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 σ t r {\displaystyle \sigma _{\mathrm {tr} }} is defined in terms of an (azimuthally symmetric and momentum independent) differential cross section d σ d Ω ( θ ) {\displaystyle {\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}(\theta )} by
σ t r = ∫ ( 1 − cos θ ) d σ d Ω ( θ ) d Ω = ∬ ( 1 − cos θ ) d σ d Ω ( θ ) sin θ d θ d ϕ . {\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
σ t r = 4 π k 2 ∑ l = 0 ∞ ( l + 1 ) sin 2 [ δ l + 1 ( k ) − δ l ( k ) ] . {\displaystyle \sigma _{\mathrm {tr} }={\frac {4\pi }{k^{2}}}\sum _{l=0}^{\infty }(l+1)\sin ^{2}[\delta _{l+1}(k)-\delta _{l}(k)].}
Explanation The factor of 1 − cos θ {\displaystyle 1-\cos \theta } arises as follows. Let the incoming particle be traveling along the z {\displaystyle z} -axis with vector momentum
p → i n = q z ^ . {\displaystyle {\vec {p}}_{\mathrm {in} }=q{\hat {z}}.}
Suppose the particle scatters off the target with polar angle θ {\displaystyle \theta } and azimuthal angle ϕ {\displaystyle \phi } plane. Its new momentum is
p → o u t = q ′ cos θ z ^ + q ′ sin θ cos ϕ x ^ + q ′ sin θ sin ϕ y ^ . {\displaystyle {\vec {p}}_{\mathrm {out} }=q'\cos \theta {\hat {z}}+q'\sin \theta \cos \phi {\hat {x}}+q'\sin \theta \sin \phi {\hat {y}}.}
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