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

48 Chapter 2
So far, we have said nothing about the value of the spherical-aberration
coefficient Cs. On the assumption of a Lorentzian (bell-shaped) field, Cs can
be calculated from a somewhat-complicated formula (Glaeser, 1952). Figure
2.13 shows the calculated Cs and focal length f as a function of the maximum
field B0, for 200kV accelerating voltage and a field half-width of a = 1.8 mm.
The thin-lens formula, Eq. (2.9), is seen to be quite good at predicting the
focal length of a weak lens (low B0) but becomes inaccurate for a strong
lens. For the weak lens, Cs � f � several mm; but for a strong lens (B0 = 2 to
3 T, as used for a TEM objective), Cs falls to about f / 4. If we take f = 2 mm,
so that Cs � 0.5 mm, and require a point resolution rs = 1 nm, the maximum
angle of the electrons (relative to the optic axis) must satisfy: Cs� 3 � rs ,
giving ��� 10 -2 rad = 10 mrad. This low value justifies our use of smallangle
approximations in the preceding analysis.
lens parameter (in mm)
10.0
5.0
2.0
1.0
0.5
thin-lens f
actual f
Cs
C c
0.2
0 1 2 3 4
B 0 (Tesla)
Figure 2-13. Focal length and coefficients of spherical and chromatic aberration for a
magnetic lens containing a Lorentzian field with peak field B0 and half-width a = 1.8 mm,
focusing 200keV electrons. Values were calculated from Eq. (2.7) and from Glaeser (1952).
10.0
1.0