Interaction of Particleswith Matter

Alfons WeberCCLRC & University of Oxford

Graduate Lecture 2004

Nov 2004 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

Nov 2004 3

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

Nov 2004 4

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

stationary charge Ze

Assume Target is non-relativistic Target does not move

Calculate Energy transferred to target (separate)

ze

Ze

br

θx

y

Nov 2004 5

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

Nov 2004 6

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

Nov 2004 7

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

Nov 2004 8

Bethe-Bloch (5) Calculate average energy loss

There must be limit for Emin and Emax

All the physics and material dependence is in the calculation of this quantities

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

Nov 2004 9

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

Nov 2004 10

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, because of polarisation

in medium

2 2 2 222max

2 20

21 ( )2 ln

2 2 2e em c m c EE Zz

Cx A I

Nov 2004 11

Energy Loss Function

Nov 2004 12

Average Ionisation Energy

Nov 2004 13

Density Correction

Density Correction does depend on material

with x = log10(p/M)

C, δ0, x0 material dependant constants

Nov 2004 14

Different Materials (1)

Nov 2004 15

Different Materials (2)

Nov 2004 16

Particle Range/Stopping Power

Nov 2004 17

Application in Particle ID Energy loss as measured in tracking

chamber Who is Who!

Nov 2004 18

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

Nov 2004 19

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

Nov 2004 20

Straggling (3)

Nov 2004 21

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

Nov 2004 22

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

Nov 2004 23

Electrons Electrons are different light

Bremsstrahlung Pair production

Nov 2004 24

Multiple Scattering Particles don’t only loose energy …

… they also change direction

Nov 2004 25

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 2

1exp

22

space

plane

plane

dN

d

dN

d

Nov 2004 26

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

Nov 2004 27

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)

Nov 2004 28

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

Nov 2004 29

Cherenkov Radiation (1) Moving charge in matter

at rest slow fast

Nov 2004 30

Wave front comes out at certain angle

That’s the trivial result!

Cherenkov Radiation (2)

1cos c n

Nov 2004 31

Cherenkov Radiation (3) 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

Nov 2004 32

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

Nov 2004 33

Threshold Counter

Particle travel through radiator Cherenkov radiation

Nov 2004 34

Differential Detectors

Will reflect light onto PMT for certain angles only β Selecton

Nov 2004 35

Ring Imaging Detectors (1)

Nov 2004 36

Ring Imaging Detectors (2)

Nov 2004 37

Ring Imaging Detectors (3) More clever geometries are possible

Two radiators One photon detector

Nov 2004 38

Transition Radiation Transition radiation is produced when a

relativistic particle traverses an inhomogeneous medium

Boundary between different materials with different n.

Strange effect What is generating the radiation? Accelerated charges

Nov 2004 39

Initially observer sees nothing

Later he seems to see two charges moving apart electrical dipole

Accelerated charge is creating radiation

Transition Radiation (2)

Nov 2004 40

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

Nov 2004 41

Transition Radiation Detector

Nov 2004 42

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

Nov 2004 43

Bibliography PDG 2004 (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!

Nov 2004 44

Plea I need feedback! Questions

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

A.Weber@rl.ac.uk