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# EM for Particle Detectors

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• pg 1
```									  Interaction of Particles
with Matter

Alfons Weber
CCLRC & University of Oxford
Nov 2004                                                          2

   Bethe-Bloch Formula
   Energy loss of heavy particles by Ionisation
   Multiple Scattering
   Change of particle direction in Matter
   Light emitted by particles travelling in
dielectric materials
   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

b              y
r
θ
x
   Assume               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
Fx            cos          cos3 
4 0 r 2        4 0b2

    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:    M=A*mp
   Mass of electron:   M=me
   But energy transfer is
p 2   Z 2 z 2e4       1 Z2
E                           
2M 2M (2 0 ) (  c) b
2      2  2
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
db
ze

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
dn            me c 2 Zz 2
E   d b    Ee (b)  2C 2          x  ln b b
bmax

bmin
db                   A                min

me c 2 Zz 2
C 2          x  ln E E
Emax

      A                min

    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
Emax                      2
 2 2  2 me c 2
me  me 
1  2       
M       M 
   Inelastic collision
Emin  I 0  average ionisation energy
   Results in the following expression

E     me c Zz       2  me c 
2     2               2   2   2
 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
E     me c 2 Zz 2  1  2 2  2 me c 2 Emax       ( ) 
 2C 2           ln                       
2

x            A 2               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                     
 x  E  E               A 2             I 02         
cut

    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    13.6 MeV   x           x 
0              z       1  0.038ln      
 cp        X0               X 0 
   Angular distributions are given by
dN    1         space 
2

      exp  
d  2 0
2      2 2   
      0 

dN          1          plane 
2

       exp  
d plane     2 0      2 2   
      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                                                                              nuclear backward
scattering (suppressed                                                                            scattering in CM
by screening)                                               electrons                     (suppressed by nuclear
at high                           form factor)
Q2

whole
atoms at                                           17
Log cross                                      low Q2
section                                       (dipole
(30                                         region)
2

electrons
log
Log pL or kL                                                         log kT
backwards in
energy transfer
CM

Log pT transfer
Example: Calculated cross section for 500MeV/c  in Argon gas.              18 7          (10 decades)
Note that this is a Log-log-log plot - the cross section varies over 20
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
   Similar, but not identical
Nov 2004                                        29

   Moving charge in matter

at rest          slow     fast
Nov 2004                                                 30

   Wave front comes out at certain angle

1
cos  c 
n

   That’s the trivial result!
Nov 2004                                                                  31

   How many Cherenkov photons are
detected? 2
z
N  L
re me c 2     ( E ) sin 2  c ( E )dE

 z2                    1 
2 
L               ( E )  1  2 2  dE
re me c               n 
          1 
 LN 0 1  2 2 
  n 
                 
with  ( E )  Efficiency to detect photons of energy E
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

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 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
Nov 2004                                                            40

   Consider relativistic particle traversing a
boundary from material (1) to material (2)
2
2
d N   z 2 
2
1                  1       
 2   2 2                 
d d        /    2  1/  2  2  1/  2 

  p                                
 p  plasma frequency

   Can be used to measure γ
Nov 2004                                   41

Nov 2004                                                          42

   Bethe-Bloch Formula
   Energy loss of heavy particles by Ionisation
   Multiple Scattering
   Change of particle direction in Matter
   Light emitted by particles travelling in
dielectric materials
   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?