Momentum-transfer cross section

In physics, and especially scattering theory, the momentum-transfer cross section (sometimes known as the momentum-transport cross section) is an effective scattering cross section useful for describing the average momentum transferred from a particle when it collides with a target. Essentially, it contains all the information about a scattering process necessary for calculating average momentum transfers but ignores other details about the scattering angle.

The momentum-transfer cross section ${\displaystyle \sigma _{\mathrm {tr} }}$ is defined in terms of an (azimuthally symmetric and momentum independent) differential cross section ${\displaystyle {\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}(\theta )}$ by $${\displaystyle {\begin{aligned}\sigma _{\mathrm {tr} }&=\int (1-\cos \theta ){\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}(\theta )\,\mathrm {d} \Omega \\&=\iint (1-\cos \theta ){\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}(\theta )\sin \theta \,\mathrm {d} \theta \,\mathrm {d} \phi .\end{aligned}}}$$

The momentum-transfer cross section can be written in terms of the phase shifts from a partial wave analysis as $${\displaystyle \sigma _{\mathrm {tr} }={\frac {4\pi }{k^{2}}}\sum _{l=0}^{\infty }(l+1)\sin ^{2}[\delta _{l+1}(k)-\delta _{l}(k)].}$$