||Dr C N Booth
Coherent Effect for Charged Particles
So far in this course, we have looked at the effect of charged particles
on individual atoms, and seen that this leads to excitation (exploited
in scintillation counters) and ionisation
of atomic electrons. However, when we looked at the Bethe-Bloch formula
for energy loss
we introduced the "density effect", δ,
which arises from a screening of the particle's field due to the polarisation
of the medium as a whole. As considered in the Bethe-Bloch formula,
this primarily results in a decrease in energy loss (hence the negative
sign). However, the changing polarisation can also lead to observable
coherent effects in the medium, which can produce an observable signal.
We will look at two manifestations of this, Cherenkov and Transition
Radiation. These notes only introduce the topics and provide a summary,
with links to descriptions in the
BriefBook. For further details of the exploitation of these effects
in detectors, it is necessary to see the lecture notes.
As a particle passes through matter, the surrounding atoms polarise and
subsequently depolarise, and a weak electromagnetic wave spreads out from
the instantaneous position of the particle. For a particle travelling
more slowly than light, wave-fronts originating at different times can
never meet, and no interference is possible.
For a particle travelling faster than light, the wave-fronts do overlap,
and constructive interference is possible, leading to a significant, observable
Click here to view animations of the radiation from a moving particle.
A particle can not, of course, travel faster than the speed of light
in a vacuum. In a medium of refractive index n, the speed
of light is c/n, and there is no reason why the speed of
the particle, βc,
cannot be greater than c/n.
It can be shown that the number of photons per unit length of track is
n is a function of the frequency ν,
for most materials decreasing rapidly in the ultraviolet. Most radiation
is therefore in the visible, peaking at the blue end of the spectrum.
In the visible range, the number of photons is roughly 500 sin2θc
per centimetre of track.
A highly relativistic particle passing through a medium is observed
to emit visible light known as Cherenkov
radiation if β > 1/n. As can be seen from the above diagram, a cone
of light radiates out from each point on the particle's track.
As a special case, consider a medium with n close to 1,
n = 1 + δ
(with δ « 1).
Cherenkov radiation then only occurs for β
very close to 1, say β = 1 − ε
(with ε « 1).
The condition for Cherenkov radiation, β > 1/n,
implies ε < δ.
The number of Cherenkov photons is then
for δ > ε.
As mentioned in the introduction, Cherenkov
detectors are used primarily for identifying the type of a particle
(whose momentum or energy is, at least approximately, known), rather than
for tracking the position of the particle. The three main types of
detector are described further in the notes (available here in PostScript
and PDF forms), and summarised below.
Threshold Cherenkov detectors consist simply of a radiator and light
detector (such as a photomultiplier). Particles with a velocity above
the threshold for producing Cherenkov light are detected, while others
are not. If a gas is used as the radiator, its refractive index,
and so the threshold velocity, can be tuned by adjusting the pressure.
Differential Cherenkov detectors are slightly more sophisticated,
in that they respond to a range of velocities. This is typically
arranged by being sensitive to a certain range of Cherenkov angles, for
example by having mirrors and/or baffles between the radiator and the light
Cherenkov detectors use spherical mirrors to focus the cone of
Cherenkov light into a ring on a position-sensitive light detector or array
of detectors. In this way, the centre of the ring indicates the position
of the particle and the radius of the ring measures the Cherenkov angle
and so the velocity of the particle. Suitable light detectors include
special multiwire proportional chambers, containing a gas mixture sensitive
to visible or ultra-violet photons.
When a particle crosses between two regions of very different dielectric
constant, there is a sudden change in the polarisation of the surrounding
medium, and transient currents and fields are set up. If the relativistic γ
of the particle is » 1, then X-radiation known as Transition
Radiation can be produced, with an angular distribution which is
very strongly peaked in the direction of the particle. The amount
of radiation is proportional to the γ
of the particle, and to the plasma frequency of the medium. In practice,
it is only high energy electrons (with their very small mass) which
normally have a high enough γ
to be detected, so transition radiation detectors are typically electron
A practical detector consists of a stack of radiator foils, of
low atomic number in order to reduce absorption of the X-rays.
The radiation is then detected in a special proportional chamber.
This contains a high-Z gas such as Xenon (to absorb the photons
rapidly). The passage of any charged particle through the
chamber will also produce a small signal due to ionisation. In order
to distinguish between transition radiation and ionisation, an asymmetric
chamber is often used, with a drift space before the amplification
region. Transition radiation photons are absorbed early in this space,
so give a large, late signal, in contrast with ionisation
which gives a small signal spread out in time. (Further details of
Transition Radiation Detectors can be found in the notes,
available in PostScript
and PDF forms.)
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