Physical Chemistry , 1st ed.

diffraction of X rays by several simple powdered compounds that have vary-

ing Bravais lattices. The different planes with their different Miller indices dif-

fract X rays at different angles. (In practice, the X rays used are usually very

close to monochromatic, which means that is the same for all diffractions.

Only the dspacings between the planes and the angles of diffraction differ.)

Even for simple crystals, X-ray diffraction patterns are complicated enough

that computer analysis is useful.

The angles of X-ray diffraction can, however, be expressed in terms of the

Miller indices (which are fractions or multiples of the unit cell dimensions,

after all). The relationship between the Miller indices and the various planes

of reflection is simplest for crystals having perpendicular unit cell axes, as

might be expected. Without deriving it, the relationship between dspacing and

Miller indices of a plane of atoms is



d

1



h

a^2

2



b

k^2

 2 



c^2

2



1/2

(21.7)

This expression is useful for cubic, orthorhombic, and tetragonal crystals. For

cubic crystals, equation 21.7 can be simplified because abc. In terms of

the dspacing directly, we get

d (21.8)

The Bragg equation for a cubic crystal is therefore (for the first-order diffrac-

tion, which is typically the strongest):

 2 sin (21.9)

The angles of diffraction of (monochromatic) X rays by cubic crystals are

thus relatively easily predicted, because the Miller indices themselves are re-

stricted to whole numbers in cubic crystals. Depending on whether the crys-

tal is simple cubic, face-centered cubic, or body-centered cubic, different

planes are defined by the atoms in the crystal, and so different angles of dif-

fraction may be possible. However, the pattern of possible angles is charac-

teristic of the type of cubic crystal, because of the integral possible values of

the Miller indices. Figure 21.24 shows the relative patterns of the diffracted

X rays along with the Miller indices of the unit cell planes that diffracted the

radiation.

Because cubic crystals have such relatively simple X-ray diffractions, it is

common to rewrite equation 21.8 in the forms

h^2 k^2 ^2 

d

a

 or h^2 k^2 ^2 

d

a^2

 2 (21.10)

Because h,k, and are whole numbers, it is easy to construct a table of their

possible values, and therefore the possible values ofa/d. Crystals whose X-ray

diffractions appear at these relative values are easily identified as cubic crystals.

Table 21.3 shows such a table for comparative purposes. Because of occasional

destructive interference due to diffracted X rays being exactly 180° out of

phase, only certain combinations ofh,k, and will appear depending on

whether a crystal is primitive, face-centered, or body-centered cubic. The pat-

tern of the possible Miller indices of refracting planes is therefore specific to

a



h^2 k^2 ^2

a



h^2 k^2 ^2

748 CHAPTER 21 The Solid State: Crystals

0
, degrees

Face-centered

Body-centered

Primitive

5040302010 60

Figure 21.24 The pattern of diffracted X rays

is characteristic of the type of unit cell a crystal

has. Here, the patterns that the three cubic Bravais

lattices cause on the X-ray diffraction is illus-

trated. For all three, we are assuming a unit cell

parameter of 3.084 Å and an X-ray wavelength of

1.542 Å. Compare these patterns with the films in

Figure 21.23: can you detect any correspondences

between these two figures?