Redo problem 3 of homework 5 for extra credit. (Pair creation problem.)

We'd like to know whether a charged track is due to a e^{+}, μ^{+}, π^{+}, K^{+}, or p.
This is the task of particle identification.
There are several ways to differentiate particle types:

- differences in the way they interact -- electrons readily bremstrahl and compton scatter, pions, kaons, and protons interact strongly, muons do neither.
- sort out the possibilities by the kinematics of a decay.
- differences in mass.

The differences in the way particles interact show up in calorimetry. We sill discuss the kinematics of decays in the future. It is an extension of our discussions of special relativity.

Let's consider how to "weigh" a particle to determine its mass. From the measurement of track curvature we determine p = βγm. If we can measure another quantity that depends differently on mass and velocity, then we should be able to determine the mass from these two measurements.

- β is the speed of the particle and is found by measuring the time-of-flight (TOF) of the particle over a distance.
This works well for β<β
_{0}, where 1-β_{0}is the limit in resolution of the measurement. - The energy loss of a particle as it traverses a medium will be discussed next. It is a difficult measurement to make. It is most useful in the range of γβ<3.
- The small angle scattering of a particle traversing material is proportional to 1/β
^{2}γm. Particle identification by scattering angle though used in the past (in bubble chambers) is rarely used in modern experiments. - When the velocity of a particle exceeds the phase velocity of light in a (transparent) medium, Cerenkov radiation is emitted.
There are three ways Cerenkov radiation is used in particle identification:
- The threshold velocity for emission is β
_{t}=1/n where n is the index of refraction. The existence or lack of Cerenkov radiation then sets a lower or upper limit on a particle's velocity. - The number of photons produced per unit path length and per unit wavelength is measured.
- The light is emitted at a particular angle, the Cerenkov angle, given by
θ
_{c}= arccos(1/nβ) = Sqrt{2(1-1/nβ)} for small θ_{c}

- The threshold velocity for emission is β
- The amount of transition radiation emitted by a particle is proportional to (ln γ)
^{2}.

All of these techniques are made use of to identify a track's particle species.

As a prelude to the calculations for a particle traversing bulk matter, let's consider the situation where a particle of mass m, four-momentum p_{i}, scatters from atomic electrons which we'll treat as being at rest.
After the collision, the particle has 4-momentum p_{f} and the electron is ejected from the atom with 4-momentum p_{e}.
We can solve this situation in general, but we are specifically interested in the case of maximum momentum transfer.
This occurs when the electron momentum is in the same direction as the momentum of the incoming particle.
In this situation, conservation of energy/momentum gives:

|**p**_{i}| = |**p**_{f}| + |**p**_{e}|

and
(p_{i}^{2} + m^{2})^{½} = (p_{f}^{2} + m^{2})^{½} + (p_{e}^{2} + m_{e}^{2})^{½}

Φ_{col}(E_{i},E_{f})dE_{f} = 2Cm_{e}c^{2}dE_{f}[E_{i}/{E_{f}(E_{i} - E_{f})} - 1/E_{i}]^{2}

or by factoring out the first term in the square brackets:

Φ_{col}(E_{i},E_{f})dE_{f} = 2C {m_{e}c^{2}E_{i}^{2} dE_{f}/((E_{i} - E_{f})^{2} E_{f}^{2})} [1 - E_{f}/E_{i} + (E_{f}/E_{i})^{2}]^{2}

Φ_{col}(E_{i},E_{f})dE_{f} = 2C (m_{e}c^{2}dE_{f}/E_{f}^{2}) [1 - E_{f}/E_{i} + (E_{f}/E_{i})^{2}]^{2}

Φ_{col}(E_{i},E_{f})dE_{f} = (2Cm_{e}c^{2}/β^{2}) (dE_{f}/E_{f}^{2}) [1 - β^{2} (E_{f}/E_{f max})]

Φ_{col}(E_{i},E_{f})dE_{f} = (2Cm_{e}c^{2}/β^{2}) (dE_{f}/E_{f}^{2}) [1 - β^{2} (E_{f}/E_{m}) + ½{E_{f}/(E_{i} + mc^{2})}^{2}]

Φ_{col}(E_{i},E_{f})dE_{f} = (2Cm_{e}c^{2}/β^{2}) (dE_{f}/E_{f}^{2}) [(1 - β^{2} (E_{f}/E_{m}))(1 + (1/3)(E_{f}/E_{c})) + (1/3)(E_{f}/(E_{i} + mc^{2}))^{2}(1 + ½(E_{f}/E_{c}))]

where EThe detection of particles relies on a transfer of energy from the incoming particle to the bulk medium of detection. Consider a charged particle passing near an atom.

If the impact parameter (closest distance between the particle and the atom) is large compared to the size of the atom then we expect the atom to respond as a whole by either excitation or possibly ionization.
We also expect that the dominant interaction will arise from the particle's electric field, since the magnetic field produced by the magnetic moment decreases like 1/R^{3}, much faster than 1/R^{2} for the electric field.
And the probability of many small (low energy) photon exchanges is more likely than one high energy photon exchange.
For this situation a semi-classical treatment is valid.

If the impact parameter is of the same order as the atomic radius then the passing particle will interact directly with an electron. We are considering particles with greater than 1 MeV of kinetic energy, much larger than the typical binding energy of atomic electrons. Therefore we will treat the electrons as quasi-free. These interactions will impart a significant momentum transfer, ejecting the electron from the atom.

When the impact parameter is smaller than the atomic radius, then the particle will be defelected in its trajectory by the electric field of the nucleus. This process gives rise to multiple Coulomb scattering (MCS).

The deflection of a light particle (primarily electrons and positrons) can be accompanied by a photon of significant energy. This process is known by the German term bremstrahlung or braking radiation.

Copyright © Robert Harr 2003