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

98 Chapter 4
�
b
a
(a) (b)
Figure 4-4. Electron trajectory for elastic scattering through an angle � which is (a) less than
and (b) greater than the objective-aperture semi-angle �. The impact parameter b is defined as
the distance of closest approach to the nucleus if the particle were to continue in a straightline
trajectory. It is approximately equal to the distance of closest approach when the
scattering angle (and the curvature of the trajectory) is small.
so they experience a stronger attractive force. Because Eq. (4.11) represents
a general relationship, it must hold for � = �, which corresponds to b = a.
Therefore, we can rewrite Eq. (4.11) for this specific case, to give:
� = KZ e 2 /(E0 a) (4.12)
As a result, the cross section for elastic scattering of an electron through any
angle
greater than � can be written as:
�e = � a 2 = � [KZe 2 /(�E0 )] 2 = Z 2 e 4 /(16��0 2 E0 2 � 2 ) (4.13)
Because �e has units of m 2 , it cannot directly represent scattering probability;
we need an additional factor with units of m �2 to provide the dimensionless
number Pe(>�). In addition, our TEM specimen contains many atoms, each
capable of scattering an incoming electron, whereas �e is the elastic cross
section for a single atom. Consequently, the total probability of elastic
scattering
in the specimen is:
Pe(>�) = N �e (4.14)
where N is the number of atoms per unit area of the specimen (viewed in the
direction of an approaching electron), sometimes called an areal density of
atoms.
For a specimen with n atoms per unit volume, N = nt where t is the
specimen thickness. If the specimen contains only a single element of atomic
number A, the atomic density n can be written in terms of a physical density:
� = (mass/volume) = (atoms per unit volume) (mass per atom) = n (Au),
where u is the atomic mass unit (1.66 � 10 -27 kg). Therefore:
Pe(>�) = [�/(Au)] t � = (� t) (Z 2 /A) e 4 /(16��0 2 uE0 2 � 2 ) (4.15)
b
�
a