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

94 Chapter 4
-e -e
-e
-e
+Ze
Figure 4-1. Rutherford model of an atom (in this case, carbon with an atomic number Z = 6).
The electrostatic charge of the central nucleus (whose mass is M ) is +Ze ; the balancing
negative charge is provided by Z atomic electrons, each with charge � e .
4.1 Kinematics of Scattering by an Atomic Nucleus
The nucleus of an atom is truly miniscule: about 3 � 10 -15 m in diameter,
whereas the whole atom has a diameter of around 3 � 10 -10 m. The fraction
of space occupied by the nucleus is therefore of the order (10 -5 ) 3 = 10 -15 , and
the probability of an electron actually “hitting” the nucleus is almost zero.
However, the electrostatic field of the nucleus extends throughout the atom,
and incoming electrons are deflected (scattered) by this field.
Applying the principle of conservation of energy to the electron-nucleus
system (Fig. 4-2) gives:
mv0 2 /2 = mv1 2 /2 + M V 2 /2 (4.1)
where v0 and v1 represent the speed of the electron before and after the
“collision,” and V is the speed of the nucleus (assumed initially at rest)
immediately after the interaction. Because v0 and v1 apply to an electron that
is distant from the nucleus, there is no need to include potential energy in
Eq. (4.1). For simplicity, we are using the classical (rather than relativistic)
formula for kinetic energy, taking m and M as the rest mass of the electron
and nucleus. As a result, our analysis will be not be quantitatively accurate
for incident energies E0 above 50 keV. However, this inaccuracy will not
affect
our general conclusions.
Applying conservation of momentum to velocity components in the z- and
x-directions (parallel and perpendicular to the original direction of travel of
the
fast electron) gives:
mv0 = mv1 cos� + MV cos� (4.2)
0 = mv1 sin� � MV sin� (4.3)
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