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3. Particle Interactions particleExperimental physics particle interactions 3. in particle detectors lots of material borrowed from the excellent course of H.-C. Schultz-Coulon http://www.kip.uni-heidelberg.de/~coulon/Lectures/Detectors/ Marco Delmastro Experimental Particle Physics 1 How do we detect particles? • In order to detect a particle, it must: ! interact with the material of the detector ! transfer energy in some recognizable fashion (signal) • Detection of particles happens via their energy loss in the material they traverses multiple Charged particles Ionization, Bremsstrahlung, Cherenkov, … interactions single Photons Photo/Compton effect, pair production interactions… multiple Hadrons Nuclear interactions interactions Neutrinos Weak interactions Marco Delmastro Experimental Particle Physics 2 ExampleParticle Interactionsof particle interactions– Examples Ionization: Pair Compton production: scattering: Charged Particle Positron Electron Electron γ γ x γ γ Nucleus Electron Electron Photon Charged Electron Particle Photon Photon Positron Electron Electron Atom Nucleus Marco Delmastro Experimental Particle Physics 3 4 27. Passage of particles through matter + ] µ on Cu g / 2 100 − m µ c Bethe-Bloch Radiative V e Anderson- M - [ Ziegler d f f r r r e a a h w h d o Eµc c n p 10 i S g Radiative L Radiative n i Minimum effects losses p p ionization reach 1% o t Nuclear S losses Without δ 1 0.001 0.01 0.1 1 10 100 1000 104 105 106 βγ 0.1 1 10 100 1 10 100 1 10 100 EnergyEnergy loss Loss[ Mbye Vby/ c]ionization: Ionization –Bethe-Bloch dE/dx[GeV/c] formula[T eV/c] Muon momentum Fig. 27.1: Stopping power (= dE/dx )forpositivemuonsincopperasa ⟨− ⟩ function of βγ = p/Mc over nine2 orders2 of magnitudeCharged in momentum (12 orders For now assume: Mc ≫ mec of magnitude in kinetic energy). Solid curves indicateParticle the total stopping power. Datai.e. below energy the loss break for at heavyβγ charged0.1aretakenfromICRU49[4], particles anddata at higher energies are from Ref.≈ 5. Vertical bands indicate boundaries between [dE/dx for electrons more difficult ...] different approximations discussed in the text. The short dottedγ lines labeled “µ− ”illustratethe“Barkaseffect,” the dependence of stopping power on projectile chargeInteraction at very low dominated energies [6]. Electron by elastic collisions with electrons ... 27.2.2. Stopping power at intermediate energies : The mean rate of energy loss by moderately relativisticBethe-Bloch charged Formula heavy particles, M1/δx,iswell-describedbythe“Bethe-Bloch”equation, dE Z 1 1 2m c2β2γ2T δ(βγ) = Kz2 ln e max β2 . (27.3) − dx A β2 2 I2 − − 2 ! " # $ It describes the mean rate of energy loss in the region 0.1 < βγ < 1000 for 2 2 2 intermediate-Z materials with an accuracy of a∝ few 1/β %. ⋅ Atln (const the∼ ⋅ lowerβ γ∼) limit the projectile velocity becomes comparable to atomic electron “velocities” (Sec. 27.2.3), Marco Delmastro Experimental Particle Physics 4 February 2, 2010 15:55 4 27. Passage of particles through matter + ] µ on Cu g / 2 100 − m µ c Bethe-Bloch Radiative V e Anderson- M - [ Ziegler d f f r r r e a a h w h d o Eµc c n p 10 i S g Radiative L Radiative n i Minimum effects losses p p ionization reach 1% o t Nuclear S losses Without δ 1 0.001 0.01 0.1 1 10 100 1000 104 105 106 βγ 0.1 1 10 100 1 10 100 1 10 100 [MeV/c] [GeV/c] [TeV/c] Muon momentum Fig. 27.1: Stopping power (= dE/dx )forpositivemuonsincopperasa function of βγ = p/Mc over nine⟨− orders of⟩ magnitude in momentum (12 orders of magnitude in kinetic energy). Solid curves indicate the total stopping power. Data below the break at βγ 0.1aretakenfromICRU49[4], anddata at higher energies are from Ref.≈ 5. Vertical bands indicate boundaries between different approximations discussed in the text. The short dotted lines labeled “µ− ”illustratethe“Barkaseffect,” the dependence of stopping power on projectile charge at very low energies [6]. Bethe-Bloch Formula Bethe-Bloch27.2.2. Stopping formula power at intermediate for heavy energies particles: The mean rate of energy loss by moderately relativistic[see e.g. charged PDG 2010] heavy particles, M1/δx,iswell-describedbythe“Bethe-Bloch”equation, dE Z 1 1 2m c2β2γ2T δ(βγ) = Kz2 ln e max β2 . (27.3) − dx A β2 2 I2 − − 2 ! " # $ [⋅ρ] It describes the mean rate of energy loss in the region 0.1 < βγ < 1000 for ∼ ∼ density intermediate-Z2 materials2 with an-1 accuracy2 of a few %. At the23 lower limit the K = 4π NA re me c = 0.307 MeV g cm NA = 6.022⋅10 projectile velocity becomes comparable to atomic electron “velocities” (Sec. 27.2.3), 2 2 2 2 [Avogardo's number] Tmax = 2mec β γ /(1 + 2γ me/M + (me/M) ) 2 2 [Max. energy transfer in single collision] re = e /4πε0 mec = 2.8 fm February 2, 2010 15:55[Classical electron radius] me = 511 keV z : Charge of incident particle [Electron mass] M : Mass of incident particle β = v/c [Velocity] Z : Charge number of medium γ = (1-β2)-2 A : Atomic mass of medium [Lorentz factor] Validity: I : Mean excitation energy of medium .05 < βγ < 500 δ : Density correction [transv. extension of electric field] M > mμ Marco Delmastro Experimental Particle Physics 5 Energy Loss of Pions in Cu Energy loss of pions in Cu Minimum ionizing particles (MIP): βγ = 3-4 dE/dx falls ~ β-2; kinematic factor [precise dependence: ~ β-5/3] 2 dE/dx rises ~ ln (βγ) ; relativistic rise [rel. extension of transversal E-field] Saturation at large (βγ) due to density effect (correction δ) [polarization of medium] Units: MeV g-1 cm2 Abbildung 2.2: Energieverlust in Kupfer. Gezeigt wird der Einfluß der Schalenkor- βγ = 3 - 4 MIP looses ~ 13 MeV/cm rektur und der Dichteeffektkorrektur. Das Minimum des Energieverlustes[density of copper: liegt 8.94 bei g/cm3] 2 βγ 3...4.VordemMinimumverh¨alt sich dE/dx β− ,nachdemMinimum ≃ ∝ steigt dE/dx logarithmischMarco Delmastro an und kommt dann inExperimental den Particle S¨attigungsbereich Physics (Dichte- 6 effekt). Man beachte: Die auf der Ordinate angegebene Gr¨oße ist vielmehr 1 dE − ρ dx (nicht dE ). Quelle: Phys. Rev. D 54:S132, 1996. − dx Der fehlende Faktor 2 in der trivialen“ Ableitung kann wie folgt verstanden werden: • ” Ein kleinerer Grenzwert von Emin vergr¨oßert den Anteil des Energieverlustes von Fernst¨oßen, sodaß der Energieverlust gem¨aß Bethe-Bloch erhalten wird. Verschiedene Formulierungen (Bohr, Bethe, Bloch) unterscheiden sich in der Parame- • trisierung der Fernst¨oße, d.h. des Energieverlustes, bei der die Bindung der Elektronen in den Atomhullen¨ nicht vernachl¨assigbar ist. Mittlere Reichweite R(E) Die mittlere Reichweite eines Teilchens berechnet man wie folgt: 0 1 R(E)= − dE, (2.20) !E dE/dx wobei der Energieverlust eine Funktion der Energie selbst ist. Der Hauptteil des Ionisati- dE 2 onsverlustes erfolgt bei kleinen Teilchengeschwindigkeiten ( β− )amEndederSpur dx ∝ Bragg Peak (siehe Abb. 2.3). Genutzt¨ wird die Tatsache, daß der Hauptteil des Energie- verlustes→ am Ende der Spur liegt, bei der Krebstherapie mit energetischen Kernen. Durch geeignete Wahl der Kerngeschwindigkeit (Protonen, leichte Kerne) kann die Energie gezielt und lokal abgegeben werden. 13 UnderstandingUnderstanding Bethe-BlochBethe-Bloch 1/β2-dependence: Remember:Slower particles fell electric force of dx atomic∆ electronsp = Ffor dtlonger= timeF … ? ? ? v Z Z i.e. slower particles feel electric force of atomic electron for longer time ... Relativistic rise for βγ > 4: High energy particle: transversal electric field increases due to Lorentz transform; Ey ➙Abbildung γEy. Thus 2.2: interactionEnergieverlust cross in Kupfer.section Gezeigtincreases wird ... der Einfluß der Schalenkor- rektur und der Dichteeffektkorrektur. Das Minimum des Energieverlustes liegt bei 2 βγ 3...4.VordemMinimumverh¨alt sich dE/dx β− ,nachdemMinimum fast≃ moving ∝ particle steigtparticledE/dx logarithmisch an und kommt dann in den S¨attigungsbereich (Dichte- at rest effekt). Man beachte: Die auf der Ordinate angegebene Große ist vielmehr 1 dE Corrections: ¨ ρ dx dE − (nicht dx ). Quelle: Phys. Rev. D 54:S132, 1996. −γ grossgrows low energy : shell corrections high energy : density corrections γ = 1 Der fehlende Faktor 2 in der trivialen“ Ableitung kann wie folgt verstanden werden: • ” Marco Delmastro EinExperimental kleinerer Particle Grenzwert Physics von Emin vergr¨oßert den Anteil des Energieverlustes7 von Fernst¨oßen, sodaß der Energieverlust gem¨aß Bethe-Bloch erhalten wird. Verschiedene Formulierungen (Bohr, Bethe, Bloch) unterscheiden sich in der Parame- • trisierung der Fernst¨oße, d.h. des Energieverlustes, bei der die Bindung der Elektronen in den Atomhullen¨ nicht vernachl¨assigbar ist. Mittlere Reichweite R(E) Die mittlere Reichweite eines Teilchens berechnet man wie folgt: 0 1 R(E)= − dE, (2.20) !E dE/dx wobei der Energieverlust eine Funktion der Energie selbst ist. Der Hauptteil des Ionisati- dE 2 onsverlustes erfolgt bei kleinen Teilchengeschwindigkeiten ( β− )amEndederSpur dx ∝ Bragg Peak (siehe Abb. 2.3). Genutzt¨ wird die Tatsache, daß der Hauptteil des Energie- verlustes→ am Ende der Spur liegt, bei der Krebstherapie mit energetischen Kernen. Durch geeignete Wahl der Kerngeschwindigkeit (Protonen, leichte Kerne) kann die Energie gezielt und lokal abgegeben werden. 13 UnderstandingUnderstanding Bethe-BlochBethe-Bloch Density correction: Polarization effect ... [density dependent] ➙ Shielding of electrical field far from particle path; effectively cuts of the long range contribution ... More relevant at high γ ... [Increased range of electric field; larger bmax; ...] For high energies: ⇤/2 ln(~⌅/I)+ln⇥ 1/2 ! Abbildung− 2.2: Energieverlust in Kupfer. Gezeigt wird der Einfluß der Schalenkor- rektur und der Dichteeffektkorrektur.Density Daseffect Minimum leads to des Energieverlustes liegt bei 2 βγ 3...4.VordemMinimumverh¨alt sich dE/dx β− ,nachdemMinimum Shell correction: ≃ saturation at high energy∝ ... steigt dE/dx logarithmisch an und kommt dann in den S¨attigungsbereich (Dichte- Arises if particle velocity is closeeffekt).
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