Multiple chiral doublet bands with octupole correlations in reflection-asymmetric triaxial particle rotor model

A reflection-asymmetric triaxial particle rotor model (RAT-PRM) with a quasi-proton and a quasi-neutron coupled with a reflection-asymmetric triaxial rotor is developed and applied to investigate the multiple chiral doublet (M$\chi$D) bands candidates with octupole correlations in $^{78}$Br. The calculated excited energies, energy staggering parameters, and $B(M1)/B(E2)$ ratios are in a reasonable agreement with the data of the chiral doublet bands with positive- and negative-parity. The influence of the triaxial deformation $\gamma$ on the calculated $B(E1)$ is found to be significant. By changing $\gamma$ from 16$^\circ$ to 21$^\circ$, the $B(E1)$ values will be enhanced and better agreement with the $B(E1)/B(E2)$ data is achieved. The chiral geometry based on the angular momenta for the rotor, the valence proton and valence neutron is discussed in details.


I. INTRODUCTION
Chirality is a subject of general interests in natural science. Since the pioneering work of nuclear chirality by Frauendorf and Meng in 1997 [1], many efforts have been devoted to explore the chirality in atomic nuclei, see e.g., reviews [2][3][4][5][6][7].
Because of the observation of eight strong electric dipole (E1) transitions linking the positive-and negative-parity chiral bands [11], the MχD candidates observed in 78 Br provide the first example of chiral geometry in octupole soft nuclei and indicate that nuclear chirality can be robust against the octupole correlations. It also indicates that the chirality-parity quartet bands [2,11], which are a consequence of the simultaneous breaking of chiral and space-reflection symmetries, may exist in nuclei. The observations of MχD with octupole correlations and/or the possible chirality-parity quartet bands have brought severe challenges to current nuclear models and, thus, require the development of new approaches.
In Ref. [11], the triaxial PRM calculation has been performed to describe the positiveand negative-parity chiral doublet bands observed in 78 Br with two individual configurations πg 9/2 ⊗ νg 9/2 and πf 5/2 ⊗ νg 9/2 , respectively. The calculation supports the interpretation of the MχD with different parities [11]. However, the E1 linking transitions between the positive-parity band 1 and the negative-parity band 3 are not accessible in the triaxial PRM due to the omission of the octupole degree of freedom.
In this work, a reflection-asymmetric triaxial PRM (RAT-PRM) with both triaxial and octupole degrees of freedom is developed and applied to the MχD candidates with octupole correlations in 78 Br. The model is introduced in Sec. II, and the numerical details are presented in Sec. III. The calculated results for the doublet bands, such as energy spectra, electromagnetic transitions, and angular momentum orientations, are discussed in Sec. IV, and a summary is given in Sec. V.

II. FORMALISM
The total RAT-PRM Hamiltonian can be expressed aŝ intr. is the intrinsic Hamiltonian for valence protons (neutrons) in a reflectionasymmetric triaxially deformed potential, andĤ core is the Hamiltonian of a reflectionasymmetric triaxial rotor, which is generalized straightforwardly from the reflection-asymmetric axial rotor in Ref. [36].

The core Hamiltonian readŝ
withR k =Î k −ĵ pk −ĵ nk . Here,R k ,Î k ,ĵ pk , andĵ nk are the angular momentum operators for the core, the nucleus, the valence protons and neutrons, respectively. The moments of inertia for irrotational flow are adopted J k = J 0 sin 2 (γ − 2kπ/3). The core parity splitting parameter E(0 − ) can be viewed as the excitation energy of the virtual 0 − state [36]. The core parity operatorP can be written as the product of the single-particle parity operator π and the total parity operatorp.
To include the pairing correlations in RAT-PRM, one should replace the single-particle and the quasiparticle operators with u 2 ν + v 2 ν = 1. Furthermore, the single-particle energies ε ν should be replaced by quasiparticle energies ε ′ ν = (ε ν − λ) 2 + ∆ 2 . Therefore, the intrinsic Hamiltonian becomeŝ The HamiltonianĤ is diagonalized numerically in the symmetrized strong-coupled basis with good parity and angular momentum, whereŜ 2 =PR 2 is the reflection operator with respect to the plane perpendicular to 2-axis, |IMK = 2π + 1 8π 2 D I * M K is the Wigner function, ψ ν ± are the intrinsic wavefunctions with good parity, Hereχ ν pχ ν n Φ a is the strong-coupled intrinsic core-quasiparticle wavefunction; Φ a represents that the core has the same orientation in space as the intrinsic single-particle potential, and χ ν p(n) is the BCS quasiparticle state of the proton (neutron). The diagonalization of the HamiltonianĤ gives rise to the nuclear eigenstate, which is a composition of the strong-coupled basis with the coefficients c ν IKp . Then, the reduced electromagnetic transition probabilities can be calculated via [42] B(σλ, where σ denotes either E or M for electric and magnetic transitions, respectively, λ is the rank of the transition operator, and M σ λµ the electromagnetic transition operator. The magnetic dipole (M1) transition operator iŝ where g p , g n , and g R are the effective gyromagnetic ratios for valence proton, valence neutron, and the collective core, respectively, andĵ 1µ denotes the spherical tensor in the laboratory frame. The electric multipole transition operators contain two terms [40], which are contributions from the core and the valence particles, respectively. Here, R 0 = 1.2A 1/3 is the nuclear radius. For electric quadrupole (E2) transitions, one can safely neglect the valence particle term, since it is much smaller than the term of the core [40]. However, this is not the case for E1 transitions. Since the total center of mass remains at rest, the motion of the valence particles is influenced by the recoil of the core. This effect is of special importance for E1 transitions. Therefore, as in Ref. [42], the total moment in a one-particle transition is obtained by replacing the charge of the particle by an effective one,
As the potential energy surface is soft with respect to β 3 , β 3 = 0.02 is adopted to include the effect of octupole correlations in the present RAT-PRM calculations.
With the deformation parameters above, the reflection-asymmetric triaxial Nilsson Hamiltonian with the parameters κ, µ in Ref. [47] is solved by expanding the wavefunction by harmonic oscillator basis [48]. The Fermi energies of proton and neutron are chosen as λ p = 44.6 MeV and λ n = 47.6 MeV, corresponding to the πg 9/2 [m z = 1/2] and νg 9/2 [m z = 5/2] orbitals respectively, which are consistent with the MDC-CDFT results. The single-particle space is truncated to 13 levels, with six above and below the Fermi level. Increasing the size of the single-particle space does not influence the band structure in the present work. The pairing correlation is taken into account by the empirical pairing gap formula ∆ = 12/ √ A MeV.
The moment of inertia J 0 = 14 2 /MeV and the core parity splitting parameter E(0 − ) = 3 MeV, are adjusted to the experimental energy spectra. For the calculations of magnetic transitions, the gyromagnetic ratios for the collective rotor, protons, and neutrons are given by g R = Z/A, g p(n) = g l + (g s − g l )/(2l + 1), respectively [40,42]. bands is an indication for nuclear chirality as suggested in Ref. [55].

IV. RESULTS AND DISCUSSION
Since the octupole degree of freedom is included in the present RAT-PRM calculations, the electric dipole transition probabilities B(E1) between the positive-and negative-parity It is found that the influence of the triaxial deformation γ on the calculated B(E1) is significant. By changing γ from 16 • to 21 • (given by cranked-shell-model calculations [49]), as shown in Fig. 2, the B(E1) values will be enhanced and better agreement with the B(E1)/B(E2) data is achieved.
In order to investigate the chiral geometry, the angular momentum components for the core R k = R 2 k 1/2 , the valence proton j pk = ĵ 2 pk 1/2 , and the valence neutron j nk = ĵ 2  For the positive-parity doublet bands in Fig. 3, the angular momentum of the valence proton mainly aligns in the i-s plane, while that of the valence neutron has nearly equal components on the three axes due to its mid-shell nature. Considering the fact that the angular momentum for the core mainly aligns along the i-axis, and grows rapidly, the total angular momentum lies close to the i-s plane, which is consistent with the large energy difference between the doublet bands. For band 1, the three components of j p for the valence proton vary smoothly with the spin, while the three components of j n for the valence neutron exhibit staggering with I > 12 . For band 2, the three components of both j p and j n exhibit staggering with I ≥ 9 . These staggering behaviors might be understood from the main components of the intrinsic wavefunctionχ p(n) . It is found that these staggering behaviors are associated with the variation of the corresponding main components. Taking band 1 as an example, with I > 12 , the main component of the neutron intrinsic wavefunction varies alternately between g 9/2 [m z = 5/2] and g 9/2 [m z = 3/2].
For the negative-parity doublet bands in Fig. 4, the angular momentum of the valence proton mainly aligns in the i-s plane, and the alignment of the valence neutron along the l-axis is significant. To be more precise, j p ∼ 4 in the i-s plane, j n ∼ 2 along l-axis, and R ∼ 2 − 13 along i-axis. This is the chiral geometry for the negative-parity doublet bands. As the total angular momentum increases, R increases gradually, j n remains almost unchanged, while j p moves gradually toward the i-axis. The difference between the proton and neutron alignments may result from the fact that the Coriolis alignment effects are weaker for the neutron in the relatively low-j f 5/2 shell. It is found that for both band 3 and band 4, the three components of j p and j n vary smoothly with the spin. This is different from the case of band 1 and band 2, because the main components here are always  In Fig. 5, the calculated effective angles θ Rp , θ Rn and θ pn as functions of spin for the positive-and negative-parity doublet bands are presented. The effective angle θ pn between the angular momenta of the proton j p and neutron j n is defined as [56] cos θ pn = j p · j n j 2 p j 2 n .

(21)
A similar expression for the effective angle θ Rp (θ Rn ) between the angular momenta for the core and the valence proton (neutron) can be defined straightforwardly.
In Finally, a few remarks on the effective angles near the bandheads are appropriate. In Ref. [57], the paradox, i.e., the effective angles between any two of the angular momentum components are closed to 90 • in the regime of chiral vibration, has been clarified. This paradox is due to the fact that the angular momentum of the rotor is much smaller than those of the proton and neutron near the bandhead. Here, the three effective angles near the bandheads, in the regime of chiral vibration, are close to 90 • for positive-parity band 2, and negative-parity doublet bands 3 and 4. However, the effective angle θ pn at the bandhead for band 1 is only 70 • due to the deviation from the ideal particle-hole configuration and triaxial deformation.
It should be noted that for the negative-parity doublet bands the previous adopted configuration is πf 5/2 ⊗ νg 9/2 [11]. In the present calculations, the configuration πg 9/2 ⊗ νg 9/2 same as in Ref. [11] is adopted for the positive-parity doublet bands. After including the octupole deformation, the positive-and negative-parity bands can be simultaneously obtained by diagonalizing the RAT-PRM Hamiltonian. For the yrast band with negative parity, the configuration is found to be πg 9/2 ⊗ νf 5/2 . Further support for this subtle change in configuration for the negative-parity doublet bands may be obtained from future microscopic calculations and experimental results, for example, the three-dimensional TAC-CDFT [24] including the octupole deformation or the measurement of the g factor in the chiral bands [58].

V. SUMMARY
In summary, a reflection-asymmetric triaxial particle rotor model (RAT-PRM) with a quasi-proton and a quasi-neutron coupled with a reflective-asymmetric triaxial rotor is de- The chiral geometry and its evolution are discussed in details from the angular momentum components for the core as well as the valence proton and neutron. For the positive-parity doublet bands, in consistent with the large energy difference between the doublet bands, the total angular momentum lying close to the i-s plane. For the negative-parity doublet bands, the chiral geometry is constructed by the angular momenta of the valence proton along the