Transcript EM for Particle Detectors

Interaction of Particles
with Matter
Alfons Weber
CCLRC & University of Oxford
Graduate Lecture 2004
Nov 2004
2
Table of Contents

Bethe-Bloch Formula


Multiple Scattering


Change of particle direction in Matter
Cerenkov Radiation


Energy loss of heavy particles by Ionisation
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
ze
r

Assume



b
θ
y
x
Ze
Target is non-relativistic
Target does not move
Calculate

Energy transferred to target
(separate)
Nov 2004
5
Bethe-Bloch (2)

Force on projectile
Zze2
Zze2
3
Fx 
cos


cos

2
2
4 0 r
4 0b

Change of momentum of target/projectile
Zze2 1
p   dtFx 

2 0  c b


Energy transferred
p 2
Z 2 z 2e4
1
E 

2M 2M (2 0 )2 (  c)2 b2
Nov 2004
6
Bethe-Bloch (3)

Consider α-particle scattering off Atom



Mass of nucleus:
Mass of electron:
M=A*mp
M=me
But energy transfer is
p 2
Z 2 z 2e4
1 Z2
E 


2
2
2
2M 2M (2 0 ) (  c) b
M

Energy transfer to single electron is
2 z 2 e4
1
Ee (b)  E 
mec 2 (4 0 )2  2 b2
Nov 2004
7
Bethe-Bloch (4)


Energy transfer is determined by impact
parameter b
Integration over all impact parameters
b
ze
db
dn
 2 b  (number of electrons / unit area )
db
NA
=2 b  Z
x
A
Nov 2004
8
Bethe-Bloch (5)

Calculate average energy loss
bmax
me c 2 Zz 2
dn
bmax
E   d b
Ee (b)  2C 2
x  ln b b
min
d
b

A
bmin
me c 2 Zz 2
Emax
C 2
x  ln E E
min

A


e2
with C  2 N A 
2 
 4 0 me c 

There must be limit for Emin and Emax

All the physics and material dependence is in
the calculation of this quantities
Nov 2004
9
Bethe-Bloch (6)

Simple approximations for

From relativistic kinematics
2 2  2 me c 2
2 2
2
Emax 

2


m
c
e
2
me  me 
1  2
 
M

M 
Inelastic collision
Emin  I0  average ionisation energy

Results in the following expression
 2 2  2 mec 2 
mec 2 Zz 2
E
 2C 2
 ln 

x

A
I0


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
me c 2 Zz 2  1  2 2  2 me c 2 Emax 
E
  ( ) 
2
 2C 2
  ln 

  
2
x

A 2 
I0
2
2 

with


ε screening correction of inner electrons
δ density correction, because of polarisation
in medium
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
1
 1

f ( ) 
exp   (  e   ) 
2
 2

E  E
with  
me c 2 Zz
C 2
x
 A

Better to use Vavilov distribution
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
δ-Ray


If one excludes δ-rays, the average
energy loss changes
Equivalent of changing Emax
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
 E 
me c 2 Zz 2  1  2 2  2 me c 2 Ecut 
 2C 2
  ln 



2

A 2 
I0

 x  E  Ecut

Ecut    (  ) 
  1 

 
2 
 Emax  2
2
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
 x 
13.6 MeV
x 
0 
1  0.038ln 

 cp
X0 
 X 0 
X 0  radiation length
z
Angular distributions are given by
2
  space

dN
1

exp  
2
2 


d  2 0
2

0 

dN
d plane
2



1
plane

exp  
2 


2

2 0
0 

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)
nuclear small angle
scattering (suppressed
by screening)
electrons
at high
Q2
nuclear backward
scattering in CM
(suppressed by nuclear
form factor)
whole
atoms at
low Q2
(dipole
region)
Log cross
section
(30
decades)
17
2
Log pL orlog kL
energy transfer
(16 decades)
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!
electrons
backwards in
CM
18 7
log kT
Log pT transfer
(10 decades)
Nov 2004
28
Signals from Particles in Matter

Signals in particle detectors are mainly
due to ionisation



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Direct light emission by particles travelling
faster than the speed of light in a medium


Gas chambers
Silicon detectors
Scintillators
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
Cherenkov Radiation (2)

Wave front comes out at certain angle
cos c 

That’s the trivial result!
1
n
Nov 2004
31
Cherenkov Radiation (3)

How many Cherenkov photons are
detected?
2
N  L
z
re me c 2
2

(
E
)
sin
 c ( E )dE

 z2

1 
L
 ( E )  1  2 2  dE
2 
re me c
  n 

1 
 LN 0 1  2 2 
  n 


with  ( E )  Efficiency to detect photons of energy E
L  radiator length
re  electron radius
Nov 2004
32
Different Cherenkov Detectors

Threshold Detectors


Differential Detectors


Yes/No on whether the speed is β>1/n
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
Transition Radiation (2)



Initially observer sees
nothing
Later he seems to
see two charges
moving apart
 electrical dipole
Accelerated charge is
creating radiation
Nov 2004
40
Transition Radiation (3)

Consider relativistic particle traversing a
boundary from material (1) to material (2)

d N
z 2 
1
1
 2   2 2
 2
2
2
2 


d d   

/




1/



1/

 p

2
2
 p  plasma frequency

Total energy radiated

Can be used to measure γ
2
Nov 2004
41
Transition Radiation Detector
Nov 2004
42
Table of Contents

Bethe-Bloch Formula


Multiple Scattering


Change of particle direction in Matter
Cerenkov Radiation


Energy loss of heavy particles by Ionisation
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


Lecture notes of Chris Booth, Sheffield


http://www.shef.ac.uk/physics/teaching/phy311
R. Bock, Particle Detector Brief Book


Especially Rossi
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…
[email protected]