In addition to the handouts from the "Review of Particle Properties" this material is covered in Perkins, chapter 11, section 5.

Recall from last lecture:

The Bethe-Bloch formula for energy loss:

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

Mean multiple Coulomb scattering angle:

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


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.


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

Copyright © Robert Harr 2003