Possible chiral doublets in $^{60}$Ni

The open problem on whether or not the chirality exists in doublet bands M1 and M4 in light-mass even-even nucleus $^{60}$Ni is studied by adopting the recently developed fully quantal four-$j$ shells triaxial particle rotor model. The corresponding experimental energy spectra, energy differences between doublet bands, and the available $B(M1)/B(E2)$ values are successfully reproduced. The analyses on the basis of the angular momentum components, the azimuthal plots, and the $K$-plots suggest that the chiral modes exist at $I\geq 12\hbar$ in doublet bands M1 and M4.

For the latter, the experimental evidence of chiral doublet bands was first observed in the A ∼ 130 mass region, and then followed by the A ∼ 100, 190, and 80 mass regions. These observations show that the nuclear chirality is not a specific phenomenon that exists in only one nucleus or one mass region.
Both of the two fundamental goals and all of relevant observations mentioned above encourage us to search for new candidates with chirality or MχD in new mass regions. In Ref. [37], we explored the MχD in A ∼ 60 mass region by the adiabatic and configurationfixed constrained CDFT for cobalt isotopes. It was found that there are high-j particle(s) and hole(s) configurations with prominent triaxially deformed shapes in these isotopes, which suggests the possibility of chirality or multiple chirality in A ∼ 60 mass region. However, the experimental energy spectra and electromagnetic transition in these isotopes are rather rare at present.
We note that in Ref. [48], the fully microscopic self-consistent tilted axis cranking covariant density functional theory (TAC-CDFT) was applied to investigate the observed dipole bands M1, M2, M3, and M4 in even-even nucleus 60 Ni [49]. It was mentioned that bands M1 and M4 might be the possible candidates for chiral doublet bands. However, due to the mean-field approximation, the TAC can only give the description for the band M1. After that, there is neither further theoretical nor experimental work to investigate bands M1 and M4 in 60 Ni. Therefore, whether the chirality exists in the bands M1 and M4 or not is still an open problem.
The aim of the present work is to investigate the chirality in doublet bands M1 and M4 in 60 Ni in a fully quantal model. As a quantal model coupling the collective rotation and the single-particle motions, the particle rotor model (PRM) has been widely used to describe the chiral doublet bands and achieved major successes [24]. In contrast to the TAC approach, PRM describes a system in the laboratory frame. The total Hamiltonian is diagonalized with total angular momentum as a good quantum number, and the energy splitting and quantum tunneling between the doublet bands can be obtained directly. Moreover, the basic microscopic inputs for PRM can be obtained from the constrained CDFT [9, 31, 38-40, 50, 51]. Various versions of PRM have been developed to investigate the chiral doublet bands with different kinds of configurations [1,9,16,23,38,39,47,[51][52][53][54][55][56][57][58][59][60][61][62][63][64]. To describe the doublet bands M1 and M4 in 60 Ni with four quasi-particle configuration [48,49], a four-j shell PRM is needed. Such version of PRM has already been developed very recently and applied to describe the MχD in 136 Nd [42].
In this letter, the four-j shell PRM will be applied to study the energy spectra and the electromagnetic transition probabilities of the doublet bands M1 and M4 in 60 Ni, and to explore the open problem on whether or not the chirality exists in this doublet by examining their angular momentum geometries.
The formalism of PRM with four-j shell can be found in Ref. [42]. The total wave function of PRM Hamiltonian is expanded into the strong coupling basis where |IMK is the Wigner function 2I+1 8π 2 D I M K , |φ is the product of the proton and neutron states those sitting in the four-j shells, and the c Kφ is the expansion coefficient obtained by diagonalizing the PRM Hamiltonian. With the obtained wave functions, the reduced transition probabilities B(M1) and B(E2) can be calculated [65]. In addition, one can also calculate the expectation values of the angular momentum components [42,62], the probability distributions of the total angular momentum in the intrinsic reference frame (azimuthal plot) [56,66,67], and the probability distributions of the total angular momentum components on the three principle axes (K-plot) [62,66], Based on these analyses, one can study the angular momentum geometries systematically to make an unambiguous judgment whether or not the chiral geometry exists in the doublet bands.
In the PRM calculations for the doublet bands M1 and M4 in 60 Ni, the configuration [48,49] is adopted. The deformation parameters β = 0.27 and γ = 19 • for this configuration at the bandhead were obtained from the microscopic self-consistent TAC-CDFT calculations [48]. With the rotation, the β value decreases smoothly, while γ value shows a smoothly increasing tendency. In the PRM calculation, the deformation parameters are fixed. To reproduce the energy spectra better, we use a deformation of β = 0.27 and γ = 22 • . The moment of inertia J 0 = 9.0 2 /MeV and Coriolis attenuation factor ξ = 0.98 are adopted according to the experimental energy spectra. For the electromagnetic transitions, the empirical intrinsic quadrupole moment Zβ, and gyromagnetic ratios for rotor g R = Z/A and for nucleons g p(n) = g l + (g s −g l )/(2l + 1) (g l = 1(0) for protons (neutrons) and g s = 0.6g s (free)) [68] are adopted.
The calculated energy spectra for the bands M1 and M4 in 60 Ni are presented in Fig. 1(a), together with the corresponding data. The experimental energy spectra are reproduced excellently by the PRM calculations. Such a good agreement can be more clearly seen by showing the energy level scheme in Fig. 1(d).
Being a fully quantal model, PRM is capable of reproducing the energy splitting ∆E between the doublet bands for the whole observed spin region. This is illustrated in Fig. 1 The ∆E increases firstly from I = 9 to 11 , and then decreases up to I = 14 . At I = 15 , an increasing trend is observed once again in the PRM results. It is known that in an ideal chiral system with the particle-hole configuration π(1h 11/2 ) 1 ⊗ ν(1h 11/2 ) −1 and a rotor with the deformation parameter γ = 30 • , the ∆E is small (less than 400 keV) and shows a trend that decreases firstly and then increases [1,56]. Here, at I ≥ 12 , the ∆E shows the similar variation trend, giving a hint that chirality might exist in bands M1 and M4. The large ∆E (higher than 400 keV) could be owed to the small triaxial deformation [47,54]. Therefore, it would be very interesting to extend the spectrum of band M4, which only reaches to I = 13 currently, to higher spins to further verify the theoretical calculations.
In Fig. 1(c The spontaneous chiral symmetry breaking or chiral geometry is realized by the total angular momentum lying outside the three principal planes in the intrinsic frame [1]. In order to visualize the angular momentum geometry in the intrinsic frame, the azimuthal plots [56,66,67], i.e., profiles P(θ, ϕ) for the orientation of the angular momentum on the (θ, ϕ) plane calculated by PRM using Eq. (4) are shown in Fig. 3 for the doublet bands M1 and M4 in 60 Ni at I = 9, 11, 12, 14, and 15 . We emphasize that θ is the angle between the total spin I and the l-axis, and ϕ is the angle between the projection of I onto the si-plane and the s-axis. As is shown in Fig. 3, the azimuthal plots are symmetric with respect to ϕ = 0 • . This is because the broken chiral symmetry in the intrinsic frame has been restored in PRM.
For I = 9 (bandhead) and 11 (kink of ∆E), the profiles for the orientation of the angular momentum for band M1 have only one single peak at (θ ∼ 45 • , ϕ = 0 • ), which suggests that the angular momentum stays within the sl-plane. Instead, the profiles for band M4 peak at (θ = 90 • , ϕ = 0 • ), corresponding to a principal axis rotation with respect to s-axis. Such two orientations do not form a chiral geometry. Therefore, not chirality is shown for I = 9-11 . This is also the reason why the energy splitting ∆E increases at this spin region, as shown in Fig. 1(b).
For I = 12 , the profile for band M1 still peaks at ϕ = 0 • . However, that for band M4 shows a node around (θ ∼ 60 • , ϕ = 0 • ) with the onset of two peaks locating at (θ ∼ 45 • , ϕ ∼ For I = 15 , the peaks for band M1 move toward to (θ ∼ 60 • , ϕ ∼ 90 • ) and (θ ∼ 60 • , ϕ ∼ −90 • ), namely in the il-plane and close to i-axis. This is mainly driven by the gradual increasing of i-components of the rotor, and valence neutron g 9/2 particle, and valence proton f 7/2 hole angular momenta, as presented in Fig. 2 attain vibration character again, which is now with respect to il-plane. As a consequence, their energy difference ∆E, as shown in Fig. 1(b), increases [69,70].
To further understand the evolution of the chirality with spin, in Fig. 4, the K-plots, i.e., For I = 9-11 , the peaks of K l locate around K l = 6 for band M1, while at K l = 1 for band M4. The K i -distribution peaks at K i = 1 for band M1, while spreads widely for band M4. The K s -distributions of both bands peak at large K s value. All of these are in accordance with the features observed in the azimuthal plots shown in Fig. 3. Namely, the angular momentum of band M1 stays within the sl-plane, while that of band M4 aligns along s-axis.
For I = 12 , K l distribution of band M4 shows a rapid change, which is caused by the discontinuous variation of the l-component of rotor angular momentum, as shown in Fig. 2.
The K i -distribution spreads around K i = 0 for band M1, whereas it almost vanishes for band M4. This is in accordance with the interpretation of the chiral vibration with respect to the sl-plane where the zero-phonon state (band M1) is symmetric with respect to K i = 0 and the one-phonon state (band M4) is antisymmetric.
For I = 13 and 14 , the K i -distributions of bands M1 and M4 are rather similar. The position of peak of K l -(K s ) distribution for band M1 is a bit smaller (larger) than those for band M4. Such differences lead the different azimuthal plots shown in Fig. 3 and lead that the energy splitting of bands M1 and M4 is a bit large (∼ 400 keV), though they are, in fact, in the static chirality region.
For I = 15 , the K l -and K i -distributions for bands M1 and M4 are similar. However, for the K s -distribution, they are different. The most probable value spreads at K s ∼ 1 for band M1, while appears at K s ∼ 11 for band M4. This further supports the appearance of second chiral vibration with respect to il-plane.
In summary, the open problem on whether or not the chirality exists in doublet bands M1 and M4 in light-mass even-even nucleus 60 Ni is studied by adopting the recently developed fully quantal four-j shells triaxial particle rotor model. The corresponding experimental energy spectra, energy differences between the doublet bands, and the available B(M1)/B(E2) values are successfully reproduced. The analyses based on the angular momentum components, the azimuthal plots, and the K-plots suggest that the chiral modes exist at I ≥ 12 .
Namely, there is no indication of chirality at I ≤ 11 . A chiral vibration appears at I = 12 , then changes to nearly static chirality at I = 14 , and finally evolves to another type of chiral vibration at I = 15 .
Further experimental efforts on extending the level scheme and extracting electromagnetic transition data for band M4 are highly demanded to obtain solid evidence. According to current investigation, we would also like to attract more experimental and theoretical efforts on the investigation of chirality or multiple chirality in the A ∼ 60 mass region and even in the lighter-mass region.