We present the finite amplitude method (FAM), originally proposed in Ref. [17], for superfluid systems. A Hartree-Fock-Bogoliubov code may be transformed into a code of the quasi-particlerandom-phase approximation (QRPA) with simple modifications. This technique has advantages over the conventional QRPA calculations, such as coding feasibility and computational cost. We perform the fully self-consistent linear-response calculation for a spherical neutron-rich nucleus 174 Sn, modifying the HFBRAD code, to demonstrate the accuracy, feasibility, and usefulness of the FAM.

The self-consistent mean-field quantum mechanical solution of a vortex and a nucleus immersed in a sea of free neutrons, a scenario representative of the inner crust of neutron stars, is presented for the first time. Because of quantal size effects the phase space for vortices inside the nucleus is essentially zero, so that the vortex core opens up and surrounds the nucleus. As a consequence, pinned configurations (in which a vortex becomes anchored to the nucleus) are favored at low and high densities in the inner crust. This result is qualitatively different from that obtained in all previous models, which predict pinning at intermediate densities. In the inner crust of a neutron star atomic nuclei are immersed in a sea of free superfluid neutrons permeated by an array of vortices. It has been proposed that the glitches, sudden irregularities in the rotation rate observed in many neutron stars, can be viewed as "vorticity jumps," equivalent to "flux jumps" in a superconducting magnet [1]. To make quantitative comparisons with existing data, one needs to develop detailed models of vortex line configurations and of vortex dynamics (see, e.g., Ref.[2]). In turn, the latter crucially depends on whether the vortex anchors itself to the nuclei or not, a phenomenon controlled by the so-called pinning energy. For example, one needs to know the pinning energy as a function of density to address the question of the global stability of a pinned lattice, and to interpret precession observations [3].To make a reliable calculation of the pinning energy two questions are to be answered: 1. How important are quantal, finite-size (proximity) effects?, 2. Which role is played by polarization effects on the pairing gap of the system? Although a quantitative estimate of the pinning energy requires that one gives an answer to both questions, qualitatively the two issues are quite independent of each other.In the following, we present the results of a calculation of the vortex-nucleus interaction in the inner crust that represents a clear improvement on previous work [4-6], because (a) it is based on quantal mean-field theory (Ref.[4] was based on Ginzburg-Landau equations, Ref.[5] used a fixed WoodsSaxon potential, whereas Ref.[6] was based on a semiclassical approximation) and (b) it only assumes axial symmetry for the neutron density (Ref.[5] assumed cylindrical symmetry). Our findings turn out to be qualitatively different from all previous ones. In particular, by comparing them with those obtained within the framework of the semiclassical approximation [6], we are able to answer for the first time the aforementioned question 1.Arguably, the most accurate (and recent) answer to question 2 indicates that polarization effects play a minor role, being much smaller than previously estimated [7]. In keeping with this result, we have used a pairing interaction that reproduces the pairing gap calculated with a bare force in uniform neutron matter.Our calculations were performed by solving the mean-field Hartree-Fock-Bogoliubov (HFB) e...