Interaction of Electron and Photons with Matter

Transcription

1 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,2) 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.

2 Interactions of Electron and Photons with Matter de dx (X 1 ) 1 E Electrons Positrons Moller (e ) Bhabha (e + ) Positron annihilation Ionization Lead (Z = 82) Bremsstrahlung electron/positrons E (MeV) (cm 2 g 1 ) Critical energy E c ~ 7 MeV Cross section (barns/atom) 1Mb 1kb 1b σ p.e. σ Rye a ligh σ Compo t n Lead (Z = 82) experimental σ tot photons κ nuc 1 mb 1 ev 1 kev 1 MeV 1 GeV 1 GeV Photon Energy κ e First important observation is that for energies above about 1 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)

3 Bremsstrahlung (Lowest Order) γ γ e e e e Z i Z f Z i Z f Pair Production (Lowest Order) γ e + e - γ e - e + Z i Z f Z i Z f

4 Energy Loss of Electrons in Matter Positrons Lead (Z = 82).2 de dx (X 1 ) 1 E 1..5 Electrons Moller (e ) Ionization Bremsstrahlung.15.1 (cm 2 g 1 ) Bhabha (e + ).5 1 Positron annihilation E (MeV) Critical energy E c ~ 7 MeV

5 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 1 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.

6 Energy Loss of Electrons and Positrons Total energy loss come from collisions and from radiation which dominates at high energies: de de = + de dx dx dx tot rad 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 T e /2 where T e 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:

7 Ionization Energy Losses for e ± 2 de 2 2 Z 1 τ ( τ + 2) C = 2πNrmc a e e ρ ln + F( τ) δ dx A β 2( I / mec ) 2 with τ representing the kinetic energy of the incoming electron, in units of m e c 2 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 2-4%. (see Fig 2.11 in Leo) In addition, energy-loss fluctuations are larger than for heavy particles

8 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 muons by ( m µ / m e ) 2 ~ 4. 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 1mchν ξ = e E = E + hν 1/3 EEZ Frequency of emitted γ ξ = represents complete screening; ξ 1 corresponds to no screening [Relevant at high energy]

9 For the limiting cases of no screening and complete screening the cross sections are ξ 1 ξ ~ dν ε σ α 2 ε 2EE 1 d = 4Z 2 r e 1 ln f( Z) 2 ν 3 mch e ν 2 ν ε ε σ = 2 2 d α ε 1/3 d 4Z re 1 + ν ln(183 Z ) f( Z) ( ) ε fz ( ) a 1 + a a +.83a.2 a a= Z/137 = E/ E get energy loss due to radiation by integrating dσ over the allowable energy range de ν dσ = N ν ( ν) ν h E, d dx dν rad Where N is the number of atoms / cm 3 (density of scattering centers) = N a ρ / A and ν = E / h

10 so we can write de 1 ν dσ = NE Φ with Φ = ν ( ν) ν h E, d rad rad dx E dν rad From before note that dσ/dν ν -1 so that Φ rad is ~ independent of ν (i.e. it is soley a property of the material). For our two limiting cases we have [no screening] Φ = α 2 2 2E 1 4Zre ln fz ( ) rad 2 mc e 3 [complete screening] Φ = 4Zrα + e ln(183 Z ) fz ( ) rad /3 1 At intermediate values the integration must be done numerically

11 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.

12 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 Z 2 replaced by Z. So to account for these interaction we can just replace Z 2 with Z(Z+1) in all expressions.

13 Radiation Length de We had, for the radiation energy loss: = N Φ raddx. E Consider the high-energy limit where collisional energy losses can be ignored Φ rad is ~ independent of E, so we can write: E x X = E e where x is the distance traveled and X = 1/(NΦ rad ) is the radiation length 1 X ρna + 2 1/3 4 ZZ ( 1) re α ln(183 Z ) fz ( ) A ( ) X g/cm ZZ ( + 1)ln(287 / Z) 2 A When we measure material thickness (t) in units of radiation lengths we have de E dt This expression is roughly independent of material type.

14 Definition of Critical Energy 2 de/dx X (MeV) Copper X = g cm 2 E c = MeV Rossi: Ionization per X = electron energy Brems E Total Ionization Exact bremsstrahlung Brems = ionization 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 E c 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 dx rad = E/ X

15 Interactions of Photons with Matter As for electron / positron interactions, at all but the lowest energies, the interaction cross-section for photons is dominated by a single process (which is related to the bremsstrahlung energy loss process of e ± ) Cross section (barns/ atom) 1 Mb 1 kb σ p.e. σ Rayleigh (a) Carbon (Z = 6) - experimental σ tot Note that as soon as a photon has undergone such an interaction it is no longer a photon, but instead is an electron positron pair which will in turn lose energy via bremsstrahlung producing a photon which then pair produces etc 1 b 1 mb 1 Mb σ p.e. σ Compton (b) Lead (Z = 82) - experimental σ tot κ nuc κ e discuss this in a moment Electromagnetic Showers Cross section (barns/ atom) 1 kb 1 b σ Compton σ g.d.r. κ nuc κ e 1 mb 1 ev 1 kev 1 MeV 1 GeV 1 GeV Photon Energy

16 Photon Interactions in Lead Cross-sections plotted, not energy loss 1 Mb σ p.e. (b) Lead (Z = 82) - experimental σ tot σ p.e. = atomic photoelectric effect σ Raleigh = Raleigh (coherent) scattering σ Compton = Incoherent scattering k nuc = pair production in nuclear field k e = pair production electron field σ g.d.r = Photonuclear interactions (nuclear breakup) Cross section (barns/ atom) 1 kb 1 b σ Compton σ g.d.r. κ nuc κ e See also Fig. 2.7 in Wigmans 1 mb 1 ev 1 kev 1 MeV 1 GeV 1 GeV Photon Energy

17 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. 1 1 Absorption length λ (g/cm 2 ) H C Si Fe Sn Pb We will see that this is given by λ = 9 7 X ev 1 ev 1 kev 1 kev 1 kev 1 MeV 1 MeV 1 MeV 1 GeV 1 GeV 1 GeV Photon energy

18 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 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 1 MeV (in Pb) (most notable the Giant Dipole Resonance)

19 Pair Production γ 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). Photon must have an energy of at least 2m e c 2 = 1.22 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 2 1mchν ξ = e Eγ = E+ + E 1/3 EEZ + Energy of electron

20 As we did previously, for bremsstrahlung, consider the limiting cases of no screening and complete screening. The cross sections are then ξ 1 ξ ~ E+ + E + 2 EE + /3 2EE + 1 dσ = 4Z re αde+ ln f( Z) 3 2 ( hν) mc e hν 2 + σ = 2 2 de + + α EE + 1/3 EE d 4Z re E E + ν 3 ln(183 Z ) f( Z) ( h ) 3 9 As for bremstrahlung, the interaction here can also take place in the Coulomb field of the atomic electrons. This is again accounted for by making the replacement Z 2 Z(Z+1). These can be integrated to yield cross-sections as was done for bremsstrahlung For the complete screening scenario, we have /3 7 1 = Nσ 4 Z( Z + 1) Nr α = pair e λ ln(183 Z ) f( Z) 9 9 X

21 Bremsstrahlung / Pair Production Cross Sections PDG expressions for bremsstrahlung and pair production cross-sections [complete screening case ξ ~ ] (X N A /A) ydσ LPM /dy GeV 1 GeV 1 TeV 1 TeV 1 TeV Bremsstrahlung 1 PeV 1 PeV y = k/e y = Fraction of initial electron energy carried away by γ dσ A = y y y = k/ E dk X N k 3 3 a A Integrate y 1 gives σ = 1 XN a (X N A /A) dσ LPM /dx TeV 1 TeV 1 TeV Pair production 1 EeV 1 PeV 1 PeV 1 PeV x = E/k x = Fraction of initial photon energy carried away by e - dσ A 4 = + A = 4 1 x x 2 1 x(1 x) de X N 3 X N 3 a Note symmetry in x, 1-x 7 A Integrate x 1 gives σ = 9 XN a a

22 NaI P Pb Fe Ar H 2 O C H Photon energy (MeV) Probability P that a γ of energy E will convert into an e + e - pair in various materials.

23 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 E. 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 photons is N = 2 t each with an equal share of the energy E(t) = E /2 t. The shower stops when the average particle energy reaches the critical energy

24 e γ e 1 X Absorber Active medium (scintillation or ionization)

25 Mean energy at depth of shower maximum Depth of shower maximum in radiation lengths Number of particles at shower maximum Et ( ) t N max max max E = = E t max 2 1 E ln ln 2 E = E = E c c c 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 3 GeV electromagnetic cascade in iron.

26 Monte Carlo Simulation of Electromagnetic Cascade (1/E ) de/dt Energy Photons 1/6.8 Electrons 3 GeV electron incident on iron t = depth in radiation lengths Number crossing plane Distribution well fit by gamma dist. a 1 bt de ( bt) e dt = Eb γ Γ() a a, b depend on material tmax = ( a 1)/ b = 1. (ln y + C ) i = e, γ C =.5, C =+.5, y = E/ E e i c 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 1 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.

27 ATLAS Most of the volume of ATLAS is the (blue) muon spectrometer (tracker)

28 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 R = X E / E [where E s = 21 MeV] M s On average, only 1% of the deposited energy lies outside the cylinder with radius R M. About 99% is contained within 3.5 R M 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.) c

29 Electromagnetic Shower Profiles Expressed in terms of X and ρ M the development of electromagnetic showers is approximately material independent. See Wigmans section and Figure 2.12 and 2.14 showing longitudinal profiles for 1 GeV electron showers in aluminum, iron and lead. Differences are attributable to low-energy effects, differences in E c for different materials. E c = 7 MeV for Pb, 22 MeV for Fe and 43 MeV for Al and so shower maximum is deeper in higher Z (lower E c ) materials. This explanation of the longer tail in Pb is similar. Explanation for differences in the transverse profiles are similar

30

31