Interaction of Particles with Matter Alfons Weber STFC & University of Oxford Graduate Lecture 2009.
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Transcript of Interaction of Particles with Matter Alfons Weber STFC & University of Oxford Graduate Lecture 2009.
Interaction of Particleswith Matter
Alfons WeberSTFC & University of Oxford
Graduate Lecture 2009
Dec 2009 Alfons Weber 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
Dec 2009 Alfons Weber 3
Dec 2009 Alfons Weber 4
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
Dec 2009 Alfons Weber 5
Bethe-Bloch (1) Consider particle of charge ze, passing a
stationary charge Ze
Assume Target is non-relativistic Target does not move
Calculate Momentum transfer Energy transferred to target
ze
Ze
br
θx
y
Dec 2009 Alfons Weber 6
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
Dec 2009 Alfons Weber 7
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
Dec 2009 Alfons Weber 8
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
Dec 2009 Alfons Weber 9
Bethe-Bloch (5) Calculate average energy loss
There must be limits material dependence is in the calculation
of the limits
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
Dec 2009 Alfons Weber 10
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
Dec 2009 Alfons Weber 11
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 (polarisation in medium)
2 2 2 222max
2 20
21 ( )2 ln
2 2 2e em c m c EE Zz
Cx A I
Dec 2009 Alfons Weber 12
Energy Loss Function
/ stopping powerE
x
Dec 2009 Alfons Weber 13
Average Ionisation Energy
Dec 2009 Alfons Weber 14
Density Correction
Density Correction does depend on material
with x = log10(p/M)
C, δ0, x0 material dependant constants
Dec 2009 Alfons Weber 15
Different Materials (1)
Dec 2009 Alfons Weber 16
Different Materials (2)
Dec 2009 Alfons Weber 17
Particle Range/Stopping Power
Dec 2009 Alfons Weber 18
Energy-loss in Tracking Chamber
Dec 2009 Alfons Weber 19
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
Dec 2009 Alfons Weber 20
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
Dec 2009 Alfons Weber 21
Straggling (3)
Dec 2009 Alfons Weber 22
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
Dec 2009 Alfons Weber 23
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
Dec 2009 Alfons Weber 24
Electrons Electrons are different light
Bremsstrahlung Pair production
Dec 2009 Alfons Weber 25
Dec 2009 Alfons Weber 26
Multiple Scattering Particles don’t only loose energy …
… they also change direction
Dec 2009 Alfons Weber 27
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 4
1exp
22
space
plane
plane
dN
d
dN
d
Dec 2009 Alfons Weber 28
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
Dec 2009 Alfons Weber 29
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)
Dec 2009 Alfons Weber 30
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
Dec 2009 Alfons Weber 31
Moving charge in dielectric medium Wave front comes out at certain angle
Cherenkov Radiation
1cos c n
slow fast
Dec 2009 Alfons Weber 32
Cherenkov Radiation (2) 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
Dec 2009 Alfons Weber 33
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
Dec 2009 Alfons Weber 34
Threshold Counter
Particle travel through radiator Cherenkov radiation
Dec 2009 Alfons Weber 35
Differential Detectors
Will reflect light onto PMT for certain angles only β Selection
Dec 2009 Alfons Weber 36
Ring Imaging Detectors (1)
Dec 2009 Alfons Weber 37
Ring Imaging Detectors (2)
Dec 2009 Alfons Weber 38
Ring Imaging Detectors (3) More clever geometries are possible
Two radiators One photon detector
Dec 2009 Alfons Weber 39
Transition Radiation Transition radiation is produced, when a
relativistic particle traverses an inhomogeneous medium
Boundary between different materials with different diffractive index n.
Strange effect What is generating the radiation? Accelerated charges
Dec 2009 Alfons Weber 40
22 vq
vacuummedium
Before the charge crosses the surface,apparent charge q1 with apparent transverse vel v1
After the charge crosses the surface,apparent charges q2 and q3
with apparent transverse vel v2 and v3
11 vq
33 qv
Transition Radiation (2)
Dec 2009 Alfons Weber 41
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
Dec 2009 Alfons Weber 42
Transition Radiation Detector
Dec 2009 Alfons Weber 43
ATLAS TRTracker
ATLAS Experimen
t
Inner Detector:pixel, silicon and straw
tubes
Combination of Central Tracker and TR for electron
identification
Dec 2009 Alfons Weber 44
Atlas TRT (II)
Dec 2009 Alfons Weber 45
Atlas TRT (III)
TRT senses ionisation transition radiation
only electron produce TR in radiator
e± / π separationElectrons with
radiator
Electrons without radiator
Bod -> J/Ko
s
High threshold hits
Dec 2009 Alfons Weber 46
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
Dec 2009 Alfons Weber 47
Bibliography
This lecture http://www-pnp.physics.ox.ac.uk/~weber/teaching
PDG 2008 (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!
Dec 2009 Alfons Weber 48
Plea I need feedback! Questions
What was good? What was bad? What was missing? More detailed derivations? More detectors? More… Less…