Basic Concepts of Charged
Particle Detection:
Part 2
David Futyan
Charged Particle Detection 2
1
Overview
Lecture 1:
Concepts of particle detection: what can we detect?
Basic design of particle detectors
Energy loss of charged particles in matter: Bethe Bloch formula
Lecture 2:
Energy loss through Bremsstrahlung radiation (electrons)
Momentum measurement in a magnetic field
Multiple Coulomb scattering - effect on momentum resolution
Interaction of photons
David Futyan
Charged Particle Detection 2
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Energy Loss of Electrons and Positrons
Electrons lose energy through ionization as for heavy charged particles,
but due to small mass additional significant loss through
bremsstrahlung radiation.
Total energy loss:
dE
dE
dE


dX tot dX rad dX ion

David Futyan
Charged Particle Detection 2
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Energy Loss Through Ionization for Electrons
Ionization loss for high energy electrons (»1MeV) can be approximated
by Bethe Bloch formula with =1, z=1:
 2mec 2 2Tmax 
dE
2
2 Z 1

 4N A re mec  ln
1
2
dX ion
A 2
I

Approximate and only valid for high energy. Full treatment requires
modification of the Bethe Bloch formula due to:

Small electron mass: assumption that incident particle is undeflected
during collision process is not valid
Collisions are between identical particles - Q.M. effects due to
indistinguishability must be taken into account.
e.g. see Leo p.37
NB. Tmax=Te/2, where Te = KE of incident electron
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Charged Particle Detection 2
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Bremsstrahlung Radiation
Emission of e.m. radiation arising from scattering in the E field of a
nucleus in the absorber medium.
Classically, can be seen as radiation due to acceleration of e+ or e- due
to electrical attraction to a nucleus.
Radiative energy loss dominates for electrons for E > few 10s of MeV.
David Futyan
Charged Particle Detection 2
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Energy Loss Through Bremsstrahlung
2
dE
183
2 Z

 4N A re
E ln 1/ 3
dX rad
A
Z
e2
 = Fine structure constant:  
hc4 0

dE

Note that dX
rad
m 2
e

m 
  40000 
 
 

1
 2
m
e2
(recall re  4 m c 2 )
0
e
1

Bremsstrahlung only significant for electrons/positrons

For E < ~1TeV, electrons/positrons are the only particles in which
radiation contributes significantly to energy loss.
David Futyan
Charged Particle Detection 2
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Radiation Length
Define Radiation Length, X0:
dE
E


dX rad X 0

E  E 0ex / X 0
where
2
1
183
2 Z
 4N A re
ln 1/ 3
X0
A Z
Units of X0: g cm-2
Divide by density  to get X0 in cm
the mean distance over which a high-energy
 Radiation length is the 
electron loses all but 1/e of its energy by bremsstrahlung.

e.g. Pb: Z=82, A=207, =11.4 g/cm3:
X0 ≈ 5.9 g/cm2
Mean penetration distance: x = X0/ = 5.9/11.4 = 5.2mm
David Futyan
Charged Particle Detection 2
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Comparison with Energy Loss Through Ionization
Compare:
dE
 Z 2E
dX rad
dE
 Z ln E
dX ion
Rapid rise of radiation loss with electron energy
 all energy of electron

Almost
can be radiated in one or two photons! In
contrast ionization loss quasi-continuous along path of particle.
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Charged Particle Detection 2
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Critical Energy
Critical energy is energy for which:
dE
dE

dX rad dX ion

ECsolid liq 

ECgas 
610MeV
Z 1.24
710MeV
Z 1.24

E.g. EC for electrons in Cu(Z=29): ~20 MeV
Energy loss through bremsstrahlung dominates for E > few 10s of MeV.
e.g. electrons in LHC events: tens of GeV  bremsstrahlung
completely dominates.
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Charged Particle Detection 2
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Example: Bremsstrahlung in CMS
Electron must traverse ~1X0 of material in the inner tracker (13 layers of
Si strip detectors) before it reaches the electromagnetic calorimeter.
On average, about 40% of electron energy is radiated in the tracker
Spray of deposits in the ECAL - must be combined to give calorimeter
energy measurement.
Momentum at vertex should be determined from track curvature before
1st bremsstrahlung emission.
David Futyan
Charged Particle Detection 2
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Full Energy Loss Spectrum for Muons
-dE/dX for positive muons over 9 orders of magnitude in momentum:
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Charged Particle Detection 2
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Momentum Measurement in a Magnetic Field
mv 2
 Bqv  pT  qBr  pT (GeV /c)  0.3Br
r
0.3BL
L
 


 sin 

pT
2r
2 2
2

2
2


    1   r
s  r  rcos  r1 1

2    2 4  8

David Futyan

0.3BL
s
8 pT
Charged Particle Detection 2

e.g. s = 3.75 cm
for pT=1 GeV/c,
L=1m and B=1T
12

Momentum Measurement Error
Determination of sagitta from 3 measurements:
x
x1  x3
s  x2 
2
2
1
s
3
s 
3
x
2
Momentum resolution:
p
s
3/2 x
T


pT
s
s


p
3
8
T
T
 x
pT meas
2 0.3BL2
p
p

T
pT
meas
 x pT
BL2
Momentum resolution degrades linearly with increasing momentum, and
improves quadratically with the radial size of tracking cavity.

 Gluckstern, NIM 24
For N equidistant measurements,
one obtains (R.L.
(1963) 381):
 pT
 p
720
 x T2
pT meas 0.3BL N  4
e.g. (pT)/pT = 0.5% for pT=1 GeV/c, L=1m, B=1T, x = 200 m and N=10
David Futyan

Charged Particle Detection 2
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Multiple Coulomb Scattering
In addition to inelastic collisions with atomic electrons (i.e. ionization Bethe Bloch), charged particles passing through matter also suffer
repeated elastic Coulomb scattering from nuclei.
Elastic Coulomb scattering produces a change in the particle direction
without any significant energy loss.
Change in direction caused by multiple Coulomb scattering degrades
the momentum measurement.
David Futyan
Charged Particle Detection 2
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Single Scattering
Individual collisions governed by Rutherford scattering formula:
2


d
1
2 2 2 me c
 z Z re 

4
d
 p  4 sin ( /2)
Does not take into account spin effects or screening
Although single large angle scattering can occur for very small impact

parameter, probability that a single interaction will scatter through a
significant angle is very small due to 1/sin4(/2) dependence.
For large impact parameter (much more probable), scattering angle is
further reduced w.r.t. Rutherford formula due to partial screening of
nuclear charge by atomic electrons.
David Futyan
Charged Particle Detection 2
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Multiple Coulomb Scattering
As a particle passes through a thickness of material, combination of a
very large number of small deflections results in a significant net
deviation - multiple coulomb scattering
Small contributions combine randomly to give a Gaussian probability
distribution:
 plane
1 
P( )d 
Plane of
incident
particle
( to B field)

e
2
d

where
RMS
 plane
RMS
  plane





(Gaussian)
plane is projection of
true space scattering
angle onto plane of
incident particle. RMS
 plane 
David Futyan
1 RMS
 space
2
Charged Particle Detection 2
plane
16
Effect of Multiple Scattering on Resolution
Approximate relation (PDG):
RMS
0   plane

13.6MeV
L
z
pc
X0
i.e. 0 
1
p
L
X0
Radiation length of absorbing material
Charge of incident particle

Apparent sagitta due to multiple scattering
(from PDG):
s plane
L 0

4 3
Contribution to momentum resolution from multiple scattering:
p

p

MS
splane
s
0.05

B LX 0
0.3BL2
using s 
8 pT
i.e.
p
p

MS
1
B LX 0
Independent of p!
David Futyan

Charged Particle Detection 2
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Effect of Multiple Scattering on Resolution
Estimated Momentum Resolution
vs pT in CMS
(p)/p
(p)/p meas.
total error
(p)/p MS
p
Example:
pT = 1 GeV/c, L = 1m, B = 1 T,
= 10, x = 200m:
p
p
N
 0.5%
meas
For detector filled with Ar, X0 = 110m:
p

David Futyan
p
 0.5%
MS
Charged Particle Detection 2
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Momentum Measurement Summary
p

T
pT
meas
 x pT
p
BL2
p

MS
1
B LX 0
Tracking detector design:

High B field. e.g. CMS: 4 Tesla

Large size e.g. CMS tracker radius = 1.2m
Low Z, low mass material. Gaseous detectors frequently chosen e.g.
ATLAS Ar (91% of gas mixture) X0=110m
David Futyan
Charged Particle Detection 2
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Interaction of Photons
No E field => inelastic collisions with atomic electrons and
bremsstrahlung which dominate for charged particles do not occur for
photons
3 main interations:
1) Photoelectric effect (dominant for E<100keV):
 Photon is absorbed by an atomic electron with the subsequent
ejection of the electron from the atom.
2) Compton scattering (important for E~1MeV):
 Scattering of photons on free electrons
(atomic electrons effectively free for
E >> atomic binding energy)
3) Pair production (dominant for E>5MeV)
 Photon is converted into an electron-positron pair
David Futyan
Charged Particle Detection 2
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Interaction of Photons
Result of these 3 interactions:
1) Photons (x-rays, -rays) much more penetrating in matter than
charged particles
 Cross-section for the 3 interactions much less than inelastic
collision cross-section for charged particles
2) A beam of photons is not degraded in energy as it passes through a
thickness of matter, only in intensity
 The 3 processes remove the photon from the beam entirely
(absorbed or scattered out).
 Photons which pass straight through have suffered no interaction
so retain their original energy, but no. of photons is reduced.
Attenuation is exponential w.r.t. material thickness:
I(x)  I0 exp(x)
Absorption coefficient
David Futyan

Charged Particle Detection 2
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e+e- Pair Procution
For energies > a few MeV, pair production is the dominant mechanism:
In order to conserve energy and momentum, pair production can only
occur in the presence of a 3rd body, e.g. an atomic nucleus.
e.g. CMS ECAL: PbW04 crystals - dense material with heavy nuclei
In order to create the pair, photon must have energy E>2mec2 i.e.
E>1.022MeV.
David Futyan
Charged Particle Detection 2
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Electron-Photon Showers
Combined effect of pair production for photons and bremsstrahlung for
electrons is the formation of electron-photon showers.
Shower continues until energy of e+e- pairs drops below critical energy
See lectures on Calorimetry (Chris Seez)
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