1. Calorimeters Chapter 4 Chapter 4 Electromagnetic Showers

2. Calorimeters Chapter 4 Basic Considerations In the previous sections all relevant electromagnetic Interactions were introduced For Electrons/Positrons a)Ionization b)Bremsstrahlung (Annihilation for Positrons) For Photons a)Photoelectric Effect b)Compton Scattering c)Pair Production A primary particle looses energy due to one of these Interactions in a given absorber material The produced secondaries in turn do interact with the Material and produce themselves further particles Development of an Electromagnetic Cascade

3. Calorimeters Chapter 4 A Very Simple Shower Relevant for photons in the MeV Range -> Nuclear Physics Nuclear  With E = 3370 keV In 65 Ga A B C C D 3378 200 511 1174 511 200 A)Pair Prodruction e- and e+: continous loss by ionization B) Annihilation of e+ C) Compton Scattering D) Photoelectric absorption e+ E-

4. Calorimeters Chapter 4 High Energetic Particles Energy loss of electron by Bremsstrahlung: Photons convert into e + e - -Pairs t=x/X 0 Simple Shower Model (see Longo for detailed discussion) - Energy Loss after t: E 1 =E o /2 - Photons -> materialize after t E ± =E 1 / 2 Number of particles after t: N(t) = 2 t Each Particle has energy Shower continues until particles reach critical energy (see p. 16) where t max = ln(E 0 /  c ) Shower Maximum increases logarithmically with Energy of primary particle (Important for detector design !!!) For high energetic particles Shower contain thousands Of particles

5. Calorimeters Chapter 4 Shower Depth GEANT4 Simulation of e - in copper Shower Maximum Increases Logarithmically with Energy ~3.5*log 10 (E) In qualitative agreement with our simple model

6. Calorimeters Chapter 4 Energy Distribution of Shower Particles e < 4 MeV e < 1 MeV e > 20 MeV Shower contains mostly of soft particles with energy < 4 MeV Compton Electrons and Photo electrons Increase with Z for constant energy Reason: Number of soft photons increase as shower Develops Majority of shower photons In Compton and Photo Effect Regime

7. Calorimeters Chapter 4 Scaling Variables I - Radiation Length (Not only) For Bremsstrahlung: Spectrum of radiated photons: Define:  rad is independent of the energy of the radiated Photon and only a function of the the material Radiation Length X 0 : Distance after which a particle has lost 1/e due to radiation With X 0 = 1/N  rad  Absorption Coefficient Photons i.e. Photons travel a longer distance before they interact Some Values: X 0,Air =30 420 cm X 0,Al = 8.9 cm X 0,Pb = 0.56 cm

8. Calorimeters Chapter 4 Longitudinal Shower Profiles Addendum: X 0 Characterizes longitudinal Extension of elm. Shower. Shape of Shower profiles Material Independent = Energy loss Independent of material.. as function of X 0 Differences: - Pair Production by emitted Photons increases with Z and extend to much lower energies - Critical Energy Z-dependent e.g. 43 MeV for Al 7 MeV for Lead - X 0 looses meaning at low energies

9. Calorimeters Chapter 4 Composition of longitudinal Profile

10. Calorimeters Chapter 4 Lateral Shower Extensions 2 Effects 1) Electrons (and positrons) undergo multiple scattering in the Coulomb Field of absorber nuclei  mean = 0 with standard deviation  rms Primary e  For projected Scattering angle distribution 2) Isotropic production of secondaries by Photoelectric effect and Compton Scattering Low energetic component of shower Expect different behaviour in region ‘far’ away from shower axis Transversal extension of shower characterized by Molière Radius 90% of shower energy is contained within  M Weak/no Z dependence Typical Values for  c,  M :  c [MeV] R M [cm] Pb 7.2 1.6 NaJ 12.5 4.4 Air 87 7400

11. Calorimeters Chapter 4 Lateral Shower Profiles - Core (I.e. region close to the shower axis is ‘empty’ in early stages of shower Multiple Scattering of High energetic particles - Core gets ‘populated’ By low energetic component As for long. profile 10 Gev e- in Cu Exponential behaviour with two slopes in radial distribution: Steep falloff close to the shower axis - high energetic component Less steep falloff far from shower axis - low energetic component

12. Calorimeters Chapter 4 Composition of Lateral Profile