Thanks to recent advancess in radioactive ion beam technology, we are now in the process of exploring the very limits of nuclear binding, namely those regions of the periodic chart in the neighborhood of the particle drip lines [1,2,3,4]. Several new structure features have already been uncovered in these studies, including the neutron halo, and others have been predicted.
In contrast to stable nuclei within or near the valley of beta stability, a proper theoretical description of weakly-bound systems requires a very careful treatment of the asymptotic part of the nucleonic density. This is particularly true in the description of pairing correlations near the neutron drip line, for which the correct asymptotic properties of quasiparticle wave functions and of one-particle and pairing densities is essential. In the framework of the mean-field approach, the best way to achieve such a description is to use the Hartree-Fock-Bogoliubov (HFB) theory in coordinate-space representation [5,6,7].
Such an approach presents serious difficulties, however, when applied to deformed nuclei. On the one hand, for finite-range interactions the technical and numerical problems arising when a two-dimensional mesh of spatial points is used are so involved that reliable self-consistent calculations in coordinate space should not be expected soon. On the other hand, for zero-range interactions existing approaches [8,9] are able to include only a fairly limited pairing phase space. The main complication in solving the HFB equations in coordinate space is that the HFB spectrum is unbounded from below, so that methods based on a variational search for eigenstates cannot be easily implemented. Because of this and other difficulties, one has to look for alternative solutions.
In principle, such an alternative solution is well known in the form of the configurational representation. In this approach, the system of partial differential HFB equations are solved by expanding the nucleon quasiparticle wave functions in an appropriate complete set of single-particle wave functions. In many applications, an expansion of the HFB wave function in a large harmonic oscillator (HO) basis of spherical or axial symmetry provides a satisfactory level of accuracy. For nuclei at the drip lines, however, expansion in an oscillator basis converges much too slowly to describe the physics of continuum states [7], which play a critical role in the description of weakly-bound systems. Oscillator expansions produce wave functions that decrease too steeply in the asymptotic region at large distances from the center of the nucleus. As a result, the calculated densities, especially in the pairing channel, are too small in the outer region and do not reflect correctly the pairing correlations of such nuclei.
In two recent works [10,11], a new transformed harmonic oscillator (THO) basis, based on a unitary transformation of the spherical HO basis, was discussed. This new basis derives from the standard oscillator basis by a local-scaling point coordinate transformation [12,13,14], with the precise form dictated by the desired asymptotic behavior of the densities. The transformation preserves many useful properties of the HO wave functions. Using the new basis, characteristics of weakly-bound orbitals for a square-well potential were analyzed and the ground-state properties of some spherical nuclei were calculated in the framework of the energy density functional approach[11]. It was demonstrated in [10] that configurational calculations using the THO basis present a promising alternative to algorithms that are being developed for coordinate-space solution of the HFB equations.
In the present work, we develop the THO basis for use in HFB equations of axially-deformed weakly-bound nuclei. Our main goal here is to present and test these new theoretical methods. As specific applications, we repeat previous calculations performed for the chain of Mg isotopes [8], but for different effective interactions, and then report a preliminary study of light, neutron-rich nuclei near the drip line. Extensive calculations throughout the mass table, together with a more detailed analysis of the pairing interaction, will be presented in a future publication.
The structure of the paper is the following. The THO basis for deformed nuclei is introduced in Sec. 2. In Sec. 3 we present an outline of the HFB theory and discuss several features of particular relevance to our investigation. Results of calculations are given in Sec. 4, and conclusions are presented in Sec. 5.