3737

TheThe Cherenkov Cherenkov effecteffectA charged particle traveling in a dielectric mediumwith n>1 radiates Cherenkov radiationif its velocity is larger than thephase velocity of light v>c/n or β > 1/n (threshold)

The emission is due to an asymmetric polarizationof the medium in front and at the rear of the particle,giving rise to a varying electric dipole momentum.Some of the particle energy is converted into light. A coherent wave front isgenerated moving at velocity v at an angle Θc

If the media is transparent the Cherenkov light can be detected.If the particle is ultra-relativistic β~1 Θc = const and has max value

In water Θc = 43˚, in ice 41˚ nct

tnc

AC

ABc ββθ

1cos ===

Wave frontWave front

Charged particleAA

CC

BB

491=dx

dNγ

3838

TheThe Cherenkov Cherenkov effecteffectThe intensity of the Cherenkov radiation (number of photons per unit length ofparticle path and per unit of wave length)

Number of photons/L and radiation Wavelength depends on charge and velocity of particleSince the intensity is proportional to1/λ2 short wavelengths dominate

Using light detectors (photomultipliers) sensitive in 400-700 nm for an ideally100% efficient detector in the visible

Energy loss is about 104 lessthan 2 MeV/cm in water fromionization but directionaleffect

491=dx

dNγ

€

d2Ndxdλ

=4π 2z2e2

hcλ21− 1

n2β 2

=2πz2

λ2α sin2ΘC

α =2πe2

hc

€

dNγ

dx= dλ

λ1

λ2∫d2Nγ

dxdλ= 2πz2α sin2ΘC

dλλ2λ1

λ2∫ = 2πz2α sin2ΘC1λ12 −

1λ22

= 393z2α sin2ΘC photons / cm490 z2 sinΘc

λπ

λν

λπλλ

λchc

hE

dxd

Nd

cdE

d

dxd

Nd

dxdE

Nd

hh

22

2222

===

==

€

1λ1−1λ2

3939

Transition RadiationTransition Radiation

Radiation is emitted when a fast charged particle crosses the boundarybetween two media with different indices of refraction (even below Cherenkovthreshold)Radiation is due to a coherent superposition of radiation fields generated bypolarization of the medium (given a charge moving towards the 2 mediainterface it can be considered together with its mirror charge an electric dipolewhose field strength varies with time. The time dependent dipole field causesthe emission of electromagnetic radiation). Coherence is assured in a smallregion whose extension is called coherent length or formation zone. Anobservable amount of X-rays can be emitted when a particle with γ >>1 crossesthe boundary of a macroscopically thick medium. The number of emittedphotons can be enhanced by radiators consisting of several boundaries.

Theory: Ginzburg, Frank 1946

4040

Transition radiationTransition radiationWhen a particle with charge ze crosses the boundary between vacuum and amedium with plasma frequency ωp the energy radiated is

Where the plasma frequency of the gas of electrons of the material is

And the emitted photon energy is

Photons are emitted in a cone of half-aperture θ ∼ 1/γ and the emission ispeaked in the forward directionThe radiation yield drops fast for ω>γωp and diverges logaritmically at lowenergies. For a particle with γ = 1000 the radiated photons are in soft X-rayrange between 2-20 keV. The number of emitted photons above an energy isNγ ∝ z2 α. The radiation probability is of order of α = 1/137 hence the necessityto have many boundaries

€

ν p =ω p

2π=

Nee2

πme

€

hν p = h ZρNAe2

πmeA= 4πNAreh2c 2

ZρA

≈ 28.8eV ZρA

4141

Transition RadiationTransition RadiationWhen the interface is between 2 media of plasma frequencies ωp1,2 the energyradiated by the particle of charge ze at the boundary per unit solid angle andunit frequency is

Where θ is the angle between the particle and the emitted photon.Three regions can be identified as a function of γ:

1) γ << 1/Y1 ⇒ low yield

2) 1/Y1 << γ << 1/Y2 ⇒ log increase with γ (used for PID)

3) γ >> 1/Y2 ⇒ saturation (yield is constant)The total emitted energy in the forward direction is proportional to z2γ

€

d2WdνdΩ

≈z2hαπ

θ 21

γ−2 + θ 2 +Υ12 −

1γ−2 + θ 2 +Υ2

2

2

Υ1,2 =ω p,1,2

ω

€

W =dWd(hν)

d(hν )0

∞

∫ ≈ z2γ αh3

ν p,1 −ν p,2( )2

ν p,1 + ν p,2

€

d2Wd(hν)dΩ

≈z2α6π

γY1( )4

€

d2Wd(hν)dΩ

≈2z2απ

ln(γY1) −1[ ]

4242

Emission in radiators and TR detectorsEmission in radiators and TR detectorsFor a foil of thickness l1:Interference producesa threshold effect in γ

For foil thickness << the yield is strongly suppressed

For large ω in order to have radiation emission

When many thin layers of material are joined together the number of photonscan be increased and the limiting factor is reabsorption in the radiator itself. In aTR detector, to minimize the photoelectric cross section (∝Z5) materials withsmall Z (eg Li) must be chosen while the γ detectors can be gas detectors withlarge Z (eg Xe). TR detectors are characterized by a threshold

€

d2Wd(hν )dΩ

foil

= 4sin2 ϕ12

d2Wd(hν)dΩ

Interference term

€

ϕ1 ≈(γ−2 +Y1

2)ωl12c

€

Z1(ω) =2c

(γ−2 +Y12)ω

Formation zone

€

l1 ≥ Z1(ω) ≈2cγ 2

ω

4343

An example: calibration ofAn example: calibration ofMACRO TRDMACRO TRD

TRD exposed to a pion/electronbeam with various crossing angles.Number of hits in proportionalcounters vs γ of incident particles.Below the threshold onlyionization contributes, for103 < γ < 104 TR is the maincontribution and the number of hitsincreases logaritmically with γ. Forlarger values it saturates

4444

Suggested readingsSuggested readingsTextbooks:Konrad Kleinknecht, Detectors for Particle Radiation (2nd edition) Cap 1C. Leroy and PG Rancoita, Principles of Radiation Interaction in Matter and

detection, World Scientific, 2004 (Cap 2)M.S. Longair, High Energy Astrophysics, Third ed., Vol 1, Particles, photons

and their interactions (Cap 2-4)

Online:

http://pdg.lbl.gov/2002/passagerpp.pdf

R.K. Bock & A. Vasilescu - The Particle Detector BriefBook, Springer 1998http://physics.web.cern.ch/Physics/ParticleDetector/BriefBook/http://www.shef.ac.uk/physics/teaching/phy311/#bookshttp://www-physics.lbl.gov/%7Espieler/physics_198_notes_1999/index.htmlhttp://besch2.physik.uni-

siegen.de/%7Edepac/DePAC/DePAC_tutorial_database/grupen_istanbul/grupen_istanbul.html