Electromagnetic Showers

The energy loss mechanisms governed by the electromagnetic force are well-described by the theory of Quantum Electro-Dynamics and can be calculated with high precision. It is therefore possible to make relatively accurate predictions and develop reliable electromagnetic shower models. Such models (electromagnetic and hadronic, see Section 3.1.3) are used in Monte-Carlo simulations to parametrize how showers evolve within different absorber materials.

Electromagnetic cascades are initiated by electrons (positrons) or photons, but the nature of the first electromagnetic interaction is intrinsically different. For high energetic electrons, the most probable process is the creation of a photon through bremsstrahlung, whereas for high energetic photons, the relevant process is the production of an electron- positron pair. Below a photon energy of∼1MeV, which is equivalent to the rest mass energy of an electron-positron pair, the dominant processes are Compton scattering and the Photo Effect. The cross sections of the respective processes depend, among others, on the electron density (and therefore on the atomic number Z) of the traversed absorber.

Pair Production: For energies of > 1MeV, a photon may produce an electron-

positron pair in the electromagnetic field of an absorber nucleus. It becomes the dominant process above 5−10MeV in the most common absorbers. The cross section for Pair Production σpair rises with energy and reaches a plateau at photon

energies of the order of 1GeV.

Compton Scattering: Compton scattering plays a role in an intermediate energy

range between ∼ 100keV−5MeV. The cross section σcompton scales with the

Absorber ρ [g/cm3_{] X} 0 [cm] RM [cm] λI [cm] λI/X0 ∆Eioniz./λI [MeV/cm] Fe 7.87 1.76 1.69 16.8 9.55 90.9 W 19.3 0.35 0.93 9.6 27.43 162 Pb 11.3 0.56 1.6 17 30.36 173

Table 3.1: Key properties of different absorber materials relevant for calorimetry [38]. The average energy loss of minimum ionizing hadrons is shown by the rightmost column. See text for more details.

(O(1GeV)) impinging photon or electron is absorbed within the electromagnetic cascade by Compton scattering underlines its importance [38]. In the process, a photon scatters with one atomic shell electron passing a fraction of its energy to it. In the following, the electron is unbound and continues the passage - like the scattered photon - through the absorber. In electromagnetic cascades, the photon energy is reduced in a sequence of Compton scatterings until it can be absorbed by the Photo Effect.

Photoelectric Effect: The Photo Effect is dominant at low photon energies. Within

the most common absorbers, the cross section σphoto for the Photo Effect exceeds

all other effects at photon energies below 0.1−1MeV (e.g. for < 0.7MeV in Uranium and for <0.1MeV in Iron) [38]. Note that σphoto decreases rapidly with

increasing energy as E−3_{. In the process, the photon is absorbed by the atomic}

shell of an absorber atom which gets excited and releases this excess by the emission of an electron.

Scaling Variables of Electromagnetic Showers

The dimensions of an electromagnetic shower within a certain absorber material are characterized by the radiation length X0 and the Moli`ere Radius RM. X0 is defined as

the average distance z over which the energy of a high energetic electron or positron (>1GeV) is reduced to 1/e= 36.8 % due to bremsstrahlung:

E =E0·exp( −z X0

) (3.2)

In the case of photons, it is 7_{/9th of the mean free path before another pair production}

process occurs [38]. The radiation length X0 depends on the atomic number Z of the

traversed absorber. In iron it amounts to 1.76cm, while it is only 0.56cm in lead (see Table 3.1).

The Moli`ere Radius RM describes the transverse extension of electromagnetic showers.

In an empirical parametrization it can be estimated as:

RM = 21.2MeV·

X0

Ec

whereEcis the critical energy. On average,90 %of the total energy of an electromagnetic

shower is contained within a cylinder with radiusRM around the lateral impact position

of the impinging particle. Note thatRM scales differently in different absorber materials

relative toX0. While the radiation length is more than 3 times longer for iron compared

to lead, the Moli`ere Radius is approximately equal.

The Development of Electromagnetic Showers

When a highly energetic electron (or positron) enters a dense absorber it may radiate thousands of photons through bremsstrahlung. While the majority of those photons is low-energetic and gets absorbed, a small fraction carries a substantial amount of energy. Such photons create further electrons and positrons through pair production, which in turn generate more photons through bremsstrahlung. This multiplication process initiates the electromagnetic cascade. The shower energy is deposited through the ionization of the traversed medium.

At a certain depth after the starting point of the electromagnetic cascade, the number of shower particles created per unit length and consequently the amount of deposited energy reaches a maximum and decreases afterwards. At lower energies, photons are more likely to produce one instead of two particles through Compton scattering or the Photo Effect. Electrons (or positrons) tend to deposit their energy through the ionization of the surrounding medium instead of generating new photons. Low energetic positrons annihilate with electrons of the absorber atoms and generate two photons (Eγ = 511keV) which loose their energy through a series of Compton scatterings ending

with the photoelectric absorption. Eventually, the remaining low energetic electrons get absorbed by the traversed medium and the electromagnetic cascade ceases. Figure 3.3 (left) shows a simplified schematic of the development of an electromagnetic shower.

Note that a high fraction of the total shower energy is deposited through relatively low energetic electrons and positrons (e.g. ∼40 % for energies of <1MeV and∼65 %

for <4MeV in uranium). Furthermore, one finds that approximately one quarter of the total energy is deposited by shower positrons and three quarters by electrons [38]. Together, this corroborates the assumption that most of the shower energy is deposited through electrons created through Compton scattering and the Photo Effect and that these are the most abundant processes in electromagnetic cascades.