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

194 Appendix
The variable � represents the direction of the force, relative to the x-axis
(Fig. A-2). From the geometry of triangle ANE, we have: r cos� =AN�b if
� is small, where b is the impact parameter of the initial trajectory.
Substituting for r in Eq. (A.8) gives:
(KZe 2 /b 2 )cos 3 � = Fx = m(dvx/dt) (A.9)
where m = �m0 is the relativistic mass of the electron, and we have applied
Newton's second law of motion in the x-direction. The final x-component vx
of electron velocity (resulting from the deflection) is obtained by integrating
Eq. (A.9) with respect to time, over the whole trajectory, as follows:
vx = � (dvx/dt) dt = � [(dvx/dt)/(dz/dt)] dz
= � [(dvx/dt)/vz] (dz/d�) d� (A.10)
In Eq. (A.10), we first replace integration over time by integration over the
z-coordinate of the electron, and then by integration over the angle �.
Because z = (EN) tan ��b tan �, basic calculus gives dz/d� �b(sec 2 �).
Using this relationship in Eq. (A.10) and making use of Eq. (A.9) to
substitute for dvx/dt , vx can be written entirely in terms of � as the only
variable:
vx = � [(Fx/m)/vz] b(sec 2 �) d� = � [KZe 2 /(bmvz) cos 3 �] sec 2 � d� (A.11)
From the vector triangle in Fig. A-2 we have: vz = v1 cos ��v1 for small �
and also, for small �, v1 � v0, as elastic scattering causes only a very small
fractional
change in kinetic energy of the electron. Because sec � = 1/cos �,
vx = � [KZe 2 /(bmv0)] cos� d� = [KZe 2 /(bmv0)] [sin�]
= 2 [KZe 2 /(bmv0)] (A.12)
where we have taken the limits of integration to be � = ��/2 (electron far
from the nucleus and approaching it) and � = +�/2 (electron far from the
nucleus
and receding from it).
Finally, we obtain the scattering angle � from parallelogram (or triangle)
of velocity vectors in Fig. A-2, using the fact that �� tan � = vx/vz � vx/v0 to
give:
�� 2KZe 2 /(bmv0 2 ) = 2KZe 2 /(�m0v0 2 b) (A.13)
Using a non-relativistic approximation for the kinetic energy of the electron
(E0 mv 2 � /2), Eq. (A.13) becomes:
��K Z e 2 /(E0b) (A.14)