Graduate Course: Quantum Physics

Jennifer Neakrase, Jennifer Neal and John Venables, Dept of Physics and Astronomy, Arizona State University, Tempe, Arizona 

Photoelectrons, Compton and Inverse Compton Scattering

The Photoelectric and Compton effects are closely related. The Compton effect is introduced in Gasiorowicz (in section 1.3, pages 7-9 in the 3rd edition, or pages 11-13 in the 2nd edition). The text describes the experimental discovery of the effect discovered by Arthur H. Compton - radiation of a given wavelength (in X-rays) sent through a foil was scattered in a manner inconsistent with classical radiation theory. If one is dealing with elastic scattering,the system can be understood quantitatively as Thomson scattering. However, the Compton effect can be understood as photons scattering inelastically off individual electrons.

In Compton scattering the incoming photon scatters off an electron that is initially at rest. The electron gains energy and the scattered photon has a frequency less than that of the incoming photon. This process is illustrated in the following figure.

Einstein's photoelectric discussion in 1905, and his other work including special relativity, led physicists to the notion of photons. Arthur Compton and Debye both provided in 1922 a very simple mathematical framework for these photons. Energy is conserved in a collision between a photon and an electron. In the original photoelectric effect, the photon energy of the photon is of the same order as the energy binding an electron to a nucleus, a few eV. Thus, when the photon strikes the electron it imparts only enough energy to eject that electron. However, if the energy of the photon is large compared to the binding energy of the electron, one could make the approximation that the electron as free.

For example, x-ray photons have an energy value of several keV. So, both conservation of momentum and energy could be observed. To show this, Compton scattered x-ray radiation off a graphite block and measured the wavelength of the x-rays before and after they were scattered as a function of the scattering angle. He discovered that the scattered x-rays had a longer wavelength than that of the incident radiation.

The original figure from Compton (1923a, figure 4) is shown above. Compton was able to account for and derive the correct expression for the shift in wavelength. Therefore, he empirically proved that light could be regarded as a particle in these experiments. The original references are: A. H. Compton, Phys. Rev. 21, 483 (1923a); 22, 409 (1923b). A more recent historical survey is A. H. Compton, Am. J. Phys. 29, 817 (1961). A web page giving mathematical details, and details about Compton's life and 1927 Nobel Prize is on the Wolfram site. An outline of the maths is given below.

The energy (E) of a particle is related to its mass (m) and momentum (P), via the relavisitic formula

E2 = (Pc)2 + m2c4,

where c is the speed of light. Since the mass of a photon is zero, its energy is E = Pc. The energy may also be defined as E = hn, where h is Planck's constant and n is frequency. Using these relations, the momentum a photon is related to its wavelength l, P = h /l.

Compton argued that the shift in wavelength is a result of a single photon imparting momentum to a single electron; thus the theory is derived from the laws of conservation of energy and momentum. Consider a photon with energy E0 and momentum P0, and a stationary electron with rest energy mc2. When the photon collides with the electron, the electron recoils with energy Ee and momentum Pe. The scattered photon will have an energy E and momentum P. By conservation of energy and momentum:

Ee + E = mc2 + E0 and Pe + P = P0

Combining energy and momentum conservation in 2 dimensions (see text books) using these equations yields:

l - l0 = (h/mc)(1- cosq).

The shift in wavelength is related only to the mass of the electron and the backscattered angle. The shift has no relation to the energy of the incident photon. The Compton effect can also be expressed as a shift in energy between the incident and scattered photon.

E - E0 = (E0E/(mc2))(1- cosq).

For a 180 backscattered photon, the shift in energy between E0 and E is

E0 - E = E0(2E0/(2E0 +(mc2)).

In astrophysics inverse Compton scattering is actually more important than Compton scattering. Inverse Compton scattering, illustrated in the figure below, takes place when the electron is moving, and has sufficient kinetic energy compared to the photon. In this case net energy may be transferred from the electron to the photon.

The inverse Compton effect is seen in astrophysics when a low energy photon (e.g. of the cosmic microwave background) bounces off a high energy (relativistic) electron. Such electrons are produced in supernovae and active galactic nuclei.

A good reference for Compton scattering and inverse Compton scattering in the astrophysical regime is in "Radiative Processes in Astrophysics" by George B. Rybicki and Alan P. Lightman. In their Chapter 7, Rybicki and Lightman derive the equations for Compton scattering and then move into a treatment of inverse Compton scattering. This topic will not be studied further in the present Quantum Physics course, but is typically treated in the Astronomy course "The Interstellar Medium".

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