Interaction of particles with matter (lecture 1) 1/58 Johann Collot collot@in2p3.fr http://lpsc.in2p3.fr/collot UdG
A brief review of a few typical situations is going to greatly simplify the subject. Mean free path of a particle, i.e. average distance travelled between two consecutive collisions in matter : λ= 1 σ n where : total interaction cross-section of the particle number of scattering centers per unit volume n n = N A for a monoatomic element of molar mass M and example : specific mass ρ . M Avogadro number N A (charged particles) Electromagnetic interaction : 1 m (neutrons ....) Strong interaction : 1cm Weak interaction : 15 m ≃ 0,1 light year (neutrinos) 10 A practical signal ( >100 interactions or hits ) can only come from electromagnetic interaction Particle detection proceeds in two steps : 1) primary interaction 2) charged particle interaction producing the signals 2/58 Johann Collot collot@in2p3.fr http://lpsc.in2p3.fr/collot UdG
typical examples : photon detection Compton scattering 2 1 Signal is induced by electrons e - 3 e - e + Pair production 3/58 Johann Collot collot@in2p3.fr http://lpsc.in2p3.fr/collot UdG
neutral pion detection : A π 0 decays into two photons with a mean lifetime of 8.5 10 -17 s. e + 1 e - 0 e - 2 e + 4/58 Johann Collot collot@in2p3.fr http://lpsc.in2p3.fr/collot UdG
neutrino detection : A 2800 MW nuclear power station produces 130 MW of neutrinos ! A detector of 1 m 3 located 20 m away from the reactor core can detect 100 neutrinos/h. e e liquid scintillator ( CH x + 6 L i ) nuclear reactor e e e p e 6 L i t n e e ~ 0 e p n e Charged particles produce light in the target scintillator n th L i 6 t 4,8 MeV 5/58 Johann Collot collot@in2p3.fr http://lpsc.in2p3.fr/collot UdG
Interaction of charged particles with matter For heavy particles ionization and excitation are the dominant processes producing energy loss. Particle P of Z charge state * atom Excitation : followed by : Z atom atom Z atom * P P Ionization : Z atom atom e Z - P P Ionization + excitation : Z atom atom Z * e - P P 6/58 Johann Collot collot@in2p3.fr http://lpsc.in2p3.fr/collot UdG
Maximal kinetic energy transferred to an ionized electron : m 0 v m e = v c p = m 0 v − 1 / 2 = 1 − 2 atome E Z − 2 = 1 − 1 / 2 Hypothesis : V > <v e > = Z α c , speed of deepest atomic orbit electrons where α is the fine structure constant : α = 1/137 . 2 max − m e = 2 m e 2 One may show (exercise) that : T e (In natural units , c=ħ=1) max = E e E CM / m 0 2 1 / 2 2 m e 2 2 m e E where : E CM = m 0 total energy in center-of-mass frame 7/58 Johann Collot collot@in2p3.fr http://lpsc.in2p3.fr/collot UdG
Two cases : m 0 >> m e , i.e. the incoming particle is not an electron and if its energy is not too big 2 2 2 = ( m 0 2 + m e 2 + 2 m e E with E =γ m 0 ( E CM / m 0 ) )≃ 1 2 m 0 m 0 m 0 2 γ m e proton E p < 50 GeV , muon E μ < 500 MeV (medium energy range) ≪ 1 m 0 2 γ then : max = E e max − m e = 2 m e β 2 T e the incoming particle is an electron m 0 = m e due to undistinguishibility of electrons, max transferable max =( E − m e ) T e energy = T max / 2 If the incoming particle is not an electron then in practice m 0 >> m e . 8/58 Johann Collot collot@in2p3.fr http://lpsc.in2p3.fr/collot UdG
Stopping power of heavy particles by excitation and ionization in matter. Average energy loss by a charged particle (other than an electron) in matter. Bethe and Bloch formula (see Nuclei and particles, Émilio Segré, W.A. Benjamin ; Principles of Radiation Interaction in Matter and Detection, C. Leroy and P.G. Rancoita, World Scientific ; Introduction to experimental particle physics, R. Fernow ) Stopping power or mean specific energy loss charge of incoming particle density effect correction Z of medium at high energy 2 Z 2 ln 2 m e 2 2 T e max 2 − − C e − dE dx [ MeV 0.3071 z 1 2 ] = − Z − 1 2 2 2 g / cm A g mol I Atomic shell correction at low mean excitation energy energy Atomic mass of medium (not covered in this lecture, see Leroy & Rancoita) Surface mass density of medium dx = ρ dl (or mass thickness of medium) 9/58 Johann Collot collot@in2p3.fr http://lpsc.in2p3.fr/collot UdG
show that : 2 2 = (β γ) β 2 1 +(β γ) 10/58 Johann Collot collot@in2p3.fr http://lpsc.in2p3.fr/collot UdG
Particle identification in Alice TPC 11/58 Johann Collot collot@in2p3.fr http://lpsc.in2p3.fr/collot UdG
12/58 Johann Collot collot@in2p3.fr http://lpsc.in2p3.fr/collot UdG
few remarks : − dE non relativistic particles − 5 / 3 - for βγ < 1 : dx ~ - for βγ ~ 3-4 : − dE is minimal over a large energy plateau . A particle in this state dx is called a minimum ionizing particle (MIP) MIP In media composed of light elements : − dE ≃ 2 MeV dx -2 g cm − dE - for βγ > 4 : relativistic increase of as ln wich is tempered by -δ/2 correction. dx - I : mean excitation and ionization energy , I = 15 eV for atomic H and 19.2 eV for H 2 I = 41.8 for He I = 15 Z 0.9 eV for Z > 2 2 At medium energy : 2 m e max = 2 m e 2 m 0 ≪ 1 T e 2 [ ln 2 m e 2 2 2 − 2 Z 2 − C e − dE dx [ MeV 2 ]= 0.3071 A g ⋅ z − Z ] I g / cm 13/58 Johann Collot collot@in2p3.fr http://lpsc.in2p3.fr/collot UdG
Density effect correction When energy increases, stopping power decreases to a minimum (1/β 2 dependance) and then starts rising again due to logarithmic term. In fact, the max. transverse electric field increases as γ but its influence is screened by nearby atoms beyond a distance of 70 (A(g)/ρ(g)Z) 1/2 Å (shown by Bohr). This density effect tempers the relativistic rise. Studies have be carried-out by Sernheimer, Peierls, Berger & Seltzer (see Leroy & Rancoita). The density correction effect term, δ is given by : S 1 : δ= 2ln (β γ)+ C + a [ β γ ) ] md S 1 ln ( 10 ) ln ( 10 1 2 S 0 : δ=δ 0 ( β γ for for S 0 <β γ< 10 10 β γ< 10 S 0 ) 10 C =− 2ln ( I S 1 : δ= 2ln (β γ)+ C for where : )− 1 β γ> 10 h ν p ν p = √ nr e c² with in which n is the density of electrons and r e the classical radius of e - : r e = 2.82 fm π show that at very high energy : S 1 β γ> 10 h ν p ≃ 28.7 √ ρ( g / cm³ ) Z eV 2 Z max 2.A ( g )[ ln ( 2 m e T e A ( g ) −( dE dx )[ MeV z 2 ]= 0.3071 2 )− 1 ] g / cm ( h ν p ) 14/58 Johann Collot collot@in2p3.fr http://lpsc.in2p3.fr/collot UdG
15/58 Johann Collot collot@in2p3.fr http://lpsc.in2p3.fr/collot UdG
Restricted energy loss knock-on electron (delta ray) generated by a 180 GeV muon as observed by the experiment GridPix at CERN SPS. High energy transfers generate delta rays that may espace the detector if it is too thin. So average energy deposits are very often much smaler thant predicted by B&B. If T 0 is the average maximal delta ray energy that can be absorbed in the detecting medium, a better estimate of the average deposited energy is given by : 2 γ 2 Z 2 T 0 2 ln ( 2 m e β 2 T 0 − C e 2 )−δ(γβ) )−β −( dE dx )[ MeV 0.3071 z ( 1 2 ] = 2 ( 1 + Z ) 2 γ − 1 ) 2 2 2 g / cm A ( g mol β 2 m e β I At extremely high energies, when β γ> 10 S 1 , stopping power reaches a constant called Fermi plateau. 2 Z 2.A ( g ) ln ( 2 m e T 0 −( dE dx )[ MeV z 2 ]= 0.3071 2 ) g / cm ( h ν p ) 16/58 Johann Collot collot@in2p3.fr http://lpsc.in2p3.fr/collot UdG
Fermi plateau measured in silicon 17/58 Johann Collot collot@in2p3.fr http://lpsc.in2p3.fr/collot UdG
Delta rays (secondary electrons) The differential probability to generate a delta ray of kinetic energy T is given by : 2 Z dw ( T , E ) F ( T ) z 2 g − 1 cm − 1 = 0.3071 MeV dTdx 2 2 2.A ( g )β T F(T) is a spin-dependent factor. 2 T for spin-0 particles F ( T )= F 0 ( T )=( 1 −β ) T max 2 for spin-1/2 particles F ( T )= F 1 / 2 ( T )= F 0 ( T )+ 1 2 ( T E ) T m e 2 T m e F ( T )= F 1 ( T )= F 0 ( T )( 1 + 1 2 )+ 1 3 ( T ( 1 + 1 for spin-1 particles E ) 2 ) 3 2 m 0 m 0 18/58 Johann Collot collot@in2p3.fr http://lpsc.in2p3.fr/collot UdG
Delta rays (secondary electrons) For T << T max and T << m 0 2 / m e , 2 Z dw ( T , E ) z 1 2 g − 1 cm − 1 = 0.3071 2 MeV dTdx 2 2.A ( g )β T This allows to compute an approximate probability to generate a delta ray of kinetic energy greater than T s in a thin absorber of mass thickness x : 2 Z z 1 show this expression w ( Ts, E, x )≃ 0.3071 x 2 T s 2.A ( g )β 19/58 Johann Collot collot@in2p3.fr http://lpsc.in2p3.fr/collot UdG
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