Electromagnetic Shower

Electron-photon cascades can be produced in the atmosphere through the interaction of primary γ rays with atoms in the air. The main air-shower interactions are bremsstrahlung and pair production processes (Figure 3.2).

Bremsstrahlung is the radiation associated with the acceleration of elec-trons in the electrostatic fields of the nucleus of atoms. The radiation length, X0 [g/cm²], is the distance over which the initial energy of a cascade particle is reduced by a factor of 1/e by bremsstrahlung. In atmosphere the radiation length is 37 g cm⁻².

The Pair production process is one of the main particle production me-chanisms in electromagnetic showers. Photons having energies higher than twice the rest mass of an electron (rest mass m_e of an electron is 0.511 MeV, where c=¯h=1) are converted into an electron-positron pair in the presence of a nucleus, which guaranties the energy and momentum conservation. As for bremsstrahlung process, the radiation lengthfor the pair production pro-cess, Xpair, can be defined. If the radiation lengths for pair production and bremsstrahlung for ultra relativistic electrons are compared, it can be found that X_pair ' X₀. This reflects the similarity of the bremsstrahlung and pair production according to quantum electrodynamics. This means that γ-ray induced cascades start to develop in the atmosphere at rather high altitu-des (10 - 20 km) above the ground level (the total atmospheric depth is

gamma ray

Abbildung 3.2: Main air-shower interactions.

1030 g cm⁻²). The secondary electrons or positrons produced through pair-production mechanism may interact further with ambient nuclei giving off photons via bremsstrahlung. The photons produced in bremsstrahlung pro-cesses may again pair-produce into secondary electrons. Thus finally a cas-cade of electrons and photons develops in the atmosphere.

While the shower propagates, the number of cascade particles increases, whereas the energy per particle simultaneously decreases. The average energy per particle is given as

E = E₀ exp and the characteristic radiation length, respectively. The atmospheric depth can be expressed in terms of the atmospheric height, h, as follows:

p(h) = p₀ exp − h h₀

!

, (3.2)

where p₀ = 1030 g cm⁻², and h₀ = 8 km. The cascade grows exponentially and finally reaches its maximum of the order of 10³ particles for a 1 TeV γ-ray-induced shower. The shower maximumoccurs at the atmospheric height between 7 and 10 km above the sea level.

At further stages of the cascade development the ionizing collisions with atomic electrons becomes a dominant process for the remaining cascade par-ticles. The ionization depends mainly on the electronic binding energy of atoms in the absorbing material (i.e. the air) and not on the energy of the cascade particle. The critical energy, E_c, at which the radiation loss equals the collision loss is∼80 MeV for air. For cascade particles with energies below Ec the collision-ionization loss becomes dominant over the radiation loss.

Abbildung 3.3: A model of electromagnetic shower according to Heitler [85].

The figure is taken from [21].

The small transverse momentum of the secondary electrons inγ-ray sho-wers cause the electromagnetic cascades to be beamed along the direction of the primary photon. Multiple Coulomb-scattering of cascade electrons deter-mine the lateral distribution of the shower. The radial spread is deterdeter-mined by the radiation length and the angular deflection per radiation length at the cri-tical energy. This spread is of order one Moliere unitX_M = 21(X₀E_c) ' 9.6 g cm². Therefore, the shower particles are moving within a cone around the shower axis. This cone has a radius of 80 - 120 m at sea level, and it contains 90%of the total energy of the shower.

The Heitler model ([85]) is a simplified picture (Figure 3.3) of the lon-gitudinal shower development. According to this model, the electromagnetic cascade begins with pair-production and continues in the second step with bremsstrahlung. This two-step process repeats itself until the critical energy, E_c, is reached. After that the production processes begin to diminish because of lack of energy. Therefore, it is expected that the number of particles pro-duced in the second step is doubled. As a result, the total amount of particles in the shower is

N = 2ⁿ , (3.3)

after n radiation lengths. At each step the energy is equally distributed bet-ween the secondary particles. So, the energy of the shower particles after n radiation lengths can be expressed as follows

E_n = E₀ 2⁻ⁿ , (3.4)

which is illustrated in Figure 3.3. It was assumed that after n_max radiation lengths the energy drops down to the critical energy, Ec. The maximum

number of radiation lengths, n_max is thus

Using Equations 3.3 and 3.5, the maximum number of particles, N_max, pro-duced after n_max radiation lengths can be found as

N_max ≈ E₀

E_c , (3.6)

where (1/3)N_max are photons and (2/3)N_max are electrons and positrons.

The position of the shower maximum in the atmosphere can be found in terms of n_max and the radiation length X₀, which is given as

Xmax = nmaxX0 . (3.7)

Substituting Equation 3.5 into Equation 3.7 the atmospheric depth of shower maximum can be deduced as

X_max = X₀

Using Equation 3.6, and Equation 3.8, one can also show X_max = X₀

ln 2 ln (N_max) . (3.9) The relation of number of particles to the atmospheric depth for showers with different energies is given in Figure 3.4.

Although the simplified shower model by Heitler explains the basic pro-perties of the shower rather well, a more detailed description of the shower dif-fers from this model. It is known as theNishimura-Kamata-Greisen-Formalism.

More detailed theory of longitudinal shower development can be found in [76].

3.1.3 Differences between Hadron- and Gamma-ray