bei48482_FM

370 Chapter Ten

(Sec. 3.5) directed at the crystal from the outside. [When is near a, 2a, 3a,... in

length, Eq. (10.17) no longer holds, as discussed later.] An electron of wavelength

undergoes Bragg reflection from one of the atomic planes in a crystal when it approaches

the plane at an angle , where from Eq. (2.13)

n  2 asin n1, 2, 3,... (10.18)

It is customary to treat the situation of electron waves in a crystal by replacing

by the wave number kintroduced in Sec. 3.3, where

Wave number k (10.19)

The wave number is equal to the number of radians per meter in the wave train it de-

scribes, and is proportional to the momentum pof the electron. Since the wave train

moves in the same direction as the particle, we can describe the wave train by means

of a vector k. Bragg’s formula in terms of kis

Bragg reflection k n1, 2, 3,... (10.20)

Figure 10.40 shows Bragg reflection in a two-dimensional square lattice. Evidently

we can express the Bragg condition by saying that reflection from the vertical rows of

ions occurs when the component of kin the xdirection, kx, is equal to n a. Simi-

larly, reflection from the horizontal rows occurs when kyna.

Let us consider first electrons whose wave numbers are sufficiently small for them

to avoid reflection. If kis less than a, the electron can move freely through the lattice

in any direction. When ka, they are prevented from moving in the xor ydirec-

tions by reflection. The more kexceeds a, the more limited the possible directions

of motion, until when kasin 45 2 athe electrons are reflected, even

when they move diagonally through the lattice.

n

asin

p



2 

Positive ions

k

θ

θ

kx

k =a sin nπθ

kx = k sin θ

=naπ

a

a

Figure 10.40Bragg reflection from the vertical rows of ions occurs when kxna.

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