Physical Principles of Electron Microscopy: An Introduction to TEM ...

192 Appendix
cathode vacuum
+
+
+
+
+
+
-e +e
(a)
+
z
(b)
vacuum vacuum
Figure A-1. (a) Electric field lines and surface-charge distribution produced by an electron
located just outside the surface of a conductor. (b) Equivalent field lines produced by an
electrostatic dipole operating in vacuum.
This total force can be written as F = (�e)E(z) = (�e)(–dV/dz) where E(z) is
the total field and V is the corresponding electrostatic potential. The potential
can therefore be obtained by integration of Eq. (A.2):
V = (1/e) � F dz = (K/4) (e/z) + Ee z (A.3)
The potential energy �(z) of the electron is therefore:
�(z) = (�e)(V) = �(K/4) (e 2 /z) � eEe z (A.4)
and is represented by the dashed curve in Fig. 3-4 (page 63). The top of the
potential-energy barrier corresponds to a coordinate z0 at which the slope
d�/dz and therefore the force F are both zero. Substituting z = z0 and F = 0 in
Eq. (A.2) results in z0 2 = (K/4)(e/Ee), and substitution into Eq. (A.4) gives
�(z0) = � e 3/2 K 1/2 Ee 1/2 = ��� (A.5)
From Eq. (3.1), the current density due to thermionic emission is now:
Je = A T 2 exp[� (�� ��) / kT ]
= A T 2 exp(� � / kT ) exp(�� / kT ) (A.6)
In other words, the applied field increases Je by a factor of exp(��/kT).
Taking Ee = 10 8 V/m, �� = 6.1 � 10 -20 J = 0.38 eV, and exp(��/kT) = 11.5
for T = 1800 K. This accounts for the order-of-magnitude increase in current
density Je (see Table 3.2) for a ZrO-coated Schottky source compared to a
LaB6 source, which has a similar work function and operating temperature.
z
z
-e