Chiral geometry and rotational structure for $^{130}$Cs in the projected shell model

The projected shell model with configuration mixing for nuclear chirality is developed and applied to the observed rotational bands in the chiral nucleus $^{130}$Cs. For the chiral bands, the energy spectra and electromagnetic transition probabilities are well reproduced. The chiral geometry illustrated in the $K~plot$ and the $azithumal~plot$ is confirmed to be stable against the configuration mixing. The other rotational bands are also described in the same framework.

1 Spontaneous chiral symmetry breaking in atomic nuclei has attracted intensive theoretical and experimental studies since its first prediction by Frauendorf and Meng [1] in 1997. It takes place when the angular momentum of a triaxial nucleus has non-vanishing components on all of the three intrinsic principle axes. These three components, contributed by the valence proton(s), the valence neutron(s), and the rotating core, respectively, can form a left-or a right-handed configuration.
The angular momentum projection (AMP) approach, which restores the broken rotational symmetry in the mean field wavefunctions, is a promising tool for the microscopic description to the nuclear system. The AMP approach could be based on mean field descriptions from Nilsson+BCS [38] to various density functionals [39][40][41][42][43], and have been applied to various problems as summarized in a recent review [44]. As one of the implementations of the AMP approach, the projected shell model [38] has been used in attempts to understand the chiral doublet bands [45,46]. The observed energy spectra and electromagnetic transitions for the doublet bands are well reproduced in these calculations. However, it was found difficult to give an illustration for the underlying chiral geometry. The difficulty lies in the fact that the angular momentum geometry is defined in the intrinsic frame, while the angular momentum projected wave functions are written in the laboratory frame. This difficulty is overcomed recently by the introduction of the K plot and the azimuthal plot [47]. The K plot and the azimuthal plot provide probability distributions of the orientation related quantities in the intrinsic frame, as demonstrated by the chiral doublet bands in 128 Cs [47]. In the angular momentum projected framework, the chiral geometry can also be illustrated by the 2 root-mean-square values of the angular momentum components in the intrinsic frame [48].
The calculation in Ref. [47] is done by blocking the lowest πh 11/2 and the fourth νh 11/2 orbitals, and the configuration mixing is neglected. For a better description to the nuclear system, the theoretical framework in Ref. [47] needs to be generalized to include various configurations, i.e., to the projected shell model. In order to have a better understanding of the chiral doublet bands, it is necessary to confirm whether the chiral geometry is stable against configuration mixing. Another advantage is that the projected shell model can provide a simultaneous description for all rotational bands in one nucleus on the same foot.
In this work the projected shell model with configuration mixing for nuclear chirality is developed and applied to 130 Cs [10]. The chiral geometry for the chiral doublet bands is illustrated in terms of the K plot and the azimuthal plot, in a similar way as Ref. [47]. The effect of the configuration mixing on the chiral geometry and the description of the other rotational bands are discussed.
The frame work of the projected shell model is based on the standard pairing plus quadrupole Hamiltonian [49], which includes a spherical single-particle part and the separable two body interactions, i.e., the quadrupole-quadrupole interaction, the monopole pairing, and the quadrupole pairing.
The intrinsic ground state of an odd-odd nucleus can be denoted as |Φ ν 0 π 0 , where ν 0 and π 0 are single-particle orbitals blocked. The state |Φ ν 0 π 0 is determined by the variation principle with the constraints on the quadrupole momentes and the average particle numbers: and can be written as a two-quasiparticle state on top of a quasiparticle vacuum of the even-even core |Φ 0 : By the variation in Eq. (2), the state |Φ ν 0 π 0 and the vacuum |Φ 0 are obtained, as well as corresponding quasiparticle operators {β + ν , β + π }, with ν and π the single-particle orbitals. Various two-quasiparticle states can be constructed as {|Φ κ } = {β + ν β + π |Φ 0 }, in which κ specifies different two-quasiparticle configurations. The effect of the configuration mixing, which was neglected in Ref. [47], can be then taken into account.

3
The two-quasiparticle states |Φ N,Z,κ with good particle number N and Z can be projected from |Φ κ : The symmetry restored basis is constructed by the angular momentum projection: The diagonalization of the Hamiltonian in the projected basis (5) leads to the Hill-Wheeler equation: in which σ specifies different eigen states of the same spin I. By solving Eq. (6) the eigenenergies E Iσ and the wave functions are obtained. With the wave functions (7) the electromagnetic transitions and other physical quantities can be calculated.
In order to examine the configuration mixing, it is necessary to know the weights of different two-quasiparticle configurations |Φ κ in the wave function (7). These weights could be obtained by resorting the concept of collective wave functions in the generator coordinate method (GCM), and regarding {K, κ} in Eq. (7) as generator coordinates. The corresponding generating functions are the projected basis in Eq. (5), and the corresponding norm matrix elements write The collective wave functions g Iσ (K, κ) can be obtained by the square root of the norm matrix (8) and the function f Iσ Kκ : and they are proved to satisfy the normalization and orthogonal relation In the GCM, the collective wave functions are understood as probability amplitudes of the generator coordinates. Therefore, the weight of the configuration |Φ κ in the state |Ψ σ IM could be written as: The collective wave functions (9) are also used in the calculations of the K plot and the azimuthal plot, which have been introduced in Ref. [47].
In the following, the five rotational bands observed in 130 Cs [50][51][52], including the chiral doublet bands, are investigated by the present approach. The parameters in the Hamiltonian (1) are taken from Ref. [53]. The quadrupole deformation parameters (β, γ) are constrained to be (0.20, 30 • ). This choice of (β, γ) is the same as that adopted in the calculation for 128 Cs in Ref. [47]. It agrees reasonably with the deformations (0.19, 39 • ) used in the projected shell model calculation in Ref. [53].
Different from the calculation in Ref. [47], where the lowest πh 11/2 and the forth νh 11/2 orbitals are blocked, here the two-quasiparticle configurations β + ν β + π |Φ 0 are constructed using orbitals ν and π from the N = 4 and N = 5 major shells. The quasiparticle energy cutoff E ν + E π ≤ 3.5MeV is adopted for the two-quasiparticle configurations, in which E ν and E π are the quasiparticle energies of the orbitals ν and π, respectively. There are 60 two-quasiparticle configurations with positive parity and 68 with negative parity taken into account in the configuration space. Half of them can be projected onto even K values and the other half can be projected onto odd K values [54]. Therefore the dimension of the projected basis (5) for spin I is 30(2I + 1) for the positive parity subspace and 34(2I + 1) for the negative parity subspace.
The calculated energy spectra of the rotational bands (denoted as bands A-E) in 130 Cs are shown in Figs. 1 (a) and (b), in comparison with the available data [50][51][52].
As shown in Fig. 1 (a), the calculated yrast and yrare bands with positive parity agree well with the experimental chiral doublet bands, band A and band B, including the near degeneracy between the partner bands. Note that the configuration mixing, which is absent in the calculation in Ref. [47], is taken into account in the present calculation. The present results demonstrate that the near degeneracy between the chiral doublet bands is persistent even with the configuration mixing.
The calculated yrast and yrare bands with negative parity agree reasonably with the experimental bands D and C, as shown in Fig. 1 (b). The levels with I ≥ 18 in band D are overestimated, which might be due to a crossing with four-quasiparticle configurations and is beyond the present two-quasiparticle configuration space.
The calculated lowest band with the dominating configuration νg 7/2 πh 11/2 suggested in Ref. [51] lies around 1 MeV above the experimental band E. The dominating configuration(s) will be explained in the following. One may notice that the moment of inertia for band E is larger than those for bands C and D, which suggests a larger deformation for band E.
In this sense, band E is beyond the present scope. Therefore we exclude band E from the following discussions.   [50,51] while the previously assigned one is νh 11/2 πg 7/2 . This difference is due to the strong mixing between the g 7/2 and the d 5/2 orbitals as shown in Fig. 2.
In Fig. 2  almost independent of spin, as is seen in Fig. 1(c). In fact, the spin independence of S(I) has been recognized as one of the criteria for the chiral doublet bands [4,55]. For bands C and D, the strong configuration mixing leads to the increasing signature splitting with spin [56] as shown in Fig. 1(d). Moreover, it is noted in Fig. 1 that the amplitudes of S(I) increase with spin, which is due to the increasing configuration mixing.
In the following we focus our discussion on the properties of the chiral doublet bands A and B.    Fig. 3, which have been reproduced with a single configuration [47], are persistent even when the configuration mixing is taken into account.
The angular momentum geometry of the chiral doublets A and B can be illustrated by the K plot and the azimuthal plot, as in Ref. [47]. The K plots, i.e., the K distributions results obtained by the PRM, which have been discussed in Refs. [26,58].
As seen in the K plot in Fig. 4, the evolution of the chiral modes can be exhibited.
For spins I = 10 and 11 , the K plots shown in Fig For spins I = 18 and 19 , the K s -and K l -distributions become broad, and the peaks of the K i -distributions become sharp at K i ∼ I. Both features suggest that the angular momenta move close to the i-axis. Therefore the static chirality disappears and the aplanar rotation is replaced by the principle axis rotation.
In Fig. 5, the profiles for the orientation of the angular momentum on the (θ, φ) plane, the azimuthal plots, are shown for the same spins as in Fig. 4. The definitions of the angles (θ, φ) can be found in Ref. [47]. For spins I = 18 and 19 , the peaks for the azimuthal plots for band A move toward (θ ∼ 80 • , φ ∼ 20 • ) and (θ ∼ 80 • , φ ∼ 160 • ), namely close to the i-axis. This is in accordance with the dominance of rotation around the i-axis reflected in the K plot discussed above, suggesting the disappearance of chiral geometry and the onset of principle axis rotation.
The peaks for the azimuthal plots for band B locate at (θ ∼ 75 • , φ ∼ 35 • ) and (θ ∼ 75 • , φ ∼ 145 • ), which are similar to those at I = 14, 15 but approaching the i-axis. Thus the chiral geometry is weakened. In general, the evolution of the angular momentum geometry in the azimuthal plot is consistent with those in the K plot.
By blocking the lowest πh 11/2 and the fourth νh 11/2 orbitals, the chiral geometry in K plot and azimuthal plot has been discussed in Ref. [47]. Figures 4 and 5 confirm that the chiral geometry in K plot and azimuthal plot can be stable against configuration mixing.
In summary, the projected shell model with configuration mixing for nuclear chirality is developed and applied to the observed rotational bands in the chiral nucleus 130 Cs. Both the chiral and achiral bands are described on the same foot. For the chiral bands, the energy spectra, S(I), B(M1) and B(E2) are well reproduced. The chiral geometry is demonstrated in the K plot and the azithumal plot. The stability of the chiral geometry against the configuration mixing is confirmed. As another advantage, the other rotational bands are described simultaneously with the same Hamiltonian.