Multiple chiral doublets in four-$j$ shells particle rotor model: five possible chiral doublets in $^{136}_{~60}$Nd$_{76}$

A particle rotor model, which couples nucleons in four single-$j$ shells to a triaxial rotor core, is developed to investigate the five pairs of nearly degenerate doublet bands recently reported in the even-even nucleus $^{136}$Nd. The experimental energy spectra and available $B(M1)/B(E2)$ values are successfully reproduced. The angular momentum geometries of the valence nucleons and the core support the chiral rotation interpretations not only for the previously reported chiral doublet, but also for the other four candidates. Hence, $^{136}$Nd is the first even-even candidate nucleus in which the multiple chiral doublets exist. Five pairs of chiral doublet bands in a single nucleus is also a new record in the study of nuclear chirality.

Chiral rotation is an exotic rotational mode in a nucleus with triaxial ellipsoidal shape.
However, chiral doublet bands were rarely observed in even-even nuclei. The general opinion for this is that the multi-quasiparticle configurations become more complex and involve at least two valence protons and two valence neutrons. In Ref. [19], two doublet bands were observed in the even-even isotopes 110,112 Ru and interpreted as soft chiral vibrations.
Very recently, five pairs of nearly degenerate doublet bands were reported in even-even nucleus 136 Nd, which were discovered in a high-statistics experiment performed with the highefficiency Jurogam II array [20]. It was demonstrated that the chiral partners of strongly populated bands in the triaxial nucleus are present close to yrast, as in the case of the odd-odd and odd-even nuclei, but are far weaker than the yrast partners and therefore not easy to observe. The observed five pairs of nearly degenerate bands were investigated by the constrained and tilted axis cranking covariant density functional theory (TAC-CDFT) [21][22][23][24][25]: one of them is revealed to be a chiral doublet, and the other four are chiral candidates [20]. These observations shed new lights on the investigations of chiral doublets in even-even nuclei. If the four chiral candidates are finally confirmed, then they will constitute a multiple chiral doublet (MχD), a phenomenon predicted by covariant density functional theory (CDFT) [21,[26][27][28][29] and particle rotor model (PRM) [30][31][32][33], and observed experimentally [34][35][36]. The future identification of such bands in 136 Nd will hopefully open a campaign of measurements for other even-even triaxial nuclei, in which the chirality or the MχD phenomenon could exist.
In Ref. [20], as mentioned above, the observed doublet bands in 136 Nd were investigated in the framework of the TAC-CDFT [22][23][24][25], which is a fully microscopic approach, but cannot describe the energy splitting and the quantum tunneling between the chiral doublet bands. The aim of the present work is to investigate the chirality of 136 Nd in the framework of PRM. PRM is a quantal model coupling the collective rotation and the single-particle motions. In contrast to the TAC approach, it describes a system in the laboratory frame.
The total Hamiltonian are 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 [10,21,25,[34][35][36]53].
Various versions of PRM have been developed to investigate the chiral doublet bands with different kinds of configurations. For example, the PRM with one-particle-one-hole configuration was used to describe the chirality in odd-odd nuclei [1,33,37,38,54]. To simulate the effects of many valence nucleons, pairing correlations were introduced and PRM with two quasiparticles configuration was developed [39,[55][56][57][58][59][60]. To describe the odd-A nuclei, the many-particle-many-hole versions of PRM with nucleons in two single-j shells [40,61] or three single-j shells [34,35,53,62], have been developed. It is noted that the unpaired nucleon configurations of the doublet bands in the even-even nucleus 136 Nd involve four different single-j shells. Such PRM is still unavailable.
In this letter, a PRM that couples nucleons in four single-j shells to a triaxial rotor core is developed and applied to study the energy spectra, the electromagnetic transition probabilities, as well as the angular momentum geometries for the observed doublet bands in 136 Nd.
The formalism of the PRM in the present work is an extension of that in Ref. [40], where the many-particle-many-hole version of PRM with two single-j shells was developed. The total Hamiltonian of PRM is expressed aŝ with the collective rotor Hamiltonian where the indexes k = 1, 2, and 3 refer to the three principal axes of the body-fixed frame.
TheR k andÎ k denote the angular momentum operators of the core and of the total nucleus, respectively, and theĴ k is the total angular momentum operator of the valence nucleons. The moments of inertia of the irrotational flow type [63] are adopted, i.e., with γ the triaxial deformation parameter. In addition, the intrinsic Hamiltonian is written as where ε i,ν is the single particle energy in the i-th single-j shell provided by Here, the plus or minus sign refers to particle or hole, and the coefficient C is proportional to the quadrupole deformation β as in Ref. [64].
The single particle state and its time reversal state are expressed as where Ω is the projection of the single-particle angular momentum j along the 3-axis of the intrinsic frame and restricted to . . . , −3/2, 1/2, 5/2, . . . due to the time-reversal degeneracy, and α denotes the other quantum numbers. For a system with 4 i=1 N i valence nucleons (N i denotes the number of the nucleons in the i-th single-j shell), the intrinsic wave function is given as with n i + n ′ i = N i and 0 ≤ n i ≤ N i . The total wave function can be expanded into the strong coupling basis where |IMK is the Wigner function 2I+1 8π 2 D I M K . The basis states are symmetrized under the point group D 2 , which leads to K − 1 i ) being an even integer.
It is noted that due to the inclusion of many-particle-many-hole configurations with four single-j shells, the size of the basis space is rather large. For example, for the calculations of band D1 in 136 Nd (see its configuration in Table I There are five pairs of doublet bands in 136 Nd (labeled as bands D1-D5), in which three of them (bands D1, D2, and D5) are positive parity. Besides, there is a dipole band (labeled as band D6), which has no partner band. In the PRM calculations for these bands, the unpaired nucleon configurations are consistent with those in Ref. [20] and the corresponding quadrupole deformation parameters (β, γ) are obtained from triaxial constrained CDFT calculations [21]. The moments of inertia J 0 and Coriolis attenuation factors ξ are adjusted to reproduce the trend of the energy spectra. The corresponding details are listed in Table I. In addition, 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)) [63] are adopted.
The calculated energy spectra for the bands D1-D6 in 136 Nd are presented in Fig. 1, together with the corresponding data. The experimental energy spectra are reproduced excellently by the PRM calculations. Being a quantum model, PRM is able to reproduce the energy splitting for the whole observed spin region. It is seen that except for D2, the trend and amplitude for the energy splitting between partner bands are reproduced well.
For D2 and D5, the energy differences between the doublet bands decrease gradually with the spin. In detail, for D2, the energy splitting is ∼ 360 keV at I = 21 , and finally goes to ∼ 110 keV at I = 25 . The PRM calculations underestimate this energy separation.
For bands D3 and D4, their B(M1)/B(E2) values are similar. They exhibit a trend that first increase and then decrease with increasing spin. Moreover, their quasi-particle alignments show pronounced similarity in a wide interval of rotational frequency, shown in Ref. [20]. It seems that they were a MχD built on identical configuration as in 103 Rh [35].
However, as their spectra are interweaved each other at several spins, this possibility is excluded. In the calculations, we use a configuration with three single-j shells to describe D3 and a configuration with four single-j shells to describe D4, shown in Table I At I = 17 , the two proton holes contribute angular momenta ∼ 2 . At I = 21 , the two proton holes contribute ∼ 5 . Such difference causes the energy difference between the doublet bands at this spin region ∼ 400 keV as shown in Fig. 1.
For the bands D4 and D4-C, as shown in Fig. 6, similar aplanar orientation of the angular momenta of the rotor, the particles, and the holes can be observed. This supports that D4 and D4-C are chiral doublets. As discussed previously, there is a band-crossing at I = 19 , and the adopted configuration is only suitable for describing the data above band-crossing.