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

82 Chapter 3
A more correct procedure for finding the minimum is to differentiate �r
with respect to � and set the total derivative to zero, corresponding to zero
slope at the minimum of the curve, which gives: �* = 0.67 (�/Cs) 1/4 .
An even better procedure is to treat the blurring terms rs and �x like
statistical errors and combine their effect in quadrature:
(�r) 2 � (rs) 2 + (�x) 2 � (Cs� 3 ) 2 + (0.6 �/�) 2
Taking the derivative and setting it to zero then results in:
(3.12)
�* = 0.63 (�/Cs) 1/4 (3.13)
As an example, let us take E0 = 200 keV, so that � = 2.5 pm, and Cs � f /4 =
0.5 mm, as in Fig. 2-13. Then Eq. (3.13) gives �* = 5.3 mrad, corresponding
to an objective aperture of diameter D � 2�*f � 20 �m. Using Eq.(3.12), the
optimum resolution is �r* � 0.29 nm.
Inclusion of chromatic aberration in Eq. (3.12) would decrease �* a little
and increase �r*. However, our entire procedure is somewhat pessimistic: it
assumes that electrons are present in equal numbers at all scattering angles
up to the aperture semi-angle �. Also, a more exact treatment would be
based on wave optics rather than geometrical optics. In practice, a modern
200 kV TEM can achieve a point resolution below 0.2 nm, allowing atomic
resolution under suitable conditions, which include a low vibration level and
low ambient ac magnetic field (Muller and Grazul, 2001). Nevertheless, our
calculation has illustrated the importance of lens aberrations. Without these
aberrations, large values of � would be possible and the TEM resolution,
limited only by Eq. (3.10), would (for sin �� 0.6) be �x ��� 0.025 nm,
well below atomic dimensions.
In visible-light microscopy, lens aberrations can be made negligible.
Glass lenses of large aperture can be used, such that sin � approaches 1 and
Eq. (3.10) gives the resolution as �x � 300 nm as discussed in Chapter 1. But
as a result of the much smaller electron wavelength, the TEM resolution is
better by more than a factor of 1000, despite the electron-lens aberrations.
Objective stigmator
The above estimates of resolution assume that the imaging system does not
suffer from astigmatism. In practice, electrostatic charging of contamination
layers (on the specimen or on an objective diaphragm) or hysteresis effects
in the lens polepieces give rise to axial astigmatism that may be different for
each specimen. The TEM operator can correct for axial astigmatism in the
objective (and other imaging lenses) using an objective stigmator located just
below the objective lens. Obtaining the correct setting requires an adjustment