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

Electron Optics 45
image plane. When spherical aberration is present, electrons arriving at an
appreciable distance x from the axis are focused to a different point F1
located
at a shorter distance f1 from the center of the lens.
We might expect the axial shift in focus (�f = f – f1 ) to depend on the
initial x-coordinate of the electron and on the degree of imperfection of the
lens focusing. Without knowing the details of this imperfection, we can
represent
the x-dependence in terms of a power series:
�f = c2 x 2 + c4 x 4 + higher even powers of x (2.11)
with c2 and c4 as unknown coefficients. Note that odd powers of x have been
omitted: provided the magnetic field that focuses the electrons is axially
symmetric, the deflection angle � will be identical for electrons that arrive
with coordinates +x and –x (as in Fig. 2-11). This would not be the case if
erms involving x or x 3 t
were present in Eq. (2.11).
From the geometry of the large right-angled triangle in Fig. 2-11,
x = f1 tan�� f tan�� f � (2.12)
Here we have assumed that x