Interaction of light charged particles with matter Ionization

Interaction of light charged particles with matter Ionization losses – electron loss energy as it ionizes and excites atoms Scattering – scattering by Coulomb field of nucleus, by field of electrons Bremsstrahlung radiation – enough high energy, accelerated motion of charged particle → emission of electromagnetic radiation, ultrarelativistic energies – pair production through virtual photon Cherenkov radiation – charged particle moving faster then light at given material emits electromagnetic radiation in the range of visible light– minimal ionization losses Scattering is induced by interaction with atomic nuclei ( ~ f(Z 2) ) and electrons at atomic cloud ( ~ f(Z) ) (difference from heavy particles – in this case mainly interaction with nuclei), energy losses mainly by interaction with electrons at atomic cloud Electromagnetic shower– very high energies Motion of electrically charged particles in magnetic and electric fields

Energy ionization losses Mostly relativistic ↔ electrons and positrons are light particles They will transfer big part of their energy during ionization Interaction of electrons – interaction of identical particles → ΔEMAX = E/2 Interaction of positrons – they are not identical particles as electrons - anihilation on path end – production of 1. 022 Me. V energy Ionization losses determination – energy losses Procedure of derivation of equation for ionization losses: 1) Classical derivation for nonrelativistic heavy particles 2) Quantum derivation for nonrelativistic particles 3) Relativistic corrections and corrections on identity of particles for electrons

Bethe - Bloch formulae Classical derivation (assumption of nonrelativistic velocity and ΔE <<E ): Change of momentum: Electric force acts on particle: Constant connected to SI unit system, often is putted equal to one Impact parameter b is changed during scattering only slightly: influence of F|| on momentum change are negated (second half negates first) Zobrazení síly F┴ F pro elektron, Influence has only: x F|| v případě iontu b We express path by velocity: dx = v·dt je přitažlivá If velocity v during interaction with one electron changes only slightly, transferred momentum is: Kinetic energy of electron after interaction with ionizing particle

Path of particle passage through matter Δx: Let have thin cylinder (annulus cross-section (b, b+db): b b+db Number of electrons at cylinder: where ne – is electron density at material Total energy losses at cylinder: Mention: where ΔNe – number of electrons at cylinder Energy losses in the whole roll If charge of material atoms is Z, number of electrons ne = Z·n 0, where n 0 – atom density at material. We express it by material density ρ Avogardo constant NA and atomic mass A: and then:

In the case of integration limits 0 and ∞ we obtain divergent integral. Limits for integration are not in the reality 0 and ∞ but bmin and bmax: Maximal energy is transferred during head collision, electron obtains energy: because maximal transferred momentum We use relation between transferred energy and impact parameter: Minimal transferred momentum depends on mean ionization potential of electrons at atom I, is and (work achieved through passage must be larger then ionization potential) and corresponding impact parameter is: We determine integral: where: and then: Constant connected to SI unit system, often is expressed as equal one Main dependency on particle velocity Main dependency on material properties Weak dependency on particle velocity and material properties

Relativistic corrections: Maximal transferred momentum: Reduction of particle electric field in the direction of flight by factor (1 -β 2) and in the perpendicular direction increasing by factor We will obtain on the end: We obtain early derived equation for v << c This formulae is for electrons even more complex: In the case of electron → identical particles → maximal transferred energy ΔEMAX = E/2 E ~ up to hundreds Me. V → light particle losses are 1000 times lower than for heavy E ~ Ge. V → ionization losses of light and heavy particles are comparable

Example of ionization losses for some particles (taken from D. Green: The physics of particle detector)

Elastic scattering 1) Single scattering 2) Few scatterings 3) Multiple scattering Heavy particles – important only for scattering on atomic nuclei Light particles – important also for scattering on electrons Single scattering in the electric field of nucleus – described by Rutheford scattering: 1) Heavy particles – scattering to small angles → path is slightly undulated 2) Light particles – scattering to large angles → range is not defined (for „lower energies“) Mean quadratic deviation from original direction quadratic value of scattering angle : depends on mean (simplified classical derivation for „heavy particles“ – small scattering angles)

→ 0: We determine and then : then is determined: where Nroz is number of scatterings: Resulting value: 1) Strong dependency on momentum: 2) Strong dependency on velocity 1/v 4 Important scattering properties: 3) Strong dependency on mass 1/m 2 4) Strong dependency on particle charge: Zion 2 5) Strong dependency on material Z Z 2

Bremsstrahlung radiation Accelerated charged particle emits electromagnetic radiation Energy emitted per time unit: Acceleration is given by Coulomb interaction: Dependency on material charge: ion charge: and mass: Rozdíl v náboji iontu malý, v hmotnosti mnohem větší: For proton and electron: For muon and electron is same ratio 2. 6·10 -5 Radiation losses show itself in „normal situation“ only for electrons and positrons For ultrarelativistic energies also for further particles

On the base of quantum physics we obtain for energy losses for electron (positron) Zion = 1: Description is equivalent pair production description: (it is similar calculation and result as for pair production – see gamma ray interaction) where for mention: Course of function F(E, Z) depends on energy (E 0 – initial electron energy) and if it is necessary count screening of electrons: E ≈ hν 0 – eigenfrequency of atom → interaction with atom – screening has not influence E >> hν 0 – interaction with nucleus → screening is necessary count according to, where electron interacts with nucleus: Low energy → strong field near nucleus is necessary High energy → weak field further from nucleus is enough – there is maximum of production Without screening : where: Complete screening : and F(E, Z) in the case without screening depends on E only weakly and in the case of complete screening does not depend on E:

Radiation losses linearly proportional to energy: For radiation length : Energy losses of electron (if they are only radiation losses): Critical energy EC: For electron and positron is EC > mec 2 → v ≈ c for v → c is valid Fion(E) = f(ln. E): ! Let approve! Air Al Pb EC [Me. V] 80 40 7, 6

Total energy losses Total losses are given by ionization and radiation losses: Electron range, absorption Protons Electrons Well defined range does not exist Rextrap - extrapolated path – point fo linear extrapolation crossing We obtain exponential dependency for spectrum of beta emitter Schematic comparison of different quantities for protons and electrons

Ultrarelativistic energies Radiation losses by bremsstrahlung radiation and pair production prevail for ultrarelativistic energies also for muons (taken from D. Green: The physics of particle detector) Electromagnetic shower creation – see gamma ray interaction

Angular and energy distributions of bremsstrahlung photons Angular distribution: Depends on electron (other particle) energy, does not depend on emitted photon energy Mean angle of photon emission: E→ ∞ ΘS → 0 Photons are emitted to narrow cone to the direction of electron motion, preference of forward angles increases with energy Energy distribution: Maximal possible emitted energy – kinetic energy of electron

Synchrotron radiation Similar origin as bremsstrahlung – it is generated during circular motion of relativistic charged particles on accelerators (synchrotrons). Influence of acceleration → emission of electromagnetic radiation Synchrotron radiation is not connected with material – lower acceleration → it has lower energy Acting force is Lorentz force: Energy losses: Classical centripetal acceleration: a=v 2/R Energy losses: Relativistic centripetal acceleration: Energy losses:

Cherenkov radiation Particle velocity in the material v > c’ = c/n (n – index of refraction) → emission of Cherenkov radiation: Results of this equation: 1) Threshold velocity exists βmin = 1/n. For βmin emission is in the direction of particle motion. Cherenkov radiation is not produced for lower velocities. 2) For ultrarelativistic particles cos Θmax = 1/n. 3) For water: n = 1. 33 → βmin = 0. 75, for electron EKIN = 0. 26 Me. V cosΘmax = 0. 75 → Θmax= 41. 5 o

Transition radiation Passage of charged particle through boundary of materials with different index of refraction → emission of electromagnetic radiation (discovery of Ginsburg, Frank 1946) Creation of dipole in boundary zone → dipole, elmg. field changes in time → emission of elmg. radiation: material Energy emitted by one transition material/vacuum: + + vacuum e- High energy electron emits transition radiation plasma frequency: ħωP ≈ 14 e. V (for Li), 0, 7 e. V (air) 20 e. V (for polyethylene) Number of photons emitted on boundary (is very small, necessity of many transitions): Energy of emitted photons 10 – 30 ke. V Emission sharply directed to the particle flight direction: Radiators of transition radiation: material with small Z, reabsorption increases with ~ Z 5 Good combination of radiators and X-ray detectors