Phy5645/Cross Section Relation

The differential cross section is related to the scattering amplitude through

${\frac {d\sigma (\theta )}{d\Omega }}=|f_{k}(\theta )|^{2}.$

Since $|f|^{2}=(\Re e\,f)^{2}+(\Im m\,f)^{2}\geq (\Im mf)^{2},$

we obtain

${\frac {d\sigma (\theta )}{d\Omega }}\geq (\Im m[f_{k}(\theta )])^{2}.$

On the other hand, the optical theorem states that

$\sigma ={\frac {4\pi }{k}}\Im m[f_{k}(0)],$

so that

${\frac {d\sigma (0)}{d\Omega }}\geq {\frac {k^{2}\sigma ^{2}}{16\pi ^{2}}}.$

From this, it follows that $\sigma \leq {\frac {4\pi }{k}}{\sqrt {\frac {d\sigma (0)}{d\Omega }}}.$

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