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 zig-zag
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 Θ0=
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.
-
Yung-Su Tsai, Pair production and bremsstrahlung of charged leptons, Reviews of Modern Physics, vol. 46, no. 815, 1974
