Momentum-transfer cross section explained

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_tr

is defined in terms of an (azimuthally symmetric and momentum independent) differential cross section

	d\sigma
	d\Omega

(\theta)

\begin\sigma_ &= \int (1 - \cos \theta) \frac (\theta) \, \mathrm \Omega \\&= \iint (1 - \cos \theta) \frac (\theta) \sin \theta \, \mathrm \theta \, \mathrm \phi.\end

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

Explanation

The factor of

1-\cos\theta

arises as follows. Let the incoming particle be traveling along the

-axis with vector momentum

\vec_\mathrm = q \hat.

Suppose the particle scatters off the target with polar angle

\theta

and azimuthal angle

\phi

plane. Its new momentum is

\vec_\mathrm = q' \cos \theta \hat + q' \sin \theta \cos \phi\hat + q' \sin \theta \sin \phi\hat.

For collision to much heavier target than striking particle (ex: electron incident on the atom or ion),

q'\backsimeqq

\vec_\mathrm \simeq q \cos \theta \hat + q \sin \theta \cos \phi\hat + q \sin \theta \sin \phi\hat

By conservation of momentum, the target has acquired momentum $\Delta \vec = \vec_\mathrm - \vec_\mathrm = q (1 - \cos \theta) \hat - q \sin \theta \cos \phi\hat - q \sin \theta \sin \phi\hat .$

Now, if many particles scatter off the target, and the target is assumed to have azimuthal symmetry, then the radial (

and

) 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

\begin\Delta \vec_\mathrm &= \langle \Delta \vec \rangle_\Omega \\&= \sigma_\mathrm^ \int \Delta \vec(\theta,\phi) \frac (\theta) \, \mathrm \Omega \\&= \sigma_\mathrm^ \int \left[q (1 - \cos \theta) \hat{z} - q \sin \theta \cos \phi\hat{x} - q \sin \theta \sin \phi\hat{y} \right ] \frac (\theta) \, \mathrm \Omega \\&= q \hat \sigma_\mathrm^ \int (1 - \cos \theta) \frac (\theta) \, \mathrm \Omega \\[1ex]&= q \hat \sigma_\mathrm / \sigma_\mathrm\end

where the total cross section is

\sigma_\mathrm = \int \frac (\theta) \mathrm \Omega .

Here, the averaging is done by using expected value calculation (see

	d\sigma
	d\Omega

(\theta)/\sigma_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_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 having the dimension of inverse length is defined as a function of energy and scattering angle : $q = \frac$

Notes and References

Zaghloul. Mofreh R.. Bourham, Mohamed A. . Doster, J.Michael . 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. April 2000. 268. 4–6. 375–381. 10.1016/S0375-9601(00)00217-6 . 2000PhLA..268..375Z .
Book: Bransden. B.H.. Joachain. C.J.. Physics of atoms and molecules. 2003. Prentice-Hall. Harlow [u.a.]. 978-0582356924. 584. 2..