06 lecture calorimetry em - desy.deschleper/lehre/det_dat/ss_2018/06_lecture_calorimetry_em.pdf ·...
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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)