In the first Born approximation, inelastic cross section depends only on velocity and the magnitude of the charge on the incident particle. Hence, an electron and a positron at the same velocity should have identical stopping powers, which should be the same as that of a proton at that velocity. In practice, there is some difference in the case of an electron because of the indistinguishability of the incident and atomic electrons. In describing an ionization caused by an incident electron, the more energetic of the two emergent electrons is called, by convention, the primary. Thus, maximum energy loss (ignoring atomic binding) is half the incident energy. Incorporating this effect, the stopping number of an electron is given by a complicated expression that involves a different arrangement of the parameters found in the stopping number of heavy charged particles; i.e.,
This stopping-power formula has a wide range of validity, from approximately a few hundred electron volts to a few million electron volts in materials of low atomic number. For low velocities, the Born approximation gradually breaks down, and highly excited states begin to be inaccessible to transitions by virtue of small maximum energy transfer. Yet, with some corrections the electron-stopping-power formula may be extended down to about 50 eV. Below that value any stopping-power formula is of doubtful validity, even though it is certain that most of the energy is still being lost to electronic states down to a few eV of energy.
On the high-velocity side, relativistic effects increase electron-stopping power from about 1,000,000 eV upward. Except for the term δ attributable to polarization screening, the relativistic stopping power tends to infinity as the electron velocity approaches the speed of light (v/c = β → 1). One-half of the stopping power, called the restricted stopping power, is numerically equal to the linear energy transfer and changes smoothly to a constant value, called the Fermi plateau, as the ratio β approaches unity. The other half, called the unrestricted stopping power, increases without limit, but its effect at extreme relativistic velocities (those very near the speed of light) becomes small compared with energy loss by nuclear encounters.
At extremely high velocities an electron loses a substantial part of its energy by radiative nuclear encounter. Lost energy is carried by energetic X rays (i.e., bremsstrahlung). The ratio of energy loss by nuclear radiative encounter to collisional energy loss (excitation and ionization) is given approximately by the incident electron energy (E) in units of 1,000,000 eV times atomic number (Z) divided by 800; i.e., EZ/800. For a large class of mediums (atomic number equal to or greater than 8; i.e., that for oxygen), the electron stopping is dominated by bremsstrahlung radiation for energies greater than 100 MeV.
Cherenkov radiation
When the speed of a charged particle in a transparent medium (air, water, plastics) is so high that it is greater than the group velocity of light in that medium, then a part of the energy is emitted as Cherenkov radiation, first observed in 1934 by Pavel A. Cherenkov, a Soviet physicist. Such radiation rarely accounts for more than a few percent of the total energy loss. Even so, it is invaluable for purposes of monitoring and spectroscopy. Cherenkov radiation is spread over the entire visible region and into the near ultraviolet and near infrared. The direction of its propagation is confined within a cone, the axis of which is the direction of electron motion.
Energy-transfer mechanism
At the low-velocity end of its path, an electron continues to excite electronic levels of atoms or molecules until its kinetic energy falls below the lowest (electronically) excited state (see ). After that it loses energy mainly by exciting vibrations in a molecule. Such a mechanism proceeds through the intermediary of temporary negative ion states, for direct momentum-transfer collisions are very inefficient. In a condensed medium (liquid, solid, or glass) very low-energy (less than 1 eV) electrons continue to lose energy by a process called phonon emission and by interaction with other low-frequency intermolecular motions of the medium.
An electron and a singly charged heavy particle with the same velocity have about equal stopping powers. Because of the small mass of the electron, however, the relative retardation (decrease in velocity per unit path length) is much more for it. This larger retardation for an electron means that, if an electron and a heavy particle start with the same velocity, the electron will have a much smaller range. Electron tracks show much more straggling and scattering compared with that of a heavy particle. The first effect results from the fact that the electron can lose a large fraction of its energy in a single encounter; the second is the result of small mass. A power law may be used to connect range and energy of electrons in a given medium—i.e., the range is proportional to energy raised to a power n; as in the case of a heavy particle, the index n is slightly less than two at high energies. At low energies the relationship is such that the exponent is one or less. Many formulas and tables are available for stopping powers and for ranges of electrons as well as of heavy particles over a wide range of energies.