Symbolic-Numeric Algorithm for Calculations in Geometric Collective Model of Atomic Nuclei

Abstract

We developed a symbolic–numeric algorithm involving a set of effective symbolic and numerical procedures for calculations of low lying energy spectra and eigenfunctions of atomic nuclei. The eigenfunctions are expanded over the orthonormal noncanonical $U(5) {\supset } O(5) {\supset } O(3)$ basis in Geometric Collective Model. We give implementation of the algorithm and procedures in Wolfram Mathematica. We present benchmark calculations of energy spectrum, quadrupole moment and the reduced upwards transition probability B(E2) for the nucleus $^{186}$Os.

Appendices

A Appendix. Sets of Input Parameters for Atomic Nuclei

To denote approximately a range of applicability of the GCM code and to make it more friendly for users, we will accompany it by the sets of input files with the values of sets of parameters for atomic nuclei given in the papers  [9, 12, 20].

For example, we present some of them in Tables 11, 12, 13, 14, 15, and 16. In Table 11 the macroscopic potential parameters are given. The value of $C_2$ is increased as we approach to double closed shell. Even the potential depends more on the quadratic term over $\beta$, it is not completely quadratic even if one approaches very close the double closed shell. Because of the great similarity, the authors only depict the PES of the $^{298}$114 and $^{304}$120 in Figs. 29 and 30 in Ref. [9]. The PES is perfectly spherical, thus, the spectrum will be that of a five-dimensional oscillator: The energy scales as $\hbar \sqrt{C_{2}/B_{2}}$. The first excited state is a $2^+$ state at the energy $\hbar \sqrt{C_{2}/B_{2}}$ and at twice this energy, there are three degenerate states with spin and parity $0^+$, $2^+$ and $4^+$. The first 3+ state is three times the energy of the first $2^+$ state. For completeness, in Fig. 31 in Ref. [9], the authors depict the spectrum of the $^{298}$114 nucleus as predicted by the GCM [9].

Table 16. The values of the phenomenological potential parameters for $^{184}$W are determined by fitting [12].

The only parameter, which cannot be deduced is the collective mass $B_2$ of the geometrical model [8]. This parameter has to be adjusted to, e.g., a particular state in the ground state band. Also assuming for neighboring nuclei the same value of $B_2$ is in general far more accurate than using the Cranking Model. For the case of nuclei in the island of stability, one will use a generic value, i.e., results will scale with $B_2$ (as it is pointed out in page 128 in Ref. [9]).

B Appendix. Boundary Value Problem for GCM Model

The equation of geometric collective model (GCM) with respect to components $\varPhi _{nK}^{L}=\varPhi _{nK}^{L}(\beta ,\gamma )$ and eigenvalue $E_{n}^{L}$ (in MeV), $\bar{B}_{2}=2B_{2}/\sqrt{5}$ in ($10^{-42}$MeV s$^2$) and $C_{2}$ in (MeV) are mass and stiffness parameters, variable $\beta$ in (fm), reads as

$$\begin{aligned}&(T(\beta ,\gamma ) {+} T^{L}_{K}(\beta ,\gamma ) {+}\hat{V}(\beta ,\gamma ){-} E_{n}^{L})\varPhi _{nK}^{L}( \beta ,\gamma )=\sum _{K'=K\pm 2 even}\!\!\!\!\!\! V_{KK'}^{L}(\beta ,\gamma )\varPhi _{vK'}^{L}(\beta ,\gamma ), \\&T( \beta ,\gamma )=\frac{\hbar ^{2}}{2\bar{B}_{2}}\left( {-}\frac{1}{\beta ^{4}} \frac{\partial }{\partial \beta }\beta ^{4}\frac{\partial }{\partial \beta }{-}\frac{1}{\beta ^{2}\sin (3\gamma )} \frac{\partial }{\partial \gamma }\sin (3\gamma )\frac{\partial }{\partial \gamma }\right) {+}\mathcal{K}(\beta ,\gamma ),\nonumber \\&T^{L}_{K}( \beta ,\gamma )= +\frac{\hbar ^2}{2\bar{B}_{2}} \left[ (L(L+1)-K^{2})\left( \frac{2 \bar{B}_{2}}{4J_{1}}+ \frac{2 \bar{B}_{2}}{4J_{2}}\right) +\frac{K^{2} 2 \bar{B}_{2}}{2J_{3}}\right] ,\nonumber \\&V^{L}_{KK'}(\bar{\beta },\gamma )={-}\frac{\hbar ^2}{2\bar{B}_{2}}\left[ \frac{2\bar{B}_{2}}{8J_{1}}{-} \frac{2\bar{B}_{2}}{8J_{2}}\right] C_{KK'}^{L},\, \; C_{KK'}^{L}=\delta _{K'K{-}2}C_{KK{-}2}^{L}{+}\delta _{K'K{+}2}C_{KK{+}2}^{L},\nonumber \\&C_{KK-2}^{L}=(1+\delta _{K2})^{1/2}[(L+K)(L-K+1)(L+K-1)(L-K+2)]^{1/2},\nonumber \\&C_{KK+2}^{L}=(1+\delta _{K0})^{1/2}[(L-K)(L+K+1)(L-K-1)(L+K+2)]^{1/2}, \nonumber \end{aligned}$$

(40)

and the moments of the inertia denoted as ${J_{k}=4\bar{B}_{(k)}\beta ^{2}\sin ^{2}(\gamma {-}\frac{2}{3}k\pi )}$, where $k=1,2,3$ and $\bar{B}_{(k)}= \bar{B}_{2}$ is a mass parameter, with potential function $\hat{V}(\beta ,\gamma )$ from (3), (4) and (5), and input set of parameters from Tables 11, 12, 13, 14, 15 and 16 in Appendix A, and additional kinetic function $\mathcal{K}(\beta ,\gamma )$ determined in  [11, 14, 17, 20, 23]. The bounded components $\phi _{vK}^{L}$ are subjected to homogeneous Neumann or Dirichlet boundary conditions at the boundary points of interval $\gamma =0$ and $\gamma =\pi /3$ for zero or odd values of L (for details of boundary conditions on interval of the $\beta$ variable see [18, 19, 23]), and orthonormalization conditions (see Eq. (15))

$$\begin{aligned} \int _{\beta =0}^{\beta _{max}}\int _0^{\pi /3} \sum _{K even}\varPhi _{n'K}^{L}(\beta ,\gamma )\varPhi _{nK}^{L}(\beta ,\gamma ) \sin (3\gamma )d\gamma \beta ^{4}d \beta = \delta _{n'n}. \end{aligned}$$

(41)

The BVP (40)–(41) will be solved by the FEM implemented in the CAS code.