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

TEM Specimens and Images 99
P e (>�)
0
with screening
no screening
Figure 4-5. Elastic-scattering probability Pe(>�), as predicted from Eq. (4.15) (solid curve)
and after including screening of the nuclear field (dashed curve). Pe(>0) represents the total
probability of elastic scattering, through any angle.
Equation (4.15) indicates that the number of electrons absorbed at the anglelimiting
(objective) diaphragm is proportional to the mass-thickness (�t) of
the
specimen and inversely proportional to the square of the aperture size.
Unfortunately, Eq. (4.15) fails for the important case of small �; our
formula predicts that Pe(>�) increases toward infinity as the aperture angle is
reduced to zero. Because any probability must be less than 1, this prediction
shows that at least one of the assumptions used in our analysis is incorrect.
By using Eq. (4.10) to describe the electrostatic force, we have assumed that
the electrostatic field of the nucleus extends to infinity (although diminishing
with distance). In practice, this field is terminated within a neutral atom, due
to the presence of the atomic electrons that surround the nucleus. Stated
another way, the electrostatic field of a given nucleus is screened by the
atomic electrons, for locations outside that atom. Including such screening
complicates the analysis but results in a scattering probability that, as � falls
to zero, rises asymptotically to a finite value Pe(>0), the probability of elastic
scattering
through any angle; see Fig. 4-5.
Even after correction for screening, Eq. (4.15) can still give rise to an
unphysical result, for if we increase the specimen thickness t sufficiently,
Pe(>�) becomes greater than 1. This situation arises because our formula
represents a single-scattering approximation: we have tried to calculate the
�