Multiple scattering of 5-30 keV electrons in evaporated metal films II: Range-energy relations

Abstract

The ranges of electrons in evaporated films of aluminium, copper, silver and gold have been measured for incident energies E₀ between 5 and 20 keV. From electron transmission measurements the maximum range and the extrapolated range were determined. Energy loss measurements gave the mean range and the Thomson-Whiddington range, corresponding to the mean energy E_m and the most probable energy E_p of the beam, respectively. The mean range, which is directly comparable with that given by the Bethe energy loss law, is found not to be identical with the maximum range as assumed by some authors. In view of its dependence on the experimental conditions, the maximum range appears to have little practical and no theoretical value, whereas the extrapolated range R_x is a reproducible quantity. The extrapolated mass-range ρR_x is proportional to E₀^3/2 and is almost independent of atomic number in the energy range studied. A relationship between R_x and the Lenard absorption coefficient is found to exist. The Thomson-Whiddington `constant' does not have a unique value even for a given element at a given value of E<sub>0</sub> because the relation between thickness and E<sub>p</sub><sup>2</sup> is not linear for very thin films. The form of this relation agrees with that predicted by the Landau energy loss expression, but the experimental values of E<sub>p</sub> are 25-30% below the theoretical values. A square law is, however, strictly valid for the mean energy E<sub>m</sub> and gives an unambiguous definition of the range. Differences between the rate of fall of E<sub>m</sub><sup>2</sup> with thickness and that predicted by the Bethe energy loss law are qualitatively explained in terms of the variation of elastic scattering with Z and E<sub>0</sub>. The mean energy loss is always greater than predicted by the Bethe law, whilst the most probable loss is less than predicted by the Landau expression and always much less than the mean loss. In the thinnest films the magnitude of the most probable loss is about 15% greater than that expected from the characteristic (`plasma') energy loss; in thicker films it is three to five times greater. In general the differently defined ranges bear a simple relation to each other and to the `ultimate' range R_B found by integrating the Bethe energy loss expression, but departures from this scaling rule are found below 10 keV. Extrapolation of the (R_x, R_B) plot yields an intercept similar to that found in phosphors by Ehrenberg and King, but an order of magnitude smaller. The fact that both R_x and R_max are appreciably greater than R_B at low energies, especially in Al, is explained as due to the neglect of straggling of energy losses in deriving the Bethe expression.