Interaction of charged (heavy*) particles in matter

* exclude e- (very light showers)

Lecture 2

Dominant process for energy loss is the electromagnetic interaction with electrons of atoms.

Interaction with nuclei negligible for energy loss, but they play a role in the angular deflection (multiple scattering).

Particle M: mass, v: velocity (v=βc)Material refractive index: n, dielectric constant: ε = ε1 + i ε2 , 1

2 n

dispersion absorption

C

p

EP

kP

'

''

p

EP

Virtual or real photon

Assumption: pkEEM

1) Energy-momentum conservation: Cvkexercise cos...

2) Dispersion relation:

222 ck 1cos C

c

v

With: n

cccm

and

m

mc

v mC /1cos

3 Regions, depending on the photon:

1) Below excitation energy (optical region)

Cherenkov radiation1

1,

mC forreal

real

2) Excitation energy (2eV … 5keV) (resonance region)

Photons are virtual, no radiation but ionization by exchange of virtual photons.

21 i complex with in general, can oscillate >11,0 12

3) Above excitation energy (X-ray region) Little absorption:

Photons are virtual, no radiation except at discontinuitiesTransition radiation

1,1 12

3 Regions, depending on the photon energy:

2

1 10 100 1K 10K 100K Photon energy [eV]

1

1 10 100 1K 10K 100K Photon energy [eV]

1

Optical Resonance X-ray

Cherenkov radiation

Ionization Transition radiation

Re(ε)>1, Im(ε)=0

Re(ε)>oscillates, Im(ε)>0

Re(ε)<1, Im(ε) ->0(vacuum properties, ε=1)

Transverse range of virtual photons relativistic riseIn the resonance region: What is the transverse range of the virtual photons?

How far can they ionize?

Dispersion relation:

222 ck

2-dimensionl (z: longitudinal, y=transverse) from: v

kz

Cvk cos

2

2

2

2

2

2

2

22

2

22

2

2222

1

1

11

m

m

my

zyzy

with

vc

v

vk

vck

ck

ckkk

2 cases:

1) (faster than speed of light in medium, water, quartz,…) ky and kz

real -> a real wave (Cherenkov radiation) 1m

2) (slower than speed of light in medium, gas) ky imaginary damping

Attenuation length:

Attenuation length increases linearly with !->Expansion of transverse field!-> relativistic rise in energy loss!

1m

0

exp)(exp)(expy

yvtz

vitrki

)1(11

1

2

2

20

c

k

v

k

iy mm

my

)(exp trki

mm

Transverse range of virtual photons, relativistic rise:

2)

2 cases:

)1(1 2

2

0

c

y

(optical region) because for the optical region.

If y0 increases until where Cherenkov radiationsets in

11 1 02

1m 1m 0y

(above ionization threshold)

y0 increases with until up to a maximum:

This is the “relativistic rise” increased transverse range -> interaction with more atoms effect saturates when:

11

m

1

1

)1(1

max

0

2

2

0

cy

cy

Transverse range of virtual photons, relativistic rise:

Fermi-Plateau

This saturation (described by ε) is classically due to the polarisation of the medium, which will screen the effect of remote atoms.

This effect is larger in denser media

“Density effect” -> see density correction in Bethe-Bloch

1

1

1)( sat

1)( sat

Allision-Cobb equation for energy loss of charged heavy particles

Differential cross-section per electron, per energy loss dE (=photon energy) of the charged particle (photo absorption model)

)(1

')'(1

2ln

)(

)1(

1ln

)(

2

12

2

0

22

22

2

2

2

42

1

22

cZN

dEZ

E

E

E

mc

EZ

E

EZ

E

dE

d

E

Relativistic rise; polarization effect; saturation

Rutherford scattering; Photoelectric emission of electron from atomic level.

Compton scattering; Prod. of “δ-ray” emission (knock-on electrons)

Cherenkov & transition radiation.

Fine structure constantPhoton absorption cross section for energy ERelative velocityComplex dielectric constantAtomic numberAtomic density

1371)(E

cv

21

ZANN 0

)1arg( 2

2

2

1 i

Observations:1) Optical region -> Cherenkov (transition) radiation only!2) kinematic factor, slower particles has more time to interact !3) Non-relativistic theory (slow particles) only term II. and III.

Energy loss through ionization described by first 3 terms integration of the first two (neglect δ-rays) over dE between I (average ionization potential) and maximum energy of struck electron, gives Bethe-Bloch formula:

021

2222 cmT eMax

2

2

22

2

22

)1(

4ln

4

I

cm

A

ZNcmr

dX

dE eAee

:mass per areaxX

2gcm

dx

dE

dX

dE

Bethe-Bloch formulaMean rate of energy loss through ionization:

2

2ln

2

11 2

2

222

2

I

Tcm

A

ZK

dX

dE Maxe

Constant

Classical electron radius

Atomic number

Atomic massAverage ionization potential

224 cmrNK eeA

2

0

2

4 cm

er

e

e

Z

A

I

222

2

222

2)/(/21

2

cm

MmMm

cmT e

ee

eMax

(max. kin. Energy transferred to free electron in single collision)

emM 2

2)1(

2ln

1 2

2

22

2

I

cm

A

ZK

dX

dE e

Bethe-Bloch formulaDiscussion:• Function of β only independent of M! -> dE/dX used for particle

identification. P can be measured in B-field -> v from dE/dX -> M calculated• Valid for βc larger than that of orbital electrons (~Zαc), small enough that

radiative effects do not dominate.• Dependence on material small ~Z/A , except for hydrogen• I Mean excitation energy, trickiest part!

For elements heavier than oxygen.

Minimum ionization atIn typical high energy physics experiments, most particles are minimum

ionizing (MIP) or above.Relativistic rise from extension of transverse range counteracted by,Density effect limitation of transverse range through polarization of medium

large correction for high density 50%-70% relativistic rise for noble gases much less for liquids and solids.

ZeVI )110(

95.0,6.32.3)(1,5.33 2

Energy loss in Si

Minimal ionizing 1 Mip = on average 32500 e- - hole pairs in 300 μm Si

Proton heavier than μ-> βγ smallerfor same E

Cherenkov radiation detectorA charged particle with velocity β in a medium with refractive index n may emit light

along a conical wave front.

The angle (Cherenkov cone half angle) of emission is given by:

And the number of photo electrons detected in a given device:

The detection efficiency and the Cherenkov angle depend on the photon energy E. As the typical energy dependent variation of n is small, the integral can be split into the mean value for the sin2θ and the detection efficiency.

)(

1cos

nC

dEEEcmr

zLN C

ee

PE )(sin)( 2

2

22

ε(E): Efficiency for collection and

detection of Cherenkov lightL: Length of the detector radiatorn(λ): Wavelength dependant refractive indexN0: Detector quality factor

dEcmr

zN

ee

2

22

0CPE LNN 2

0 sin with

Cherenkov detectors

Cherenkov angle for different radiators in LHCb RICH

• Each radiator is optimized for a different momentum region

• Particle identification PID is based on the measurement of the Cherenkov angle.

CC

tan

Small θC ensures good velocityResolution (i.e. mass) resolution.But, NPE grows with θC and ΔθC , so The detector is inefficient if they are tooSmall.

RICH in LHCb

Transition radiation detector TRD

• Transition radiation is electromagnetic radiation emitted when a charged particle traverses a medium with a discontinuous refractive index, e.g. the boundaries between vacuum and a dielectric layer.

• Radiation energy per medium to vacuum boundary transition

• Build a detector with many transitions (foil stack) to enhance signal. Typical photon energy is in the x-ray region. Angular emission is (stay close to track).

• Particle must traverse a minimum distance, the so-called formation zone Zf in order to efficiently emit TR , Zf(air)~mm, Zf(CH2)~20um, important for detector radiator design.

PW 3

1 W

Only high energetic e+/- emit TR of detectable intensity and can be used for PID.

e

eP

m

eN

0

2

Plasma frequency ( for plastic radiators)eVP 20

/1

TRD Xenon straw tubes for x-ray detection, traverse the full length

Radiator, fleece based

Particle

fleece removed for illustration