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##### Electrons

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**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**• 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**• The radiation length (in cm) for some common materials**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**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**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**• 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