﻿ The Compton Effect in the Scattering of Photons by Electrons
San José State University

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Thayer Watkins
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The Compton Effect in the Scattering
of Photons by Electrons

In the early 1920's Arthur Compton was engaged in experiments involving the scattering of high powered radiation, Xrays and gamma rays, from crystals and other materials. The perception of the time was that the frequency/wave length of the radiation was not changed by the scattering process. It was Arthur Compton's contribution to science to demonstrate experimentally and explain theoretically that the scattered radiation underwent a diminishment in energy; i.e., a decrease in frequency and an increase in wavelength. The crucial experiment involved the scattering of radiation by free electrons.

Theoretically it was not obvious how to establish the results of the interaction of electromagnetic waves and material particles. If the interaction involved only particles then the conservation of momentum and energy would determine the end result. But it was not immediately clear until the development of quantum mechanics what the energy and momentum of a wave is.

Albert Einstein in his explanation of the photoelectric effect established that the energy of a photon is hν, where h is Planck's constant and ν is the frequency of the radiation. Others, including Louis de Broglie, established that the momentum of a photon is hν/c.

Let ν0 be the frequency of the incident radiation and νθ that of the scattered radiation, where θ is the angle of the scattered radiation.

The electron recoils from the interaction with some velocity v. The relativistic formula for the momentum of a particle of rest mass m0 is

#### m0βc/(1−β²)½

where c is the speed of light.

Suppose the interaction of a photon and an electron are as follows:

The momentum vectors can be rearranged into a triangle as shown below.

The conservation of momentum requires that

#### [m0βc/(1−β²)½]² = (hν0/c)² + (hνθ/c)² + 2(hν0/c)(hνθ/c)cos(θ)

The kinetic energy of the electron after its recoil is relativistically

#### m0c²[1/(1−β²)½ − 1]

where β is equal to the electron velocity relative to the speed of light c.

The conservation of energy then requires

#### hν0 = hνθ + mc²[1/(1−β²)½ − 1]

The solution to the two equations is

#### φ = νθ/ν0 = 1 + 2α·sin²(½θ) and β = 2α·sin(½θ)[(1+(2α+α²)sin²(½θ))½/ (1+2(α+α²)sin²(½θ)]

This solutions makes use of the trigonometric identity that (1−cos(θ))/2 is equal to sin(½θ).

The effect of the interaction on the wave length of the radiation is given by

#### λθ = λ0 + (2h/(m0c))sin²(½θ)

(To be continued.)