must he recognized that 3~ the traditional presentation of the subject may well blinker our thinking to some of the Figure 3. a. The W-S unit cell (D44 of a body-centered tetragonal lattice where a > c. b. The W-S unit cell that exists in the (D4h) of a body-centered tetragonal lanice where a < c. c. The W-S unit cell (Oh)of a body-centered cubic definition of a unit cell. It is lanice. d. The W-S unit cell (Oh)of aface-centered cubic lanice. only t h e volume a n d t h e atomic contents of a primitive unit cell that are fixed for any Literature Cited given crystal structure. here are no restrictions on the 1. Kettle, S. F. A ; NDIT~Y, L. J. J. C h m . Educ. 1993. 7 0 , 9 5 9 4 6 3 , nature of its boundarv surface. Conventionallv. of course. 2. Wigner. E: Seitr. F Phys. Re". 1933.43,804-810. the unit cell is houndeb by three pairs of paralcel planes: 1t 3. Vainshtain, B. K Modern Cr/slollogrophy I Springer-Verlsg: Berlin. 1981; p 154. is a parallelepiped. However, as we have seen with many D. ~ ~ i d ~ l : 1986: 13. a. E"@. P oametne of the W-S cells, there can be many more pairs of facets 5. Kitfel, C. lntrodudion to Solid State Physics. 6th ed.: Why: New York, 1986: p 12. than this. There is not even a requirement that the bound6 ~oster.G.E Spocp~mupsomiThevRopm=nmtlons;kadermc:NewYork, 1957;p22. ary surfaces of the unit cell be planar. They can be convol.Turrell, G. InframdandRomon SpctrnofCrysfil1s;Aeademie: London, 1972;Appenluted and even re-entrant although it is difficult to condir F, p 350. ceive of any physical situation in which it would be useful 8. Bums,G.:Glazer. A. M. Spoee Gmupa for Solid Stars Seirnllsis. 2nd ed.: Academic: to invoke such a surface. Those who have attempted to deBoston, 1990; Appendix 3. p 290. duce unit cells in some of the drawings of the Dutch artist 9. Kettle, S. E A. Symmetry ondStruetum; Wiley: Chichesier and New York, 1985. Maurits Escher (11)will be well aware of this because for 10. Kettle. S. F. A ; Norrby, L. J.J. Chem. Educ IsW),67,1022-1028. many of his pictures curved unit cell edges are more evill.M.CEshrr HisLiPnndCompIe~GrophieWork Loeher.J. L.. Ed.:HanyN.Abramn: NewYork. 1931. dent than are straight-line edges.
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