1.3 Interaction of Radiation with Matter 1 By Archana SHARMA CERN Geneva Switzerland March 2009 Troisieme Cycle EPFL Lausanne, Switzerland Gaseous Particle.

1.3 Interaction of Radiation with Matter1

Gaseous Particle Detectors:

1.3 Interaction of Radiation with Matter

Chapter I 5th March 2009 1.1 Introduction 1.2 Units and Definitions, Radiation Sources 1.3 Interaction of Radiation with Matter

Chapter II 12th March 2009 2.1 General Characteristics of gas detectors, Electronics for HEP detectors2.2: Transport Properties 2.3: Wire-based Detectors

Tool

Interaction with atomic electrons. Particle loses energy; atoms are excited or ionized.

Interaction with atomic electrons. Particle loses energy; atoms are excited or ionized.

Interaction with atomic nucleus. Particle undergoes multiple scattering. Could emit a bremsstrahlung photon.

Interaction with atomic nucleus. Particle undergoes multiple scattering. Could emit a bremsstrahlung photon.

If particle’s velocity is greater than the speed of light in the medium -> Cherenkov Radiation. When crossing the boundary between media, ~1% probability of producing a Transition Radiation X-ray.

If particle’s velocity is greater than the speed of light in the medium -> Cherenkov Radiation. When crossing the boundary between media, ~1% probability of producing a Transition Radiation X-ray.

Electromagnetic Interaction of Particles with MatterElectromagnetic Interaction of Particles with Matter

Material with atomic mass A and density ρ contains n atoms

Material with atomic mass A and density ρ contains n atoms

Probability, p of incoming particle hitting an atom

Probability, p of incoming particle hitting an atom

A volume with surface S and thickness dx contains N=nSdx atoms

A volume with surface S and thickness dx contains N=nSdx atoms

Probablity that a particle hits exactly one atom between x and (x + dx)

Probablity that a particle hits exactly one atom between x and (x + dx)

Mean free path Mean free path Average collisions/cmAverage collisions/cm

S

dx

Cross-sectionCross-section

Differential cross-section is the cross-section from an incoming particle of energy E to lose an energy between E and E’

Differential cross-section is the cross-section from an incoming particle of energy E to lose an energy between E and E’

Total cross-sectionTotal cross-section

Probability (P(E)) that a particle of energy, E, loses between E’ and E’ + dE’ in a collision

Probability (P(E)) that a particle of energy, E, loses between E’ and E’ + dE’ in a collision

Average number of collisions/cm causing an energy loss between E’ and E’+dE’

Average number of collisions/cm causing an energy loss between E’ and E’+dE’

Average energy loss per cmAverage energy loss per cm

Differential Cross-sectionDifferential Cross-section

Linear stopping power (S) is the differential energy loss of the particle in the material divided by the differential path length. Also called the specific energy loss.


Part

icle

Data

G

roup

Stopping Power of muons in Copper

Bethe-Bloch Formula

Energy loss through ionization and atomic excitation


Stopping Power Stopping Power



Part

icle

Data

G

roup

Stopping Power of muons in Copper

Bethe-Bloch Formula



Stopping Power Stopping Power


Bethe-Bloch Formula

Describes how heavy particles (m>>me) lose energy when travelling through a material

Exact theoretical treatment difficultAtomic excitationsScreeningBulk effects

Phenomenological description

Describes how heavy particles (m>>me) lose energy when travelling through a material

Exact theoretical treatment difficultAtomic excitationsScreeningBulk effects

Phenomenological description

Part

icle

Data

G

roup

Bethe-Bloch Formula

m – electronic massv – velocity of the particle (v/c = )N – number density of atomsI – ‘Effective’ atomic excitation energy – average value found empiricallyGas is represented as a dielectric medium through which the particle propagates And probability of energy transfer is calculated at different energies – Allison Cobb


ze

Ze

br

θx

y

A very roughBethe-Bloch Formula

Consider particle of charge ze, passing a stationary charge ZeAssume

Target is non-relativisticTarget does not move

CalculateMomentum transfer Energy transferred to target

Consider particle of charge ze, passing a stationary charge ZeAssume

Target is non-relativisticTarget does not move

CalculateMomentum transfer Energy transferred to target

11

2

0

1

2x

Zzep dtF

c b

Projectile force

Change of momentum of target/projectile

Energy transferred

2 23

2 20 0

cos cos4 4x

Zze ZzeF

r b

2 2 2 4

2 2 20

1

2 2 (2 ) ( )

p Z z eE

M M c b

Bethe-Bloch Formula

Consider α-particle scattering off Atom– Mass of nucleus: M=A*mp

– Mass of electron: M=me

But energy transfer is

Energy transfer to single electron is

2 2 2 4 2

2 2 20

1

2 2 (2 ) ( )

p Z z e ZE

M M c b M

2 4

2 2 2 20

2 1( )

(4 )ee

z eE b E

m c b

Bethe-Bloch Formula


Energy transfer is determined by impact parameter b

Integration over all impact parametersb

dbze

2 (number of electrons / unit area )

=2 A

dnb

dbN

b Z xA

Bethe-Bloch Formula

1.3 Interaction of Radiation with MatterDec 2008 Alfons Weber 14

There must be limits Dependence on the material is in the calculation

of the limits of the impact parameters

max

max

min

min

max

min

2 2

2

2 2

2

2

20

dd ( ) 2 ln

d

ln

with 24

bbe

e bb

EeE

Ae

m cn ZzE b E b C x b

b A

m c ZzC x E

A

eC N

m c

Bethe-Bloch FormulaCalculate average energy loss


Simple approximations for– From relativistic kinematics

– Inelastic collision

Results in the following expression

min 0 average ionisation energyE I

2 2 2 22

20

22 lne em c m cE ZzC

x A I

2 2 22 2 2

max 2

22

1 2

ee

e e

m cE m c

m mM M

Bethe-Bloch Formula


This was a very simplified derivation– Incomplete– Just to get an idea how it is done

The (approximated) true answer is

with– ε screening correction of inner electrons– δ density correction (polarisation in medium)

2 2 2 222max

2 20

21 ( )2 ln

2 2 2e em c m c EE Zz

Cx A I

Bethe-Bloch Formula


Energy Loss FunctionEnergy Loss Function

1 10 100 1000 10000

1.6

1.5

1.4

1.3

1.2

1.1

Minimum ionizing particles (mips)

Relativistic Rise

Fermi Plateau

Rel

To m

ips


Diff

ere

nt

Mate

rials

Diff

ere

nt

Mate

rials


Different Materials (2)Different Materials (2)

20

Average Ionisation EnergyAverage Ionisation Energy

Few eV to few tens of eVFew eV to few tens of eV


Density Correction depends on material

with– x = log10(p/M)

– C, δ0, x0 material dependant constants

Density CorrectionDensity Correction2 2 2 22

2max2 2

0

21 ( )2 ln

2 2 2e em c m c EE Zz

Cx A I


Part

icle

Range/S

topp

ing

Pow

er

Part

icle

Range/S

topp

ing

Pow

er


Energy-loss in Tracking ChambersEnergy-loss in Tracking Chambers

The Bethe Bloch Formula tool for Particle IdentificationThe Bethe Bloch Formula tool for Particle Identification


Mean energy loss Actual energy loss will scatter around the

mean valueDifficult to calculate

– parameterization exist in GEANT and some standalone software libraries

– Form of distribution is important as energy loss distribution is often used for calibrating the detector

StragglingStraggling


Energy LossIs a statisticalprocess

Simple parameterisation– Landau function

2

2

1 1( ) exp ( )

22

with e

f e

E E

m c ZzC x

A





Energy loss distribution is not Gaussian around mean.In rare cases a lot of energy is transferred to a single electron

If one excludes δ-rays, the average energy loss changesEquivalent of changing Emax

δ-raysδ-rays


Some detectors only measure energy loss up to a certain upper limit Ecut

– Truncated mean measurement– δ-rays leaving the detector

2 2 2 22

2 20

2

max

212 ln

2

( ) 1

2 2

cut

e e cut

E E

cut

m c m c EE ZzC

x A I

E

E

Restricted dE/dxRestricted dE/dx


Electrons are different light– Bremsstrahlung– Pair production

ElectronsElectrons


Multiple ScatteringMultiple Scattering

Particles not only lose energy …

but also they also change direction


Average scattering angle is roughly Gaussian for small deflection angles

With

Angular distributions are given by

00 0

0

13.6 MeV1 0.038ln

radiation length

x xz

cp X X

X

2

2 20 0

2

200

1exp

2 4

1exp

22

space

plane

plane

dN

d

dN

d

Multiple ScatteringMultiple Scattering


Multiple scattering and dE/dx are normally treated to be independent from each

Not true– large scatter large energy transfer– small scatter small energy transfer

Detailed calculation is difficult, but possible– Allison & Cobb

Correlation bet dE/dx and MSCorrelation bet dE/dx and MS

Integrate the Bethe-Bloch formula to obtain the range

Integrate the Bethe-Bloch formula to obtain the rangeUseful for low energy hadrons and muons with momenta below a few hundred GeV

Useful for low energy hadrons and muons with momenta below a few hundred GeV

Radiative Effects important at higher momenta. Additional effects at lower momenta.

Radiative Effects important at higher momenta. Additional effects at lower momenta.

RangeRange

Electrons: bremsstrahlung

Photons: pair production

ppn

np

pn

n n

ppn

pn

e

γe

Characteristic amount of matter traversed for these interactions is the radiation length (X0)

Characteristic amount of matter traversed for these interactions is the radiation length (X0)

ppn

np

pn

n n

ppn

pn

e

e

γ

Presence of nucleus required for the conservation of energy and momentum

Presence of nucleus required for the conservation of energy and momentum

Photon and Electron InteractionsPhoton and Electron Interactions

also

Energy Loss in Lead

Radiation LengthRadiation Length

Mean distance over which an electron loses all but 1/e of its energy through bremsstralung

Mean distance over which an electron loses all but 1/e of its energy through bremsstralung

7/9 of the mean free path for electron-positron pair production by a high energy photon

7/9 of the mean free path for electron-positron pair production by a high energy photon

A charged particle of mass M and charge q=Z1e is deflected by a nucleus of charge Ze (charge partially shielded by electrons)

A charged particle of mass M and charge q=Z1e is deflected by a nucleus of charge Ze (charge partially shielded by electrons)The deflection accelerates the charge and therefore it radiates bremsstrahlung

The deflection accelerates the charge and therefore it radiates bremsstrahlung

Partial screening of nucleus by electrons

Energy Loss by electronsEnergy Loss by electrons

Elastic scattering of a nucleus is described byElastic scattering of a nucleus is described by

Energy loss through bremsstrahlung is proportional to the electron energy

Energy loss through bremsstrahlung is proportional to the electron energy

Ionization loss is proportional to the logarithm of the electron energy

Ionization loss is proportional to the logarithm of the electron energy

Critical energy (Ec) is the energy at which the two loss rates are equal

Critical energy (Ec) is the energy at which the two loss rates are equal

Electron in Copper: Ec = 20 MeVMuon in Copper: Ec = 400 GeV!

Electron in Copper: Ec = 20 MeVMuon in Copper: Ec = 400 GeV!

Electron Critical EnergyElectron Critical Energy

1. Atomic photoelectric effect

2. Rayleigh scattering3. Compton scattering of an

electron4. Pair production (nuclear

field)5. Pair production (electron

field)6. Photonuclear interaction

1. Atomic photoelectric effect

2. Rayleigh scattering3. Compton scattering of an

electron4. Pair production (nuclear

field)5. Pair production (electron

field)6. Photonuclear interaction

Light element:Carbon

Heavy element:Lead

At low energies the photoelectric effect dominates; with increasing energy pair production becomes increasingly dominant.

At low energies the photoelectric effect dominates; with increasing energy pair production becomes increasingly dominant.

Energy Loss by electronsContributing ProcessesEnergy Loss by electronsContributing Processes

photo electric cross section

Strong dependence of ZAt high energies ~ Z5

photo electric cross section

Strong dependence of ZAt high energies ~ Z5

Probability that a photon interaction will result in a pair production

Differential Cross-sectionDifferential Cross-section

Total Cross-sectionTotal Cross-section

Photon Pair ProductionPhoton Pair Production

What is the minimum energy for pair production?

What is the minimum energy for pair production?

A high-energy electron or photon incident on a thick absorber initiates an electromagnetic cascade through bremsstrahlung and pair production

A high-energy electron or photon incident on a thick absorber initiates an electromagnetic cascade through bremsstrahlung and pair production

Longitudinal Shower Profile

Longitudinal development scales with the radiation length

Longitudinal development scales with the radiation lengthElectrons eventually fall beneath critical energy and then lose further energy through dissipation and ionization

Electrons eventually fall beneath critical energy and then lose further energy through dissipation and ionization

Measure distance in radiation lengths and energy in units of critical energy

Measure distance in radiation lengths and energy in units of critical energy

Electromagnetic cascadesElectromagnetic cascades

Visualization of cascades developing in the CMS electromagnetic and hadronic calorimeters

Visualization of cascades developing in the CMS electromagnetic and hadronic calorimeters

Electromagnetic cascadesElectromagnetic cascades

For muons the critical energy (above which radiative processes are more important than ionization) is at several hundred GeV.

For muons the critical energy (above which radiative processes are more important than ionization) is at several hundred GeV.Ionization

energy loss

Pair production, bremsstrahlung and photonuclear

Mean rangeMean range

Muon Energy LossMuon Energy Loss

Muon critical energy for some elements

Critical energy defined as the energy at which radiative and ionization energy losses are equal.

Critical energy defined as the energy at which radiative and ionization energy losses are equal.

Muon Energy LossMuon Energy Loss

Luis Alvarez used the attenuation of muons to look for chambers in the Second Giza Pyramid

He proved that there are no chambers present

Luis Alvarez used the attenuation of muons to look for chambers in the Second Giza Pyramid

He proved that there are no chambers present

Muon Tomography

X-Ray Radiography for airport security


Signals in particle detectors are mainly due to ionisationAnd excitation in a sensitive medium – gasAlso:Direct light emission by particles travelling faster than

the speed of light in a medium– Cherenkov radiation

Similar, but not identical– Transition radiation

Signals from Particles in a Gas DetectorSignals from Particles in a Gas Detector


Moving charge in dielectric mediumWave front comes out at certain angle

1cos c n

slow fast

Cerenkov RadiationCerenkov Radiation


How many Cherenkov photons are detected?

22

2

2

2 2 2

0 2 2

( )sin ( )d

1( ) 1 d

11

with ( ) Efficiency to detect photons of energy

radiator length

electron radius

ce e

e e

e

zN L E E E

r m c

zL E Er m c n

LNn

E E

L

r

Cerenkov Radiation (2)Cerenkov Radiation (2)


Transition radiation is produced, when a relativistic particle traverses an inhomogeneous medium– Boundary between different materials

with different diffractive index n.

Strange effect– What is generating the radiation?– Accelerated charges

Transition RadiationTransition Radiation

50

22 vq

vacuummedium

Before the charge crosses the surface,apparent charge q1 with apparent transverse vel v1

After the charge crosses the surface,apparent charges q2 and q3 with apparent transverse vel v2 and v3

11 vq

33 qv

Transition Radiation (2)Transition Radiation (2)


Consider relativistic particle traversing a boundary from material (1) to material (2)

Total energy radiated

Can be used to measure

22 2

22 2 2 2 2 2 2

d 1 1

d d / 1/ 1/

plasma frequency

p

p

N z

Transition Radiation (3)Transition Radiation (3)

From Interactions to DetectorsFrom Interactions to Detectors




Multiwire Proportional ChamberMultiwire Proportional Chamber


Multiwire Proportional Chamberand derivativesMultiwire Proportional Chamberand derivatives


Key Points: Lecture 1-3 Energy loss by heavy particlesMultiple scattering through small anglesPhoton and Electron interactions in

matterRadiation LengthEnergy loss by electronsCritical EnergyEnergy loss by photonsBremsstrahlung and pair production

Electromagnetic cascadeMuon energy loss at high energyCherenkov and Transition Radiation


Exercise: Lecture 1-3 • Estimate the range of 1 MeV alphas in

• Aluminium

• Mylar

• Argon

• Indicate major interaction processes in:

• 1 MeV in Al

• 10 MeV in Argon

• 100 keV in Iron

• 1 MeV in Al

1.3 Interaction of Radiation with Matter 1 By Archana SHARMA CERN Geneva Switzerland March 2009 Troisieme Cycle EPFL Lausanne, Switzerland Gaseous Particle.

Documents

matter slide

interaction of radiation

average energy loss

energy atoms

incoming particle of

differential energy

power slide

target slide