Calculation of the first four moments of electronic energy loss of protonsin aluminium

We present a novel scheme for calculating electronic energy losses and thei r higher moments which allows to fully account for the electronic structure of solids. Typical structures in solid materials are found in metals with their half filled conduction bands, and in insulators which exhibit a compl etely filled valence band, and above this a forbidden band gap. The conduct ion electrons in a metal resemble most closely a free electron gas which wo uld fill a Fermi sphere in velocity space (or in momentum space) homogeneou sly from nu = 0 to nu = nu (F). The core electrons are separated from the f reely available phase space by distinct band gaps. For protons slowing-down in metals, this means that conduction electrons can accept small energy tr ansfers which are inacceptable for core electrons or valence electrons in i nsulators. This leads to a completely different stopping behavior of metals , particularly at low velocities. In metals we expect that all energy loss moments increase strictly proportional to nu (n) starting at zero velocity i.e., the energy loss proportional to nu (1), the energy loss straggling pr oportional to nu (2), etc. We perform the stopping calculations first for t he metal Al, where the theoretical input is well known, and where most meas urements have been performed. The comparison of theory and experiments show s agreement within the experimental straggle.