The exact Klein-Nishina total cross section per electron

PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · Nuclear Physics · Radiation interactions, detection and dosimetry

The Nucleus Resolved, Ch Radiation in matter, computed: Bethe, Klein-Nishina and what they leave out

Problem

The exact Klein-Nishina total cross section per electron, with

\alpha_{KN} = E_\gamma/m_ec^2

, is quoted as

\begin{aligned}\sigma_{\rm KN} = 2\pi r_e^2\Bigg(&\frac{1+\alpha_{KN}}{\alpha_{KN}^2}\Bigg[\frac{2(1+\alpha_{KN})}{1+2\alpha_{KN}}\\ &\qquad - \frac{1}{\alpha_{KN}}\ln(1+2\alpha_{KN})\Bigg]\\ &+ \frac{\ln(1+2\alpha_{KN})}{2\alpha_{KN}} - \frac{1+3\alpha_{KN}}{(1+2\alpha_{KN})^2}\Bigg),\end{aligned}

where

r_e = 2.818\,\mathrm{fm}

is the classical electron radius and

m_ec^2 = 0.511\,\mathrm{MeV}

. Calculate

\sigma_{\rm KN}

per electron for photons of energy

300\,\mathrm{keV}

, in barns (

1\,\mathrm{b} = 100\,\mathrm{fm}^2