Multiple scattering for particles in the matter.

In addition to
inelastic collisions with the atomic
electrons, particles passing through
matter suffer repeated elastic Coulomb
scattering from nuclei although
with a smaller probability.
Considering that usually nuclei have mass greater than the incoming particle, the energy transfer is negligible but each scattering centre adds a small deviation to the incoming particle’s trajectory also. Even if this deflection is small the sum of all the contribution adds a random component to the particle’s path which proceeds with a zigzag path (see Figure 1.). As result, incoming beam after a thickness of material shown a divergence greater than the initial.
·
Fig 1. Effect of Multiple Coulomb
Scattering.
Three situations
can be considered:
1. Single scattering. When the thickness is extremely small and the probability to have more than one interaction is negligible. This situation is well described by the Rutherford formula:
2.
Plural
scattering. When the number of Coulomb
scattering increases but remains
under few tens of interactions.
This is the most difficult case
to deal with, several works have
been done by different authors (see
[1] for further information).
3.
MMultiple
scattering. When the thickness increases
and the number of interactions become
high the angular dispersion can
be modelled as Gaussian.
Referring to multiple scattering, that is the most common situation, naming Θ the solid angle into which is concentrated the 98% of the beam after a thickness X of material, if we define Θ_{0}= Θ/√2 as the projection of Θ on a plane, the angular dispersion can be calculated by the relation:
where p is the momentum and Xo
is the radiation length. This last
quantity is characteristic of the
material and can be found tabulated
by Y.S. Tsai [2] or can be used
the approximated formula
References

E. Keil, E.Zeitler, and W. Zinn, Zeitschrift für Naturforschung A, vol. 15A, no. 1031, 1960.

YungSu Tsai, Pair production and bremsstrahlung of charged leptons, Reviews of Modern Physics, vol. 46, no. 815, 1974