Transformed Harmonic Oscillator Basis

Going away from the beta stability valley towards particle drip lines, the Fermi energy becomes very small and the nucleonic densities and fields acquire large spatial extensions due to the coupling to the particle continuum. In this region of weakly bound nuclei, the asymptotic behavior of nuclear densities has an effect on nuclear properties. Consequently, when performing calculations for drip-line systems, it is important to have a firm grasp on physics at large distances. The recently developed HFB-THO technique based on the transformed harmonic oscillator (THO) method [27,28,22] is very helpful in this respect: it is fast, efficient, and easy to implement.

Figure 3: Comparison of the neutron densities (in logarithmic scale) calculated for the deformed nucleus $^{110}$Zn using coordinate-space 2D calculations (solid squares) with the configurational calculations based on THO (open squares) and HO (open circles) basis [29]. Each point corresponds to one Gauss-integration node in the $z-\rho$ plane, and the results are plotted as functions of the distance from the origin, $r=\sqrt {z^2+\rho ^2}$.

Figure 3 shows the neutron density of the deformed nucleus $^{110}$Zn obtained in two configurational calculations based on expansions in the harmonic oscillator (HO) and THO bases [27,28,22] compared to full-fledged 2D coordinate-space calculations [30,29] with the box boundary conditions. Every point in the figure corresponds to the value of the neutron density at a given Gauss-integration node in the $z-\rho$ plane. Since the nucleus is deformed, and there are always several nodes near a sphere of the same radius $r=\sqrt {z^2+\rho ^2}$, there can be seen some scatter of points corresponding to different densities in different directions. While the significant deviation from the correct decaying behavior is seen in the HO results, the THO expansion agrees very well with the deformed coordinate-space method. Other promising techniques that can alternatively be used in this context are the Gaussian-expansion basis method [31] and the Berggren expansion method [32].