Momentum-transfer cross section

In physics, and especially scattering theory, the momentum-transfer cross section (sometimes known as the momentum-transport cross section[1]) 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 [2]

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}}.}

For collision to much heavier target than striking particle (ex: electron incident on the atom or ion), q ′ ⋍ q {\displaystyle q'\backsimeq q} so

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} }\simeq q\cos \theta {\hat {z}}+q\sin \theta \cos \phi {\hat {x}}+q\sin \theta \sin \phi {\hat {y}}}

By conservation of momentum, the target has acquired momentum

Δ p → = p → i n − p → o u t = q ( 1 − cos ⁡ θ ) z ^ − q sin ⁡ θ cos ⁡ ϕ x ^ − q sin ⁡ θ sin ⁡ ϕ y ^ . {\displaystyle \Delta {\vec {p}}={\vec {p}}_{\mathrm {in} }-{\vec {p}}_{\mathrm {out} }=q(1-\cos \theta ){\hat {z}}-q\sin \theta \cos \phi {\hat {x}}-q\sin \theta \sin \phi {\hat {y}}.}

Now, if many particles scatter off the target, and the target is assumed to have azimuthal symmetry, then the radial ( x {\displaystyle x} and y {\displaystyle y} ) components of the transferred momentum will average to zero. The average momentum transfer will be just q ( 1 − cos ⁡ θ ) z ^ {\displaystyle q(1-\cos \theta ){\hat {z}}} . If we do the full averaging over all possible scattering events, we get

Δ p → a v g = ⟨ Δ p → ⟩ Ω = σ t o t − 1 ∫ Δ p → ( θ , ϕ ) d σ d Ω ( θ ) d Ω = σ t o t − 1 ∫ [ q ( 1 − cos ⁡ θ ) z ^ − q sin ⁡ θ cos ⁡ ϕ x ^ − q sin ⁡ θ sin ⁡ ϕ y ^ ] d σ d Ω ( θ ) d Ω = q z ^ σ t o t − 1 ∫ ( 1 − cos ⁡ θ ) d σ d Ω ( θ ) d Ω = q z ^ σ t r / σ t o t {\displaystyle {\begin{aligned}\Delta {\vec {p}}_{\mathrm {avg} }&=\langle \Delta {\vec {p}}\rangle _{\Omega }\\&=\sigma _{\mathrm {tot} }^{-1}\int \Delta {\vec {p}}(\theta ,\phi ){\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}(\theta )\,\mathrm {d} \Omega \\&=\sigma _{\mathrm {tot} }^{-1}\int \left[q(1-\cos \theta ){\hat {z}}-q\sin \theta \cos \phi {\hat {x}}-q\sin \theta \sin \phi {\hat {y}}\right]{\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}(\theta )\,\mathrm {d} \Omega \\&=q{\hat {z}}\sigma _{\mathrm {tot} }^{-1}\int (1-\cos \theta ){\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}(\theta )\,\mathrm {d} \Omega \\[1ex]&=q{\hat {z}}\sigma _{\mathrm {tr} }/\sigma _{\mathrm {tot} }\end{aligned}}}
where the total cross section is
σ t o t = ∫ d σ d Ω ( θ ) d Ω . {\displaystyle \sigma _{\mathrm {tot} }=\int {\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}(\theta )\mathrm {d} \Omega .}

Here, the averaging is done by using expected value calculation (see d σ d Ω ( θ ) / σ t o t {\displaystyle {\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}(\theta )/\sigma _{\mathrm {tot} }} as a probability density function). Therefore, for a given total cross section, one does not need to compute new integrals for every possible momentum in order to determine the average momentum transferred to a target. One just needs to compute σ t r {\displaystyle \sigma _{\mathrm {tr} }} .

Application

This concept is used in calculating charge radius of nuclei such as proton and deuteron by electron scattering experiments.

To this purpose a useful quantity called the scattering vector q having the dimension of inverse length is defined as a function of energy E and scattering angle θ:

q = 2 E ℏ c sin ⁡ ( θ / 2 ) [ 1 + 2 E M c 2 sin 2 ⁡ ( θ / 2 ) ] 1 / 2 {\displaystyle q={\frac {{\frac {2E}{\hbar c}}\sin(\theta /2)}{\left[1+{\frac {2E}{Mc^{2}}}\sin ^{2}(\theta /2)\right]^{1/2}}}}

References

  1. ^ Zaghloul, Mofreh R.; Bourham, Mohamed A.; Doster, J.Michael (April 2000). "Energy-averaged electron–ion momentum transport cross section in the Born approximation and Debye–Hückel potential: Comparison with the cut-off theory". Physics Letters A. 268 (4–6): 375–381. Bibcode:2000PhLA..268..375Z. doi:10.1016/S0375-9601(00)00217-6.
  2. ^ Bransden, B.H.; Joachain, C.J. (2003). Physics of atoms and molecules (2. ed.). Harlow [u.a.]: Prentice-Hall. p. 584. ISBN 978-0582356924.