Physical Chemistry , 1st ed.

a, b.To determine the crystal structure we construct the following table:

d(from Bragg’s law; Å) 1/d^2 (Å^2 )

13.7 3.25 0.0947

15.9 2.81 0.127

22.8 1.99 0.253

27.0 1.70 0.346

28.3 1.63 0.376

33.2 1.41 0.503

36.6 1.29 0.601

37.8 1.26 0.630

Taking the ratio of the first two reciprocals:



0

0

.0

.1

9

2

4

7

7

0.746

which is close enough to 0.75: the crystal is face-centered cubic.

c.To determine the unit cell dimension, we can use equation 21.10:

(h^2 k^2 ^2 ) 

d

a^2

 2

Table 21.3 indicates that the first diffraction, occurring at 13.7°, must have

the Miller indices (111). We therefore substitute, using the corresponding d

spacing from the table above:

(1^2  12  12 ) 

(3.2

a

5

2

Å)^2



Solving for the unit cell dimension:

a5.63 Å

As it turns out, this is the unit cell parameter for sodium chloride, which has

a face-centered cubic unit cell. This kind of procedure is applicable to any

cubic crystal.

For other types of unit cells, the pattern of X-ray diffractions depends on

the exact (nonperpendicular) angles that the unit cell’s sides make with each

other. A general discussion that is applicable to all such crystals is impossible

because of this. However, there is still a relationship between the dspacings of

the atomic planes and the Miller indices; it is just not as simple as for cubic

cells. The equation relating the dspacing with Miller indices, unit cell dimen-

sions, and unit cell angles is complex and will not be considered here.

Details of experimental X-ray crystallography can be obtained from texts on

analytical chemistry or experimental physical chemistry.

Finally, we need to consider the fact that in an X-ray diffraction pattern, not

all diffractions have the same intensity. (See Figure 21.23 as an example.) This

might be considered unusual: the X rays going in have a given intensity; why

is it that the X rays coming out have different intensities?

The answer lies in part with the atoms that are contributing to the individ-

ual diffractions. Not all atoms scatter X rays with the same efficiency. The abil-

ity of an atom to scatter X rays is directly related to the electron density of the

atom. (It is this very concept that makes X rays useful for medical purposes: tis-

sues composed of heavier materials, like bone, scatter X rays more than other

750 CHAPTER 21 The Solid State: Crystals