Physics: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 [math]\displaystyle{ \sigma_{\mathrm{tr}} }[/math] is defined in terms of an (azimuthally symmetric and momentum independent) differential cross section [math]\displaystyle{ \frac{\mathrm{d} \sigma}{\mathrm{d} \Omega} (\theta) }[/math] by [math]\displaystyle{ \begin{align} \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{align} }[/math]

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

Explanation

The factor of [math]\displaystyle{ 1 - \cos \theta }[/math] arises as follows. Let the incoming particle be traveling along the [math]\displaystyle{ z }[/math]-axis with vector momentum [math]\displaystyle{ \vec{p}_\mathrm{in} = q \hat{z}. }[/math]

Suppose the particle scatters off the target with polar angle [math]\displaystyle{ \theta }[/math] and azimuthal angle [math]\displaystyle{ \phi }[/math] plane. Its new momentum is [math]\displaystyle{ \vec{p}_\mathrm{out} = q' \cos \theta \hat{z} + q' \sin \theta \cos \phi\hat{x} + q' \sin \theta \sin \phi\hat{y}. }[/math]

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

By conservation of momentum, the target has acquired momentum [math]\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} . }[/math]

Now, if many particles scatter off the target, and the target is assumed to have azimuthal symmetry, then the radial ([math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math]) components of the transferred momentum will average to zero. The average momentum transfer will be just [math]\displaystyle{ q (1 - \cos \theta) \hat{z} }[/math]. If we do the full averaging over all possible scattering events, we get [math]\displaystyle{ \begin{align} \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{align} }[/math] where the total cross section is [math]\displaystyle{ \sigma_\mathrm{tot} = \int \frac{\mathrm{d} \sigma}{\mathrm{d} \Omega} (\theta) \mathrm{d} \Omega . }[/math]

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

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