Physical Chemistry Third Edition

1160 28 The Structure of Solids, Liquids, and Polymers

is a scattering center at each lattice point. Each scattering center diffracts incident

electromagnetic radiation, sending a spherical electromagnetic wave out in all direc-

tions. The condition for constructive interference in a particular direction is that two

waves from adjacent diffraction centers have crests and troughs at the same locations.

If the distance between the planes is equal tod, and the wavelength of the radiation is

λ, the condition for constructive interference is that the extra distance traveled by the

wave diffracted from the second layer is an integral number (n) of wavelengths. The

integernis called theorderof the reflection. Trigonometry gives the condition:

nλ 2 dsin(θ) (28.1-1)

where θ is the angle between the plane and the direction of the radiation.
The Bragg equationisnamedafter Equation (28.1-1) is called theBragg equation.
SirWilliamHenry Bragg,1862–1942,
andhis sonWilliamLawrence Bragg,
1890–1971,whojointly receivedthe
1915 Nobel Prize inphysics for their
studies inX-raydiffraction.

At first glance, it might seem that the two angles labeledθin the diagram would not

have to be equal. However, one cannot consider just two atoms as scattering centers.

In order for the scattering from other pairs of atoms in the same two planes to produce

constructive interference, the two angles must be equal, so that the diffraction condition

is similar to a reflection from the planes of atoms. Diffracted X-ray beams are therefore

sometimes called “reflections.”

Exercise 28.3

By drawing a replica of Figure 28.5 and drawing incident and diffracted rays from other pairs of

atoms, show that if the two angles labeledθin Figure 28.5 are equal, all of the diffracted beams

interfere constructively if the Bragg condition is satisfied.

Any planes specified by Miller indices can diffract X-rays, so we append the Miller

indices to the distance between the planes in Eq. (28.1-1):

nλ 2 dhklsin(θ) (28.1-2)

This equation is the same as

λ 2 dnh,nk,nlsin(θ) (28.1-3)

For example, the distance between the (200) planes is half as great as the distance

between the (100) planes, so that the second-order (n2) diffraction from the (100)

planes is at the same wavelength as the first-order (n1) diffraction from the (200)

planes.

The diffraction of X-rays by a crystal lattice is more complicated than we have

indicated. In some cases the diffracted beams interfere destructively and are not seen.

This destructive interference is calledextinction. Some extinction rules are:^1

1. For a primitive lattice: no extinctions.


2. For a face-centered lattice: all three Miller indices must be even, or all three must
   be odd to avoid extinction.


3. For a body-centered lattice: the sum of the three Miller indices must be an even
   integer to avoid extinction.

(^1) M. J. Buerger,Contemporary Crystallography, McGraw-Hill Book Co., New York, 1970, chapter 5.