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

Recall from last lecture:

Experimental Examples (continued)

Particle Identification

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:

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.

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

Kinematics of Scattering with Atomic Electrons

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 pi, scatters from atomic electrons which we'll treat as being at rest. After the collision, the particle has 4-momentum pf and the electron is ejected from the atom with 4-momentum pe. 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:

|pi| = |pf| + |pe|
(pi2 + m2)½ = (pf2 + m2)½ + (pe2 + me2)½

Scattering Probabilities

Møller scattering

Φcol(Ei,Ef)dEf = 2Cmec2dEf[Ei/{Ef(Ei - Ef)} - 1/Ei]2

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

Φcol(Ei,Ef)dEf = 2C {mec2Ei2 dEf/((Ei - Ef)2 Ef2)} [1 - Ef/Ei + (Ef/Ei)2]2

Bhabha scattering

Φcol(Ei,Ef)dEf = 2C (mec2dEf/Ef2) [1 - Ef/Ei + (Ef/Ei)2]2

Scattering of spin 0 particles of mass m

Φcol(Ei,Ef)dEf = (2Cmec22) (dEf/Ef2) [1 - β2 (Ef/Ef max)]

Scattering of spin ½ particles of mass m

Φcol(Ei,Ef)dEf = (2Cmec22) (dEf/Ef2) [1 - β2 (Ef/Em) + ½{Ef/(Ei + mc2)}2]

Scattering of spin 1 particles of mass m

Φcol(Ei,Ef)dEf = (2Cmec22) (dEf/Ef2) [(1 - β2 (Ef/Em))(1 + (1/3)(Ef/Ec)) + (1/3)(Ef/(Ei + mc2))2(1 + ½(Ef/Ec))]
where Ec = m2c2 / me.

Energy loss formula

The 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/R3, much faster than 1/R2 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