1. Common to all charged particles ffl ionization (ο kev!) ffl Coulomb scattering from nuclei (ο kev!) ffl Cerenkov effect ffl transition radiation 2.

Transcription

1 interactions of particles with matter 1 lectromagnetic interactions of particles with Electromagnetic matter October 9, 2000 Abstract This document is a brief review to the main mechanisms of electromagnetic interactions of charged particles and photons with matter, pertinent in calorimetry

2 1. Common to all charged particles ffl ionization (ο kev!) ffl Coulomb scattering from nuclei (ο kev!) ffl Cerenkov effect ffl transition radiation 2. Muons ffl (e+,e-) pair production (ο 100GeV!) ffl bremsstrahlung (ο 100GeV!) ffl nuclear interaction (ο 1TeV!) 3. Electrons and positrons ffl bremsstrahlung (ο 10MeV!) ffl e+ annihilation 4. Photons ffl gamma conversion (ο 10MeV!) ffl incoherent scattering (ο 100keV!ο 10MeV ) ffl photo electric effect (ψ ο 100keV ) ffl coherent scattering (ψ ο 100keV ) C

3 at = n 1 + n 2 + = N ρw 1 A1 n N ρw 2 + A2 + μ : absorption, attenuation coefficients..etc.. lossary 1 ff cross section per atom. (cm 2 =atom) n at = N ρ=a number of atoms per unit of volume. (atoms=cm 3 ) Φ = n at ff number of interactions per unit of length. (1=cm) ± : macroscopic cross section = 1=Φ mean free path, interaction length, etc.. (cm) t = xρ mass-thickness, mass/surface.. (g=cm 2 ) Φ=ρ nb of interactions per (mass/surface). (1=(g=cm 2 )) μ=ρ : mass attenuation coefficient..etc.. X 0 =ρ radiation length, expressed in mass/surface (g=cm 2 ) de=dt energy loss per (mass/surface) (MeV=(g=cm 2 ))

4 1 nization Ionization

5 basic mechanism is an inelastic collision of the moving charged The with the atomic electrons of the material, ejecting off an particle μ + atom! μ + atom + + e each individual collision, the energy transfered to the electron is In But the total number of collisions is large, and we can well small. 2 nization Ionization electron from the atom : define the average energy loss per (macroscopic) unit path length.

6 Bethe-Bloch formula gives the mean rate of energy loss by The relativistic charged particles (other than electrons). moderately 2mc 2 fi 2 fl 2 T max 3 nization Mean rate of energy loss» de (fi) 2ßr2mc2 z 2 p n el 2 e = fi dx I 2 ln 2fi 2 ffi(fi)

7 mc 2 n el z p T max el = Z n at = Z N avρ n A 2 (fl 2 1) 2mc + 2flm=M + (m=m) 2 1 nization 4 where r e classical electron radius: e 2 =(4ßffl 0 mc 2 ) energy-mass of electron electrons density in the material charge of the incident particle maximum kinetic energy transferable to a free electron I mean excitation energy in the material ffi density effect function T max =

8 exists a variety of phenomenological approximations for I, There simplest being I = 10eV Z the nization 5 mean excitation energy See [ICRU84] or [PDG00] for up to date recommended values I/Z (ev) ICRU 37 (1984) Barkas Berger Z

9 is a correction term which takes into account of the reduction in ffi loss due to the so-called density effect. This becomes energy at high energy because media have a tendency to important polarised as the incident particle velocity increases. As a become the atoms in a medium can no longer be considered as consequence, To correct for this effect the formulation of Sternheimer isolated. 6 nization the density effect [Ster71] is generally used.

10 mean energy loss can be described by the Bethe-Bloch formula The if the projectile velocity is larger than that of orbital electrons. only the low-energy region where this is not verified, a different kind In parameterisation should be used. of ICRU Report 49 [ICRU93] for a detailed discussion of See corrections. low-energy nization 7 low energies For instance: ffl Andersen and Ziegler [Ziegl77] for 0:01 < fi < 0:05 ffl Lindhard [Lind63] for fi < 0:01

11 nization 8 Stopping power [MeV cm 2 /g] Lindhard- Scharff µ Anderson- Ziegler Nuclear losses Bethe-Bloch Minimum ionization µ + on Cu Radiative effects reach 1 Radiative losses Without δ βγ E µc [MeV/c] [GeV/c] [TeV/c] Muon momentum

12 9 nization 10 de/dx (MeV g 1 cm 2 ) minimum ionization βγ = p/mc 0.1 H 2 liquid He gas Fe Sn Pb Muon momentum (GeV/c) Al C de/dx min (MeV g 1 cm 2 ) H 2 gas: 4.10 H 2 liquid: 3.97 Solids Gases log 10 (Z) Pion momentum (GeV/c) Proton momentum (GeV/c) 0.5 H He Li Be B CNONe Fe Sn Z

13 = (de=dx): x gives only the average energy loss by h Ei There are fluctuations. Depending of the amount of ionization. in x the distribution of E can be strongly asymmetric matter the Landau tail). (! large fluctuations are due to a small number of collisions with The energy transfers. large 10 nization Fluctuations in energy loss

14 nization 11 The figure shows the energy loss distribution of 3 GeV electrons in 5 mm of an Ar/CH4 gas mixture [Affh98].

15 when (de=dx): x fl T max the distribution becomes nearly Only Gaussian: 12 nization ((de=dx): x)=t max» 0:01 Landau distribution < 10 Vavilov distribution 10 Gaussian limit

16 total (zigzag) pathlength of an ionizing particle of kinetic Mean T: energy 1 dffl (dffl=dx) nization 13 Energy-Range relation Z ffl=t R(T ) = ffl=tlow= Pb Fe C R/M (g cm 2 GeV 1 ) H 2 liquid He gas βγ = p/mc Muon momentum (GeV/c) Pion momentum (GeV/c) Proton momentum (GeV/c) Because of multiple Coulomb scatterings, the mean range, defined as the straight-line thickness, is smaller than R(T).

17 on E lead to fluctuations on the actual range Fluctuations (straggling). 14 nization WORL RUN NR 1 EVENT NR / 9/ 0 WORL RUN NR 1 EVENT NR / 9/ 0 penetration of e (16 MeV) and proton (105 MeV) in 10 cm of water. e e e e e e e e e e e e e e e e e e e e p p p p pp p p p p p p p p p p 1 cm 1 cm

18 energy per unit length are deposit towards More end of trajectory rather at its beginning. the nization 15 Bragg curve. proton 105 MeV in water: Bragg peak MeV/mm 5 p 105 MeV 4 e- 16 MeV X(mm) Edep along X(MeV/bin)

19 differential cross-section for producing an electron of kinetic The T, with I fi T cut» T» T max, can be written: energy 16 nization Energetic ffi rays and truncated energy loss rate may wish to take into account separately the high-energy One electrons produced above a given threshold T cut (miss knock-on detection, explicit simulation...).» dff z2 e mc2 p Z 2ßr2 2 1 T 2 fi = dt 2 T 1 + fi T 2 max T 2E 2 (the last term only for particle of spin 1/2)

20 the truncated total cross-section for emitting such electrons Then, is: dff dt dt electrons must be excluded from the mean energy loss count. Those truncated energy loss rate is: The de dx = 2ßr T<Tcut mc2 n 2e z 2 p el 2mc 2 fi 2 fl 2 T cut fluctuations on the truncated energy loss are smaller, since the The energy transfers are excluded. large 17 nization =T max Z T ff(e;t cut» T» T max ) = T =T cut fi 2» fi 2 ln ffi I 2 + T cut 1 max T

21 nization 18 delta rays WORL RUN NR 1 EVENT NR / 9/ MeV electrons, protons, alphas in 1 cm of Aluminium 0.1cm

22 nization 19 muon : number of ffi-rays per cm in Aluminium Tables for MUON + in Aluminium DRAY X-sec (1/cm) muon kinetic energy (GeV)

23 incident e =+ the Bethe Bloch formula must be modified For of the mass and identity of particles (for e ). because use the Moller or Bhabha cross sections [Mess70] and the One de/dx formula [ICRU84, Selt84]. Berger-Seltzer 20 nization Incident electrons and positrons

24 de dx T cut 2mc2 (fl + 1) I 2 fi max maximum energy transfer: fl 1 for e +, (fl 1)=2 for e nization 21 truncated Berger-Seltzer de/dx formula = 2ßr 2 e mc2 1 n el fi 2 cut T<T» + F ± (fl 1;fi up ) ffi ln where cut for ffi ray energy c T cut =mc 2 fi fi up min(fi c ;fi max ) The functions F ± are given in [Selt84].

25 nization 22 de/dx due to ionization (Berger-Seltzer formula) Tables for ELECTRON in Aluminium LOSS (MeV/cm) electron kinetic energy (GeV)

26 dff dffl 2 e Z 2ßr 2 (fl 1) fi» (fl 1) 2 fl ffl dff 2ßr2 e Z (fl 1) = dffl 1 + ffl 1 nization 23 differential cross section For the electron-electron (Möller) scattering we have: = 1 2fl 1 fl 2 ffl 1 ffl 2fl 1 fl 2 1 and for the positron-electron (Bhabha) scattering:» 1 ffl 2 B 1 ffl + B 2 B 3 ffl + B 4 ffl 2 2 fi

27 nization 24 where E = energy of the incident particle fl = E=mc 2 y = 1=(fl + 1) 1 = 2 y 2 B 2 = (1 2y)(3 + y 2 ) B 3 = (1 2y) 2 + (1 2y) 3 B 4 = (1 2y) 3 B ffl = T=(E mc 2 ) with T the kinematic energy of the scattered electron. The kinematical limits for the variable ffl are: cut T mc 2» ffl» 1 2 E for e e ffl 0» ffl» 1 for e + e ffl 0 =

28 Ionization 24-1 References [Bark62] W. H. Barkas. Technical Report 10292,UCRL, August [Lind63] J. Lindhad and al., Mat.-Fys. Medd 33, No 14 (1963) [Mess70] H.Messel and D.F.Crawford. Pergamon Press,Oxford,1970. [Ster71] R.M.Sternheimer and al. Phys.Rev. B3 (1971) H.H. Andersen andj.f. Ziegler, The stopping and ranges of [Ziegl77] in matter (Pergamon Press 1977) ions S.M. Seltzer and M.J. Berger, Int J. of Applied Rad. 35, 665 [Selt84] (1984) [ICRU84] ICRU Report No. 37 (1984) [ICRU93] ICRU Report No. 49 (1993) [Affh98] K. Affholderbach at al. NIM A410, 166 (1998) C

29 Ionization 24-2 D.E. Groom et al. Particle Data Group. Rev. of Particle Properties. [PDG00] Eur. Phys. J. C15,1 (2000) C

30 Multiple Coulomb scattering Coulomb scattering 1 ultiple

31 2 z2 p Z2 e mc ultiple Coulomb scattering 2 Single Coulomb scattering Single Coulomb deflection of a charged particle by a fixed nuclear target. θ The cross section is given by the Rutherford formula dff 1 dω = r 4 fip sin 4 =2

32 ultiple Coulomb scattering 3 Multiple Coulomb scattering Charged particles traversing a finite thickness of matter suffer repeated elastic Coulomb scattering. The cumulative effect of these small angle scatterings is a net deflection from the original particle direction. t(l) θ(l) r(l) l If the number of individual collisions is enough (> 20) the multiple Coulomb scattering angular distribution is gaussian at small angles and like Rutherford scattering at large angles. The Moli ere theory reproduces rather well this distribution. [Mol48, Bethe53]

33 Coulomb scattering 4 ultiple

34 r l X 0» l X 0 Coulomb scattering 5 ultiple Gaussian approximation The central part of the spatial angular distribution is approximately exp dω ( ) dω = 1 P 2 0 2ß» with 13.6 MeV 0 = z 1 + 0:038 ln fi pc where l=x 0 is the thickness of the medium measured in radiation lengths X 0.

35 formula of 0 is from a fit to Moli ere distribution. It is This to» 10 for 10 3 < l=x 0 < 10 2 accurate ultiple Coulomb scattering 6 note: the appearance of X 0 in the formula is only for convenience. Others formulas for 0 have been developed, starting from the Moli ere theory. [Lynch91] related quantities ffl lateral displacement r(l) ffl true (or corrected) path length t(l) ffl projected angular deflection proj (l) they are correlated random variables, for instance needed in Monte Carlo simulation. t(l) θ(l) r(l) l

36 ultiple Coulomb scattering 7 Energy dependence WORL RUN NR 1 EVENT NR / 9/ 0 10 ß + of 200 MeV and 1 GeV crossing 10 cm of Aluminium. 1 cm

37 ultiple Coulomb scattering 8 10 cm of Aluminium. Field 5 tesla. top: 10 e (300 MeV): energy loss fluctuations only (no muls) WORL RUN NR 1 EVENT NR / 9/ 0 bottom: 10 e + (300 MeV): multiple scattering only (no eloss fluct) 1 cm

38 Coulomb scattering 9 ultiple Others models for simulation Several models of multiple Coulomb scattering simulation algorithms have been proposed, not necessarily based on the Moli ere theory. (See the references in [PDG00] ) For instance : ffl J.M. Fernandez-Verea et al. : a "mixed" (detailed + condensed) model. [Fer93] ffl L.Urban : a condensed model based on Lewis theory.[urb00]

39 ultiple Coulomb scattering 10 backscattering of low energy electrons Because of its small mass, electron can have large deflection by scattering from nuclei. For low energy incident electron beam, the ratio of electrons which are backscattered out of the detector may be important (albedo).

40 ultiple Coulomb scattering 11 albedo : The incident beam is 10 electrons of 600 kev entering in 50 μm of Tungsten. WORL RUN NR 1 EVENT NR / 9/ 0 4 electrons are transmitted, 2 are backscattered. 10 µ

41 D.E. Groom et al. Particle Data Group. Rev. of Particle Properties. [PDG00] Eur. Phys. J. C15,1 (2000) Multiple Coulomb scattering 11-1 References [Mol48] Moli ere, Z. Naturforsch. 3a, 78 (1948) [Bethe53] H.A.Bethe, Phys. Rev. 89, 1256 (1953) [Lynch91] G.R.Lynch and O.I.Dahl, NIM B58,6 (1991) [Fer93] J.M. Fernandez-Verea et al. NIM B73, 447 (1993) [Urb00] L. Urban C

42 »Cerenkov radiation radiation 1 erenkov

43 »Cerenkov radiation a material with refractive index n, a charged particle emits In if its velocity is greater than the local phase velocity of photons charged particle polarizes the atoms along its trajectory. The time dependent dipoles emit electromagnetic radiations. These v < c=n the dipole distribution is symmetric around the particle If and the sum of all dipoles vanishes. position, v > c=n the distribution is asymmetric and the total time If dipole is non nul, thus radiates. dependent erenkov radiation 2 light.

44 erenkov radiation 3 mechanism of the» Cerenkov radiation [Grupen96].

45 = 1 cos fin 1» fi < 1 =) 0» < arccos 1 n n number of photons produced per unit path length and per The interval of the photons is energy 1 2 n 2 (ffl) fi erenkov radiation 4 The Huyghens construction gives immediately : Thus :» 2 d ffz2 N = dx dffl 2 (ffz)2 = sin mc 2 e r 1 μhc in which fi n(ffl) > 1 In the X-ray region n(ffl) ß 1. There is no X-ray»Cerenkov emission.

46 energy lost by the charged particle due to»cerenkov emission is The compared to collision loss, even in gas : small ο 10 1 to 10 3 MeV=(g=cm 2 ) radiation 5 erenkov

47 »Cerenkov radiation 5-1 References J.D.Jackson, Classical Electrodynamics, John Wiley and [Jackson98] (1998) Sons C. Grupen, Particle Detectors, Cambridge University Press [Grupen96] (1996) D.E. Groom et al. Particle Data Group. Rev. of Particle Properties. [PDG00] Eur. Phys. J. C15,1 (2000) C

48 Bremsstrahlung 1 remsstrahlung

49 fast moving charged particle is decelerated in the Coulomb field A atoms. A fraction of its kinetic energy is emitted in form of real of a few tens MeV, bremsstrahlung is the dominant process for Above and e+ in most materials. It becomes important for muons (and e- remsstrahlung 2 Bremsstrahlung photons. probability of this process is / 1=M 2 (M: masse of the particle) The / Z 2 (atomic number of the matter). and pions) at few hundred GeV. γ e-

50 differential cross section is given by the Bethe-Heitler formula The corrected and extended for various effects: [Heitl57], remsstrahlung 3 differential cross section ffl the screening of the field of the nucleus ffl the contribution to the brems from the atomic electrons ffl the correction to the Born approximation ffl the polarisation of the matter (dielectric suppression) ffl the so-called LPM suppression mechanism ffl... See Seltzer and Berger for a synthesis of the theories [Sel85].

51 of the energy of the projectile, the Coulomb field of the Depending can be more on less screened by the electron cloud. nucleus screening parameter measures the ratio of an impact parameter A the projectile to the radius of an atom, for instance given by a of screening functions are introduced in the Bethe-Heitler Then, formula. remsstrahlung 4 screening effect Thomas-Fermi approximation or a Hartree-Fock calculation. Qualitatively: ffl at low energy! no screening effect ffl at ultra relativistic electron energy! full screening effect

52 projectile feels not only the Coulomb field of the nucleus The Ze), but also the fields of the atomic electrons (Z electrons (charge bremsstrahlung amplitude is roughly the same in both cases, The the charge. except the electron cloud gives an additional contribution to the Thus proportional to Z (instead of Z 2 ). bremsstrahlung, 5 remsstrahlung electron-electron bremsstrahlung of charge e).

53 derivation of the Bethe-Heitler formula is based on The theory, using plane waves for the electron. If the perturbation violated for the initial and/or final velocity (low energy) the is waves would be used instead of the plane waves. Coulomb correct for this, a Coulomb correction function is introduced in To Bethe-Heitler formula. the remsstrahlung 6 Born approximation validity of the Born approximation: fi fl ffz

54 dff dk 4ff r 2 e 1 k remsstrahlung 7 high energies regime : E fl mc 2 =(ffz 1=3 ) Above few GeV the energy spectrum formula becomes simple : ß Tsai ρ 4 Z 4 y + y2 2 [L rad f(z)] + ZL 0 ff rad (1) 3 3 where k energy of the radiated photon ; y = k=e ff fine structure constant r e classical electron radius: e 2 =(4ßffl 0 mc 2 ) L rad (Z) ln(184:15=z 1=3 ) (for Z 5) L 0 rad (Z) ln(1194=z2=3 ) (for Z 5) f(z) Coulomb correction function

55 suppression mechanism, which is not shown in the dielectric 1. formula 8 remsstrahlung limits of the energy spectrum k min = 0 : In fact the infrared divergence is removed by the For k=e» 10 4 : dff=dk becomes proportional to k [Antho96] k max = E mc 2 ß E : In this limit, the screening is incomplete, and the expression 1 of the cross section is not completely accurate.

56 de = n at dx dff k dk de = E (3) 0 X dx remsstrahlung 9 mean rate of energy loss due to bremsstrahlung max ßE Z k dk (2) k min =0 nat is the number of atoms per volume. The integration immediately gives: with: 1 def = 4ff r 2 e n Φ at 2 Z [L rad f(z)] + ZL 0 Ψ rad 0 X

57 r e remsstrahlung 10 Radiation Length The radiation length has been calculated by Y. Tsai [Tsai74] 1 = 4ff r 2 e n Φ at 2 Z [L rad f(z)] + ZL 0 radψ 0 X where ff fine structure constant classical electron radius n at number of atoms per volume: N av ρ=a L rad (Z) ln(184:15=z 1=3 ) (for Z 5) L 0 rad (Z) ln(1194=z2=3 ) (for Z 5) f(z) Coulomb correction function f(z) = a 2 [(1 + a 2 ) 1 + 0: :0369a 2 + 0:0083a 4 0:002a 6 ] with a = ffz

58 per unit path length due to the bremsstrahlung increases loss with the initial energy of the projectile. linearly is the exponential attenuation of the energy of the projectile This radiation losses. by 11 remsstrahlung main conclusion: The relation 3 shows that the average energy equivalent: xx0 E(x) = E(0) exp

59 total mean rate of energy loss is the sum of ionization and The bremsstrahlung. critical energy E c is the energy at which the two rates are The Above E c the total energy loss rate is dominated by equal. remsstrahlung 12 critical energy bremsstrahlung. de/dx X 0 (MeV) Copper X 0 = g cm 2 E c = MeV Rossi: Ionization per X 0 = electron energy Brems E Total Ionization Exact bremsstrahlung Brems = ionization Electron energy (MeV) E c (MeV) MeV Z Solids Gases 710 MeV Z H He Li Be B CNONe Fe Sn Z

60 remsstrahlung 13 de/dx for electrons Tables for ELECTRON in Aluminium LOSS (MeV/cm) electron kinetic energy (GeV)

61 remsstrahlung 14 e 1 GeV in 1 meter of Aluminium. WORL RUN NR 1 EVENT NR / 9/ 0 Brems counted as continuous energy loss versus cascade development 10 cm

62 the path length, almost all the energy can be emitted in one along two photons. Thus, the fluctuations on energy loss by or 15 remsstrahlung fluctuations : Unlike the ionization loss which is quasicontinuous bremstrahlung are large.

63 de dx dff k dk the truncated total cross-section for emitting hard photons Then, is: k cut dff dk remsstrahlung 16 Energetic photons and truncated energy loss rate may wish to take into account separately the high-energy One emitted above a given threshold k cut (miss detection, photons explicit simulation...). Those photons must be excluded from the mean energy loss count. cut Z k dk (4) k<k cut = n at k min =0 nat is the number of atoms per volume. max ßE Z k ff(e;k cut» k» k max ) = dk (5)

64 the high energies regime, one can use the complete screened In 1 of dff=dk to integrate 5. This gives: expression k cut 17 remsstrahlung for E fl k cut : ff br (E;k k cut ) ß 4 3» 1 E ln 0 X at n (6) The bremsstrahlung total cross section increases as ln E. p taking into account the LPM effect leads to ff br / 1= E when fact, (in E > E lpm ) the average angle of emission of the photon is mc2 = E which is independent of the energy of the emitted photon.

65 remsstrahlung 18 number of interactions per mm in Lead (cut 10 kev) electron kinetic energy (ln(mev))

66 19 remsstrahlung WORL RUN NR 1 EVENT NR 1 11/ 9/ 0 e 200 MeV in 10 cm Aluminium 1 cm

67 remsstrahlung 20 e 200 MeV in 10 cm Aluminium (cut: 1 MeV, 10 kev). Field 5 tesla WORL RUN NR 1 EVENT NR 2 11/ 9/ 0 WORL RUN NR 1 EVENT NR 2 11/ 9/ 0 1 cm 1 cm

68 bremsstrahlung cross section is proportional to 1=m 2. Thus The critical energy for muons scales as (m μ =m e ) 2, i.e. near the TeV. the dominates other muon interaction processes in the Bremsstrahlung of catastrophic collisions (y 0:1 ), at "moderate" muon region high energies (E 1 TeV) this process contributes about 40 At average muon energy loss. to 21 remsstrahlung Bremsstrahlung by high energy muon energies - above knock on electron production kinematic limit. See [Keln97] for up-to-date review and formulae.

69 the bremsstrahlung process the longitudinal momentum transfer In the nucleus to the electron can be very small. For E fl mc 2 from k 2 2fl the uncertainty principle requires that the emission take Thus, over a comparatively long distance : place v is called the formation length for bremsstrahlung in vacuum. f is the distance of coherence, or the distance required for the It and photon to separate enough to be considered as electron particles. If anything happens to the electron or photon separate remsstrahlung 22 formation length ([Antho96]) and E fl k : k(mc2 ) 2 q long ο ο k) 2E(E 2μhcfl2 ο f v k (7) while traversing this distance, the emission can be disrupted.

70 suppression is due to the photon interaction with the electron The (Compton scattering) within the formation length. gas magnitude of the process can be evaluated using classical The [Ter72]. electromagnetism ffl(k) = 1 (μh! p =k) 2 can be shown that the formation length in this medium is It reduced: 2 k 2μhcfl 2 + (flμh! p ) 2 (8) k remsstrahlung 23 dielectric suppression mechanism A dielectric medium is characterized by his dielectric function: with μh! p = p 4ßn el r 3 e mc2 =ff (n el is the electron density of the medium = Znat:) f m ο

71 effect is factorized in the bremsstrahlung differential cross The section: dff = S d(k) dk 24 remsstrahlung» dff dk (9) T sai S d is called the suppression function : d (k) def = f m(k) S v (k) = k 2 f k 2 + (flμh! p ) 2 (10) S d (k) is vanishing for k 2 fi (flμh! p ) 2 i.e. : k fi μh! p E mc 2 (ο 10 4 ; 10 5 in all materials) (11) Then, S d (k) ß 2 k p ) 2 (12) (flμh! which cancel the infrared divergence of the Bethe-Heitler formula.

72 electron can multiple scatter with the atoms of the medium The it is still in the formation zone. If the angle of multiple while ms, is greater than the typical emission angle of the scattering, photon, br = mc 2 =E, the emission is suppressed. emitted Writing 2 ms > 2 br E lpm = ff 4ß (mc 2 ) 2 k < E 1 2 f v (k) X 0 fl E (13) lpm E remsstrahlung 25 Landau-Pomeranchuk-Migdal suppression mechanism In the gaussian approximation : 2 ms = 2ß ff where f v is the formation length in vacuum, defined in equation 7. show that suppression becomes signifiant for photon energies below a certain value, given by E lpm is a characteristic energy of the effect : X 0 ο (7:7 T ev=cm) X 0 (cm) (14) μhc

73 multiple scattering increases q long by changing the direction The the momentum of the electron. One can show : and On the other hand the uncertainty principle says : q long = μhc=f m f m (k) ß 2μhcfl2 r k Elpm E 2 if k E lpm fi E 2 (15) (k=e) lpm ) (E=E remsstrahlung 26 k 2 2fl»1 + (mc2 ) 2 2μhc E lpm f m (k) q long = where f m is the formation length in the material. These two equations can be solved in f m : k Hence, the suppression function S lpm : s lpm (k) def = f m(k) S v (k) = f r Elpm k 2 E (16)

74 LPM and dielectric mechanism both reduce the effective The length. The suppressions do not simply multiply. The formation S = d S S 2 lpm 27 remsstrahlung total suppression total suppression follows [Gal64] S

75 28 remsstrahlung

76 remsstrahlung 29 e- 10 Gev in Pb. Bremsstrahlung: gamma spectrum evts x no LPM 0.1 with LPM log10(e(mev)) energy distribution of photons at z=zplot

77 remsstrahlung 30 E146 (SLAC) data [Antho96]

78 Bremsstrahlung 30-1 References W. Heitler The Quantum theory of radiation, Oxford University [Heitl57] Press (1957) [Gal64] V.M.Galitsky and I.I.Gurevich, Nuovo Cimento 32, 1820 (1964) M.L. Ter-Mikaelian High Energy Electromagnetic Processes in [Ter72] Media, John Wiley and Sons (1972) Condensed Y.S. Tsai, Rev. Mod. Phys. 46,815 (1974) [Tsai74] Tsai, Rev. Mod. Phys. 49,421 (1977) Y.S. [Sel85] S.M. Seltzer and M.J. Berger, MIN 80, 12 (1985) [Antho96] P. Anthony et al., Phys. Rev. Lett. 76,3550 (1996) D.E. Groom et al. Particle Data Group. Rev. of Particle Properties. [PDG00] Eur. Phys. J. C15,1 (2000) C

79 R.P.Kokoulin, A.A.Petrukhin. Phys. Atomic Nuclei, 60 S.R.Kelner, 576. (1997) Bremsstrahlung 30-2 S.R.Kelner, R.P.Kokoulin, A.A.Petrukhin. Preprint MEPhI [Keln97] Moscow, 1995; CERN SCAN , S.R.Kelner, R.P.Kokoulin, A.Rybin. Geant4 Physics Reference Manual, Cern (2000) C

80 (e + ; e ) annihilation + ;e ) annihilation 1 e

81 ffl incoming e + outgoing e + ;e ) annihilation 2 e (e + ; e ) annihilation into two photons e + + e! fl + fl (need two fl for momentum conservation, if the e is assumed to be free). Theoretically, (e + ;e ) annihilation is related to Compton scattering by crossing symmetry: ffl outgoing fl incoming fl

82 ff(z; E) = Zßr2 e fl = E=mc 2 + fl ff nr (Z; E) ο Zßr2 e p 1 2 fl fl + 3 p 1 fl2 e + ;e ) annihilation 3 total cross section per atom " # fl 2 + 4fl + 1 ln fl + 1 fl 2 1 E = total energy of the incident positron r e = classical electron radius The cross section decreases with increasing E. The nonrelativistic limit is: fi The angles of emitted photons are determined by momentum conservation.

83 e + ;e ) annihilation 4 number of interactions per cm in Aluminium Tables for POSITRON in Aluminium ANNI X-sec (1/cm) positron kinetic energy (GeV)

84 + ;e ) annihilation 5 e The annihilation in fly is not the dominant process. Most of the time the positron comes at rest and does a positronium with the electron. The positronium decays in two-photon (in 0:125 nanosec) or three-photon state (in 142 nanosec:) The (e + ;e ) can also annihilate in a single photon: the other photon is absorbed by the recoil mucleus. However this mechanism is suppressed by a factor ff 4.

85 e + ;e ) annihilation 6 e + 30 MeV in 10 cm Aluminium. Annihilation in fly (left), at rest (right). WORL RUN NR 1 EVENT NR 2 12/ 9/ 0 WORL RUN NR 1 EVENT NR 6 12/ 9/ 0 γ γ e + e + γ γ 1 cm 1 cm

86 (e + ; e ) pair creation by + ;e ) pair creation by photon 1 e photon

87 is the transformation of a photon into an (e + ;e ) pair in the This field of atoms (for momentum conservation). Coulomb create the pair, the photon must have at least an energy of To 2 (1 + m=m rec ). 2mc E fl few tens MeV, (e + ;e ) pair creation is the dominant For for the photon, in all materials. process e + ;e ) pair creation by photon 2 (e + ; e ) pair creation by photon (e + ;e ) pair production is related to bremsstrahlung Theoretically, crossing symmetry: by incoming e outgoing e + ffl outgoing fl incoming fl ffl γ e+ e-

88 differential cross section is given by the Bethe-Heitler formula The corrected and extended for various effects: [Heitl57], e + ;e ) pair creation by photon 3 differential cross section ffl the screening of the field of the nucleus ffl the pair creation in the field of atomic electrons ffl the correction to the Born approximation ffl the LPM suppression mechanism ffl... See Seltzer and Berger for a synthesis of the theories [Sel85].

89 of the energy of the projectile, the Coulomb field of the Depending can be more on less screened by the electron cloud. nucleus screening parameter measures the ratio of an impact parameter A the projectile to the radius of an atom, for instance given by a of screening functions are introduced in the Bethe-Heitler Then, formula. e + ;e ) pair creation by photon 4 screening effect Thomas-Fermi approximation or a Hartree-Fock calculation. Qualitatively: ffl at low energy! no screening effect ffl at ultra relativistic electron energy! full screening effect

90 projectile feels not only the Coulomb field of the nucleus The Ze), but also the fields of the atomic electrons (Z electrons (charge for bremsstrahlung, the amplitude is roughly the same in both As except the charge. cases, the electron cloud gives an additional contribution to the pair Thus proportional to Z (instead of Z 2 ). creation, in the Coulomb field of an electron, the threshold is : However, fl 4mc 2. E recoil electron may be ejected from the atom, thus the final The can be a triplet (e + ;e ;e ). The kinetic energy of this e is state e + ;e ) pair creation by photon 5 triplet creation in the electron field of charge e). small.

91 derivation of the Bethe-Heitler formula is based on The theory, using plane waves for the electron. If the perturbation violated for the e + and/or e velocity (low energy photon) the is waves would be used instead of the plane waves. Coulomb correct for this, a Coulomb correction function is introduced in To Bethe-Heitler formula. the e + ;e ) pair creation by photon 6 Born approximation validity of the Born approximation: fi fl ffz

92 dff de 4ff r 2 e 1 k e + ;e ) pair creation by photon 7 high energies regime : k fl mc 2 =(ffz 1=3 ) Above few GeV the energy spectrum formula becomes simple : ß T sai ρ 4 x) Z x(1 2 [L rad f(z)] + ZL 0 ff rad (1) 1 3 where energy of the incident photon k E total energy of the created e + (or e ) ; x = E=k L rad (Z) ln(184:15=z 1=3 ) (for Z 5) L 0 rad (Z) ln(1194=z2=3 ) (for Z 5) f(z) Coulomb correction function

93 partition of the photon energy between e + and e is flat at low The (k» 50 MeV) and increasingly asymmetric with energy. energy e + ;e ) pair creation by photon 8 energy spectrum limits: E min = mc 2 : no infrared divergence. E max = k mc 2. For k > TeV the LPM effect reinforces the asymmetry.

94 the high energies regime, one can use the complete screened In 1 of dff=de to compute the total cross section. expression n at Z Emax ßk E min ß0 dff de 1 (3) X 0 at n pair creation total cross section is approximately constant The few GeV, for at least 4 decades (then, LPM effect). above average angle of emission of the electron, respective to the the photon is incident e + ;e ) pair creation by photon 9 ff(k) = de (2) which gives: ff pair (k) ß 7 9 is the number of atoms per volume. mc2 = k which is independent of the energy of the emitted electron.

95 e + ;e ) pair creation by photon 10 number of interactions per cm in Aluminium Tables for GAMMA in Aluminium PAIR X-sec (1/cm) photon energy (GeV)

96 + ;e ) pair creation by photon 11 e fl 200 MeV in 10 cm Aluminium. Field 5 tesla WORL RUN NR 1 EVENT NR 2 12/ 9/ 0 e γ e + 1 cm

97 to the LPM mechanism, the (e + ;e ) pair creation is reduced Due [PDG00] : for E(k E) > k E lpm E lpm = ff 4ß ) k > 4E lpm (mc 2 ) 2 () x(1 x) > E lpm lpm =k E x) x(1 e + ;e ) pair creation by photon 12 Landau-Pomeranchuk-Migdal suppression mechanism (4) k where: x = E=k E lpm is a characteristic energy of the effect : X 0 ο (7:7 T ev=cm) X 0 (cm) (5) μhc The suppression function S lpm is: s s E lpm k E) E(k (6) S lpm (E) =

98 + ;e ) pair creation by photon 13 e

99 (e + ;e ) pair creation by photon 13-1 References W. Heitler The Quantum theory of radiation, Oxford University [Heitl57] Press (1957) Y.S. Tsai, Rev. Mod. Phys. 46,815 (1974) [Tsai74] Tsai, Rev. Mod. Phys. 49,421 (1977) Y.S. [Sel85] S.M. Seltzer and M.J. Berger, MIN 80, 12 (1985) [Antho96] P. Anthony et al., Phys. Rev. Lett. 76,3550 (1996) D.E. Groom et al. Particle Data Group. Rev. of Particle Properties. [PDG00] Eur. Phys. J. C15,1 (2000) C

100 Compton scattering scattering 1 ompton

101 fl + e! fl 0 + e 0 ompton scattering 2 Compton scattering The Compton effect describes the scattering off quasi-free atomic electrons : Each atomic electron acts as an independent cible; Compton effect is called incoherent scattering. Thus: cross section per atom = Z cross section per electron The inverse Compton scattering also exists: an energetic electron collides with a low energy photon which is blue-shifted to higher energy. This process is of importance in astrophysics. Compton scattering is related to (e + ;e ) annihilation by crossing symmetry.

102 T = k k 0 limits = 0 : k 0 max = k T min = 0 ffi max = ß 2 ompton scattering 3 kinematic Assuming the initial electron free and at rest, the kinematic is given by energy-momentum conservation of two-boby scattering. k k θ T φ k k 2 mc k 0 = where» = 1 +»(1 cos ) cot ffi = (1 +»)tan( =2) = ß : k 0 min = k 1 2»+1 T max = k 2» 2»+1 ffi min = 0

103 r e» k=mc 2 ffl + 1 ffl 2» 1 ffl 2# ompton scattering 4 energy spectrum Under the same assumption,the unpolarized differential cross section per atom is given by the Klein-Nishina formula [Klein29] : " dff 0 = ß r2 e mc 2 dk Z 2» 1 1 ffl + 2 ffl» (1) ffl where k 0 energy of the scattered photon ; ffl = k 0 =k classical electron radius

104 Z k 0 max =k» 2ß 2 2» Z 2» 2 ff(k) 3 e r = 2» k! 1 : ff goes to 0 : ff(k) ο ß r 2 e ff! 8ß 3 r2 e»3 + 9» 2 + 8» + 2 ln(2» 4 + 1) 3 +» 2 + 4» + 4» ompton scattering 5 total cross section ff(k) = 0 dff =k=(2»+1) dk 0 dk0 min k limits ln 2» Z» Z (classical Thomson cross section) k! 0 :

105 ompton scattering 6 low energy limit In fact, when k» 100 kev the binding energy of the atomic electron must be taken into account by a corrective factor to the Klein-Nishina cross section: dff 0 = dk» dff 0 dk S(k; k 0 ) KN See for instance [Cullen97] or [Salvat96] for derivation(s) and discussion of the scattering function S(k,k). As a consequence, at very low energy, the total cross section goes to 0 like k 2. It also suppresses the forward scattering. At X-rays energies the scattering function has little effect on the Klein-Nishina energy spectrum formula 1. In addition the Compton scattering is not the dominant process in this energy region.

106 ompton scattering 7 number of interactions per cm in Aluminium Tables for GAMMA in Aluminium COMP X-sec (1/cm) photon energy (GeV)

107 scattering 8 ompton WORL RUN NR 1 EVENT NR 16 12/ 9/ 0 WORL RUN NR 1 EVENT NR 13 fl 10 MeV in 10 cm Aluminium: Compton scattering 12/ 9/ 0 γ e e γ 1 cm 1 cm

108 scattering 9 ompton WORL RUN NR 1 EVENT NR 4 fl 10 MeV in 10 cm Aluminium: Compton scattering 12/ 9/ 0 WORL RUN NR 1 EVENT NR / 9/ 0 γ e e e e e e e e e 1 cm 1 cm

109 ompton scattering 10 Compton scattering : fl 10 MeV in Aluminium. Compton edge: energy spectrum of scattered photon (left) and emitted e (right)

110 ht (k)i n at μ = abs k μ abs Z k k 0 min ompton scattering 11 absorption, diffusion, attenuation Only a fraction of the energy of the incident photon is transferred to the recoil electron, which is generally stopped into the material. Thus this energy is absorbed by the material. The mean kinetic energy of the electron is : ht (k)i = dff T 0 dk0 dk Then the absorption coefficient of Compton scattering is defined as (n at is the nb of atoms per volume) is the mass absorption coefficient. ρ

111 Z k k 0 min k 0 def = n at ff = μ abs + μ sca att μ hk 0 i n at! = sca μ k scattering 12 ompton Similar, from the mean energy of the diffused photon one defines the scattered coefficient of Compton scattering dff 0 dk0 dk hk 0 i = The attenuation coefficient of Compton scattering is and similar relations for the mass coefficients.

112 scattering 13 ompton Rayleigh scattering Rayleigh scattering is the scattering of photons by an atom as a whole : all the electrons of the atom participate in a coherent manner. It is an elastic collision: no energy transfer from photon to atom (no ionisation nor excitation). At x-rays and fl-rays energy region, Rayleigh scattering is small compared to the photo electric effect, and can be generally neglected.

113 D.Cullen et al. Evaluated photon library 97, UCRL-50400, [Cullen97] Rev.5 (1997) vol.6, F.Salvat et al. Penelope, Informes Técnicos Ciemat 799, [Salvat96] (1996) Madrid Compton scattering 13-1 References [Klein29] O.Klein and Y.Nishina Z.Phys.52,853 (1929) J.H.Hubbell et al. Rad. Phys. Chem. vol50, 1 (1997) C

114 absorption 1 hotoelectric Photoelectric absorption

115 fl + atom! atom + + e hotoelectric absorption 2 Photoelectric absorption A bound electron can absorb completely the energy of a photon : The electron is ejected with kinetic energy T = k B s. (k: energy of the incident photon, B s : binding energy of the corresponding subshell). The nucleus absorbs the recoil momentum. The cross section per shell can be parametrized [Biggs87] : ff s = r 2 e ff4 Z 5 f 1 a(k) k with f : nonsimple function of 1=k, and 1» a(k)» 4.

116 The total cross section has discontinuities at k = B s hotoelectric absorption 3 (absorption edge). There are several parametrizations and tables of the cross sections. See [Cullen97, Biggs87]. If k > B K the absorption occurs mainly on the K-shell (80 of the cases). The electron is emitted forward in the direction of the incident photon at high k, and perpendicular to the photon at low k [Sauter31, Gavri61]. Following the photoabsorption in the K-shell, characteristic X-rays or Auger electrons are emitted [Perkin91].

117 absorption 4 hotoelectric 1 Mb (a) Carbon (Z = 6) experimental σ tot Cross section (barns/atom) 1 kb 1 b σ p.e. σ coherent 10 mb σ incoh σ nuc κ N κ e

118 absorption 5 hotoelectric

119 hotoelectric absorption 6 attenuation ff tot = ff pair + ff comp + ff phot + ff rayl μ = n at ff tot! beam of monoenergetic photons is attenuated in intensity (not in A energy) according : I(x) = I(0) exp( μ x) = I(0) exp( x= ) WORL RUN NR 1 EVENT NR / 9/ 0 Below : 20 photons, 5 MeV, entering 10 cm of Al. 4 exit unaltered. 1 cm

120 hotoelectric absorption 7 20 photons, 400 kev, entering 10 cm of water. (compare with e and protons) WORL RUN NR 1 EVENT NR / 9/ 0 γ e e e e e e γ γ γ e e γ e e e e γ e e e γ e γ γ γ γ γ e e e e γ e γ γ e γ γ γ 1 cm

121 hotoelectric absorption 8 Macroscopic cross sections for photon in water. (! mean free path) Tables for GAMMA in Water PHOT X-sec (1/cm) PAIR X-sec (1/cm) Tot X-sec (1/cm) COMP X-sec (1/cm) RAYL X-sec (1/cm) Mean free path (cm) photon energy (GeV)

122 hotoelectric absorption Absorption length λ (g/cm 2 ) H C Si Fe Sn Pb ev 100 ev 1 kev 10 kev 100 kev 1 MeV 10 MeV 100 MeV 1 GeV 10 GeV 100 GeV Photon energy

123 D.Cullen et al. Evaluated photon library 97, UCRL-50400, [Cullen97] Rev.5 (1997) vol.6, F. Biggs and R.Lighthill, Sandia Laboratory SAND [Biggs87] (1987) D.E. Groom et al. Particle Data Group. Rev. of Particle Properties. [PDG00] Eur. Phys. J. C15,1 (2000) Photoelectric absorption 9-1 References J.H.Hubbell et al. Rad. Phys. Chem. vol50, 1 (1997) [Sauter31] F. Sauter, Ann. Physik. 11, 454 (1931) [Gavri61] M. Gavrila, Phys. Rev. 124, 1132 (1961) [Perkin91] S.T. Perkin et al. UCRL-50400,30 (1991) [Hub00] C

124 Electromagnetic cascades cascades 1 lectromagnetic

125 development of cascades induced by electrons or photons is The by bremsstrahlung of electrons/positrons and (e + ;e ) governed creation by photons, until the energy of secondary pair fall below the critical energy E c. electrons/positrons the electrons and positrons lose their energy preferentially by Then halting the cascade. ionization, cascades 2 lectromagnetic Electromagnetic cascades

126 lectromagnetic cascades 3 fl 200 MeV in 1 X 0 Aluminium. left: pair only; right: pair + brem WORL RUN NR 1 EVENT NR 2 12/ 9/ 0 WORL RUN NR 1 EVENT NR 1 21/ 9/ 0 e γ e + 1 cm 1 cm

127 lectromagnetic cascades 4 fl 200 MeV in 2 X 0 Aluminium. Pair + brem WORL RUN NR 1 EVENT NR 5 21/ 9/ 0 1 cm

128 the first radiation length, the mean longitudinal profile of Beyond energy deposition is well described by a gamma function the dt = E 0 const t a exp( bt) with t = x=x 0 (1) the other hand, a simple model of shower development On can predict the maximum of the distribution : ([Leo94]) lectromagnetic cascades 5 longitudinal profile in homogeneous material [Long75] : de a and b are fit parameters dependent on the material. max a t = k ln E0 = c E b

129 lectromagnetic cascades 6 e 10 GeV in PbWO4 Simulation and gamma-fit of the longitudinal profile [Melo99] 1/E0 de/dt t = X/X0

130 lectromagnetic cascades 7 left: e 1 GeV, 10 GeV, 100 GeV, 1 Tev in PbWO4 right: e 10 GeV; profile and its intrinsic fluctuations 1/E0 de/dt /E0 de/dt t = X/X t = X/X0

131 lectromagnetic cascades 8 e 5 GeV in PbWO4 cumulative longitudinal profile and its intrinsic fluctuations (! leakage) PbWO4 e- 5 GeV PbWO4 e- 5 GeV Ecumul/E0 100 rms/e L/Radl cumul longit energy dep ( of E inc) L/Radl rms on cumul longit Edep ( of E inc)

132 lectromagnetic cascades 9 e 1 GeV in Water data [Cran69] and simulation of the longitudinal profile H2O e- 1 GeV G4-G3-data comparison (1/E0)(dE/dRadl) G4 G3 data longit energy profile ( of E inc) L/Radl (1/E0)(dE/dRadl) G4 G3 data radial energy profile ( of E inc) R/Radl

133 lateral spread of an electromagnetic shower is mainly caused The multiple scattering. by MeV 21 X 0 c E average, 90 of the shower energy is contained in a (semi On cylinder of radius R m, in any material. infinite) radial distribution is well represented by the sum of two This gaussian. The distribution is nearly independent of the incident energy E 0 lectromagnetic cascades 10 radial profile in homogeneous material The Moli ere radius is defined : R m =

134 lectromagnetic cascades 11 radial profile left: e 10 GeV in PbWO4, simulation and two-gaussian fit. right: 1 GeV and 1 TeV profiles 1/E0 de/du 1 1/E0 de/du u = Rho/X u = Rho/X0

135 lectromagnetic cascades 12 At high enough energies, the LPM effect can cause signifiant elongation of electromagnetic cascades... apparently, not yet at 10 TeV... Fe e- 10 TeV LPM - no LPM comparison (1/E0)(dE/dX0) 9 8 lpm 7 no lpm L/X0 longit energy profile ( of E inc)

136 Electromagnetic cascades 12-1 References [Long75] E. Longo and I. Sestili, NIM 128, 283 (1975) W. Leo, Techniques for particle physics experiments, Springer- [Leo94] (1994) Verlag [Cran69] J. and H. Crannel, Phys. Rev (1969) [Melo99] F. Melot, rapport de stage, Lapp(Annecy) (1999) D.E. Groom et al. Particle Data Group. Rev. of Particle Properties. [PDG00] Eur. Phys. J. C15,1 (2000) C

137 Direct (e + ; e ) pair creation by muon (e + ;e ) pair creation by muon 1 irect

138 of a (e + ;e ) pair by virtual photon in the Coulomb field Creation the nucleus (for momentum conservation). of irect (e + ;e ) pair creation by muon 2 Direct (e + ; e ) pair creation by muon μ + nucleus! μ + e + + e + nucleus e- e+ µ µ

139 TeV muon energies, pair creation cross section exceeds those of At muon interaction processes in a wide region of energy other 100 MeV» ffl» 0:1 E μ energy loss for pair production increases linearly with Average energy, and in TeV region this process contributes over 50 muon (e + ;e ) pair creation by muon 3 irect It is one of the most important processes of muon interaction. transfers : to the total energy loss rate.

140 contribution to the total cross section is given by transferred Main energies: 5MeV» ffl» 0:01 E μ contribution to average muon energy loss is determined mostly The region: by 10 3 E μ» ffl» 0:1 E μ to adequatly describe the number of pairs produced, average Thus, loss and stochastic energy loss distribution one need to energy with a sufficient accuracy the differential cross section reproduce in a wide range of energy transfers: behaviour 5MeV» ffl» 0:1 E μ irect (e + ;e ) pair creation by muon 4 energy transfers

141 (e + ;e ) pair creation by muon 5 irect

142 irect (e + ;e ) pair creation by muon de/dx (MeV g 1 cm 2 ) Fe pair Fe brems Fe nucl Fe total Fe ion Muon energy (GeV)

143 differential cross section is given by Kokoulin et al. [Koko71]. The includes : It (e + ;e ) pair creation by muon 7 irect differential cross section ffl screening of the field of the nucleus ffl correction for finite nuclear size ffl contribution from the atomic electrons [Keln97] ffl... See [Koko71] for a complete discussion.

144 Z ρmax me m μ 2 irect (e + ;e ) pair creation by muon 8 differential cross section The differential cross section per atom can be written as : dff = 4 ff2 r 2 e dffl v 1 ffl [Z(Z + )] F (Z; E; ffl) (1) 3ß with " # dρ (2) F (Z; E; ffl) = 0 Φ μ (v; ρ) Φ e (v; ρ) + where ffl = ffl + + ffl = total energy of the created pair; v = ffl=e ρ = (ffl + ffl )=ffl = asymmetry coefficient; The functions Φ e ; Φ μ ; can be found in [Koko00].

145 ffl min = 4m e c 2»1 6(m μc 2 ) 2 (e + ;e ) pair creation by muon 9 irect limits max = E 3p e ffl 4 m μ c 2 Z 1=3 ρ min = 0 r ρ max = E(E ffl) ffl min 1 ffl

146 n at Z fflcut Z fflmax ffl cut irect (e + ;e ) pair creation by muon 10 Energetic pairs and truncated energy loss rate may wish to take into account separately the high-energy pairs One above a given threshold ffl cut (miss detection, explicit emitted simulation...). Those pairs must be excluded from the mean energy loss count. de dx ffl<ffl cut = n at ffl dff dffl min ffl dffl (3) is the number of atoms per volume. Then, the truncated total cross-section for emitting hard pairs is: dff ff(e; ffl cut» ffl» ffl max ) = dffl (4) dffl

147 = mc2 ο1000 TeV the LPM suppression mechanism may have an Above effect. (e + ;e ) pair creation by muon 11 irect The muon deflection angle is of the order of: E

148 irect (e + ;e ) pair creation by muon 12 number of interactions per cm in Iron. (cut 100 MeV) Tables for MUON - in Iron BREM X-sec (1/cm) MUNU X-sec (1/cm) PAIR X-sec (1/cm) muon kinetic energy (GeV)

149 irect (e + ;e ) pair creation by muon 13 1 TeV muon in 10 meter of Fe (field 5 tesla). WORL RUN NR 1 EVENT NR / 9/ 0 direct pair creation only 100 cm

150 irect (e + ;e ) pair creation by muon meter of Fe : muons 100 GeV, 1 TeV, 5 TeV. right : direct pair creation + brems WORL RUN NR 1 EVENT NR / 9/ 0 WORL RUN NR 1 EVENT NR / 9/ 0 left : brems only 100 cm 100 cm

151 Direct (e + ;e ) pair creation by muon 14-1 References [Lohm85] W.Lohmann,R.Kopp,R.Voss, Cern Report (1985) R.P.Kokoulin and A.A.Petrukhin, Proc. 11th Intern. Conf. on [Koko69] Rays, Budapest, 1969 [Acta Phys. Acad. Sci. Hung.,29, Cosmic Suppl.4,p.277, 1970]. R.P.Kokoulin and A.A.Petrukhin, Proc. 12th Int. Conf. on [Koko71] Rays, Hobart, 1971, vol.6, p Cosmic [Keln97] S.R.Kelner, Phys. Atomic Nuclei, 61 (1998) 448. S.R.Kelner, R.P.Kokoulin, A.Rybin. Geant4 Physics Reference [Koko00] Cern (2000) Manual, D.E. Groom et al. Particle Data Group. Rev. of Particle Properties. [PDG00] Eur. Phys. J. C15,1 (2000) C

152 Muon photonuclear interaction photonuclear interaction 1 uon

153 uon photonuclear interaction 2 Muon photonuclear interaction The inelastic interaction of muons with nuclei is important for high muon energies E 10 TeV, at relatively high energy transfers ν=e 10 2, in particular, in light materials, from view point of the detector response for high energy muons, muon propagation and muon induced hadronic background. µ (Q2, ν) µ

154 Q 2 fi 1 GeV 2 uon photonuclear interaction 3 de/dx : Average energy loss for this process is almost lineary increasing with energy, and at TeV muon energies constitutes about 10 in standard rock. differential cross section : The main contribution to the cross section dff=dν and energy loss is given by low Q 2 region : Most widely used are the expressions given by Borog and Petrukhin [BOR75] and Bezrukov and Bugaev [BEZ81]. Results of these authors agree within 10 for differential cross section and within about 5 for the average energy loss (if the same photonuclear cross section fffln is used in calculations).

155 photonuclear interaction 4 uon Theoretical estimates show that inelastic muon scattering gives, along with multiple Coulomb scattering, appreciable contribution to muon deflection (and dominates at large angles). see [Koko00] for a review of Borog and Petrukhin cross section.

156 V.V.Borog and A.A.Petrukhin, Proc. 14th Int.Conf. on Cosmic [BOR75] Munich,1975, vol.6, p Rays, V.V.Borog, V.G.Kirillov-Ugryumov, A.A.Petrukhin, Sov. J. [BOR77] Phys., 25, 1977, p.46. Nucl. L.B.Bezrukov and E.V.Bugaev, Sov. J. Nucl. Phys., 33, 1981, [BEZ81] p.635. S.R.Kelner, R.P.Kokoulin, A.Rybin. Geant4 Physics Reference [Koko00] Cern (2000) Manual, Muon photonuclear interaction 4-1 References C