Quantum corrections to the semiclassical Hartree-Fock theory of a harmonically trapped Bose gas

An analytical approach is proposed to investigate the mechanism of implantation of size selected clusters into graphite, in order to explain the origin of linear variation of measured penetration depth with momentum or energy of incident cluster. In agreement with experimental observations, the cluster experiences, during its penetration, a force which consists in a component proportional with cluster velocity and a constant component. Expressions of these forces were obtained in the frame work of this approach. Regardless of whether the cluster breaks down into single atoms on the surface or not, there is evidence for existence of a wave generated under impact of cluster on the surface. Under the assumption that the cluster does not break up at impact on the surface, the penetration depth depends on the cross-section between the cluster and the surface, the cluster velocity and the properties of graphite. When the cluster fragments upon the impact on the surface, the generated wave is followed by a collective motion (??collective cascade??) of displaced atoms of target, including the constituents of cluster themselves, due to the transfer of cluster momentum. Thus, it is these displaced atoms which penetrate in the medium. During this collective penetration, some constituents of cluster can reach a certain depth which may be considered as the range of the deepest implanted constituents of cluster. It is shown that, the depth of penetration depends on the initial radius of cluster, its velocity and the properties of graphite. In addition, the depth varies non linearly with cluster velocity, for small clusters (n ?? 7), while for large clusters (n ?? 13), it varies (i) linearly with cluster velocity (or momentum) when the force proportional with speed of cluster is dominant. (ii) Linearly with the square of cluster velocity (or energy) if the constant force becomes dominant. It is shown that, a mechanism based on a collective motion of displaced atoms including the constituents of cluster themselves, induced by transfer of cluster momentum to the medium, permits to explain the behavior of measured depth of implanted clusters into graphite. This collective motion involves only one free parameter for all clusters of the same nature which are used as projectiles in the same experiment.