Electrons

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

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 • 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 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. 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