Klein-Nishina Formula
The Klein-Nishina formula provides an accurate prediction of the angular distribution of x-rays and gamma-rays which are incident upon a single electron. The Klein-Nishina formula describes incoherent or Compton scatter.
More precisely, the Klein-Nishina formula provides the differential cross section with respect to solid angle of scatter, and it accounts for factors such as radiation pressure and relativistic quantum mechanics. For a photon of energy <math>E_\gamma<math>, the differential cross section is:
<math> \frac{d\sigma}{d\Omega} = 0.5 r_e^2 (P(E_\gamma,\theta) – P(E_\gamma,\theta)^2 \sin^2(\theta) + P(E_\gamma,\theta)^3) <math>
where <math>\theta<math> is the angle of scatter; <math>r_e<math> is the classical electron radius; <math>m_e<math> is the mass of an electron; and <math>P(E_\gamma,\theta)<math> is the ratio of photon energy before and after the collision:
<math> P(E_\gamma,\theta) = \frac{1}{1 + \frac{E_\gamma}{m_e c^2}(1-\cos\theta)} <math>
The value <math>d\sigma/d\Omega<math> is the probability that a photon will scatter into the solid angle defined by <math>d\Omega = 2 \pi \sin \theta d\theta<math>.
The Klein-Nishina formula was derived in 1929 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. Previous to 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 performed showed significant deviations from the results predicted by the Thomson cross section. Further scattering experiments performed agreed perfectly with the predictions of the Klein-Nishina formula.
Note that if <math>E_\gamma << m_ec^2<math>, <math> \frac {E_\gamma}{m_ec^2} \rightarrow <math> 0 and the Klein-Nishina formula reduces to the classical Thomson expression.
The final energy of the scattered photon, <math>E_\gamma'<math>, is entirely dependent upon scatter angle and the original photon energy, and so it can be computed without the use of the Klein-Nishina formula:
<math> E_\gamma'(E_\gamma,\theta) = E_\gamma \cdot P(E_\gamma, \theta) <math>