||Dr C N Booth
Interaction of a Charged Particle with Matter
Most particle detectors make use of the effect charged particles have on
the matter they traverse. The interaction of charged particles with
matter is covered in considerable detail in the lecture handouts, PostScript
versions of which can be seen here and here,
while PDF versions can be seen here and here.
The aim of this summary is not to reproduce the mathematics but to give
a brief overview of the physical principles involved in calculating both
how much energy a particle loses as it passes through matter and where
that energy goes.
The result of the calculation, for particles heavier than an electron,
is known as the Bethe Bloch equation:
A relativistic charged particle interacts with the electrons and nuclei
close to its trajectory via electromagnetic effects. A reasonable
approximation to the result can be obtained using simple electrostatics.
The time the particle is close enough to a "target" electron or nucleus
to affect it is sufficiently short that it is usually reasonable to use
the impulse approximation. This assumes that the target does not
move significantly during the collision, so that the integrated effect
parallel to the direction of motion of the "projectile" particle is zero, and momentum
transfer is purely transverse. It is easiest also to assume that
a small enough amount of energy is transferred in each interaction that
the target particle remains non-relativistic.
The momentum transfer in an individual interaction then depends on the
mass and charge of the target particle, the charge and velocity of the
projectile particle and the impact parameter of the collision.
Almost all energy is lost to atomic electrons, which are more numerous
and much lighter than the nuclei, despite their lower charge.
The overall energy loss of a particle passing through a slice of matter
is then obtained by integrating over the distribution of impact parameters
involved. Conservation of energy, atomic and quantum effects limit
the range of impact parameter over which the integral is physically meaningful.
Further corrections are required for two forms of screening:
The "shell correction" is a small adjustment due to the screening of inner
atomic electrons by outer ones. (It can often be ignored.)
The "density effect" is caused by polarisation of the medium as a whole,
and reduces the effective field of the particle at large impact parameters.
It is important in denser materials (hence the name) and for large values
of the relativistic &gamma of the projectile.
|| is the
mean energy loss in passing through thickness Δx,
||me is the mass of the electron,
||Z and A are the atomic number and mass of the medium
||z is the charge (in units of electronic charge) of the projectile
||β and γ are the usual relativistic parameters,
||ρ is the density of the medium,
||I0 is its mean ionisation energy,
||ε is the shell correction and
||δ is the density effect.
(For an alternative formulation and description, with different notation,
see the BriefBook.)
The energy loss therefore only depends on the incoming particle's velocity,
and not directly on its mass, as is shown in the figure below.
Energy loss in argon, as a function of particle mass and momentum;
the vertical scale gives the relative increase above the minimum of ionisation.
(With apologies for the poor quality of the figure!)
The energy loss therefore shows four regions:
a rapid decrease proportional to 1/β2
at lower velocities (mostly off the left of this plot);
a minimum at E ~ 3 Mc2 (i.e. γ ~ 3);
a slow logarithmic "relativistic rise", proportional to ln(γ)
a plateau as ionisation is limited by the density effect.
Note that the changes in energy loss
above the minimum are very much less than those in the low velocity region.
For this reason, any particle with γ above 3 is often known as a minimum ionising particle, or mip.
for very low density materials, such as gases, the energy loss eventually
levels out at 50% above minimum when γ
for solids, the plateau is only 10% above minimum, and may be reached with γ as low as 10.
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