Multiple Chirality in Nuclear Rotation: A Microscopic View

Covariant density functional theory and three-dimensional tilted axis cranking are used to investigate multiple chirality in nuclear rotation for the first time in a fully self-consistent and microscopic way. Two distinct sets of chiral solutions with negative and positive parities, respectively, are found in the nucleus 106Rh. The negative-parity solutions reproduce well the corresponding experimental spectrum as well as the B(M1)/B(E2) ratios of the transition strengths. This indicates that a predicted positive-parity chiral band should also exist. Therefore, it provides a further strong hint that multiple chirality is realized in nuclei.


Introduction
Chirality is a well-known phenomenon in many fields, such as chemistry, biology, molecular and particle physics, etc. In nuclear physics, chirality was originally suggested by Frauendorf and Meng in 1997 [1]. It represents a novel feature of triaxial nuclei rotating around an axis which lies outside the three planes spanned by the principal axes of the triaxial ellipsoidal density distribution; i.e., aplanar rotation. As depicted in Fig. 1, the short, intermediate, and long principal axes of a triaxial nucleus form a screw with respect to the angular momentum vector J; resulting in left-and right-handed systems, which correspond to the same magnitude of the angle φ but with opposite signs. The experimental evidence for MχD bands has been reported in 133 Ce [15] and 103 Rh [16].
Recently, octupole correlations between MχD bands in 78 Br have also been observed [18].
On the theoretical side, the triaxial particle-rotor model (PRM) [1,23,24,25,26] has been extensively used in studies of chiral doublet bands. However, such analyses are all phenomenological and are fitted to the data in one way or another. The tilted axis cranking approach allows one to study nuclear chiral rotation based on a microscopic mean field. For nuclear chirality, a three-dimensional tilted axis cranking (3DTAC) calculation is required [2], in which the cranking axis could lie outside the three planes spanned by the principal axes; i.e., both θ and φ in Fig. 1 could be nonzero. The chiral solutions of the 3DTAC approach have been examined based on a Woods-Saxon potential combined with the Shell correction method [27] or a Skyrme-Hartree-Fock mean field [28].
In particular, the former was also used to interpret the likelihood of two chiral bands with different configurations in the observed band structures of 105 Rh [8,9].
The nuclear density functional theories (DFTs) provide a fully self-consistent mean field which depends entirely on a universal energy density functional based on the effective nuclear interactions and, thus, would provide a further strong test of the multiple chirality in nuclei. Moreover, the covariant version of DFT exploits basic properties of QCD at low energies, in particular, the presence of symmetries and the separation of scales [29].
It provides a consistent treatment of the spin degrees of freedom, includes the complex interplay between the large Lorentz scalar and vector self-energies induced at the QCD level [30], and naturally provides the nuclear currents induced by the spatial parts of the vector self-energies, which play an essential role in rotating nuclei.
To describe nuclear rotation, covariant DFT has been extended with the cranking method [31,32,33,34]. In particular, the two-dimensional tilted axis cranking covariant DFT [35] has been successfully used to describe the magnetic rotation bands [35,36], antimagnetic rotation bands [37,38], transitions of nuclear spin orientation [39], and linear alpha cluster bands [40], and has demonstrated high predictive power [33,34]. Therefore, in the present work, we report on the first application of the 3DTAC method based on covariant density functional theory for nuclear chirality.

Theoretical Framework
Covariant DFT starts from a Lagrangian, and the corresponding Kohn-Sham equations have the form of a Dirac equation with effective fields S(r) and V µ (r) derived from this Lagrangian [41,42,43,44,45]. In the 3DTAC method, these fields are triaxially deformed, and the calculations are carried out in the intrinsic frame rotating with a constant angular velocity vector ω, pointing in an arbitrary direction in space: Here,Ĵ is the total angular momentum of the nucleon spinors, and the fields S and V µ are connected in a self-consistent way to the nucleon densities and current distributions, which are obtained from the single-nucleon spinors ψ k [35,38]. The iterative solution of these equations yields single-particle energies, expectation values for the three components Ĵ i of the angular momentum, total energies, quadrupole moments, transition probabilities, etc. The magnitude of the angular velocity ω is connected to the angular momentum quantum number I by the semiclassical relation Ĵ · Ĵ = I(I + 1), and its orientation is determined by minimizing the total Routhian self-consistently (see below).
In this work, the point-coupling Lagrangian PC-PK1 [46] is adopted, and the calculations are free of additional parameters. The present study focuses on chirality in the odd-odd nucleus 106 Rh [7], which has been regarded as one of the best known examples for chiral doublet bands [3]. The Dirac equation [Eq. (1)] is solved in a three-dimensional Cartesian harmonic oscillator basis with 10 major shells.
For 106 Rh, we first solve iteratively Eq. (1) by placing the nucleons self-consistently in the single-particle orbitals according to their energies starting from the bottom of the well. This leads automatically to the ground-state configuration where one neutron particle sits at the bottom of the h 11/2 shell, and three proton holes are at the top of the g 9/2 shell, but two of them are anti-aligned. In short, we denote this configuration as πg −1 9/2 ⊗ νh 1 11/2 . With increasing cranking frequency ω, the occupation of the protons remains unchanged up to ω = 0.7 MeV. However, as shown in Fig. 2 by the arrow at high frequencies, the last occupied neutron in the gd shell with positive parity can easily drop to the h 11/2 shell with negative parity. Therefore, this provides another configuration labeled as πg −1 9/2 ⊗ νh 2 11/2 (gd) 1 with the two neutrons aligning at the bottom of the h 11/2 4 shell.
Apart from the high-j particles and holes involved in these two configurations, it is also found that their deformation parameters (β, γ) are respectively (0. 25 Single-neutron energy (MeV)

Results and discussion
To examine the possible presence of chiral geometry, it is crucial to check the calculated orientation angles θ and φ of the total angular momentum J in the intrinsic frame. Phi (deg) (circles) and πg −1 is always larger than 45 • since the angular momentum alignment along the short axis, mainly contributed by the neutron particle(s) in the h 11/2 shell, is remarkably larger than that along the long axis contributed mainly by the proton hole in the g 9/2 shell. At low frequencies, the azimuth angle φ vanishes; providing a planar rotation, where the angular momentum lies in the short-long plane (see Fig. 1). Above some finite frequencies, however, the φ values become nonzero and, thus, the rotation becomes chiral. Therefore, a transition between planar and chiral rotation has been found for both configurations, and the corresponding transition points are ω ∼ 0.46 MeV for the πg −1 9/2 ⊗ νh 1 11/2 configuration and ∼ 0.66 MeV for the πg −1 9/2 ⊗ νh 2 11/2 (gd) 1 one, respectively. It should be noted that TAC gives only the classical orientation, around which the angular momentum J can execute a quantal motion. In the planar regime (φ = 0), the angular momentum vector J oscillates around the planar equilibrium into the left-and right-handed sectors; leading to so-called chiral vibrations [4]. As a result, two separate bands are expected to be observed, corresponding to the first two vibrational states.
In the chiral regime, the energy differences between the chiral twin bands could remain due to tunneling between the left-and right-handed sectors. This can be seen from Fig. 4, where the total Routhian curves are shown as functions of φ ω for both configurations at their largest frequencies. Here, the azimuth angle φ ω and the polar angle θ ω are used to represent the orientation of the angular velocity ω. The total Routhian curves are 6 determined by minimizing the total Routhian with respect to θ ω for each given value of φ ω . It is found that the orientation of ω is parallel to the angular momentum vector J at the position of the lowest Routhian for both configurations. This indicates that self-consistency has been fully achieved in the present calculations. It should be noted that the Routhians at ±φ ω are degenerate and, thus, one can actually find two degenerate minima of the Routhian for each configuration. They correspond to the left-and right-handed sectors, respectively. As shown in Fig. 4, the minima of the total Routhian are rather soft in the φ ω direction. The barrier of the Routhian at φ ω = 0 is only several tens of keV in magnitude for both configurations. This indicates that tunneling between the left and the right-handed sectors could be substantial, and a strong degeneracy of the chiral twin bands is, thus, not expected.
A pair of negative-parity chiral bands has been observed in 106 Rh [7], and the data on the excitation energies, rotational frequencies, and B(M 1)/B(E2) ratios are given in Fig. 5 in comparison with the calculated results, based on the negative-parity configuration πg −1 9/2 ⊗νh 1 11/2 . At the present mean-field level, this does not take into account either the chiral vibrations nor the tunneling between the left-and right-handed sectors. Therefore, the energy splitting between the two partner bands cannot be calculated. However, it can be clearly seen in Fig. 5(a)   and B(M 1)/B(E2) ratios for the configuration πg −1 9/2 ⊗ νh 1 11/2 in 106 Rh, as a function of the angular momentum in comparison with the data for the chiral bands observed in Ref. [7]. The excitation energies are renormalized to the ground state. 8 band can be reproduced well. For its partner band, further extensions going beyond the mean field by using, for instance, the methods of the random phase approximation [12] or the collective Hamiltonian [47,48] will be required in the framework of DFTs.
As shown in Fig. 5(b), the experimental rotational frequencies ω for the two partner bands behave very similarly with respect to the angular momentum, and they agree well with the calculated results. In particular, the data indicate a kink around I = 12 , where the slope of the ω vs spin curve changes abruptly. In comparison with the calculated results, it is found that the appearance of this kink is due to the change from the planar to the chiral solutions around ω ∼ 0.46 MeV. In the planar regime, the angular momentum is generated in the plane spanned by the short and long axes, while in the chiral one, it increases along the intermediate axis, which has larger moments of inertia. Note that the observed energy separation between the twin bands is almost constant at ∼ 300 keV, regardless of the planar-chiral transition. This is consistent with the rather soft Routhian curve obtained in the chiral regime (Fig. 4), which indicates a substantial tunneling between the left and right-handed minima, and this may explain the lack of degeneracy in the chiral regime.
The B(M 1) and B(E2) transition probabilities can be calculated in the semiclassical approximation from the magnetic and quadrupole moments, respectively [1]. Here, the magnetic moments are derived from the relativistic electromagnetic current operator [49] with the Dirac effective mass scaled approximately to the nucleon mass by introduction of a factor 0.58 that accounts for the so-called back-flow effects, which have been calculated in infinite nuclear matter by a Ward identity [50,51]. It can be seen from Fig. 5(c) that the calculated B(M 1)/B(E2) ratios are in a good agreement with the data [7].
Note that the deformation parameters (β, γ) change only slightly along the band from  and it is based on the positive-parity configuration πg −1 9/2 ⊗ νh 2 11/2 (gd) 1 . Note that this configuration is different from that with positive parity reported in Ref. [19], where the two neutrons in the h 11/2 shell are anti-aligned due to the imposed time-reversal symmetry. However, the breaking of this symmetry in the present cranking calculations enables the alignment of the two valence neutrons. This provides the angular momentum along the short axis, and plays a crucial role in the generation of the chiral band.
The calculated angular momenta, excitation energies, and the reduced transition probabilities are listed in Table 1. Features, such as the kinks in the ω-I relation and the B(M 1)/B(E2) ratios, similar to those seen in the negative-parity band, are found for this band as well. Quantitatively, the excitation energy of this positive-parity band is roughly 1 MeV higher than that of the negative-parity one at any given rotational frequency. The angular momentum I also becomes higher because of the alignment of the two h 11/2 neutrons, and its linear increase with frequency is marked by a steep rise of slope above ω = 0.66 MeV. This is due to the abrupt change of the φ values (see It is worthwhile to mention that similar calculations have also been carried out for the neighboring nucleus 105 Rh, in which two chiral bands are observed based on the configurations πg −1 9/2 ⊗ νh 1 11/2 (gd) 1 and πg −1 9/2 ⊗ νh 2 11/2 , respectively [8,9]. It is found that the experimental data for both bands can be reproduced well by the present approach of 3DTAC-CDFT. For the relation between the angular momentum and rotational frequency, the present calculations provide a similar, or even better, agreement with the data in comparison with the results given by the previous shell correction tilted axis cranking method [8,9]. In fact, several candidates of chiral bands have also been observed in other Rh isotopes. In particular, the phenomenon of multiple chirality based on the same configurations has been reported in 103 Rh [16]. Therefore, it would be interesting to perform a systematical investigation of nuclear chirality in Rh isotopes in the future. This will be discussed together with a detailed introduction of the formalism of 3DTAC-CDFT in a forthcoming paper.

Summary
In summary, covariant density functional theory and three-dimensional tilted axis cranking are used to investigate multiple chirality in nuclear rotation for the first time in a fully self-consistent and microscopic way. Two distinct sets of chiral solutions based on the respective configurations πg −1 9/2 ⊗ νh 1 11/2 and πg −1 9/2 ⊗ νh 2 11/2 (gd) 1 have been uncovered in the nucleus 106 Rh, which are the first examples for multiple chirality found by means of covariant DFT. A transition between planar and chiral rotation has been found for both configurations, while in the chiral regime, the tunneling between the left and the right-handed orientations could be substantial due to the soft Routhians. The calculated energy spectrum and B(M 1)/B(E2) ratios for the negative-parity band are in good agreement with the corresponding experimental data. This demonstrates the predictive power of the present investigation and, thus, the potential for observation of the other predicted positive-parity chiral band appears to be good.