Calorimetry I Electromagnetic Calorimeters 6.1 Allgemeine Grundlagen Funktionsprinzip

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Calorimetry I Electromagnetic Calorimeters 6.1 Allgemeine Grundlagen Funktionsprinzip – 1

!IntroductionIn der Hochenergiephysik versteht man unter einem Kalorimeter einen Detektor, welcher die zu analysierenden Teilchen vollständig absorbiert. Da- durch kann die Einfallsenergie des betreffenden Teilchens gemessen werden. Calorimeter: ! Die allermeisten Kalorimeter sind überdies positionssensitiv ausgeführt, um dieDetector Energiedeposition for energy measurement ortsabhängig via total absorption zu messen of particles und sie ... beim gleichzeitigen DurchgangAlso: most calorimeters von mehreren are position Teilchen sensitive den to measure individuellen energy depositionsTeilchen zuzuordnen. depending on their location ... ! Ein einfallendes Teilchen initiiert innerhalb des Kalorimeters einen Teilchen- schauerPrinciple of (eine operation: Teilchenkaskade) aus Sekundärteilchen und gibt so sukzessive seineIncoming ganze particle Energie initiates and particle diesen shower Schauer ... ab. DieShower Zusammensetzung Composition and shower dimensions und die depend Ausdehnung on eines solchen SchauersSchematic ofhängen vonparticle der type Art and des detector einfallenden material ... Teilchens ab (e±, Photon odercalorimeter Hadron). principle Energy deposited in form of: heat, ionization, particle cascade (shower) excitation of atoms, Cherenkov light ... Different calorimeter types use different kinds of theseBi lsignalsd rec toh tmeasures: Gro totalbes energy Sch e...ma incident particle Important:eines Teilchenschauers in einem (homogenen) Kalorimeter Signal ~ total deposited energy detector volume [Proportionality factor determined by calibration] M. Krammer: Detektoren, SS 05 Kalorimeter 2 Introduction

Energy vs. momentum measurement: 1 Calorimeter: E Gas detector: p p [see below] E ⇠ pE [see above] p ⇠

e.g. ATLAS: e.g. ATLAS:

E 0.1 p 4 5 10 pt E ⇡ pE p · ·

i.e. σE/E = 1% @ 100 GeV i.e. σp/p = 5% @ 100 GeV

At very high energies one has to switch to calorimeters because their resolution improves while those of a magnetic spectrometer decreases with E ... Shower depth: E Calorimeter: L ln Shower depth nearly energy independent [see below] ⇠ Ec i.e. calorimeters can be compact ... [Ec: critical energy] Compare with magnetic spectrometer: p/p p/L2 Detector size has to grow quadratically to maintain⇠ resolution Introduction

Further calorimeter features:

Calorimeters can be built as 4π-detectors, i.e. they can detect large for small θ particles over almost the full solid angle

⇥ 2 ⇥ 2 ⇥ 2 Magnetic spectrometer: anisotropy due to magnetic ﬁeld; remember: ( p/p) =( pt/pt) +( /sin )

Calorimeters can provide fast timing signal (1 to 10 ns); can be used for triggering [e.g. ATLAS L1 Calorimeter Trigger]

Calorimeters can measure the energy of both, charged and neutral particles, if they interact via electromagnetic or strong forces [e.g.: γ, μ, Κ0, ...] Magnetic spectrometer: only charged particles!

Segmentation in depth allows separation of hadrons (p,n,π±), from particles which only interact electromagnetically (γ,e) ...

... Electromagnetic Showers

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%] Abbildung 8.2: Entwicklung eines elektromagnetischen Sch

• Nur die Prozesse

werden ber • Electromagnetic ShowersAuf der StreckeAnalytic Shower Model Electromagnetic shower

• ücksichtigt ( Reminder: Das Photon materialisiert nach Simple shower model: [from Heitler] Dominant processes • X at high energiesF ... Only two dominant interactions: Folgende Gr 0 ür • Pair verliertproduction das K and Bremsstrahlung ... PhotonsF : Pair production γ ür E> =Kern).➛ + + − Electrons : Bremsstrahlung γ + Nucleus Nucleuse + e + e Electromagnetic Shower E [PhotonsX0 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 • Pairößen production: 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 • 7 183 K Transversale Breite des Schauers ⇥ 4 r2Z2 ln Bremsstrahlung:Shower developmentdurch Bremsstrahlung governed die Hby X0 ... Simplification: pair e 1 K ⇡ 9 Z 3 + ✓ ◆ After a distance2 X0 electrons remain with auers (Monte CarloEγ = Simulation) Ee ≈ E0/2 dE Z 183dE E+ e + Nach Durchlaufen der Schichtdicke 7 A X only (1/e)th of their2 primary energy ... [Ee looses half the energy] =4NA re E ln 1 = e len Teilchen = [X0: radiation length] E + [in cm or g/cm2] dx0 A · + − Zdx3 X0 + 9 NAX0 ,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≈ nbE0/2 e t r γ − [Energy shared by e+/e–] Absorption x/X= 0 coefficient: ➛ E = E0e Assume: E NA 7 E After passage of one 0X0 electron ... with initial particle energy E0 µ = n⇥ = ⇥pair = E has> E onlyc : no(1/e) energyth of2 its primaryloss by energy ionization/excitation ... A · 9 X0 ± =E < Ec : energy loss only[i.e. via 37%] ionization/excitation E 1 2 älfte 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: 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 Electromagnetic Showers 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 Some Useful 'Rules 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 Design of a Calorimeter

• TransverseSimplified modelShower [Heitler containment]: shower development governed by X0 ‣- RM(Pb) ≈ 1.6 cm ⎫ e loses [1 - 1/e] = 63% ofuse energy high-Z in material 1 Xo (Brems.) the mean free path of⎬ a is 9/7 Xo (pair prod.) ‣ RM(C) ≈ 22 cm γ Lead%%absorbers%in%cloud%chamber% ⎭ • LongitudinalAssume: shower containment (and realistic compactness) E > Ec : no energy loss by ionization/excitation ‣ L(95% in Pb) ≈ 26 X0 → L ≈ 13 cm ⎫ E < Ec : energy loss only via ionization/excitationuse high-Z material ⎬ ‣ L(95% in C) ≈ 17 X0 → L ≈ 170 cm Simple shower model: ⎭ • Signalt generation and measurement After shower max is reached: • 2 particles after t [X0] use low-Zonly ionization,material Compton, photo-electric • chargeeach with collection energy from E/2 ionisationt ‣ → (mean free path of electrons) • Stops if E < critical energy ε ‣ light collection from scintillationC → transparent medium (low- or high-Z) • Number of particles N = E/εC • Homogeneous• Maximum ator sampling calorimeter cost, performance, detector integration…

Roman Kogler 30 Calorimetry Electromagnetic Showers

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 Electromagnetic Shower Proﬁle ElectromagneticLongitudinal shower Shower proﬁles Shape (longitudinal) 8.1 Electromagnetic calorimeters 235

Depth [X0] 600 Longitudinal proﬁle 5000 MeV

Energy deposit of electrons as a function of depth in a Parametrization: block of copper; integrals normalized to same value [EGS4* calculation] [Longo 1975]

] 400 0 Depth of shower maximum increases dE ⇥t X logarithmically with energy = E0 t e

dt [MeV/

E t t ln( 0/Ec) d

max / / α,β : free parameters 2000 MeV E tα : at small depth number of d 200 secondaries increases ... 1000 MeV –βt Energy deposit per cm [%] e : 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]

Depth [cm] More exact *EGS = Electron Gamma Shower [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/

6 t d

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 contained in a cylinder around the shower axis whose radius is R(95%) = 2RM almost independently of the energy of the incident particle. The dependence of the containment radius on the material is taken into account by the critical energy and radiation length appearing in Eq. (8.11). TransversalLateral Shower Shape proﬁle

Molière Radii

Transverse proﬁle at different shower depths ....

Up to shower maximum broadening mainly due to multiple scattering ...

Characterized by RM: [90% shower energy within RM]

21 MeV Energy deposit [a.u.] RM = X0 Ec Beyond shower maximum broadening mainly due to low energy photons ...

Radial distributions of the energy deposited by 10 GeV electron showers in Copper [Results of EGS4 simulations]

Distance from shower axis [RM] 16 Homogeneous Calorimeters

★ In a homogeneous calorimeter the whole detector volume is ﬁlled by a high-density material which simultaneously serves as absorber as well as as active medium ...

Signal Material

Scintillation light BGO, BaF2, CeF3, ...

Cherenkov light Lead Glass

Ionization signal Liquid nobel gases (Ar, Kr, Xe)

★ Advantage: homogenous calorimeters provide optimal energy resolution

★ Disadvantage: very expensive

★ Homogenous calorimeters are exclusively used for electromagnetic calorimeter, i.e. energy measurement of electrons and photons Example: CMS Crystal Calorimeter

Homogeneous Calorimeters CMS electromagnetic calorimeter Chapter 4

Electromagnetic Calorimeter

4.1 Description of the ECAL In this section, the layout, the crystals and the photodetectors of the Electromagnetic Calor- imeter (ECAL) are described. The section ends with a description of the preshower detector which sits in front of the endcap crystals.Example:Two important CMS Crystalchanges haveCalorimeteroccurred to the geometry and configuration since the ECAL TDR [5]. In the endcap the basic mechanical unit, the “supercrystal,” which was originally envisaged to hold 6 6 crystals, is now a 5 5 unit. × × The lateral dimensions of the endcap crystals have been increased such that the supercrystal Homogeneousremains Calorimeterslittle changed in size. This choice took advantage of the crystal producer’s abil- ity to produce larger crystals, to reduce the channel count. Secondly, the option of a barrel preshower detector, envisaged for high-luminosity running only, has been dropped. This simplificationCMSallows mor electromagnetice space to the tracker, but requir calorimeteres that the longitudinal vertices of H γγ events be found with the reconstructed charged particle tracks in the event. → Scintillator : PBW04 [Lead Tungsten] Photosensor : APDs [Avalanche Photodiodes] 4.1.1 The ECAL lay out and geometry 4.1. Description of the ECAL 147 The nominal geometry of theNumberECAL (the of crystals:engineering ~ specification)70000 is simulated in detail in The barrel part of the ECAL covers the pseudorapidity range η < 1.479. The barrel granu- | | larity is 360-fold in φ and (2 85)-fold in η, resulting in a total of 61 200 crystals.The truncated- × the GEANT4/OSCAR model. There are 36 identical supermodules, 18 in each half barrel, each pyramid shaped crystals are mounted in a quasi-projective geometry so that their axes make Light output: 4.5 photons/MeV a small angle (3o) with the respect to the vector from the nominalcoveringinteraction vertex,20◦ ininbothφ. The barrel is closed at each end by an endcap. In front of most of the the φ and η projections. The crystal cross-section corresponds to approximately 0.0174 × 0.0174 in η-φ or 22 22 mm2 at the front face of crystal, and 26fiducial26 mm2 at theregionrear face. Theof each endcap is a preshower device. Figure 4.1 shows a transverse section ◦ × × crystal length is 230 mm corresponding to 25.8 X0. through ECAL. The centres of the front faces of the crystals in the supermodules are at a radius 1.29 m. The crystals are contained in a thin-walled glass-fibre alveola structures (“submodules,” as shown in Fig. CP 5) with 5 pairs of crystals (left and right reflections of a single shape) per submodule. The η extent of the submodule corresponds to a trigger tower. To reduce the number of different type of crystals, the crystals in each submodule have the same shape. There are 17 pairs of shapes. The submodules are assembled into modules and there are Barrel ECAL (EB) 4 modules in each supermodule separated by aluminium webs. The arrangement of the 4 modules in a supermodule can be seen in the photograph shown in Fig. 4.2. 1.653 y = = 1.479 Preshower (ES) = 2.6 = 3.0 Endcap z ECAL (EE)

Figure 4.1: Transverse section through the ECAL, showing geometrical conﬁguration.

Figure 4.2: Photograph of supermodule, showing modules.

The thermal screen and neutron moderator in front of the crystals are described in the model, as well as an approximate modelling of the electronics, thermal regulation system and mechanical structure behind the crystals. 146 The endcaps cover the rapidity range 1.479 < η < 3.0. The longitudinal distance between | | the interaction point and the endcap envelop is 3144 mm in the simulation. This location takes account of the estimated shift toward the interaction point by 2.6 cm when the 4 T magnetic ﬁeld is switched on. The endcap consists of identically shaped crystals grouped in mechanical units of 5 5 crystals (supercrystals, or SCs) consisting of a carbon-ﬁbre alveola × structure. Each endcap is divided into 2 halves, or “Dees” (Fig. CP 6). Each Dee comprises 3662 crystals. These are contained in 138 standard SCs and 18 special partial supercrystals on the inner and outer circumference. The crystals and SCs are arranged in a rectangular Sampling Calorimeters

Scheme of a sandwich calorimeter Principle: passive absorber Alternating layers of absorber and shower (cascade of secondaries) active material [sandwich calorimeter] incoming particle Absorber materials: [high density]Simple shower model Iron (Fe) ! ConsiderLead only (Pb) Bremsstrahlung and (symmetric) pair productionUranium (U) active layers [For compensation ...]

! AssumeActive X0 materials: ! !pair Plastic scintillator ! After t XSilicon0: detectors Liquid ionization chamber ! t N(t)Gas = detectors2 t Electromagnetic shower ! E(t)/particle = E0/2

! Process continues until E(t)

tmax ! E(tmax) = E0/2 = Ec

! tmax = ln(E0/Ec)/ln2

! Nmax " E0/Ec

5 Sampling Calorimeters

★ Advantages: By separating passive and active layers the different layer materials can be optimally adapted to the corresponding requirements ... By freely choosing high-density material for the absorbers one can built very compact calorimeters ...

Sampling calorimeters are simpler with more passive material and thus cheaper than homogeneous calorimeters ...

★ Disadvantages: Only part of the deposited particle energy is actually detected in the active layers; typically a few percent [for gas detectors even only ~10-5] ... Due to this sampling-ﬂuctuations typically result in a reduced energy resolution for sampling calorimeters ... Sampling Calorimeters

Scintillators as active layer; Possible setups signal readout via photo multipliers

Absorber Scintillator Scintillator (blue light) Light guide Scintillators as active layer; wave length shifter to convert light Photo detector

Wavelength shifter

Charge ampliﬁer Absorber as electrodes Ionization chambers between absorber HV plates Electrodes

Argon Analogue Active medium: LAr; absorber signal embedded in liquid serve as electrods Sampling Calorimeters

Example: ATLAS Liquid Argon Calorimeter Response and Linearity

Simplified model [Heitler]: shower development response = average signal per unit of deposited energy” governed by X0 e.g. # photoelectrons/GeV, picoCoulombs/MeV, etc e- loses [1 - 1/e] = 63% of energy in 1 Xo (Brems.) the mean free path of a γ is 9/7 Xo (pair prod.) A linear calorimeter has a constant response Lead%%absorbers%in%cloud%chamber%

Assume :

E > Ec : no energy loss by ionization/excitation E < Ec : energy loss only via ionization/excitation

Simple shower model:

t After shower max is reached: • 2 particles after t [X0] only ionization, Compton, photo-electric • each with energy E/2t In general: • Stops if E < critical energy ε Electromagnetic calorimetersC are linear • Number ! ofAll particles energy deposited N = E/εC through ionization/excitation of absorber • MaximumHadronic at calorimeters are not … (later)

Roman Kogler 37 Calorimetry Energy Resolution

CalorimeterSimplified modelenergy [ Heitlerresolution]: shower determined development by ﬂuctuations Differentgoverned effects by X 0have different energy dependence - e loses– quantum, [1 - 1/e] sampling = 63% offluctuations energy in 1 Xo σ/E ( Brems~ E-1/2 .) the mean free path of a is 9/7 Xo (pair prod.) -1/4 (*) – shower leakage γ σ/E ~ constantLead%%absorbers%in%cloud%chamber% or E

– electronic noise σ/E ~ E-1 Assume: – structural non-uniformities σ/E = constant E > Ec : no energy loss by ionization/excitation Add in quadrature: σ2 = σ2 + σ2 + σ2 + σ2 + ... E < Ec : energy loss only via ionization/excitationtot 1 2 3 4 (*) different for longitudinal and transverse leakage Simple shower model: After shower max is reached: t example: • 2 particles after t [X0] only ionization, Compton, photo-electric • each with energy E/2t ATLAS EM calorimeter

• Stops if E < critical energy εC

• Number of particles N = E/εC • Maximum at

Roman Kogler 40 Calorimetry Energy Resolution

Shower ﬂuctuations: [intrinsic resolution]

Ideal (homogeneous) calorimeter without leakage: energy resolution limited Energyonly Resolution by statistical ﬂuctuations of the number N of shower particles ... Energyi.e.: resolution

Simplified model [Heitler]: shower developmentE N pN 1 E Ideally, if all shower particles counted: E ~ N, σ ~ √N =~ √E with N = E : energy of primary particle governed by X0 E / N ⇡ N pN W In practice: W : mean energy required to e- loses [1 - 1/e] = 63%p of energy in 1 Xo (Brems.) absolute: = a E b cE E W produce 'signal quantum' the mean free path of a is 9/7 Xo (pair prod.) a γ b E / E Lead%%absorbers%in%cloud%chamber% relative: = c r Examples: Assume: E pE E Resolution improves due to correlations Silicon detectors : W ≈ 3.6 eV a:E >stochastic Ec : no energy term loss by ionization/excitation Gas detectors : W ≈ 30 eV intrinsic statistical shower fluctuationsbetween ﬂuctuations (Fano factor; see above) ... E < Ec : energy loss only via ionization/excitation Plastic scintillator : W ≈ 100 eV sampling fluctuations signal quantum fluctuations (e.g. photo-electron statistics) Simple shower model: E FW b: noise term After shower max is reached: t E / E [F: Fano factor] • 2readout particles electronic after t [X noise0] onlyr ionization, Compton, photo-electric • eachRadio-activity, with energy pile-up E/2t fluctuations c:• constantStops if termE < critical energy ε inhomogeneities (hardware orC calibration) • Numberimperfections of particles in calorimeter N = E/ε constructionC (dimensional variations, etc.) • Maximumnon-linearity at of readout electronics fluctuations in longitudinal energy containment (leakage can also be ~ E-1/4) fluctuations in energy lost in dead material before or within the calorimeter

Roman Kogler 41 Calorimetry IntrinsicIntrinsic Energy ResolutionEnergy Resolution of EM calorimeters

HomogeneousSimplified model calorimeters: [Heitler]: shower development signal = sum of all E deposited by charged particles with E > Ethreshold governed by X0 If eW- losesis the mean[1 - 1/e] energy = 63% required of energy to produce in 1 Xo a ‘signal(Brems quantum’.) (eg an electron-ion pair in thea noble mean liquid free or path a ‘visible’ of a photon is 9/7 inXo a (paircrystal) prod.) ! mean number of ‘quanta’ produced γ Lead%%absorbers%in%cloud%chamber% is n = E/W 〈n〉 = E / W h i TheAssume intrinsic: energy resolution is given by the ﬂuctuations on n. E > Ec : no energy loss by ionization/excitation σE / E = 1/√ n1 = 1/ √ (EW / W) E < Ec : energy loss only via= ionization= /excitation E pn r E i.e.Simple in a semiconductor shower model: crystals (Ge, Ge(Li), Si(Li)) Wt = 2.9 eV (to produce e-hole pair) After shower max is reached: • 2 particles after t [X0] only ionization, Compton, photo-electric • each! 1 MeV with γenergy = 350000 E/2 telectrons ! 1/√ n = 0.17% stochastic term In• addition, Stops ifﬂuctuations E < critical on energy n are εreducedC by correlation in the production of consecutive• Number e-hole of particles pairs: theN = Fano E/εC factor F FW • Maximum at = σE / E = √ (FW / E) E E r For GeLi γ detector F ~ 0.1 ! stochastic term ~ 0.05%/√E[GeV]

Roman Kogler 42 Calorimetry Example:Example: CMS CMS ECAL ECAL Resolutionresolution

Simplified model [Heitler]: shower development governed by X0 e- loses [1 - 1/e] = 63% of energy in 1 Xo (Brems.) the mean free path of a is 9/7 Xo (pair prod.) γ Lead%%absorbers%in%cloud%chamber%

Assume:

E > Ec : no energy loss by ionization/excitation E < Ec : energy loss only via ionization/excitation

Simple shower model: Relatively large size of sampling After shower max is reached: • 2t particles after t [X ] term (3%): 0 only ionization, Compton, photo-electric • each with energy E/2t PbWO4 rather weak scintillator • Stops if E < critical energy ε ‣ 4500 photos / 1 GeV C • Number of particles N = E/ε • Fano factor of 2 for crystal / APDC • combinationMaximum at Still: sampling term 3 times smaller than for ATLAS ECAL!

Roman Kogler 43 Calorimetry EnergyResolution Resolution of Sampling Calorimeters

Simplified model [Heitler]: shower development Sampling fluctuations: governed by X0 eAdditional- loses [1 contribution- 1/e] = 63% to ofenergy energy resolution in 1 Xo in ( Bremssampling.) calorimeters due theto fluctuationsmean free ofpath the ofnumber a is of9/7 (low-energy) Xo (pair prod.) electrons crossing active layer ... γ Lead%%absorbers%in%cloud%chamber% Increases linearly with energy of incident particle and fineness of the Assumesampling: ... E > Ec : no energyE loss Nbych ionization: charged particles/excitation reaching active layer E < NEch : energy loss onlyNmax via : ionizationtotal number of/excitation particles = E/E c c / Ec tabs tabs : absorber thickness in X0

Reasoning: Energy deposition dominantly due to low energy electrons; Simple shower model: range of these electrons smaller than absorber thickness tabs; only few electronsAfter reach shower active layer max ... is reached: • 2t particles after t [X ] Resulting 0 Fraction f ~ 1/tabs onlyreaches ionization, the active Compton, medium ... photo-electric • energyeach resolution: with energy E/2t Semi-empirical: • Stops if E < critical energy ε E t C E [MeV] t E Nch c abs E =3.2% c · abs • NumberE / N of particles/ E N = E/εC E F E [GeV] ch r s · • MaximumChoose: Ec smallat (large Z) where F takes detector threshold tabs small (ﬁne sampling) effects into account ...

Roman Kogler 44 Calorimetry Resolution of Sampling Calorimeters Energy Resolution Simplified model [Heitler]: shower development

governedMeasure by X0 energy resolution .. - Kanale e losesof [1 a sampling- 1/e] = 63%calorimeter of energy for in 1 Xo (BremsGeV .) the meandifferent free absorber path of thicknesses a is 9/7 Xo (pair prod.) γ Lead%%absorbers%in%cloud%chamber% tabs : absorber thickness in X0 Assume:

E >D E : cabsorber : no energy thickness loss by in ionization mm /excitation E < Ec : energy loss only via ionization/excitation

Simple shower model: t After shower max is reached: • Sampling2 particles after t [X0] only ionization, Compton, photo-electric • contribution:each with energy E/2t • Stops if E < Ecritical[MeV] energyt ε E =3.2% c · abs C Sampling Fluktuationen Fluctuations • NumberE ofs particlesF E [GeV] N = E/ε · C Photo-electronPhotoelektron−Statistik Statistics + Leakage + Leakage • Maximum at D [mm] Best choice: Ec small (large Z) tabs small (fine sampling) Abbildung 8.9: Gemessene Energieauﬂ ¨osung eines Sampling–Kalorimeters f ur¨ verschiedene DickenRoman des KoglerPb–Absorbers [109]45 Calorimetry

8.6 Ortsaufl ¨osung

Neben der Energie m öchte man h äufig den Ort bestimmen, an dem ein Photon auf das Kalorimeter tri fft. Dies gelingt bei senkrechtem Auftre ffen auf den Z ähler ( pointing“) der ” Photonen dadurch, daß man die endliche transversale Breite des Schauers ausnutzt. Abh ängig vom Auftre ffort variiert die Pulsh öhe im benachbarten Z ähler. Da die Breite eines Schauers näherungsweise durch RM gegeben ist (Abb.8.5), muß der Durchmesser des Z ählers typischer- weise < 2RM sein. Die Ortsaufl ösung ist durch die transversale Granularit ät des Kalorimeters festgelegt. Zus ätzlich spielen transversale Schauerfluktuationen eine Ro lle, f ur¨ hinreichend große Energien gilt (siehe Abb.8.10): 1 σx . ∼ √E

Ort

E [MeV]

Abbildung 8.10: Gemessene Ortsauﬂ ¨osung eines Sampling–Kalorimeters [110]

163