Lawrence Suchow New Jersey Institute of Technology Newark, New Jersey 07102
Other Views of Unit Cells
The conventional manner of depicting the unit cells of crystalline materials with atoms a t the corners is probahly the best way if only one representation is to be chosen. There are, however, other ways to draw them in order to demonstrate various concepts more clearly. I t should be remembered that the origin of a unit cell may be chosen to occur at any point and the axes then drawn from that point in the usual directions and with the proper lengths. Unit cells so drawn can be used to show very easily the number of atoms per cell as well as the structural symmetry of compounds, and this will be done in this paper. In some cases, extracellular atoms will be added in order to make certain points about symmetry, and in one case the origin will not be shifted at all though extracellular atoms will be drawn in. For simplicity andbecause of their common occurrence, all structures discussed herein will be cubic, with simple, body-centered, and face-centered cases included. The unit cell origins will, for convenience, be taken to be a t the lower left front corners of the cubes, and positive directions will he considered to be up, right, and away from the viewer. Elements Figure l(a) is a conventional drawing of a simple cubic unit cell of an element (the only known example being one form of Po). I t illustrates that the atoms can he thought of as being a t all the corners of the cube, and it can then he shown that since only one-eighth of each of the eight atoms is within the cell, there is one atom per unit cell volume. In Figure l(h), the origin of the cell has been shifted to a position one-half the distance along the hody diagonal (i.e., the [ I l l ] direction). This illustrates the cube content very simply because there is now only one whole atom, which is a t the cube center. Figure 2(a) is a conventional representation of a hodycentered cubic unit cell of an element (e.g., K, V, Cr). There is a whole atom at its center and, since it can he shown once more that the eight corner atoms really contribute only one to the cell, there are, overall, two atoms per unit cell. By shifting the origin of the cell one-quarter the distance along the body diagonal (again [ I l l ] ) , Figure 2(b) results and one immediately sees two whole atoms per unit cell situated along the hody diagonal. Figure 3(a) is a conventional depiction of a face-centered cubic unit cell (e.g., Cu, Ag, one form of Ni). It is usually proved that it contains four atoms by showing that the eight corner atoms contribute one atom (as in the cases above) and the six face atoms three more because each of these is one-half within the cell. If, as in Figure 3(h), the cell origin is shifted to a position one-quarter the distance
la1
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Figure 1. Unit cell of a simple cubic element (a) as usually depicted, and (b) with wigin shifted half the distance along the cube diagonal.
226 / Journal of Chemical Education
along the body diagonal ( [ I l l ] ) , all four atoms are entirely within the cell. Compounds The different views of elemental unit cells presented in Figures l(h), 2(b), and 3(h) permit an easy count of atoms per unit cell but make the symmetry more difficult to ascertain. In compounds, however, shifting cell origins or making other changes can be a great aid in seeing the symmetry. Many authors have used the expression, "interpenetrating lattices," to call attention to the symmetry in a unit cell, but this is not really satisfactory and even fails in one case to he discussed below (the fluorite structure). Rather than seeking out simpler lattices which interpenetrate, it makes more sense crvstalloeranhicallv to show that. in " . of atoms are ;encompound structures, [dentical tered on nositions such as the elemental atom positions of Figures lia), 2(a), and 3(a). As with atoms in the elemental bases, then, a compound is simple cubic if these groupings are centered only on the unit cell corners, body-centered cubic if there is a group a t the body center which is identical with those at the corners, and face-centered cubic if there are groupings on all the faces which are identical with those on the corners. In all these cases, the spatial orientation of all groups must be the same in any given unit cell. In Figure 4(a) is the usual representation of the simple cubic CsCl structure and it can be shown by arguments such as those employed above that each cell contains one formula unit. In Figure 4(b) the origin has been moved onequarter of the distance along the body diagonal and sufficient atoms have been added so that there is a formula unit in the shape of an unsymmetrical dumbbell along [ I l l ] centered on each of the eight cuhe corners. I t should he pointed out that, in this as well as in subsequent drawings, the lines joining atoms do not represent special bonds, but rather merely serue to isolate formula unit groupings for
Figure 2. Unit cell of s body-centered cubic element (a) as usually depicted, and (b) with origin shined one-quarter the distance along the cube diagonal.
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Figure 3. Unit cell of a face-centered cubic element (a) as usually depicted. and (b) with origin shined one-quarter the distance along the cube diagonal.
0 cs @cl Figure 4. Unit cell of CsCl (a1 as usually depicted, and (b) with origin shined one-quarter the distance along the cube diagonal.
symmetry purposes. In Figure 4(b) one can also see that one full Cs and one full Cl are in the unit cell. Finally, i t should be quite clear from this drawing why the all too common textbook classification of the CsCl structure as hody-centered cubic is incorrect. Going on to the face-centered cubic NaCl (rocksalt) structure, the usual form of its unit cell appears in Figure 5(a), from which it can be demonstrated by the type of reasoning used above that each cell contains four formula units of NaCI. In Figure 5(b) the origin has been moved one-quarter of the distance along the cell edge (i.e., the [loo] direction) and atoms added so that there is an unsymmetrical Na-C1 dumbbell parallel to [loo] centered on each of the corners and face centers of the unit cell. If one wished to draw a unit cell of NaCl containing four full Na and four full C1 atoms, it would he necessary only to shift the cell origin to any suitable point (e.g., at one-quarter the distance along the body diagonal). The usually seen form of the face-centered cubic ZnS
IaJ
0
Na
0 Cl
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Figure 5. Unit cell of NaCl (rocksan) (a) as usually depicted, and (b) with origin shifted one-quarter the distance along the cube edge.
0zn 8s Figure 6. Unit cell of cubic ZnS (zincblends or sphalerite) (a) as usually depicted, and (b) with origin shined ane-eighth the distance along the cube diagonal.
Figure 7. Unit cell of CeF2 (fluorite) (a) as usually depicted, and (b) wnh CaF2 ''units?' shown along the [iil cube ] diagonal.
(zinchlende) structure is given in Figure 6(a), while Figure 6(b) is a unit cell drawn with the origin moved one-eighth of the distance along the cube diagonal. Once again atoms are added where necessary and the Zn-S formula units are drawn as unsymmetrical dumbbells, here aligned with [lll].There is a dumhbell centered on each corner and face center of the unit cell. In Figure 6(a), one can show that there are on average four formula units of ZnS per unit cell, but in Figure 6(b) there are actually four whole Zn and four whole S atoms within the cell. I t is interesting to note that the four Zn atoms in Figure 6(a) are in the same positions as the four atoms in Figure 3(b), though the two cells are viewed from different directions. The last example to he offered here is face-centered cubic CaF2 (fluorite), whose conventionally taken unit cell appears in Figure 7(a). If this were to be considered in terms of interpenetrating lattices, it would seem to consist of a face-centered cubic lattice of Ca2+ ions and a simplecubic F- ion lattice. thoueh the latter lattice would consist of cuhelets only one-eigh& the volume of the unit cell. 1t is not, therefore. obvious from such consideration that the fluorite structure is face-centered. However, on drawing the unit cell as in Fieure 7(b). in which there is no shift in origin but fluoride ions have 'been added outside the cell, one immediately sees the linear F-Ca-F formula units. Because of the relatively large number of atoms required for the alternate view of the CaFz cell and because the F-Ca-F rods are quite long, drawing the rods along the [ I l l ] body diagonal, which is elongated by the perspective chosen for the drawing, results in some confusion. One can, however, select any of the (111) directions, and Figure 7(b) is the case where the linear F-Ca-F groups are taken to be along the [111] direction. With the unit cell perspective employed, foreshortening occurs in this direction and greater clarity results. If desired, such modification could also be applied to Figures 4(b) and 6(b). Since the origins of the unit cells in Figures 7(a) and (b) are the same, there are in both cases only fractional Ca2+ ions actually inside the cells along with eight whole F- ions, and the presence of four CaF2 formula units per unit cell can he calculated by the usual method. Alternatively, by thinking in terms of whole CaFz units a t all the corners and face centers, one can also arrive a t the figure four just as in the case of the face-centered cubic unit cell of an element (Figure 3(a)). If one wished, however, the origin could be shifted one-eighth the distance along the body diagonal so that only four full Ca2+ and eight full F- ions would be within the cell. This would of course mean that the linear F-Ca-F groups would he attached to the corners and face centers at points between the Ca and one of the two F's rather than a t the Ca, but the symmetry would still be obvious because all the groups would be displaced in the same way.
Volume 53,Number 4. April 1976
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