Page 31 - Electrical Properties of Materials

14 The electron as a particle

differential equations are then converted into algebraic equations, and by mak-
ing the determinant of the coefficients zero we get the dispersion equation. I
shall not go through the detailed derivation here because it would take up a
great deal of time, and the resulting dispersion equation is hardly more com-
plicated than eqn (1.50). All that happens is that ω in the ωτ term is replaced
by ω ± ω c . Thus, the dispersion equation for transverse electromagnetic waves
in the presence of a longitudinal d.c. magnetic field is

iσ
1/2
1/2
k = ω(μ ) 1+ , (1.57)
ω {1–i(ω ± ω c )τ}
where
e
ω c = B 0 . (1.58)
m

The plus and minus signs give circularly polarized electromagnetic waves
rotating in opposite directions. To see more clearly what happens, let us split
the expression under the square root into its real and imaginary parts. We get

2 2 2 1/2
ω τ (1 – ω c /ω) ω τ

p p 1
k = ω (μ ) 1– +i . (1.59)
2 2
2 2
1+(ω – ω c ) τ ω 1+(ω – ω c ) τ
This looks a bit complicated. In order to get a simple analytical expression, let
us confine our attention to semiconductors where ω p is not too large and the
applied magnetic field may be large enough to satisfy the conditions,
ω c ω p and ω c τ 1. (1.60)

We intend to investigate now what happens when ω c is close to ω.The
second and third terms in eqn (1.59) are then small in comparison with unity;
so the square root may be expanded to give
2
√ i ω τ 1
p
k = ω μ 1+ . (1.61)
2 2
2 ω 1+(ω – ω c ) τ
The attenuation of the electromagnetic wave is given by the imaginary part
of k. It may be seen that it has a maximum when ω c = ω. Since ω c is called
∗
∗ the cyclotron frequency this resonant absorption of electromagnetic waves
After an accelerating device, the
cyclotron, which works by accelerating is known as cyclotron resonance. The sharpness of the resonance depends
particles in increasing radii in a fixed strongly on the value of ω c τ, as shown in Fig. 1.7, where Im k, normalized
magnetic field.
to its value at ω/ω c = 1, is plotted against ω/ω c . It may be seen that the
The role of ω c τ is really analogous resonance is hardly noticeable at ω c τ =1.
to that of Q in a resonant circuit. The curves have been plotted using the approximate eqn (1.61); neverthe-
For good resonance we need a high less the conclusions are roughly valid for any value of ω p . If you want more
value of ω c τ. accurate resonance curves, use eqn (1.59).
Why is there such a thing as cyclotron resonance? The calculation from the
dispersion equation provides the figures, but if we want the reasons, we should
look at the following physical picture.