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

We detect particles using sensors made from bulk matter: gases, liquids, and solids. Let's consider the general types of processes that can lead to "signals" in bulk matter, and how the travel of a particle is effected. I'll limit the discussion to relativistic particles, since this is generally the case in particle physics.

Process | Description |
---|---|

ionization | passing particle ionizes an atom, leaving behind an electron and positive ion. |

scintillation | passing particle excites (or ionizes) an atom which emits light (visible or u.v.) as it returns to an unexcited state |

showers (electromagnetic and hadronic) | cascade of secondary particles created when particles that interact electromagnetically and strongly pass through high A/Z material. |

Cerenkov radiation | emission of light (visible and u.v.) when the speed of a particle, βc, exceeds the speed (phase velocity) of light in a medium, c/n where n is the index of refraction (analogy of sonic boom for a vehicle exceeding the speed of sound) |

transition radiation | emission of light (visible and u.v.) when a particle passes from one medium to another of differing EM properties |

In the list of interactions I neglect several novel ones, for simplicity. These include phonon excitations in a crystal at low temperatures, and radio frequency Cerenkov radiation.

particle | lifetime |
---|---|

e^{+}, e^{-} | infinite |

p, pbar | infinite |

γ | infinite |

π^{+}, π^{-} | 2.6×10^{-8} s |

K^{+}, K^{-} | 1.2×10^{-8} s |

K^{0}_{S} | 0.89×10^{-10} s |

K^{0}_{L} | 5.2×10^{-8} s |

Λ, Λbar | 2.6×10^{-10} s |

Σ^{±}, Ξ, Ω | about 10^{-10} s |

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.

The Bethe-Bloch formula:

-dE/dx = Kz^{2}(Z/A)(1/β^{2})[(1/2)ln(2m_{e}c^{2}β^{2}γ^{2}T_{max}/I^{2}) - β^{2} - 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, T_{max} is given by

T_{max} = (2m_{e}c^{2}β^{2}γ^{2})/(1 + 2γm_{e}/M + (m_{e}/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.

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)^{-1/2}θ_{0}^{-1} exp(θ^{2}/2θ_{0}^{2})dθ

with θ
θ_{0} = (13.6MeV/βcp)z(x/X_{0})^{1/2} [1 + 0.038ln(x/X_{0})].

The quantity XHigh 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> = E_{0} exp(-X/X_{0})

where XAbsorption 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.

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, λ_{I}, for materials used in particle physics experiments are tabulated in the handout.

Copyright © Robert Harr 2005