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

96 Chapter 4
�
(a) elastic
z
x
m
m
(b) inelastic
Figure 4-3. Hyperbolic trajectory of an incident electron during (a) elastic scattering (the case
of a large-angle collision is indicated by the dashed curve) and (b) inelastic scattering.
protons + neutrons in the nucleus), essentially the atomic weight of the atom.
For a hydrogen atom (A = 1), m/M � 1/1836 and Eq. (4.8) gives E/E0 <
0.002, so for 180-degree scattering only 0.2% of the kinetic energy of the
electron is transmitted to the nucleus. For other elements, and especially for
smaller scattering angles, this percentage is even lower. Scattering in the
electrostatic field of a nucleus is therefore termed elastic, implying that the
scattered electron retains almost all of its kinetic energy.
4.2 Electron-Electron Scattering
If the incident electron passes very close to one of the atomic electrons that
surround the nucleus, both particles experience a repulsive force. If we
ignore the presence of the nucleus and the other atomic electrons, we have a
two-body encounter (Fig. 4-3b) similar to the nuclear interaction depicted in
Figs. 4-2 and 4-3a. Equations (4.1) to (4.6) should still apply, provided we
replace the nuclear mass M by the mass m of an atomic electron (identical to
the mass of the incident electron, as we are neglecting relativistic effects). In
this
case, Eq. (4.6) becomes:
E/E0 = 1 + v1 2 /v0 2 � 2 (v1/v0) cos� (4.9)
Because Eq. (4.9) no longer contains the factor (m/M), the energy loss E can
be much larger than for an elastic collision, meaning that scattering from the
electrons that surround an atomic nucleus is inelastic. In the extreme case of
a head-on collision, v1 � 0 and E � E0 ; in other words, the incident electron
loses all of its original kinetic energy. In practice, E is in the range 10 eV to
50 eV for the majority of inelastic collisions. A more correct treatment of
inelastic scattering is based on wave-mechanical theory and includes the
collective
response of all of the electrons contained within nearby atoms.