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

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

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

Let the incoming particle be traveling along the

-axis with vector momentum

Suppose the particle scatters off the target with polar angle

and azimuthal angle

cos ⁡ θ

sin ⁡ θ cos ⁡ ϕ

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

≃ q cos ⁡ θ

+ q sin ⁡ θ cos ⁡ ϕ

By conservation of momentum, the target has acquired momentum

= q ( 1 − cos ⁡ θ )

− q sin ⁡ θ cos ⁡ ϕ

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

) components of the transferred momentum will average to zero.

The average momentum transfer will be just

If we do the full averaging over all possible scattering events, we get

( θ , ϕ )

− q sin ⁡ θ cos ⁡ ϕ

where the total cross section is

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

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.

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:

sin ⁡ ( θ