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 MeV one considers this to be Bhabha or Moller scattering

2. 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

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 b, g colliding with the Coulomb field of N charges Ze/V, the energy loss is

4. 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
The expression on the previous slide is for the case of high energy electrons where complete screening by atomic electrons occurs

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

6. Radiation Length

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)

8. Radiation Length

9. Radiation Length
The radiation length (in cm) for some common materials

10. Critical Energy Bremsstrahlung Ionization
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

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

12. Critical Energy

13. Critical Energy

14. Electron Energy Loss Pb
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
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. EGS At 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