In
physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
, and especially
scattering theory
In physics, scattering is a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including particles and radiat ...
, the momentum-transfer cross section (sometimes known as the momentum-''transport'' cross section) is an effective
scattering
In physics, scattering is a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including particles and radiat ...
cross section useful for describing the average
momentum transfer
In particle physics, wave mechanics, and optics, momentum transfer is the amount of momentum that one particle gives to another particle. It is also called the scattering vector as it describes the transfer of wavevector in wave mechanics.
In ...
red 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
In physics, the cross section is a measure of the probability that a specific process will take place in a collision of two particles. For example, the Rutherford cross-section is a measure of probability that an alpha particle will be deflect ...
by
The momentum-transfer cross section can be written in terms of the phase shifts from a
partial wave analysis as
Explanation
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
This concept is used in calculating
charge radius of nuclei such as
proton
A proton is a stable subatomic particle, symbol , Hydron (chemistry), H+, or 1H+ with a positive electric charge of +1 ''e'' (elementary charge). Its mass is slightly less than the mass of a neutron and approximately times the mass of an e ...
and deuteron by
electron scattering
Electron scattering occurs when electrons are displaced from their original trajectory. This is due to the electrostatic forces within matter interaction or, if an external magnetic field is present, the electron may be deflected by the Lorentz ...
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 :
References
{{reflist
Momentum
Scattering theory