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Interaction of Particles with Matter Alfons Weber CCLRC & University of Oxford Graduate Lecture 2004 Nov 2004 2 Table of Contents Bethe-Bloch Formula Energy loss of heavy particles by Ionisation Multiple Scattering Change of particle direction in Matter Cerenkov Radiation Light emitted by particles travelling in dielectric materials Transition radiation Light emitted on traversing matter boundary Nov 2004 3 Bethe-Bloch Formula Describes how heavy particles (m>>me) loose energy when travelling through material Exact theoretical treatment difficult Atomic excitations Screening Bulk effects Simplified derivation ala MPhys course Phenomenological description Nov 2004 4 Bethe-Bloch (1) Consider particle of charge ze, passing a stationary charge Ze ze b y r θ x Assume Ze Target is non-relativistic Target does not move Calculate Energy transferred to target (separate) Nov 2004 5 Bethe-Bloch (2) Force on projectile Zze2 Zze2 Fx cos cos3 4 0 r 2 4 0b2 Change of momentum of target/projectile Zze2 1 p dtFx 2 0 c b Energy transferred p 2 Z 2 z 2e4 1 E 2M 2M (2 0 )2 ( c) 2 b2 Nov 2004 6 Bethe-Bloch (3) Consider α-particle scattering off Atom Mass of nucleus: M=A*mp Mass of electron: M=me But energy transfer is p 2 Z 2 z 2e4 1 Z2 E 2M 2M (2 0 ) ( c) b 2 2 2 M Energy transfer to single electron is 2 z 2 e4 1 Ee (b) E mec 2 (4 0 )2 2 b2 Nov 2004 7 Bethe-Bloch (4) Energy transfer is determined by impact parameter b Integration over all impact parameters b db ze dn 2 b (number of electrons / unit area ) db NA =2 b Z x A Nov 2004 8 Bethe-Bloch (5) Calculate average energy loss bmax dn me c 2 Zz 2 E d b Ee (b) 2C 2 x ln b b bmax bmin db A min me c 2 Zz 2 C 2 x ln E E Emax A min e2 with C 2 N A 2 4 0 me c There must be limit for Emin and Emax All the physics and material dependence is in the calculation of this quantities Nov 2004 9 Bethe-Bloch (6) Simple approximations for From relativistic kinematics 2 2 2 me c 2 Emax 2 2 2 2 me c 2 me me 1 2 M M Inelastic collision Emin I 0 average ionisation energy Results in the following expression E me c Zz 2 me c 2 2 2 2 2 2C 2 ln x A I0 Nov 2004 10 Bethe-Bloch (7) This was just a simplified derivation Incomplete Just to get an idea how it is done The (approximated) true answer is E me c 2 Zz 2 1 2 2 2 me c 2 Emax ( ) 2C 2 ln 2 x A 2 2 I0 2 2 with ε screening correction of inner electrons δ density correction, because of polarisation in medium Nov 2004 11 Energy Loss Function Nov 2004 12 Average Ionisation Energy Nov 2004 13 Density Correction Density Correction does depend on material with x = log10(p/M) C, δ0, x0 material dependant constants Nov 2004 14 Different Materials (1) Nov 2004 15 Different Materials (2) Nov 2004 16 Particle Range/Stopping Power Nov 2004 17 Application in Particle ID Energy loss as measured in tracking chamber Who is Who! Nov 2004 18 Straggling (1) So far we have only discussed the mean energy loss Actual energy loss will scatter around the mean value Difficult to calculate parameterization exist in GEANT and some standalone software libraries From of distribution is important as energy loss distribution is often used for calibrating the detector Nov 2004 19 Straggling (2) Simple parameterisation Landau function 1 1 f ( ) exp ( e ) 2 2 E E with me c 2 Zz C 2 x A Better to use Vavilov distribution Nov 2004 20 Straggling (3) Nov 2004 21 δ-Rays Energy loss distribution is not Gaussian around mean. In rare cases a lot of energy is transferred to a single electron δ-Ray If one excludes δ-rays, the average energy loss changes Equivalent of changing Emax Nov 2004 22 Restricted dE/dx Some detector only measure energy loss up to a certain upper limit Ecut Truncated mean measurement δ-rays leaving the detector E me c 2 Zz 2 1 2 2 2 me c 2 Ecut 2C 2 ln x E E A 2 I 02 cut Ecut ( ) 1 2 Emax 2 2 Nov 2004 23 Electrons Electrons are different light Bremsstrahlung Pair production Nov 2004 24 Multiple Scattering Particles don’t only loose energy … … they also change direction Nov 2004 25 MS Theory Average scattering angle is roughly Gaussian for small deflection angles With 13.6 MeV x x 0 z 1 0.038ln cp X0 X 0 X 0 radiation length Angular distributions are given by dN 1 space 2 exp d 2 0 2 2 2 0 dN 1 plane 2 exp d plane 2 0 2 2 0 Nov 2004 26 Correlations Multiple scattering and dE/dx are normally treated to be independent from each Not true large scatter large energy transfer small scatter small energy transfer Detailed calculation is difficult but possible Wade Allison & John Cobb are the experts Nov 2004 27 Correlations (W. Allison) nuclear small angle nuclear backward scattering (suppressed scattering in CM by screening) electrons (suppressed by nuclear at high form factor) Q2 whole atoms at 17 Log cross low Q2 section (dipole (30 region) decades) 2 electrons log Log pL or kL log kT backwards in energy transfer CM (16 decades) Log pT transfer Example: Calculated cross section for 500MeV/c in Argon gas. 18 7 (10 decades) Note that this is a Log-log-log plot - the cross section varies over 20 and more decades! Nov 2004 28 Signals from Particles in Matter Signals in particle detectors are mainly due to ionisation Gas chambers Silicon detectors Scintillators Direct light emission by particles travelling faster than the speed of light in a medium Cherenkov radiation Similar, but not identical Transition radiation Nov 2004 29 Cherenkov Radiation (1) Moving charge in matter at rest slow fast Nov 2004 30 Cherenkov Radiation (2) Wave front comes out at certain angle 1 cos c n That’s the trivial result! Nov 2004 31 Cherenkov Radiation (3) How many Cherenkov photons are detected? 2 z N L re me c 2 ( E ) sin 2 c ( E )dE z2 1 2 L ( E ) 1 2 2 dE re me c n 1 LN 0 1 2 2 n with ( E ) Efficiency to detect photons of energy E L radiator length re electron radius Nov 2004 32 Different Cherenkov Detectors Threshold Detectors Yes/No on whether the speed is β>1/n Differential Detectors βmax > β > βmin Ring-Imaging Detectors Measure β Nov 2004 33 Threshold Counter Particle travel through radiator Cherenkov radiation Nov 2004 34 Differential Detectors Will reflect light onto PMT for certain angles only β Selecton Nov 2004 35 Ring Imaging Detectors (1) Nov 2004 36 Ring Imaging Detectors (2) Nov 2004 37 Ring Imaging Detectors (3) More clever geometries are possible Two radiators One photon detector Nov 2004 38 Transition Radiation Transition radiation is produced when a relativistic particle traverses an inhomogeneous medium Boundary between different materials with different n. Strange effect What is generating the radiation? Accelerated charges Nov 2004 39 Transition Radiation (2) Initially observer sees nothing Later he seems to see two charges moving apart electrical dipole Accelerated charge is creating radiation Nov 2004 40 Transition Radiation (3) Consider relativistic particle traversing a boundary from material (1) to material (2) 2 2 d N z 2 2 1 1 2 2 2 d d / 2 1/ 2 2 1/ 2 p p plasma frequency Total energy radiated Can be used to measure γ Nov 2004 41 Transition Radiation Detector Nov 2004 42 Table of Contents Bethe-Bloch Formula Energy loss of heavy particles by Ionisation Multiple Scattering Change of particle direction in Matter Cerenkov Radiation Light emitted by particles travelling in dielectric materials Transition radiation Light emitted on traversing matter boundary Nov 2004 43 Bibliography PDG 2004 (chapter 27 & 28) and references therein Especially Rossi Lecture notes of Chris Booth, Sheffield http://www.shef.ac.uk/physics/teaching/phy311 R. Bock, Particle Detector Brief Book http://rkb.home.cern.ch/rkb/PH14pp/node1.html Or just it! Nov 2004 44 Plea I need feedback! Questions What was good? What was bad? What was missing? More detailed derivations? More detectors? More… Less… A.Weber@rl.ac.uk