Nuclear Instruments and Methods in Physics Research B48 (1990) 29-33 North-Holland 29 zyxwvutsr STOPPING POWER AND ENERGY LOSS SPECTRUM OF A DENSE MEDIUM OF BOUND ELECTRONS Ali BELKACEM and Peter SIGMUND * Physics Division, Argonne National Laboratory Argonne, IL 60439, USA We have derived the dielectric function ~(k, w) in the Lindhard approximation for a medium consisting of electrons individually bound by harmonic forces. This function is expressible in terms of a hypergeometric series and approaches well-known results in the limits of negligible binding, large momentum transfer, and long wavelength, respectively. The stopping power of the medium for a moving point charge scales very well with the shifted resonance frequency a, = (& + w$)‘/* (oe = oscillator frequency; w n = plasma frequency) that follows from classical dispersion theory. The discrete excitation levels of an isolated harmonic oscillator are increasingly shifted and broadened with increasing density of the medium. The results for both the excitation spectrum and the stopping power differ noticeably from free-electron behavior even at a rather high electron density. 1. Introduction Electronic excitations induced by charged particles are described with considerable rigor by the Bethe the- ory [l]. Although this theory was derived originally for a dilute medium, i.e., an atomic or molecular gas, some important features, such as the stopping power formula, remain valid for condensed media, provided that perti- nent parameters such as oscillator strengths are rede- fined accordingly [2-41. However, quantitative evalua- tions from first principles have concentrated on the case of an isolated atom [5] or ion [6]. Electronic excitations in solids and liquids can con- veniently be modelled by replacing the penetrated medium by a homogeneous electron gas [2,7,8]. This implies that the medium is characterized by an electron density N which determines a plasma frequency wP and a Fermi speed ur. In such a model, possible differences between metals and semiconductors are eliminated from the beginning. More generally, the question may be asked how the discrete excitation spectrum of an atom translates into the continuous excitation spectrum of an electron gas. In the present communication, we consider the penetration of a heavy point charge (typically a proton) through a medium containing an arbitrary density of bound electrons. The emphasis is laid on providing a model that contains the above cases of a dilute gas and a homogeneous electron gas as simple limiting cases, * Argonne Fellow 1988-89; present (permanent) address: Physics Department, Odense University, 5230 Odense, Den- mark. 0168-583X/90/$03.50 0 Elsevier Science Publishers B.V. (North-Holland) rather than finding a comprehensive picture allowing for all the complexities of electron structure in a con- densed medium. In the simplest explicit version of the Bethe theory, the target atom is modelled as a harmonic oscillator [9], in much the same way as in Bohr’s classical theory of particle stopping [lo]. The essential parameter in this theory is the resonance frequency o0 of the individual oscillator. From classical dispersion theory, it is well known that in a dense medium consisting of harmonic oscillators, the resonance frequency is shifted toward a0 = (&Jo’ + cozpy2, where wP is the plasma frequency /4nNe2/m. One may expect this shifted resonance frequency also to play a role in a quantum theory of electron excitation. Although identified commonly with the free-electron gas, the Lindhard description [2] is not restricted to this case. The general scheme is readily applicable also to bound electrons. For the specific case of harmonic binding, the dielectric function is no more complicated than that of the Fermi gas, as will be shown below. A system of harmonically bound electrons should exhibit the limiting properties of the free-electron gas and an isolated harmonic oscillator for high and low electron density, respectively. Therefore, it must be sui- table to study variations in the excitation spectra with increasing electron density as well as the relevance of eq. (1) in the quantum theory of particle stopping. In the present study, we restrict our attention to the lowest order in the projectile charge e,. In a separate paper, some aspects have been analysed to the next order in e, (Barkas effect) [ll]. I. EXCITATION, STOPPING