Interactions of Electrons

1. From Mauricio Barbi, TSI’07 lectures A B electrons 3 1.95 heavy charged particles 4 2 Interactions of Particles with Matter Interactions of Electrons In the high energy limit(β1), the energy loss for both “heavy” charged particles and electrons/positrons can be approximated by Where The second terms indicates that the rate of relativistic rise for electrons is slightly smaller than for heavier particles. This provides a criterion for identification between charged particles of different masses. Phys 521A

2. Electron energy loss • Fractional energy loss is dominated by radiation for E>10’s of MeV • Dependence of critical energy on Z will be shown later • Very low energy behavior differs for e+ and e- but differences are small compared to total Phys 521A

3. Bremsstrahlung • Radiation in Coulomb field (primarily nuclear field) • Electron energy loss (dE/dx)rad = -E/X0 => E(x) = E0 exp(-x/X0) where x is depth (in g/cm2) and X0 is the radiation length • Total power radiated by electrons: • Limiting cases: • parallel acceleration (bremsstrahlung) ~ γ6; • perpendicular acceleration (synchrotron) ~ γ4 Phys 521A

4. Emitted photons from Bremsstrahlung • Cross section vs. photon energy dσ/dk ~ 1/k, but distribution shifts to larger k for ultra-relativistic (TeV) electrons • Radiated photons peaked strongly forward:(this peaks at 2.5 mrad for 100 MeV electrons, 0.25 mrad for 1 GeV electrons) photons / radiation length fractional photon energy Phys 521A

5. Critical energy • Energy at which average electron energy losses from ionization and from bremsstrahlung are equal • Crudely given by Ec = (610 MeV)/(Z+1.24) • Helpful in parameterizing electromagnetic showers Critical energy vs Z Phys 521A

6. Multiple Coulomb Scattering • Angular deflections mostly from nuclear Coulomb field (Rutherford scattering); very large angles unlikely, but possible • Multiple small scatters  rms scattering angle • Scattering independent in each plane; spatial angle described by 2D Gaussian Phys 521A

7. Multiple Coulomb Scattering • Empirical formula for projected rms scattering angle: where x=thickness, X0=radiation length, z, β, p= charge, speed and momentum of particle • Non-Gaussian tails due to hard (large angle) scatters • The non-Gaussian tails are problematic for tracking and for simulation Phys 521A

8. Cherenkov radiation(see talk by Ratcliff at http://rich2007.ts.infn.it/invitedtalks.php) • Coherent (Cherenkov) radiation of particles with superluminal speed important (for detection, not Eloss) • Photons emitted at angle where n is the index of refraction of the medium and β is the speed of the charged particle. • Threshold: β>1/n • Radiation is prompt(unlike scintillation) • Group index ng(l) = n(l)-l dn(l)/dlcan be strong fn of l(dispersive media) Phys 521A

9. η Cherenkov radiation II • Photon direction not orthogonal to cone ½ angle: • Relevant speed is group velocity • Flux • In practice, photoelectron statistics are typically • Can measure both angle of emitted photon (w.r.t. track direction) and propagation time tp = Lng/c Phys 521A

10. Cherenkov radiation III(from Blair Ratcliff) • Dependence on β implies discrimination between different mass particles for a given momentum; for particles well above the velocity threshold, Refractive Indices N=1.474 (Fused Silica) N=1.27 (C6F14 CRID) N=1.02 (Silica Aerogel) N=1.001665 (C5F12/N2 Mix) N=1.0000349 (He) s[qc(tot)] u l n ▲ solidliquidfoamgasgas p/K separation-limiting case 2 mrad 1 mrad 0.5 mrad 0.1 mrad Phys 521A

11. Photon interaction in matter • In the <10eV range atomic and molecular excitation is predominant • From ~10eV to ~100keV cross-section dominated by photoelectric effect (peaks correspond to atomic or molecular levels) • In ~0.01-100 MeV range, Compton scattering is important • At energies above 2me, pair production takes over (nuclear field dominates); pair production is related to bremsstrahlung (two e, two γ) • Pair production cross-section at high energy is σ = (7/9) (A/X0NA)(recall X0 is radiation length) Phys 521A

12. From Mauricio Barbi, TSI’07 lectures Interactions of Particles with Matter Interactions of Photons Photoelectric effect • Can be considered as an interaction between a photon and an atom as a whole • ThresholdE > Eb, binding energy of electron • Electron is ejected with energyE - Eb • Discontinuities in the cross-section due to discrete energies Eb of atomic electrons (strong modulations at E=Eb; L-, K-edges, …) • Dominant process at low  energy (<MeV)  Gives low energy electrons p.e. cross-section in Pb Kleinknecht E Phys 521A

13. From Mauricio Barbi, TSI’07 lectures   εL   Atomic electron atom εK e Free electron Interactions of Particles with Matter Interactions of Photons Compton scattering  A photon with energy E,in scatters off an (quasi-free) atomic electron  A fraction of E,in is transferred to the electron  The resulting photon emerges with E,out < E,in and at different direction Using conservation of energy and momentum: The energy of the outgoing photon is: , where E Phys 521A

14. From Mauricio Barbi, TSI’07 lectures  + e-  e+ + e- + e-  + nucleus e+ + e- + nucleus Interactions of Particles with Matter Interactions of Photons Pair Production  An electron-positron pair can be created when (and only when) a photon passes by the Coulomb field of a nucleus or atomic electron  this is needed for conservation of momentum. Threshold energy for pair production at E = 2mc2 near a nucleus. E = 4mc2near an atomic electron  Pair production is the dominant photon interaction process at high energies. Cross- section from production in nuclear field is dominant. First cross-section calculations made by Bethe and Heitler using Born approximation (1934). Phys 521A

15. From Mauricio Barbi, TSI’07 lectures Interactions of Particles with Matter Interactions of Photons Pair Production Photon pair conversion probability (attenuation length is 9/7 X0) Cross-section independent ofphoton energy (once well abovethreshold), ~ Z2 P=54% http://pdg.lbl.gov Phys 521A