calorimetry - 2

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Calorimetry - 2Calorimetry - 2

Mauricio BarbiUniversity of Regina

TRIUMF Summer InstituteJuly 2007

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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

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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:

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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


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

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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

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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


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


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

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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

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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)

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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

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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)

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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

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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

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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

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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

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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

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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.

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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

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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

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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

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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

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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

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Electromagnetic CalorimetersHomogeneous CalorimetersHomogeneous Calorimeters

BaBar EM Calorimeter

CsI as active/passive material Total of ~6500 crystals

EM calorimeter

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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

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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

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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

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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

calorimetry - 2

Documents

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