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other energy losses, such as ionisation or radiative losses described in Section 4.3. However, this effect plays a crucial role in the detection of neutrinos by the IceCube detector.

4.3

Propagation of particles through the ice

4.3.1

Energy losses of particles

When particles travel through matter they lose their energy via various interaction processes. Ionisation, Bremsstrahlung, pair production and photo-nuclear interac- tions are the main contributors to the losses [114]. The total energy loss is a sum of the individual losses

− dE dx = ( dE dx ) I +( dE dx ) B +( dE dx ) P P +( dE dx ) N (4.6) The fraction of energy lost through each of these mechanisms depends on the particle type and its energy. The mechanisms relevant to Cherenkov neutrino detectors are presented in the following sections.

4.3.2

Muons in ice

For muons with energies above 1 GeV losses due to ionisation have a weak energy dependence. On the other hand, the losses due to Bremsstrahlung, pair production and photo-nuclear interactions together form radiative losses, which are rising with energy. Therefore, the energy losses can be simplified [114] as

−dE

dx = aI(E) + bR(E)· E, (4.7) where aI(E) corresponds to ionisation losses and bR(E) = bB(E) + bP(E) + bN(E)

characterise radiative losses. Different components of muon energy losses as a func- tion of its kinetic energy are depicted in Figure 4.2. Parameters aI, bB, bP, bN have

rather weak energy dependence and, therefore, can be assumed constant for the energy range relevant to this work. Then, Equation (4.7) can be simplified as

− dE_dx ≈ a + b · E, (4.8) where a≈ 2 MeV/cm and b ≈ 3.4 · 10−6 _cm−1 _{[114]. In this way the average range}

(track length) R of a muon with energy E is calculated as R = 1 b · ln ( b a · E + 1 ) . (4.9)

Ranges of individual muons have rather significant variations from Equation (4.9) due to the stochastic nature of radiative losses. The average distance travelled by

10-2 ₁₀-1 ₁₀0 ₁₀1 ₁₀2 ₁₀3 ₁₀4

Muon kinetic energy [ GeV ]

100

101

- dE/dx [ MeV / cm ]

Critical energy

Radiative losses are 1%

Muons in ice with

ρ

.

/

Ionisation losses

Total radiative losses

Total losses

Figure 4.2: Muon energy losses in the Antarctic ice as a function of kinetic energy. The solid curve depicts total energy losses, while dashed and dash-dotted lines represent ioni- sation and radiative losses, respectively. Additionally, the energy of minimum ionisation, critical energy and the energy where radiative losses are 1% of the total losses are shown in the figure. The data from [114] are used for the figure.

a muon track in Antarctic ice is about 47 meters at 10 GeV and 1.6 km at 500 GeV. This length is much larger than typical sizes of cascades produced by other particles, as described in the next sections. For muons, the critical energy, where radiative losses start to dominate, is about 1.03 TeV, which is outside of the energy range considered in this work.

4.3.3

Electromagnetic showers

Electrons, positrons and photons can be produced directly in neutrino interactions, as well as in interactions or decays of the secondary particles. Electrons lose their energy by emission of hard photons in Bremsstrahlung processes, while photons produce e+_e− _{pairs when travelling through matter. These processes happen repet-}

itively and collectively lead to the formation of electromagnetic (EM) showers. The development of EM showers stops when the energies of individual particles reach their critical energy Ec. Its value for water and ice is approximately 77 MeV with a

slight difference for electrons and positrons [23] due to the differences in the anni- hilation cross-section. Another important variable that describes the development of the shower is the radiation length X0, which defines the distance after which the

energy of an electron or a positron is reduced by a factor of 1/e. Also, it charac- terises 7

9 of the mean free path for pair production of high-energy photons. For ice,

the radiation length is approximately 39.3 cm [23].

The longitudinal profile of EM showers can be parametrised [23] by a gamma distribution

dE

dt = E· b ·

(b_{· t)}a−1_{exp(−b · t)}

4.3. PROPAGATION OF PARTICLES THROUGH THE ICE 55 where t = x/X0, and the factors a and b can be fitted from direct particle propaga-

tion simulations described in [115] as

a≈ 2.01 + 0.63 ln E [ GeV ], b ≈ 0.63 (e+_{, e}−

),

a_{≈ 2.83 + 0.58 ln E [ GeV ], b ≈ 0.64} (γ). (4.11) The value tmax corresponding to the maximal energy deposition is given by

tmax =

a_{− 1}

b . (4.12)

Therefore, Equations (4.10) and (4.12) lead to a logarithmic growth of the cascade size as a function of its energy. EM showers are significantly smaller than muon tracks of comparable energy (see Section 4.3.2). For a 10 GeV photon-induced cas- cade, 90% of the energy is deposited within 4.2 meters, while for 500 GeV this length is only 6 meters. Electron-induced showers are slightly smaller due to differences in the point of the first interaction.

The particles in EM showers travel mainly along the direction of the initial particle, leading to the Cherenkov light emission that is peaked at the Cherenkov angle, as shown in the left part of Figure 4.3.

4.3.4

Hadronic showers

As discussed in Section 4.1, hadrons are almost always produced in neutrino interac- tions with ice at the energies relevant to this work. Hadrons interact strongly with nucleons of matter and produce secondary particles. Variability of the produced particles, their interactions and decays lead to large shower-to-shower variations in longitudinal profiles and Cherenkov light emission. For example, π0 _{particles pro-}

duced in the interactions decay almost immediately into a pair of photons creating

1.0 0.5 0.0 0.5 1.0 cosΦ 10-3 10-2 10-1 100 1 N dN d Cherenkov angle

Electron-induced shower

π+-_{induced shower,E=10 GeV}

π+-_{induced shower,E=1000 GeV}

101 ₁₀2 ₁₀3 ₁₀4 Energy [ GeV ] 0.4 0.5 0.6 0.7 0.8 0.9 1.0 F factor

F factor, π+-induced showers

68% variation

Figure 4.3: (Left) Angular distributions of Cherenkov light produced by electron- and π+-induced showers. Two energies are depicted for π+_{showers to demonstrate the energy} dependence of the angular distribution, while energies of EM showers have little impact on their Cherenkov light angular profiles. (Right) The factor F (E), which describes the brightness of hadronic showers compared to EM ones, and its 68% variation for π+_-induced showers. Parametrisations from [115] are used for both figures.

Table 4.1: The values used to parametrise hadronic light output of charged pions by Equations (4.13) and (4.14). The values for other particles can be found in [115].

Particle Es [ GeV ] f0 m σ0 γ

π+ _{0.156 0.273 0.158 0.406 1.018}

π− _{0.134 0.287 0.153 0.433 1.056}

an electromagnetic component of hadronic showers. Charged pions decay producing muons, and produce less Cherenkov light. Also, hadronic showers can have neutral particles or heavy hadrons with energies below the Cherenkov threshold, leading to a component that does not emit any light.

On average hadronic showers have slightly larger sizes than EM cascades of the same energy. For example, a 10 GeV shower produced by π+ _{deposits 90% of energy}

within 5.6 meters, for 500 GeV this size grows to 8 meters. Different hadrons produce showers with slightly different longitudinal and lateral distributions.

Similarly to EM showers, the light from hadronic showers is also peaked around the Cherenkov angle. However, the light is more smeared due to larger masses and the variability of individual hadrons in the showers. The angular distributions of the Cherenkov light produced by showers induced by π+ _{with energies of 10 and}

1000 GeV are shown in Figure 4.3. Detailed parametrisations of the shower light outputs for various types of hadrons can be found in [115, 116].

Hadrons are heavier than electrons and, therefore, have a higher energy threshold for Cherenkov light emission. The brightness of hadronic showers is given by Thadron,

a total length of all charged particles above the threshold for light emission. It is useful to introduce a factor F (E), which defines a relative brightness of hadronic showers compared to EM cascades. This factor can be parametrised [115, 116] as

F (E) = Thadron TEM = 1_{− (1 − f}0) ( E Es )−m (4.13) with a variance given by

σF(E) = σ0(ln E)−γ. (4.14)

Values of parameters Es,f0, m, σ0 and γ are different for hadronic showers initiated

by different particles. The corresponding values for charged pions are given in Ta- ble 4.1, while parametrisations for other particles can be found in [115]. The factor F and its variation as a function of energy for showers induced by π+ _{are shown in}

the left part of Figure 4.3.