Interaction of Charged Particles with Matter

Charged particles change their energy and direction in matter. There are several pro-cesses contributing to these effects. All of them are based on the same phenomena which is the electromagnetic interaction with electrons and nuclei. The electromag-netic interaction is responsible for elastic and inelastic particle scattering, ionization and excitation of atoms and bremsstrahlung. Cherenkov and transition radiation are among those processes as well but result in a negligible energy loss and do not change the particles direction. On the contrary, ionization is the dominating process with the main contribution to the energy loss for heavy charged particles. For electrons and positrons bremsstrahlung comes into play as dominating process. The heavier the particle, the more crucial is the energy loss due to inelastic scattering. In such a hard collision a certain fraction of the kinetic energy is transferred causing an ionization.

Thereby so-calledδ electrons are produced. These are free electrons which are able to ionize further atoms. The energy loss by a heavy spinless particle can be deter-mined by the Bethe-Bloch equation [23]. Considering a particle with atomic number Z, velocityv =βcand kinetic energy, the energy loss is given by

− dE

dx

ion

=Kz²Z A

1 β²

1

2ln2m_ec²β²γ²_max

I² −β²− δ 2

, (1.4)

withK = 4πN_Ar_e²m_ec² = 0.307075 MeV/(g/cm²),γ =E/M c²and whereIis the average ionization potential andδa correction on the density-effect. In this notation z is the charge in units of the electron charge,Ais the atomic mass andM the mass of the incoming particle, whiler_e is the classical electron radius andm_eis the elec-trons mass. The parameter I is correlated with Z and in the order of eV, while the parameterδcan be considered as extension to the original Bethe-Bloch equation and gets dominant for very low and relativistic energies.

For different materials the ionization energy loss rate as function of the relativistic particle velocity is shown in Fig. 1.19. For low energies the various dependences

Figure 1.19: Ionization energy-loss rate in different materials [24].

have a common fast increase, as 1/β². In the region around βγ ≈ 3 the energy-loss rate has a wide minimum and increases slowly from there. Particles

kinemati-cally located near the minimum of the energy-loss rate are calledMinimumIonizing Particles (MIPs). The ionization losses of MIPs are almost the same for most materi-als and within a range between1−2 MeV/(g/cm²). This is the reason why usually large absorbers with high densities are mandatory to stop MIPs. A well-known ex-ample for MIPs are cosmic muons which penetrate a large amount of material.

As already mentioned, the emission of Bremsstrahlung gives a significant contribu-tion to the energy loss for light charged particles like electrons and positrons. This process plays an important role in the development of an electromagnetic shower which will be discussed later. The contribution to the energy loss is only relevant for electrons and positrons because the cross section of bremsstrahlung is dependent on the mass, as1/m². The radiation emission energy loss of an electron or positrons with a massm_eat an initial energyE₀ is given by

dE

dx =N E₀Φ_rad, (1.5)

with

Φ_rad =







4Z²r²_eα ln_m^2E0

ec² −¹₃ −f(Z)

form_ec² E₀ 137m_ec²Z^−1/3 4Z²r²_eα ln 183Z^−1/3

+ ₁₈¹ −f(Z)

forE₀ 137m_ec²Z^−1/3

(1.6) whereα = Z/137, N is the number of atoms percm³, Z is the atomic number of the material andf(Z)is a correction function which takes into account the Coulomb interaction of the emitting electron in the field of the nucleus. Furthermore, Eq. 2.5 distinguishes two cases: no screening and a completely screened electric field of the nucleus by the surrounding bound electrons. In general, bremsstrahlung depends on the screening caused by atomic electrons because it occurs in the Coulomb field of the nucleus. But bremsstrahlung can also occur in the field of atomic electrons. In that case the termZ² in Eq. 2.5 has to be replaced byZ(Z + 1).

The radiation lengthX₀ characterizes radiation losses and gives a more convenient

way to formulate Eq. 1.5. In terms ofX₀ the equation can be expressed as

− dE

dx

rad

= E X0

. (1.7)

Hence, the radiation length is defined as the mean pathlength in a layer of material af-ter which the electron energy decreases to1/eof its initial energy. It can be regarded as the natural unit of absorber thickness. Values of X₀ dependent on the material can be calculated by solving Eq. 1.7. In a common approximation [25] the radiation length is given by

X0 = 716.4 g/cm² A Z(Z + 1) ln

287/√

Z , (1.8)

which was found to be accurate within2.5%for almost every material.

Most scintillator materials are mixtures or compounds for which the radiation length can be approximated by

X₀ = 1

PρiX₀ⁱ , (1.9)

whereρiis the mass fraction of thei^thcomponent with its radiation lengthX₀ⁱ. A comparison of the specific radiation losses,−(dE/dx)_rad, and the ionization losses,

−(dE/dx)_ion, respectively Eq. 1.7 and Eq. 1.4, reveals that at high energies the lin-ear rise of the radiative loss dominates the logarithmic rise of the ionization loss.

For absorbers with high atomic number Z the effect sticks out even more, since it contributes quadratically to specific radiation losses. The energy at which specific radiation losses and ionization losses are equal is called critical energyE_c. For solids and liquids the critical energy for electrons can be approximated [23] and is given by

E_c= 610 MeV

Z+ 1.24. (1.10)

The Cherenkov effect causes additional energy losses in a radiator via emission of coherent Cherenkov light. The effect was already discussed for a radiator such as

fused silica in thePANDA-DIRC detector (see paragraph 1.2.1.3). The effect is not negligible for the lead tungstate, PbWO4 (PWO-II) crystals which are used in the PANDAEMC. The detected energy information of a PWO-II crystal shows a small fraction originated from Cherenkov radiation. If a particle with chargez·eirradiates a medium, the number of photons emitted per distance can be calculated by

dN

dx = 2πz²α²sin²θ_c Z λ2

λ1

1

λ²dλ, (1.11)

where the Cherenkov angleθ_c is defined by Eq. 1.3. The evaluation of the integral over the range of sensitivity of a typicalAvalanchePhotoDiode (APD) used in the PANDAEMC leads to

dN dx

850 nm

350 nm

= 771·z²sin²θc

photons

cm . (1.12)

Hence, in case of PWO-II, a rather small number of only a few hundred of photons per cm originate from Cherenkov radiation. This corresponds to approximately a fifth of the overall light output at+18^◦Cfor a MIP in PWO-II.

Furthermore, the energy loss due to elastic scattering on nuclei is negligible low as well. In that case, the mass of the absorber atom is much higher than the mass of the incoming particle which results in low momenta transfer. The contribution becomes only relevant at very low velocities β < 10⁻³ and is important for the detection of neutrons.

Above all, a nuclear reaction can occur, if a particle is able to overcome the Coulomb barrier. However, the cross section for a nuclear reaction is generally small compared to electromagnetic cross sections. Therefore, the mean free pathlength of the nuclear reactionΛ_{N R}, which is defined as the distance after which the number of particles is reduced by 1/e because of nuclear reactions, is larger than the electromagnetic radiation lengthX₀. For this reason, electromagnetic processes are dominant.