Interaction of Particles with Matter

with Matter

Alfons Weber

CCLRC & University of Oxford Graduate Lecture 2004

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>>m_e) 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

 Energy transferred to target (separate)

ze

Ze r b

θ x

y

Bethe-Bloch (2)

2

0

1

x 2 p dtF Zze

 c b



 



 Force on projectile

 Change of momentum of target/projectile

 Energy transferred

2 2

3

2 2

0 0

cos cos

4 4

x

Zze Zze

F r  b 

 

 

2 2 2 4

2 2 2

0

1

2 2 (2 ) ( )

p Z z e

E M M   c b

   

Bethe-Bloch (3)

 Consider α-particle scattering off Atom

 Mass of nucleus: M=A*m_p

 Mass of electron: M=me

 But energy transfer is

 Energy transfer to single electron is

2 2 2 4 2

2 2 2

0

1

2 2 (2 ) ( )

p Z z e Z

E M M  c b M

    

2 4

2 2 2 2

0

2 1

( ) (4 )

e

e

E b E z e

m c   b

  

Bethe-Bloch (4)

 Energy transfer is determined by impact parameter b

 Integration over all impact parameters

b db

ze

2 (number of electrons / unit area )

=2 ^A

dn b

db

b Z N x A



 

 

 

Bethe-Bloch (5)

 Calculate average energy loss

 There must be limit for E_min and E_max

 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

2 0

d d ( ) 2 ln

d

ln

with 2

4

b e b

e b

b

e E

E

A

e

n m c Zz

E b E b C x b

b A

m c Zz

C x E

A C N e

m c

 

 

 

   

 

 

  

 



Bethe-Bloch (6)

 Simple approximations for

 From relativistic kinematics

 Inelastic collision

 Results in the following expression

min 0 average ionisation energy E  I 

2 2 2 2 2

2

0

2 m c^e ln 2 m c^e

E Zz

x C A I

  



 

   

2 2 2

2 2 2

max 2

2 2

1 2

e

e

e e

E m c m c

m m

M 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, because of polarisation in medium

2 2 2 2 2

max 2

2 2

0

2

1 ( )

2 ln

2 2 2

e e

m c m c E

E Zz

x C A I

    

 



   

        

Energy Loss Function

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

Application in Particle ID

 Energy loss as measured in tracking chamber

 Who is Who!

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 ( )

2 2

with

e

f e

E E

m c Zz

C 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 E_max

δ-Ray

Restricted dE/dx

 Some detector only measure energy loss up to a certain upper limit E_cut

 Truncated mean measurement

 δ-rays leaving the detector

2 2 2 2 2

2 2

0

2

max

2

2 1 ln

2

1 ( )

2 2

cut

e e cut

E E

cut

m c m c E

E Zz

x C 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

0

0 0

0

13.6 MeV

1 0.038ln radiation length

x x

cp z X X

X

 

  

    

 

 



2

2 2

0 0

2 2 0 0

1 exp

2 2

1 exp 2 2

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 k_L

2

18

17

7

log k_T

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 p_L or energy transfer

(16 decades)

Log p_T 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

Cherenkov Radiation (1)

 Moving charge in matter

at rest slow fast

 Wave front comes out at certain angle

 That’s the trivial result!

Cherenkov Radiation (2)

cos _c 1

 n

 

Cherenkov Radiation (3)

 How many Cherenkov photons are detected? ₂

2 2

2

2 2 2

0 2 2

( )sin ( )d

( ) 1 1 d

1 1

with ( ) Efficiency to detect photons of energy radiator length

electron radius

c e e

e e

e

N L z E E E

r m c

L z E E

r m c n

LN n

E E

L r



  

 









 

   

 

 

 

 

 

 











Different Cherenkov Detectors

 Threshold Detectors

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

 Differential Detectors

 β_max > β > β_min

 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 Selecton

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 n.

 Strange effect

 What is generating the radiation?

 Accelerated charges

 Initially observer sees nothing

 Later he seems to see two charges moving apart

 electrical dipole

 Accelerated charge is creating radiation

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

2 2 2

2

2 2 2 2 2 2 2

d 1 1

d d / 1/ 1/

plasma frequency

p p

N z  

        



 

   

     



Transition Radiation Detector

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

 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!

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