Inelastic Scattering Cross Sections for 20-keV Electrons in Al, Be, and Polystyrene

Abstract

Measurements are reported of the inelastic scattering cross sections associated with the characteristic energy losses of 20-keV electrons transmitted through thin films of Al, Be, and polystyrene. These measurements have been made using two techniques which overcome several sources of systematic error in previous measurements. For materials which have narrow characteristic loss lines (such as Al), it is possible to establish the form of the differential cross section and the generalized oscillator strength for the \ensuremath{\approx} 15-eV Al plasmon energy loss for scattering angles between 0 and 20 mrad, the cutoff angle. The differential cross section has been used in repeated two-dimensional folding calculations to correct the intensity measurements of the multiple plasmon losses at the larger scattering angles where unambiguous intensity measurements could not be made. For materials which have broad characteristic loss peaks, such as polystyrene, it is not possible to establish the variation of generalized oscillator strength $f(q)$ with momentum transfer $q$. If it can be assumed that $f(q)\ensuremath{\approx}f(0)$ for small $q$, a cross section for a given loss can be obtained from an energy-loss spectrum measured at one scattering angle. The cross-section measurement accuracy is improved by a comparison of two energy-loss spectra, that of a standard (such as Al) and that of a material of unknown loss cross section, which have been obtained at zero angle under the same measurement conditions. Cross sections of (1.5\ifmmode\pm\else\textpm\fi{}0.4)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}18}$ ${\mathrm{cm}}^{2}$ and (1.5\ifmmode\pm\else\textpm\fi{}0.5)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}18}$ ${\mathrm{cm}}^{2}$ have been measured for the excitation of the \ensuremath{\approx} 15-eV and \ensuremath{\approx} 19-eV plasmon energy losses in Al and Be, respectively, using the former measurement technique. These values are in good agreement with the cross sections expected theoretically. The second measurement technique has been used to obtain cross sections of (2.4\ifmmode\pm\else\textpm\fi{}0.8)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}20}$ ${\mathrm{cm}}^{2}$ and (1.0\ifmmode\pm\else\textpm\fi{}0.2)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}18}$ ${\mathrm{cm}}^{2}$ for the 7-eV and 21-eV energy losses, respectively, in polystyrene. The corresponding oscillator strengths $f(0)$ are in good agreement with those deduced recently by LaVilla and Mendlowitz. Values were also obtained for the Al loss-dispersion constants, $\ensuremath{\Delta}E(\ensuremath{\theta})=14.8+(1.39\ifmmode\pm\else\textpm\fi{}0.07)\ifmmode\times\else\texttimes\fi{}{10}^{4}{\ensuremath{\theta}}^{2}+(1.2\ifmmode\pm\else\textpm\fi{}0.2)\ifmmode\times\else\texttimes\fi{}{10}^{7}{\ensuremath{\theta}}^{4}$ (energy loss $\ensuremath{\Delta}E$ in eV, scattering angle $\ensuremath{\theta}$ in radians), and for the critical wave vector for plasmon excitation ${k}_{c}=1.46\ifmmode\pm\else\textpm\fi{}0.07$ ${\mathrm{\AA{}}}^{\ensuremath{-}1}$.