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Electrons
Electrons lose energy primarily through ionization and radiation
Bhabha (e+e-→e+e-) and Moller (e-e-→e-e-) scattering also contribute
When the energy loss per collision is above 0.255 MeV one considers this to be Bhabha or Moller scattering
radiationionizationtotal dx
dE
dx
dE
dx
dE
2
Ionization LossIonization (collision) loss is given
by the Bethe-Bloch equation with two modifications Small electron mass means the
incident electron has significant recoil as it passes through material
Electrons are identical particles
The result is similar in appearance to Bethe-Bloch
3
Radiation Loss
Bremsstrahlung is an important process for x-ray production
Jackson gives a semi-classical derivation For a particle of charge ze, mass M, and initial
velocity , colliding with the Coulomb field of N charges Ze/V, the energy loss is
23/1
2
2
2222 233
ln3
16Mc
mZ
M
Mc
ez
c
eNZ
dx
dE
erad
4
Radiation LossSince bremsstrahlung depends on the
strength of the electric field felt by the electron, the amount of screening from atomic electrons plays an important role The effect of screening is parameterized using
The expression on the previous slide is for the case of high energy electrons where complete screening by atomic electrons occurs
screening complete toscorrespond 0
1003/1
2
ZEE
hvcmefinal
einitial
e
5
Screening
The screening parameter is related to the Fermi-Thomas model where one takes the form of the Coulomb potential to be
At large impact parameters screening effects from the atomic electrons causes the potential to fall off faster than 1/r
3/1
0
2
4.1
exp
Zaa
a
r
r
zZerV
6
Radiation Length
0/0
0
1
3/1
2
2
2222
0
0
2
is solution and length radiation thecalled is
233ln
3
16
writecan we, Since
Xx
e
rad
eExE
X
mZ
M
Mc
ez
c
eZNX
X
E
dx
dE
EγMc
7
Radiation Length
The radiation length X0 is The mean free path over which a high energy
electron’s energy is reduced by 1/e 7/9 of the mean free path for pair production
There are a number of empirical formulas for the radiation length But usually one takes it from a table (e.g.
those found at pdg.lbl.gov)
pairX 9
70
10
Critical Energy
Bremsstrahlung Energy loss dE/dx~ E
Ionization Energy loss dE/dx ~ ln E
Critical energy is that energy where dE/dxionization=dE/dxradiation
An oft-quoted formula is
24.1
610
Z
MeVEc
11
Critical Energy
An alternative definition of the critical energy is from Rossi
This form is somewhat more useful in describing EM showers
This form and the first definition are equivalent if
ce
ionization
EEdX
dE
0
00 X
E
X
E
dx
dE ce
radiation
15
Electron Range
As with protons and alphas, the electron range can be calculated in the CSDA approximation There will be contributions from ionization
and radiation CSDA range values can be found at NIST The CSDA range is the mean range for an
average electron but the fluctuations are large
Also the CSDA range does not include nuclear scattering contributions
19
Electron Range
While protons and alphas have a (more or less) well-defined range, the small electron mass produces significantly more scattering Backscattering can occur as well
20
EGS
The following plots come from the EGS Monte Carlo For a demo see
http://www2.slac.stanford.edu/vvc/egs/advtool.html
EGS was originally developed by SLAC but is now maintained by NRCC Canada (EGSnrc) and KEK in Japan (EGS4)
MCNP is a competitive Monte Carlo model One difference is that in MCNP many
interactions are summarized by random sampling at the end of each step while in EGS some interactions are modeled individually
23
EGSAt low Z, the agreement with experiment
is better than a percent ~5% disagreement at higher Z (Pb e.g.)
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