Calculation of the first four moments of electronic energy loss of protons in aluminium

We present a novel scheme for calculating electronic energy losses and their 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 completely filled valence band, and above this a forbidden band gap. The conduction electrons in a metal resemble most closely a free electron gas which would fill a Fermi sphere in velocity space (or in momentum space) homogeneously from nu = 0 to nu = nu (F). The core electrons are separated from the freely available phase space by distinct band gaps. For protons slowing-down in metals, this means that conduction electrons can accept small energy transfers which are inacceptable for core electrons or valence electrons in insulators. 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 proportional to nu (2), etc. We perform the stopping calculations first for the metal Al, where the theoretical input is well known, and where most measurements have been performed. The comparison of theory and experiments shows agreement within the experimental straggle.