interaction of elementary particles with mattereisenhar/teaching/... · particle physics detectors,...

II/1Particle Physics Detectors, 2010 Stephan Eisenhardt

Interactionof Elementary Particles

with Matter

Cross section & mean free pathInteraction of charged particlesInteraction of neutral particles


Cross SectionTotal and Differential cross section on single target:

,– Ns = scattered particles / time– F = flux = Ninc / (area · time)– [d(σ)] = 1 barn = 10-28 m2

Mass thickness of target = ρ·t (aka. ‘surface density’)– ρ = density [g/cm3]– t = thickness [cm]– [d(ρ·t)] = g/cm2

Differential cross section per unit mass of extended, but thin target:

– A = target area– δx = target thickness

Probability of interaction for single particle per unit mass of extended, but thin target:

– Ntot = total number of scattered particles / time into all angles

∫ ΩΩ=

dddE σσ )(

Ω=Ω

Ω ddN

xFAE

dd s

ρδσ 1),(

xFANtot σρδ= xxw σρδδ =⇒

Ω=Ω

Ω ddN

FE

dd s1),(σ


Mean Free PathProbabilities of:

– having an interaction between x and dx: w dx– not having an interaction between x and dx: P(x+dx) = P(x)(1- w dx)– not having an interaction after distance x: P(x) ‘survival prob.’

Exponential decay of survival probability:

Mean free path λ (mean distance without interaction):

Survival probability:

Interaction probability:

Interaction probability in dx after survival of travel through x:

)()( wxexP −=

ρσλ 11

)(

)(===

∫∫

wdxxP

dxxxP

⎟⎠⎞

⎜⎝⎛ −

⋅− == λρσx

x eexP )()(⎟⎠⎞

⎜⎝⎛ −

⋅− −=−= λρσx

x eexP 11)( )(int

λρσ λρσ dxedxedxxP

xx ⎟

⎠⎞

⎜⎝⎛ −

⋅− =⋅=+ )()(


Interactions in MatterCharged Particles:Passage of charged particles through matter:

– Energy loss: inelastic collisions with atomic electrons very frequent• soft: exitation (scintillation)• hard: ionisation (knock-on electrons (‘δ-rays’))

– Deflection: elastic scattering from nuclei frequent– Bremsstrahlung:– Photon Emission: (Cherenkov & Transition radiation)– Nuclear reactions: (neutrons, alphas)– Weak interactions: (neutrinos)

Divided into two classes:– heavy particles: μ, π, K, p, d, α, …– particles of electron mass: e-, e+

Neutral Particles:Photons

– Photo electric effect– Compton Scattering– Pair production

Neutrons– nuclear reactions


Interactions of Charged ParticlesCollisions with nuclei not important (me<<mN). Collisions with atomic electrons of absorber material

Dispersion relation:

Optical region: Cherenkov

Absorptive region: excitation, ionisation

X-ray region: Transition radiation

212

222 0122 εεεε

εω

λππνω inck

nck

nc

+===−===

βεβθεεε 11cosreal11 21 ==<<>

n

02 >>ε

1 1 21 <<< εε

e-

θ

khh ,ω

0,mvr

Schematically !

1

ε

ωoptical absorptive X-rayCherenkovradiation ionisation transition

radiation

regime:effect:

Re ε

Im ε


Classical Energy Loss (Bohr)Consider:

– heavy particle (M>>me) with charge ze and velocity v– quasi-free atomic electron at impact parameter b– restriction: interaction short, i.e. electron static

Calculate momentum transfer: current seen by electron

Calculate momentum transfer: energy gained by electron

Energy lost to all electrons in volume dV = 2πb db dx with electron density Ne

bvze

vdxeEdteEFdtI

22=

=== ∫∫ ∫ ⊥⊥ bzedxEzebdxE

Gauss 242 =⇒ ∫∫ = ⊥⊥ ππ

22

422 22

)(bvmez

mIbE

ee

==Δ

min

max2

42

2

42

ln44)()(bbN

vmez

dxdEdx

bdbN

vmezdVNbEbdE e

ee

ee

ππ=−⇒=Δ=−


Classical Energy Loss (Bohr)Integration limits: naïve approach fails

– b=0: infinite energy loss– b=∞: contradicts short interaction time

bmin: maximum energy transfer in head-on collision

bmax: electron bound with orbital frequency ν → interaction time short wrt. τ=1/ν– relativistic interaction time: t~b/γv– upper limit for averaged electron frequencies <ν>:

Bohr’s classical energy loss formula:

Reasonable description for heavy particles like the α-particle or heavier nucleiFails proton or lighter particles due to quantum effects

2

2

min22

2min

2

42

22vm

zebvmbvmez

ee

e γγ =⇒=

νγ

ντ

γvb

vb

=⇒=≤ max1

νγπ

2

32

2

42

ln4ze

vmNvm

ezdxdE e

ee

=−


Bethe-Bloch FormulaCorrect quantum-mechanical calculation leads to the energy loss dE/dx [MeV/cm] due to ionisation caused by heavy (M≥mμ) charged particles:

with maximum energy transfer in single collision Tmax:

and experimental mean excitation potential I:

and its corrections δ(log10(βγ)) (density effect) and C(I,βγ) (shell corrections)

⎥⎦

⎤⎢⎣

⎡−−−=−

ZC

ITcmz

AZcmrN

dxdE e

eeA 22ln4 2

2

max222

21

2

222 δββγ

βρπ

gcmMeVcmrN eeA

222 1535.02 =π

222

2

222

222max 2

121

2 βγβγ

βγ cm

Mm

Mm

cmT eee

e ≅+++

=

– NA = Avogadro’s number– re = classical electron radius– me = electron mass [g]– c = velocity of light

– ρ = density of absorber [g/cm3]– Z = atomic number of absorber– A = atomic weight of absorber

– z = charge of incident particle [e-]– M = mass of incident particle– β = v/c of incident particle– γ = 1 / √1- β2 of incident particle

– often used convention: x =: ρx [g/cm2]→ ‘mass stopping power’

dE/dx: [MeV/g cm2]eVZeVZ

ZIZ

eVZZ

IZ

108.5876.9:13

712:13

19.1 ≈+=≥

+=<

−


Not valid for e-, e+: mproj=mtarget → Bremsstrahlung

small β: (kinematic factor)– dE/dx falls like ∝ 1/β2 (more precise ∝ β-5/3)– shell correction small

βγ ≈ 4: (minimum ionising, MIP)– dE/dx constant

βγ >> 1: (relativistic rise)– dE/dx rises like ∝ ln γ2

– density larger for lighter materials (gases)

dE/dx rather independent of Z (except H2)Mixed materials: (Bragg’s rule)

– add mass stopping powers by weight contributions

– fairly good approximation

Bethe-Bloch Formula

2121 cm MeV g.dxdE −≈

solid line: Allison and Cobb, 1980dashed line: Sternheimer (1954)data from 1978 (Lehraus et al.)

δ correction

C correction

L+⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

22

2

11

11dxdEw

dxdEw

dxdE

ρρρ


Not valid for e-, e+: mproj=mtarget → Bremsstrahlung

small β: (kinematic factor)– dE/dx falls like ∝ 1/β2 (more precise ∝ β-5/3)– shell correction small

βγ ≈ 4: (minimum ionising, MIP)– dE/dx constant

βγ >> 1: (relativistic rise)– dE/dx rises like ∝ ln γ2

– density larger for lighter materials (gases)

dE/dx rather independent of Z (except H2)Mixed materials: (Bragg’s rule)

– add mass stopping powers by weight contributions

– fairly good approximation

Bethe-Bloch Formula

2121 cm MeV g.dxdE −≈

solid line: Allison and Cobb, 1980dashed line: Sternheimer (1954)data from 1978 (Lehraus et al.)

δ correction

C correction

L+⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

22

2

11

11dxdEw

dxdEw

dxdE

ρρρ

Two GammaTPC

e

μπ

Kp


dE/dx Scaling & RangeScaling Law:

– T = kinetic energy

– scaling from particle 1(M1,T1)to particle 2(M2,T2)

Range:– Tmin = minimum energy for dE/dx valid– energy loss statistical → range straggling

→ mean range Rmean & extrapolated range R

– scaling: different particles in same medium– scaling: same particles in different medium

222 )1()( McTMTfzfz

dxdE

−=⎟⎠⎞

⎜⎝⎛′==− γβ

2

12

2

12

121

22

22 )(

MMTT

MMT

dxdE

zzT

dxdE

=⎟⎟⎠

⎞⎜⎜⎝

⎛−=−

∫−

⎟⎠⎞

⎜⎝⎛+= 0

min

1

min00 )()(T

TdE

dxdETRTR

⎟⎟⎠

⎞⎜⎜⎝

⎛=

2

1212

2

21

1

222 )(

MMTR

zz

MMTR

2

1

1

2

2

1

AA

RR

ρρ

=


RangeDependence on absorber:Dependence on particle mass:

– in Al, for heavy particles


Electrons & PositronsEnergy loss due to collision and radiation:

Critical energy EC:

Energy loss due to collision → Bethe-Bloch has to be modified:– mproj=mtarget → large angle scattering– e- scatter off e- → indistinguishable

Electrons:

Positrons:

radcoltot dxdE

dxdE

dxdE

⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛

)()( Crad

Ccol

EdxdEE

dxdE

⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛

22

2

21

222

2)(

2)2(ln14

cmT

ZCF

IT

AZcmrN

dxdE

e

eeeeA

col

=⎥⎦

⎤⎢⎣

⎡−−−

+=⎟

⎠⎞

⎜⎝⎛− τδττ

βρπ

2max eTT =

2

2

2

)1(

2ln)12(81)(

+

+−+−=

τ

τ

βτer

F

⎟⎟⎠

⎞⎜⎜⎝

⎛+

++

++

+−= 32

2

)2(4

)2(10

21423

122ln2)(

τττβτF

24.1MeV710

24.1MeV610 gasliqsolid

+=

+=+

ZE

ZE cc


Energy loss in Cu

BremsstrahlungPhoton emission in nucleon field:

electron-electron Bremsstrahlung:– Z2 → Z(Z+1)

Important only for e+and e-:

Radiation length X0:– path length were ⟨E⟩=E0/e

( )

( )40000=

⎟⎟⎠

⎞⎜⎜⎝

⎛−

⎟⎟⎠

⎞⎜⎜⎝

⎛−

±

±

μBrems

Brems

dxdE

edxdE

.183ln43

1022 corr

ZErZN

dxdE

eA +=− αZ,A

E0

E

hν = E0-E

00

00

XXeEEXdx

EdE −==−

⎥⎦

⎤⎢⎣

⎡++≅

gcmcorr

Zr

ANZZ

X eA

22

0

.183ln)1(413

1αρ


Interactions of PhotonsPhoton detection:

– photons must create charged particles or transfer energy to charged particles

As photon beam traversing matter:– energy is not degraded but intensity is attenuated mainly by:

• Photoelectric effect• Compton scattering• Pair production

xeIxI μγ

−= 0)(

⎥⎦

⎤⎢⎣

⎡=

+++=

gcm

AN

iAi

pairComptonphoto

2

σρμ

μμμμ K

– I0 = incident beam intensity– x = absorber thickness– μ = absorption coefficient– μ/ρ = mass absorption coefficient– σ = cross section


Photoelectric EffectDominates at low photon energies:

– Eγ < 10eV … 1 MeV

Electron energy:– Ee = hν - B.E. (B.E. = binding energy)

Nucleus absorbs recoil momentum:

Increase sharply at:– Eγ ≈ B.E. of e in atomic K- and L-shell

Cross section:– EK < Eγ < mec2

– Eγ >> mec2

22e

27

54 r123

32cm

EZ

e

Kphoto

γεε

απσ ==

εαπσ 14 542 Zre

mphoto

e =5Zphoto ∝σ

−+ +→+ eatomatomγ

e-

X+X


Compton ScatteringAssume electron as quasi-free

Energy of scattered photon:

Cross section:– non-relativistic limit: Eγ < mec2

no energy transfer forThomson & Rayleigh scattering

– relativistic energies: Klein-Nishina formula: Eγ >> mec2

Atomic Compton cross section:

⎟⎠⎞

⎜⎝⎛ +≅ ε

εσσ 2ln

211

83

Thec

ec

atomicc Z σσ ⋅=

( )γγγ θε cos11

1−+

=′ EE

mb6653

8 2eTh === re

cπσσ

2cmE

e

γε = '' ee +→+ γγ

θ γ

Eγ = hν

E’γ = hν’

e-


Pair ProductionFor momentum conservation:

– Coulomb field of nucleus (electron) needed to absorb recoil– minimum energy:

Related to Bremsstrahlung by substitutionCross-section:

– low energy limit: 2 << ε << 137/Z1/3

– high energy limit: ε >> 137/Z1/3

Mean free path: λpair– Radiation length X0

22 cmE e≥γ

Independent of energy

nucleuseenucleus +→+ −+γZ

e+

e-

pairAAepair N

AXN

AZ

Zrλ

ασ 1197183ln

974

0

22

31 ≈≈⎟

⎠⎞

⎜⎝⎛≈

⎟⎠⎞

⎜⎝⎛≈ εασ 2ln

974 22Zrepair

⎥⎦⎤

⎢⎣⎡= 207

9cm

gXpairλ


Electron-Photon ShowerHigh energy electron or photon starts shower:

– photon emission and pair production alternate– statistical process– in average: equal split of energy between particles– until energy drops below critical energy EC where loss due to collision halt the cascade

After t radiation lengths:– number of particles (e-, e+, γ):

– average energy:

Maximum penetration depth:– assume abrupt stop at EC:

– maximum number of particles:

tN 2≅

t

EtE2

)( 0≅

2ln

ln

2)(

0

max

0max max

C

Ct

EE

t

EEtE

=

=≅

CEEN 0

max ≅


NeutronsNeutron detection:

– only strong interaction: needs ~10-15m distance to nuclei → much rarer process– → neutrons are very penetrating– large scattering angles

Processes: σtot = Σσi– high energy hadron shower: np → X , nn → X En >100MeV– elastic scattering off nuclei: A(n,n)A dominant for En O(MeV)– inelastic scattering: A(n,n’)A* +γ , A(n,2n’)B+γ En >1MeV– radiative neutron capture: n+(Z,A)→ γ+(Z,A+1) low vn and resonances– nuclear reactions: (n,p), (n, d), (n, α), … En O(eV)…O(keV)– fission: (n, f) En thermal

Mean free path (like for photons):

Attenuation:

⎥⎦⎤

⎢⎣⎡= 2cm

gA

Ntot

A σρλ

λx

eNxN−

= 0)(


Table of Material Properties

interaction of elementary particles with mattereisenhar/teaching/... · particle physics detectors,...

Documents