The development of experimental facilities that accelerate radioactive ion beams [1,2] has opened up a window to many nuclei that were heretofore inaccessible. With these new facilities and the new detector technology that is accompanying them, it is becoming possible to study the properties of nuclei very far from the valley of beta stability, all the way out to the particle ``drip lines" and perhaps even beyond.
Much work is now in progress to develop appropriate theoretical tools for describing nuclei in these exotic regimes [3]. A proper theoretical description of such weakly-bound systems requires a careful treatment of the asymptotic part of the nucleonic density. An appropriate framework for these calculations is Hartree-Fock-Bogoliubov theory, solved in coordinate representation [4,5,6]. This method has been used extensively in the treatment of spherical systems but is much more difficult to implement for systems with deformed equilibrium shapes [7,8,9].
In the absence of reliable coordinate-space solutions to the deformed HFB equations, it is useful to consider instead the configuration-space approach, whereby the HFB solution is expanded in a single-particle basis. One approach has been to use a truncated basis composed partly of discrete localized states and partly of discretized continuum and oscillating states [10,7,8]. Because of the technical difficulties in implementing this method, it has typically been restricted to include states in the continuum up to at most several MeV. As a consequence, such an approach should not be able to describe adequately the spatial properties of nuclear densities at large distances.
An alternative possibility is to expand in a basis of spatially localized states. Expansion in a harmonic oscillator (HO) basis is particularly attractive because of the simple properties of oscillator states. There have been many configuration-space HFB+HO calculations reported, either employing Skyrme forces or the Gogny effective interaction [11,12,13,14], or using a relativistic Lagrangian [15,16]. This methodology has proven particularly useful when treating nuclei in or near the valley of stability. For nuclei at the drip lines, however, the HFB+HO expansion converges slowly as a function of the number of oscillator shells [6], producing wave functions that decrease too steeply at large distances. The resulting densities, especially in the pairing channel, are artificially reduced in the outer region and do not reflect correctly the pairing correlations of these weakly-bound nuclei.
A related approach that has recently been proposed is to instead expand the quasiparticle HFB wave functions in a complete set of transformed harmonic oscillator (THO) basis states [19,17,18], obtained by applying a local-scaling coordinate transformation (LST) [20,21,22] to the standard HO basis. The THO basis preserves many useful properties of the HO wave functions, including its simplicity in numerical algorithms, while at the same time permitting us to incorporate the appropriate asymptotic behavior of nuclear densities.
Applications of this new HFB+THO methodology have been reported both in the non-relativistic [19,18] and relativistic domains [17]. In all of these calculations, specific global parameterizations were employed for the scalar LST function that defines the THO basis. There are several limitations in such an approach, however. On the one hand, any global parameterization of the LST function will of necessity modify properties throughout the entire nuclear volume, in order to improve the asymptotic density at large distances. This is not desirable, however, since the HFB+HO results are usually reliable in the nuclear interior, even for weakly-bound systems. In addition, because of the need to introduce matching conditions between the interior and exterior regions, a global LST function will invariably have a very complicated behavior, especially around the classical turning point, making it difficult to simply parameterize. Perhaps most importantly, the minimization procedure that is needed in such an approach to optimally define the basis parameters is computationally very time consuming, especially when a large number of shells is included, making it very difficult to apply the method systematically to nuclei across the periodic table.
In the present work, we propose a new prescription for choosing the THO basis. For a given nucleus, our new prescription requires as input the results from a relatively simple HFB+HO calculation, with no variational optimization. The resulting THO basis leads to HFB+THO results that almost exactly reproduce the coordinate-space HFB results for spherical [5] and axially deformed [10] nuclei and are of comparable quality to those of the former, more complex, HFB+THO methodology .
Because the new prescription requires no variational optimization of the LST function, it can be readily applied in systematic studies of nuclear properties. As the first such application, we carry out a detailed study of nuclei between the two-particle drip lines throughout the periodic table, using the Skyrme force SLy4 [23] and volume pairing [19]. In order to restore good particle number, we apply the Lipkin-Nogami (LN) prescription [24,25,26,27,28,29] followed by exact particle-number projection (PNP) [30].
The structure of the paper is the following. In Sec. 2, we briefly review the HFB and LN methods, noting several features particular to its coordinate and configurational representation. In Sec. 3, we introduce the THO basis and then formulate our new prescription for the LST function. The results of systematic calculations of even-even nuclei are reported in Sec. 4, with special emphasis on those nuclei that are at the neutron drip line and just beyond. Conclusions and thoughts for the future are presented in Sec. 5.