Triaxial angular momentum projection and configuration mixing calculations with the Gogny force

We present the first implementation in the $(\beta ,\gamma )$ plane of the generator coordinate method with full triaxial angular momentum and particle number projected wave functions using the Gogny force. Technical details about the performance of the method and the convergence of the results in both the symmetry restoration and the configuration mixing parts are discussed in detail. We apply the method to the study of ${}^{24}\mathrm{Mg}$; the calculated energies of excited states as well as the transition probabilities are compared to the available experimental data, showing a good overall agreement. In addition, we present the RVAMPIR approach, which provides a good description of the ground and $\gamma$ bands in the absence of strong mixing.

Figure 1

(a) Potential energy surfaces along the $\beta$ deformation for particle number projection and particle number and angular momentum projection (${}^{24}\mathrm{Mg}$). The bullets correspond to the excitation energies for the different GCM levels $(I,\sigma )$ with their positions at ${\mathrm{\beta ̄}}^{I\sigma }={\sum }_{\beta }\beta |F^{I\sigma }(\beta )|{}^2$. The energy is normalized to the GCM ground-state energy ($0_1^{+}$). (b) Collective wave functions for the $(\sigma =1)$ GCM levels. The values of the ordinate axis are displaced by 0.05 with increasing angular momentum. (c) Same as (b) but for the $(\sigma =2)$ GCM levels and $2_3^{+}$ state. Positive and negative values of $\beta$ correspond to prolate ($\gamma =0^{°}$) and oblate ($\gamma ={180}^{°}$) shapes, respectively.

Figure 2

Axial PNAMP-GCM excitation spectra of ${}^{24}\mathrm{Mg}$ obtained considering 7 shells (left) and 11 shells (right) in the calculations.

Figure 3

Orientations of the intrinsic deformation as a function of the $\gamma$ parameter. $\gamma =0^{°},{120}^{°},{240}^{°}$ and $\gamma ={60}^{°},{180}^{°},{300}^{°}$ correspond to axial symmetric prolate and oblate shapes, respectively.

Figure 4

Expectation values of the total angular momentum operator calculated with angular momentum projected states $|\mathit{IK}\rangle$ as a function of the $\beta$ deformation ($\gamma ={50}^{°}$) and for different sets of integration points in the Euler angles $(a,b,c)$ [red circles, $S_2=(16,16,32)$; black filled boxes, $S_1=(6,16,12)$]. The top and bottom panels correspond to $I=2$ and $I=6$, and their corresponding $K$ values, respectively.

Figure 5

Expectation values of the total angular momentum operator between angular momentum projected states $I=0,2,4,6,8;K=0$ and $I=3;K=2$ as a function of the $\beta$ deformation and for different sets of integration points in the Euler angles and orientation of the intrinsic axes [red circles, $S_2=(16,16,32)$; black filled boxes, $S_1=(6,16,12)$ with $\gamma ={50}^{°}$; blue filled diamonds, $S_1=(6,16,12)$ with $\gamma ={170}^{°}$].

Figure 6

(Left) Excitation energies and $B(E2)$ (in $e^2$ fm${}^4$ units) values for the states before and after $K$ mixing, $|\mathit{IMK}\rangle$ and $|\mathit{IM}\rangle ,$ respectively, with $\beta =0.625$ and $\gamma ={60}^{°}$. (Right) Excitation energies and $\mathit{BE}(2)$ values for the state $|\mathit{IMK}=0\rangle$ with $\beta =0.625$ and $\gamma ={180}^{°}$.

Figure 7

Mesh of points using constant step in $\beta$ and $\gamma$ in degrees (top left) or triangle division (top right) and the corresponding calculated VAP-PN potential energy surfaces (lower panels) transformed to the interval $\gamma$ $\epsilon [0^{°},{60}^{°}]$. The energy is normalized to the minimum of the PES ($-196.01$ MeV) and the contour lines are divided in 1-MeV (black dashed lines) and 2-MeV steps (continuous magenta lines).

Figure 8

PNAMP potential energy surfaces including $K$ mixing in the $(\beta ,\gamma )$ plane, with $\gamma$ in degrees, for $I=0$–8 and the first eigenvalues in $K$ space. The PES are normalized to the minimum of the surfaces ($-200.74$, $-199.43$, $-194.04$, $-196.61$, $-190.86$, $-192.27$, $-186.09$, and $-185.33$ MeV for $I=0$, 2, 3, 4, 5, 6, 7, and 8, respectively). The contour lines are divided in 1-MeV (black dashed lines) and 2-MeV steps (continuous magenta lines). States with projected norm less than ${10}^{-6}$ are removed.

Figure 9

Potential energy contour plots for ${}^{24}\mathrm{Mg}$ in the $(\beta ,\gamma )$ plane, with $\gamma$ in degrees, for $\gamma =0^{°}$–${360}^{ˆ}$ in different approaches and angular momenta normalized to the corresponding minima. The contour lines are divided in 1-MeV (black dashed lines) and 2-MeV steps (continuous magenta lines). (Left panels) Top: Particle number projection (VAP), $E_{\min }=-196.02$ MeV; bottom: PNAMP approach for $I=0\hslash$, $E_{\min }=-200.74$ MeV. (Middle panels) $\mathit{IK}$-projected energies according to Eq. (26). Top: $I=2,K=0$, $E_{\min }=-199.42$ MeV; bottom: $I=2,K=2$, $E_{\min }=-198.78$ MeV. (Right panels) Lowest eigenvalues of the PNAMP approach. Top: $I=2_1$, $E_{\min }=-199.43$ MeV; bottom: $I=2_2$, $E_{\min }=-195.18$ MeV.

Figure 10

RVAMPIR spectrum and transition probabilities.

Figure 11

GCM-PNAMP energies ($I=2$) as a function of the corresponding norm eigenvalue, normalized to the highest value, used as cutoff in the definition of the natural basis [Eq. (17)].

Figure 12

Calculated excitation energies and reduced transition probabilities $B(E2)$ (in $e^2{\text{fm}}^4$) in ${}^{24}\mathrm{Mg}$ obtained using axially symmetric (left) and triaxial (middle) GCM-PNAMP approaches compared to the experimental values (right). The widths of the arrows are proportional to the corresponding $B$ values. The experimental values are taken from Ref. [68].

Figure 13

Spectroscopic quadrupole moments as a function of the angular momentum $I$ calculated for the states in the first (black bullets), second (blue diamonds), and third (magenta triangles) bands and for the vibrational-rotational collective model with $K=0$ (red boxes) and $K=2$ (black circles).

Figure 14

GCM-PNAMP collective wave functions $|F^{I;\mathit{NZ};\sigma }(\beta ,\gamma )|{}^2$ in the $(\beta ,\gamma )$ plane, with $\gamma$ in degrees, for the ground-state (left), second (middle), and third (right) bands, respectively. Contour lines are separated in 0.01 units.