Transcript Interactions of Particles with Matter

Interaction of Particles
with Matter
Alfons Weber
STFC & University of Oxford
Graduate Lecture 2009
Table of Contents

Bethe-Bloch Formula


Multiple Scattering


Light emitted by particles travelling in
dielectric materials
Transition Radiation
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Dec 2009
Change of particle direction in Matter
Cerenkov Radiation
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
Energy loss of heavy particles by Ionisation
Light emitted on traversing matter boundary
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Bethe-Bloch Formula


Describes how heavy particles (m>>me)
loose energy when travelling through
material
Exact theoretical treatment difficult




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Atomic excitations
Screening
Bulk effects
Simplified derivation ala MPhys course
Phenomenological description
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Bethe-Bloch (1)

Consider particle of charge ze, passing a
stationary charge Ze
ze
r

Assume



θ
y
x
Ze
Target is non-relativistic
Target does not move
Calculate

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Dec 2009
b
Momentum transfer
Energy transferred to target
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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
Dec 2009
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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
Dec 2009
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Bethe-Bloch (4)


Energy transfer is determined by impact
parameter b
Integration over all impact parameters
b
ze
Dec 2009
db
dn
 2 b  (number of electrons / unit area )
db
NA
=2 b  Z
x
A
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Bethe-Bloch (5)

Calculate average energy loss
bmax
me c 2 Zz 2
dn
b
E   d b
Ee (b)  2C 2
x  ln b bmax
min
db

A
bmin
me c 2 Zz 2
Emax
C 2
x  ln E E
min

A


e2
with C  2 N A 
2 
4

m
c
0 e



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There must be limits
material dependence is in the calculation
of the limits
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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 Zz
2  mec 
E
 2C 2
 ln 

x

A
I0


2
Dec 2009
2
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Bethe-Bloch (7)

This was just a simplified derivation

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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
 2C 2
  ln 
x

A 2 
I 02

  ( ) 
2

  
2
2 

with

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ε screening correction of inner electrons
δ density correction (polarisation in medium)
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Energy Loss Function
E
/   stopping power
x
Dec 2009
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Average Ionisation Energy
Dec 2009
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Density Correction

Density Correction does depend on
material
with

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Dec 2009
x = log10(p/M)
C, δ0, x0 material dependant constants
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Different Materials (1)
Dec 2009
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Different Materials (2)
Dec 2009
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Particle Range/Stopping Power
Dec 2009
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Energy-loss in Tracking Chamber
Dec 2009
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Straggling (1)



So far we have only discussed the mean
energy loss
Actual energy loss will scatter around the
mean value
Difficult to calculate

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Dec 2009
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
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Straggling (2)

Simple parameterisation

Landau function
1
 1

f ( ) 
exp   (  e   ) 
2
 2

E  E
with  
me c 2 Zz
C 2
x
 A

Dec 2009
Better to use Vavilov distribution
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Straggling (3)
Dec 2009
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δ-Rays

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Energy loss distribution is not Gaussian
around mean.
In rare cases a lot of energy is transferred
to a single electron
δ-Ray
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If one excludes δ-rays, the average
energy loss changes
Equivalent of changing Emax
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Restricted dE/dx
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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
Dec 2009
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Electrons

Electrons are different light

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Dec 2009
Bremsstrahlung
Pair production
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Dec 2009
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Multiple Scattering

Particles don’t only loose energy …
… they also change direction
Dec 2009
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MS Theory

Average scattering angle is roughly
Gaussian for small deflection angles
With
 x 
13.6 MeV
x 

Angular distributions are given by

0 
1  0.038ln 

 cp
X0 
 X 0 
X 0  radiation length
z
2



dN
1
space

exp  
2
2 


d  2 0
4

0


dN
d plane
Dec 2009
2
  plane

1

exp  
2 


2

2 0
0


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Correlations


Multiple scattering and dE/dx are normally
treated to be independent from each
Not true
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Detailed calculation is difficult, but
possible
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Dec 2009
large scatter  large energy transfer
small scatter  small energy transfer
Wade Allison & John Cobb are the experts
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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 or
log kL
energy transfer
(16 decades)
electrons
backwards in
CM
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!
Dec 2009
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log kT
Log pT transfer
(10 decades)
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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
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Cherenkov radiation
Similar, but not identical
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Dec 2009
Gas chambers
Silicon detectors
Scintillators
Transition radiation
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Cherenkov Radiation


Moving charge in dielectric medium
Wave front comes out at certain angle
slow
fast
1
cos c 
n
Dec 2009
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Cherenkov Radiation (2)

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
Dec 2009
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Different Cherenkov Detectors
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Threshold Detectors
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Differential Detectors
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βmax > β > βmin
Ring-Imaging Detectors
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Dec 2009
Yes/No on whether the speed is β>1/n
Measure β
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Threshold Counter

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Dec 2009
Particle travel through radiator
Cherenkov radiation
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Differential Detectors
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Dec 2009
Will reflect light onto PMT for certain
angles only  β Selection
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Ring Imaging Detectors (1)
Dec 2009
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Ring Imaging Detectors (2)
Dec 2009
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Ring Imaging Detectors (3)

More clever geometries are possible

Dec 2009
Two radiators  One photon detector
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Transition Radiation

Transition radiation is produced, when a
relativistic particle traverses an
inhomogeneous medium


Strange effect


Dec 2009
Boundary between different materials with
different diffractive index n.
What is generating the radiation?
Accelerated charges
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Transition Radiation (2)
q1
Dec 2009
v1
vacuum
medium
v3
Before the charge crosses
the surface,
apparent charge q1 with
apparent transverse vel v1
q3
q2
v2
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After the charge crosses
the surface,
apparent charges q2 and q3
with apparent transverse
vel v2 and v3
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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
2
 p  plasma frequency

Total energy radiated

Can be used to measure γ
Dec 2009
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Transition Radiation Detector
Dec 2009
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ATLAS TRTracker
ATLAS
Experiment
Inner Detector:
pixel, silicon and straw tubes
Combination of Central Tracker and
TR for electron identification
Dec 2009
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Atlas TRT (II)
Dec 2009
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Atlas TRT (III)

Electrons with radiator

only electron produce
TR in radiator
e± / π separation
Electrons without radiator

TRT senses


Bod -> J/yKos
ionisation
transition radiation
High threshold hits
Dec 2009
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Table of Contents

Bethe-Bloch Formula


Multiple Scattering


Light emitted by particles travelling in
dielectric materials
Transition radiation

Dec 2009
Change of particle direction in Matter
Cerenkov Radiation


Energy loss of heavy particles by Ionisation
Light emitted on traversing matter boundary
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Bibliography

This lecture


PDG 2008 (chapter 27 & 28) and
references therein


Dec 2009
http://www.shef.ac.uk/physics/teaching/phy311
R. Bock, Particle Detector Brief Book


Especially Rossi
Lecture notes of Chris Booth, Sheffield


http://www-pnp.physics.ox.ac.uk/~weber/teaching
http://rkb.home.cern.ch/rkb/PH14pp/node1.html
Or just
it!
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Plea


I need feedback!
Questions








Dec 2009
What was good?
What was bad?
What was missing?
More detailed derivations?
More detectors?
More…
Less…
[email protected]
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