�
function of the accelerating potential U a and the spacing s. This
is given by
j = ρ(x) v(x),
(5.117)
where ρ is the space charge density, and v is the particle speed.
Charge conservation dictates that the current density j is independent of x. The electrostatic potential U (x) is governed by Poisson’s
equation, which is given in one dimension as
d
2
ρ(x)
U (x) = −
.
(5.118)
dx 2
f 0
The particle speed is given by energy conservation as
2 e U (x)
v =
.
(5.119)
m
Substituting, we obtain a differential equation for the potential
U (x) as
U
�� (x) =
α ,
(5.120)
U (x)
where we have defined the constant α as
α ≡ −
j
f 0
m
2 e
.
(5.121)
We now make use of the fact that
d
d
d
U
� 2 = 2 U
� U
�� ,
= U
�
.
(5.122)
dx
dx
dU
This yields the differential equation
d U
� 2 = 2 α
dU
√
U
.
(5.123)
This is integrated immediately to yield
U
� 2 = 4 α
√
U + const.
(5.124)
At this point we invoke the condition that the field is zero at
the emission surface at cutoff. Mathematically, this equilibrium

328
Chapter 5. Electron emission from solids