SURFACE-MODES OF LIQUID-HE-4

This method of correlated basis functions is applied at the variationa l level to give an optimized description, at zero temperature, of the structure and elementary excitations of liquid He-4 in the geometry of a half-space. A trial ground-state wave function of Hartree-Jastrow f orm is assumed, and the Feynman ansatz is adapted to construct trial e lementary excitations based on this variational ground state. Function al variation of the energy expectation value with respect to ground an d excited trial states leads, in conjunction with the Bogoliubov-Born- Green-Kirkwood-Yvon relations and the hypernetted-chain (HNC) equation s, to coupled Euler-Lagrange equations consisting of (i) a modified Ha rtree equation, (ii) a paired-phonon equation, and (iii) a renormalize d Bogoliubov eigenvalue equation. These relations and equations provid e for simultaneous optimal determination of (i) the density profile, t he chemical potential, and the Hartree inhomogeneity factor, (ii) the anisotropic two-body pseudopotential and two-body spatial distribution function, and (iii) the wave functions and energies of the Feynman ex citations as functions of the momentum parallel to the surface plane. In the numerical calculation reported, the bulk liquid density is take n equal to the experimental value at saturation. Since the correspondi ng Jastrow variational treatment of the bulk liquid does not produce a self-bound system at this density, an external potential is introduce d to stabilize the surface, its strength being adjusted so that the ca lculated chemical potential matches the experimental saturation value. The calculation yields dispersion relations for two distinct branches of bound surface states, extending from the continuum of liquid state s at small wave numbers to the continuum of liquid states close to the wave number characteristic of a bulk roton. The two branches are dist inguished by the number of nodes (zero or one) of the corresponding wa ve functions in the surface region. At small wave numbers, the wave fu nctions of the lowest-lying surface states penetrate exponentially int o the bulk liquid to a characteristic depth proportional to wavelength . These modes are associated with surface phonons and capillary waves, being driven by the external potential (renormalized by correlation e ffects due to the strong internal forces) and by the surface tension. The spectrum of surface excitations of the first branch follows the hy drodynamic dispersion relation in the range of wave numbers 0 less-tha n-or-equal-to q less-than-or-equal-to 0.5 angstrom-1. Employing a spec ialized version of the renormalized Bogoliubov equation, analytic expr essions are derived that permit evaluation of the speed of surface sou nd and the surface-tension coefficient in terms of quantities generate d by the microscopic calculation. In the opposite regime of large wave numbers corresponding to the atomic scale, q greater-than-or-equal-to 1 angstrom-1, the wave functions of the first branch are centered at a local density approaching that of the bulk liquid. The dispersion cu rves of both branches appear to terminate by merging with the bulk exc itation curve near the roton minimum, in conformity with the interpret ation of the bound surface states in this wave-number range as trapped rotons.