Physical Chemistry , 1st ed.

interference condition. Figure 21.17 shows that the same wavelength of X rays

as in Figure 21.16b will be diffracted at a different angle because now nequals

2: this is called second-order diffraction. There is also third-order diffraction

(n3), fourth-order diffraction (n4), and so on. The first-order diffrac-

tion is usually the most obvious, although depending on the crystal the zeroth-

order diffraction (n0, simple reflection) may also be very obvious.

The following examples illustrate the use of Bragg’s law.

Example 21.5

A simple cubic lattice has atoms that are spaced by 2.77 Å. If the diffraction is

caused by planes of atoms that are 2.77 Å apart, at what angle are X rays diffracted

in the first order and the second order if they have a wavelength of 1.82 Å?

Solution

Notice that the X-ray wavelength and the distance between diffracting planes

are given in the same units, angstroms. This is important, since these are the

only quantities in Bragg’s law that have units. For first-order diffraction, we

use equation 21.6:

 2 dsin

1.82 Å2(2.77 Å) sin

The only variable is the angle :



2 

1.8

2

2

.7

Å

7Å

sin

Again, notice that the units of Å cancel algebraically. We get

sin 

0.329

Taking the inverse of the sine function of both sides, we find that



19.2°

So the angle in Figure 21.16b would be 19.2° in this case. At all other angles,

destructive interference takes place and virtually no intensity of X rays is de-

tectably diffracted.

For the second-order diffraction, we must use n2:

2  2 dsin

(It was assumed that nis 1 in the first part, so it did not show up in the ex-

pression for Bragg’s law.) We get

2 1.82 Å2(2.77 Å) sin

^2

2





1

2

.

.

8

7

2

7

Å

Å

sin

sin 

0.657

Again, note that the units cancel. Taking the inverse sine of both sides, we find



41.1°

Notice that the angle of the second-order diffraction (41.1°) is notdouble the

angle of the first-order diffraction (19.2°). That’s because the order of dif-

fraction is dependent on the sine of the angle, not the angle itself.

742 CHAPTER 21 The Solid State: Crystals

Net

reflection

In phase

(b) Constructive interference



No net

scattering

Out of phase

(a) Destructive interference





d

Figure 21.16 (a) At any random angle, X rays

that reflect off sequential layers of atoms in a

crystal destructively interfere to yield, ultimately,

no net intensity of refracted X rays. (b) If the re-

lationship between angle, X-ray wavelength, and

spacing between the layers of atoms is just right,

there are an integral number of wavelengths be-

tween the distance traveled by X rays and there is

constructiveinterference: a strong diffraction of

X-radiation occurs. The relationship between an-

gle, X ray wavelength, and dspacing is called the

Bragg equation. This equation includes the possi-

bility that there may be more than one wave-

length between adjacent reflections; this is the

orderof the diffraction.

Figure 21.17 When there are two (or more)

additional wavelengths between adjacent reflections

of X rays, it is considered second- (or higher-)

order diffraction. In this case, the variable nin

the Bragg equation equals 2.