Particle shower

Types

There are two basic types of showers. Electromagnetic showers are produced by a particle that interacts primarily or exclusively via the electromagnetic force, usually a photon or electron. Hadronic showers are produced by hadrons (i.e. nucleons and other particles made of quarks), and proceed mostly via the strong nuclear force.

Electromagnetic showers

An electromagnetic shower begins when a high-energy electron, positron or photon enters a material. At high energies (above a few MeV), in which the photoelectric effect and Compton scattering are insignificant, photons interact with matter primarily via pair production — that is, they convert into an electron-positron pair, interacting with an atomic nucleus or electron in order to conserve momentum. High-energy electrons and positrons primarily emit photons, a process called bremsstrahlung. These two processes (pair production and bremsstrahlung) continue, leading to a cascade of particles of decreasing energy until photons fall below the pair production threshold, and energy losses of electrons other than bremsstrahlung start to dominate. The characteristic amount of matter traversed for these related interactions is called the radiation length ${\displaystyle X_0}$. ${\displaystyle X_0}$ is both the mean distance over which a high-energy electron loses all but 1/e of its energy by bremsstrahlung and 7/9 of the mean free path for pair production by a high energy photon. The length of the cascade scales with ${\displaystyle X_0}$; the "shower depth" is approximately determined by the relation

${\displaystyle X=X_0\frac{\ln (E_0/E_{\mathrm{c}})}{\ln 2},}$

where ${\displaystyle X_0}$ is the radiation length of the matter, and ${\displaystyle E_{\mathrm{c}}}$ is the critical energy (the critical energy can be defined as the energy in which the bremsstrahlung and ionization rates are equal. A rough estimate is ${\displaystyle E_{\mathrm{c}}=800\mathrm{M}\mathrm{e}\mathrm{V}/(Z+1.2)}$). The shower depth increases logarithmically with the energy, while the lateral spread of the shower is mainly due to the multiple scattering of the electrons. Up to the shower maximum the shower is contained in a cylinder with radius < 1 radiation length. Beyond that point electrons are increasingly affected by multiple scattering, and the lateral size scales with the Molière radius ${\displaystyle R_{\mathrm{M}}}$. The propagation of the photons in the shower causes deviations from Molière radius scaling. However, roughly 95% of the shower are contained laterally in a cylinder with radius ${\displaystyle 2R_{\mathrm{M}}}$.

The mean longitudinal profile of the energy deposition in electromagnetic cascades is reasonably well described by a gamma distribution:

${\displaystyle \frac{dE}{dt}=E_{0}b\frac{(bt{)}^{a-1}{e}^{-bt}}{\mathrm{\Gamma }(a)}}$

where ${\displaystyle t=X/X_0}$, ${\displaystyle E_0}$ is the initial energy and ${\displaystyle a}$ and ${\displaystyle b}$ are parameters to be fitted with Monte Carlo or experimental data.

Hadronic showers

The physical processes that cause the propagation of a hadron shower are considerably different from the processes in electromagnetic showers. About half of the incident hadron energy is passed on to additional secondaries. The remainder is consumed in multiparticle production of slow pions and in other processes. The phenomena which determine the development of the hadronic showers are: hadron production, nuclear deexcitation and pion and muon decays. Neutral pions amount, on average to 1/3 of the produced pions and their energy is dissipated in the form of electromagnetic showers. Another important characteristic of the hadronic shower is that it takes longer to develop than the electromagnetic one. This can be seen by comparing the number of particles present versus depth for pion and electron initiated showers. The longitudinal development of hadronic showers scales with the nuclear interaction length:

${\displaystyle \lambda =\frac{A}{N_A{\sigma }_{\mathrm{a}\mathrm{b}\mathrm{s}}}}$

The lateral shower development does not scale with λ.