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

The Transmission Electron Microscope 81
Besides producing contrast in the TEM image, the objective aperture
limits the amount of image blurring that arises from spherical and chromatic
aberration. By restricting the range of scattering angles to values less than �,
given by Eq. (3.9), the loss of spatial resolution is limited to an amount rs �
Cs� 3 due to spherical aberration and rc � Cc � (�E/E0) due to chromatic
aberration. Here, Cs and Cc are aberration coefficients of the objective lens
(Chapter 2); �E represents the spread in kinetic energy of the electrons
emerging from the specimen, and E0 is their kinetic energy before entering
the specimen. Because both rs and rc decrease with aperture size, it might be
thought that the best resolution corresponds to the smallest possible aperture
diameter. But in practice, objective diaphragm gives rise to a diffraction effect that
becomes more severe as its diameter decreases, as discussed in Section 1.1,
leading to a further loss of resolution �x given by the Rayleigh criterion:
�x � 0.6 �/sin�� 0.6 �/� (3.10)
where we have assumed that � is small and is measured in radians.
Ignoring chromatic aberration for the moment, we can combine the effect
of spherical aberration and electron diffraction at the objective diaphragm by
adding the two blurring effects together, so that the image resolution �r
(measured in the object plane) is
�r � rs + �x � Cs� 3 + 0.6 �/� (3.11)
Because the two terms in Eq. (3.11) have opposite �-dependence, their sum
is represented by a curve that displays a minimum value; see Fig. 3-13. To a
first approximation, we can find the optimum aperture semi-angle �*
(corresponding to smallest �r) by supposing that both terms make equal
contributions at � = �*. Equating both terms gives (�*) 4 = 0.6 �/Cs and
results in �* = 0.88 (�/Cs) 1/4 .
blurring
measured
in the
object
plane
0
��
�r
C s � �
�x
�
Figure 3-13. Loss or resolution due to spherical aberration (in the objective lens) and
diffraction (at the objective aperture). The solid curve shows the combined effect.