1

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

8

Radiation Length

9

Radiation Length

The radiation length (in cm) for some common materials

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

12

Critical Energy

13

Critical Energy

14

Electron Energy LossPb

Note y-axis scale

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

16

Electron Range

Al

17

Electron Range

Pb

18

Electron Range

Soft Tissue

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

21

EGS vs MCNP

22

EGS

Valid for electron/photon energies from 1 keV – 100 GeV

23

EGSAt low Z, the agreement with experiment

is better than a percent ~5% disagreement at higher Z (Pb e.g.)

24

Electron Range

10 MeV electrons on 5cm x 5cm water

25

Electron Range

1 MeV electrons on 0.5cm x 0.5cm water

26

Electron Range

1 MeV electrons on 0.25cm x 0.25cm aluminum

27

Electron Range

100 keV electrons on 0.025cm x 0.025cm water