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

The Transmission Electron Microscope 65
Table 3-1. Operating parameters of four types of electron source*
Type of source Tungsten
thermionic
LaB 6
thermionic
Schottky
emission
Material W LaB6 ZrO/W W
� (eV) 4.5 2.7 2.8 4.5
Cold field
emission
T (K) 2700 1800 1800 300
E (V/m) low low � 10 8
>10 9
Je (A/m 2 ) � 10 4
� 10 6
� 10 7
� 10 9
� (Am -2 /sr -1 ) � 10 9
� 10 10
� 10 11
� 10 12
ds (�m) � 40 � 10 � 0.02 � 0.01
Vacuum (Pa) < 10 -2
< 10 -4
< 10 -7
� 10 -8
Lifetime (hours) � 100 � 1000 � 10 4
�10 4
�E (eV) 1.5 1.0 0.5 0.3
* � is the work function, T the temperature, E the electric field, Je the current density,
and � the electron-optical brightness at the cathode; ds is the effective (or virtual) source
diameter, and �E is the energy spread of the emitted electrons.
Although the available emission current Ie decreases as we proceed from
a tungsten-filament to a field-emission source, the effective source diameter
and the emitting area (As = �ds 2 /4) decrease by a greater factor, resulting in
an increase in the current density Je. More importantly, there is an increase
in a quantity known as the electron-optical brightness � of the source,
defined as the current density divided by the solid angle � over which
electrons are emitted:
� = Ie /(As �) = Je /� (3.2)
Note that in three-dimensional geometry, solid angle replaces the concept of
angle in two-dimensional (Euclidean) geometry. Whereas an angle in radians
is defined by � = s/r, where s is arc length and r is arc radius (see Fig. 3-3a),
solid angle is measured in steradians and defined as � = A/r 2 where A is the
area of a section of a sphere at distance r (see Fig. 3-3b). Dividing by r 2
makes
� a dimensionless number, just like � .
If electrons were emitted in every possible direction from a point source,
the solid angle of emission would be � = A/r 2 = (4�r 2 )/r 2 = 4� steradian. In
practice, � is small and is related in a simple way to the half-angle � of the
emission cone. Assuming small �, the area of the curved end of the cone in
Fig.
3-3c approximates to that of a flat disk of radius s, giving:
�� (�s 2 )/r 2 = �� 2
(3.3)