2. Passage of Radiation Through Matter
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1 2. Passage of Radiation Through Matter
2 Passage of Radiation Through Matter: Contents Energy Loss of Heavy Charged Particles by Atomic Collision (addendum) Cherenkov Radiation Energy loss of Electrons and Positrons Multiple Coulomb Scattering The Interaction of Photons: Photo electric effect Compton scattering Pair production Electron-photon shower etc. The Interaction of Neutrons
3 Relativistic Variables γ β = v c E = = Mc β Velocity in units of speed of light Lorentz factor p Mc = γ 2 1
4 Addendum: 2.1 Energy Loss of Heavy Charged Particles At around β 0.96 Most relativistic particles considered MIPs
5 Addendum: 2.1 Energy Loss of Heavy Charged Particles (cont.): Chemical Compounds and Bragg-Kleeman Rule: Mixtures If accurate values are desired: direct measurements, but good approximation is given by the 1 de ρ dx = i w ρ i i de dx w i : weight of element i or atomic fraction ρ: atomic density More explicitly: w i =a i A i /A m, A i : atomic weight, A m =Σa i A i Mass stopping power a i : Number of atoms of the ith element in the molecule M i
6 Addendum: 2.1 Energy Loss of Heavy Charged Particles (cont.): Channeling Important exception to Bethe-Bloch formula Occurs in materials having a spatially symmetric atomic structure, i.e. crystals Only occurs at an incident angle θ < certain critical angle w.r.t. a symmetry axes of the crystal θ c Particle suffers a series of correlated small-angle scatterings: Slowly oscillating trajectory, wavelength of the trajectory is generally many lattice lengths long Channeling greatly reduces energy loss (particle encounters less electrons than it normally would) In general: θ c 1 o for β 0.1
7 Addendum: 2.1 Energy Loss of Heavy Charged Particles (cont.): Range Range: penetration depth/distance of a particle in a material before it loses all its energy well defined number Same for all identical particles with the same initial energy in the same type of material: R( Material, Particle type, E) Experimental determination: passing a beam of monoenergetic particles through different thicknesses of material measure ratio of transmitted to incident particles range number distance curve Spread of ranges because energy loss is not continuous but statistical in nature range straggling
8 Calculated range-energy curves of different heavy particles in aluminium (num. int. of BBF): Addendum: 2.1 Energy Loss of Heavy Charged Particles (cont.): Range T min : minimum energy at which de/dx formula is valid R 0 (T min ): empirically determined constant accounting for remaining low energy behavior of de/dx results accurate within a few percent R ~ E b At not too high energies: -de/dx ~ β -2 ~ T -1 R~ T 2 T: kinetic energy - Range energy relations are extremely useful in Particle energy measurements - Detector sizes - Thickness of radiation shielding
9 Energy Loss of Electrons and Positrons Like heavy charged particles, electrons and positrons suffer collisional energy loss when passing through matter However because of their small mass: energy loss from emission of EM radiation caused by scattering in the electric field of a nucleus (bremsstrahlung, means braking radiation ) Total energy loss: de dx de de = + dx dx tot rad coll
10 Energy Loss of Electrons and Positrons (cont.): Collision Loss Bethe-Bloch formula needs to be modified for two reasons: 1. small mass of electrons/positrons 2. for electrons the collisions are between identical particles (in particular maximum energy transfer becomes W max =T e /2, T e : kinetic energy of incident e +, e - ) τ: kinetic energy in units of m e c 2
11 Energy Loss of Electrons and Positrons (cont.): Radiation Loss: Bremsstrahlung E < few hundred GeV, e- and e+ are the only particles for which radiation contributes substantially to energy loss Why? Cross-section: σ ~ r e2 = (e 2 /mc 2 ) 2 Several effects need to be considered when calculating the energy loss: For example screening from the atomic electrons impact parameter and atomic number Z play an important role: define screening parameter ξ ~ 1/E 0 Z 1/3, ξ 0: complete screening, ξ >> 1: no screening
12 Energy Loss of Electrons and Positrons (cont.): de dx coll ~ ln E, Z de dx rad ~ E, Z 2 Note: electron-electron bremstrahlung In the field of the atomic electrons, cross-sections essentially the same as for radiation loss in the field of the nucleus only that Z 2 is replaced by Z contribution can be taken into account by replacing Z 2 with Z(Z+1) in cross-section formulas
13 Energy Loss of Electrons and Positrons (cont.): Other important parameters Critical Energy: energy loss by radiation depends strongly on absorber for each material a critical energy E c can be defined as: de dx de = dx rad coll for E=E c Approximation: E 800 c Z + MeV 12.
14 Energy Loss of Electrons and Positrons (cont.): Other important parameters Radiation length L rad : another quantity characteristic for the absorber, distance over which the electron energy is reduced by a Factor of 1/e At high energies, where collision loss can be ignored: E = E exp 0 x L rad Useful approximation: 2 g cm A L = / rad ZZ ( + 1)ln( 287 / Z)
15 Energy Loss of Electrons and Positrons (cont.): Other important parameters Critical Energy, alternative definition: Here: X 0 : radiation length From PDG
16 Energy Loss of Electrons and Positrons (cont.): Range of Electrons Greater susceptibility to multiple scattering by nuclei range of electrons is generally very different from the calculated path length obtained from integration of de/dx formula Also: energy loss of electrons fluctuates much more than for heavy charged particles (much greater energy transfer per collision, emission of bremstrahlung) great range straggeling
17 Energy Loss of Electrons and Positrons (cont.): Absorption of β Electrons Continuous spectrum of β decay electron energies I = I exp( µ x) 0 µ: β absorption constant Number-distance curve for Beta decay electrons from 185 W
18 Multiple Coulomb Scattering Repeated elastic Coulomb scatterings of charged particles from nuclei Smaller probability than inelastic collisions with the atomic electrons Rutherford formula 2 2 dσ dω = 1 4 Zze pv 1 4 sin ( θ / 2) Large for small angles Multiple small angle scatters Gaussian + Rutherford tails
19 Multiple Coulomb Scattering (cont.) Coulomb scattering distribution is well represented by the theory of Molière Define: rms 1 θ0 = θplane = θ 2 rms space θ 0 For the central 98% of the projected angular distribution: 136. MeV = z x/ Lrad +. ln( x/ L ) βcp [ rad ]
20 Large-angle deflections of electrons along their track Backscattering
21 Electron-photon shower: Shower Size Longitudinal development: difficult (MC) Transversal: Moliere radius R M =21MeV*L rad /E c Containment: 1 R M 90% 3.5 R M 99%
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