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• Comment: Fails WP:GNG, requires significant coverage in multiple independent secondary sources. Dan arndt (talk) 06:24, 2 December 2024 (UTC)

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Energy distribution of the Compton electron

Three examples of Compton electron energy distributions

A quantity of frequent relevance is the energy distribution of the Compton electron $T={E_{\gamma }}-{E_{\gamma '}}$. This can be derived from the Klein-Nishina cross section ${d\sigma }/{d\Omega }$ by a change of variable

${\frac {d\sigma }{dT}}={\frac {d\sigma }{d\Omega }}{\frac {d\Omega }{d\theta }}{\frac {d\theta }{dT}}$.

The result can be conveniently expressed as

${\frac {d\sigma }{dT}}={\frac {\pi r_{e}^{2}}{m_{e}c^{2}\upsilon ^{2}}}{\Biggl [}{2+{\frac {\epsilon ^{2}}{\upsilon ^{2}(1-\epsilon )^{2}}}+{\frac {\epsilon }{1-\epsilon }}{\Bigl (}{\epsilon -{\frac {2}{\upsilon }}}{\Bigr )}}{\Biggr ]}{;}\qquad \upsilon =T/{E_{\gamma }}$

This formula is valid for $0\leqslant T\leqslant E_{\gamma }{\biggl (}{\frac {2\upsilon }{1+2\upsilon }}{\biggr )}$. The upper value is the Compton Edge.