Reconciling Particle-Beam and Optical Stopping-Power Measurements in Silicon

A swift, charged particle passing through matter loses energy to electronic excitations \textit{via} the electro-magnetic transients experienced by atoms along its path. Bethe related this process to the matter's frequency-dependent dielectric function $\varepsilon (\hbar \omega )$ through the energy-loss function, Im[-1/$\varepsilon (\hbar \omega )$]. The matter's response may be summarized by a single parameter, the mean excitation energy, or $I$ value, that combines the optical excitation spectrum and excitation probability. Formally, ln $I$ is the mean of ln $\hbar \omega $ weighted by the energy-loss function. This provides an independent optical check on particle energy-loss experiments. However, a persistent disagreement is found for silicon: direct particle-beam studies yield 173.5\textless $I$\textless 176 eV, but a fit to the stopping-power of 36 elements suggests 165 eV. An independent determination from optical data in 1986 gave 174 eV supporting the higher values. However, recent x-ray measurements disclosed short comings in the 1986 optical data: 1. Measurements by Ershov and Lukirskii underestimated the L-edge strength, and 2. A power-law extrapolation overestimated the K-edge strength. We have updated these data and find $I=$ 162 eV, suggesting that silicon's recommended $I$ value should be reconsidered. While this 5{\%} change in $I$ value changes the stopping power by only 1{\%}, it is significant for precision measurements with Si detectors.