Interaction of Particles with Matter Alfons Weber STFC & University of Oxford Graduate Lecture 2009

Interaction of Particleswith Matter

Alfons WeberSTFC & University of Oxford

Graduate Lecture 2009

Dec 2009 Alfons Weber 2

Table of Contents Bethe-Bloch Formula

Energy loss of heavy particles by Ionisation Multiple Scattering

Change of particle direction in Matter Cerenkov Radiation

Light emitted by particles travelling in dielectric materials

Transition Radiation Light emitted on traversing matter boundary



Bethe-Bloch Formula

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

Exact theoretical treatment difficult Atomic excitations Screening Bulk effects

Simplified derivation ala MPhys course Phenomenological description


Bethe-Bloch (1) Consider particle of charge ze, passing a

stationary charge Ze

Assume Target is non-relativistic Target does not move

Calculate Momentum transfer Energy transferred to target

ze

Ze

br

θx

y


Bethe-Bloch (2)

2

0

1

2x

Zzep dtF

c b

Force on projectile

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 (3) 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 (4) Energy transfer is determined by impact

parameter b Integration over all impact parameters

bdb

ze

2 (number of electrons / unit area )

=2 A

dnb

dbN

b Z xA


Bethe-Bloch (5) Calculate average energy loss

There must be limits material dependence is in the calculation

of the limits

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 (6) 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 (7) This was just a 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


Energy Loss Function

/ stopping powerE

x


Average Ionisation Energy


Density Correction

Density Correction does depend on material

with x = log10(p/M)

C, δ0, x0 material dependant constants


Different Materials (1)


Different Materials (2)


Particle Range/Stopping Power


Energy-loss in Tracking Chamber


Straggling (1) So far we have only discussed the mean

energy loss Actual energy loss will scatter around the

mean value Difficult to calculate

parameterization exist in GEANT and some standalone software libraries

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


Straggling (2) Simple parameterisation

Landau function

Better to use Vavilov distribution

2

2

1 1( ) exp ( )

22

with e

f e

E E

m c ZzC x

A


Straggling (3)


δ-Rays 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 changes

Equivalent of changing Emax

δ-Ray


Restricted dE/dx Some detector 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


Electrons Electrons are different light

Bremsstrahlung Pair production



Multiple Scattering Particles don’t only loose energy …

… they also change direction


MS Theory 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


Correlations 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

Wade Allison & John Cobb are the experts


Correlations (W. Allison)

Example: Calculated cross section for 500MeV/c in Argon gas. Note that this is a Log-log-log plot - the cross section varies over 20 and more decades!

log kL

2

18

17

7

log kT

whole atoms at low Q2 (dipole region)

electrons at high

Q2

electrons backwards in

CM

nuclear small angle scattering (suppressed

by screening)

nuclear backward scattering in CM

(suppressed by nuclear form factor)

Log pL or energy transfer

(16 decades)

Log pT transfer (10 decades)

Log cross

section (30

decades)


Signals from Particles in Matter Signals in particle detectors are mainly

due to ionisation Gas chambers Silicon detectors Scintillators

Direct light emission by particles travelling faster than the speed of light in a medium

Cherenkov radiation Similar, but not identical

Transition radiation


Moving charge in dielectric medium Wave front comes out at certain angle

Cherenkov Radiation

1cos c n

slow fast


Cherenkov Radiation (2) How many Cherenkov photons are

detected?2

22

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


Different Cherenkov Detectors Threshold Detectors

Yes/No on whether the speed is β>1/n Differential Detectors

βmax > β > βmin

Ring-Imaging Detectors Measure β


Threshold Counter

Particle travel through radiator Cherenkov radiation


Differential Detectors

Will reflect light onto PMT for certain angles only β Selection


Ring Imaging Detectors (1)


Ring Imaging Detectors (2)


Ring Imaging Detectors (3) More clever geometries are possible

Two radiators One photon detector


Transition Radiation 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


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 (3)

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

Total energy radiated

Can be used to measure γ

22 2

22 2 2 2 2 2 2

d 1 1

d d / 1/ 1/

plasma frequency

p

p

N z


Transition Radiation Detector


ATLAS TRTracker

ATLAS Experimen

t

Inner Detector:pixel, silicon and straw

tubes

Combination of Central Tracker and TR for electron

identification


Atlas TRT (II)


Atlas TRT (III)

TRT senses ionisation transition radiation

only electron produce TR in radiator

e± / π separationElectrons with

radiator

Electrons without radiator

Bod -> J/Ko

s

High threshold hits


Table of Contents Bethe-Bloch Formula

Energy loss of heavy particles by Ionisation Multiple Scattering

Change of particle direction in Matter Cerenkov Radiation

Light emitted by particles travelling in dielectric materials

Transition radiation Light emitted on traversing matter boundary


Bibliography

This lecture http://www-pnp.physics.ox.ac.uk/~weber/teaching

PDG 2008 (chapter 27 & 28) and references therein

Especially Rossi Lecture notes of Chris Booth, Sheffield

http://www.shef.ac.uk/physics/teaching/phy311

R. Bock, Particle Detector Brief Book http://rkb.home.cern.ch/rkb/PH14pp/node1.html

Or just it!


Plea I need feedback! Questions

What was good? What was bad? What was missing? More detailed derivations? More detectors? More… Less…

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Interaction of Particles with Matter Alfons Weber STFC & University of Oxford Graduate Lecture 2009