Interaction of Electron and Photons with Matter
In addition to the references listed in the first lecture (of this part of the course) see also “Calorimetry in High Energy Physics” by Richard Wigmans. (Oxford University Press,2000) This is actually an excellent book, which I would encourage you all to have a look at at some point. I will show some plots from this in today’s lecture. Interactions of Electron and Photons with Matter
0.20 = Positrons Lead (Z = 82) Lead (Z 82) − σ tot 1Mb σ experimental Electrons p.e.
) 1.0 ) 1 0.15 1 − 0 −
g σ
( Ryea ligh 2 photons dx dE 0.10 (cm 1 E Ionization
− 1kb − 0.5 Moller (e ) κ ncu + Bhabha (e ) electron/positrons 0.05 Cross section (barns/atom) σ κ Compo t n e Positron annihilation 0 1b 1 10 100 1000 V E (MeV) Photon Energy
Critical energy Ec ~ 7 MeV 10 mb 10 eV 1 keV 1 MeV 1 GeV 100 Ge First important observation is that for energies above about 10 MeV (here in Pb) the energy loss mechanisms are in each case dominated by a single process, bremsstrahlung for the electrons (and positrons) and pair production for the photons.
Note also that these processes are related (so their amplitudes are related) Bremsstrahlung (Lowest Order)
γγ e e e e
Zi Zf Zi Zf
Pair Production (Lowest Order)
e+ e- γγ e- e+
Zi Zf Zi Zf Energy Loss of Electrons in Matter
0.20 Positrons Lead (Z = 82)
Electrons
) 1.0 ) 1 0.15 1 − 0 − g
X Bremsstrahlung ( 2 dx dE 0.10 (cm 1 E Ionization − − 0.5 Moller (e ) + Bhabha (e ) 0.05
Positron annihilation 0 1 10 100 1000 E (MeV)
Critical energy Ec ~ 7 MeV Electron/Positron Interactions with Matter
In general electrons and positrons lose energy in matter in almost identical ways. There are however some small differences: In collisions with atomic electrons we have: Møller scattering for electrons (two lowest-order diagrams in QED) Bhabha scattering for positrons (two lowest-order diagrams in QED) These contributions are sizeable only for low energies (below about 10 MeV) and are never dominant. Differences are visible in the plot on the previous slide (where annihilation component of Bhabha scattering is plotted separately). Small difference in the ionization energy losses for electrons and positrons is also attributable to differences between the Møller and Bhabha scattering cross-sections The size of the momentum transfer is what determines whether the interaction leads to ionization or just excitation. Energy Loss of Electrons and Positrons
Total energy loss come from collisions and from radiation which dominates at high energies: ⎛⎞dE ⎛⎞ dE ⎛⎞ dE ⎜⎟=+ ⎜⎟ ⎜⎟ ⎝⎠dxtot ⎝⎠ dx rad ⎝⎠ dx coll
Collision losses similar to those for heavy charged particles with some differences: For electrons need to account for indistinguishability of final state electrons in scattering processes For positrons need to account for annihilation effects Kinematics are different: maximum allowed energy transfer in a single collision is
Te/2 where Te is the kinetic energy of the electron or positron. Accounting for these effects one can obtain a version of the Bethe-Bloch equation for the dE/dx losses of electrons and positrons: Ionization Energy Losses for e± πρ τδ
⎡ 2 ⎤ dE22 Z1(2)β ττ+ C −=2ln()2Nrmcae e 222⎢ + F −−⎥ dx A⎣ 2( I / me c ) 2 ⎦
2 with τ representing the kinetic energy of the incoming electron, in units of mec
F(τ ) = F(β,τ ) takes different forms for electrons and positrons (see Leo). In both cases F(τ ) is a decreasing function of τ
Thus, as before (for heavy charged particles) the rate of collision energy loss rises logarithmically with energy, and linearly with Z.
Range straggling for electrons worse than for heavy charged particles since multiple Coulomb scattering much worse for light particles. Can increase the path length of a single particle from 20-400%. (see Fig 2.11 in Leo) In addition, energy-loss fluctuations are larger than for heavy particles Bremsstrahlung
At energies below a few hundred GeV, only electrons and positrons lose significant amounts of energy through radiation. Emission probability varies as m-2, so losses for electrons exceed those for 2 muons by ( mµ / me ) ~ 40000. Bremsstrahlung depends on the strength of the electric field seen by the particle (usually the nuclear electric field) so screening due to atomic electrons needs to be accounted for. Cross-section is therefore dependent not only on the electron energy, but also on the impact parameter and the Z of the material. Effects of screening are parameterized in the following quantity:
2 100mche ν Frequency of emitted γ ξ ==+1/3 EEh0 ν EEZ0
ξ = 0 represents complete screening; ξ 1 corresponds to no screening [Relevant at high energy] For the limiting cases of no screening and complete screening the cross sections are σαε
22dνε⎛⎞ 2 21⎡ 2EE0 ⎤ ξ 1 dZrσα=+−−−41e ⎜⎟ ε⎢ ln()2 fZ⎥ νν⎝⎠32⎣ mche ⎦ ν 22dν ⎧⎛⎞ 22εε− 1/3 ⎫ ξ ~ 0 dZr=+−−+4e ⎨⎜⎟ 1⎡⎤ ln(183 Z ) fZ ( ) ⎬ ⎩⎭⎝⎠39⎣⎦
−1 fZ( )≅++ a22⎡⎤( 1 a) 0.20206 − 0.0369 a 2 + 0.0083 a 4 − 0.002 a 6 a = Z /137ε = E / E ⎣⎦0
get energy loss due to radiation by integrating dσ over the allowable energy range
ν ⎛⎞dE0 dσ −=Nhν ν () E,νν d ⎜⎟ ∫0 0 ⎝⎠dxrad d
3 Where N is the number of atoms / cm (density of scattering centers) = Na ρ / A
and ν0 = E0 / h so we can write
ν ⎛⎞dE1 0 dσ −=ΦΦ=NEwith hν ν () E ,νν d ⎜⎟ 00rad rad ∫0 ⎝⎠dxrad E0 d
-1 From before note that dσ/dν ∝ ν so that Φrad is ~ independent of ν (i.e. it is soley a property of the material).
For our two limiting cases we have
[no screening] [complete screening]
22 ⎛⎞2E0 1 22⎛⎞− 1/3 1 Φ=4ln()Zrα −− fZ Φ=4ln(183)()Zre α ⎜⎟ Z + − fZ rad e ⎜⎟2 rad 18 ⎝⎠mce 3 ⎝⎠
At intermediate values the integration must be done numerically Comparing the form of the radiation energy loss to that of the collisional energy loss we can make the following observations: The ionization energy loss rises logarithmically with energy and linearly with Z The bremsstrahlung energy loss rises ~ linearly with energy and quadratically with Z. This explains the dominance of radiation energy loss at all but the lowest energies.
A further difference has to do with statistical fluctuations. Collisional energy loss is typically quasi-continuous, coming from a large number of small-energy-loss collisions. In bremsstrahlung there is a high probability for almost all of the energy to be emitted in a small number (one or two) photons. The corresponding statistical fluctuations therefore have a larger effect. Contribution from Atomic Electrons
There will of course also be a contribution to the radiation losses from interactions with the electromagnetic field of the atomic electrons. The calculation for these interaction is similar to that for the interaction with nuclei, and give a similar result except with the Z2 replaced by Z. So to account for these interaction we can just replace Z2 with Z(Z+1) in all expressions. Radiation Length
dE We had, for the radiation energy loss: −=Φ Ndx . E rad Consider the high-energy limit where collisional energy losses can be ignored
Φrad is ~ independent of E, so we can write:
x − X where x is the distance traveled and EEe= 0 0 X0 = 1/(NΦrad ) is the radiation length
2 1 ⎡⎤ρNa 21/3− 716.4 g/cm A ≅+4ZZ ( 1) re α () ln(183 Z ) − fZ ( ) ⎢⎥ X0 ≅ XA0 ⎣⎦ ZZ(+ 1)ln(287 / Z )
When we measure material thickness (t) in units of radiation lengths we have dE − E This expression is roughly independent of material type. dt 0 Definition of Critical Energy
200 Copper = −2 X0 12.86 g cm = 100 Ec 19.63 MeV
g al n 70 t lu To h (MeV) Rossi: ra 0 50 st E s
X Ionization per X 0 ≈ × 40 = electron energy m bre dx / 30 ct Brems xa dE E Ionization 20
Brems = ionization
10 2 5 10 20 50 100 200 Electron energy (MeV)
Mentioned the critical energy earlier. There are two definitions. One is the one we mentioned earlier. The other, due to Rossi, defines Ec as the energy at which the ionization energy loss per radiation length is equal to the electron energy. The two are equivalent in the approximation that dE = EX/ 0 dx rad Interactions of Photons with Matter
(a) Carbon (Z = 6) σ As for electron / positron interactions, at 1 Mb - experimental tot all but the lowest energies, the interaction σ cross-section for photons is dominated by p.e. a single process (which is related to the 1 kb bremsstrahlung energy loss process of e±) σ Rayleigh Cross section (barns/atom) Note that as soon as a photon has undergone 1 b κ nuc such an interaction it is no longer a photon, but σ κ Compton e instead is an electron positron pair which will in 10 mb turn lose energy via bremsstrahlung producing (b) Lead (Z = 82) σ - experimental tot a photon which then pair produces etc…… 1 Mb σ p.e.
Electromagnetic Showers 1 kb
κ nuc
Cross section (barns/atom) σ g.d.r. 1 b discuss this in a moment σ κ Compton e
10 mb 10 eV 1 keV 1 MeV 1 GeV 100 GeV Photon Energy Photon Interactions in Lead
Cross-sections plotted, not energy loss (b) Lead (Z = 82) σ - experimental tot 1 Mb σ p.e.
σp.e. = atomic photoelectric effect
σRaleigh = Raleigh (coherent) scattering 1 kb σCompton = Incoherent scattering
knuc = pair production in nuclear field κ ke = pair production electron field nuc
Cross section (barns/atom) σ σ = Photonuclear interactions g.d.r. g.d.r 1 b (nuclear breakup) σ κ Compton e
10 mb 10 eV 1 keV 1 MeV 1 GeV 100 GeV Photon Energy See also Fig. 2.7 in Wigmans Attenuation Length
For electrons and positrons we talk about the radiation length to quantify the expected energy loss over a certain distance in material. For photons interaction via pair production, once the photon has interacted it is no longer a photon, so we talk in this case about the photon attenuation length.
100
10
) Sn 2 1 Fe Pb cm / Si We will see that this (g 0.1 λ H C is given by 0.01 0.001 9
Absorption length –4 10 λ = X0 –5 10 7
–6 10 10 eV 100 eV 1 keV 10 keV 100 keV 1 MeV 10 MeV 100 MeV 1 GeV 10 GeV 100 GeV Photon energy Interactions of photons with matter
As we have seen previously, at very low energies the photoelectric effect dominates
Other processes at low energy: Compton scattering γe Æ γe well understood from QED. If photon energy is high wrt binding energies then atomic electrons can be treated as unbound. Rayleigh scattering: scattering of photons by the atoms as a whole. All electrons in the atom participate in a coherent manner. This is also called coherent scattering. There is no associated energy loss, only a change in the direction of the photon. Photo-nuclear interactions causing nuclear breakup. Contribute mainly in a small energy regions around 10 MeV (in Pb) (most notable the Giant Dipole Resonance) Pair Production γ Æ e+ e-
Cannot happen in free space due to conservation of 4-momentum. Need to conserve 4-momentum through interaction with nuclear field. As was the case with bremsstrahlung, there is also a small contribution from interaction with the electric field of the atomic electrons (see plot. σ much smaller).
2 Photon must have an energy of at least 2mec = 1.022 MeV
As this process involves interactions with the nuclear field in much the same way as the bremsstrahlung process (remember, the processes are similar) the screening of the nuclear charge must again be accounted for. This time the (very similar) screening parameter is
ξ 2 100mche ν Energy of electron ==+1/3 EEEγ + − EEZ+ − ⎫ ⎬ 0 X ⎤ ⎥ ⎦ 1 − e ⎡⎤ ⎣⎦ − ⎡ ⎢ ⎣ −− 2 e can also take place in the Coulomb ⎡⎤ ⎣⎦ 32 − 2/32 νν 21/3 hmch e () 2 −− − + ++ ⎛⎞ ⎜⎟ ⎝⎠ 22 ++ + ⎩⎭ ⎧ ⎨ 3 E E EE EE ++ + + ν . h dE EE EE () 3 9 4( ) 1) () ln(183 e e 99 22 22 2 2 1/3 Z(Z+1) 4ln() 4Æ ln(183 ) ( ) 2 =−− =++−+ Z pair σα σα σα dZrdE fZ dZr EE Z fZ NZZNrZfZ =+ −= 1
17λ 71 ~ 0 ξ ξ As for bremstrahlung, the interaction her field of the atomic electrons. This is again accounted for by making replacement These can be integrated to yield cross-sections as was done for bremsstrahlung For the complete screening scenario, we have As we did previously, for bremsstrahlung, consider the limiting cases of no screening and complete screening. The cross sections are then Bremsstrahlung / Pair Production Cross Sections
PDG expressions for bremsstrahlung and pair production cross-sections
[complete screening case ξ ~ 0]
10 GeV y = Fraction of initial electron energy carried away by γ 1.2 Bremsstrahlung 100 GeV /dy 1 TeV LPM
σ 0.8 dAσ ⎛⎞44 2 10 TeV =−+=yy ykE/ yd ⎜⎟ ) dk X0 Na k ⎝⎠33 /A A 100 TeV N 0.4 0 X ( 1 PeV ⎛⎞ 10 PeV A 0 Integrate y 0Æ1 gives σ = 1⎜⎟ 0 0.25 0.5 0.75 1 ⎜⎟XN y = k/E ⎝⎠0 a
x = Fraction of initial photon energy carried away by e- 1.00 Pair production
/dx dA44 A 0.75 σ ⎛⎞⎛⎞2 1 TeV =−+=−−⎜⎟⎜⎟11(1)xx x x LPM dE X N⎝⎠⎝⎠33 X N σ 10 TeV 00aa d
) 0.50
/A 100 TeV A 1 EeV Note symmetry in x, 1-x N
0 0.25 X ( 1 PeV ⎛⎞ 100 PeV 10 PeV 7 A 0 Integrate x 0Æ1 gives σ = ⎜⎟ 0 0.25 0.5 0.75 1 9 ⎜⎟XN x = E/k ⎝⎠0 a 1.0
0.9
0.8 NaI
0.7 Ar C Pb 0.6 H2O H P 0.5 Fe
0.4
0.3
0.2
0.1
0.0 1 2 5 10 20 50 100 200 500 1000 Photon energy (MeV)
Probability P that a γ of energy E will convert into an e+e- pair in various materials. Electromagnetic Showers (Cascades)
Consequence of the dominance of bremsstrahlung and pair production processes at high energies. The electromagnetic shower will continue until the charged particles drop below the critical energy and give up the remainder of their energy via atomic collisions. Usual illustration: suppose we start with an energetic photon of energy E. Ignore the slight difference between the radiation length and the photon attenuation length and make a statistical argument:
Start with an energetic photon of energy E0. After one radiation length it emits an electron positron pair that share its energy equally. Each of these travels one radiation length before giving up half of its energy to a bremsstrahlung photon. And so on ……. At a depth of t radiation lengths the total number of electrons positrons and t t photons is N = 2 each with an equal share of the energy E(t) = E0/2 . The shower stops when the average particle energy reaches the critical energy γ e− e−
1 X 0
Absorber Active medium (scintillation or ionization) E Mean energy at depth of shower maximum 0 Et()max == Ec 2t max
1 E0 Depth of shower maximum in radiation lengths tmax = ln ln2 Ec
E0 Number of particles at shower maximum N max = Ec
This is of course a rather simplified model. One also needs to remember that this simple model describes a mean behaviour that does not account for statistical fluctuations, which are important. Use more sophisticated Monte Carlo methods to examine for example (next slide) the longitudinal shower profile for a 30 GeV electromagnetic cascade in iron. Monte Carlo Simulation of Electromagnetic Cascade
0.125 100 Distribution well fit by gamma dist. 30 GeV electron 0.100 incident on iron 80 dE() btabt−−1 e = Eb0 0.075 60 dtΓ() a dE/dt
) Energy 0 0.050 40 /E a, b depend on material
(1 Photons × 1/6.8
0.025 20 Number crossing plane Electrons tabmax =−(1)/
0.000 0 =×1.0 (lnyC + ) ie = ,γ 0 5 10 15 20 i t = depth in radiation lengths CCyEEec=−0.5,γ =+ 0.5, = /
See also Figure 2.9 in Wigmans for longitudinal profiles of electron-induced electromagnetic showers in copper. Amount of material required to contain a factor of 10 more energy is rather small. Thickess of material required for containment goes like ln E. Important for detector design….tracker size often scales linearly with E. ATLAS
Most of the volume of ATLAS is the (blue) muon spectrometer (tracker) Transverse Shower Development
As shower progresses, the lateral size will also increase due to a variety of effects: finite opening angle between e+e- in pair production emission of bremsstrahlung photons away from the longitudinal axis (which can then travel some distance before interacting). multiple scattering of electrons and positrons The transverse shower dimensions are most conveniently measured in terms of the Moliere radius [where E = 21 MeV] RXEEM = 0 sc/ s On average, only 10% of the deposited energy lies outside the cylinder with radius RM. About 99% is contained within 3.5 RM
See also Figure 2.13 in Wigmans (transverse shower development in copper)
Both longitudinal and transverse shower development are important issues in calorimeter design, which you will learn about later in the course (for example longitudinal and transverse containment, out of cone energy in jet-finding algorithms, choices about segmentation etc.) Electromagnetic Shower Profiles
Expressed in terms of X0 and ρM the development of electromagnetic showers is approximately material independent. See Wigmans section 2.1.6 and Figure 2.12 and 2.14 showing longitudinal profiles for 10 GeV electron showers in aluminum, iron and lead.
Differences are attributable to low-energy effects, differences in Ec for different materials.
Ec = 7 MeV for Pb, 22 MeV for Fe and 43 MeV for Al and so shower maximum is deeper in higher Z (lower Ec) materials. This explanation of the longer tail in Pb is similar. Explanation for differences in the transverse profiles are similar