Momentum-transfer cross section

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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 by

.

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

Explanation[edit]

The factor of arises as follows. 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 plane. Its new momentum is

.

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

By conservation of momentum, the target has acquired momentum

.

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

.
.

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. One just needs to compute .

Application[edit]

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:

References[edit]

  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.