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Physics Recalls

1. – Physics recalls

1.1 – Energy loss by electrons: bremsstrahlung

1.2 – Electromagnetic and hadronic showers

1.3 – Particle in a magnetic field


Physics Recalls
1. – Physics Recalls.

In this section we collect together brief descriptions of other types of process by which particles can interact and deposit energy in materials, as well as radiative processes associated with the passage of particles through magnetic fields.

2.1 – Energy loss by electrons: bremsstrahlung

2.2 – Electromagnetic and hadronic showers

2.3 – Particle in a magnetic field


1.1. - Energy loss by electrons: bremsstrahlung.

Electrons lose energy by ionising atoms in the medium, as does any charged particle. The equation for the rate is similar to that for heavier incident particles, but it is no longer valid to assume that the incident electron is unaffected by the exchange of photons with other electrons. There are also quantum effects arising from the fact that the incident particle is identical to the atomic electrons with which it interacts.

Bethe gives a modified form of the Bethe-Bloch relation for electrons

There is another energy loss process for electrons which is insignificant for heavier particles: bremsstrahlung. This is the process whereby an electron is accelerated abruptly by the strong electric field very close to a nucleus and in so accelerating it radiates a photon.

The Feynman diagram is as shown. It has three vertices, one for the emission of a real photon and the other two for the virtual photon exchanged with the atomic nucleus. The nucleus has charge number Z and contributes a factor Z2 to the cross section for the process. There is a factor  from each vertex, so the cross section Z23.

The lighter a particle is the greater is its acceleration, so the amplitude of the emitted radiation is inversely proportional to the mass and the cross section to the square of this

If  is the energy of the emitted photon then we want a value for d/d. (This  is nothing to do with dielectric constant). We would guess that

since this is dimensionally correct.

Integrating over the range of photon energies gives the energy loss per unit distance moved by the fast electron.

The maximum energy loss per encounter, max ,  is the kinetic energy of the electron T. The minimum min is essentially zero. As in the case of the Bethe-Bloch equation there are a few numerical factors this naive treatment has missed and in fact

One reason for the difference is that at large distances, the nuclear charge is screened by the atomic electrons, a rather significant effect.

As the rate of energy loss is proportional to the kinetic energy, [image] Consequently [image].

Here X0 is a constant, the distance in which the electron's kinetic energy is reduced by a factor e. It is known as the radiation length and is given by

The spectrum of energy  for emitted photons is expected from the above treatment to be flat,

up to the maximum energy max = T.  Here  is the frequency of the emitted photons, so approximately equal energy is emitted into equal intervals of d. The spectrum does not actually cut off sharply at the value T, but rolls off from various effects such as screening.

As also shown in our discussion, the intensity of the radiation emitted is inversely proportional to the square of the particle mass. Electrons are very much lighter than other particles and for energies of interest bremsstrahlung is significant only for electrons. At the very highest energies available today, 100 GeV or more, bremsstrahlung can be seen also for muons in thick absorbers. For high-energy hadrons, strong interaction collisions are a much more important source of energy loss, giving rise to showering as discussed below. We can define a critical energy Ec, above which the rate of energy loss due to bremsstrahlung dominates the ionisation loss. For electrons Ec varies from below 10 MeV in heavy metals (Pb, W, U) to around 300 MeV in hydrogen. For muons it is in the region of 100s of GeV.

The process of bremsstrahlung may be visualised in the photon exchange picture. A charged particle is at all times surrounded by a cloud of virtual photons. If the particle is abruptly accelerated to one side then a photon in the cloud may become detached and carry on along the particle’s path, to appear as a forward going real photon.


1.2. - Electromagnetic and hadronic showers

High-energy electrons (or photons) will produce electromagnetic showers.

•	A high-energy electron produces photons by bremsstrahlung.

•	These photons still have high energy and cause pair production, to give electron-positron pairs.

•	These in turn produce more photons through bremsstrahlung and if their energy is still high enough these photons may give more electron-positron production ...

A shower of electrons/positrons and photons develops and continues to grow until the energy of its constituents falls too low to give any more interactions, after which the shower dies away. If all the electrons and positrons give some form of signal along their path this gives an estimate of the total energy of the initial electron or photon. Such an electromagnetic calorimeter can equally detect charged electrons or neutral photons, as it does not matter at which stage of the process the shower begins.

The total energy of all the charged particles in the shower gives a measure of the total energy of the particle which initiated it. In a typical electromagnetic shower, the maximum rate of energy deposition per unit length occurs at a depth into the absorber of roughly 5X0. A total absorber thickness of 25–30 radiation lengths is sufficient to contain most of the energy of a shower.

High-energy hadrons entering a block of absorber initiate showers or cascades of strong interaction processes. Again, as for electromagnetic showers, each interaction of a fast particle with a nucleus produces several secondary particles, which can interact further and produce yet more, lower energy products. The typical depth of material characterising this process is called the hadronic interaction length I, and is related to the cross section I for strong interactions with the nuclei of the material (which, at high energy, is roughly independent of the incident particle type and energy):

For materials other than hydrogen, I is several times larger than X0. The factor increases with Z, from about 2 for carbon, 4 for aluminium and 10 for iron and copper, to 20–30 for heavy metals. Finally we note that the multiple strong interaction processes involved in hadronic showers can produce numbers of neutral pions, π0. These particles have a short lifetime and decay electromagnetically, producing photons. These will then initiate electromagnetic cascades as discussed above, within the hadronic shower.


1.3. - Particle in a magnetic field

In a magnetic field B a moving charged particle experiences a Lorentz force:

where e is the charge on the particle and v its velocity. The force is in the plane perpendicular to B. It is also at right angles to the particle’s velocity, so the path bends without any change in speed.

We look at the particle’s path projected onto the plane perpendicular to the field lines. The component of v in this plane is v, so that the force can be written evB, and the momentum component is p. If the momentum changes by a small amount dp, the result is a change in direction by a small angle:

Further, d  = (vdt)/, where  is the radius of curvature of the track. This gives:

and as a consequence

This is a general result and is in fact relativistically correct. A charged particle moving through a uniform magnetic field follows a track which curves in a plane perpendicular to the field, with a radius of curvature proportional to the momentum of the particle.