SURFACE-MODES OF LIQUID-HE-4

This method of correlated basis functions is applied at the variational 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 form is assumed, and the Feynman ansatz is adapted to construct trial elementary excitations based on this variational ground state. Functional variation of the energy expectation value with respect to ground and excited trial states leads, in conjunction with the Bogoliubov-Born-Green-Kirkwood-Yvon relations and the hypernetted-chain (HNC) equations, to coupled Euler-Lagrange equations consisting of (i) a modified Hartree equation, (ii) a paired-phonon equation, and (iii) a renormalized Bogoliubov eigenvalue equation. These relations and equations provide for simultaneous optimal determination of (i) the density profile, the 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 excitations as functions of the momentum parallel to the surface plane. In the numerical calculation reported, the bulk liquid density is taken equal to the experimental value at saturation. Since the corresponding Jastrow variational treatment of the bulk liquid does not produce a self-bound system at this density, an external potential is introduced to stabilize the surface, its strength being adjusted so that the calculated 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 states at small wave numbers to the continuum of liquid states close to the wave number characteristic of a bulk roton. The two branches are distinguished by the number of nodes (zero or one) of the corresponding wave functions in the surface region. At small wave numbers, the wave functions of the lowest-lying surface states penetrate exponentially into 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 effects due to the strong internal forces) and by the surface tension. The spectrum of surface excitations of the first branch follows the hydrodynamic dispersion relation in the range of wave numbers 0 less-than-or-equal-to q less-than-or-equal-to 0.5 angstrom-1. Employing a specialized version of the renormalized Bogoliubov equation, analytic expressions are derived that permit evaluation of the speed of surface sound and the surface-tension coefficient in terms of quantities generated 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 curves of both branches appear to terminate by merging with the bulk excitation curve near the roton minimum, in conformity with the interpretation of the bound surface states in this wave-number range as trapped rotons.