ELECTROMAGNETIC SHOWERS. Paolo Lipari. Lecture 2

Transcription

1 ELECTROMAGNETIC SHOWERS Paolo Lipari Lecture 2 Corsika school 26/Nov/2008

2 Bremsstrahlung Pair Creation

3 BREMSSTRAHLUNG Fully ionized free nucleus (approximation of infinite mass) High Energy Limit (Full screening)

4 PAIR PRODUCTION Fully ionized free nucleus (approximation of infinite mass) High Energy Limit (Full screening)

5 Radiation Length: Meaning: Length where the energy of an electron is reduced to E/e 7/9 of the mean free path of photons

6 From Particle Data Book

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8 The SPLITTING FUNCTIONS

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10 ELECTROMAGNETIC SHOWERS Pair Production Brems strahlung Radiation Length (Energy independent) Vertices : theoretically understood (and scaling)

11 AVERAGE LONGITUDINAL EVOLUTION for a PURELY ELECTRO-MAGNETIC SHOWER Two functions of energy and depth

12 Possible Generalizations: 3-Dimensional treatment. Hadronic Showers: add other components

13 SYSTEM of INTEGRO-DIFFERENTIAL EQUATIONS that describe the evolution with t of for a given initial condition.

14 Variation with t of the number of photons with energy E

15 Variation with t of the number of photons with energy E =

16 Electrons

17 Electrons

18 Electrons

19 Electrons 2 divergent e -> e contributions. Their combination is finite.

20 Approximation A

21 INTRODUCTION of ENERGY LOSS of ELECTRONS for COLLISIONS

22 INTRODUCTION of ENERGY LOSS of ELECTRONS for COLLISIONS

23 Bethe-Bloch formula Very Simple Approximation: ENERGY INDEPENDENT LOSS Critical energy

24 Insert the collision energy loss in the shower equations: General treatment: Energy variation Law

25 =

26 =

27 Approximation A

28 Approximation B

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30 Bruno Rossi 1933 Eritrea East West effect

31 Kenneth Greisen NCAR Texas 1971 after discovery of 200 MeV photons from the Crab Nebula

32 Shower Equations in Approximation A (neglect electron ionization losses) No Parameters with the Dimension of Energy

33 Solutions to the shower equations. Initial Condition: Photon of energy E0 Electron of energy E0

34 Let us consider an electron population that is has the spectral shape of an unbroken power law and no photons: Study the Shower evolution using approximation A

35 Initial condition Electron and Photon population remain a power law of same slope Only the normalizations are a function of the depth t Depth evolution

36 Coefficients Ke, (t) are linear combinations of two exponential

37 One controls the (faster) convergence to an s-dependent gamma/e ratio (large and negative) A second exponential describes the (slower ) evolution of the two population with a constant ratio.

38 S=1 Special spectrum Equal amount of energy per decade of E

39 S=1 Special spectrum Equal amount of energy per decade of E depth-independent solution

40 What can we say about: Without explicit calculation? Spectrum E-2 equal power per decade of E Pair Production and Bremsstrahlung redistribute the energy but nothing can change

41 What can we say about: Spectrum flatter than E-2 power per decade of E grows with E

42 What can we say about: Spectrum steeper than E-2 power per decade of E decreases with E

43 Insert functional form of the solution in the shower equation. Obtain simple quadratic equation connecting s

44 Time derivative Example of one term

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48 Stability Ratio for the Photon/Electron Ratio

49 t-slope and E-slope are connected Integral Electron Spectrum Evolution Can deduce the AGE (and spectral shape)

50 Approximate simple analytic expression proposed by Greisen

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52 Include the effects of electron ionization losses de/dt = constant= Approximation B critical energy

53 s=1 Approximation B elementary solution e

54 Include the effects of electron ionization losses de/dt = constant= critical energy Elementary Solutions that develop as exponential in depth Limit for small x

55 Equations for the correction functions

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59 Rossi and Greisen show how the shape of the elementary functions can be obtained as a Power Series

60 Earlier results

61 Concept of : Shower AGE Shower Longitudinal Development Often used but (in my view) unsatisfactory definition Shower at maximum: s=1 Shower before maximum s < 1 Shower after maximum s > 1 S=

62 Age as a function of t/tmax

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66 The Spectral shape of the electron Spectrum is determined (on very good approximation) by the shower Age The Photon Spectral; shape is (in good Approximation) also determined by the shower Age Calculated first by Rossi, Greisen in 1941 The Ratio photon/electron is determined by the shower Age Model Independent Definition of AGE

67 Go Beyond the Elementary Solutions Exponential Depth Dependent Realistic initial condition Depth Evolution of Monochromatic Photon E0 Monochromatic Electron E0 Age is a function of depth: s(t)

68 Monochromatic Photon

69 Monochromatic Photon Critical energy

70 Write initial condition as a superposition of power law component Inverse Mellin transform

71 Write initial condition as a superposition of power law component Inverse Mellin transform Depth Evolution

72 For a given E0, E, t what is the parameter s that dominate? Solution of this equation

73 For a given E0, E, t what is the parameter s that dominate? Solution of this equation

74 Monochromatic Photon

75 Monochromatic Photon. Approximation A,B

76 Different Energy : Same Age (Shower Maximum)

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78 What About the Universality of Longitudinal Development?

79 Age and Longitudinal Development

80 Age and Longitudinal Developmen t S=

81 Age and Longitudinal Developmen t S= Differential Equation

82 Differential Equation Boundary Condition Solution : Greisen Profile

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84 For real showers the longitudinal development is not identical to the Greisen Profile and fluctuates from shower to shower Violations of the Universality

85 For real showers the longitudinal development is not identical to the Greisen Profile and fluctuates from shower to shower Violations of the Universality General Model Independent Definition of Age