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 Motion of electrically charged particles in magnetic and electric fields Electromagnetic shower– very high energies

Energy ionization losses Interaction of electrons – interaction of identical particles → ΔE MAX = E/2 Interaction of positrons – they are not identical particles as electrons - anihilation on path end – production of MeV energy Mostly relativistic ↔ electrons and positrons are light particles They will transfer big part of their energy during ionization 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 Ionization losses determination – energy losses

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

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

Limits for integration are not in the reality 0 and ∞ but b min and b max : In the case of integration limits 0 and ∞ we obtain divergent integral. Maximal energy is transferred during head collision, electron obtains energy: because maximal transferred momentum We use relation between transferred energy and impact parameter: Main dependency on particle velocity Main dependency on material properties Weak dependency on particle velocity and material properties Constant connected to SI unit system, often is expressed as equal one 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:

Relativistic corrections: In the case of electron → identical particles → maximal transferred energy ΔE MAX = E/2 We obtain early derived equation for v << c 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: This formulae is for electrons even more complex: E ~ up to hundreds MeV → light particle losses are 1000 times lower than for heavy E ~ GeV → 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 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 depends on mean quadratic value of scattering angle : (simplified classical derivation for „heavy particles“ – small scattering angles) Heavy particles – important only for scattering on atomic nuclei Light particles – important also for scattering on electrons

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

Bremsstrahlung radiation Accelerated charged particle emits electromagnetic radiation Energy emitted per time unit: Acceleration is given by Coulomb interaction: 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 Dependency on material charge: ion charge: and mass: Rozdíl v náboji iontu malý, v hmotnosti mnohem větší:

Course of function F(E,Z) depends on energy (E 0 – initial electron energy) and if it is necessary count screening of electrons: Without screening : Complete screening : where: (it is similar calculation and result as for pair production – see gamma ray interaction) On the base of quantum physics we obtain for energy losses for electron (positron) Z ion = 1: Description is equivalent pair production description: where for mention: 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: 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

For radiation length : Critical energy E C : Radiation losses linearly proportional to energy: Energy losses of electron (if they are only radiation losses): For electron and positron is E C > m e c 2 → v ≈ c E C [MeV] Air 80 Al 40 Pb 7,6 for v → c is valid F ion (E) = f(lnE):! Let approve!

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

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

Angular and energy distributions of bremsstrahlung photons Depends on electron (other particle) energy, does not depend on emitted photon energy Mean angle of photon emission: E→ ∞  Θ S → 0 Photons are emitted to narrow cone to the direction of electron motion, preference of forward angles increases with energy Angular distribution: 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 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 β min = 1/n. For β min emission is in the direction of particle motion. Cherenkov radiation is not produced for lower velocities. 2) For ultrarelativistic particles cos Θ max = 1/n. 3) For water: n = 1.33 → β min = 0.75, for electron E KIN = 0.26 MeV cosΘ max = 0.75 → Θ max = 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) e-e- + + vacuummaterial Creation of dipole in boundary zone → dipole, elmg. field changes in time → emission of elmg. radiation: Energy emitted by one transition material/vacuum: Number of photons emitted on boundary (is very small, necessity of many transitions): High energy electron emits transition radiation plasma frequency: ħω P ≈ 14 eV (for Li), 0,7 eV (air) 20 eV (for polyethylene) Emission sharply directed to the particle flight direction: Radiators of transition radiation: material with small Z, reabsorption increases with ~ Z 5 Energy of emitted photons 10 – 30 keV Good combination of radiators and X-ray detectors