7 Particle Detectors and Detector Systems
317
direction for backward transition radiation. N is the total number of emitted photons.
Equation (7.40) is plotted in Fig. 7.29b.
The energy radiated from a single surface, assuming ε 0 → ε, is given by:
W =
1
3
αZ
2 ω p γ
(7.41)
where ω p is the plasma frequency.
7.5.1 Plasma Frequency
The influence of the plasma frequency was shown in the saturation of the relativistic
rise expressed by the Bethe-Bloch formula, Chap. 2 and [82], due to the polarisation
of the medium:
δ
2
= ln
ω p
I
+ ln βγ −
1
2
(7.42)
where I and ω p are respectively the mean excitation energy and the plasma
frequency of the medium and δ is the density correction.
The plasma frequency, ω p , is the natural frequency of density oscillations of free
electrons and its value depends only weakly on the wavelength. Longitudinal plasma
waves are resonant at ω p . Transverse electromagnetic waves are absorbed below ω p .
If ω < ω p , the index of refraction has an imaginary part and the electromagnetic
waves are attenuated or reflected. If ω ω p , the index is real and a metal becomes
transparent. For large ω one can write
n
2
= 1 −
ω p
ω
2
(7.43)
The plasma frequency is given as:
ω
2
p =
NZe 2
ε 0 m
(7.44)
and depends only on the total number, NZ, of free electrons per unit volume. The
plasma frequency can be approximated with:
ω p (eV) 28.8
ρ(g/cm 3 ) · z
A
(7.45)
where z is the effective number of free electrons per unit volume. Table 7.5 gives
the corresponding calculated and measured wavelength, λ p , for alkali metals. z = 1
for alkali, group 1a, metals. The calculated plasma energies in Si, Ge and InSb are