APPENDIX H
A. H. COMPTON’S THEORETICAL DERIVATION OF THE CHANGE IN THE WAVE-LENGTH
OF ETHER-WAVES BECAUSE OF SCATTERING BY FREE ELECTRONS
Imagine, as in Fig. 42A, that an X-ray quantum of frequency is scattered by an electron of mass . The momentum of the incident ray will be , where is the velocity of light and is Planck’s constant, and that of the scattered ray is at an angle with the initial momentum.
Fig. 42
The principle of the conservation of momentum accordingly demands that the momentum of recoil of the scattering electron shall equal the vector difference between the momenta of these two rays, as in Fig. 42B. The momentum of the electron, , is thus given by the relation where is the ratio of the velocity of recoil of the electron to the velocity of light. But the energy in the scattered quantum is equal to that of the incident quantum less the kinetic energy of recoil of the scattering electron, i.e.,
We thus have two independent equations containing the two unknown quantities and . On solving the equations we find where or, in terms of wave-length instead of frequency, Substituting the accepted values of , , and ,