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

66 Chapter 3
�
r
r
s
r
�
(a) (b) (c)
r
r
area A
Figure 3-5. (a) Two-dimensional diagram defining angle � in radians. (b) Three-dimensional
diagram defining solid angle � in steradians. (c) Cross-section through (b) showing the
relationship between solid angle � and the corresponding half-angle � of the cone.
Because image broadening due to spherical aberration of an electron lens
increases as � 3 , the ability of a demagnifying lens to create a high current
density within a small-diameter probe is helped by keeping � small and
therefore by using an electron source of high brightness. In the scanning
electron microscope, for example, we can achieve better spatial resolution by
using a Schottky or field-emission source, whose brightness is considerably
above that of a thermionic source (see Table 3-1).
Equation (3.3) can also be used to define an electron-optical brightness �
at any plane in an electron-optical system where the beam diameter is d and
the convergence semi-angle is �. This concept is particularly useful because
� retains the same value at each image plane, known as the principle of
brightness conservation. In other words,
� = Je/� = I [�(d/2) 2 ] -1 (�� 2 ) -1 = source brightness = constant (3.4)
Note that Eq. (3.4) is equivalent to saying that the product �d is the same at
each plane. Brightness conservation remains valid when the system contains
a diaphragm that absorbs some of the electrons. Although the beam current I
is reduced at the aperture, the solid angle of the beam is reduced in the same
proportion.
The last row of Table 3-1 contains typical values of the energy spread �E
for the different types of electron gun: the variation in kinetic energy of the
emitted electrons. In the case of thermionic and Schottky sources, this
energy spread is a reflection of the statistical variations in thermal energy of
electrons within the cathode, which depends on the cathode temperature T.
In the case of a field-emission source, it is due to the fact that some electrons
are emitted from energy levels (within the tip) that are below the Fermi
level. In both cases, �E increases with emission current Ie (known as the
�
r
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r
s