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 $\sigma _{{{\mathrm {tr}}}}$ is defined in terms of an (azimuthally symmetric and momentum independent) differential cross section ${\frac {{\mathrm {d}}\sigma }{{\mathrm {d}}\Omega }}(\theta )$ by

$\sigma _{{{\mathrm {tr}}}}=\int (1-\cos \theta ){\frac {{\mathrm {d}}\sigma }{{\mathrm {d}}\Omega }}(\theta ){\mathrm {d}}\Omega$

$=\int \int (1-\cos \theta ){\frac {{\mathrm {d}}\sigma }{{\mathrm {d}}\Omega }}(\theta )\sin \theta {\mathrm {d}}\theta {\mathrm {d}}\phi$.

The momentum-transfer cross section can be written in terms of the phase shifts from a partial wave analysis as ^[2]

$\sigma _{{{\mathrm {tr}}}}={\frac {4\pi }{k^{2}}}\sum _{{l=0}}^{\infty }(l+1)\sin ^{2}[\delta _{{l+1}}(k)-\delta _{l}(k)].$

Explanation

The factor of $1-\cos \theta$ arises as follows. Let the incoming particle be traveling along the $z$-axis with vector momentum

${\vec {p}}_{{\mathrm {in}}}=q{\hat {z}}$.

Suppose the particle scatters off the target with polar angle $\theta$ and azimuthal angle $\phi$ plane. Its new momentum is

${\displaystyle {\vec {p}}_{\mathrm {out} }=q'\cos \theta {\hat {z}}+q'\sin \theta \cos \phi {\hat {x}}+q'\sin \theta \sin \phi {\hat {y}}}$.

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

${\displaystyle {\vec {p}}_{\mathrm {out} }\simeq q\cos \theta {\hat {z}}+q\sin \theta \cos \phi {\hat {x}}+q\sin \theta \sin \phi {\hat {y}}}$

By conservation of momentum, the target has acquired momentum

${\displaystyle \Delta {\vec {p}}={\vec {p}}_{\mathrm {in} }-{\vec {p}}_{\mathrm {out} }=q(1-\cos \theta ){\hat {z}}-q\sin \theta \cos \phi {\hat {x}}-q\sin \theta \sin \phi {\hat {y}}}$.

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

$\Delta {\vec {p}}_{{\mathrm {avg}}}=\langle \Delta {\vec {p}}\rangle _{\Omega }$.

${\displaystyle =\sigma _{\mathrm {tot} }^{-1}\int \Delta {\vec {p}}(\theta ,\phi ){\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}(\theta )\mathrm {d} \Omega }$.

${\displaystyle =\sigma _{\mathrm {tot} }^{-1}\int \left[q(1-\cos \theta ){\hat {z}}-q\sin \theta \cos \phi {\hat {x}}-q\sin \theta \sin \phi {\hat {y}}\right]{\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}(\theta )\mathrm {d} \Omega }$

$=q{\hat {z}}\sigma _{{\mathrm {tot}}}^{{-1}}\int (1-\cos \theta ){\frac {{\mathrm {d}}\sigma }{{\mathrm {d}}\Omega }}(\theta ){\mathrm {d}}\Omega$

$=q{\hat {z}}\sigma _{{\mathrm {tr}}}/\sigma _{{\mathrm {tot}}}$

where the total cross section is

$\sigma _{{\mathrm {tot}}}=\int {\frac {{\mathrm {d}}\sigma }{{\mathrm {d}}\Omega }}(\theta ){\mathrm {d}}\Omega$.

Here, the averaging is done by using expected value calculation (see ${\displaystyle {\frac {\mathrm {d} \sigma }{\mathrm {d} \Omega }}(\theta )/\sigma _{\mathrm {tot} }}$ 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 $\sigma _{{\mathrm {tr}}}$.

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:

${\displaystyle q={\frac {{\frac {2E}{\hbar c}}\sin(\theta /2)}{{[1+{\frac {2E}{Mc^{2}}}\sin ^{2}(\theta /2)}]^{1/2}}}}$

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.