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V Encuentro Sud Americano de Colisiones Inelásticas en la Materia LPA stopping power of swift ions in solids. Modeling the inhomogeneous electron gas J. M. Fernández-Varea 1 , C. D. Denton 2 , I. Abril 2 and R. Garcia-Molina 3 1 Facultat de Física (ECM and ICC), Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain 2 Departament de Física Aplicada, Universitat d’Alacant, Apartat 99, E-03080 Alacant, Spain 3 Departamento de Física-CIOyN, Universidad de Murcia, Apartado 4021, E-30080 Murcia, Spain email address corresponding author: jose@ecm.ub.es The stopping of swift ions in solids is still a subject of intense research in spite of the many decades elapsed since the pioneering works of Bohr, Bethe and others. There are numerous theories that describe the energy loss of a heavy charged particle (charge Z 1 , velocity v) in a free-electron gas (FEG) of uniform density ρ, such as the dielectric formalism or non-linear methods [1,2]. If the projectile penetrates a medium with a varying local electron density, e.g. a crystalline solid, the local-plasma approximation (LPA) lets us express the stopping power as where V is the volume of the unit cell. We find it convenient to split ρ(r) into contributions from atomic inner shells (cores) and weakly-bound (valence) electrons, i.e. difficult to implement. As an alternative, the ab initio TB-LMTO method [4,5] has been used occasionally to model the stopping of swift ions in channeling conditions [6]. In the present work, Roothaan-Hartree- Fock wave function [7] are employed to model the electron density of the cores. In turn, the program TB-LMTO-ASA [4,5] is adopted to generate the 3D valence-electron densities. These methods are used to calculate the electron densities of solid Al, Si, Cu, Ag, and Au. In order to avoid computing the LPA stopping power due to valence electrons as a 3D integral, the probability distribution function p(ρ v ) is first extracted from ρ v (r) because then Figure 1 shows the p(ρ v ) distributions of solid Al, Si, and Cu. Neglecting the overlap between the two electron densities, the LPA stopping power may be written as Electrons in the cores are barely disturbed by the solid-state environment. Hence, the corresponding density is spherically symmetrical and can be evaluated straightforwardly using atomic wave functions. On the other hand, the spatial distribution of valence electrons is strongly affected by aggregation effects. and may display a non-negligible anisotropy. The 3D electron density can be calculated with the Dawson- Stewart-Coppens formalism. However, so far this method has found only limited application (see e.g. reference [3]), possibly because it is Figure 1. Probability distribution function p(ρ v ) of solid Al, Si, and Cu, extracted from the 3D valence-electron densities calculated with the TB-LMTO-ASA program. 44 Valparaíso, Chile
V Encuentro Sud Americano de Colisiones Inelásticas en la Materia As an application of the presented methods we investigate the stopping power of H and He ions in the aforementioned solids. For the sake of simplicity we resort to the well-known relation [1,2,8,9]) which is the low-velocity form of non-linear theory. The transport cross section is evaluated numerically from phase shifts determined in a self-consistent way so as to satisfy the Friedel sum rule [1,2,8,9]. The LPA friction coefficient Q=S/v is found to change 4% in the case of slow H and He ions in Al compared to the values obtained assuming a constant density for the valence electrons. In turn, for H and He in Si the change amounts to 12% and 16%, respectively. These trends are to be expected in sight of the valence electron density distributions displayed in figure 1. Work is in progress to explore further consequences of the presented decomposition of the local electron density. In particular, the stopping power at higher velocities is being studied with the non-linear model of Nagy and Apagyi [10]. [8] P. M. Echenique, R. M. Nieminen, J. C. Ashley, and R. H. Ritchie, Phys. Rev. A 33, 897 (1986). [9] J. Calera-Rubio, A. Gras-Martí, and N. R. Arista, Nucl. Instrum. Meth. B 93, 137 (1994). [10] I. Nagy and B. Apagyi, Phys. Rev. A 58, R1653 (1998). References [1] P. M. Echenique, F. Flores, and R. H. Ritchie, Solid State Phys. 43, 229 (1990). [2] Adv. Quantum Chem., volumes 45 and 46 (2004). [3] J. Sillanpää, J. Peltola, K. Nordlund, J. Keinonen, and M. J. Puska, Phys. Rev. B 63, 134113 (2001). [4] O. K. Andersen and O. Jepsen, Phys. Rev. Lett. 53, 2571 (1984). [5] http://www.fkf.mpg.de/andersen/ [6] J. E. Valdés, P. Vargas, C. Celedón, E. Sánchez, L. Guillemot, and V. A. Esaulov, Phys. Rev. A 78, 032902 (2008). [7] C. F. Bunge, J. A. Barrientos, and A. V. Bunge, At. Data Nucl. Data Tables 53, 113 (1993). 45 Valparaíso, Chile
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