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). to orbital Man beachte: Die auf der Ordinate angegebene Gr¨oße ist vielmehr 1 dE − ρ dx velocity of electrons, i.e. βc ~ (nichtve. dE ). Quelle: Phys. Rev.Shell D 54:S132, correction 1996. are − dx in general small ... Assumption that electron is at rest breaks down ... Capture process is possible ... 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 Energieverlustes8 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 EnergyEnergy loss Loss of of (heavy) Charged charged Particles particles 6 27. Passage of particles through matter

10 Dependence on 8 Mass A

) 6 2 H2 liquid

Charge Z m

c 5 1

of target nucleus − g

4 V

e He gas M

Minimum ionization: ( 3

x C d -2 / Al ca. 1 - 2 MeV/g cm E 2 Fe -2 d Sn [H2: 4 MeV/g cm ] − Pb

1 0.1 1.0 10 100 1000 10 000 βγ = p/Mc

0.1 1.0 10 100 1000 Muon momentum (GeV/c)

Marco Delmastro Experimental Particle Physics 9 0.1 1.0 10 100 1000 Pion momentum (GeV/c)

0.1 1.0 10 100 1000 10 000 Proton momentum (GeV/c) Figure 27.2: Mean energy loss rate in liquid (bubble chamber) hydrogen, gaseous helium, carbon, aluminum, iron, tin, and lead. Radiative effects, relevant for muons and pions, are not included. These become significant for muons in iron for βγ > 1000, and at lower momenta for muons in higher-Z absorbers. See Fig. 27.21.∼

as a function of βγ = p/Mc is shown for a variety of materials in Fig. 27.4. The mass scaling of dE/dx and range is valid for the electronic losses described by the Bethe-Bloch equation, but not for radiative losses, relevant only for muons and pions. For a particle with mass M and momentum Mβγc, Tmax is given by

2 2 2 2mec β γ Tmax = 2 . (27.4) 1+2γme/M +(me/M ) In older references [2,7] the “low-energy” approximation

February 2, 2010 15:55 IdentifyingdE/dx and particlesParticle Identification by dE/dx

180 Measured energy loss 140 [ALICE TPC, 2009]

100 TPC Signal [a.u.]

60 Bethe-Bloch

Remember: 20 dE/dx depends on β! 0.1 0.2 1 2 Momentum [GeV] Marco Delmastro Experimental Particle Physics 10 ALICE TPC

Marco Delmastro Experimental Particle Physics 11 die Plasmafrequenz, die die kollektiven Elektronenschwingungen in dem Me- dium beschreibt. Damit ergibt sich aus (2.36) fur das Einsetzen der Sattigung ¨ ¨ !"# des Energieverlust: 2ω βγ . (2.39) ≈ ωP

Wegen ωp √ne wird die S¨attigung bei dichteren Medien fruher¨ erreicht. ∼ Parameter der Bethe-Bloch-Formel fur¨ Mischungen und Verbindungen:

Fur¨ verschiedene Komponenten0121001210 i in einem Medium werden die dE/dx-Werte mit den Gewichtsanteilen wi gewichtet!"#$$%&'($#)*+(,"(-%&%-.-(%#&%/,"%#& summiert (dabei werden atomare Korrekturen vernachl¨assigt): 345(06678 dE dE = w (2.40) ρdx i ρ dx i i i ! " # Fur¨ Verbindungen ergibt sich der Gewichtsfaktor mit der Anzahl ai des Atoms i mit dem Atomgewicht Ai in der Verbindung:

aiAi wi = (2.41) j ajAj Fur¨ die Parameter in der BBF lassen$ sich fur¨ Mischungen und Verbindungen Effektivwerte berechnen:

Zeff = aiZi (2.42) i ! Aeff = aiAi (2.43) i !1 ln Ieff = aiZi ln Ii (2.44) Zeff $%&'(()*++,-.+/-&'/0'12'(3 i ,-.-/.0--1 1 ! δ = a Z δ (2.45) eff Z i i i eff i ! Ceff = aiCi (2.46) i ! 2.1.2 Statistische Fluktuationen der dE/dx-Verteilung 0121001210 9&*+':(;#//(<;.=".,"%#&/ Die Bethe-Bloch-FormeldE/dx gibt den Fluctuations mittleren Energieverlust pro Wegl¨ange dE/dx an. Tats¨achlich ist der EnergieverlustdE/dx fluctuations aber ein statistischer Prozess mit Fluktuationen: der Energieverlust ∆E auf!"#$%&'#(%)%*)+$,#-(./(01##2,#"+)#3%&,4$%(5 einer Wegstrecke ∆x setzt sich aus vielen kleinen Bei- $&)6%$##Bethe-Bloch.#2"#&)%$2&'#+7#)62*8"%,,# describes mean energy09 loss; measurement via tr¨agen δEn,dieeinzelnenIonisations-oderAnregungsprozessen entsprechen, zusam- energy loss ΔE in !a material of thickness Δx with men: # # "#> "43;%$#+7#*+''2,2+", N " " ∆E = δE δN : number of collisions (2.47) n δE : energy loss in a single collision n=1 " :$+;&;2'2)<#(2,)$2;4)2+"#+7#,2"='%#%"%$=<#)$&",7%$,! ?&"(&4#@&2'9 $$# # Statistische Fluktuationen treten sowohl fur¨ die Anzahl N als auch fur¨ die jeweils # ! abgegebene Energie δE auf. Wenn die δE statistisch unabh¨angigIonization sind, loss δ giltE nach n distributedA%&,+"#7+$#(%B%"(%"*%9# statistically ... dem “zentralen Grenzwertsatz” der Statistik, dass ∆E fur¨ N 3+3%")43#)$&",7%$#normalverteilt →∞so-called *'&,,2*&''<9#*%")$&'2)<#+7#*+''2,2+"#C;D1EF ist, mit einer Varianz, die N mal die Varianz der EinzelprozesseEnergy ist. loss 'straggling' rel. probability Ionization by close collisions 2&'3*45%4$674%8868*7'9%#: production of δ-electrons Complicated *problem ... " " % $()#! 14 Thin absorbers:*+ ! Landau distribution ! +% ! ! + + !'& #%"! Standard Gauss with mean energy loss E0 ! + tail towards high* energies* *+ due to δ-electrons! " " !'& ! % $()# ! ! ! *#%"! *+ *#%"! #%"! %'& Below excitation threshold Excitations Ionization by distant collisions ! $()#! " see! also! Allison & Cobb energy transfer δE [Ann. Rev.' &Nucl. Part.#%"! Sci. 30 (1980) 253.]

Marco Delmastro Experimental Particle Physics 12

$%&'(()*++,-.+/-&'/0'12'(3 ,-.-/.0--1

!"#$%&' (")* + Mean particle range

Marco Delmastro Experimental Particle Physics 13 EnergyEnergy loss Loss of of electrons Electrons

Bethe-Bloch formula needs modification

Incident and target electron have same mass me Scattering of identical, undistinguishable particles

dE Z 1 m 2c2⇥2T = K ln e + F (⇥) dx A 2 2I2 ⌧ el.  [T: kinetic energy of electron]

Wmax = ½T

Remark: different energy loss for electrons and positrons at low energy as positrons are not identical with electrons; different treatment ...

Marco Delmastro Experimental Particle Physics 14 !"#

>=?=>>=?=> !"#$%& '()) *& +$#,))-$./'0"%

12*344'2"56&#) 7#8'5*89%&6%3:8;7*6$8%;8'5*8#&<6*78%;8'5*8BremsstrahlungBremsstrahlung and Radiation Length ":4%2:*283"'*27"6

! )" # ! $ %! Bremsstrahlung $%# arises" if particles ! A (( ' & are accelerated" &' $ in Coulomb field of nucleus * ! # ! )* ' (" !$ # ! JKL 2 dE z2Z2 1 e2 183 E @ =4NA 2 E ln 1 2 dx A 4⇤⇥ mc 3 / m =;;*<'8%#6>82*6*?"#'8;%28* "#$8&6'2"A2*6"'7?74'7<8 BC,---8D*EF✓ 0 ◆ Z i.e. energy loss proportional to 1/m2 ➙ main relevance for electrons ... ! ;%28*6*<'2%#4G )" # ! $%# ... or ultra-relativistic muons (( ' +% " &' $ )* Consider' electrons:# # )" " 2 dE Z 2 183 * ) , " =4NA re E ln 1 x/X0 dx A · "Z 3 " % E = E e )* , " " 0 dE E A After passage of one X0 electron has = with X0 = ' 2 2 183 th dx X0 4NA Z re ln 1 lost all but (1/e) of its energy , " Z 3 ! ! $%# [Radiation length in g/cm2] / [i.e. 63%] (( '# +% &' $ 2"$7"'7%#86*#)'58H).<3 I # Marco Delmastro # Experimental Particle Physics 15

12-#$34.)).%#3(23("#3$.56.-6("3'#"%-/36"3,.-#$6.'3-/#3,#."3#"#$%&33(23."3#'#7-$("3*#.,36)3$#507#53*&3 .32.7-($3#338-(39:;3(23-/#36"765#"-3#"#$%&<3=

$%&'(()*++,-.+/-&'/0'12'(3 ,-.-+./--0

>=?=>>=?=> @$6-67.'3!"#$%&

)" )" )" )" )" +" * +" * )* $ 4+%!1 )* $ -./01230./ 2#&(&2!40.).#5/062 )* 3.3 )* $.77010./ )* +2)0230./

7$)&*!(&$)0.).#5/04$**0$10.4.2(#$)! :21 .$"5%6 !""#$%&'!(&$)*+ "$ 890'$-&1&2!(&$)0$10:: # $,!(

#.2$&4 ;'!%<;&)=> -$"5%6 &)-&*(&)5?&*@!A4.0"!#(&24.*0,BC3 " 1.70) )70890) ,-.)*&(/0.11.2(3 $ # $,!( $%&'(()*++,-.+/-&'/0'12'(3 ,-.-+./--0

!"#$%&' (")* + 20 27. Passage of particles through matter

10 GeV 1.2 Bremsstrahlung y

d 100 GeV /

M 1 TeV P L

σ 0.8 d 10 TeV y

)

A /

A 100 TeV N 0.4 0

X ( 1 PeV 10 PeV CriticalBremsstrahlung Energy – Critical Energy0 0 0.25 0.5 0.75 1 y = k/E

Figure 27.11: The normalized bremsstrahlung cross section kdσLP M /dk in Critical energy: lead versus the fractionaldE photon energydE y = k/E.TheverticalaxishasunitsdE of photons per radiation length.= + dx Tot dx Ion dx Brems dE dE ✓ ◆ ✓ ◆ ✓ ◆ (E ) = (E ) dx c dx c Brems Ion 200 Copper −2 X0 = 12.86 g cm Approximation: 100 Ec = 19.63 MeV ) g V l n e 70 a u ot l M h 710 MeV ( T

a Gas Rossi: r 0 t Ec = 50 s X E s Z +0.92 Ionization per X0

× ≈ m 40 = electron energy s e x r m b d e / t 610 MeV 30 r c Sol/Liq E B a x E = d c Z +1.24 E Ionization 20

Brems = ionization

Example Copper: 10 2 5 10 20 50 100 200 Ec ≈ 610/30 MeV ≈ 20 MeV Electron energy (MeV)

Figure 27.12: Two definitions of the critical energy Ec. Marco Delmastro Experimental Particle Physics 16 incomplete, and near y =0,wheretheinfrareddivergenceisremovedby the interference of bremsstrahlung amplitudes from nearby scattering centers

February 2, 2010 15:55 TotalTotal Energy LossLoss ofof ElectronsElectrons 27. Passage of particles through matter 19

from PDG 2010 0.20 Positrons Lead (Z = 82)

Electrons

) 1.0 ) 1 0.15 1 − 0 Møller − X Bremsstrahlung g ( e– e– 2

E x m c d d ( 0.10 1 E Ionization − 0.5 Møller (e−) e– e– Bhabha (e+) 0.05 e+ e+ Positron annihilation e+ γ 0 1 10 100 1000 e– E (MeV) Bhabha Figure 27.10: Fractional energy loss per radiation length in lead as a e– γ Fractional energy loss per radiation length in lead function of electron oras positron a function energy. of electron Electron or positron (positron) energy scattering is consideredAnnihilation as ionization when the energy loss per collision is below 0.255 Marco Delmastro Experimental Particle Physics 17 MeV, and as Møller (Bhabha) scattering when it is above. Adapted from Fig. 3.2 from Messel and Crawford, Electron-Photon Shower Distribution Function Tables for Lead, Copper, and Air Absorbers,PergamonPress, 2 1970. Messel and Crawford use X0(Pb) = 5.82 g/cm ,butwehavemodified the figures to reflect the value given in the Table of Atomic and Nuclear 2 Properties of Materials (X0(Pb) = 6.37 g/cm ).

At very high energies and except at the high-energy tip of the bremsstrahlung spectrum, the cross section can be approximated in the “complete screening case” as [38] 2 4 4 2 2 dσ/dk =(1/k)4αr ( y + y )[Z (Lrad f(Z)) + ZL′ ] e { 3 − 3 − rad (27.26) + 1 (1 y)(Z2 + Z) , 9 − } where y = k/E is the fraction of the electron’s energy transfered to the radiated photon. At small y (the “infrared limit”) the term on the second line ranges from 1.7% (low Z)to2.5%(highZ)ofthetotal.Ifitisignoredandthefirstline simplified with the definition of X0 given in Eq. (27.22), we have

dσ A 4 4 2 = 3 3 y + y . (27.27) dk X0NAk − This cross section (times k)isshownbythetopcurveinFig.27.11.! " This formula is accurate except in near y =1,wherescreeningmaybecome

February 2, 2010 15:55 4 27. Passage of particles through matter EnergyEnergy loss Loss for – Summarymuons Plot for Muons PDG 2010

+

] 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

Marco Delmastro Fig. 27.1: Stopping power (= ExperimentaldE/dx Particle)forpositivemuonsincopperasa Physics 18 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].

27.2.2. Stopping power at intermediate energies : The mean rate of energy loss by moderately relativistic charged 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 intermediate-Z materials with an accuracy of a few %. At the∼ lower∼ limit the projectile velocity becomes comparable to atomic electron “velocities” (Sec. 27.2.3),

February 2, 2010 15:55 Cherenkov radiation

Particles moving in a medium with speed larger than speed of light in that medium will loose energy by emitting electromagnetic radiation ! Charged particle polarize medium generating an electrical dipole varying in time ! Every point in trajectory emits a spherical EM wave, waves constructively interfere…

refractive index dielectric constant

Marco Delmastro Experimental Particle Physics 19 CherenkovCherenkov Radiationradiation: –radiators Radiators

Parameters of Typical Radiator

Medium n βthr θmax [β=1] Nph [eV-1 cm-1]

LuftAir 1.000283 0.9997 1.36 0.208

Isobutan 1.00127 0.9987 2.89 0.941

WasserWater 1.33 0.752 41.2 160.8

Quartz 1.46 0.685 46.7 196.4

Note: Energy loss by Cherenkov radiation very small w.r.t. ionization (< 1%). Example: [Proton with Ekin =1 GeV passing through 1 cm water ] Visible light only! β = p/E ≈ 0.875; cosθC = 1/nβ = 0.859 ➛ θC = 30.8° [E = 1 - 5 eV; λ = 300 - 600 nm] 2 2 -1 -1 -1 -1 d N/(dE dx) = 370 sin θC eV cm ≈ 100 eV cm

2 ➛ ΔEloss = d N/(dE dx) ΔE Δx ΔEloss < 1.25 keV = 2.5 eV⋅100 eV-1 cm-1⋅5 eV ⋅1 cm = 1.25 keV

Marco Delmastro Experimental Particle Physics 20 Identifying particles with Cherenkov radiation

Marco Delmastro Experimental Particle Physics 21 Typische Werte dN Phot Pth n =1+δ Θ [ ] β γ = 2 c dx cm th · th mc 3 0 H2 1+0.14 10− 0.96 0.32 59.8 · 3 0 N2 1+0.3 10− 1.4 0.7 40.8 · 3 0 Freon 13 1 + 0.72 10− 2.2 1.7 26.3 · 0 Plexiglas 1.49 47.8 630 0.91 Aerogel 1.005 ... 1.05 11.40 ... 17.80 45 ... 107 3 ... 5

Einige Schlußfolgerungen: In Plexiglas gilt (E 2 eV ) • γ ≈ ion,Anr dE 2 106 eV dx Min = · cm 2 103 . ! dE" eV ≈ · dx C˘ 800 cm ! " Wenn man C–Strahlungˇ zum Teilchennachweis nutzen will, dann muß man also sicher- stellen, daß die angeregten Atome der durchlaufenen Materie kein Szintillationslicht aussenden. Wenn man β fur¨ ein geladenes Teilchen messen will, dann muß man eine Mindestan- • zahl von emittierten Cerenkov–Photonenˇ fordern. Da die Cerenkov–Photonenˇ statistisch emittiert werden, kann man die Fluktuationen mit Hilfe der Poisson–Statistik be- schreiben. Die Wahrscheinlichkeit, kein Cerenkov–Photonˇ nachzuweisen, ist ne w0 = e− , n =Photonenzahl=#n (C˘) ε (Licht) η (Quantenausbeute des Photomultipliers) e γ ∗ ∗ ! 0.8 0.1 ... 0.2 # $% & # $% & Die Wahrscheinlichkeit,CherenkovCherenkov Radiation einradiation:Cerenkov–Photonˇ –momentum Momentum nachzuweisen dependenceDependence (Abb.3.2), ist dann ∞ max θ / θ = N/N n = n N θ

1.0 π p 0.5 0.5 K

0.1

0 0 5 10 15 20 25 p [GeV]

Abbildung 3.2:Cherenkov Relative angle Zahlgrows der with Photonen β and reaches als asymptotic Funktion value der for β T= eilchenenergie1 Number of photons 2 [θmax = arccos (1/n); N∞ = x⋅370/cm (1-1/n )]

Marco Delmastro Experimental Particle Physics 22 ne w =1 e− . − Forderung w 0.999 ! n 6.9 ! n " 100. ≥ e ≥ γ Fur¨ Freon braucht man dann etwa eine L¨ange von 0.5 m fur¨ den Radiator. Anwendung fand ein solcher Z¨ahler am SFM–Detektor [63].

48 LHCbCherenkov RICH Radiation – Application

Measurement of Cherenkov angle: Use medium with known refractive index n ➛ β

Principle of:

RICH (Ring Imaging Cherenkov Counter) DIRC (Detection of Internally Reflected Cherenkov Light) DISC (special DIRC; e.g. Panda)

LHCb RICH Event [December 2009]

LHCb RICH

Marco Delmastro Experimental Particle Physics 23 Transition radiation Transition radiation occurs if a relativistic particle (large γ) passes the boundaries between two media with different refraction indices… Transition Radiation – Properties ! Intensity of radiation is logarithmically proportional to γ

Angular distribution strongly forward peaked [Interference; coherence condition] θ ≤ 1/γ Plasma frequency [from Drude model] Coherent radiation is generated only over a very small formation length D = γc/ωp ρmax = γv/ω [transversal range ...... with large polarization] Volume element from which coherent 2 V = π D ρ radiation is emitted ... max

Maximum energy of radiated photons – Emax = γhωp limited by plasma frequency ... [ X-Rays ➛ large γ !! ] [results from requiring V ≠ 0 ➛ ω = γωp]

3 Typical values: CH2: ħωp = 20 eV; γ = 10 D = 10 μm [ Air: ħωp = 0.7 eV ] [d > D: absorption dominates] Marco Delmastro Experimental Particle Physics 24 4.3 Ubergangsstrahlendetektoren¨

Ein Ubergangsstrahlendetektor¨ besteht aus zwei Komponenten: IdentifyingTransitionRadiator Radiationparticles Dunne¨ with– FolienApplication transition aus Materialien radiation mit kleinem Z Nachweisger¨at Proportionalkammern a) Anode Detection Principle: γ-Absorption [Electron-ID] Radiator ! 0 +

TR TR + Electron e dE/dxdE/dx Wires Radiator foils

Detector Signal: – b) δ-electronTR !!•Elektronen DetectorTR should be sensitive to[Transition 3 ≤ E Radiation]γ ≤ 30 keV. dE/dx ! Gaseous detectors 5 • In gas σphoto effect ∝ Z Q>Q • Gases with high Z are required S ! e.g. Xenon (Z=54) Drift time

Marco Delmastro Experimental Particle Physics 25 Driftzeit

Abbildung 4.1: a) Schema der Meßapparatur; b) mit einem FADC gemessene Ladung in einer Driftkammer als Funktion der Driftzeit Typische Resultate zeigen die nachfolgenden Abb.4.1–4.3. Drei Informationen k¨onnen fur¨ die Trennung hochrelativistischer Teilchen (γ ≫ 1), d.h. Elek- tronen genutzt werden. • Die Ladung der Photonen wird in der Nachweiskammer lokal deponiert, bei Verwen- dung eines Flash ADC (FADC) kann diese Information fur¨ die Analyse genutzt werden (Abb.4.1b). • Die Absorption der TR–Quanten ist breit um die Teilchenspur verteilt (Abb.4.2). • Die gesamte, um ein Elektron in einer Driftkammer deponierteLadungistgr¨oßer als die von Pionen, weil zur ublichen¨ Ionisation der Beitrag der TR–Quanten hinzukommt (Abb.4.3). Ubergangsstrahlendetektoren¨ werden heute vorwiegend zur Identifizierung der Elektronen, d.h. zur Abtrennung von Muonen,¨ Pionen, ... verwendet. Bei einer Nachweiswahrscheinlich- − keit von 95 % ... 90 % fur¨ e ist die Verwechslungswahrscheinlichkeit von Pionen etwa 1 % ... 10 %. Diese Rejektionsrate ist nicht sehr groß, aber in Kombination mit anderen Meß- ger¨aten (RICH, Schauerz¨ahler) fuhrt¨ dies zu guten Identifizierungswahrscheinlichkeiten von Elektronen (totale Rejektionsrate bis 10−4)[35].

64 ATLASTransition Transition Radiation Radiation – ATLAS asTracker Example

Barrel Straw Tube Tracker assembly with interspace filled with foam ➛ Tracking & transition radiation

End-cap assembly

Marco Delmastro Experimental Particle Physics 26 IdentifyingTransition Radiationparticles with– ATLAS transition as Example radiation

0.25 ATLASATLAS Preliminarypreliminary

0.2 ElectronElectron candidates candidates GenericGeneric tracks tracks 0.15 FitElectrons to data (MC) Electrons Generic tracks (MC) from Conversions 0.1

High threshold probability Mainly Pions 0.05 TRTTRT endcap endcap 0 10 102 −factor 10 3 104 1 10 1 10 Pion momentum (GeV) Electron momentum (GeV)

Marco Delmastro Experimental Particle Physics 27 InteractionInteractions of of photons Photons withwith Mattermatter

I Characteristic for interactions of photons with matter:

A single interaction I - dI removes photon from beam !

Possible Interactions dI = µIdx Photoelectric Effect [ µ:absorptioncoecient ] Compton Scattering Pair Production depends on E, Z, ρ Rayleigh Scattering (γA ➛ γA; A = atom; coherent) Thomson Scattering (γe ➛ γe; elastic scattering) ➛ Beer-Lambert law: Photo Nuclear Absorption (γΚ ➛ pK/nK) µx Nuclear Resonance Scattering (γK ➛ K* ➛ γK) I(x)=I e Delbruck Scattering (γK ➛ γK) 0 Hadron Pair production (γK ➛ h+h– K) with =1/µ =1/n⇥ [meanfreepath]

Marco Delmastro Experimental Particle Physics 28 InteractionInteractions of of photons Photons withwith Mattermatter 24 27. Passage of particles through matter

from PDG 100

10

) Sn 2 1

m Fe Pb c / Si g (

0.1 λ

h t H C g

n 0.01 e l

n o i t

p 0.001 r o s b

A – 4 10

– 5 10

– 6 10 10 eV 100 eV 1 keV 10 keV 100 keV 1 MeV 10 MeV 100 MeV 1 GeV 10 GeV 100 GeV Photon energy

Marco DelmastroFigure 27.16: The photon massExperimental attenuation Particle Physics length (or mean free path) λ = 29 1/(µ/ρ)forvariouselementalabsorbersasafunctionofphotonenergy.The mass attenuation coefficient is µ/ρ,whereρ is the density. The intensity I remaining after traversal of thickness t (in mass/unit area) is given by I = I0 exp( t/λ). The accuracy is a few percent. For a chemical compound or − mixture, 1/λeff elements wZ /λZ,wherewZ is the proportion by weight of the element with≈ atomic number Z.Theprocessesresponsibleforattenuation are given in Fig. 27.10.! Since coherent processes are included, not all these processes result in energy deposition. The data for 30 eV

27.4.4. Energy loss by photons : Contributions to the photon cross section in a light element (carbon) and a heavy element (lead) are shown in Fig. 27.14. At low energies it is seen that the photoelectric effect dominates, although Compton scattering, Rayleigh scattering, and photonuclear absorption also contribute. The photoelectric cross section is characterized by discontinuities (absorption edges) as thresholds for photoionization of various atomic levels are reached. Photon attenuation lengths for a variety of elements are shown in Fig. 27.16, and data for 30 eV

February 2, 2010 15:55 22 27. Passage of particles through22 matter27. Passage of particles through matter

475&B7).#%&4!#9&F5 475&B7).#%&4!#9&F5 ,&-. <&(=>()?1(%$7@&σ$#$ ,&-. <&(=>()?1(%$7@&σ$#$

σ>A(A σ>A(A

,&/. ,&/.

σ σG7+@(?*" G7+@(?*"

Interactions of Photons withB)#CC&C(D$?#%&&4.7)%CE7$#15 Matter InteractionB)#CC&C(D$?#%&&4.7)%CE7$#15 of photons with matter ,&. ,&. 22 27. Passage of particles through matterκ%HD κ%HD σ Photonσ B#1>$#%Total Cross Sections κ" B#1>$#% κ" ,0&1. ,0&1.

4.5&6(78&4475&B7).#%&4!&9&:;5!#9&F5 4.5&6(78&4!&9&:;5 ,&-. Carbon<&(=>()?1(%$7@& (Z σ= 6) Lead (Z = 82) <&(=>()?1(%$7@&σ$#$$#$ <&(=>()?1(%$7@&σ$#$ ,&-. ,&-. σ>A(A σ>A(A Photo effect σ>A(A σ σG7+@(?*" G7+@(?*" 1 MeV ,&/. ,&/. ,&/. Pair production σG7+@(?*" 1 MeV

B)#CC&C(D$?#%&&4.7)%CE7$#15 κ%HD κ%HD B)#CC&C(D$?#%&&4.7)%CE7$#15 B)#CC&C(D$?#%&&4.7)%CE7$#15 σ σ ,&. *A8A)A *A8A)A Rayleigh,&. ,&. scattering κ σ κ" σB#1>$#% κ%HD" B#1>$#% σB#1>$#% κ" ,0&1. ,0&1. ,0&1. ,0&(2 ,&/(2 ,&-(2 ,&3(2 ,00&3(2 ,0&(2 ,&/(2 ,&-(2 ,&3(2 ,00&3(2 !"#$#%&'%()*+ !"#$#%&'%()*+ 4.5&6(78&4!&9&:;5 Pair Production Figure 27.14: Photon total cross sections<&(=>()?1(%$7@&Figure as aσ function$#$ 27.14: ofPhoton energy total in carbon cross sections as a function of energy in carbon ,&-. σ Compton scattering and lead,photo showing effect the contributions>A(A of differentand lead, processes: showing the contributions of different processes: Marco Delmastro Experimental Particle Physics 30 σp.e. =Atomicphotoelectriceffect (electronσp.e. =Atomicphotoelectrice ejection, photon ffect (electron ejection, photon absorption)σG7+@(?*" absorption) σRayleigh =Rayleigh(coherent)scattering–atomneitherionizednorexcited,&/. σRayleigh =Rayleigh(coherent)scattering–atomneitherionizednorexcited σCompton =Incoherentscattering(ComptonscatteringoσCompton =Incoherentscattering(Comptonscatteringoff an electron) ff an electron) κnuc =Pairproduction,nuclearfield κnuc =Pairproduction,nuclearfield κ%HD B)#CC&C(D$?#%&&4.7)%CE7$#15 κe =Pairproduction,electronfield σ*A8A)Aκe =Pairproduction,electronfield σg.d.r. =Photonuclearinteractions,mostnotablytheGiantDipole,&. σg.d.r. =Photonuclearinteractions,mostnotablytheGiantDipole σ κ" Resonance [48].B#1>$#% In these interactions, theResonance target nucleus [48]. is In these interactions, the target nucleus is broken up. broken,0&1. up. ,0&(2 ,&/(2 ,&-(2 ,&3(2 ,00&3(2 !"#$#%&'%()*+ 27.4.3. Critical energy : An electron loses27.4.3. energyCritical by bremsstrahlung energy : An electronat a loses energy by bremsstrahlung at a Figure 27.14: Photon total cross sections as a function of energy in carbon and lead, showing the contributions of different processes: February 2, 2010 15:55 February 2, 2010 15:55 σp.e. =Atomicphotoelectriceffect (electron ejection, photon absorption) σRayleigh =Rayleigh(coherent)scattering–atomneitherionizednorexcited σCompton =Incoherentscattering(Comptonscatteringoff an electron) κnuc =Pairproduction,nuclearfield κe =Pairproduction,electronfield σg.d.r. =Photonuclearinteractions,mostnotablytheGiantDipole Resonance [48]. In these interactions, the target nucleus is broken up.

27.4.3. Critical energy : An electron loses energy by bremsstrahlung at a

February 2, 2010 15:55 Pair production

Air

Marco Delmastro Experimental Particle Physics 31 ElectromagneticElectromagnetic showersShowers

Reminder:

Dominant processes at high energies ...

Photons : Pair production Electrons : Bremsstrahlung X0

Pair production:

7 2 2 183 Bremsstrahlung: ⇥pair 4 re Z ln 1 ⇡ 9 Z 3 ✓ ◆ 2 dE Z 2 183dE E 7 A =4NA r E ln = = [X0: radiation length] e 1 2 dx A · Zdx3 X0 9 NAX0 [in cm or g/cm ]

Absorption x/X0 coefficient: ➛ E = E0e

NA 7 After passage of one X0 electron µ = n⇥ = ⇥pair = has only (1/e)th of its primary energy ... A · 9 X0 [i.e. 37%]

Marco Delmastro Experimental Particle Physics 32 20 27. Passage of particles through matter

10 GeV 1.2 Bremsstrahlung y

d 100 GeV /

M 1 TeV P L

σ 0.8 d 10 TeV y

)

A /

A 100 TeV N 0.4 0

X ( 1 PeV 10 PeV 0 0 0.25 0.5 0.75 1 y = k/E

Figure 27.11: The normalized bremsstrahlung cross section kdσLP M /dk in lead versus the fractional photon energy y = k/E.Theverticalaxishasunits ElectromagneticElectromagnetic showersShowers of photons per radiation length.

200 Copper −2 Further basics: X0 = 12.86 g cm 100 Ec = 19.63 MeV ) g V l n e 70 a u ot l Critical Energy [see above]: M h

( T

a Rossi: r 0 t 50 s X E s Ionization per X0

× ≈ m 40 = electron energy s e x r dE dE m b d e / t (E ) = (E ) 30 r c c c E B a x dx dx d Brems Ion E Ionization 20 Brems = ionization Approximations: 10 2 5 10 20 50 100 200 Gas 710 MeV Sol/Liq 610 MeV E = Ec = Electron energy (MeV) c Z +0.92 Z +1.24 Figure 27.12: Two definitions of the critical energy Ec. with: dE dE Z E incomplete,dE and near y =0,wheretheinfrareddivergenceisremovedbyE dE E · = & c =const. dx dx 800 MeV the interferencedx of bremsstrahlungX dx amplitudes⇡ X from nearby scattering centers Brems Ion Brems 0 Ion 0 ✓ ◆ ✓ ◆ February 2, 2010 15:55 Transverse size of EM shower given by 21 MeV RM = X0 RM : Moliere radius radiation length via Molière radius Ec Ec : Critical Energy [Rossi] [see also later] X0 : Radiation length

Marco Delmastro Experimental Particle Physics 33 ElectromagneticElectromagnetic showersShowers

Typical values for X0, Ec and RM of materials used in calorimeter

X0 [cm] Ec [MeV] RM [cm]

Pb 0.56 7.2 1.6 Scintillator (Sz) 34.7 80 9.1 Fe 1.76 21 1.8 Ar (liquid) 14 31 9.5 BGO 1.12 10.1 2.3 Sz/Pb 3.1 12.6 5.2 PB glass (SF5) 2.4 11.8 4.3

Marco Delmastro Experimental Particle Physics 34 Abbildung 8.2: Entwicklung eines elektromagnetischen Sch

• Nur die Prozesse

werden ber • Auf derA StreckeAnalytic simple Showershower Model model

• ¨ucksichtigt ( Das Photon materialisiert nach Simple shower model: [from Heitler]

• X F Only two dominant interactions: Folgende Gr 0 ¨ur • Pair verliertproduction das K and Bremsstrahlung ... F γ ¨ur E> =Kern).➛ + + − γ + Nucleus Nucleuse + e + e Electromagnetic Shower E [Photons absorbed via pair production]+ K • ϵ [Monte Carlo Simulation] Zahl der Teilchen im Schauer K • ≤ tritt kein Energieverlust durch Ionisation/Anregunge auf. + Nucleus ➛ Nucleus + e + γ Lage des Schauermaximums ϵ • ¨oßen sind bei der Beschreibung eines Schauers von Interesse [Energy loss of electrons via Bremsstrahlung]→ Wir messen die longitudinalenLongitudinalverteilung Komponenten des Schauers des Schauers in S im Raumverlieren die Elektronen Energie e − → Use • K Transversale Breite des Schauers Shower developmentdurch Bremsstrahlung governed die Hby X0 ... Simplification: K + After a distance X0 electrons remain with auers (Monte CarloEγ = Simulation) Ee ≈ E0/2 + e + Nach Durchlaufen der Schichtdicke th [Ee looses half the energy] X only (1/e) of their primary energy ... e len Teilchen E + 0 + − + ,dPhoton i eE produces n1 e r g e ie ev-pair o nPafter o 9/7X s i0 t ≈ rX0 o ... nue n dE l e k tE re o≈ E nb0/2 e t r γ − [Energy shared by e+/e–] = Assume: E E 0 ... with initial particle energy E0 E > Ec : no energy2 loss by ionization/excitation ± =E < Ec : energy loss only via ionization/excitation E 1 Marco Delmastro 2 Experimental Particle Physics 35 ¨alfte seiner Energie nur

durch Ionisation/Anregung. t betr t = ¨agt in unserem einfachen Modell die Zahl der schnel- x N X ( t 0 )=2 ¨agt :

156 t , trahlungsl

¨angen: 8.1 Electromagnetic calorimeters 231

8.1 Electromagnetic calorimeters

8.1.1 Electron–photon cascades The dominating interaction processes for spectroscopy in the MeV energy range are the photoelectric and Compton effect for photons and ionisa- tion and excitation for charged particles. At high energies (higher than 100 MeV) electrons lose their energy almost exclusively by bremsstrahlung while photons lose their energy by electron–positron pair production [1] (see Sect. 1.2). The radiation losses of electrons with energy E can be described by the simplified formula: dE E = , (8.1) − dx X ! "rad 0 where X0 is the radiation length. The probability of electron–positron pair production by photons can be expressed as

dw 1 x/λprod 9 = e− , λprod = X0 . (8.2) dx λprod 7 A convenient measure to consider shower development is the distance normalised in radiation lengths, t = x/X0. The most important properties of electron cascades can be understood in a very simplified model [2, 3]. Let E0 be the energy of a photon incident on a bulk of material (Fig. 8.1). + After one radiation length the photon produces an e e− pair; electrons and positrons emit after another radiation length one bremsstrahlung photon each, which again are transformed into electron–positron pairs. Let us assume that the energy is symmetrically shared between the particles at

Sketch of simple shower development AAnalytic simple Showershower Model model E0

Simple shower model: [continued] E 2 E 4 E 8 E 16 0 / 0/ 0/ 0/

Shower characterized by: 0 12345678t [X0] Fig. 8.1. Sketch of a simple model for shower parametrisation. Number of particles in shower Longitudinal components; Location of shower maximum measured in radiation length ... x Longitudinal shower distribution ... use: t = X Transverse shower distribution 0

Number of shower particles Total number of shower particles after depth t: with energy E1: E E t t1 log ( 0/E1) 0 N(t)=2 N(E0,E1)=2 =2 2 = E1 Number of shower particles Energy per particle at shower maximum: after depth t: tmax E0 E0 t N(E0,Ec)=Nmax =2 = E = = E0 2 Ec N(t) · Shower maximum at: N(E0,E1) E0 E E / ➛ t =log ( 0/E) tmax ln( 0/Ec) 2 /

Marco Delmastro Experimental Particle Physics 36 AnalyticA simple Shower shower model Model

Simple shower model: [continued] Longitudinal shower distribution increases only logarithmically with the primary energy of the incident particle ...

Some numbers: Ec ≈ 10 MeV, E0 = 1 GeV ➛ tmax = ln 100 ≈ 4.5; Nmax = 100 E0 = 100 GeV ➛ tmax = ln 10000 ≈ 9.2; Nmax =10000 Relevant for energy measurement (e.g. via scintillation light): total integrated track length of all charged particles ...

tmax 1 µ with t0: range of electron with energy Ec T = X0 2 + t0 Nmax X0 · · [given in units of X0] µ=0 Xtmax E0 = X0 (2 1) + t0 X0 · · Ec E log E0/Ec E0 0 = X0 (2 2 1) + t0 X0 (1 + t0) X0 E0 · · E · Ec ⇥ Marco Delmastro Experimental Particle Physics c 37 As only electrons Energy proportional contribute ... E0 to track length ... T = X0 F [ with F<1 ] Ec · · EMElectromagnetic shower longitudinal Shower developmentProfile 8.1 Electromagnetic calorimeters 235

600 Longitudinal profile 5000 MeV Parametrization: [Longo 1975]

] 400 0 dE ⇥t X = E0 t e

dt [MeV/

t d

/ α,β : free parameters 2000 MeV E tα : at small depth number of d 200 secondaries increases ... 1000 MeV e–βt : at larger depth absorption dominates ... 500 MeV

Numbers for E = 2 GeV (approximate): 0 α = 2, β = 0.5, tmax = α/β 0 5 10 15 20 t [X0]

More exact [Longo 1985] 100 with: 1 ⇥t dE (⇥t) e 1 E0 = E0 ⇥ ➛ tmax = =ln + Ce Ce = 0.5 [γ-induced] dt · · () ⇥ Ec10 lead ✓] ◆ 0 Ce = 1.0 [e-induced] [Γ: Gamma function] X iron aluminium 1 [MeV/

t d Marco Delmastro Experimental Particle Physics 38 /

E d 0.1

0.01 0 5152025303510

t [X0]

Fig. 8.4. Longitudinal shower development of electromagnetic cascades. Top: approximation by Formula (8.7). Bottom: Monte Carlo simulation with EGS4 for 10 GeV electron showers in aluminium, iron and lead [11].

Figure 8.6 shows the longitudinal and lateral development of a 6 GeV electron cascade in a lead calorimeter (based on [12, 13]). The lateral width of an electromagnetic shower increases with increasing longitudinal shower depth. The largest part of the energy is deposited in a relatively narrow shower core. Generally speaking, about 95% of the shower energy is con- tained in a cylinder around the shower axis whose radius is R(95%) = 2RM almost independently of the energy of the incident particle. The depen- dence of the containment radius on the material is taken into account by the critical energy and radiation length appearing in Eq. (8.11). ElectromagneticEMLongitudinal shower showerlongitudinal Shower profiles Shape development (longitudinal)

Depth [X0]

Energy deposit of electrons as a function of depth in a block of copper; integrals normalized to same value [EGS4* calculation] Depth of shower maximum increases logarithmically with energy E t ln( 0/Ec) max / Energy deposit per cm [%]

Depth [cm] *EGS = Electron Gamma Shower

Marco Delmastro Experimental Particle Physics 39

6 16 27. Passage of particles through matter

Eq. (27.14) describes scattering from a single material, while the usual problem involves the multiple scattering of a particle traversing many different layers and mixtures. Since it is from a fit to a Moli`ere distribution, it is incorrect to add the individual θ0 contributions in quadrature; the result is systematically too small. It Analytic Shower Modelis much more accurate to apply Eq. (27.14) once, after finding x and X0 for the EM shower transversecombined development scatterer. Lynch and Dahl have extended this phenomenological approach, fitting Gaussian distributions to a variable fraction of the Moli`ere distribution for Transverse shower developmentarbitrary scatterers ... [35], and achieve accuracies of 2% or better. Multiple coulomb scattering Opening angle for bremsstrahlung and pair production 2 2 x (m/E) = 1/2 h i⇡ x/2 Small contribution as me/Ec = 0.05

Ψplane yplane Multiple scattering splane deflection angle in 2-dimensional plane ... θplane k 2 2 2 Figure 27.9: Quantities used to describe multiple Coulomb scattering. The k = m = k h i h i particle is incident in the plane of the figure. m=1 X 2 13.6MeV/c x Assuming the approximate range of electrons The nonprojected (space) and projected (plane) angular distributions are given h i⇡ p X0 [β = 1] to be X0 yields lateral extension: R =〈θ〉⋅X0 ... p r approximately by [33] In 3-dimensions extra factor √2: 2 1 θspace 21MeV RM =2 exp x=X0 2 X0dΩ , X0 (27.15) 2πθ0 ⇥ ⇤⎧− 2θ0· ⎫ EC 2 19.2MeV/c x 3d Molière⎪ Radius;⎪ ⎩⎪ ⎭⎪ h i ⇡ p X0 [β = 1] characterizes lateral shower spread ... r 2 p 1 θplane exp dθplane , (27.16) √2πθ − 2θ2 Marco Delmastro Experimental Particle Physics 0 ⎧ 0 ⎫ 40 ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ 2 2 2 where θ is the deflection angle. In this approximation, θspace (θplane,x + θplane,y), where the x and y axes are orthogonal to the direction≈ of motion, and dΩ dθplane,x dθplane,y.Deflectionsintoθplane,x and θplane,y are independent and identically≈ distributed.

February 2, 2010 15:55 r/RM

1 e+,e− γ e−

!1

t

Abbildung 8.3: Schematische Entwicklung eines elektromag netischen Schauers

EM shower transverse profile Electromagnetic Shower Profile z/X0

Abbildung 8.4: Longitudinalverteilung der Energiedeposi tion in einem elektromagnetischen Schauer f ur¨ zwei Primenergy¨arenergien deposit der Elektronen Transverse profile [arbitrary unites]

Parametrization:

dE r/R r/ = e M + ⇥e min dr α,β : free parameters

RM : Molière radius λmin : range of low energetic photons ... Inner part: coulomb scattering ... Electrons and positrons move away from shower axis due to multiple scattering ... Outer part: low energy photons ... r/RM r/ Rr/RMM Photons (and electrons) produced in isotropic processes (Compton scattering,Abbildung photo-electric 8.5: Transversalverteilung effect) move away from der Energie in einem elektromagnetischen Schauer in shower axis; predominant beyond shower maximum, particularly in high-Z absorber media... unterschiedlichen Tiefen gemessen Shower gets wider at larger depth ... 159

Marco Delmastro Experimental Particle Physics 41 6.2.1 Elektromagnetische Schauer Schauerprofil – 2

6.2.1 ElekLotnrgoitumdinaaleg unnde trtainssvcehrsaele Scchahueareuntweicrklung einer durch 6!GeV/c Elektronen EMau sshowergelösten el eprofilesktromagnet ischen Kaskade in einem Absorber aus Blei. SchauerproElectromagneticfil – 2 Shower Profile Bild links: lineare Skala. – Bild rechts: hablogarithmische Skala

Longitudinal and transversal shower profile nen Longitudinale und transversale Schauerentwickluforng a e6i nGeVer d electronurch 6 !inG eleadV/c absorber Elektro ... ausgelösten elektromagnetischen Kaskade in ein[left:em linear Ab scale;sorb right:er alogarithmicus Ble iscale]. energy deposit Bild links:[arbitrary linea unites]re Skala. – Bild rechts: hablogarithmische Skala

energy deposit [arbitrary unites]

0]

longitudinal shower depth [X

lateral shower width [X0]

0]

Quelle: C. Grupen, Teilchendetektoren, B.I. W issenschaftsverlag, 1993 longitudinal shower depth [X

lateral shower width [X0] MMarco. K rDelmastroamm er: Dete Experimental Particle Physics 42 ktoren, SS 05 Kalorimeter 36

Quelle: C. Grupen, Teilchendetektoren, B.I. W issenschaftsverlag, 1993

M. Krammer: Detektoren, SS 05 Kalorimeter 36 EMSome showers Useful in 'Rules a nutshell of Thumbs'

Problem: Calculate how much Pb, Fe or Cu 180A g is needed to stop a 10 GeV electron. Radiation length: X0 = 3 2 2 Pb : Z = 82 , A = 207, ρ = 11.34 g/cm Z cm 3 Fe : Z = 26 , A = 56, ρ = 7.87 g/cm 3 Cu : Z = 29 , A = 63, ρ = 8.92 g/cm 550 MeV Critical energy: Ec = [Attention: Definition of Rossi used] Z E 1.0 e– induced shower Shower maximum: t =ln max 1.0 γ induced shower Ec 0.5 { Longitudinal energy containment: L(95%) = tmax +0.08Z +9.6[X0]

Transverse Energy containment: R(90%) = RM R(95%) = 2RM

Marco Delmastro Experimental Particle Physics 43 KL Hadron shower Hadronic Showers μ Hadronic showers KS

ν π0

Shower development: N

1. p + Nucleus ➛ Pions + N* + ... π0 2. Secondary particles ... ν undergo further inelastic collisions until they fall below pion production threshold Mean number of μ 3. Sequential decays ... secondaries: ~ ln E n ➛ π0 γγ: yields electromagnetic shower Typical transverse Fission fragments ➛ β-decay, γ-decay momentum: pt ~ 350 MeV/c Neutron capture ➛ fission Spallation ...

Cascade energy distribution: [Example: 5 GeV proton in lead-scintillator calorimeter] Substantial electromagnetic fraction Ionization energy of charged particles (p,π,μ) 1980 MeV [40%] Electromagnetic shower (π0,η0,e) 760 MeV [15%] fem ~ ln E Neutrons 520 MeV [10%] [variations significant] Photons from nuclear de-excitation 310 MeV [ 6%] Non-detectable energy (nuclear binding, neutrinos) 1430 MeV [29%]

5000 MeV [29%]

Marco Delmastro Experimental Particle Physics 44 Hadronic Showers Hadronic showers 12 40. Plots of cross sections and related quantities

Hadronic interaction:

at high energies also diffractive contribution Cross Section: 10 2

total tot = el + inel ⇓

For substantial energies !"#$$%$&'()#*%+,-. pp σinel dominates:

10 elastic el 10 mb ⇡ 2/3 [geometrical cross section] inel A Plab GeV/c

/ -1 2 3 4 5 6 7 8 10 1 10 10 10 10 10 10 10 10 2/3 ∴ tot = tot(pA) tot(pp) A √s GeV 1.9 2 10 102 103 104 [σtot slightly grows· with √s] Total proton-proton cross section Hadronic interaction length: [similar for p+n in 1-100 GeV range]

10 2 1 A 1 /3 ⇓ = = A [for √s ≈ 1 – 100 GeV]total int 2/3 ⇤tot n ⇤pp A NA⇥ · · !"#$$%$&'()#*%+,-. −p p 1/3 35 g/cm2 A Interaction length characterizes both, elastic · 10 longitudinal and transverse profile of which yields: hadronic showers ... Plab GeV/c N(x)=N0 exp( x/int) a -1 2 3 4 5 6 7 8 10 1 10 10 10 10 10 10 10 10 Figure 40.11: Total and elastic cross sections for pp and pp collisions as a function of laboratory beam momentum and total center-of-mass Remark: In principle one should distinguish between collision energy. Corresponding computer-readable data files may be found at http://pdg.lbl.gov/current/xsect/.(CourtesyoftheCOMPASgroup, IHEP, Protvino, August 2005) length λW ~ 1/σtot and interaction length λint ~ 1/σinel where the latter considers inelastic processes only (absorption) ...

Marco Delmastro Experimental Particle Physics 45 HadronicHadronic vs.Showers EM showers

Comparison hadronic vs. electromagnetic shower ... 20 250 GeV 250 GeV [Simulated air showers] proton photon

15

10 altitude above sea level [km]

5

0 0 +5 lateral shower width [km] lateral shower width [km]

Marco Delmastro Experimental Particle Physics 46 HadronicHadronic vs.Showers EM showers

Hadronic vs. electromagnetic Some numerical values for materials interaction length: typical used in hadron calorimeters

A λint [cm] X0 [cm] X0 4 ⇠ Z2 int A /3 ➛ X 1/3 0 ⇠ int A Szint.Scint 79.4 42.2 ⇠

int X0 a LAr 83.7 14.0 [λint/X0 > 30 possible; see below] Fe 16.8 1.76 Typical [EM: 15-20 X0] Longitudinal size: 6 ... 9 λint Pb 17.1 0.56 [95% containment] Typical Transverse size: one λint [EM: 2 RM; compact] [95% containment] U 10.5 0.32

Hadronic calorimeter need more depth C 38.1 18.8 than electromagnetic calorimeter ...

Marco Delmastro Experimental Particle Physics 47 HadronicHadronic showerShowers development

Hadronic shower development: But: [estimate similar to e.m. case] Only rough estimate as ... Depth (in units of λint): energy sharing between shower particles x fluctuates strongly ... t = part of the energy is not detectable (neutrinos, int binding energy); partial compensation possible (n-capture & fission) Energy in depth t: spatial distribution varies strongly; different E range of e.g. π± and π0 ... E(t)= & E(t )=E n t max thr h i [with Ethr ≈ 290 MeV] electromagnetic fraction, i.e. fraction of energy E deposited by π0 ➛ γγ increases with energy ... Ethr = tmax n f f 0 ln E/(1 GeV) h i em ⇡ ⇠ Shower maximum: Number of particles Explanation: charged hadron contribute to electromagnetic lower by factor Ethr/Ec fraction via π–p ➛ π0n; the opposite happens only rarely as compared to e.m. shower ... E π0 travel only 0.2 μm before its decay ('one-way street') ... n tmax = Intrinsic resolution: E worse by factor √Ethr/Ec At energies below 1 GeV hadrons loose their h i thr energy via ionization only ... E ln ( /Ethr) t = Thus: need Monte Carlo (GEISHA, CALOR, ...) max ln n h i to describe shower development correctly ...

Marco Delmastro Experimental Particle Physics 48 HadronicHadronic showerShowers longitudinal development

Longitudinal shower

development: Strong peak near λint ... followed by exponential decrease .... Shower depth: t 0.2ln(E/GeV) + 0.7 15 max ⇡ L95 = tmax +2.5att 0.3 with (E/GeV ) att ⇡ 10 Example: 300 GeV pion ... tmax = 1.85; L95 = 1.85 + 5.5 ≈ 7.4 [95% within 8λint; 99% within 11 λint]

5

Number of nuclei [arbitrary units] 95% on average Longitudinal shower profile for 300 GeV π– interactions in a block of uranium measured from the induced 99Mo radioactivity ...

0 1 2 3 4 5 6 7 8 9 10

Depth [λint]

Marco Delmastro Experimental Particle Physics 49 6.3.1 Hadronische Schauer Schauerprofil

Longitudinale Schauerentwicklung von Longitudinale und transversale (geladenen) Pionen in Wolfram für 3 Schauerentwicklung einer durch versch. Einfallsenergien. Die Linien 10!GeV/c Pionen (!–) ausgelösten sind das Ergebnis von Monte-Carlo- hadronischen Kaskade in Eisen. Simulationen, die Punkte Meßdaten. Shower profile HadronicHadronic showerShowers transverseof 10development GeV/c pions in iron

Transverse shower profile depth # Shower particles Typical transverse momenta of secondaries: p 350MeV /c ... h ti' Lateral extend at shower maximum: R95% ≃ λint ... Electromagnetic component leads to relatively well-defined core: R ≃ RM ... Exponential decay after shower maximum ...

Lateral shower position [cm] Lateral profile for 300 GeV π– 238 Quelle beider Bilder : C. Grupen, T[targeteilch ematerialndet ektU]oren, B.I. W issenschaftsverMeasurementlag, 1993 from induced [measured at depth 4 λint] radioactivity: 99Mo (fission): neutron induced ... [energetic neutron component] M. Krammer: Detektoren, SS 05 Kalorimeter 237U: mainly produced via 238U(γ,n)237U ... 48 [electromagnetic component] 0 More π ‘s and γ in core 239Np: from 239U decay ... Energetic neutrons and [thermal neutrons] charged pions form a wider core Ordinate indicates decay rate Thermal neutrons generate broad tail of different radioactive nuclides ...

Marco Delmastro Experimental Particle Physics 50 Detectors are imperfect… • Detection efficiency not instrumented region

passive region ! M = P(entering active region) • Upstream material, entrance windows, … active region ! R = P(generating signal) signal • Interaction cross sections, response, fluctuations, … ! D = P(signal gets registered)

upstream material upstream passive region • Readout properties, thresholds, …

not instrumented • Acceptance region ! Instrumented/reactive region of the phase space (e.g. pseudorapity, azimuthal angle, but readout also energy/momentum) • dynamic range

Marco Delmastro Experimental Particle Physics 51 Detectors are imperfect… • Dead Time • Response linearity ! Delay between particle entrance in detector and signal acquisition, in which the detector is rendered inactive (detector + readout) ! Dead time efficiency

• Response resolution

• Trigger ! Detector cannot acquire signals continously…

Marco Delmastro Experimental Particle Physics 52