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

Mathematical Derivations 193
A.2 Impact Parameter in Rutherford Scattering
Here we retrace Rutherford’s derivation of the relation between the impact
parameter b and the scattering angle � , but replacing his incident alpha
particle by an electron and assuming that the scattering angle is small (true
for most electrons, if their incident kinetic energy is high). Figure A-2 shows
the hyperbolic path of an electron deflected by the electrostatic field of an
unscreened atomic nucleus. At any instant, the electron is attracted toward
the nucleus with an electrostatic force given by Eq. (4.10):
F = K Ze 2 /r 2
(A.7)
where K = 1/(4��0) as previously. During the electron trajectory, this force
varies in both magnitude and direction; its net effect must be obtained by
integrating over the path of the electron.
However, only the x-component Fx of the force is responsible for angular
deflection and this component is
Fx = F cos� = (KZe 2 /r 2 ) cos� (A.8)
V 0
B
z
�
A
V z
E
b
r
�
V x
V 1
Figure A-2. Trajectory of an electron E with an impact parameter b relative to an unscreened
atomic nucleus N, whose electrostatic field causes the electron to be deflected through an
angle �, with a slight change in speed from v0 to v1 .
N
x