For an incident unpolarized photon of energy , the differential cross section is:[2]
where
- is the classical electron radius (~2.82 fm, is about 7.94 × 10−30 m2 or 79.4 mb)
- is the ratio of the wavelengths of the incident and scattered photons
- is the scattering angle (0 for an undeflected photon).
The angular dependent photon wavelength (or energy, or frequency) ratio is
as required by the conservation of relativistic energy and momentum (see Compton scattering). The dimensionless quantity expresses the energy of the incident photon in terms of the electron rest energy (~511 keV), and may also be expressed as , where is the Compton wavelength of the electron (~2.42 pm). Notice that the scatter ratio increases monotonically with the deflection angle, from (forward scattering, no energy transfer) to (180 degree backscatter, maximum energy transfer).
In some cases it is convenient to express the classical electron radius in terms of the Compton wavelength: , where is the fine structure constant (~1/137) and is the reduced Compton wavelength of the electron (~0.386 pm), so that the constant in the cross section may be given as:
If the incoming photon is polarized, the scattered photon is no longer isotropic with respect to the azimuthal angle. For a linearly polarized photon scattered with a free electron at rest, the differential cross section is instead given by:
where is the azimuthal scattering angle. Note that the unpolarized differential cross section can be obtained by averaging over .
Low energy
For low energy photons the wavelength shift becomes negligible () and the Klein–Nishina formula reduces to the classical Thomson expression:
which is symmetrical in the scattering angle, i.e. the photon is just as likely to scatter backwards as forwards. With increasing energy this symmetry is broken and the photon becomes more likely to scatter in the forward direction.
The differential cross section may be integrated to find the total cross section.
In the low energy limit there is no energy dependence and we recover the Thomson cross section (~66.5 fm2):
The Klein–Nishina formula was derived in 1928 by Oskar Klein and Yoshio Nishina, and was one of the first results obtained from the study of quantum electrodynamics. Consideration of relativistic and quantum mechanical effects allowed development of an accurate equation for the scattering of radiation from a target electron. Before this derivation, the electron cross section had been classically derived by the British physicist and discoverer of the electron, J.J. Thomson. However, scattering experiments showed significant deviations from the results predicted by the Thomson cross section. Further scattering experiments agreed perfectly with the predictions of the Klein–Nishina formula.[citation needed]
Weinberg, Steven (1995). The Quantum Theory of Fields. Vol. I. pp. 362–9.
- Evans, R. D. (1955). The Atomic Nucleus. New York: McGraw-Hill. pp. 674–676. OCLC 542611.
- Melissinos, A. C. (1966). Experiments in Modern Physics. New York: Academic Press. pp. 252–265. ISBN 0-12-489850-5.
- Klein, O.; Nishina, Y. (1994). "On the Scattering of Radiation by Free Electrons According to Dirac's New Relativistic Quantum Dynamics". In Ekspong, Gösta (ed.). The Oskar Klein Memorial Lectures, Vol. 2: Lectures by Hans A. Bethe and Alan H. Guth with Translated Reprints by Oskar Klein. Singapore: World Scientific. pp. 113–139. Bibcode:1994okml.book.....E.