Density profiles in a quantum Coulomb fluid near a hard wall
Résumé
Equilibrium particle densities near a hard wall are studied for a quantum fluid made of point charges which interact via Coulomb potential without any regularization. In the framework of the grand-canonical ensemble, we use an equivalence with a classical system of loops with random shapes, based on the Feynman-Kac path-integral representation of the quantum Gibbs factor. After systematic resummations of Coulomb divergences in the Mayer fugacity expansions of loop densities, there appears a screened potential phi. It obeys an inhomogeneous Debye-Hukel equation with an effective screening length which depends on the distance from the wall. The formal solution for phi can be expanded in powers of the ratios of the de Broglie thermal wavelengths lambda(alpha)'s of each species alpha and the limit of the screening length far away from the wall. In a regime of low degeneracy and weak coupling, exact analytical density profiles are calculated at first order in two independent parameters. Because of the vanishing of wave-functions close to the wall, density profiles vanish gaussianly fast in the vicinity of the wall over distances lambda(alpha)'s, with an essential singularity in Planck constant (h) over bar. When species have different masses, this effect is equivalent to the appearance of a quantum surface charge localized on the wall and proportional to (h) over bar at leading order. Then, density profiles, as well as the electrostatic potential drop created by the charge-density profile, also involve a term linear in (h) over bar and which decays exponentially fast over the classical Debye screening length xi(D). The corresponding contribution to the global surface charge exactly compensates the charge in the very vicinity of the surface, so that the net electric field vanishes in the bulk, as it should.