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

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

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

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

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

Radiation Length

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)

Radiation Length

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

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

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

Critical Energy

Critical Energy

Electron Energy Loss Pb Note y-axis scale

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

Electron Range Al

Electron Range Pb

Electron Range Soft Tissue

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

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

EGS vs MCNP

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

EGS At low Z, the agreement with experiment is better than a percent ~5% disagreement at higher Z (Pb e.g.)

Electron Range 10 MeV electrons on 5cm x 5cm water

Electron Range 1 MeV electrons on 0.5cm x 0.5cm water

Electron Range 1 MeV electrons on 0.25cm x 0.25cm aluminum

Electron Range 100 keV electrons on 0.025cm x 0.025cm water