Electrons
 Electrons lose energy primarily through
ionization and radiation
 dE 
 dE 
 dE 


 


 dx total  dx ionization  dx  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
1
Ionization Loss
Ionization (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
2
Radiation Loss
 Bremsstrahlung is an important process for xray production
 Jackson gives a semi-classical derivation

dE
dx rad
For a particle of charge ze, mass M, and initial
velocity b, g colliding with the Coulomb field of N
charges Ze/V, the energy loss is
16 2  e2  z 2e2 2  233M 
2



  NZ  
ln
g
Mc

2 
1/ 3


 3
 c  Mc   Z me 
3
Radiation Loss
 Since 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
100mec 2hv
 e
EinitialE efinal Z 1/ 3
  0 corresponds tocompletescreening

The expression on the previous slide is for the case
of high energy electrons where complete screening
by atomic electrons occurs
4
Screening
 The screening parameter is related to the
Fermi-Thomas model where one takes the form
of the Coulomb potential to be
zZe 2
 r
V r  
exp  
r
 a
a  1.4a0 Z
1/ 3
 At large impact parameters screening effects
from the atomic electrons causes the potential
to fall off faster than 1/r
5
Radiation Length
Since γMc  E , we can write
dE
E

dx rad
X0
2
1
16 Z 2e2  z 2e2   233M 

 ln 1 / 3 
X0   N
2 
c  Mc   Z me 
 3

X 0 is called theradiationlengthand solutionis
2
E  x   E0 e
x / X0
6
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
7
X 0   pair
9
 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)
7
Radiation Length
8
Radiation Length
 The radiation length (in cm) for some common
materials
9
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
610MeV
Ec 
Z  1.24
10
Critical Energy
 An alternative definition of the critical energy is
from Rossi
dE
 Ee  Ec
dX 0 ionization

This form is somewhat more useful in describing EM
showers
 This form and the first definition are equivalent
if
Ee Ec
dE


dx radiation X 0 X 0
11
Critical Energy
12
Critical Energy
13
Electron Energy Loss
Pb

Note y-axis scale
14
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
15
Electron Range
 Al
16
Electron Range
 Pb
17
Electron Range
 Soft Tissue
18
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
19
EGS
The following plots come from the EGS
Monte Carlo



For a demo see
http://www2.slac.stanford.edu/vvc/egs/adv
tool.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
20
EGS vs MCNP
21
EGS
 Valid for electron/photon energies from 1 keV
– 100 GeV
22
EGS
 At low Z, the agreement with experiment is
better than a percent

~5% disagreement at higher Z (Pb e.g.)
23
Electron Range
 10 MeV
electrons
on 5cm x
5cm
water
24
Electron Range
 1 MeV
electrons
on 0.5cm
x 0.5cm
water
25
Electron Range
 1 MeV
electrons
on
0.25cm x
0.25cm
aluminum
26
Electron Range
 100 keV
electrons
on
0.025cm
x
0.025cm
water
27