Interactions of Particles/Radiation with Matter
ESIPAP : European School in Instrumentation for Particle and Astroparticle Physics
Lucia Di Ciaccio - ESIPAP IPM - January 2018 1 Non-exhaustive list of « Particles/Radiation » and « Matter »
PARTICLES RADIATION 4 • 2He α radiation MATTER • e ± β± radiation • γ e.m, X, γ radiation • μ, γ, e±, π, ν ,p … cosmic radiation
PARTICLES <--> RADIATION 2 aspects of the same « entity »
De Broglie relation λ = h/p
( h = Planck constant)
MATTER PET • detectors ( research, medical app.,..) • humain tissus/body ( medical app.) • electronic circuits • Louvre paintings • beauty cream, potatos, …
Lucia Di Ciaccio - ESIPAP IPM - January 2018 2 Motivation
§ The interaction between particles & matter is at the base of several human activities § Plenty of applications not only in research and not only in Particle & Astroparticle Very important for particle detection ! § In order to detect a particle, the latter must interact with the material of the detector, and produce ‘a (detectable) signal’
Charged particle electrical signal + - + - + - gas
Charged particle
photodetector
The understanding of particle detection requires the knowledge of the Interactions of particles & matter 3 Lucia Di Ciaccio - ESIPAP IPM - January 2018 Brief outline and bibliography
Two lectures + two tutorials § Interaction of charged particles « heavy » (mPa >>me ) « light » (mPa ~ me ) § Interaction of neutral particles Photons Neutral Hadrons: n, π0, …
§ Radiation detection and measurement, G.F. Knoll, J. Wiley & Sons § Experimental Techniques in High Energy Nuclear and Particle Physics, T. Ferbel, World Scientific § Introduction to experimental particle physics, R. Fernow, Cambridge University Press § Techniques for Nuclear and Particle Physics Experiments , W.R. Leo, Springer-Verlag § Detectors for Particle radiation, K. Kleinknecht, Cambridge University Press § Particle detectors, C. Grupen, Cambridge monographs on particle physics § Principles of Radiation Interaction in Matter and Detection, C. Leroy, P.G. Rancoita, World Scientific
§ Nuclei and particles, Emilio Segré, W.A. Benjamin “The classic “ § High-Energy Particles, Bruno Rossi, Prentice-Hall
Lucia Di Ciaccio - ESIPAP IPM - January 2018 4 Also: Particle Data Group http://pdg.lbl.gov/2017/reviews/rpp2017-rev-passage-particles-matter.pdf
For ‘professionals’(*) : GEANT4 (for GEometry ANd Tracking) (Platform for the simulation of the passage of the particles through the matter Using Monte Carlo simulation)
My slides have been inspired by :
Hans Christian Schultz-Coulon’s lectures Johann Collot @ ESIPAP 2014
(*) more exists: Fluka Garfield (simulation gas detector)
5 Lucia Di Ciaccio - ESIPAP IPM - January 2018 Interaction Cross Section (σ) definition Target parameters: σ characterises the probability of a given interaction process n ≡ number of target particles M = target mass S0 N have interacted Amol = molar mass ρ = target density N0 particles X N0- N on area S0 23 -1 (thin) target NA= 6.022 10 mol (Avogadro number)
Number of interactions per number of target particles in unit time σ ≡ Incident flux
Number of interactions per number of target particles in unit time = (1/n) * dN/dt
Incident flux = (1/S0) * dN0 /dt Interaction probability
σ ≡ [(1/n) * dN/dt] / [1/S0 * dN0 /dt ] = (dN/ dN0 * (S0 / n )
where n = (M/Amol) NA = (ρ V) (NA/Amol) = ρ (S0 X) (NA/Amol)
σ doesn’t depend from S0 6 Lucia Di Ciaccio - ESIPAP IPM - January 2018 Cross section (σ)
σ ≡ ( interaction probability) * (S0 /n) n ≡ number of target particles . . . . σ ≡ area of a small disk . . . around a target particle X target S0
1 barn = 10-28 m2 = 10-24 cm2 [ σ ] = [ l ]2 --> σ is measured in m2 or barn 1 mbarn = 10-27 cm
Oder of magnitude of cross sections : ‘strong interaction’
48 -22 2 Neutron of ~ 1 eV on 113Cd σ = 100 barn = 10 cm ‘weak interaction’
See also Neutrino of ~ 1 GeV on p σ= 10-38 cm2 Marco Delmastro lectures Lucia Di Ciaccio - ESIPAP IPM - January 2018 7 Mean free path λ (*) λ = Average distance traveled between two consecutive interactions in matter 1 Another way of expressing the λ ≡ probability of a given process σ nv σ total interaction cross-section
nv number of scattering centers per unit volume nv = (ρ NA)/Amol
Order of magnitudes: Electromagnetic interaction : λ < ~ 1 μm N0 N(X) = number of Strong interaction : λ > ~ 1 cm surviving particles at thickness X Weak interaction : λ > ~ 1015 m N(X) A practical signal ( > 100 interactions or ‘hits’ ) results from the electromagnetic interaction -x/λ N(X) = N0e N0/e Particle detection proceeds in two steps : 1) primary interaction 2) charged particle interaction producing the signals X= λ X (*) Also: λ = absorption length, interaction length, attenuation length, …then σ is the cross-section for the corresponding process (see later) Lucia Di Ciaccio - ESIPAP IPM - January 2018 8 Examples: detection of photons(γ), π0(2γ), neutrons(n), neutrinos (ν) Signals are induced by e.m. interactions of charged particles in detectors
Detector Detector γ τ (π0) ~ 10-16 s Compton γ e+ scattering γ 0 e- π e- e- Pair γ e+ γ production e+ e- _ ν e Detector
_ νe
_ νe Nuclear reactor _ + νe p à n e Lucia Di Ciaccio - ESIPAP IPM - January 2018 9 Useful relations of relativistic Kinematics and HEP units
1 m0 ≡ rest mass • p = m 0 γ v γ ≡ β ≡ v/c γ ≡ Lorentz boost √1- β2 m ≡ m0 γ
2 • Kinetic energy Ek = (γ-1) m0 c
2 2 2 • Total energy E = √ (pc) + (m0 c )
2 2 2 2 • Total energy E = Ek + m0 c = m0 γ c = m c γ =E/(m0 c ) E = m c2 « equivalence mass & energy » Units : [E] = eV [m] = eV/c2 [p] = eV/c
See Marco Delmastro [l] [c] = [l] = [t] lectures « Natural units » h =1 [t] c = 1 [E] = [p] = [m] = [ν] = [t]-1 Lucia Di Ciaccio - ESIPAP IPM - January 2018 10 Outline: main interaction processes
§ Charged particle interactions • 1) Ionization: inelastic collision with electrons of the atoms lecture • 2) Bremsstrahlung: photon radiation emission by an accelerated charge e.m. interactions 2nd • 3) Multiple Scattering: elastic collision with nucleus • 4) Cerenkov & transition radiation effects: photon emission ( • 5) Nuclear interactions (p, π, Κ): processes mediated by strong interactions )
lecture § Neutral particle interactions • Photons : e.m. + - 1rst • photoelectric and Compton effects, e e pair production interactions
• High energy neutral hadrons with > τ ~ 10-10 s ( n, K0, ..) : • nuclear interactions • Moderate/low energy neutrons : • scattering (moderation), absorption, fission • Neutrinos : • processes mediated by weak interactions not treated here After the interaction the particles loose their energy and/or change direction or ‘disappear’
Lucia Di Ciaccio - ESIPAP IPM - January 2018 11 Neutral particle interactions
• Photons • High energy neutral hadrons with > τ ~ 10-10 s ( n, K0, ..) :
• Moderate/low energy neutrons
Lucia Di Ciaccio - ESIPAP IPM - January 2018 12 Interactions of photons (γ)
PC - - γ : particles with mγ = 0, qγ = 0, J (γ) = 1
Since qγ = 0 , the photons are indirectly detected : in their interactions they produce electrons and/or positrons which subsequently interact (e.m.) with matter.
Main processes : 1. Photoelectric effect 2. Compton scattering 3. e+ e- pairs production Eγ
Photons may be absorbed (photoelectric effect or e+e- pair creation) or scattered (Compton scattering) through large deflection angles.
à difficult to define a mean range à an attenuation law is introduced :
−µx µ absorption coefficient I0 I(x) I(x)= I 0e N atoms/m3 A masse molaire
NA 1 NA nombre Avogadro µ = N σ = ρ σ ≡ ρ density A λ σ Photon cross section € See also slide 8 λ Mean free path or x absorption lenght Lucia Di Ciaccio - ESIPAP IPM - January 2018 13 γ Absorption lenght (λ’ ≡ 1/µ’) , , -2 −µx −µx x’ ≡ x ρ [ x’ ] = [m] [ l ] I(x)= I 0e I(x)= I 0e [ µ’ ] = [m]-1 [ l ]2 µ’ ≡ µ/ρ [ λ ] = [ m ] [ l ]-2 λ’ ≡ 1/µ’ € €
Lucia Di Ciaccio - ESIPAP IPM - January 2018 14 1.Photoelectric effect
§ The energy of the γ is transferred to the σphotoe electron and the γ disappeares Absorption edges § Energy of the final electron:
Ee = Eγ - Eelectron binding energy = hν - Eb
Eb = EK or EL or EM etc… 5 ~Z /Eγ
This effect can take place only on bounded electrons since the process (on ‘free’ electrons) γ e --> e Eγ photon energy cannot conserve the momentum and energy
Lucia Di Ciaccio - ESIPAP IPM - January 2018 15 1.Photoelectric effect
2 I = ionisation potential § At « low » energy ( I 0<< Eγ << mec ): 0 α fine structure constant 2 re = α/(mec )= classical electron 2 radius § At « high » energy ( Eγ >> mec ): σphotoe
Example:
-10 aB = 0.53 10 m I0 = 13.6 eV
For Eγ = 100 KeV σ (Fe) = 29 barn σ (Pb) = 100 barn Rayleigh scattering
§ At low energy (E γ <100 keV) , the photoelectric effect dominates the total photon cross section Eγ ~Z5/E 3.5 5 γ ~Z /Eγ 2 mec Lucia Di Ciaccio - ESIPAP IPM - January 2018 16 Atom de-excitation (after photoelectric effect)
Following the emission of a “photoelectron”, the atom is in an excited state De-excitation occurs via two effects (time scale: ~10-16 s) § Fluorescence: Atom*+ à Atom *+ + γ à X rays Observed § Auger effect Atom*+ à Atom *++ + e- ! Auger electron first by Lisa Meitner Auger electrons deposit energy locally (small energy < ~ 10 KeV)
wk
Fluorescence yield : Prob (fluorescence)
wk= Prob (florescence) + Prob (Auger)
Lucia Di Ciaccio - ESIPAP IPM - January 2018 17 General definition of fluorescence
Emission of light by a substance that has absorbed light or other electromagnetic radiation. Energy levels in a molecule :
Important for scintillators !
Lucia Di Ciaccio - ESIPAP IPM - January 2018 18 2. Compton scattering
Scattering of γ on « free » electrons γ + e --> γ’ + e
In the matter electrons are bounded. When the γ energy , Eγ >> binding electron energy the electron can be considered as free. e Eγ Eγ ’ = 2 γ 1+ (Eγ/mec ) (1-cos θγ )
γ’ 2 (1-cos θ ) (Eγ/mec ) e e Ek = Eγ - Eγ’ = Eγ Kinetic energy of the outgoing electron Ek : 2 1+ (Eγ/mec ) (1-cos θγ )
Initial photon can give all its γ Forward scattering θγ = 0 --> Eγ’ = Eγ’max = Eγ Ee= 0 energy to final photon
e γ Backward scattering θγ = π --> Eγ’ = Eγ’min à Ek is max The photon cannot be completely absorbed Lucia Di Ciaccio - ESIPAP IPM - January 2018 19 Compton Edges
• γ Backward scattering θγ = π γ + e --> γ’ + e
e e Eγ’ min Ek max Eγ = Eγ’ + Ek
2 Transfer of complete Eγ 2 E /mec E = E e = E γ γ energy to e via γ’ min 2 k max γ 2 1+ 2 Eγ /mec 1+ 2 Eγ /mec Compton scattering is not possible
Ex. : ‘Compton N Eγ = h ν = 1 MeV e Edges’ Eγ’ min = 0.2 MeV e Ek max = 0.8 MeV Important for single photon detection: the photon is not completely absorbed, the scattered electron misses a small amount of Initial energy
e Ek Lucia Di Ciaccio - ESIPAP IPM - January 2018 20 Compton Cross Section Klein-Nishina Formula (LO QED):
σc= σa + σs
σC 2 @ Small photon energy (Eγ << mec ) 2 σ = 8π/3 r 2= 0.66 barn σC = σTh (1- Eγ/(mc )) Th e (σTh ) = Thomson σ ) 2 @ Large photon energy (Eγ >> mec )
σC ~ (ln Eγ)/Eγ = E’ /E σs γ γ σc Cross section per atom:
Eγ
Lucia Di Ciaccio - ESIPAP IPM - January 2018 21 me 3. Pair production: γà e+ e-
Called also photon conversion For energy-momentum conservation this process cannot take place in ‘vacuum’, an interaction with an electromagnetic field is necessary
Pair production in the field of the nucleus Pair production in the field of an electron (smaller probability ~ 1/Z)
2 Threshold process : Eγ > 2 mec ( 1+ me/mx) mx = mN mx = me
Kinetic energy transferred to the “target” (nucleus or electrons)
Lucia Di Ciaccio - ESIPAP IPM - January 2018 22 First experimental observation of a positron
direction of the high-energy photon
Production of an electron-positron pair by a high-energy photon in a Pb plate
Lucia Di Ciaccio - ESIPAP IPM - January 2018 23 e+ e- pair production cross-section
σPairs
In the high energy regime
(Eγ -->∞ )
accurate to within a few percent down to energies as Threshold = 1.022 MeV low as 1 GeV, particularly for high-Z materials.
Eγ
X0 = Rises above threshold radiation lenght Pair production is
the leading effect at reaches saturation for high energy large Eγ [screening effect]
Lucia Di Ciaccio - ESIPAP IPM - January 2018 24 Normalized e+ e- pair production cross section
LPM =Landau–Pomeranchuk–Migdal cross section. k = Eγ E = Ee
Eγ =
fractional electron energy x = E/k = Ee/Eγ
Lucia Di Ciaccio - ESIPAP IPM - January 2018 25 γ total cross section
Several effects take place (not discussed here): Rayleigh Scattering Photo Nuclear Interactions (giant dipole resonance).
Lucia Di Ciaccio - ESIPAP IPM - January 2018 26 Dependence on Z et on E
Z=53
Z=6
Eγ = 0.1 MeV in C (Z=6) Compton effect is dominant in I (Z=53) Photoelectric effect is dominant
Eγ = 1 MeV Compton effect is dominant
Eγ = 10 MeV in C (Z=6) Compton effect is dominant in I (Z=53) pair production is dominant
Lucia Di Ciaccio - ESIPAP IPM - January 2018 27 Electromagnetic showers See M. Delmastro lectures Dominant processes for photons (and electrons) at very high energies
Einc 1 e- tmax= ln - Ec 0.5 gamma
Einc L 95% tmax + 0.08 Z + 9.6 [X0]
L 95% = longitudinal shower containment
tmax =depth in ratio length units, where the max energy is deposited
Ein= incoming photon energy
Ec= critical energy Also electrons can start e.m. showers 28 Lucia Di Ciaccio - ESIPAP IPM - January 2018 Hadron collisions and interaction lengths The total cross section for very high energy hadrons is expressed as:
σT = σelastic + σinelastic The inelastic part of the total cross-section is susceptible to induce a hadron shower (increase of particles multiplicity) Two mean lengths are introduced: : § nuclear collision length
§ nuclear interaction length
See M. Delmastro slides for more details
95% containment of a hadronic shower is for a thickness of :
L 95%(in units of λI) 1+1.35 ln (E(GeV)) à ~ 10 interaction lengths are needed to contain a 1 TeV hadronic shower
In high A materials λ I > X0 This explains why hadron calorimeters are after installed electromagnetic Lucia Di Ciaccio - ESIPAP IPM - January 2018 29 Lucia Di Ciaccio - ESIPAP IPM - January 2018 30 Lucia Di Ciaccio - ESIPAP IPM - January 2018 31 Neutron interactions
Electric charge of the neutron n : qn = 0
The n interacts via « strong interaction » with nuclei (short range force ~ 10-13 cm)
Classification of neutrons:
Cold or ultracold neutrons En < 0.025 eV Thermal or slow neutrons En ~ 0.025 eV Intermediate neutrons En ~ 0.025 eV ÷ 0.1 MeV Fast neutrons En ~ 0.1 ÷ 10-20 MeV High energy neutrons En > 20 MeV
Additional classification:
Slow neutrons (absorbed) En < ~ 0.5 MeV Fast neutrons En > ~ 0.5 MeV E = 0.5 MeV = ‘cadmium cutoff’
Main interaction processes of n: scattering (elastic and inelastic), absorption, fission hadron shower production depending on the neutron energy
Lucia Di Ciaccio - ESIPAP IPM - January 2018 32 Neutron interactions
A A (*) Scattering with nuclei : n + ZX -> ZX + n Elastic à important for moderation Inelastic Absorption & Nuclear reactions:
A A n + ZX -> Z-1Y + p
A A-3 4 n + ZX -> Z-2Y + 2He
A A-1 n + ZX -> ZX + 2n
A A+1 n + ZX -> ZX + γ radiative capture of n
A A1 A2 Fission: n + ZX -> Z2Y + Z2Y + n + n +… fission
Cross section ≈ 1/vn ( more probable for low energy) + resonant peaks
Hadron shower En > ~ 100 MeV
Lucia Di Ciaccio - ESIPAP IPM - January 2018 33 Cross section of low energy neutrons (n)
Neutron cross section on H2O, paraffine and protons
n intermediate Fast n
σtot(barn)
1 barn = 10-24 cm2 = 10-28 m2
Radius of the nucleon R ≈ 10-15 ÷ 10-14 m En 1 barn ≈ R2
Lucia Di Ciaccio - ESIPAP IPM - January 2018 34 34 Low energy neutron (n) cross section
‘résonances' NB : Very different scales on the vertical axis σtot(barn)
238U σtot(barn)
27Al
1/v
En(eV) En(eV) Lucia Di Ciaccio - ESIPAP IPM - January 2018 35 Charged particle interactions
• 1) Ionization: inelastic collision with electrons of the atoms • 2) Bremsstrahlung: photon radiation emission by an accelerated charge • 3) Multiple Scattering: elastic collision with nucleus • 4) Cerenkov & transition radiation effects: photon emission ( • 5) Nuclear interactions (p, π, Κ): processes mediated by strong interactions )
Lucia Di Ciaccio - ESIPAP IPM - January 2018 36 1) Inelastic collision with electrons of the atoms
Main e.m. process for heavy (MPa >> me) charged particles Pa (ex. μ)
• ionisation e- e- e- e- e- e- +7e e- e- e- e- +7e e- e- e- e- Pa
+ - Pa + atom ---> atom + e + Pa
e- e- e- e- • excitation e- e- e- +7e e- e- +7e e- e- e- e- P e-
Pa + atom ---> atom* + Pa atom + γ § Both processes together ( ionization & excitation) can also happen § Inelastic collisions on nucleus(N) are much less frequent (since the energy transfer depends inversely on the target mass and mN >> me )
§ The particle Pa looses a bit of its energy (in each of the many collisions), its directions is ~ unchanged. Lucia Di Ciaccio - ESIPAP IPM - January 2018 37 Average energy loss per unit of lenght (- dE/dx) of Pa due to inelastic collisions with electrons of the atom
Analytic formula: Bethe & Bloch formula
let’us derive here a simplified ‘semi-relativistic’ expression for - dE/dx
Lucia Di Ciaccio - ESIPAP IPM - January 2018 38 Simple computation of the average energy loss of particle Pa (derivation of the B&B formula) Assumptions: § Electrons considered free and initially at rest Heavy particle § P a moving slightly during interaction (v ~ const) § Heavy particle undeflected § Electric force acting on e (during dt = dx/v): dp F = E = electric field generated by Pa dt
(*) Δp = F dt = e E dt = e E dx/v E_ from Gauss law: Φ ( E ) = 4π ze _ | _ | | _ | _ I S E_ | 2 π b dx = 4π ze
Momentum transferred to one electron: Δp = 2z e2 / (b v) 2 z2 e4 2 Kin. Energy transferred to one electron: ΔE = Δp /(2 me) = (non-rel.) 2 2 me v b
(*) Δp // effects average to 0 (symmetry)
Lucia Di Ciaccio - ESIPAP IPM - January 2018 39 Simple computation of the average energy loss
2 z2 e4 ΔE = 2 2 me v b b Heavy particle § Effect of the interaction of Pa with the electrons in dV M (energy loss by Pa): 4π z2 e4 db - dE (b) = ΔE Nc dV = Nc dx 2 dV ≈ 2 π b db dx me v b
Nc = (ρ NAZ)/Amol= number of electrons per unit of volume
2 4 dE 4π z e bmax - = Nc ln 2 De Broglie wavelength of electron dx me v bmin (after an head-on collision ve ≈ 2*particle velocity )
bmin from De Broglie wavelenght bmin = λe = h/pemax ~ h/(2 me γ v)
Lucia Di Ciaccio - ESIPAP IPM - January 2018 40 Simple computation of the average energy loss
2 4 dE 4π z e bmax - = Nc ln b 2 Heavy particle dx me v bmin M
“Collision time” bmax from “adiabatic invariance” : the perturbation should occur in a time short compared Electron period in atom to the period τ of the bound electron
Particle velocity b/(γ v) < τ where τ = 1/νe b = ( γ v )/ν max e Orbital frequency
2 4 2 2 2 dE 4π z e 2 mec β γ - = Nc ln 2 I = h νe dx me v h νe
Close to the Bethe&Bloch formula ( within a factor ~ 2)
Lucia Di Ciaccio - ESIPAP IPM - January 2018 41 Average energy loss by a charged particle ( mPa>> me ) in matter
Incident charged ‘heavy’ particle P of energy E, M matter (e.x. gaz of a detector) a dx Bethe-Bloch formula (B & B) 2 2 2 2 dE Z z 2 me c β γ Tmax C - = K ρ [ ln - 2 β2 - δ - 2 ] dx A β2 I2 Z
2 2 -1 2 2 K = 4 π NA re me c = 0.307 MeV g cm r = classic radius of electron = α/(m c2) = 2.8 fm e e 2 2 m = electron mass = 511 KeV 2meβ γ e max z = charge of incident particle in unit of e Tmax = Ee - me = 2 β = particle speed in unit of c (ECM/M) 2 γ = 1/ √1- β 2 2 2 Tmax ~ 2 mec β γ for γ << mPa / (2 me ) Tmax = maximum Kin energy transferred in a collision) ρ = density of the matter Z, A = atomic number, atomic weight of the matter I = effective excitation potential of the matter Difficult to compute --> obtained from dE/dx I (eV) = (12+ 7/Z) Z (Z ≤ 12) I (eV) = (9.76 + 58.8 Z -1.19) Z (Z ≥ 12) Lucia Di Ciaccio - ESIPAP IPM - January 2018 42 Shell (C) and Density(δ) effect corrections
C = Relevant at low energy. Small correction. The particle velocity ~ orbital velocity of e à the assumption that atomic electrons initially are at rest breaks. Takes into account binding energy. The energy loss is reduced. The capture process of the particle is possible
δ = “Density effect”. Relevant at high energy. The electric field of the particle polarise the atoms of the matter à The energy loss is reduced since shielding of electrical field far from the particle path à moderation of the relativistic rise It depends on the particle speed and on the matter density Density effect leads to “saturation” at high energy _ For high energy: hωp = “Plasma energy”
+ + _ + + _ _ + _
P _ _ a _
_ Polarisation effect (correction ) + +
+ δ + à reduction of the energy loss
Lucia Di Ciaccio - ESIPAP IPM - January 2018 43 Stopping power or mean specific energy loss = dE/dx (MPa >> me)
Few remarks:
♦ dE/dx = f( β ) ~ 1/β 2
♦ dE/dx doesn’t depend on MPa ~ ln βγ ♦dE/dx ∝ z2 (particle charge)
u On vertical l’axis -dE/(ρ dx) (MeV cm2)/g Why?
β γ = 3-4 Tcut (Minimum Ionizing Particle= MIP) restricted energy loss discussed later See Marco Delmastro lectures for explanation of 1/β 2 and ln βγ
Lucia Di Ciaccio - ESIPAP IPM - January 2018 44 Stopping power
* If material thickness is measured in ρ dx (g/cm2) à on vertical l’axis -dE/(ρ dx) (MeV cm2)/g à the dependence on the material is reduced since ~ Z/A ~ 0.5
1 - 2 MeV g-1 cm2 « minimum of ionisation » β γ A-Z
Z
Lucia Di Ciaccio - ESIPAP IPM - January 2018 45 Stopping power at the minimum of ionization in greater detail
Lucia Di Ciaccio - ESIPAP IPM - January 2018 46 Use of dE/dx for particle identification
1 • p = m γ c β γ ≡ √1- β2
Measuring independently p and γ β one can extract m à particle identification
p d K μ π Plot from PEP4-9 Time Projection Chamber (TPC) e @SLAC (late ’70)
Lucia Di Ciaccio - ESIPAP IPM - January 2018 47 Knock-on electrons or delta(δ) rays or secondary electrons High energy transfers generates secondary electrons (delta rays) 2-metre hydrogen Distribution (prob.) of δ with kinetic energies T I : bubble chamber B=1.7 T
K = 0.307 F(T) = Spin dependent factor β, mPa = speed and mass of primary particle x = “mass thickness” (ρ*t) ~ 70 cm Spin 0
Spin 1/2 K- beam, 4.2 GeV Spin 1 Pa Pa
2 For T << Tmax & T << mPa /me & F(T) = 1 : approximate probability to generate a δ with T > Ts in a thin absorber of thickness x:
Lucia Di Ciaccio - ESIPAP IPM - January 2018 48 Delta(δ) rays in Micro Pattern Gas Detector
Pa
μ
e
δ electron knocked out by a 180 GeV muon, measured with a GridPix detector (Micro Pattern Gas Detector)
δ rays produce ionization. This is called secondary to distinguish from the primary (impinging particle) For a β ≈ 1 particle, on average one collision with T > 10 keV along a path length of 90 cm of Ar gas δ rays are ~ rare, why to care? Lucia Di Ciaccio - ESIPAP IPM - January 2018 49 Restricted energy loss § δ rays that may escape the detector if it is too thin à The average energy deposits are very often much smaller than predicted by Bethe & Bloch
If the energy transferred is restricted to ! “restricted energy loss”
The difference between the restricted energy loss formula and the B & B is given by the contribution of the (escaping) δ rays
S1 At very high energies ( β γ > 10 , S1 ~ 2-5 ) the stopping power reaches a constant called “Fermi plateau”:
h νp = cut _ hωp = “Plasma energy”=
S1 , hνp = “density effect” parameters
Lucia Di Ciaccio - ESIPAP IPM - January 2018 50 Density effect parameters
Lucia Di Ciaccio - ESIPAP IPM - January 2018 51 -dE/dx μ on
‘Fermi plateau’
Δp = most probable energy loss (see later) x = mass thickness
Lucia Di Ciaccio - ESIPAP IPM - January 2018 52 -dE/dx Fluctuations ! Energy straggling
§ Bethe-Bloch formula describes mean energy loss per unit of lenght. The actual energy loss ΔE in a material of thickness x is: § N number of collisions N § δE energy loss in a single one collision ΔE = Σ δEn (β) n=1 δE stochastic fluctuations à energy straggling ( besides it depends on β of the particle)
Complex subject first studied by L. Landau and then by P.V. Vavilov No general exact solutions, few approximate formulas help to estimate it. Introduce : Significance parameter : K K =
mean energy loss = NB: ΔE depends on thickness x in thickness ρdx 53 Lucia Di Ciaccio - ESIPAP IPM - January 2018 ΔE distribution Ø Thin absorbers ( K << 1 ): ΔE = Most Probable § MP Landau distribution. Not analytic, useful approximation : value See previous slide
§ Improved (I) generalized energy loss distribution : convolution of a Landau with a Gaussian (takes better into account distant collisions)
σI = inelastic collisions cross section
Ø Thick absorbers ( K >> 1 ): The distribution tends to a Gaussian
Lucia Di Ciaccio - ESIPAP IPM - January 2018 54 Energy loss ( Δ ) distribution
Δ = energy loss x = thickness x =
Δp = most probable value
Lucia Di Ciaccio - ESIPAP IPM - January 2018 55 Stopping power of a compound medium
ρ = density of element i • For a compound of f elements: i dE = stopping power of element i dE dE ρ dx f i - = ∑ wi dx 1 dx ρ ρi wi = mass fraction of element i
wi = (Ni Ai )/Am Ni = number of atoms of element i Ai = atomic weight of element i Am = molar mass of compound
Am = ∑ Ni Ai • It is also possible to use effective quantities (empirical):
Zeff = ∑ Ni Zi
Aeff = ∑ Ni Ai
ln Ieff = ( ∑ Ni Zi ln Ii )/ Zeff
δeff = (∑ Ni Zi δi )/ Zeff
Ceff = ∑ Ni Ci
Lucia Di Ciaccio - ESIPAP IPM - January 2018 56 Particle Range in matter : R § Charged particles ionize, lose energy until their energy is (almost) zero. The distance to this point is called the range of the particle: R ~0 ~0 dx dE -1 R (E ) = dE = dE k dE dx E k Ek § This expression ignores the Coulomb scattering (producing a zig-zag trajectory of P) § The mean range
Range of heavy charged particles • R may be used to evaluate the particle energy
b b ~ 1.75 for E < minimum R ∝ Ek k ionisation • Scaling laws • Particle 1 and 2, in same material
2 M2 z1 R (E ) ∝ R (E * M /M ) 2 k2 2 1 k1 1 2 M1 z2
• Same particle in two different materials(1,2))
R1 ρ2 √A1 = R(cm) R2 ρ1 √ A2
• Compounds: Rcompound = Am / ∑ (Ni Ai /Ri)
Lucia Di Ciaccio - ESIPAP IPM - January 2018 58 Mean Particle Range
§ If the medium is thick enough, a particle will progressively decelerate while increasing its stopping power ( β-5/3) until it reaches a maximum (called the Bragg peak).
§ Possibility to precisely deposit dose at well defined depth dependent on Ebeam (Remember also dependence on z2)
Applications: Tumor therapy
59 Lucia Di Ciaccio - ESIPAP IPM - January 2018 Heidelberg Ion-Beam Therapy Center (HIT)
~ 50 centers around the world Lucia Di Ciaccio - ESIPAP IPM - January 2018 60 Stopping power of e± by ionization and excitation in matter For e± the Bethe-Bloch formula must be modified since: 1) the change in direction of the particle was neglected; for e± this approximation is not valid (scattering on particle with same mass) 2) Pauli Principle : the incoming and outgoing particles are the identical particles
dE Z z2 τ 2 (τ+2 ) C 2 2 - = 2 π NA re me c ρ [ ln + F(τ) - δ - 2 ] 2 2 2 2 dx A β ( I / me c ) Z 1 2 2 (τ /8) -(2 τ +1) ln 2 τ = - 1 = Ek/(mc ) For electrons: F(τ) = 1-β 2 + √1-β 2 (τ +1)2 β 2 14 10 4 For positrons : F(τ) = 2 ln2 - 23 + + + 12 τ+2 (τ+2 ) 2 (τ+2 )2 e± loose more energy wrt heavier particles since they interact with particles of the same mass § When a positron comes to a rest it annihilates : e+ + e- → γ γ of 511 keV each § A positron may also undergo an annihilation in flight: with a cross section : Lucia Di Ciaccio - ESIPAP IPM - January 2018 61 2. Bremsstrahlung. Mean radiative energy loss.
2 § An accelerated (or decelerated) charged particle ( Pa ) Nucleous F=Qq/r emits electromagnetic radiation ( γ ) (Z,A) § Very fundamental process ! P § Here the process takes place in the Coulomb a γ field of the nucleus. The amount of screening from a=F/m 2 dv the atomic electrons plays an important role dσ ∝ Z 3 v N = atoms/cm (N= ρ NA/A) § Relevant in particular for e± due to their small Z = atomic number mass E0 = Initial energy of particle Pa v =E / h dE 0 o dσ ν = E /h − ( ) =N hv dv =NE φ(Z 2 ) 0 0 ∫ 0 hν = energy of emitted γ dx brem 0 dv ~ d σ = Differential cross section of d ν the bremsstralung process
If Pa = electron: If E >> m c2 et E << 137 m c2 /Z 1/3 φ (Z2) = 4α Z2 r 2 ln (2E /m c2 -1/3-f(Z)) 0 e 0 e e 0 e α = 1/137 2 1/3 2 2 2 -1/3 If E0 >> 137 mec / Z φ (Z )= 4α Z re ln (183 Z -1/18 - f(Z)) 2 re= α/(mec ) See W.R. Leo f(Z)= Coulomb correction Lucia Di Ciaccio - ESIPAP IPM - January 2018 62 2. Bremsstrahlung – Energy Spectrum
Normalized bremsstrahlung cross section vs y (= k / E0) where k = Eγ à y = fraction of the electron energy (E0) transferred to the radiated γ
K dσ/dk = ν dσ/dν (for given E0)
For high energy E0 (small y):
E0 =
Formula accurate except for y =1 and y = 0
see PDG for further details y (= k / E0) LPM =Landau–Pomeranchuk–Migdal cross section.
Lucia Di Ciaccio - ESIPAP IPM - January 2018 63 Bremsstrahlung. Mean radiative energy loss
For a particle of charge z and mass m:
brem brem
§ Relevant in particular for e± due to their small mass
§ Shown so far is the mean energy loss due interaction in the field of the nucleus § Contribution also from radiation which arises in the fields of the atomic electrons. § Cross section are given by the above formula but replacing Z2 with Z. § T he overall contribution can be approximated by replacing Z2 by Z (Z+1) in all the above formulas
Lucia Di Ciaccio - ESIPAP IPM - January 2018 64
Comparison -dE/dx Bremsstrahlung vs ionisation/excitation
/g) 2
total collisions inélastiques
dx
(MeV cm dE
Energy (MeV)
§ The average energy loss due to ionisation/excitation dE increases with the log of the energy and linearly ∝ Z/A, 1/β 2 ln E dx with Z : ion./excit.
§ The average energy loss of due to brem increases linearly dE 2 2 2 ∝ Z /A, E, 1/m with energy ans linearly with E and Z : dx brem
Energie loss due to brem is a discrete process: results from the emission of ~ 1 γ ou 2 γ --> fluctuations
Lucia Di Ciaccio - ESIPAP IPM - January 2018 65 Critical energy (Ec)
§ The relevance of bremsstralung wrt ionisation depends on the critical energy (Ec) of the particle Pa in the material
§ The critical energy (Ec) is the energy at which the ionization stopping power is equal to the mean radiative energy loss.
@ E = Ec @ E > Ec @ E < Ec dE dE dE dE dE dE = > < dx dx dx dx dx dx brem. ion brem. ion brem. ion
± For e in : Pb Ec = 9.5 MeV For liquid and solids: Ec ~ 610 MeV/(Z+1.24) Cu Ec = 24.8 MeV
Fe Ec = 27.4 MeV For gas Ec ~ 710 MeV/( Z + 0.92) Al Ec = 51 Mev
For other particles Ec would scale according to the square of their masses with respect to the electron mass. Lucia Di Ciaccio - ESIPAP IPM - January 2018 66 Radiation lenght X0
dE dE N = atoms/cm3 For E >> Ec ≈ dE dx dx - brem. = N E0 φ tot dx dE -x/X0 - = N φ dx E = E0 e X0 ≡ 1/(N φ ) E E
X0 ≡ 1/(N φ ) ≡ radiation lenght ≡ distance after which an high energy electron has E0/e lost 1/e of his energy by radiation x X0
Mean radiated energy of an electron over a path x in the medium: brem.
Lucia Di Ciaccio - ESIPAP IPM - January 2018 67 Radiation lenght X0
Pb = 0.56 cm
X0 Fe = 1.76 cm Air = 30050 cm
’ X’0 ≡ X0 ρ
Expressing the mean radiated energy in unit of X0’ à The probability of the process becomes less dependent on the material
Pour un composé de N éléments :
1 1 w = fraction in mass of element i = ∑ w i i i X0 X0i X0i = radiation lenght of element i
Lucia Di Ciaccio - ESIPAP IPM - January 2018 68 For electrons
Lucia Di Ciaccio - ESIPAP IPM - January 2018 69 Electron interactions in copper : higher energies dE dE dE = + dx dx dx tot ioniz brem
Ec
Lucia Di Ciaccio - ESIPAP IPM - January 2018 70 Interactions of electrons in lead: a more complete picture
Moller scattering e- e- à e- e- Bhabha scattering e+ e- à e+ e- Positron annihilation e+ e- à 2γ
Lucia Di Ciaccio - ESIPAP IPM - January 2018 71 Total energy lost by a muon (µ) per unit length in copper
Bethe-Bloch formula
dE 1 ∝ dx β 2
“kinematic term”
density effect “minimum of ionization” βγ ≈ 3-4 “relativistic rise” dE ∝ ln β 2γ 2 dx At very low energy the Bethe-Bloch formula is not valid since the speed of the interacting particle is ~ speed of electrons in the atoms. For βγ < 0.05 there are only phenomenological
fitting formulae Lucia Di Ciaccio - ESIPAP IPM - January 2018 72 3. Elastic scattering with nuclei
A charged particle Pa traversing a medium is deflected many times (mainly) by small-angles essentially due to Coulomb scattering in the electromagnetic field of the nuclei.
Pa The energy loss ( or transferred to the nuclei ) Pa is small ( mnucleus >> mPa) therefore neglected, ( nucleus + e ) The change of direction is important.
§ A single collision is described by the Rutherford formula (ignores spin and screening effects)
2 dσ 2⎛ mec ⎞ 1 = 4zZre ⎜ ⎟ d ⎜ p ⎟ 4 Pa Ω ⎝ β ⎠ sin θ 2 ϑ ϕ
dΩ = sinϑ dϑ dϕ § Multiple scattering: Ncollisions > 20 The particle follows a zig-zag trajectory Deflection angles are described by the Molière theory
H. A. Bethe” “Molière's Theory of Multiple Scattering” Phys. Rev. 89, 1256 – Published March 1953
Lucia Di Ciaccio - ESIPAP IPM - January 2018 73 3. Multiple scattering through small angles (< ~100)
For small scattering angles, the t distribution of ϑx ≈ Gaussian
P ϑspace a x 1 ϑ 2 2 x prob(ϑx) dϑx = exp(- ϑx /(2 σ0 )) dϑx √2π σ0
2 2 2 y ϑ space = (ϑx +ϑy ) 2 2 2 ( similar for ϑy and ϑspace = ϑx +ϑy )
Where: t = medium thickness 13.6 MeV t t ρ = matter density σ0 = |z| 1+0.038 ln β p X0 X0 X0 = radiation lenght β, p = speed/c and momentum t of the incident particle
Particles emerging from the the medium are also d laterally shifted : 1 drms = t σ ϑx √3 0
Lucia Di Ciaccio - ESIPAP IPM - January 2018 74 Momentum resolution Multiple scattering impacts the measurement of the momentum Assume B | v particle:
2 Mv /R = q | v ^ B | p = B R (q = 1)
The momentum is measured from R, which is obtained from L and s
B ¤
Φ ~ L/R
The precision on the momentum will depend on the precision on the track reconstruction and also on the multiple scattering that the particle undergoes
Lucia Di Ciaccio - ESIPAP IPM - January 2018 75 Momentum resolution p = q B R ~ B L2 to momentum resolution
(β=1) d
Φ
Due to multiple scattering Lucia Di Ciaccio - ESIPAP IPM - January 2018 76 3. Back-scattering of electrons
* increases with Z of the material
* is relevant for low energy electrons Number of backscattered electrons = η Number of incident electrons
η η
Kinetic energy (MeV) Energie cinétique(MeV) Effect to take into account when building a detector for low energy electrons (< ~ 10 MeV) 77 Lucia Di Ciaccio - ESIPAP IPM - January 2018 77 4.Cherenkov light emission
Radiation emitted when a charged particle crosses a medium Charged at a speed > than the phase velocity of light in the medium P ϑ vparticle > c/n
n = refracting index § The medium is electrically polarized by the particle's electric field (oscillating dipoles) § When the particle travels fast this effect is left in the wake of the particle. § The emitted energy radiates as a coherent shockwave Cherenkov emission (e) TRIGA reactor γ c/n t 1 ϑ P cos ϑ = P βn β c t Lucia Di Ciaccio - ESIPAP IPM - January 2018 78 4.Cerenkov light emission
§ Number of photons N emitted per unit path length and unit of wave length strongly peaked at short λ z2 (λ)
§ Number of photons per unit path length is:
z2
Assuming n ~ const over the wavelength region detected n = refracting index z2
in λ range 350-500 nm (photomultiplier sensitivity range),
dE/dx due to Cherenkov radiation is small compared to ionization loss (< 1%) and much weaker than scintillating output. It can be neglected in energy loss of a particle, but is Important for particle detection Lucia Di Ciaccio - ESIPAP IPM - January 2018 79 Cerenkov angle 1 cos ϑ = βn
Photon yield
Lucia Di Ciaccio - ESIPAP IPM - January 2018 80 4.Cerenkov light emission
Lucia Di Ciaccio - ESIPAP IPM - January 2018 81 4.Transition radiation
§ When a relativistic charged particle crosses a boundary between media of different dielectric properties radiation is emitted, mostly in the X- ray domain The electric field generated by the particle is different on the two sides
https://arxiv.org/pdf/1111.4188v1.pdf ε ε0
Pa
§ The radiation is emitted in a cone at an angle cos θ = 1/γ § The probability of radiation per transition surface is low ~ 1/2 α (fine structure constant)
AMS detector:
Lucia Di Ciaccio - ESIPAP IPM - January 2018 82 4.Transition radiation
§ The energy of radiated photons Pa= e increases as a function of particle momentum
Useful for particle identification
p(GeV)
Lucia Di Ciaccio - ESIPAP IPM - January 2018 83 Lucia Di Ciaccio - ESIPAP IPM - January 2018 84 END
Lucia Di Ciaccio - ESIPAP IPM - January 2018 85