hv hv
-= -cs+ p cos 8e (A-2)
c c
hv sin B = pc sin 8e (A-3)
pc can be written in terms of T by invoking the "law of invariance":
p~c= T(T+moc2 (A-4)
In which mo is the electron's rest mass.
As a result of solving these equations algebraically, one can derive the following
Compton's relations
hv
hv = (A-5)
1+(hvlm0C2)(1-COS 8)
co~ 1hymC)a (A-6)
21
T, = hv hv = hv 1-+2 i2 Bi2 (A-7)
To obtain the fraction of the scattered photons in a given direction, Klein and Nishina
(reference) have carried out a quantum-mechanical treatment of the problem using the Dirac
equation for the electron and have obtained the equation
derKN 02 hy'hv hv .
~ -v +-si2 (A-8)
da 2 hv hv h
Equations similar to those for the Compton photon distribution can be obtained for the
Compton electron distribution. Since the probability that an electron will be scattered into the
solid angle daZ situated in the direction 8e is the same as the probability that a primary quantum