When an electron passes through matter, it emits photons from bremsstrahlung which in turn may create electron positron pairs. This is an iterative process where a primary particle creates secondary particles which create more particles. Hence, this is called a particle shower or cascadeand is important for neutrino detection in IceCube. The main features can be explained in a simplistic model formulated by Walter Heitler, in which photons, electrons, and positrons are the only involved particles interacting via the electromagnetic force [200]. A primary electron with energy E0interacts via bremsstrahlung after one radiation length X0 and produces a high- energy photon. In this simple model, the number of particles is doubled after each propagation length, and the energy is split equally among all particles. This multiplication process stops when the energy per particle crosses the critical energy Ec, below which the radiative energy losses due to bremsstrahlung become smaller than the ionization losses. For electrons in ice, this is Ec≃ 72 MeV, which can be obtained by setting ⟨dE/dX⟩brems=⟨dE/dX⟩ion.
In this model, the number of particles is given by N(t) = 2t and the energy by E(t) = E 0/2t with t = X/X0. The maximum number of particles is Nmax = E0/Ec at the maximum shower depth tmax = log2(E0/Ec). Although the Heitler model has simplistic assumptions, it shows two important features that are phenomenologically true for all particle showers. First, the maximum number of shower particles increases linearly with the primary energy Nmax ∼ E0. This is important in IceCube as all charged particles of the shower emit Cherenkov light (in the limit β ≃ c, see Section 3.1.4). Consequently, the Cherenkov light yield scales linearly with the primary energy. Second, the maximum shower depth scales logarithmically with the primary energy tmax ∼ log E0. This is important in IceCube because it means that all particle showers over a wide energy range are extremely small (a few meters) with respect to the dimensions of the detector (one kilometer). This feature causes the event topologies in IceCube to be very different for secondary electrons and muons (see Section3.3.1).
In a more realistic model than the one by Heitler, the longitudinal energy loss profile of a particle shower is parametrized by
dE
dt = E0b
(bt)a−1e−bt
Γ (a) , (3.5)
describing a steeply rising edge and a slow decrease after the maximum [78]. The shower maximum can be calculated via tmax = (a− 1)/b. The dimensionless parameters a and b are determined experimentally and depend on the target material (see below). The transversal energy loss profile is characterized by the Molière radius RM ≃ 21 MeV X0/Ec, which on average contains 90% of the total deposited shower energy.
In IceCube, there are two distinct types of particle showers: electromagnetic cascades and hadronic cascades. An electromagnetic cascade can generally be described more precisely, because the interactions only involve electrons, positrons, and photons interacting via the elec- tromagnetic force. In contrast, calculations for hadronic cascades are more complex, because many different secondary particles can be produced and the cross sections involved have larger uncertainties. Secondaries can be baryons or mesons and interact via the strong, electromag- netic, or weak forces. The production of neutral pions is an important feature of hadronic cascades as the immediate decay π0→ 2γ feeds an electromagnetic subpart of the shower. Both electromagnetic and hadronic showers can be approximately described in the same way in IceCube. The longitudinal energy loss profile in Equation 3.5 is used for both electromagnetic and hadronic cascades with different values for the shower parameters a and b. Originally, they were determined experimentally for electromagnetic cascades in water [201]. In a more recent study using GEANT4 simulations, these shower parameters have been fitted for different primary particles inducing electromagnetic or hadronic showers [202]. For an electromagnetic cascade induced by an electron, they are a = 2.02 + 0.63 log(E0/GeV) and b = 0.63, and for a hadronic cascade induced by a charged pion, they are a = 1.81 + 0.39 log(E0/GeV) and b = 0.34. Note that the electromagnetic radiation length in Equation 3.4 is used for both showers, although the nuclear interaction length for hadronic showers is generally larger than that. However, only the parts of the shower that emit Cherenkov light are detectable in IceCube. The light yield of hadronic cascades is lower than that of electromagnetic cascades. One reason is that a considerable fraction of neutral particles or slowly moving charged particles do not emit Cherenkov light. Furthermore, the production threshold of hadrons is higher than of electrons, positrons, and photons. The amount of energy that is proportional to the Cherenkov light yield is called the visible energy. It is calculated by convolving the energy of the primary particle with the relative light yield of hadronic cascades compared to electromagnetic cascades. The relative light yield is parametrized by
f = 1.− (E0/0.399 GeV)−0.130(1.− 0.467), (3.6)
and increases with the primary energy due to the growing ratio of π0. It is between ∼ 60 − 95 % within the energy range of IceCube [203].
0 5 10 15 20
Distance from Vertex [X0]
0.00 0.05 0.10 0.15 0.20 0.25 1 E0 d E d X [ 1 ]X0
Longitudinal Electromagnetic Shower Profile
1 GeV 10 GeV 100 GeV 1 TeV 10 TeV 100 TeV 1 PeV 0 5 10 15 20
Distance from Vertex [X0]
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 1 E0 d E d X [ 1]X0
Longitudinal Hadronic Shower Profile
1 GeV 10 GeV 100 GeV 1 TeV 10 TeV 100 TeV 1 PeV 0 2 4 6 8 10 12 14 16
Distance from Vertex [m]
0.0 0.1 0.2 0.3 0.4 0.5 0.6 1 E0 d E dx [ 1]m
Longitudinal Electromagnetic Shower Profile
1 GeV 10 GeV 100 GeV 1 TeV 10 TeV 100 TeV 1 PeV 0 2 4 6 8 10 12 14 16
Distance from Vertex [m]
0.00 0.05 0.10 0.15 0.20 0.25 1 E0 d E dx [ 1]m
Longitudinal Hadronic Shower Profile
1 GeV 10 GeV 100 GeV 1 TeV 10 TeV 100 TeV 1 PeV
Figure 3.4: Longitudinal shower profiles for visible energy losses of an electromagnetic cascade (left) and a hadronic cascade (right). Shown is the relative differential deposition of visible energy for different primary energies as a function of the distance from the shower vertex using the radiation length in Equation 3.4, the longitudinal shower parametrization in Equation 3.5, and the light scale factor for hadronic cascades in Equation 3.6. Note the different scale of the axes due to the substantially lower light yield of hadronic cascades.
As an example, the longitudinal energy loss profiles for electromagnetic and hadronic cascades are shown in Figure3.4for different primary energies. It can be seen that the shower maximum is displaced from its origin by a few meters, depending on energy. The shower elongation extends with increasing energy, whereas hadronic cascades are more elongated than electromagnetic cascades. It can also be seen that the relative amount of visible energy in a hadronic cascade is generally lower than in an electromagnetic cascade due to the neutral particles in the shower not being visible. However, the visible energy increases with the shower energy due to the light scale factor that accounts for a larger contribution of electromagnetic subshowers.