Microscopic benchmark study of triaxiality in low-lying states of ${}^{76}\mathrm{Kr}$

We report on a seven-dimensional generator coordinate calculation in the two deformation parameters $\beta$ and $\gamma$ together with projection on three-dimensional angular momentum and two particle numbers for the low-lying states in ${}^{76}\mathrm{Kr}$. These calculations are based on covariant density functional theory. Excellent agreement is found with the data for the spectrum and the electric multipole transition strengths. This answers the important question of dynamic correlations and triaxiality in a fully microscopic way. We find that triaxial configurations dominate both the ground state and the quasi $\gamma$ band. This yields a different picture from the simple interpretation in terms of “coexistence of a prolate ground state with an oblate low-lying excited state,” which is based on the measured sign of spectroscopic quadrupole moments. This study also provides for the first time a benchmark for the collective Hamiltonian in five dimensions. Moreover, we point out that the staggering phase of the $\gamma$ band is not a safe signature for rigid triaxiality of the low-energy structure.

Figure 1

Potential-energy surfaces in the $(\beta ,\gamma )$ plane for ${}^{76}\mathrm{Kr}$: (a) constrained $\mathrm{RMF}+\mathrm{BCS}$, (b) $\mathrm{RMF}+\mathrm{BCS}$ with zero-point corrections, (c) with PNP, and (d) with both PNP and 3DAMP ($J=0$). Two neighboring contour lines are separated by 0.25 MeV.

Figure 2

Low-lying spectra and $B\left(E2\right)$ values (in $e^2$ fm${}^4$) of ${}^{76}\mathrm{Kr}$. Results from (b) the full relativistic GCM calculation with PNP and 3DAMP are compared with (c) 5DCH results (cf. Fig. 9 of Ref. [17]) and with (a) experimental data from Ref. [14].

Figure 3

Same as in Fig. 2. (a) The results from triaxial relativistic GCM calculations with 3DAMP and PNP are compared with (b) axial calculations with 1DAMP and PNP and with (c) axial calculations with 1DAMP only.

Figure 4

Square of the collective wave functions in the $(\beta ,\gamma )$ plane for the ground-state band of ${}^{76}\mathrm{Kr}$ calculated (a) by full $\mathrm{GCM}+\mathrm{PNDAMP}$ and (b) by 5DCH.

Figure 5

Same as in Fig. 4 but for the quasi-$\beta$ band calculated (a) by full $\mathrm{GCM}+\mathrm{PNDAMP}$ and (b) by 5DCH.

Figure 6

Same as in Fig. 4 but for the quasi-$\gamma$ band calculated (a) by full $\mathrm{GCM}+\mathrm{PNDAMP}$ and (b) by 5DCH.

Figure 7

Comparison between the results of full GCM and the 5DCH calculations for the low-lying states in ${}^{76}\mathrm{Kr}$. The same underlying EDF (PC-PK1) is used.

Figure 8

Staggering behavior of $\gamma$ band in ${}^{76}\mathrm{Kr}$ from the full GCM and 5DCH calculation in comparison with the data. The $S\left(4\right)$ and $S\left(5\right)$ values from the DF ($\gamma ={30}^{\circ }$) and WJ models are indicated by horizontal dashed lines.