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Passage of Particles Through Matter

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Passage of Particles Through Matter C4: Particle Physics Major Option Passage of Particles Through Matter Author: William Frass Lecturer: Dr. Roman Walczak Michaelmas 2009 1 Contents 1 Passage of charged particles through matter3 1.1 Energy loss by ionisation.....................................3 1.1.1 Energy loss per unit distance: the Bethe formula...................3 1.1.2 Observations on the Bethe formula...........................5 1.1.3 Behaviour of the Bethe formula as a function of velocity...............5 1.1.4 Particle ranges......................................7 1.1.5 Back-scattering and channelling.............................8 1.1.6 The possibility of particle identification........................9 1.1.7 Final energy dissipation.................................9 1.2 Energy loss by radiation: bremsstrahlung........................... 10 1.2.1 In the E-field of atomic nuclei.............................. 10 1.2.2 In the E-field of Z atomic electrons.......................... 10 1.2.3 Energy loss per unit distance and radiation lengths.................. 11 1.2.4 Particle ranges...................................... 11 1.3 Comparison between energy loss by ionisation and radiation................. 12 1.3.1 The dominant mechanism of energy loss........................ 12 1.3.2 Critical energy, EC .................................... 12 1.4 Energy loss by radiation: Cerenkovˇ radiation......................... 14 2 Passage of photons through matter 15 2.1 Absorption of electromagnetic radiation............................ 15 2.1.1 Linear attenuation coefficient, µ ............................ 15 2.1.2 Partial cross-sections................................... 15 2.2 Photoelectric effect........................................ 15 2.2.1 Summary......................................... 15 2.2.2 Work function...................................... 15 2.2.3 Cross-section....................................... 16 2.3 Compton scattering....................................... 16 2.3.1 Summary......................................... 16 2.3.2 Compton shift formula.................................. 17 2.3.3 Cross-section....................................... 18 2.4 Pair-production.......................................... 18 2.4.1 Summary......................................... 18 2.4.2 Reaction rate....................................... 19 2.4.3 Cross-section....................................... 19 2.5 Overall cross-section for photons................................ 20 2.6 Comparative summary of photon attenuation phenomena.................. 20 3 Passage of hadrons through matter 21 3.1 Hadron-nucleus interactions................................... 21 3.2 Quasi-elastic interactions.................................... 21 3.3 Inelastic interactions....................................... 22 3.3.1 Inelastic cross-section.................................. 22 3.3.2 Absorption length, λ ................................... 22 4 Development of showers in matter 23 4.1 Electromagnetic showers..................................... 23 4.1.1 Modelling an electromagnetic shower.......................... 23 4.1.2 Total shower depth.................................... 24 4.2 Hadronic showers......................................... 24 4.2.1 Modelling a hadronic shower.............................. 24 4.2.2 Total shower depth.................................... 25 2 1 Passage of charged particles through matter 1.1 Energy loss by ionisation A charged particle moving through different types of matter will interact with it in different ways, losing energy in various ways. • Solids, liquids and gases: the charged particle will ionise the material's atoms. • Semiconductors: the charged particle will produce electron-hole pairs. • Scintillator: the charged particle will lead to the production of light. • Superheated liquid: the charged particle will leave a trail of bubbles. In the limit that all electrons in the material's atoms can be modelled as free, the kinetic energy (EKE) loss per unit distance travelled (dx), known as the stopping power is: dE 1 − / (1) dx β2 Here, β = v=c. This relationship can be understood by looking at the Bethe formula1, derived in the following section. 1.1.1 Energy loss per unit distance: the Bethe formula Charged particles like alpha-particles, protons and pions, scatter electrons according to Rutherford scattering. Therefore scattering of the charged particles by free electrons is the same process, just in the rest frame of the electrons. • The Rutherford scattering formula is improved by the Mott scattering formula for electrons of momentum p and velocity v and particles with charge Ze: 2 2 dσ Zα~c 4 θ v 2 θ = cosec 1 − 2 sin (2) dΩ Mott 2pv 2 c 2 • Convert the scattering angle θ into a momentum transfer q: q θ = p sin (3) 2 2 θ q sin = (4) 2 2p θ q 2 sin2 = (5) Figure 1: Momentum triangle. 2 2p 4 θ 2 θ • Using (5), the cosec =2 and sin =2 terms in the Mott scattering formula, (2), can be replaced: 2 4 2 2 dσ Zα~c 16p v q = 4 1 − 2 2 (6) dΩ Mott 2pv q c 4p 1The term "Bethe-Bloch formula" actually refers to the modified Bethe formula that includes Bloch's Z4 correction. 3 • Now convert the differential cross-section, dσ=dΩ into dσ=dq2: Replace the differential solid angle, dΩ, in (6) using dΩ = 4π sin(θ)dθ: dσ Zα c2 16p4 v2 q2 = 4π ~ 1 − (7) sin(θ)dθ 2pv q4 c2 4p2 The operator sin(θ)dθ can be obtained by differentiating (5) in the form q2 = 4p2 sin2(θ=2) to give: dq2 θ θ 1 θ θ = 4p2 sin cos = 4p2 sin + = 2p2 sin(θ) (8) dθ 2 2 2 2 2 dq2 sin(θ)dθ = (9) 2p2 Substituting (9) into (7) and cancelling terms yields: dσ Zα c2 v2 q2 = 8π ~ 1 − (10) dq2 q2v c2 4p2 • Now transfer to the frame in which the electron is at rest, with the charged particle moving towards it: pelectron ! pnucleus = mevγ (11) 2 2 2 dσ Zα~c v q 2 = 8π 2 1 − 2 2 (12) dq q v c 4(mevγ) • The momentum transfer squared is the same in both frames and is related to the energy transfer, δ as follows: 2 2 q = 2me δ ! dq = 2me dδ (13) 2 2 dσ Zα~c v 2meδ = 8π 1 − 2 2 (14) 2me dδ 2meδv c 4(mevγ) 2 dσ Zα~c meδ = 16πme 1 − 2 2 2 (15) dδ 2meδv 2mec γ • So the cross-section for energy loss in the range δ ! δ + dδ is: 2 2 dσ 4π Zα~c δ v = 1 − 2 1 − 2 (16) dδ me δv 2mec c • If the charged particle loses energy −dE in a distance dx in a material containing n atoms of atomic number Zmaterial per unit volume then: number of electrons energy loss −dE = × (17) per unit volume per unit volume energy loss −dE = nZ × dx (18) material per unit area dE Z δmax dσ − = nZmaterial δ dδ (19) dx δmin dδ 4 • Substitute (16) into (19) and integrate: Z δmax 2 2 dE 4π Zα~c δ v − = nZmaterial δ 1 − 2 1 − 2 dδ (20) dx δmin me δv 2mec c 2 Z δmax 2 dE 4π Zα~c 1 1 v − = nZmaterial − 2 1 − 2 dδ (21) dx me v δmin δ 2mec c 2 2 dE 4π Zα~c δmax δmax − δmin v − = nZmaterial ln − 2 1 − 2 (22) dx me v δmin 2mec c • At large δ, the electrons can be assumed to be free; kinematics shows that: 2 2 δmax = 2mev γ (23) • However, this is not true at small δ. At low δ the momentum and energy transfer can lead to excitation as well as ionisation: it is therefore necessary to integrate over δ and q with cross- sections that depend on the atomic structure of the material in question. • The Bethe formula avoids this problem by defining a mean excitation potential, I, as follows: 0:9 δmin = I = 16Z (24) • The final result is... 2 2 2 dE 4π Zα~c 2mev v − = nZmaterial ln 2 2 − 2 (25) dx ionisation me v I(1 − v =c ) c Remember: Z is the charge (in units of e) of the charged particle striking the material, which has atomic number or effective charge Zmaterial . 1.1.2 Observations on the Bethe formula The stopping power (dE=dx) has the following general properties: • It is independent of the mass of the incident particle. • It depends on the charge of the incident particle squared (Z2). • Treating the slowly varying logarithmic term as a constant, it can be seen that dE=dx / c2=v2 or dE=dx / 1/β2, as noted above. 1.1.3 Behaviour of the Bethe formula as a function of velocity Low velocity regime: βγ < 3:5 In general the stopping power (dE=dx) falls off as 1=v2 in accordance with (1). Slow moving positively charged particles can capture the material's negative electrons, reducing the inci- dent particle's effective charge, Z, thus reducing dE=dx. Negative incident particles can suffer stripping of their electrons, reducing their value of Z. Minimum ionisation loss: βγ ≈ 3:5 The 1=(1 − β2) term in the logarithm halts the general decrease in stopping power. It stops decreasing somewhere in the range of βγ = 3 ! 4: this is known as the minimum ionisation loss. 5 The relativistic rise: βγ > 3:5 Beyond the minimum ionisation loss dE=dx begins to rise again, in fact the Bethe formula predicts it should rise indefinitely. In addition, as the incident charged particle reaches relativistic energies its transverse electric field in- creases, meaning more of the material's atoms are within range of the particle's electric field, leading to greater ionisation energy loss. This is known as the relativistic rise. Density effect However, the slope actually does shallow out because of the density effect: long distance atomic elec- trons are screened from the electric field of the incident particle by the dielectric effect of the intervening atoms in the material. Stopping Power of H2, He, C, Al, Fe, Sn and Pb 10 8 ) 6 2 H2 liquid cm 5 1 − 4 He gas 3 (MeV g C Al 2 Fe dE/dx Sn − Pb 1 0.1 1.0 10 100 1000 10 000 βγ = p/Mc 0.1 1.0 10 100 1000 Muon momentum (GeV/c) 0.1 1.0 10 100 1000 Pion momentum (GeV/c) 0.1 1.0 10 100 1000 10 000 Proton momentum (GeV/c) Figure 2: Mean energy loss rate divided by density in: liquid (bubble chamber) hydrogen, gaseous helium, carbon, aluminium, iron, tin, and lead.
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