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Pair Productionproduction If Hν > 2Mc2 ≈ 1.02 Mev the Production of a Pair of Electron and Positron Becomes Possible Close to a Charged Massive Object (Eg

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Pair Productionproduction If Hν > 2Mc2 ≈ 1.02 Mev the Production of a Pair of Electron and Positron Becomes Possible Close to a Charged Massive Object (Eg PairPair productionproduction If hν > 2mc2 ≈ 1.02 MeV the production of a pair of electron and positron becomes possible close to a charged massive object (eg. a nucleus) which takes away the amount of momentum needed to preserve momentum conservation during the interaction with the Coloumb field of the massive object itself. The probability of pair production is a slowly increasing function of energy while the Compton probability decreases rapidly with increasing energy Exercise 3: calculate the threshold energy of the photon Eth = ? Compton: For small energy the cross section reduces to Thompson one 8 ε → 0 ⇒ σ → σ = πr2 C Th 3 e hν ε = 2 3 2ln(2ε) +1 mc ε >>1⇒ σ → σ C 8 Th ε At high energy pair production dominates At high energy pair production€ dominates (Bethe-Heitler) € 2 2 σ pair (E) ≈ αZ re ln Eγ γ At low energies photoelectric absorption 26 on atomic shells € PairPair productionproduction The differential cross section has the form Fractional energy transfer to the electron/positron (k incident γ energy) hν 1 2 2 28 2hν 218 2 −1 1<< 2 << 1/3 ⇒ σ pair = αZ re ln 2 − m atom mec Zα 9 mec 27 No screening case hν 1 2 2 28 183 2 2 −1 2 >> 1/3 ⇒ σ pair = αZ re ln 1/3 − m atom mec Zα 9 Z 27 Normalized pair prod cross section vs Fractional electron energy € for photons of variouos energy At high energies σpair > σCompton 27 ElectromagneticElectromagnetic andand hadronichadronic showersshowers 28 ElectromagneticElectromagnetic showersshowers Cascades initiated by electrons or photons discovered by Blackett-Occhialini 1933. High energy electrons lose most of their energy through bremsstrahlung, producing high energy photons that in turn produce pairs and Compton electrons. These electrons and positrons radiate new photons which in turn undergo pair production or Compton scattering. Since the bremsstrahlung and pair production cross sections become almost energy dependent for incoming electron and photon energies >>mc2/αZ1/3, the radiation length emerges as typical unit length of which a reasonable approx is 716A A Lp~9/7X0 X (g / cm2 ) = = 0 183 1 N σ Z(Z +1.3) ln + A bremss Z1/3 8 After reaching a maximum where the largest number of secondaries are € created the cascade decays slowly and the multiplication process is almost stopped when the electron energy reaches the critical energy εc. Rossi - Greisen (1941) 29 ElectromagneticElectromagnetic showersshowers + - If E0 is the energy of the incoming photon it generates after 1X0 a pair e e of equal energies. Both will emit a bremsstrahlung photon in 1X0. By continuing the process and assuming equal energy sharing the number of particles after t t t radiation lengths is N(t)≈2 and their energy will be E(t)≈E0/2 . When E~εc the multiplication stops and tmax=ln(E0/εc)/ln2. εc = critical energy where radiation and ionization loss are equal . Nmax = E0/εc=# of particles at max Toy model At very high energies the Landau-Pomeranchuk-Midgal effect reduces the pair production and bremsstrahlung cross-section and cause significant elongation of showers (quantum interference between amplitudes from different30 scattering centers) LongitudinalLongitudinal showershower developmentdevelopment Number of electrons depth dependence for showers initiated by γs of various energies. tmax=ln(E0/εc)/ln2 Nmax = E0/εc 31 ElectromagneticElectromagnetic showersshowers The longitudinal profile is described by The traverse development is described by the Moliere radius on average 10%(1%) outside cylinder of radius RM(3.5RM) 32 HadronicHadronic showersshowers Cascades initiated by hadronic interactions of nucleons or nuclei. In the first interaction about 1/2 of the energy is transferred to secondary mesons (charged and neutral - the ratio of π±/π0~2). Secondary mesons can themselves interact and generate secondary hadronic cascades until hadron energy goes below interaction threshold (~hadron mass). Neutral pions induce em cascades due to their immediate decay with decay length π 0 → γγ E 0 π −6 l 0 l 2.51 10 cm d = γ π 0 = × ⋅ m 0 π € High energy charged pions have € large decay length hence they interact, lower energy ones decay into neutrinos and muons. Int length > radiation length and also energy dependent λn = 80 g/cm2 33 λπ = 120 g/cm2 HadronicHadronic showersshowers The shower hadronic component carries a large fraction of the primary particle energy deeper than an em cascade. But the secondary multiplicity in hadronic interactions is higher than the effective multiplicity of 2 secondaries in em interactions and this faster energy dissipation compensates for the longer interaction length. For a proton and a photon of 105 GeV the p initiated shower peaks earlier than the photon initiated one (max depths at 506 and 520 g/cm2) This is due to the higher secondary multiplicity in hadronic interactions The p shower is not absorbed as easily as γ one The toy model can be applied using the interaction length 34 AtmosphericAtmospheric HadronicHadronic showersshowers Given a nucleon of energy E0 (GeV) that interacts in the atmosphere at depth λN: in the interaction it loses (1-Kel) fraction of its energy and generates <m> secondary pions, 1/3 of which are neutral. The depth of the shower maximum can be written as the sum of the depths of the em showers induced by neutral pions and the int length of the primary nucleon: 2(1− Kel )E 0 X max = X 0 ln + λN (E 0 ) max 1 m (1− Kel )E 0 ( m / 3)ε0 Ne = 2 3 ε0 And the number of electrons at maximum is: Splitting in 2 of π energy π multiplicity € € 2 2 5 Eg. λN = 80 g/cm , <m> = 12, Kel = 0.5 Xmax = 500 g/cm for a 10 GeV proton max 4 And Ne = 8 10 electrons A longitudinal development parametrization (Gaisser) X max −λ X −X1 st X − X λ − The depth of the 1 interaction X1 is distributed N (X) = N max 1 e λ e e as exp-(X / ), main source of fluctuations X max − λ as exp-(X1/λ), main source of fluctuations 35 € AtmosphericAtmospheric HadronicHadronic showersshowers β E 0 Nµ = Number of muons related to number of decaying pions: επ Where β = 0.85 = fraction of multiplicity of charged pions=ln(2/3<m>)/ln<m> ε = an effective energy = 20 GeV π € For heavy nuclei the superposition model is a good approximation: the interaction of a nuclei of mass number A and total energy E0 is equal to the interaction of A nucleons of energy E0/A And the shower depth is related to the nucleon one by A 2(1− Kel )E 0 / A A p X max = X 0 ln + λN (E 0 ) ⇒ X max = X max − X 0 ln A ( m / 3)ε0 And β € A E 0 1−β p Nµ = A = A Nµ Aεπ Showers generated by heavy nuclei generate more mouns than p showers Inclined showers develop in more tenuous atmosphere where pions and kaons36 are more€ likely to decay than interact so contain more muons..
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