calorimeters chapter 4 chapter 4 electromagnetic showers

Calorimeters Chapter 4

Chapter 4

Electromagnetic Showers


Basic Considerations

In the previous sections all relevant electromagneticInteractions were introduced

For Electrons/Positrons

a) Ionizationb) Bremsstrahlung(Annihilation for Positrons)

For Photons

a) Photoelectric Effectb) Compton Scatteringc) Pair Production

A primary particle looses energy due to one of theseInteractions in a given absorber materialThe produced secondaries in turn do interact with theMaterial and produce themselves further particles

Development of an Electromagnetic Cascade


A Very Simple Shower

Relevant for photons in the MeV Range -> Nuclear Physics Nuclear With E = 3370 keVIn 65Ga

A

B

CC

D

3378

200

511

1174

1174511

200

A) Pair Prodruction e- and e+: continous loss by ionizationB) Annihilation of e+

C) Compton Scattering D) Photoelectric absorption e+

E-


High Energetic Particles

Energy loss of electron by Bremsstrahlung:Photons convert into e+e--Pairs

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dE

dx= −EeX0

⇒ E = E0e−x /X 0

t=x/X0

Simple Shower Model (see Longo for detailed discussion) - Energy Loss after t: E1 =Eo/2

- Photons -> materialize after t E±=E1/ 2

Number of particles after t: N(t) = 2t

Each Particle has energy

Shower continues until particles reach critical energy (see p. 16) wheretmax = ln(E0/c)

Shower Maximum increases logarithmically with Energy of primary particle(Important for detector design !!!)

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E =E0

N(t)= E02−t ⇒ t = ln

E0

E

⎛

⎝ ⎜

⎞

⎠ ⎟/ln2

For high energetic particlesShower contain thousandsOf particles


Shower Depth

GEANT4 Simulation of e- in copper

Shower MaximumIncreases Logarithmically withEnergy

~3.5*log10(E)

In qualitative agreement with oursimple model


Energy Distribution of Shower Particles

e < 4 MeV

e < 1 MeV

e > 20 MeV

Shower containsmostly of soft particleswith energy < 4 MeV

Compton Electrons and Photo electrons

Increase with Z for constant energy

Reason: Number of soft photonsincrease as showerDevelopsMajority of shower photonsIn Compton and

Photo Effect Regime


Scaling Variables I - Radiation Length

(Not only) For Bremsstrahlung:

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−dE

dx

⎛

⎝ ⎜

⎞

⎠ ⎟rad

= N Eγdσ raddEγ

dEγ0

Eγ 0

∫

Spectrum of radiated photons:

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dσ raddEγ

~1

Eγf (Z)

Define:

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Φrad =1

E0

N Eγdσ raddEγ

dEγ0

Eγ 0

∫ ⇒ −dE

dx= NE0Φrad

Φrad is independent of the energy of the radiatedPhoton and only a function of the the material

Radiation Length X0: Distance after which a particle has lost 1/e due to radiation

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E = E0 exp −x

X0

⎛

⎝ ⎜

⎞

⎠ ⎟ With X0 = 1/NΦrad

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€

X0 =A

4αNAZ(Z +1)re2 ln(183Z−1/ 3)

[g /cm2]

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μ(E >> mec2) =

28

9nZ 2αre

2 ln183

Z1 / 3=

7

9X0

−1

Absorption Coefficient Photons

i.e. Photons travel a longer distance before they interact

Some Values:X0,Air=30 420 cmX0,Al = 8.9 cmX0,Pb = 0.56 cm


Longitudinal Shower Profiles

Addendum: X0 Characterizes longitudinal Extension of elm. Shower

.

Shape of Shower profilesMaterial Independent =Energy loss Independent of material

.. as function of X0

Differences:- Pair Production by emitted Photons increases with Z and extend to much lower energies- Critical Energy Z-dependent e.g. 43 MeV for Al 7 MeV for Lead - X0 looses meaning at low energies


Composition of longitudinal Profile


Lateral Shower Extensions2 Effects1) Electrons (and positrons) undergo multiple scattering in the Coulomb Field of absorber nuclei mean = 0 with standard deviation

rmsPrimary e

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rms = Θ =13.6

βcp

x

X0

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For projected Scattering angle distribution

2) Isotropic production of secondaries by Photoelectric effect and Compton Scattering Low energetic component of shower Expect different behaviour in region ‘far’ away from shower axis

Transversal extension of shower characterized by Molière Radius

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ρM = mc 2 4π /αX0

εC= 21.2MeV

X0

εc90% of shower energy is contained within ρM

Weak/no Z dependence

Typical Values for c, ρM: c[MeV] RM

[cm]Pb 7.2 1.6 NaJ 12.5 4.4Air 87 7400


Lateral Shower Profiles

- Core (I.e. region close to the shower axis is ‘empty’ in early stages of shower Multiple Scattering of High energetic particles

- Core gets ‘populated’ By low energetic component As for long. profile

10 Gev e- in Cu

Exponential behaviour with two slopes in radial distribution:Steep falloff close to the shower axis - high energetic componentLess steep falloff far from shower axis - low energetic component


Composition of Lateral Profile

calorimeters chapter 4 chapter 4 electromagnetic showers

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