Chapter 3 of Martin&Shaw.

# Experimental Methods

## Interactions of Particles with Bulk Matter

#### 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:

• 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<β0, where 1-β0 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/β2γm. 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:
1. 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.
2. The number of photons produced per unit path length and per unit wavelength is measured.
3. 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 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.

### 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 = (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.

### Energy loss formula

The Bethe-Bloch formula:

-dE/dx = Kz2(Z/A)(1/β2)[(1/2)ln(2mec2β2γ2Tmax/I2) - β2 - δ/2]

Taking a look at Fig. 26.1 in the Review of Particle Properties handout, we can see the range over which the Bethe-Bloch formula is applicable, from βγ of about 0.1 to several hundred. This plot is for μ+, but the conclusion about the range of applicability of the Bethe-Block formula is true for all heavy charged particles, μ, π, K, p, and d. I will ignore the region of βγ<0.1 as it is not generally important in particle physics experiments.

The region of βγ> few hundred is where radiative losses (bremstrahlung) turns on. For corresponds to muon momenta greater than about 100 GeV/c. Again, most experiments don't have muons of such high momentum, so that bremstrahlung by muons and heavier particles is basically ignored. (There are people now considering the effects of muon bremstrahlung in the next generation of experiments.) Bremstrahlung is an important effect for electrons.

The energy loss function has a minimum around βγ=3. The energy loss at this point is known as minimum ionizing, and a particle with this energy is called a minimum ionizing particle or MIP. This value is important, since it quantifies the smallest signals that a particular detector expects to see.

The quantity I is the mean excitation energy for the material. As can be seen in Figure 26.5, I depends on the material composition. This plot demonstrates that the I is approximately given by:

I = [(10±1)eV] Z
This relation works

For a particle of mass M and speed βc, Tmax is given by

Tmax = (2mec2β2γ2)/(1 + 2γme/M + (me/M)2).

Again looking at Figure 26.3, we see that the mean energy loss decreases for heavier elements. This counterintuitive effect is compensated for by the (normally) higher density of material composed of heavy elements.

### 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.