View
4
Download
0
Category
Preview:
Citation preview
Interaction of charged particles in matter
today electrons
Introductory remarks
• Electrons ‚feel‘ electro-magnetic and weak interaction
• Electrons couple to photons and W- and Z-Boson of weak interaction
• Electrons ‚talk‘ easily to photons, remember pair creation and
Bremsstrahlung
• Electron and photon detection are entangled, e.g. electro-magnetic
shower detection at high energies
Relevance of electrons in nuclear and particle physics some examples:
- beta decays
- conversion electrons (complementary to g-spectroscopy)
- electron beams for electron-nucleon-scattering (structure functions)
- electron-positron collisions at high energies provide clean source for
particle production and new discoveries
- electron is most abundant reaction product, often source of unwanted
and tremendous back ground
2222)/( mcere
Consider moving particle of charge ze passing by with v the stationary charge Zeze
vb
Ze
r q
y
Assumptions:
• momentum approximation: short inter-
action, only transversal moment. transfer
• target remains non-relativistic
Interaction of heavy charged
particles in matter
Bethe-Bloch
energies)(low correction shell :
energies)ic relativist (at correctiondensity : terms correction two
collision single in transferenergy max. : matter abs. of massatomic :
potential ionisation averaged :I matter abs. of number charge :
v/c : mass electron :
particle incoming of charge : cm10 x 2.81 radius electron class. :
matter absorbing ofdensity : mol10 x 6.022 constant Avogadro
13-
1- 23
C
WA
Z
m
zr
N
Z
C
I
Wvmz
A
ZcmrN
dx
dE
e
e
a
eeea
gg
g
max
2
2
2
max
22
2
222
1/1,
:
222
ln2
Energy loss of electrons and positrons via collisions with electrons on atoms and
electro-magnetic radiation (Bremsstrahlung)
collradtot dx
dE
dx
dE
dx
dE
Modified calculation of energy loss
- large scattering angle of incoming particle
- scattering of identical particles -> Q.M. interference
positrons electrons
of units inenergy kinetic
...)23(12
2ln2)(...1)(:)(
:
2)()/(2
)2(ln
12
22
2
22
2
2
22
FFF
cm
Z
CF
cmIA
ZcmrN
dx
dE
e
e
eea
coll
Collisional energy loss:
Interaction of electrons and positrons
Bremsstrahlung loss and ionisation loss for electrons and protons in copper
electron
2-4 MeV proton
> 3 GeV
Comparison of interactions
crit. energy
24 MeV
Variation in range and energy loss caused by statistics of absorbing collisions.
Energy loss distribution
- average energy loss –(dE/dx)
- energy loss straggling
- range straggling
heavy charged particle electrons
Range and energy loss
Electron range fluctuates considerably due to
multiple scattering.
Large energy transfer via collisions is possible.
Range distribution is smeared out .
Transmission of electrons in aluminium [1 g/cm2 = 3.7 mm]
Electron range
Polyethylene [1 g/cm2 = 9.0 mm]
Aluminium [1 g/cm2 = 3.7 mm]
Lead [1 g/cm2 = 0.88 mm]
Range variation below critical energy is
very energy dependent.
Electron range: low energies
– decay of 185W
185W -> 185Re + - + n
Q = 433 keV, T1/2 = 75.1 d
Neutrinos cause continuous electron
spectrum
Folding of electron energy spectrum and
range in matter shows exponential
attenuation
I = I0 exp(-mx)
– attenuation coefficient m
Exponential attenuation is not valid in
general, only for simple –decay
scheme.
Absorption of electrons from185W - decay
Absorption of -electron
Scattering cause large deflection, even
Backward angles and small energy losses.
Depending on incident angle!
Electrons leave matter in most cases even
below the detection threshold.
Reduced detection efficiency!
Strongly dependent on:
-Electron energy
- at low energies highest impact
- charge number Z of absorbing matter
Electron back scattering
h is backscattering coefficient, given for perpendicular incidence
- Feynman graphs
- Weizsäcker-Williams approximation: Bremsstrahlung
Bremsstrahlung
Electrons and positrons are particles with considerable energy loss via
Bremsstrahlung for electron energies higher than several tens of MeV.
2222)/( mcere m mass: mm=106 MeV electron mass: me=511 keV
Differential Bremsstrahlung cross section
nuclei of fieldin electrons ofn interactio ,correction Coulomb :)(
100 :Parameter screening )(),( :function screening
energyelectron initial incoming:E
radiation after energy electron :E E/E
137
1
)(ln3
1
4
)(
3
2)(ln
3
1
4
)()1(4
3/1
0
2
21
0
0
21222
Zf
EZE
hcm
ZfZZfZd
rZd
e
e
n
n
n
E0
E
Bremsstrahlung
Screening of nuclear charge depends on
(i) initial energy z~ 12-25 for 1 MeV electrons, no screening
z~ 0,1-0,3 for 100 MeV electrons, complete screening
(ii) energy of Bremsstrahlung photon vs. remaining energy varies from small numbers for soft
photons to highest numbers for head on collisions causing high energy radiation
Differential Bremsstrahlung cross section
EZE
hcm
EZE
hcm
h
ZfZZfZd
rZd
ee
e
3/1
0
2
3/1
0
2
0
0
21222
100100 :Parameter screeningImportant
energy photon :
radiation after energy electron :E
energyelectron initial incoming:E
E/E
)(ln3
1
4
)(
3
2)(ln
3
1
4
)()1(4
nn
n
n
n
E0
E
Bremsstrahlung
nnn
n
n
nnn
n
n
n
n
n
n
n
n
dEd
dh
E
h
ENEdE
d
dhN
dx
dE
ZfZd
rZd
Zfhcm
EEdrZd
rad
rad
rad
e
e
e
)(1
and A
NN )(
N energy)photon (xsection integral from lossenergy Radiativ
9)()183ln()
3
21(4
:screening complete 0For
)(2
12ln)
3
21(4
:screening no 1For
,0
00
00
a0,0
0
3/1222
2
0222
0
0
N: number of atoms/cm3
Electron interaction: Bremsstrahlung
Two solutions for limiting extrem cases:Low energies
high energies
)(18
1)183ln(4
0,137
)(3
12ln4
1,137
3/122
3/12
2
022
3/122
ZfZrZ
Zcm
Zfcm
ErZ
Zcmcm
erad
e
e
erad
ee
:screening complete E For
:screening no For
0
High energy limit is independent of electron energy E0
Bremsstrahlung
)(
18
1)183ln(4 3/122 ZfZrZ erad
Collisions vs. Bremsstrahlung
Compare collision or ionisation vs Bremsstrahlung:
- ionisation increases logarithmic in E and linear in Z
- bremsstrahlung increases linear in E and quadratic in Z
- ionisation: quasi-continuous energy loss
- bremsstrahlung: emission of one or two photons can give away
full electron energy -> huge fluctuations
Z
CF
cmIA
ZcmrN
dx
dE
e
eea
coll
2)()/(2
)2(ln
12
22
2
2
22
)(18
1)183ln(4)(
1
)(
N energy)photon (xsection integral from lossenergy Radiativ
3/122
,0
00
0,0
0
0
0
ZfZrZdEd
dh
E
NEdEd
dhN
dx
dE
erad
rad
rad
nnn
n
nnn
n
n
n
Critical energy Ec:
2.1
800
Z
MeVEEE
dx
dE
dx
dEcc
collrad
for
Typical values:
material critical energy Ec
Pb 9.51 MeV
Al 51.0 MeV
Fe 27.4 MeV
Cu 24.8 MeV
Luft 102 MeV
NaI 17.4 MeV
H2O 92 MeV
Radiation loss
radiation lengthDistance in matter related to a radiative energy loss by 1/e
)(183ln)1(41
length radiation is 1/ andmatter in distance is
)exp(
DEQ ofSolution
energy fromt independen is (ii)
ignored loss coll. (i)
: 0)(limit energtichigh
/
3/12
0
ZfZrA
NZZ
L
NLx
L
xEE
dxNEdE
ea
rad
radrad
rad
rad
rad
Radiation length
Radiation length
Typical values:
Material cm
Pb 0.56
Al 8.90
Fe 1.76
Cu 1.43
NaI 2.59
BGO 1.12
Org.Scint. 42.40
H2O 36.10
Air 30050.00
Radiation length
collradtot dx
dE
dx
dE
dx
dE
positron ...)23(12
2ln2)( electron ...1)( :)(
of unitsin energy Kin. :
2)()/(2
)2(ln
12
22
2
22
2
2
22
FFF
cm
Z
CF
cmIA
ZcmrN
dx
dE
e
e
eea
coll
Energy loss via collisions with electrons:
Kritical energy Ec:
2.1
800 for
Z
MeVEEE
dx
dE
dx
dEcc
collrad
Summary: interactions of electrons
radiation lengthRadiational energy loss by 1/e
summary
Compare collision or ionisation vs Bremsstrahlung:
- Ionisation increases logarithmic in E and linear in Z
- Bremsstrahlung increases linear in E and quadratic in Z
- Ionisation: quasi-continuos energy loss
- Bremsstrahlung: emission of one or two photons can give away
full electron energy -> huge fluctuations
Z
CF
cmIA
ZcmrN
dx
dE
e
eea
coll
2)()/(2
)2(ln
12
22
2
2
22
)(18
1)183ln(4)(
1
)(
N energy)photon (xsection integral from lossenergy Radiativ
3/122
,0
00
0,0
0
0
0
ZfZrZdEd
dh
E
NEdEd
dhN
dx
dE
erad
rad
rad
nnn
n
nnn
n
n
n
Recommended