�
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Problems
1. Perform the integral (5.17) to obtain the result (5.18).
2. Verify the result (5.19) for the limit T → 0 by repeating the
procedure of this section, making use of the fact that
ε − ζ
−1
1,
0 ≤ ε ≤ ζ
n(ε) = exp
+ 1
=
(5.20)
kT
0,
ε > ζ.
3. Derive an analytical expression for the Fermi energy ζ of a
metal based on the number of conduction band electrons per unit
volume and the density of states with respect to total energy ε.
304
Chapter 5. Electron emission from solids
5.3 Thermionic emission
In the case of zero applied electric field and elevated temperature,
the energy needed for an electron to surmount the potential barrier is thermal. This is called thermionic emission. The central
problem in this section is to calculate the emission current density
j for thermionic emission. We make use of (5.1, 5.18). The task
remains to calculate the probability D(W ) that an electron with
total energy W will be transmitted across the barrier.
The potential energy U (x) associated with the image force is given
by (5.3). This is plotted as a function of coordinate x in the direction normal to the emission surface in Figure 5.1. The surface of
the metal is at coordinate x = 0. The left region x < 0 represents
the interior of the bulk emitting material, and the right region
�
Problems
1. Perform the integral (5.17) to obtain the result (5.18).
2. Verify the result (5.19) for the limit T → 0 by repeating the
procedure of this section, making use of the fact that
ε − ζ
−1
1,
0 ≤ ε ≤ ζ
n(ε) = exp
+ 1
=
(5.20)
kT
0,
ε > ζ.
3. Derive an analytical expression for the Fermi energy ζ of a
metal based on the number of conduction band electrons per unit
volume and the density of states with respect to total energy ε.
304
Chapter 5. Electron emission from solids
5.3 Thermionic emission
In the case of zero applied electric field and elevated temperature,
the energy needed for an electron to surmount the potential barrier is thermal. This is called thermionic emission. The central
problem in this section is to calculate the emission current density
j for thermionic emission. We make use of (5.1, 5.18). The task
remains to calculate the probability D(W ) that an electron with
total energy W will be transmitted across the barrier.
The potential energy U (x) associated with the image force is given
by (5.3). This is plotted as a function of coordinate x in the direction normal to the emission surface in Figure 5.1. The surface of
the metal is at coordinate x = 0. The left region x < 0 represents
the interior of the bulk emitting material, and the right region