Approximation to the coordinate-space HFB local densities

Our goal is to try to find an approximation to the exact (coordinate-space) HFB density that is based only on information contained in the HFB+HO results. Towards that end, we make use of the WKB asymptotic solution of the single-particle Schrödinger equation for a given potential $V(r)$, assuming that beyond the classical turning point only the state with the lowest decay constant $k$=2$\kappa$ contributes to the local density. Under this assumption, the logarithmic derivative of the density can be written as

$$\left. \frac{\rho ^{\prime }(r)}{\rho (r)}\right\vert _{r\lo... ...{\cal V}} - \frac{1}{2}\frac{{\cal V}'}{\kappa^2+{\cal V}} ~,$$ (20)

where the first term comes from the three-dimensional volume element, while the next two correspond to the first- and second-order WKB solutions [34]. The reduced potential ${\cal V}$,

$${\cal V}(r) = \frac{2m}{\hbar^2} V(r) = {\cal V}_N + \frac{\ell(\ell+1)}{r^2} + \frac{2m}{\hbar^2} \frac{Ze^2}{r},$$ (21)

is the sum of the nuclear, centrifugal, and Coulomb (for protons) contributions, with $\ell$ being the single-particle orbital angular momentum.

In practical applications, it turns out that near $R_m$ the next-to-lowest quasiparticle states still contribute to the local density $\rho$ in a way that may be more important than the second-order WKB term shown in Eq. (20). Moreover, in deformed nuclei the quasiparticle states do not have good total angular momentum $\ell$, so that several quasiparticles may contribute to the asymptotic density depending on their $\ell$-content and the value of $\kappa$. Therefore, we need a practical prescription to fix a reasonable approximate asymptotic form of the density with minimal numerical effort but high reliability. This can be achieved by using in (20) a reduced potential of the form

$${\cal V}(r) = \frac{C}{r^2} + \frac{2m}{\hbar^2} \frac{Ze^2}{r},$$ (22)

where the nuclear part ${\cal V}_{N}$ (which is small around and beyond $R_m$) is neglected, and the coefficient $C$ is allowed to differ from its centrifugal barrier value $\ell(\ell+1)$. The actual value of $C$ is fixed by the requirement that the logarithmic derivative (20) coincides at the mid point $R_{m}$ with the $\ell$=0 component of the HFB+HO density, i.e., with

$$\bar{\rho}(r)=\int_0^{\pi/2} \bar{\rho}(r,\theta) P_{\ell=0}(\cos(\theta)) \sin(\theta) d \theta .$$ (23)

Next, in order to make a smooth transition from the HFB+HO density $\bar{\rho}(r)$ in the inner region to the approximate asymptotic expression (20) in the outer region, we introduce the following approximation $\tilde{\rho}$ for the logarithmic density derivative:

$$\frac{\tilde{\rho}^{\prime }(r)}{\tilde{\rho}(r)}=\left\{ \b... ...}{\kappa ^{2}+V}$& for $r\geq R_{\max }$. \end{tabular}\right.$$ (24)

The coefficients $a$ and $b$, and the power $s$, are determined from the condition that the logarithmic derivative (24) and its first derivative are smooth functions at the points $R_{\min }$ and $R_{\max }$. Note that the first derivative of (24) at $R_{\min }$ is automatically equal to zero, so that there is no need to introduce a fourth parameter to satisfy this condition.

Having determined the smooth expression for the logarithmic derivative of $\tilde{\rho}(r)$, we can derive the approximate local density distribution $\tilde{\rho}(r)$ by simply integrating Eq. (24). The result is

$$\tilde{\rho}(r)=\left\{ \begin{tabular}{lr} $\bar{\rho}(r)$&... ...^{2}+{\cal V}}}$& for$\;r\geq R_{\max }$, \end{tabular}\right.$$ (25)

where the integration constants $A$ and $B$ are determined from the matching conditions for the density at points $R_{\min }$ and $R_{\max }$, respectively. Finally, $\tilde{\rho}(r)$ is normalized to the appropriate particle number.

The approximate density (25) works fairly well for all nuclei that we have considered. This is illustrated in Fig. 1 where the approximate density $\tilde{\rho}$ (circles) is seen to be in perfect agreement with the coordinate-space HFB results.

It should be stressed that the above procedure is applicable only when the number of shells is large enough that the HFB+HO density has a minimum at the point $R_{\min }$. The minimum value of $N_{{\rm sh}}$ required to satisfy this condition depends on the particular deformations or on the nuclei considered. For the number of shells $N_{{\rm sh}}=20$ used in our calculations, the above condition is always satisfied.