Page 23 - Modern physical chemistry

12 Structure in Solids

planes cut the diagonal through the cube into equal parts. This line is the hypotenuse of
a right triangle whose sides are {2a and a long; so it is -f3a in length.
In a unit cube of the simple cubic lattice, edge a is left undivided, the diagonal {2a
is bisected, the diagonal -f3a is trisected. So the ratios of the spacings are given by
-J2 13
dlOO : d 110 : d l1 ! = 1: - [1.5]
2 3
See figure 1.8
For the face-cantered cubic lattice, figure 1.9 shows that
-J2 213
d lOO : d 110 : d1l1 = 1: - : --. [1.6]
2 3
Spacings for the body-cantered cubic lattice appear in figure 1.10, whence

dlO O : d 110 : d l1 ! = 1: -J2 : 13 [1.7]
3

Example 1.4
X-ray studies of a cubic crystal by the Bragg method yielded the ratios
dlOO : d 110 : d l1 ! = 1.000: 0.709 :1.164.

What lattice does the crystal possess?
These numbers are very close to those in (1.6):
-J2 213
1: - : - = 1.000: 0.707: 1.155.
2 3
So the crystal has a face-cantered cubic lattice.

1.10 Some Simple Crystal Types

The physical entity, the basis, associated with a lattice point may consist of one or
more atoms or ions, spread out over space. The lattice point may be located at the mean
centroid of the entity, or at the equilibrium position for one of the nuclei, or at some
other constant position with respect to the equilibrium point for the basis.
A formula unit of a strong electrolyte consists of positive and negative ions, cations
and anions. The whole unit may be associated with a lattice point. Or, one may consider
the equilibrium position for the center of each cation a lattice point, while the similar
position for each anion is a point on another interpenetrating lattice.
Sodium chloride and other alkali halides generally come out of solution as small imper-
fect cubes. In the presence of impurities, the comers may not develop, causing octahe-
dral planes to appear. These external shapes indicate that the internal lattice is cubic.
Accordingly, crystallographers early surmised that the alkali ions and the halide ions are
arranged alternately in the three-dimensional checkerboard pattern of figure 1.11.
This was first tested by W. L. Bragg. His X-ray measurements showed that NaCI, KBr,
and KI are face centered, while KCL seemed to be simple cubic. Now, a given ion inter-
acts with X rays largely through its electrons. In the first three salts, each anion contains
many more electrons than the accompanying cation; so the anion is much more effective