Reading

Chapter 3 of Martin&Shaw.

Recall

Last time we discussed types of accelerators and beams, and the properties of particles that can conceivably be measured.

Experimental Methods

Interactions of Particles with Bulk Matter

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|
and
(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 = (2Cmec2/β2) (dEf/Ef2) [1 - β2 (Ef/Ef max)]

Scattering of spin ½ particles of mass m

Φcol(Ei,Ef)dEf = (2Cmec2/β2) (dEf/Ef2) [1 - β2 (Ef/Em) + ½{Ef/(Ei + mc2)}2]

Scattering of spin 1 particles of mass m

Φcol(Ei,Ef)dEf = (2Cmec2/β2) (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.

Multiple Coulomb Scattering (MCS)

Particles lose little energy in collisions with atomic nuclei, but their trajectory will be perturbed. We say that particles are scattered when passing through matter due to electromagnetic interactions with the nucleus (Coulomb scattering). This scattering is analogous to Rutherford scattering. For single interactions, the resulting distribution of scattered angles can be calculated similarly to the Rutherford scattering formula.

Normally particles must traverse modest thicknesses of material in which they can experience many scatters. This is the case we are now interested in and it is called multiple coulomb scattering or MCS for short. After many scatters, the resulting distribution of scattered angles is roughly Gaussian:

P(θ)dθ = (2π)-1/2θ0-1 exp(-θ2/2θ02)dθ
with θ0 given by:
θ0 = (13.6MeV/βcp)z(x/X0)1/2 [1 + 0.038ln(x/X0)].
The quantity X0 is called the radiation length of the material. It characterizes electromagnetic interactions with nuclei of the material.

Electromagnetic Showers

Electrons

High energy charged particles lose energy primarily through bremstrahlung. However, except for electrons, this occurs for energies greater than about 100 GeV. For electrons, bremstrahlung becomes important for energies greater than about 10 MeV! This difference is exploited as a way to identify a charged particle as an electron.

The average energy of an electron after traversing a thickness X of a medium is:

<E> = E0 exp(-X/X0)
where X0 is the radiation length of the material. Radiation lengths of materials used in experiments are listed in the table of atomic and nuclear properties handed out today. To give you an idea of the magnitudes, they range from about 0.3cm for platinum to 650m for methane gas at 1 atm. and 20°C. The radiation length sets the scale for a detector designed to measure the energy deposited when an electron is stopped, an electromagnetic calorimeter. Typically a calorimeter must have a thickness of 10 to 20 radiation lengths.

Photons

Absorption of γ-rays in matter is dominated by Compton scattering and pair production. The length scale for these processes is again governed by the radiation length of the medium. An electromagnetic calorimeter is therefore capable of measuring the energies of electrons and photons.

Hadronic Showers

Hadrons are all particles bound by the strong force, that is, those composed of quarks, the mesons (q and qbar) and baryons (3 quarks or 3 anti-quarks). The nuclei of atoms are composed of hadrons. The collision of a hadron with a nucleus has a large cross section for inelastic interaction, of order 1 barn, with a length scale given by the interaction length of the medium. (Electrons and photons will be stopped in a relatively short distance, radiation lengths are significantly smaller than interaction lengths.) The muons have relatively small cross sections for inelastic interaction.

The interaction lengths, λI, for materials used in particle physics experiments are tabulated in the handout.

Cerenkov Radiation

When the velocity of a particle exceeds the phase velocity of light in a (transparent) medium, Cerenkov radiation is emitted. The threshold velocity for emission is βt=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 light is emitted at a particular angle, the Cerenkov angle, given by

θc = arccos(1/nβ) = Sqrt{2(1-1/nβ)} for small θc
The number of photons emitted is approximately:
N = L (α2z2)/(remec2)<sin2θc>
where <sin2θc> is the energy average of the square of the sine of the Cerenkov angle.

Copyright © Robert Harr 2005