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The Compton Effect

Wednesday, February 2, 2022

What is the Compton Effect?

When a incident photon interacts with a loosely bound free electron, both the photon and electron scatter, in what is called the Compton effect. The photon has some initial energy, EE, and momentum, pp, which is converted into energy and momentum of the scattered photon, EE' and pp', and energy and momentum for the electron, EeE_e and pep_e (we assume the electron to be at rest).

E=hf=hcλp=EcE=hf=\frac{hc}{\lambda} \newline p=\frac{E}{c}

The electron has rest energy mec2m_ec^2, and is scattered at an angle ϕ\phi below the line of incidence. The photon scatters with energy E=hc/λE'=hc/\lambda' and momentum p=E/cp'=E'/c at an angle of θ\theta above the line of incidence.

Einitial=Efinal:E+mec2=E+Eepx,initial=px,final:p=pecosϕ+pcosθpy,initial=py,final:0=pesinϕpsinθE_{initial}=E_{final} : E+m_ec^2=E'+E_e \newline p_{x, initial}=p_{x,final} : p = p_ecos{\phi}+p'cos{\theta} \newline p_{y, initial}=p_{y,final} : 0 = p_esin{\phi}-p'sin{\theta}

This system of equations is not enough alone to be solved uniquely, so the scattered photon's energy and direction are measured to eliminate EeE_e and ϕ\phi. After some algebra and rearranging, the relationship between the energy is as follows:

1E1E=1mec2(1cosθ)\frac{1}{E'} - \frac{1}{E}=\frac{1}{m_ec^2}\left(1-\cos{\theta}\right)

Or, equivalently:


h/mech/m_ec is known as the Compton wavelength of the electron, and is about 0.0024260.002426nm. This is simply the change in wavelength the electron causes, and is not an actual wavelength.

Since the scattered photon always has less energy than the incident photon (the difference being the kinetic energy of the electron), the following equation is true:


Compton Effect Experiments

Compton used an x-ray source radiating on a target atom with loosely attached valence electrons, then measured the wavelength of the scattered photons at a variety of angles around the target.