Rodney J. Sime Sacramento State College Sacramento, California
Some Models of Close Packing
Three dimensional sketches of the models in Figures 1 and 2 (or a line lattice equivalent) are often used to represent the difference between ABABAB . . . and ABCABCABC . . . close packmg of spheres. It is readily apparent that A R packing gives rise to a
Figure
1.
Examination of the model in Figure 3 reveals that the six yellow spheres (medium grey in the photograph) completely inclose an octahedral hole, while under each red sphere (darker grey in the photograph) lies a tetrahedral hole, formed with the three yellow spheres to which the red sohere is tangent. The models in Figures 3 and 4 differ only i n t h e addition of a few spheres in some octahedral holes. If the small spheres represent cations and the large ones anions (the usual case), then the structure is that of sodium chloride, a general type consisting of cubir close packed anions, containing an equal number of rations in the octahedral holes.
Figure 2.
hexagonal unit cell to which the model in Figure 1 is equivalent.' However, it is not immediately apparent that Figure 2 is of cubic symmetry. Both models in Figures 2 and 3 consist of the layers ABCA, although three spheres have been added to each of the B and C layers and six spheres removed from the top A layer in Figure 2 to arrive a t Figure 3. The face centered cube is easily seen in Figure 3. The reason that the student sees hexagons in both Figures 1 and 2 is because all the layers in both kinds of packing lie perpendirular to a six fold axis of symmetry. The rube is sitt,ing, not on a square face, hut on a body diagonal, which is a six fold axis of rotation. The hexagonal prism is sitting on a hexagonal base. A line connecting the face centering atoms in these faces is also a six fold axis of rotation. So what the student intuitively detects in common with the two kinds of packing is not the hexagonalness, but rather the six fold symmetry axis perpendicular to the direction of packing. The models mere constructed from ll/?in. stvrafoam balls. The balls in each layer were konnectkd with round toothpicks and the layers were stacked close packed. In order to bring out the symmetry of the packing, some of the balls were colored red or yellow. The layers for constructing the models in Figures 2 and 3 are shown in Figure 5 , in their proper relative orientat,ion. I BUNN, C. W., "Chemical Crystallography," Oxford University Press, London, 1945, p. 135. The crystallographergenerally finds it convenient to choose one third of the illustrated unit cell as his unit cell.
Figure 3.
Figure 4.
The ~mrdinationnumber of an ion is the number of nearest neighbors to which it is tangent. In sodium chloride earh anion is surrounded by and tangent to six anions; each cation is surrounded by and tangent to six anions, and the structure is 6:6 coordinate. I n simple close packing, however, each sphere has twelve nearest neighbors, and is twelve co: ordinate. If attention is focused on the dark sphere in the top layer in Figure 2, it isseen to be surrounded by six white nearest . . neighbors. T h e layer immediately below consists of three more nearest '! neighbors; imrnedia t e ~ y a h o the v ~ dark sphere three spheres are missing in the model, for a total of txelve. :
bF-
Figure 5.
Volume 40, Number 2, Februory 1963 / 61
The distance between layers in terms of the radii of the spheres may be readily calculated from Figure 6. The centers of four spheres in contact with each other in this close packed manner form a regular tetrahedron, the altitude h of which is the layer separation and is given by: h equals ( 8 4 3 1 3 , where s is the length of the edge of the tetrahedron. Since the length of the edge is twice the radius of a sphere, the axial ratio (2h/2r) for any hexagonal close packed structure is just (24\/6)/3or 1.633. When the renters of four close packed spheres forming a tetrahedron are located on a rectangular coordinate system, they lie on the corners of a cube illustrated in Figure 7. The calculation of the size of a tetrahedral
When s is eliminated: R,/R.
Figure 7
hole is simplified when it is realized that the radius R, of the hole and R. of the sphere equals half the body diagonal of the cube whose edge is s. Also, the sum of two radii R, equals a face diagonal. 2R, = (a&)
0.22475
+
Figure 9.
Figure 10.
For a supplementary approach to close packing, the articles by Campbell? and by Barnett3 are helpful. Pauling's monograph4 contains a wealth of material on close packing, and Sanderson's book5 is virtually a dictionary of chemical models. The author wishes to express his gratitude for the support of the National Science Foundation during the period in which this mas written. a
BARNETT, E. D., J. CHEM.EDUC., 35, 186 (1958).
' PAULING,L., "The Nature of the Chemical Bond," 2nd ed., Cornell University Press, Ithacrt, New York, 1948. i S a ~ ~ ~ R. ~ sT.,o "Teaching ~ , Chemistry with Models," D. Van Nostrand Company, Inc., Princeton, Kew Jersey, 1962.
And:
/
=
* CAMPBELL, J. A,, J. CHEM.EDUC., 34,210 (195i).
Thus:
62
4 3 3 -1
Qualitatively, this ratio is seen in Figure 8, which shows a cation nestled into a tetrahedral depression, the covering sphere having been removed. The size of an octahedral hole in terms of its radius R, and that of the sphere R. may be calculated from Figure 9. Cosor = R./(R, R,) = 1 / 9 2 so that RJR. = - 1 = 0.4142. The relative positions and sizes of the spheres, octahedral holes, and tetrahedral holes are shown in Figure 10. The view is parallel to the direction of packing.
Figure 8.
Figure 6.
=
Journal of Chemical Education
