Physical Chemistry , 1st ed.

the type of cubic unit cell a crystal has. (For additional details, consult a crys-
tallography text.) Equation 21.10, Table 21.3, and Bragg’s law are used together
to determine the size and type of unit cell a cubic crystal has. The following
example illustrates how to use these three items.

Example 21.9

An unknown crystal has some cubic unit cell. The refractions of X rays by a

powdered sample are seen at the values of 13.7°, 15.9°, 22.8°, 27.0°, 28.3°,

33.2°, 36.6°, and 37.8°. If the X rays have a wavelength of 1.5418 Å, determine

the following:

a.The dspacings for each refraction

b.The type of cubic unit cell

c.The unit cell parameter a(that is, the length of the cubic unit cell)

Solution

Table 21.3 indicates that only certain combinations of Miller indices are pos-

sible for diffracted X rays. We will perform the following steps:

1. Use Bragg’s law to calculate a dspacing for each angle of diffraction.


2. Square the dspacing and determine its reciprocal. We now have 1/d^2 ,but
   we still don’t know aor the Miller indices.


3. Take the ratio of the two lowest reciprocals.


4. Look at the entries in Table 21.3. If the ratio is 0.5 (or ^12 ), it can be ei-
   ther primitive or body-centered cubic. To determine which it is, take
   the ratio of the seventh and eighth values of 1/d^2. If the ratio is 0.875
   (or ^78 ), it is body-centered cubic. If the ratio is 0.889 (or ^89 ), it is sim-
   ple cubic.


5. If the ratio of the two lowest reciprocals is 0.75 (or ^34 ), the crystal is face-
   centered cubic.

21.6 Miller Indices 749

Table 21.3 Values of (hk) and (h^2 k^2 ^2 ) that diffract X rays in cubic crystalsa

(hk) (h^2 k^2 ^2 ), sc (h^2 k^2 ^2 ), bcc (h^2 k^2 ^2 ), fcc

100 1 — —

110 2 2 —

111 3 — 3

200 4 4 4

210 5 — —

211 6 6 —

220 8 8 8

300, 221 9 — —

310 10 10 —

311 11 — 11

222 12 12 12

320 13 — —

321 14 14 —

400 16 16 16

Source:D. P. Shoemaker, C. W. Garland, and J. W. Nibler,Experiments in Physical Chemistry, 6th ed., McGraw-Hill,

New York, 1996.)a

For each set of Miller indices, if the value (h^2 k^2 ^2 ) is listed under each type of cubic cell, then that plane of

atoms diffracts X rays. A — indicates that the plane does not diffract X rays (or experiences extinction). The pattern

of diffracted X rays therefore indicates the type of cubic unit cell a crystal has.