Interaction of Particles with Matter
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Transcript of Interaction of Particles with Matter
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…