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3-7: The Compton Effect

As learned on the preceding page, Einstein proposed the hypothesis of light quanta and succeeded in the explanation of the photoelectric effect. As a result, it has been established that light exists in the space as grains (or corpuscles or particles) with energy . The particle nature or corpuscular nature of light was more firmly established by the discovery of the Compton effect.
In 1923, A. H. Compton (USA, 1892 - 1962) discovered that the scattering of X rays by a crystal can be explained well on the basis of the particle nature of light. (Of course, the phenomenon that X rays are scattered by crystals had been well-known before.)
The scattering of X rays by a particle (an electron at present) is sometimes called Compton scattering. Compton found that, if we consider that X rays collide with the electrons in a crystal as if they were balls of billiard, Compton scattering could be explained well. To formulate this process, we have to define the momentum of a light quantum.

[Momentum of a Light Quantum]
It has been discussed that a light quantum is a "particle" with energy . This "particle" is considered to carry a momentum at the same time, because we know that light gives a pressure on the surrounding wall. How large is the amount of the momentum? Let us study this below.
Consider a container filled with light of frequency . Let the pressure given on the wall of the container by the light be P and the energy of the light per unit volume be U. We have a relation

which is proved by experiment. (This relation can also be derived from the classical theory.) From this relation, we see that the momentum p of a light quantum is given as

Namely, we can say that the momentum of a light quantum is given by dividing the energy by the light speed.
This result can be obtained by a method similar to that through which we derived the relation between the molecular motion in a gas and its pressure. Let us explain this on the other page:
3-7-A: Momentum of Light Quantum
If you feel it tedious, you may skip this.

[The Compton Effect by the Hypothesis of Light Quanta]
Compton found that, when the monochromatic X rays are illuminated on a graphite (a kind of carbon crystal), the wavelength of the scattered X rays would be longer as the scattering angle becomes larger. This cannot be explained with the classical theory.
In the classical theory consisting of Newtonian mechanics and Maxwellian electromagnetism, the incident X rays oscillate the charged particle (electron at present) and the oscillating charged particle radiates around the same frequency of electromagnetic waves (X rays). Accordingly, the frequency of the radiated (scattered) X rays must be the same as that of the incident X rays.
Compton analyzed Compton scattering as follows. He thought that the incident X rays collide against the electron in the graphite as a "particle" with the energy . and the momentum . Suppose that the energy and the momentum of the X rays scattered in the scattering angle are and , respectively.
The electron (the mass = m ) recoils with a momentum mv. The energies and momenta in Compton scattering are shown in the following figure (A), and the relation of the momentum conservation is in the following figure (B).

The momentum conservation relation is written

Since , we have

Inserting this into the above momentum conservation relation, we get

On the other hand, the energy conservation is written

Therefore, the difference between the the wavelength of the incident and scattered X rays becomes

Accordingly, the wavelength of the scattered X rays becomes longer as the scattering angle increases.
Compton's results at the scattering angles, and , are shown in the following figure. Each graph represents the intensity of X rays (ordinate) as a function of wavelength (abscissa).

As the scattering angle increases in the above figure, the intensity of X rays separates into two peaks; the right one of the longer wavelength shifts to longer region. This wavelength is well fit to the formula presented above.
The wavelength of another peak is just the same as that of the incident X-ray beam. This is that scattered by the whole atom whose mass is quite large, so that the wavelength seems not to be varied.
Thus the particle nature of X rays was completely confirmed.
Incidentally, the recoil electron could not be detected in Compton's experiment, but, a little bit later, its picture was taken with Wilson's cloud chamber.

After Einstein proposed the hypothesis of light quanta in 1905, people have been doubtful of the idea of the particle nature of light for about 20 years. However, once they look at the experimental results of the Compton effect, they cannot help convincing themselves about the theory of light quanta. Then, the particle of light (light quantum) has been called photon, which has been admitted into the brotherhood of "particles" like electrons or protons.
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