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Interactions of Fast Particles in a Medium Electromagnetic interaction: –Ionization –Radiative processes –Cerenkov radiation Hadronic interactions.

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Presentation on theme: "Interactions of Fast Particles in a Medium Electromagnetic interaction: –Ionization –Radiative processes –Cerenkov radiation Hadronic interactions."— Presentation transcript:

1 Interactions of Fast Particles in a Medium Electromagnetic interaction: –Ionization –Radiative processes –Cerenkov radiation Hadronic interactions

2 Energy Loss Processes First part considers soft interactions between a particle and the medium it is travelling through (atomic excitation and ionization) Discuss two approaches 1)Energy loss as a succession of scatters A) Classical approximation B) Relativistic treatment 2) Moving particle as source of virtual

3 Energy Loss by Scattering As a particle goes through matter it suffers many soft or glancing collisions. In each collision the particle loses energy and changes direction slightly. Consider a single collision….

4 Energy Loss as a Succession of Soft Electromagnetic Scatterings.

5 Classical EM Scattering – Impact parameter, b e m e, p M, ze b 2b

6 Classical EM Scattering – Cross-section Have just derived the relationship between impact parameter and scattering angle. In practice cant measure impact parameter – so we need to find the relationship between the scattering probability and the scattering angle.

7 Scattering Probability vs. Scattering Angle All areas equally likely to be hit. dP ~ db = d ( dP = prob of scattering) Every particle will have some scattering angle d ~b db d Use and Want to get d ( )/d …. b.db.d b

8 Classical EM Scattering – Cross-section b ~ 2z / pv, Sin ~ db/d = z / 2 pv Which gives the Rutherford Scattering Formula

9 Energy Loss by Scattering – Approximate, Non-Relativistic Model Energy transferred to target: E = ( p t ) 2 /M ~ z 2 2 /b 2 v 2 Integrating over impact parameter will add a constant, but not change dependence on z,,v. So for a single scatter: E ~ z 2 2 /v 2

10 Scattering – approx. model Almost all energy loss will be to electrons ( m e << m N ) Number of electrons that a projectile passes per unit length ~ elec = Z atoms atoms = mass /A, so elec = Z mass /A So, energy loss per unit length:

11 Kinematics of energy loss via scattering NB. Not necessarily elastic Substitute for E and p: –(Taylor expand in terms of /E to 2 nd order, valid for small /E )

12 Energy Loss via Scattering Use -q 2 =2M 2 -2EE+2ppCos and our approximation for p For small angles, Cos ~ 1 – ½ 2 Hence, for small, /E: At finite angles, the first term dominates. The second term defines a minimum q 2 for a given energy loss.

13 Towards a Model of Energy Loss by ionization Energy loss of charged particle through scattering will mainly be from scattering from electrons –m e << M nucleus, maximum energy exchange is higher for electrons than nuclei. We are interested in particles of mass M ( normally M >> m e ), charge Z, scattering from stationary electrons. However, cross-section the same as for electrons scattering from stationary particle.

14 Large Energy Loss Scatters. Close collisions (~ large q 2, small b) Large angle and/or Large q 2 can resolve electrons in atom Minimum energy exchange to consider a collision as close, min,set by some ionization energy scale of medium ( eV) Maximum energy loss ( T max ) set by kinematics

15 Small Energy Loss Scatters. Distant collisions ( ~ low q 2, large b) Low energy loss Low q 2 photons will interact with atom as a whole. Maximum energy loss set by boundary with close collisions Minimum energy loss set by smallest available excitation energy of medium

16 Rutherford Cross-section Consider the electron moving and the projectile stationary.(Close collisions) Earlier we derived the Rutherford scattering cross-section: –z = electron charge=1 –m p = electron mass –T = projectile kinetic E The complete result is:

17 Relativistic Corrections – Mott Scattering Formula As 1, Rutherford formula becomes inaccurate. Simplest modifications give Mott formula. –First order perturbation theory –Assumes no recoil of target –No spin or structure effects.

18 Mott Cross-Section as a Function of q 2 Can write Mott formula in terms of q 2, rather than –put q 2 =4p e 2 sin 2 ( /2) Hence:

19 Back to Stationary Electron and Moving Projectile Eliminate electron momentum –p e = m e Have Mott Scattering formula for moving electron. Now move back to lab frame, where electron is stationary. –For free electron at rest q 2 =2m e –Hence

20 Energy Loss from Close collisions In summary, cross-section for transfer of energy in the range to + d

21 Energy Loss Per Unit Distance So the energy –dE lost by a particle passing through distance dx in a material is: The number density,n, is n = N A / A ( =density, N A =Avogadros number,A= molar mass )

22 de/dx From Close Interactions Putting in the differential cross-section and integrating gives Recall expression for max : For small compared to M/m e

23 de/dx From Close Interactions Putting in expression for max and assuming that max - cut ~ max

24 dE/dx for Distant collisions Can no longer treat electrons as free. rather more complicated (but result looks rather similar) Get a term ln(q 2 min / q 2 cut ) q 2 min =I 2 /( Lower energy limit I, represents some typical ionisation energy –( I ~ 10Z eV for Z>10)

25 dE/dx for Distant collisions q 2 max = 2m e min (any energy exchange larger than cut is counted as close) Get: The term models the density effect (polarization of the medium)

26 dE/dx – the Bethe-Bloch Equation Adding contributions from close and distant collisions we get the Bethe- Bloch equation: Where:

27 Approximate Bethe-Bloch Equation There are different versions of the formula. At moderate energies can approximate by ignoring and putting max = ( ) 2 m e

28 Bethe-Bloch References References: –Rossi, High-Energy Particles, chap. 2 –Jackson, Classical Electrodynamics, chap. 13 –Fano, Ann. Rev. Nucl. Sci., Vol. 13 (1963), p.1 –Particle Data Group (PDG), in the WWW edition of Review of Particle Properties, via link from the course Web site

29 dE/dx – 1/ At low velocity, energy loss falls steeply with increasing energy (1/ nteracting particles have less time to see each other at higher speeds. –(Curve is actually better modelled by

30 Virtual Interaction Range vs. and Recall (from Quarks and Leptons) that the range of a virtual particle is proportional to 1/|q| –Range ~ c/|q 2 | 1/2 ( large q 2 short distances ) q 2 small for zero angle : |q| / –Hence virtual photon range c/ I.e. range increases with Range decreases with Range wont increase indefinitely – polarisation of medium.

31 Maximum Range for Close Interactions The lower limit for energy exchange, min, sets an upper limit on virtual photon range (for close interactions) |q min | min / R max 1/ |q min | / min Bigger R max more target particles to couple to – rate of interactions will increase. Recall that Since q min 2 decreases with increasing ….. Cross-section (and hence prob. of interaction) increases with increasing (relativistic rise)

32 dE/dx – Minimum Ionising Particles As 1 curve flattens off to a minimum –Minimum reached at Minimum ionising value is roughly 2 MeV g -1 cm 2

33 dE/dx – Relativistic Rise Above minimum dE/dx rises with momentum Relativistic Rise –ln( term in Bethe-Bloch From derivation can see two factors, each ln( – max rises with more close collisions –Decrease in q 2 min for distant collisions – increase in range of virtual photons.

34 dE/dx – Density / Screening Rise flattens off in solids (and to a lesser extent gasses) due to density effect –Modeled by 2 in Bethe-Bloch Comes from bulk effects such as polarization ( virtual photons are screened from distant atoms ) –Stronger in solids (e.g. copper – plotted above) than in gases.

35 dE/dx – Femi Plateau –If high-energy knock-on electrons ( -rays) are excluded, measured dE/dx reaches a constant value less than 1.5 minimum ionising (Fermi Plateau) –At very high energy, radiative energy loss processes (bremsstrahlung) become important.

36 dE/dx Data Data from gaseous track detector. –Each point from a single particle –Several energy loss samples for each point –Averaged to get energy loss –Fluctuations easily seen ( see later in course) dE/dx (keV/cm) p (GeV/c) p K e

37 dE/dx in Different Materials Z/A similar for most nuclei Effective ionization, I, varies only slowly with Z Min. ionization occurs for ~ same value of

38 Scattering of a Charged Particle by Exchange of Virtual Photons

39 Charged Particles and Virtual Photons EM interaction is mediated by exchange of photons. On-shell photons have zero mass, but from uncertainty principle: p x h/2 –The more localized the photon near a charge the larger the uncertainty in its momentum E t h/2 –The shorter the life of a photon (more confined to source charge) the larger the uncertainty in its energy.

40 Charged Particle as Photon Source View charged particle as surrounded by cloud of virtual photons. Higher energy virtual photon (more off- shell) clustered more tightly round charge. As charged particle passes through matter the virtual photons will interact with medium. Hence, study interaction of photons with matter:

41 Interaction of Photons with Matter

42 Can not study interaction of virtual photons directly, but can study interaction of real photons. Photon Interactions are interesting in their own right. Different processes important at different energy ranges.

43 Coherent Elastic Scattering Two sorts: a)Rayleigh scattering from atomic electrons b)Thompson scattering from nuclear charge. -Does not excite atoms or cause energy loss. -Doesnt leave a detectable signal in medium

44 Photo-excitation Photon absorbed by atom – exciting it into a higher energy state. Strong absorption peaks for photon energies corresponding to atomic transitions. Mainly in low energy (<10eV ) region –Not labelled on plot

45 Photoelectric Effect Photon absorbed by atom which expels and electron. Cross-section depends on atomic charge Z –At high energies varies approx. as Z 5 ( Z C =6, Z Pb = 82 )

46 Compton Scattering In Rayleigh/Thompson scattering, photon scatters off all the electrons in an atom coherently. In Compton scattering the photon interacts with a single electron –Needs a shorter wavelength (higher energy) photon Scattered electron usually has enough energy to leave atom ionization

47 Compton Scattering Atomic cross-section proportional to the number of electrons in the atom – Pb / C = 82/6 KE of scattered electron (incoming photon E, scattered through ):

48 Pair Production When a photon has an energy greater than twice the energy of an electron it can convert to an electron and a positron A photon in free space can not create a e + /e - pair ( conservation of p,E ) Can convert if third body to transfer momentum to pair production near nuclei

49 Pair Production Heavy nucleus takes less energy for a given momentum exchange Threshold energy higher for carbon than lead Can also get pair production near atomic electrons (energy threshold higher) Cross-section ~ Z 2

50 Photonuclear Absorption Photons ( ~ >10MeV) can excite resonant states in nuclei –Nuclear analogue of atomic photo-excitation. –Cross-section small overall, but with peaks in region of nuclear giant resonance. (Photons can cause nuclear photo- disintegration)

51 Photon Propagation in a Medium The effect of the medium a photon travels through can be characterised by the dielectric constant. In an dense medium a photon will interact with many atoms simultaneously. ( Depending on wavelength/energy of photon)

52 Dielectric Constant Elastic collisions at zero angle do not change the energy of a photon, but they do change its phase. –Described by the real component of Interactions resulting in photon absorption or energy loss are described by the imaginary component of

53 Dielectric Constant – Phase Velocity Dielectric constant describes the way that the interaction of photons with the medium modifies their phase velocity u – the phase velocity is not necessarily the speed at which a signal is transmitted.

54 Refractive Index and Dielectric Constant The refractive index for an optical medium and its dielectric constant are related by The dielectric constant (refractive index) is a function of the photon energy.

55 Dielectric constant as function of energy

56 Dielectric Constant – Optical Region Photon energy below threshold for exciting atomic transitions. Interaction modifies the phase velocity of the light. Absorption low –medium transparent Dielectric constant real and greater than 1 –Phase velocity < c

57 Dielectric Constant – Resonance Region Photon energy comparable with atomic excitation energy. Absorption high –medium opaque –Im large

58 Dielectric Constant – X-ray Region Photon energy well above ionization energy. –Photon sees atomic electrons as free particles. –Scattered electrons can have high energy: -rays Absorption low –Phase velocity > c

59 EM Field from a Moving Charge For a high energy particle traveling through a medium individual interactions will not affect energy significantly –View particle as traveling at constant velocity through an infinite medium. In rest-frame of particle, EM field is a static coulomb field.

60 EM Field from Moving Charge Write the EM field in the lab frame of the moving charge as a sum of plane waves:

61 Field from Moving Charge The field at a given position relative to the particle remains the same: Components of field must propagate with velocity of particle: v. k

62 Effect of the Medium In a vacuum and c related by = c k The effect of the medium is to modify the phase velocity of the photons So we have:

63 A simple 2-D model Limit ourselves to two-dimensions to keep things simple. (no Bessel functions) x y Particle track v= c

64 Wave-number, k, in 2-D Components of wave-number k Rearrange to get k y :

65 Moderate velocity – virtual photons If particle velocity, v, through the medium is less than the phase velocity: k y is imaginary I.e. virtual photons rather than free propagation. Define a transverse range, y 0 : –y 0 = i / k y k.r = k x x + k y y …. So:

66 Range of Virtual Photons EM field shows an exponential fall with transverse distance from track ( e -y ) (= Decrease in density of cloud of virtual photons away from their source) Transverse range, y 0 :

67 Range of transverse EM field Photon energy, Range of virtual photons decreases with increasing photon energy

68 Dielectric constant, In optical region of photon energies, phase velocity, u < c Fast particles can have a velocity, v>u k y real, I.e. photons are real and can propagate away from particle track Cerenkov radiation

69 Cerenkov Radiation in Popular Culture… Picture of Cerenkov radiation from the core of a water- cooled nuclear reactor. given off by fast electrons emerging from fission reactions

70 Dielectric constant, If e.g. in X-ray region of photon energies) phase velocity u>c –No particle can travel faster than u – k y always imaginary ( y 0 always real ) –No Cerenkov radiation.

71 Relativistic Rise Photon range: Range rises with momentum ( p = m ) If y 0 =c / relativistic rise

72 Density Effect Relativistic rise will not continue indefinitely due to (1- term Range of virtual photons at high photon energies tends to: As tends to 1 (material acts like the vacuum), range tends to infinity Classical view: polarisation of medium screening of charge

73 The Allison-Cobb Expression for Energy Loss References: –Kleinknecht Detectors for Particle Radiation, chap. 1 –Allison and Cobb, Ann. Rev. Nucl. Part. Sci, Vol. 70 (1980), p. 253 –Allison and Wright in Experimental Techniques in High Energy Physics, ed. T. Ferbel, p. 371

74 Steps in Allison & Cobb Derivation 1)Solve Maxwells equation in the medium to obtain a field (produced by polarisation of the medium) through which the particle moves. 2)that the particle loses energy by doing work against this field. (This is the same as the energy transfer by virtual particles, but in classical terms)

75 Steps in Allison&Cobb 3) Express energy loss as integral over angular frequency Interpret energy loss as energy transfer by photons of energy. (Quantum picture)

76 Steps in Allison&Cobb Integral over and k becomes integral over E ( = h` and p ( = h`k ) of exchanged photons (Photons are virtual so E,p not related by E=pc )

77 Steps in Allison&Cobb 4) Dont have direct data for interactions of virtual photons. use cross-sections for real photons, dispersion relations and assume that high energy virtual photons interact with free electrons - This step is the same as calculating k,

78 Steps in Allison&Cobb 5) Previous step allows us to integrate over virtual photon momentum, p Get energy loss per unit length as integral over photon energy:

79 Allison&Cobb – d dE Differential cross-section per unit energy loss: i phase ( 1 - i

80 Allison&Cobb – d dE Terms in d dE represent contributions to generic Feynman scattering diagram –Factor z 2 from coupling of photon to fast particle –As before, all terms contain 1/ slower particles spends more time in vicinity of atoms in medium higher probability of interaction – Coupling to medium at B described by photo-absorption cross section (E)

81 Allison&Cobb – d dE First two terms in d dE correspond to exchange of transversely polarized photons. Only significant at high speed ( (Real photons have to be transversely polarized)

82 Allison&Cobb – d dE Last two terms in d dE correspond to exchange of longitudinally polarized photons. Adding transversely polarized photons ~ Rutherford Mott

83 Allison&Cobb – d dE First term has ln( behaviour -(relativistic rise found in 2D model) -Eventual saturation at Fermi plateau

84 Allison&Cobb – d dE For low energy photons in optical region, so only second term contributes. In this regime, term describes Cerenkov radiation (see later)

85 Allison&Cobb – d dE Third term corresponds to photoelectric emission of electrons. Photon energy in the resonance region.

86 Allison&Cobb – d dE Fourth term describes Compton scattering of energetic photons from atomic electrons. production of -rays.

87 Features of Allison&Cobb dE/dx Overall features the same as Bethe-Bloche (of course!) Better fit to data than Bethe-Bloche … now the model gives: visible-wavelength photons produced by fast particles in a transparent medium.

88 Cerenkov Radiation For photons with energy in visible region, refractive index of transparent materials, n>1 Phase velocity, u < c Transverse component photon momentum ( k y in our 2D model) can be real (if charged particle velocity v>u) Passage of charged particle through medium produces real rather than virtual photons.

89 Cerenkov radiation Photons emitted at angle c: Cerenkov radiation (optical light) produced when >1/n wavefront light, velocity=c/n Fast particle, velocity= c

90 Cerenkov Light Used to Detect Neutrinos -SNO

91 Cerenkov Angle High refractive index low threshold for Cerenkov production Can either use threshold or angle for particle identification n=1.5 n=1.1 c

92 Intensity of Cerenkov Radiation Intensity of Cerenkov radiation described by second term in Allison and Cobb formula: –(only describes Cerenkov radiation in optical region) –In this region so –Multiply d dE by NEdE to get energy loss due to Cerenkov radiation per unit length.

93 Intensity of Cerenkov Radiation Have found dE/dx from Cerenkov Since energy of photon E = h can calculate flux of photons per unit length: ( Phase angle jumps from 0 for to when ie. At threshold )

94 Intensity of Cerenkov Radiation Number of photons per unit length increases as increases. Over range of visible light photon energies: c dN dx (arb units)

95 Intensity of Cerenkov Radiation Spectrum almost flat over optical region. Photons only emitted in the optical region. –In the resonance region there is too much absorption –In the X-ray region so u>c, so particle velocity always less than phase velocity

96 Transition Radiation Transition radiation is closely related to Cerenkov radiation. Occurs when a charged particle crosses the boundary between materials of different refractive indices.

97 Transition Radiation Can think of as diffractively broadened Cerenkov radiation (thin source) –Broadening of Cerenkov angle causes radiation at angles that would otherwise be unphysical e.g. in X-ray region Can also be thought of as apparent acceleration of the charge as it passes through the boundary –(Think of an object underwater breaking through surface)

98 Intensity of Transition Radiation Total energy flux of transition radiation: Typical photon energy: Number of photons ~ many foils Plasma frequency: In x-ray region:

99 Intensity of Transition radiation The energy given off in transition radiation is proportional to –Compare this to Cerenkov radiation where the threshold and the intensity are a function of –Transition radiation can be used to discriminate between particles of high momenta – where very close to 1 and Cerenkov detectors can not discriminate. TR intensity saturates at high values of destructive interference between faces of radiator)

100 Transition Radiation B d 0 J/ψ K s 0 ~1 TR hit ~7 TR hits Electrons with radiator Electrons without radiator Two threshold analysis MIP threshold 0.2 keV – precise tracking/drift time determination TR threshold 5.5 keV – electron/pion separation Transition radiation is produced when a charged ultra-relativistic particle crosses the interface between different media, PP (fibers or foils) & air for the TRT. TR photons are emitted at very small angle with respect to the parent-particle trajectory. Energy deposition in the TRT is the sum of ionization losses of charged particles (~2 keV) and the larger deposition due to TR photon absorption (> 5 keV)

101 dE/dx for Electrons Assumption that individual interactions transfer energy much less than particle energy is not true for electrons (much lighter than other charged particles) Quantum effects – incident particle identical to atomic electrons it is interacting with. Different behaviour of energy loss as function of momentum.

102 dE/dx for Electrons Modified Bethe-Bloch equation: T = electron energy ( c.f. simplified Bethe-Bloch for other particles: )

103 Energy Loss Processes other than Ionization Bremsstrahlung –Electromagnetic showers Hadronic Interactions –Hadronic showers Charged Particles in a Magnetic Field … Ok, so this isnt an energy loss process Synchrotron Radiation

104 Energy Loss due to Radiation Have assumed that energy exchanged per interaction is small compared to particle energy. Particle can transfer a large amount near to atomic nuclei –Nuclei are surrounded by very high energy virtual photons The particle is momentum is significantly changed by interaction –i.e. particle is accelerated Particle radiates photons Process called Bremsstrahlung –Lit. braking radiation

105 Bremsstrahlung Simplest Feynman diagrams have three vertices –Two for exchange of virtual photon between nucleus (charge Z) and particle (charge z) –One for emission of real photon Cross section: –Factor Z 2 from nucleus –Factor from vertices –Can view as separating off one of the cloud of virtual photons surrounding particle

106 Bremstrahlung The lighter the particle the greater its acceleration for a given momentum exchange – cross-section has a factor of 1/m 2 If the energy of the emitted photon is the differential cross-section is

107 dE/dx from Bremsstrahlung Integrate over photon energy to get energy lost by particle per unit length: –Where max ~ particle kinetic energy T and min ~0. ( n is the density of nuclei) –Put in factors missed out by hand-waving (eg. screening) and get (for electrons)

108 Bremsstrahlung

109 Critical Energy All high energy charged particles can loose energy through bremsstrahlung. For a given material the energy at which bremsstrahlung becomes significant relative to energy loss by ionization is dependant on the particle mass Particle energy at which bremsstrahlung overtakes ionization called the critical energy, E c (~m 2 ) –( Other definitions as well )

110 Electron Bremsstrahlung For (current) particle detectors in high energy physics, the only particle that undergoes significant bremsstrahlung is the electron. Critical energy for electrons –Heavy metals (Pb,W,U) ~ 10MeV –Hydrogen ~ 300MeV Critical energy for muons ~ 100s GeV High energy hadrons – hadronic interactions much more important.

111 Radiation Length The rate of energy loss due to bremsstrahlung is proportional to the kinetic energy: I.e. So… … where X 0 is the radiation length –Distance in which energy reduced by factor of e

112 Radiation Length Number density of nuclei, n, Avagadros number N A and density, related by – n N A A –E.g. Pb has X 0 for electrons of 6.37g/cm 2 (0.56cm) Z/A ~ constant, so X 0 Use high Z materials to shield against photons (and electrons)

113 Radiation Length Units: X 0 is often (indeed usually) given in terms of g/cm 2 –Multiply by density,, to get X o as a distance. Radiation length of mixture/compound ( w j = weight fraction) Radiation length depends on incident particle ( ~ m 2 ). 236mmuon 0.56cmelectron X 0 in leadParticle

114 Bremsstrahlung Energy Spectrum Energy spectrum ~ flat up to maximum energy max = T –Cross-section for a photon of energy –Photon energy E = h – dE = dN d ~ constant Spectrum doesnt actually have a sharp cut- off at =0 and = T : rolls off from e.g. screening of nuclear charge.

115 Bremsstrahlung Energy Spectrum Plot of scaled bremsstrahlung cross- section as a function of y =

116 Electromagnetic and Hadronic Showers

117 Electromagnetic Showers High energy electron produces photon through bremsstrahlung Photon produces e + e - through pair production

118 Electromagnetic Showers Shower of electrons/positrons and photons continues until the energy of the particles is too low for further multiplication ( E

119 Shower Profile Shower profile for a Lead/Scintillator calorimeter:

120 EM-Shower Profile Maximum number of shower particles ( shower maximum ) occurs at a depth of roughly 5X 0 into the absorber. Depth of shower maximum depends logarithmically on particle energy. In transverse direction energy deposited falls off (very approximately) exponentially. –Length scale (the Moliere radius) ~ (7g/cm 2 )(A/Z) –90% of energy contained in ~ 1 R M –R M = 1.5cm in lead

121 EM-Shower Transverse Profile Results from a test Fe calorimeter with 500MeV electrons:

122 EM-Shower Profile Example: 3.2 GeV shower in lead has 400 particles in the cascade at shower maximum, which occurs at a depth of 6X 0 ( ie. 3.36cm in lead) A total absorber thickness of 25-30X 0 is enough to absorb most of the energy in a shower.

123 Electromagnetic Showers Total energy deposited is the same for electron and photon of equal energy Total energy deposited depends linearly on the energy of incoming particle. Maximum number of shower particles ( shower max ) occurs at a depth of roughly 5X 0 into the absorber. Depth of shower max depends logarithmically on particle energy.

124 Hadronic Showers High-energy hadrons give hadronic showers. Hadron interacts with nucleus by the strong interaction. Number of particles produced in each collision ln(E) Length scale : – I is nuclear cross-section for strong interaction – I = hadronic interaction length

125 Hadronic Showers – Length Scale I is approximately independent of particle energy and type –nucleus behaves like a black ball In terms of g/cm 2 : I (35 g cm -2 )A 1/3 Depth of shower maximum depends logarithmically on incident particle energy: x shower-max / I 0.2 ln ( E/1GeV ) + 0.7

126 Hadronic Showers – Length Scale At typical HEP energies need roughly 9 I to contain average of 99% of energy in hadronic shower.

127 Hadronic and EM length scales For materials other than hydrogen I is several times larger than X Pb Cu H2H2 X 0 (g/cm 2 ) I (g/cm 2 ) Material

128 Hadronic Shower Development On average about 1/3 of the particles produced in each hadronic interaction are neutral. Mainly pions, 0 0 rapidly decays to photons. Photons initiate EM showers. For an energetic hadronic shower most of the detectable energy deposited is from e + /e - in EM-shower from decaying 0

129 Fluctuations in Hadronic Showers Fluctuation in detectable ( ionization, atomic excitation) energy deposited is greater than for electromagnetic showers: Fluctuation in neutral fraction of shower ( 0,n,etc. ) –In general response to hadron and electron of same energy is not the same (non compensating) Energy in nuclear binding effects is not detectable.

130 Fluctuations in Hadronic Showers Fraction of undetectable energy changes with particle energy: Total visible energy deposited does not depend linearly on the energy of incoming particle. Detectable energy deposited in detector depends on type of particle as well at its energy

131 Charged Particle Motion in a Magnetic Field Lorentz Force Radius of Curvature

132 Lorentz Force A particle of charge ze, and velocity v moving in a magnetic field B feels a force F given by: Force is at right angles to particles path – so direction of velocity changes but not its magnitude. In general the path is a helix, with the axis along the field lines.

133 Particle Trajectory Particle travelling a small distance through a magnetic field. Look in plane perpendicular to the field: –Compt. of velocity,momentum perp. to field =

134 Particle Trajectory Force: Change in momentum: Angular deflection: Path is part of a circle, so d = (v dt) / Hence And –p t in GeV, B in Tesla, in metres

135 Synchrotron Radiation

136 Accelerated charged particles radiate. High energy particles in a high magnetic field can radiate energetic photons: synchroton radiation In particle centre of mass frame the magnetic field in the lab frame transforms to have an electric component. – 4-vector for EM field A = (, A ) – Look at acceleration caused by this E field:

137 Synchrotron Radiation For a particle moving with velocity v in a magnetic field B: –(particle moving along x-axis. B field at an angle relative to x-axis) Resulting acceleration: –Here m e is the rest energy

138 Synchrotron Radiation Since p = mc and p =eB c 2 then m e = eB c/ – Again, is the radius of curvature in field)

139 Synchrotron Radiation - Intensity From classical electrodynamics, energy radiated from an accelerating charge: –Since p/mc at a given momentum dE/dt (p/m) 4 –dE/dt calculated in CoM frame but same in lab. frame.

140 Synchrotron Radiation – Angular Distribution Accelerated charge Dipole field In laboratory frame the boost distorts this shape ( headlights effect )

141 Synchrotron Radiation – Frequency Distribution In classical electrodynamics frequency of the emitted radiation is the same as the frequency of the accelerating force. –E.g. Accelerate electrons in an antenna in simple harmonic motion, frequency, get an EM wave of frequency Relativistic boost increases this by In relativistic circular motion get a distribution of frequency (photon energy)

142 Synchrotron Radiation – Frequency Distribution Characteristic frequency c = c/2a

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