Ex 6030: Thermionic emission of electrons
Submitted by: Dotan Davidovich
The problem:
A piece of metal (”Cathode”) is placed inside a vacuum metal tube (”Anode”). The cathode has a
work function W and surface A and is held at temperature T .
Calculate the saturation current Is (V → ∞) .
a)For T W
b)For T W
The solution:
a) Under the assumption that the electron gas inside the metal is in a state of quasi-static thermal
equilibrium we use the velocity distribution:
F(v) = 2 · L3 · (
m 3
1
) · f ( mv 2 − µ) · d3 v
2π
2
(1)
Now, understanding
qthat only incident electrons with kinetic energy larger than the potential barrier
will escape (Vz ≥ 2·(Wm+µF ) ) and using the incident flux on a wall perpendicular to the Z axis ,
we get:
Z
Z ∞
Z ∞
F(v)
m 3
2πV 0 dV 0
J = 2 · 3 · Vz = 2 · ( ) ·
Vz dVz ·
(2)
1
1
2
02
L
2π
VZ
0
e−β( 2 mvz −µ) · eβ 2 mv + 1
min
Integrating over the x − y velocities and using a dimensionless variable x = βW :
J =2·
m2
· 2π · T ·
(2π)3
Z
∞
1
2
ln(1 + e−β( 2 mvz −µ) )Vz dVz =
VZmin
4πm
· T2 ·
(2π)3
Z
∞
ln(1 + e−x )dx (3)
βW
For T W we use ln(1 + e−x ) ≈ e−x :
J=
4πmT 2 −βW
·e
(2π)3
(4)
The current density is : e · J , and for the saturation current all emitted electrons make it to the
Anode :
Is = e · J · A = e · A ·
4πmT 2 −βW
·e
(2π)3
(5)
1
b) For T W , realizing we are now in the classical regime :
µ = T ln(n · λ3 )
(6)
Subsituting into (4) we get :
J =n·(
T 1 −β 1 mvz2
)2 · e 2
2πm
(7)
2