Potential energy surfaces

Figure 1 shows the potential energy surfaces (PESs) in the (β, γ)-plane for the 12 triaxial nuclei obtained from the CDFT calculations with an effective interaction PC-PK1 [60]. For comparison, the experimental deformation parameters from previous studies [12] are also included for comparison. For ¹¹⁰Ru, a triaxial deformation of $(\beta =\mathrm{0.26,}\gamma =40)$ is observed. The predicted β and γ values are slightly smaller and larger than the experimental values of $0.310(11)$ and $29.0(4.8)$ [12], respectively. The PES of this nucleus is relatively flat in the β- and γ-directions towards the oblate side. By contrast, for ¹⁵⁰Nd, ¹⁵⁶Gd, ^166,168Er, ¹⁷²Yb, and ^182,184W, the minimum is located at $\gamma =0$, i.e., it exhibits a prolate shape, which disagrees with the triaxial deformation indicated by the experimental data. Around the calculated minimum, the curve is relatively flat along the γ-direction towards the experimental deformation parameters ($\approx 0.5\text{ MeV}$). Nevertheless, these findings underscore the limitations of the mean-field approximation used in the CDFT calculations and suggest that the additional refinements, such as those for the fluctuation effects, are necessary to improve the predictions for these nuclei (c.f. Fig. 10). For ^186-192Os, the CDFT calculations show good agreement with the experimental data for ground-state deformation parameters. The minima γ values range between 20° and 30°, indicating the importance of triaxial deformation in these nuclei.

Fig. 1Full-screen View

(Color online) Potential energy surfaces in the (β, γ)-plane for 12 nuclei calculated via CDFT. All energies are normalized with respect to the minimum (stars). The contour lines are spaced at 0.5 MeV intervals. Experimental deformation parameters from the literature [12] are included (dots) for comparison

Fig. 10Full-screen View

(Color online) Deformation parameters $\overline{\beta }$ (upper) and $\overline{\gamma }$ (lower) of the $0_1^{+}$ state in 5DCH (open circles) compared with the ground state deformation parameters in CDFT (open squares) and experimental data (solid circles) for 12 nuclei [12]. In the 5DCH results, the Δ β and Δ γ fluctuations are depicted as positive and negative error bars for $\overline{\beta }$ and $\overline{\gamma }$, respectively. The light-blue band represents the region of remarkable triaxial deformation

Moments of inertia

Using the obtained single-nucleon wave functions, energies, and occupation factors generated from the constrained self-consistent CDFT solutions, we calculate the MoIs using the IB formula for all three principal axes [26, 27, 23, 33, 36]. The calculated MoIs, denoted as ${\mathcal{J}}_{\text{CDFT}\mathrm{,}k}$, are plotted as a function of ${\beta }^2{{\displaystyle \sin }}^2\left(\gamma -2k\pi /3\right)$ and γ, as shown in Fig. 2. For comparison, the experimental values, denoted as ${\mathcal{J}}_{\text{Exp}\mathrm{,}k}$ [12], are also plotted. As shown in Fig. 1, the predicted deformation parameters obtained via CDFT are not ideally identical to the experimental parameters. This can lead to ambiguity when comparing the CDFT-predicted and experimental MoIs. To address this issue, the deformation parameters used in the CDFT calculations are constrained to be the same as the experimental parameters [12]. Finally, the obtained ${\mathcal{J}}_{\text{CDFT}\mathrm{,}k}$ are compared with ${\mathcal{J}}_{\text{rig}\mathrm{,}k}$ (1). and ${\mathcal{J}}_{\text{irr}\mathrm{,}k}$ (2).

Fig. 2Full-screen View

(Color online) Left: CDFT, experimental, and irrotational-flow MoIs relative to the rigid-body value as functions of ${\beta }^2{{\displaystyle \sin }}^2\left(\gamma -2k\pi /3\right)$ for the 1-, 2-, and 3-axes, corresponding to the m, s, and l axes, respectively. Right: CDFT and experimental MoIs relative to the irrotational flow value as functions of γ for the m, s, and l axes. The dashed lines represent the average ratio of the 12 nuclei

Figure 2 shows that both the experimental and CDFT MoIs fall between the expectations of rigid and irrotational motions. This suggests that the flow structures within realistic nuclei are neither purely irrotational nor rigid. The ${\mathcal{J}}_{\text{Exp}\mathrm{,}k}$ values are found to be 6.3, 7.4, and 10.0 times larger than the ${\mathcal{J}}_{\text{irr}\mathrm{,}k}$ values for the intermediate (m, 1-axis), short (s, 2-axis), and long (l, 3-axis) axes, respectively. Correspondingly, the ratios ${\mathcal{J}}_{\text{CDFT}\mathrm{,}k}$ are smaller at 4.5, 5.8, and 10.0, respectively. This underestimation is due to the use of the IB formula without considering the Thouless-Valatin dynamic rearrangement [61-64]. Therefore, the order of MoIs for the different models can be expressed as: ${\mathcal{J}}_{\text{irr}\mathrm{,}k}<{\mathcal{J}}_{\text{CDFT}\mathrm{,}k}<{\mathcal{J}}_{\text{Exp}\mathrm{,}k}<{\mathcal{J}}_{\text{rig}\mathrm{,}k}$.

To investigate whether ${\mathcal{J}}_{\text{CDFT}\mathrm{,}k}$ follows the properties of ${\mathcal{J}}_{\text{irr}\mathrm{,}k}$, we examine their relative MoIs as functions of γ, as shown in Fig. 3, and compare them with the experimental data [12] for all three principal axes. The scale used in Fig. 3 is normalized to 𝒥₁, which represents the MoI of the m-axis. We find that the relative ${\mathcal{J}}_{\text{CDFT}\mathrm{,}k}$ are also qualitatively consistent with ${\mathcal{J}}_{\text{irr}\mathrm{,}k}$ and in agreement with ${\mathcal{J}}_{\text{Exp}\mathrm{,}k}$. This agreement confirms the validity of the CDFT approach for describing the triaxially deformed nuclei. It is worth noting that the MoIs calculated using the cranking model based on the modified oscillator potential exhibit the same behavior as ${\mathcal{J}}_{\text{irr}\mathrm{,}k}$ [65]. These results demonstrate the importance of considering ${\mathcal{J}}_{\text{irr}\mathrm{,}k}$ to understand the collective rotational behavior of triaxially deformed nuclei. For example, the m axis exhibits the largest MoI, which leads to the appearance of transverse wobbling in the low-spin region [11, 65, 66] as well as chiral rotation when the nucleus possesses a particle-hole configuration [10, 67-70].

Fig. 3Full-screen View

(Color online) Relative MoIs for all three principal axes as functions of γ. The CDFT and experimental values are normalized to the irrotational values using the 1-axis (m axis) as a reference

Moments of inertia in ¹⁹⁰Os

Notably, as shown in Fig. 3, the calculated relative MoIs for ^186-192Os along the l axis are overestimated compared with the experimental values, which contradicts the trend for ${\mathcal{J}}_{\text{irr}\mathrm{,}k}$. To investigate this discrepancy, we further analyze the ¹⁹⁰Os case. Using the CDFT, we calculate the MoIs as a function of γ for fixed values of β = 0.1, 0.2, 0.3, and 0.5. Additionally, the individual contributions from neutrons and protons to the MoIs are also analyzed, as shown in Fig. 4. As β increases, the degree of asymmetry becoms more pronounced. The relative MoIs along the s axis remain consistent with the irrotational flow, and significant deviations occur at β = 0.1 and 0.2. Intriguingly, for the l axis MoI, a peculiar phenomenon is observed. At $\gamma =0$, the relative MoIs for β = 0.1 and 0.2 do not vanish, which exceeds the expectation for a prolate shape. Additionally, the contributions of protons and neutrons are substantially different at β = 0.2, with the proton contribution being significantly larger than that of neutrons. However, when β=0.3 and 0.5, the l axis relative MoI returns to zero.

Fig. 4Full-screen View

(Color online) Same as Fig. 3, but for ¹⁹⁰Os at β=0.1, 0.2, 0.3, and 0.5

To investigate the potential influence of β deformation on the MoI behavior with respect to γ for isotope ¹⁹⁰Os, we calculate the MoIs for both the m and l axes as a function of β, while maintaining γ fixed at 0 Å, as shown in Fig. 4. It is worth noting that when γ is set to 0 Å, 𝒥m is equal to 𝒥s. Additionally, we plot the individual contributions of neutrons and protons to the MoIs, as shown in Fig. 5. To further analyze the results, we compare them with those obtained for ¹⁶⁸Er, which exhibit good agreement with ${\mathcal{J}}_{\text{irr}\mathrm{,}k}$, as shown in Fig. 3. Figure 5 shows the 𝒥m and 𝒥l behavior for ¹⁶⁸Er, revealing an increasing trend in 𝒥m as the deformation β increases, which is expected given that the degree of the asymmetry increases. Conversely, 𝒥l aligns with the anticipated behavior for a prolate shape, namely, it approaches zero. However, for ¹⁹⁰Os, noticeable deviations are observed, particularly for 𝒥l values that do not vanish within a range of β values between 0.08–0.22 (as indicated by the vertical lines in the figure).

Fig. 5Full-screen View

(Color online) Calculated MoIs (upper) and pairing energy (lower) of the total, neutron, and proton as functions of β for ¹⁶⁸Er (left) and ¹⁹⁰Os (right). The vertical lines label the region of non-zero 𝒥l for ¹⁹⁰Os, while the horizontal lines label the vanishing of pairing energy

As mentioned previously, the MoIs in the CDFT are calculated using the IB formula, which are determined from the quasi-particle energy of the quasi-particle states in the denominator and matrix elements of the angular momentum in the quasi-particle states in the numerator. In particular, the single-particle energy levels near the Fermi surface play a decisive role in determining the MoIs. Thus, to elucidate the reason behind the non-vanishing 𝒥l values for ¹⁹⁰Os shown in Fig. 5, we investigate the corresponding single-particle energy levels of protons and neutrons for ¹⁹⁰Os as a function of β. We compare these results with those for ¹⁶⁸Er, which serve as a reference, as shown in Fig. 6. Upon examining the single-particle energy levels for protons in ¹⁹⁰Os, we observe that the energy level density near the Fermi surface in the region of $0.08\le \beta \le 0.22$ is much smaller than that of the other, which may indicate a decrease in the pairing interaction. Indeed, a quantitative study of microscopic nuclear level densities based on CDFT, as in Refs. [72, 73], would be interesting.

Fig. 6Full-screen View

(Color online) Partial proton (upper) and neutron (lower) single-particle energy levels as functions of β at $\gamma =0$ for ¹⁶⁸Er (left) and ¹⁹⁰Os (right). The dashed line denotes the Fermi surface

To study the effect of pairing interactions, we plot the pairing energy, as shown in the lower panels of Fig. 5. It can be clearly seen that in the region of $0.08\le \beta \le 0.22$, the proton-pairing energy tends to vanish, indicating the collapse of the proton pairing interaction. Consequently, the MoI of the l axis of proton does not disappear and behaves like a rigid MoI, as previously mentioned. Beyond this region, the pairing energy is a finite value and the proton 𝒥l becomes zero. However, no pairing collapse occurred for ¹⁶⁸Er, and 𝒥l is zero. It is noted that in the selected PC-PK1 interaction [60], the pairing strengths are fixed at specific values: G_n=349.5 MeV fm³ for neutrons and for protons. To further explore the potential impact of varying pairing strengths on the final outcome, we analyze the calculated MoIs and pairing energies of the total neutrons and protons as a function of β for the nucleus ¹⁹⁰Os, as shown in Fig. 7. Specifically, we maintain G_n at 349.5 MeV fm³ while adjusting G_p from 0.0 to 165.0 (half of the original G_p strength) and 660.0 MeV fm³ (twice the original G_p strength). The observations reveal that when G_p is reduced or eliminated, the proton-pairing energy is significantly diminished. Consequently, there is a corresponding increase in the proton moment of inertia 𝒥l. In comparison with the case of the original G_p strength, the proton pairing collapse occurs in a larger β region. When G_p is enhanced, the proton-pairing energy increases dramatically. This results in a vanishing proton 𝒥l. Hence, this indicates that rigid flow is a consequence of the collapse of the pairing interaction from irrotational flow. However, it should be noted that the underlying mechanism behind this phenomenon is still not fully understood. Additionally, we note that the pairing energy collapse is attributed to the fact that the present pairing correlations are treated in the BCS approximation without particle number projection. To avoid this problem, restoring particle number symmetry is necessary [3]. However, this is beyond the scope of the present study. Nevertheless, the importance of pairing correlations on the nuclear structure and also, for example, on the fragment mass distribution [74] and β-decay half-lives [75] are already well known.

Fig. 7Full-screen View

(Color online) Calculated MoIs (upper) and pairing energy (lower) of the total, neutron, and proton as functions of β for ¹⁹⁰Os with G_p= 0.0 (left), 165.0 (middle), and 660.0 MeV fm³ (right)

Low-lying energy spectra

To further assess the predictive power of CDFT for the MoIs, we investigate the energy spectra of the ground-state, β, and γ bands in the 12 triaxial nuclei using 5DCH based on CDFT inputs. As shown in Fig. 2, the calculated CDFT MoI values significantly underestimate the empirical results owing to neglecting the Thouless-Valatin corrections that are largely independent of deformation [62-64]. Therefore, we enhance the CDFT MoI values using a constant factor f (≈1.55) to fit the experimental $2_1^{+}$ state energy. The results are presented in Fig. 8 along with the available experimental data obtained from the National Nuclear Data Center (NNDC) [71]. Note that the collective potentials in the 5DCH include the zero-point energy (ZPE) corrections originating from vibrational and rotational kinetic energy [76]. We label this enhanced factor f used in the calculations for each subfigure to provide clarity. As shown in Fig. 8, the 5DCH energy spectra and the corresponding experimental data for the 12 triaxial nuclei are generally in good agreement. This agreement not only validates the effectiveness of the 5DCH but also provides further support for the overall validity of the CDFT approach for describing the nuclear dynamics of triaxially deformed nuclei.

Fig. 8Full-screen View

(Color online) Energy spectra of the ground-state, β, and γ bands for the 12 nuclei calculated via the 5DCH (open circles) in comparison with those obtained from TRM calculations (open diamonds) and available experimental data (solid symbols) from NNDC [71]. The f value represents the enhanced factor for CDFT MoIs used in the 5DCH

However, the 5DCH predictions may differ from the experimental observations in certain cases. Specifically, for ¹¹⁰Ru, the theoretical γ band energies are slightly higher than the experimental values. For ¹⁵⁰Nd, ¹⁵⁶Gd, and ¹⁶⁸Er, while the γ band is reproduced, the β-band energy is overestimated, indicating that the mass parameter along the β-direction is underestimated. For ¹⁷²Yb, the energy crossings between the β and γ bands are less evident in the calculations. For ^182,184W and ¹⁸⁶Os, the β-band slope is overestimated, indicating that the MoI in the β band is too small. Therefore, there is room for improvement in the theoretical models used in this study. For instance, the inclusion of dynamic pairing vibrations can improve the description of the 0⁺ state of the β band [77]. A more streamlined approach involves considering the enhancement of the mass parameters by a scaling factor, which is denoted by f’. For further investigation, we select the nuclide ¹⁵⁰Nd as a representative example. Figure 9 shows the obtained energy spectra of the ground-state and β bands in ¹⁵⁰Nd using the 5DCH method, juxtaposed with pertinent experimental data from NNDC [71]. The analysis results reveal that exclusively enhancing the MoIs leads to an overestimation of the excitation energy within the β band. Similarly, enhancing only the mass parameter along the β-direction, denoted as Bββ, leads to an overestimation of the slopes exhibited by the ground-state and β bands. Notably, when both the MoIs and Bββ are enhanced concurrently, the computed energy spectra for both the ground-state and β bands align favorably with the experimental observations.

Fig. 9Full-screen View

(Color online) Energy spectra for the ground-state and β bands in ¹⁵⁰Nd calculated via 5DCH (open symbols) compared with available experimental data (solid symbols) from NNDC [71]. The f and f’ values respectively represent the enhanced factors for the CDFT MoIs and mass parameter Bββ used in the 5DCH

To examine the fluctuation effects in the 5DCH, we perform rigid triaxial rotor model (TRM) [2] calculations using the experimental MoIs along the three principal axes [12] as inputs. The TRM does not consider the β degree of freedom and cannot predict the β band. Therefore, Fig. 8 shows the TRM results for the ground-state and γ bands. The TRM successfully reproduces the ground-state band energies for ¹⁵⁶Gd, ^166,168Er, ¹⁷²Yb, ^182,184W, and ¹⁸⁶Os. It also describes the γ-band behavior for these nuclei. However, this method overestimates the ground-state band energies in the high-spin region for ¹¹⁰Ru ¹⁵⁰Nd, and ^188-192Os, resulting in poorly described γ bands with higher energies and a more pronounced staggering behavior, that is, indicating the γ deformation degrees of freedom is estimated to be too rigid [78]. In contrast, the 5DCH calculations show better agreement in these cases, highlighting the importance of considering the fluctuations in the deformation degree of freedom.

Using the 5DCH wave functions, we calculate the deformation expectation $\overline{\beta }$ and $\overline{\gamma }$ and their fluctuations Δ β and Δ γ [38], and compare these values for the $0_1^{+}$ state with those obtained from the CDFT ground-state calculations and the experimental data [12], as shown in Fig. 10. While the CDFT ground-state deformation parameters deviate from the experimental values for some nuclei, the 5DCH accurately reproduces the experimental data. In particular, the nuclei predicted to have an axial shape in the CDFT calculations exhibit triaxiality in the 5DCH, emphasizing the significance of considering the fluctuations in the deformation degree of freedom. Additionally, we find that the $\overline{\beta }$ value of the 5DCH is similar to that of CDFT for all nuclei except for ¹¹⁰Ru and ¹⁵⁰Nd, suggesting a relatively small fluctuation in the β direction in these nuclei. However, only ¹¹⁰Ru and ^186-192Os exhibit notable triaxial deformation parameters ($20\le \gamma \le 40$), as indicated by the light-blue band in Fig. 10. Additionally, all nuclei display significant fluctuations in triaxial deformation ($\text{Δ}\gamma \approx 10$), which indicates a certain degree of γ softness. Therefore, the occurrence of rigid triaxially deformed ground states remains uncommon.

Furthermore, utilizing the obtained 5DCH wavefunctions, we calculate the in-band E2 transition probabilities B(E2) and compare these calculations with the available experimental data for Os isotopes ^186-192Os [71], which exhibit notable triaxiality (c.f. 10), as shown in Fig. 11. Note that E2 transition results for additional nuclei can be referenced from Refs. [27, 52, 54, 40]. Despite a slight overestimation by the 5DCH approach compared with the experimental data, the agreement is reasonable. This is attributed to the significant advantage of the 5DCH-CDFT methodology: transition probabilities are computed within the entire configuration space, thus obviating the need for effective charges. This underscores the robustness of the 5DCH approach for transition probability calculations.

Fig. 11Full-screen View

(Color online) Calculated in-band B(E2) transition probabilities of the ground-state, γ, and β bands for ^186-192Os in the 5DCH compared with the available data [71]