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Calorimetry - 2Calorimetry - 2
Mauricio BarbiUniversity of Regina
TRIUMF Summer InstituteJuly 2007
2
Principles of CalorimetryPrinciples of Calorimetry(Focus on Particle Physics)(Focus on Particle Physics)
Lecture 1:Lecture 1:i. Introductionii. Interactions of particles with matter (electromagnetic)iii. Definition of radiation length and critical energy
Lecture 2:
i. Development of electromagnetic showersii. Electromagnetic calorimeters: Homogeneous, sampling.iii. Energy resolution
Lecture 3:
i. Interactions of particle with matter (nuclear)
ii. Development of hadronic showers
iii. Hadronic calorimeters: compensation, resolution
3
Interactions of Particles with MatterInteractions of PhotonsInteractions of Photons
Last lecture:
http://pdg.lbl.gov
productionpair pair
scatteringCompton Comp
effect ricPhotoelectp.e.
-
pairCompp.e.
ee
...σσσσ
Rayleigh scattering
Compton
Pair production
Photoeletric effect
Michele Livan
Energy range versus Z for more likely process:
4
Interactions of Particles with MatterInteractions of ElectronsInteractions of Electrons
Ionization (Fabio Sauli’s lecture)
For “heavy” charged particles (M>>me: p, K, π, ), the rate of energy loss (or stopping power) in an inelastic collision with an atomic electron is given by the
Bethe-Block equation:
(βγ) : density-effect correction
C: shell correction
z: charge of the incident particle
β = vc of the incident particle ; = (1-β2)-1/2
Wmax: maximum energy transfer in one collision
I: mean ionization potential
g
cmMeV
Z
Cδβ
I
Wcm
β
z
A
ZcmrN
dx
dE eeeA 2)(2
2ln2 2
2max
222
2
222
http://pdg.lbl.gov
5
Interactions of Particles with MatterInteractions of ElectronsInteractions of Electrons
IonizationFor electrons and positrons, the rate of energy loss is similar to that for “heavy”charged particles, but the calculations are more complicate: Small electron/positron mass Identical particles in the initial and final state Spin ½ particles in the initial and final states
k = Ek/mec2 : reduced electron (positron) kinetic energyF(k,β,) is a complicate equation
However, at high incident energies (β1) F(k) constant
g
cmMeV
Z
CkF
cmI
)(kk
βA
ZcmrπN
dx
dE
e
eeA 2),,(2
2ln
12
2
2
2
222
6
Interactions of ElectronsInteractions of Electrons
IonizationAt this high energy limits (β1), the energy loss for both “heavy” charged particles and electrons/positrons can be approximate by
Where,
The second terms indicates that the rate of relativistic rise for electrons is slightly smaller than
for heavier particles. This provides a criterion for identification between charge particles of different masses.
Interactions of Particles with Matter
BγA
I
cm
dx
dE e ln2
ln22
A Belectrons 3 1.95heavy charged particles 4 2
7
Interactions of Particles with MatterInteractions of ElectronsInteractions of Electrons
Bremsstrahlung (breaking radiation)
A particle of mass mi radiates a real photon while being
decelerated in the Coulomb field of a nucleus with a
cross section given by:
Makes electrons and positrons the only significant contribution to this process for
energies up to few hundred GeV’s.
E
E
m
Z
dE
d
i
ln2
2
mi
2 factor expected since classically
radiation2
2
ii m
Fa
32
1037
e
μ
μ
em
m
dEdσ
dEdσ
8
Interactions of Particles with MatterInteractions of ElectronsInteractions of Electrons
Bremsstrahlung
The rate of energy loss for k >> 137/Z1/3 is giving by:
Recalling from pair production
The radiation length X0 is the layer thickness that reduces the electron energy by a factor e (63%)
3
1220
183ln4
1
ZαZrN
AX
eA
00
0
X
x
eEE(x)X
E
dx
dE
EZA
NαZr
dx
dE Ae
31
183ln4 22
9
Interactions of Particles with MatterInteractions of ElectronsInteractions of Electrons
Bremsstrahlung
Radiation loss in lead.
http://pdg.lbl.gov
10
Interactions of ElectronsInteractions of Electrons
Bremsstrahlung and Pair production
Note that the mean free path for photons for pair production is very similar to X0
for electrons to radiate Bremsstrahlung radiation:
This fact is not coincidence, as pair production and Bremsstrahlung have very
similar Feynman diagrams, differing only in the directions of the incident
and outgoing particles (see Fernow for details and diagrams).
In general, an electron-positron pair will each subsequently radiate a photon by
Bremsstrahlung which will produce a pair and so forth shower development.
Interactions of Particles with Matter
07
9Xλpair
11
Interactions of Particles with MatterInteractions of ElectronsInteractions of Electrons
Bremsstrahlung (Critical Energy)Another important quantity in calorimetry is the so called critical energy. One definition is that it is the energy at which the loss due to radiation equals that due to ionization. PDG quotes the Berger and Seltzer:
21
800
.Z
MeVEc
http://pdg.lbl.govOther definition Rossi
ionc
Bremcc )(E
dx
dE)(E
dx
dEE
12
Interactions of Particles with MatterSummary of the basic EM interactionsSummary of the basic EM interactions
e+ / e-
Ionisation
Bremsstrahlung
P.e. effect
Comp. effect
Pair production
E
E
E
E
E
g
dE
/dx
dE
/dx
s
s
s
Z
Z(Z+1)
Z5
Z
Z(Z+1)
13
Electromagnetic Shower DevelopmentDetecting a signal:
The contribution of an electromagnetic interaction to energy loss usually depends on the energy of the incident particle and on the properties of the absorber
At “high energies” ( > ~10 MeV): electrons lose energy mostly via Bremsstrahlung photons via pair production
Photons from Bremsstrahlung can create an electron-positron pair which can radiate new photons via Bremsstrahlung in a process that last as long as the electron (positron) has energy E > Ec
At energies E < Ec , energy loss mostly by ionization and excitation
Signals in the form of light or ions are collected by some readout system
Building a detector
X0 and Ec depends on the properties of the absorber material
Full EM shower containment depends on the geometry of the detector
14
Electromagnetic Shower DevelopmentA simple shower model A simple shower model ((Rossi-HeitlerRossi-Heitler))
Considerations:
Photons from bremsstrahlung and electron-positron from pair production produced
at angles = mc2/E (E is the energy of the incident particle) jet character
Assumptions:
λpair X0
Electrons and positrons behave identically
Neglect energy loss by ionization or excitation for E > Ec
Each electron with E > Ec gives up half of its energy to bremsstrahlung photon after 1X0
Each photon with E > Ec undergoes pair creation after 1X0 with each created particle receiving half of the photon energy
Shower development stops at E = Ec
Electrons with E < Ec do not radiate remaining energy lost by collisions
B. Rossi, High Energy Particles, New York, Prentice-Hall (1952) W. Heitler, The Quantum Theory of Radiation, Oxford, Claredon Press (1953)
15
Electromagnetic Shower DevelopmentA simple shower modelA simple shower model
Shower development:
Start with an electron with E0 >> Ec
After 1X0 : 1 e- and 1 , each with E0/2
After 2X0 : 2 e-, 1 e+ and 1 , each with E0/4.. After tX0 :
Maximum number of particles reached at E = Ec
[ X0 ]t
tt
EE(t)
eN(t)
2
2
0
2ln
Number of particles increases exponentially with t equal number of e+, e-,
E
E)EN(E
EE)Et(
0
0
2ln
12ln
ln Depth at which the energy of a shower particle equals some value E’ Number of particles in the shower with energy > E’
ct
c
EEeN
EEt
02ln
max
0max
max
2ln
ln
16
Electromagnetic Shower DevelopmentA simple shower modelA simple shower model
Concepts introduce with this simple mode:
Maximum development of the shower (multiplicity) at tmax
Logarithm growth of tmax with E0 : implication in the calorimeter longitudinal dimensions
Linearity between E0 and the number of particles in the shower
17
Electromagnetic Shower DevelopmentA simple shower modelA simple shower model
What about the energy measurement?
Assuming, say, energy loss by ionization
Counting charges:
Total number of particles in the shower:
Total number of charge particles (e+ and e- contribute with 2/3 and with 1/3)
c
ttt
t
tall E
EN 0
0
2221222 maxmax
max
ccee E
E
E
EN 00
3
42
3
2 Measured energy proportional to E0
18
Electromagnetic Shower DevelopmentA simple shower modelA simple shower model
What about the energy resolution?
Assuming Poisson distribution for the shower statistical process:
Example: For lead (Pb), Ec 6.9 MeV:
More general term:
EE
E
E
E
NE
Ec
ee
1)(
23
1)(
][
27
GeVE
%.
E
σ(E)
Resolution improves with E
E
cb
E
a
E
E
)(
Statistic fluctuations Constant term(calibration, non-linearity, etc
Noise, etc
19
Electromagnetic Shower DevelopmentA simple shower modelA simple shower model
Simulation of the energy deposit in copper as a function of the shower depth for incident electrons at4 different energies showing the logarithmic dependence of tmax with E.
EGS4* (electron-gamma shower simulation)
*EGS4 is a Monte Carlo code for doing simulations of the transport of electrons and photons in arbitrary geometries.
Longitudinal profile of an EM shower
Number of particle decreases after maximum
copper
20
Electromagnetic Shower DevelopmentA simple shower modelA simple shower model
Though the model has introduced correct concepts, it is too simple:
Discontinuity at tmax : shower stops no energy dependence of the cross-section
Lateral spread electrons undergo multiple Coulomb scattering
Difference between showers induced by and electrons
λpair = (9/7) X0
Fluctuations : Number of electrons (positrons) not governed by Poisson statistics.
21
Electromagnetic Shower DevelopmentShower ProfileShower Profile
Longitudinal development governed by the radiation length X0
Lateral spread due to electron undergoing multiple Coulomb scattering:
About 90% of the shower up to the shower maximum is contained in a cylinder of radius < 1X0
Beyond this point, electrons are increasingly affected by multiple scattering
Lateral width scales with the Molière radius ρM MeV 21,20 s
c
sM Ecmg
E
EX
95% of the shower is contained laterally in a cylinder with radius 2ρM
22
Electromagnetic Shower DevelopmentShower profileShower profile
From previous slide, one expects the longitudinal and transverse developments to scale with X0
Transverse development10 GeV electron
Longitudinal development10 GeV electron
ρM less dependent on Z than X0: ZAZEZAX Mc 1,20
EGS4 calculationEGS4 calculation
23
Electromagnetic Shower DevelopmentShower profileShower profile
Different shower development for photons and electrons
EGS4 calculation
At increasingly depth, photons carry larger fraction of the shower energy than electrons
24
Electromagnetic Shower DevelopmentEnergy depositionEnergy deposition
The fate of a shower is to develop, reach a maximum, and then decrease in number of particles once E0 < Ec
Given that several processes compete for energy deposition at low energies, it is important to understand how the fate of the particles in a shower.
Most of energy deposition by low energy e±’s.
e± (< 4 MeV)
40%
60%
e± (< 1 MeV)
e± (>20 MeV)
Ionization dominates
EGS4 calculation
25
Electromagnetic CalorimetersHomogeneous CalorimetersHomogeneous Calorimeters
Only one element as passive (shower development) and active (charge or light collection) material
Combine short attenuation length with large light output high energy resolution
Used exclusively as electromagnetic calorimeter
Common elements are: NaI and BGO (bismuth germanate) scintillators, scintillating, glass, lead-glass blocks (Cherenkov light), liquid argon (LAr), etc
Material PropertiesX0(cm) ρM (cm) a(cm) light energy resolution (%) experiments
NaI 2.59 4.5 41.4 scintillator 2.5/E1/4 crystal ballCsI(TI) 1.85 3.8 36.5 scintillator 2.2/E1/4 CLEO, BABAR, BELLELead glass 2.6 3.7 38.0 cerenkov 5/E1/2 OPAL, VENUSa= nuclear absorption length
26
Electromagnetic CalorimetersHomogeneous CalorimetersHomogeneous Calorimeters
BaBar EM Calorimeter
CsI as active/passive material Total of ~6500 crystals
EM calorimeter
27
Electromagnetic CalorimetersSampling CalorimetersSampling Calorimeters
Consists of two different elements, normally in a sandwich geometry:
Layers of Active material (collection of signal) – gas, scintillator, etc
Layers of passive material (shower development)
Segmentation allows measurement of spatial coordinates
Can be very compact simple geometry, relatively cheap to construct
Sampling concept can be used in either electromagnetic or hadronic calorimeter
Only part of the energy is sampled in the active medium.
Extra contribution to fluctuations
28
Electromagnetic CalorimetersSampling CalorimetersSampling Calorimeters
Typical elements:
Passive: Lead, W, U, Fe, etc Active: Scintillator slabs, scintillator fibers, silicon detectors, LAr, LXe, etc.
Active mediumPassive medium
hadronic
EM
EE
EE
E
E
)%8035(
)%255.7(
Typical Energy Resolution
29
Electromagnetic CalorimetersSampling CalorimetersSampling Calorimeters
ATLAS LAr Accordion Calorimeter
Ionization chamber: 1 GeV E-deposit 5 x106 e-
Accordion shape Complete Φ symmetry without azimuthal cracks better acceptance.
Test beam results, e- 300 GeV (ATLAS TDR)
cEbEaEE /)( Spatial and angular uniformity 0.5%
Spatial resolution 5mm / E1/2
30
Electromagnetic CalorimetersSampling CalorimetersSampling Calorimeters
Sampling fraction:
The number of particles we see: Nsample
0Xd
NN ee
sample
detectors absorbers
d
d = distance between active plates
Sampling fluctuation:
E
d
NE sample
sample 1 The more we sample, the better is
the resolution