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]


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 .


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:


  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.