Electromagnetic Calorimeters

Calorimetry I

M. Krammer: Detektoren, SS 05 Kalorimeter 2

6.1 Allgemeine GrundlagenFunktionsprinzip – 1

! In der Hochenergiephysik versteht man unter einem Kalorimeter einenDetektor, welcher die zu analysierenden Teilchen vollständig absorbiert. Da-durch kann die Einfallsenergie des betreffenden Teilchens gemessen werden.

! Die allermeisten Kalorimeter sind überdies positionssensitiv ausgeführt, umdie Energiedeposition ortsabhängig zu messen und sie beim gleichzeitigenDurchgang von mehreren Teilchen den individuellen Teilchen zuzuordnen.

! Ein einfallendes Teilchen initiiert innerhalb des Kalorimeters einen Teilchen-schauer (eine Teilchenkaskade) aus Sekundärteilchen und gibt so sukzessiveseine ganze Energie and diesen Schauer ab.

Die Zusammensetzung und die Ausdehnung eines solchen Schauers hängenvon der Art des einfallenden Teilchens ab (e±, Photon oder Hadron).

Bild rechts: Grobes Schemaeines Teilchenschauers ineinem (homogenen) Kalorimeter

Introduction

Calorimeter:

Detector for energy measurement via total absorption of particles ...

Also: most calorimeters are position sensitive to measure energy depositionsdepending on their location ...

Principle of operation:

detector volume

incident particle

particle cascade (shower)

Incoming particle initiates particle shower ...Shower Composition and shower dimensions depend on particle type and detector material ...

Energy deposited in form of: heat, ionization,excitation of atoms, Cherenkov light ...Different calorimeter types use different kinds ofthese signals to measure total energy ...

Important:

Signal ~ total deposited energy

[Proportionality factor determined by calibration]

Schematic of calorimeter principle

Introduction

Energy vs. momentum measurement:�E

E⇠ 1p

ECalorimeter:[see below]

Gas detector:[see above]

�p

p⇠ p

e.g. ATLAS:�E

E⇡ 0.1p

E

�p

p� 5 · 10�4 · pt

e.g. ATLAS:

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:

Calorimeter:[see below]

L ⇠ lnE

Ec

[Ec: critical energy]

Shower depth nearly energy independenti.e. calorimeters can be compact ...

Compare with magnetic spectrometer:Detector size has to grow quadratically to maintain resolution

�p/p ⇠ p/L2

Introduction

Further calorimeter features:

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

Magnetic spectrometer: anisotropy due to magnetic field; remember:

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

(⇥p/p)2 = (⇥pt/pt)2 + (⇥�/sin �)2

large for small θ

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) ...

...

µ = n⇥ = �NA

A· ⇥pair

Electromagnetic Showers

Reminder:

X0

Dominant processes at high energies ...

Photons : Pair productionElectrons: Bremsstrahlung

Pair production:

dE

dx=

E

X0

dE

dx= 4�NA

Z2

Ar2e · E ln

183Z

13

⇥pair ⇡79

✓4 �r2

eZ2 ln183Z

13

◆

=79

�

X0

Bremsstrahlung:

E = E0e�x/X0

[X0: radiation length][in cm or g/cm2]

Absorption coefficient:

After passage of one X0 electronhas only (1/e)th of its primary energy ...

[i.e. 37%]

➛

=79

A

NAX0

Abbildung

8.2: Entw

icklungeines

elektromagnetischen

Schauers(M

onteCarlo

Simulation)

•Nur

dieProzesse

γ+

K→

K+

e ++

e −

e+

K→

K+

e+

γ

werden

berucksichtigt(K

=Kern).

•Auf der

StreckeX

0verliert

dase −

durchBrem

sstrahlungdie

Halfte

seinerEnergie

E1

=

E02

•Das

Photon

materialisiert

nachX

0 , dieEnergie

vonPositron

undElektron

betragt

E±

=

E12

•Fur

E>

ϵtritt

keinEnergieverlust

durchIonisation/A

nregungauf.

•Fur

E≤

ϵverlieren

dieElektronen

Energie

nurdurch

Ionisation/Anregung.

FolgendeGroßen

sindbei der

Beschreibung

einesSchauers

vonInteresse:

•Zahl der

Teilchenim

Schauer

•Lage

desSchauerm

aximum

s

•Longitudinalverteilung

desSchauers

imRaum

•Transversale

Breite

desSchauers

Wir

messen

dielongitudinalen

Kom

ponentendes

Schauersin

Strahlungslangen:

t=

xX0

Nach

Durchlaufen

derSchichtdicke

t betragtin

unseremeinfachen

Modell die

Zahl derschnel-

lenTeilchen

N(t)

=2 t

,156

Simple shower model:[from Heitler]

Only two dominant interactions: Pair production and Bremsstrahlung ...

γ + Nucleus ➛ Nucleus + e+ + e−

[Photons absorbed via pair production]

e + Nucleus ➛ Nucleus + e + γ[Energy loss of electrons via Bremsstrahlung]

Electromagnetic Shower[Monte Carlo Simulation]

Shower development governed by X0 ...After a distance X0 electrons remain with only (1/e)th of their primary energy ...

Photon produces e+e−-pair after 9/7X0 ≈ X0 ...

Analytic Shower Model

Eγ = Ee ≈ E0/2

Simplification:

Ee ≈ E0/2

[Ee looses half the energy]

[Energy shared by e+/e–]

Assume:

E > Ec: no energy loss by ionization/excitation

E < Ec: energy loss only via ionization/excitation

Use

... with initial particle energy E0

Electromagnetic shower

µ = n⇥ = �NA

A· ⇥pair

Electromagnetic Showers

Reminder:

X0

Dominant processes at high energies ...

Photons : Pair productionElectrons: Bremsstrahlung

Pair production:

dE

dx=

E

X0

dE

dx= 4�NA

Z2

Ar2e · E ln

183Z

13

⇥pair ⇡79

✓4 �r2

eZ2 ln183Z

13

◆

=79

�

X0

Bremsstrahlung:

E = E0e�x/X0

[X0: radiation length][in cm or g/cm2]

Absorption coefficient:

After passage of one X0 electronhas only (1/e)th of its primary energy ...

[i.e. 37%]

➛

=79

A

NAX0

Electromagnetic Showers

Transverse size of EM shower given by radiation length via Molière radius

[see also later]

RM =21 MeV

EcX0 RM : Moliere radius

Ec : Critical Energy [Rossi]X0 : Radiation length

Critical Energy [see above]:

Further basics:

20 27. Passage of particles through matter

0

0.4

0.8

1.2

0 0.25 0.5 0.75 1y = k/E

Bremsstrahlung

(X0

NA/A

) yd

σ LP

M/d

y

10 GeV

1 TeV

10 TeV

100 TeV

1 PeV10 PeV

100 GeV

Figure 27.11: The normalized bremsstrahlung cross section k dσLPM/dk inlead versus the fractional photon energy y = k/E. The vertical axis has unitsof photons per radiation length.

2 5 10 20 50 100 200

CopperX0 = 12.86 g cm−2

Ec = 19.63 MeV

dE/dx

× X

0 (M

eV)

Electron energy (MeV)

10

20

30

50

70

100

200

40

Brems = ionization

Ionization

Rossi:Ionization per X0= electron energy

Tota

l

Brem

s ≈E

Exact

brem

sstr

ahlu

ng

Figure 27.12: Two definitions of the critical energy Ec.

incomplete, and near y = 0, where the infrared divergence is removed bythe interference of bremsstrahlung amplitudes from nearby scattering centers

February 2, 2010 15:55

dE

dx(Ec)

����Brems

=dE

dx(Ec)

����Ion

ESol/Liqc =

610 MeVZ + 1.24

Approximations:

dE

dx

����Brems

=E

X0

dE

dx

����Ion

⇡ Ec

X0= const.

with:

&

✓dE

dx

◆

Brems

�✓dE

dx

◆

Ion

� Z · E

800 MeV

EGasc =

710 MeVZ + 0.92

Ec =550 MeV

Z

tmax = lnE

Ec

R(95%) = 2RM

R(90%) = RM

� 1.0{

Some Useful 'Rules of Thumbs'

X0 =180A

Z2

gcm2

� 0.5� 1.0

[Attention: Definition of Rossi used]

Radiation length:

Critical energy:

Shower maximum:

e– induced shower

γ induced shower

Longitudinal

energy containment:

Transverse

Energy containment:

Problem:Calculate how much Pb, Fe or Cuis needed to stop a 10 GeV electron.

Pb : Z = 82 , A = 207, ρ = 11.34 g/cm3

Fe : Z = 26 , A = 56, ρ = 7.87 g/cm3

Cu : Z = 29 , A = 63, ρ = 8.92 g/cm3

L(95%) = tmax + 0.08Z + 9.6 [X0]

30 CalorimetryRoman Kogler

Design of a CalorimeterSimplified 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 γ is 9/7 Xo (pair prod.) Assume: E > Ec : no energy loss by ionization/excitation E < Ec : energy loss only via ionization/excitation Simple shower model: •  2t particles after t [X0] •  each with energy E/2t •  Stops if E < critical energy εC •  Number of particles N = E/εC •  Maximum at

Lead%%absorbers%in%cloud%chamber%

After shower max is reached: only ionization, Compton, photo-electric

• Transverse Shower containment

‣ RM(Pb) ≈ 1.6 cm

‣ RM(C) ≈ 22 cm

• Longitudinal shower containment (and realistic compactness)

‣ L(95% in Pb) ≈ 26 X0 → L ≈ 13 cm

‣ L(95% in C) ≈ 17 X0 → L ≈ 170 cm

• Signal generation and measurement

‣ charge collection from ionisation

‣ light collection from scintillation

• Homogeneous or sampling calorimeter

⎫⎬⎭

use high-Z material

⎫⎬⎭

use high-Z material

use low-Z material (mean free path of electrons)transparent medium (low- or high-Z)

cost, performance, detector integration…

→→

Electromagnetic Showers

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

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

Electromagnetic shower profiles (longitudinal)

6

Longitudinal Shower Shape

Depth [cm]

Energ

y d

ep

osi

t p

er

cm

[%

]

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

*EGS = Electron Gamma Shower

tmax / ln(E0/Ec)

dE

dt= E0 · ⇥ · (⇥t)��1e�⇥t

�(�)tmax =

�� 1⇥

= ln✓

E0

Ec

◆+ Ce�➛

Electromagnetic Shower Profile

Longitudinal profile

dE

dt= E0 t�e�⇥t

8.1 Electromagnetic calorimeters 235

0

0.1

0.01

1

10

100

105 15 20 25 30 35t [X0]

dE / d

t [M

eV/X

0]

lead

iron

aluminium

0

500 MeV

1000 MeV

2000 MeV

5000 MeV

0

200

400

600

105 15 20t [X0]

dE / d

t [M

eV/X

0]

Fig. 8.4. Longitudinal shower development of electromagnetic cascades. Top:approximation by Formula (8.7 ). Bottom: Monte Carlo simulation with EGS4 for10 GeV electron showers in aluminium, iron and lead [11].

Figure 8.6 shows the longitudinal and lateral development of a 6 GeVelectron cascade in a lead calorimeter (based on [12, 13]). The lateral widthof an electromagnetic shower increases with increasing longitudinal showerdepth. The largest part of the energy is deposited in a relatively narrowshower core. Generally speaking, about 95% of the shower energy is con-tained in a cylinder around the shower axis whose radius is R(95%) = 2RMalmost independently of the energy of the incident particle. The depen-dence of the containment radius on the material is taken into account bythe critical energy and radiation length appearing in Eq. (8.11).

Parametrization:[Longo 1975]

α,β : free parameters

tα : at small depth number of secondaries increases ...e–βt : at larger depth absorption dominates ...

Numbers for E = 2 GeV (approximate):α = 2, β = 0.5, tmax = α/β

More exact[Longo 1985]

[Γ: Gamma function]

with:

[γ-induced]

[e-induced]

Ce� = �0.5Ce� = �1.0

Transversal Shower ShapeLateral profile

16

Radial distributions of the energy deposited by 10 GeV electron showers in Copper

[Results of EGS4 simulations]

Transverse profileat different shower depths ....

Distance from shower axis [RM]

Molière Radii

Energ

y d

ep

osi

t [a

.u.]

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

Beyond shower maximum broadening mainly due to low energy photons ...

RM =21 MeV

EcX0

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

★

Homogeneous Calorimeters

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

Advantage: homogenous calorimeters provide optimal energy resolution

Disadvantage: very expensive

Homogenous calorimeters are exclusively used for electromagneticcalorimeter, i.e. energy measurement of electrons and photons

Signal Material

Scintillation light BGO, BaF2, CeF3, ...

Cherenkov light Lead Glass

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

★

★

★

Homogeneous Calorimeters

Example: CMS Crystal Calorimeter

CMS electromagnetic calorimeter

Homogeneous Calorimeters

Chapter 4

Electromagnetic Calorimeter

4.1 Description of the ECALIn 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 detectorwhich sits in front of the endcap crystals. Two important changes have occurred to the ge-ometry 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 supercrystalremains little 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 barrelpreshower detector, envisaged for high-luminosity running only, has been dropped. Thissimplification allows more space to the tracker, but requires that the longitudinal vertices ofH → γγ events be found with the reconstructed charged particle tracks in the event.

4.1.1 The ECAL lay out and geometry

The nominal geometry of the ECAL (the engineering specification) is simulated in detail inthe GEANT4/OSCAR model. There are 36 identical supermodules, 18 in each half barrel, eachcovering 20◦ in φ. The barrel is closed at each end by an endcap. In front of most of thefiducial region of each endcap is a preshower device. Figure 4.1 shows a transverse sectionthrough ECAL.

y

z

Preshower (ES)

Barrel ECAL (EB)

Endcap

= 1.653

= 1.479

= 2.6= 3.0 ECAL (EE)

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

146

4.1. Description of the ECAL 147

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-pyramid shaped crystals are mounted in a quasi-projective geometry so that their axes makea small angle (3o) with the respect to the vector from the nominal interaction vertex, in boththe 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 26×26 mm2 at the rear face. Thecrystal length is 230 mm corresponding to 25.8 X0.

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,” asshown in Fig. CP 5) with 5 pairs of crystals (left and right reflections of a single shape) persubmodule. The η extent of the submodule corresponds to a trigger tower. To reduce thenumber 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 are4 modules in each supermodule separated by aluminium webs. The arrangement of the 4modules in a supermodule can be seen in the photograph shown in Fig. 4.2.

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 me-chanical structure behind the crystals.

The endcaps cover the rapidity range 1.479 < |η| < 3.0. The longitudinal distance betweenthe interaction point and the endcap envelop is 3144 mm in the simulation. This locationtakes account of the estimated shift toward the interaction point by 2.6 cm when the 4 T mag-netic field is switched on. The endcap consists of identically shaped crystals grouped inmechanical units of 5×5 crystals (supercrystals, or SCs) consisting of a carbon-fibre alveolastructure. Each endcap is divided into 2 halves, or “Dees” (Fig. CP 6). Each Dee comprises3662 crystals. These are contained in 138 standard SCs and 18 special partial supercrystalson the inner and outer circumference. The crystals and SCs are arranged in a rectangular

Scintillator : PBW04 [Lead Tungsten]

Photosensor: APDs [Avalanche Photodiodes]

Number of crystals: ~ 70000Light output: 4.5 photons/MeV

Example: CMS Crystal Calorimeter

CMS electromagnetic calorimeter

Sampling Calorimeters

Simple shower model

! Consider only Bremsstrahlung and (symmetric) pair production

! Assume X0 ! !pair

! After t X0:

! N(t) = 2t

! E(t)/particle = E0/2t

! Process continues until E(t)<Ec

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

! tmax = ln(E0/Ec)/ln2

! Nmax " E0/Ec

5

Alternating layers of absorber and active material [sandwich calorimeter]

Absorber materials:[high density]

Principle:

Iron (Fe)

Lead (Pb)

Uranium (U)[For compensation ...]

Active materials:

Plastic scintillator

Silicon detectors

Liquid ionization chamber

Gas detectors

passive absorber

shower (cascade of secondaries)

active layers

incoming particle

Scheme of asandwich calorimeter

Electromagnetic shower

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 andthus cheaper than homogeneous calorimeters ...

★

Disadvantages:

Only part of the deposited particle energy is actually detected in theactive 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

Absorber Scintillator

Light guide

Photo detector

Scintillator(blue light)

Wavelength shifter

electrodes Absorber as

Charge amplifier

HV

Argon

Electrodes

Analoguesignal

Scintillators as active layer;signal readout via photo multipliers

Scintillators as active layer; wave length shifter to convert light

Active medium: LAr; absorberembedded in liquid serve as electrods

Ionization chambersbetween absorber plates

Possible setups

Sampling Calorimeters

Example:ATLAS Liquid Argon Calorimeter

37 CalorimetryRoman Kogler

Response and LinearitySimplified 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 γ is 9/7 Xo (pair prod.) Assume: E > Ec : no energy loss by ionization/excitation E < Ec : energy loss only via ionization/excitation Simple shower model: •  2t particles after t [X0] •  each with energy E/2t •  Stops if E < critical energy εC •  Number of particles N = E/εC •  Maximum at

Lead%%absorbers%in%cloud%chamber%

After shower max is reached: only ionization, Compton, photo-electric

�response = average signal per unit of deposited energy” e.g. # photoelectrons/GeV, picoCoulombs/MeV, etc

A linear calorimeter has a constant response

In general: Electromagnetic calorimeters are linear

! All energy deposited through ionization/excitation of absorber Hadronic calorimeters are not … (later)

40 CalorimetryRoman Kogler

Energy ResolutionSimplified 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 γ is 9/7 Xo (pair prod.) Assume: E > Ec : no energy loss by ionization/excitation E < Ec : energy loss only via ionization/excitation Simple shower model: •  2t particles after t [X0] •  each with energy E/2t •  Stops if E < critical energy εC •  Number of particles N = E/εC •  Maximum at

Lead%%absorbers%in%cloud%chamber%

After shower max is reached: only ionization, Compton, photo-electric

Calorimeter energy resolution determined by fluctuationsDifferent effects have different energy dependence

–  quantum, sampling fluctuations σ/E ~ E-1/2

–  shower leakage σ/E ~ constant or E-1/4 (*)

–  electronic noise σ/E ~ E-1

–  structural non-uniformities σ/E = constant Add in quadrature: σ2

tot= σ21 + σ2

2 + σ23 + σ2

4 + ...

(*) different for longitudinal and transverse leakage

example: ATLAS EM calorimeter

Energy resolution

Ideally, if all shower particles counted: E ~ N, σ ~ √N ~ √E In practice: absolute: relative: a: stochastic term

intrinsic statistical shower fluctuations sampling fluctuations signal quantum fluctuations (e.g. photo-electron statistics)

b: noise term readout electronic noise Radio-activity, pile-up fluctuations

c: constant term inhomogeneities (hardware or calibration) imperfections in calorimeter construction (dimensional variations, etc.) non-linearity 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

41 CalorimetryRoman Kogler

Energy ResolutionSimplified 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 γ is 9/7 Xo (pair prod.) Assume: E > Ec : no energy loss by ionization/excitation E < Ec : energy loss only via ionization/excitation Simple shower model: •  2t particles after t [X0] •  each with energy E/2t •  Stops if E < critical energy εC •  Number of particles N = E/εC •  Maximum at

Lead%%absorbers%in%cloud%chamber%

After shower max is reached: only ionization, Compton, photo-electric

�E

E/ �N

N⇡p

N

N=

1pN

N =E

W

�E

E/

rW

E

�E

E/

rFW

E

Energy Resolution

Shower fluctuations:[intrinsic resolution]

Ideal (homogeneous) calorimeter without leakage: energy resolution limitedonly by statistical fluctuations of the number N of shower particles ...

i.e.:

with E : energy of primary particle

W : mean energy required to produce 'signal quantum'

Examples:

Silicon detectors : W ≈ 3.6 eV

Gas detectors : W ≈ 30 eV

Plastic scintillator : W ≈ 100 eV

Resolution improves due to correlationsbetween fluctuations (Fano factor; see above) ...

[F: Fano factor]

�

E=

apE

� b

E� c

� = apE � b� cE

Intrinsic Energy Resolution of EM calorimeters

Homogeneous calorimeters: signal = sum of all E deposited by charged particles with E > Ethreshold

If W is the mean energy required to produce a ‘signal quantum’ (eg an electron-ion pair in a noble liquid or a ‘visible’ photon in a crystal) ! mean number of ‘quanta’ produced is 〈n〉 = E / W

The intrinsic energy resolution is given by the fluctuations on n.σE / E = 1/√ n = 1/ √ (E / W)

i.e. in a semiconductor crystals (Ge, Ge(Li), Si(Li)) W = 2.9 eV (to produce e-hole pair)

! 1 MeV γ = 350000 electrons ! 1/√ n = 0.17% stochastic term

In addition, fluctuations on n are reduced by correlation in the production of consecutive e-hole pairs: the Fano factor F σE / E = √ (FW / E)

For GeLi γ detector F ~ 0.1 ! stochastic term ~ 0.05%/√E[GeV]

42 CalorimetryRoman Kogler

Intrinsic Energy ResolutionSimplified 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 γ is 9/7 Xo (pair prod.) Assume: E > Ec : no energy loss by ionization/excitation E < Ec : energy loss only via ionization/excitation Simple shower model: •  2t particles after t [X0] •  each with energy E/2t •  Stops if E < critical energy εC •  Number of particles N = E/εC •  Maximum at

Lead%%absorbers%in%cloud%chamber%

After shower max is reached: only ionization, Compton, photo-electric

hni = E/W

�

E=

1pn=

rW

E

�

E=

rFW

E

Example: CMS ECAL resolution

43 CalorimetryRoman Kogler

Example: CMS ECAL ResolutionSimplified 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 γ is 9/7 Xo (pair prod.) Assume: E > Ec : no energy loss by ionization/excitation E < Ec : energy loss only via ionization/excitation Simple shower model: •  2t particles after t [X0] •  each with energy E/2t •  Stops if E < critical energy εC •  Number of particles N = E/εC •  Maximum at

Lead%%absorbers%in%cloud%chamber%

After shower max is reached: only ionization, Compton, photo-electric

Relatively large size of sampling term (3%):

• PbWO4 rather weak scintillator ‣ 4500 photos / 1 GeV

• Fano factor of 2 for crystal / APD combination

Still: sampling term 3 times smaller than for ATLAS ECAL!

44 CalorimetryRoman Kogler

Resolution of Sampling CalorimetersSimplified 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 γ is 9/7 Xo (pair prod.) Assume: E > Ec : no energy loss by ionization/excitation E < Ec : energy loss only via ionization/excitation Simple shower model: •  2t particles after t [X0] •  each with energy E/2t •  Stops if E < critical energy εC •  Number of particles N = E/εC •  Maximum at

Lead%%absorbers%in%cloud%chamber%

After shower max is reached: only ionization, Compton, photo-electric

�E

E/ �Nch

Nch/

rEc tabs

E

Energy Resolution

Sampling fluctuations:

Additional contribution to energy resolution in sampling calorimeters dueto fluctuations of the number of (low-energy) electrons crossing active layer ...

Increases linearly with energy of incident particle and fineness of the sampling ...

Nch : charged particles reaching active layerNmax : total number of particles = E/Ec

tabs : absorber thickness in X0

Reasoning: Energy deposition dominantly due to low energy electrons; range of these electrons smaller than absorber thickness tabs; only few electrons reach active layer ...

Fraction f ~ 1/tabs reaches the active medium ...Resulting

energy resolution:Semi-empirical:

�E

E= 3.2%

sEc [MeV] · tabs

F · E [GeV]where F takes detector threshold effects into account ...

Choose: Ec small (large Z) tabs small (fine sampling)

Nch /E

Ec tabs

45 CalorimetryRoman Kogler

Resolution of Sampling CalorimetersSimplified 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 γ is 9/7 Xo (pair prod.) Assume: E > Ec : no energy loss by ionization/excitation E < Ec : energy loss only via ionization/excitation Simple shower model: •  2t particles after t [X0] •  each with energy E/2t •  Stops if E < critical energy εC •  Number of particles N = E/εC •  Maximum at

Lead%%absorbers%in%cloud%chamber%

After shower max is reached: only ionization, Compton, photo-electric

D [mm]

Photoelektron−Statistik + Leakage

Sampling Fluktuationen

GeV..Kanale

Abbildung 8.9: Gemessene Energieauflosung eines Sampling–Kalorimeters fur verschiedeneDicken des Pb–Absorbers [109]

8.6 Ortsauflosung

Neben der Energie mochte man haufig den Ort bestimmen, an dem ein Photon auf dasKalorimeter trifft. Dies gelingt bei senkrechtem Auftreffen auf den Zahler (

”pointing“) der

Photonen dadurch, daß man die endliche transversale Breite des Schauers ausnutzt. Abhangigvom Auftreffort variiert die Pulshohe im benachbarten Zahler. Da die Breite eines Schauersnaherungsweise durch RM gegeben ist (Abb.8.5), muß der Durchmesser des Zahlers typischer-weise < 2RM sein. Die Ortsauflosung ist durch die transversale Granularitat des Kalorimetersfestgelegt. Zusatzlich spielen transversale Schauerfluktuationen eine Rolle, fur hinreichendgroße Energien gilt (siehe Abb.8.10):

σx ∼ 1√E

.

!O

rt

E [MeV]

Abbildung 8.10: Gemessene Ortsauflosung eines Sampling–Kalorimeters [110]

163

Energy Resolution

�E

E= 3.2%

sEc [MeV] · tabs

F · E [GeV]Sampling Fluctuations

Photo-electron Statistics + Leakage

Samplingcontribution:

Measure energy resolutionof a sampling calorimeter for

different absorber thicknesses

tabs : absorber thickness in X0

D : absorber thickness in mm

Best choice: Ec small (large Z) tabs small (fine sampling)