⇦Chinese Physics Letters, 2020, Vol. 37, No. 11, Article code 112101 Possible Candidates for Chirality in the Odd-Odd As Isotopes Chen Liu (刘晨), Shouyu Wang (王守宇)*, Bin Qi (亓斌), and Hui Jia (贾慧) Affiliations Shandong Provincial Key Laboratory of Optical Astronomy and Solar-Terrestrial Environment, School of Space Science and Physics, Institute of Space Sciences, Shandong University, Weihai 264209, China Received 6 August 2020; accepted 17 September 2020; published online 8 November 2020 Supported by the National Natural Science Foundation of China (Grant Nos. 11705102 and 11675094), the Shandong Natural Science Foundation (Grant No. JQ201701), and the Young Scholars Program of Shandong University, Weihai.
*Corresponding author. Email: sywang@sdu.edu.cn Citation Text: Liu C, Wang S Y, Qi B and Jia H 2020 Chin. Phys. Lett. 37 112101    Abstract The deformations and the corresponding configurations of the odd-odd As isotopes are investigated using the adiabatic and configuration-fixed constrained triaxial relativistic mean field (RMF) theory. Energy minima with triaxial deformations and high-$j$ particle-hole configurations are obtained in $^{72,74,76,78,80}$As, where the chiral doublet bands are possible to appear. The existence of multiple chiral doublet (M$\chi$D) is demonstrated in $^{74,76,78}$As. Based on the calculated single-particle levels, we also find possible coexistence of chiral and pseudospin symmetries in the odd-odd As isotopes. DOI:10.1088/0256-307X/37/11/112101 PACS:21.60.Jz, 21.10.Re, 27.50.+e © 2020 Chinese Physics Society Article Text Research on the nuclear physics is in the forward position of matter science. The atomic nucleus has exhibited a wealth of interesting physics, such as shell evolution,[1,2] reflection symmetry breaking,[3,4] and chirality.[5,6] Among those phenomena, the occurrence of chirality in nuclear physics was suggested in 1997 by Frauendorf and Meng.[5] The restoration of chiral symmetry in the laboratory frame may give rise to pairs of nearly degenerate $\Delta I = 1$ bands with the same parity, namely chiral doublet bands. Since the pioneering work on chirality in nuclear physics, the manifestation of nuclear chirality has become the subject of numerous experimental and theoretical studies in the last two decades. Based on the triaxial relativistic mean field (RMF) calculations, it has been suggested that multiple chiral doublet ($M\chi$D) can exist in a single nucleus.[7–15] Up to now, the chiral and $M\chi$D have been reported in the $A \approx 80$, 100, 130 and 190 mass regions (see recent reviews[6,16,17] and the references therein). The $A \sim 80$ mass region is the lightest region in the present investigations of chiral symmetry breaking and has attracted both experimental and theoretical interests. For the $A \approx 80$ mass region, the possible candidates for chirality are most likely to appear in the As, Br, and Rb isotopes considering the necessary high-$j$ particle and hole configurations. So far, only three candidate chiral nuclei ($^{78,80,82}$Br) were reported experimentally.[18–20] On the theoretical side, based on the adiabatic and configuration-fixed constrained triaxial RMF theory calculations, possible existence of chiral and $M\chi$D was demonstrated in the odd-odd Br ($Z = 35$) isotopes and Rb ($Z = 37$) isotopes.[13,14] However, systematic theoretical calculations have not been carried out for the As isotope up to date. The present work focuses on searching for the chirality in the As ($Z = 33$) isotopes. In addition, for the nuclei in the $A \approx 80$ mass region, the valence protons and neutrons may occupy the $1g_{9/2}$, $2p_{1/2}$, $1f_{5/2}$ and $2p_{3/2}$ orbits. The latter two orbits are pseudospin doublet states, i.e., single-particle states with quantum numbers ($n$, $l$, $j =l+1/2$) and ($n-1$, $l+2$, $j =l+3/2$).[21–24] The coexistence of chiral and pseudospin symmetries in one single nucleus is also expected in the $A \approx 80$ mass region. Based on the above considerations, we study the deformations and the corresponding configurations in the odd-odd $^{72,74,76,78,80,82}$As isotopes using the adiabatic and configuration-fixed constrained triaxial RMF theory. The RMF theory has been successfully used in describing the triaxial shape coexistence and possible $M\chi$D in nuclei.[25–28] The details of the formalism and numerical techniques of the triaxial RMF can be seen in Refs. [29,30]. In the present triaxial RMF calculations, the point-coupling density functional PC-PK1[30] with a basis of 12 major oscillator shells is employed. The constrained calculations with $\beta$ are carried out, in which the $\gamma$ is automatically obtained by minimizing the energy. The adiabatic constrained calculations are used to obtain the states with different configurations, and then the configuration-fixed constrained calculations are performed to obtain the energy minima with a certain configuration. The adiabatic and configuration-fixed constrained calculations mean that the nucleons always occupy the lowest single particle levels and the same combination of the single particle levels during the constraint process.[7] The calculated potential energy surfaces as a function of $\beta$ in adiabatic and configuration-fixed constrained triaxial RMF calculations for $^{72,74,76,78,80,82}$As are presented in Fig. 1, where the minima obtained in the potential energy surfaces are labeled alphabetically. As shown in Fig. 1, $^{72,74,76,78,80,82}$As have more than one states of minimum with obvious triaxial deformations ($15^{\circ} \leq \gamma \leq 45^{\circ}$), which present good examples of triaxial shape coexistence. In the present work, the states with excited energies higher than 3 MeV are not considered because they are difficult to be populated in experiment. The total energies $E_{\rm tot}$, quadrupole deformation parameters $\beta$ and $\gamma$, and their corresponding valence nucleon configurations for the states with excitation energies less than 3 MeV in the configuration-fixed constrained triaxial RMF calculations for $^{72,74,76,78,80,82}$As are listed in Table 1. The valence nucleon configurations are given by taking reference of the $^{68}$Ni ($Z = 28$, $N = 40$) core. Apart from the triaxial deformation, the proper high-$j$ particle and hole configuration is also necessary for the construction of chirality. By examining the deformations and the configurations obtained in the calculations, we find that some states are suitable for the construction of chirality. These states are marked as blue with stars in the figures. As shown in Table 1, the states C, D, E, and H in $^{72}$As have the triaxial deformations, in which the state H with the $\pi g^{1}_{9/2}\otimes\nu g^{-1}_{9/2}$ particle-hole configuration satisfies the conditions of establishing chiral doublet bands. States C, D, and E can be ruled out because they do not have the required high-$j$ particle and hole configuration. It is interesting to search for the possible chiral doublet bands in $^{72}$As. For $^{74}$As, all the states except the ground state A show triaxial deformations. However, only the states B, E, and F associated with the high-$j$ particle and hole configuration. The configurations for the states B and E/F are $\pi g^{1}_{9/2}\otimes\nu g^{-1}_{9/2}$ and $\pi (f_{5/2},p_{3/2})^{1}\otimes\nu g^{-1}_{9/2}$, respectively. These states are suitable for the construction of chirality. Therefore, $M\chi$D is expected in $^{74}$As. $^{76}$As is also a good example for triaxial shape coexistence. As shown in Table 1, triaxial deformations are exhibited in all the states except the state E. After taking the high-$j$ particle and hole configuration into consideration, we find that the states A, B, and F are suitable for the construction of chirality. Thereinto, the configurations of the states A and F are $\pi g^{1}_{9/2}\otimes\nu g^{-1}_{9/2}[\frac{5}{2}]$ and $\pi g^{1}_{9/2}\otimes\nu g^{-1}_{9/2}[\frac{7}{2}]$, respectively. Here we expect two pairs of chiral doublet bands based on the $\pi g^{1}_{9/2}\otimes\nu g^{-1}_{9/2}$ configuration, thereby forming $M\chi$D in $^{76}$As. The same type of $M\chi$D has also been theoretically predicted in $^{78}$Rb.[13] In addition, the state B with the $\pi (f_{5/2},p_{3/2})^{1}\otimes\nu g^{-1}_{9/2}$ configuration also shows triaxial deformation. Chiral doublet bands based on the $\pi (f_{5/2},p_{3/2})^{1}\otimes\nu g^{-1}_{9/2}$ configuration are also expected in $^{76}$As.
cpl-37-11-112101-fig1.png
Fig. 1. The energy surfaces in adiabatic (open circles) and configuration-fixed (solid lines) constrained triaxial RMF calculations with PC-PK1 for $^{72,74,76,78,80,82}$As. The minima in the energy surfaces for fixed configuration are represented as stars and labeled alphabetically. Their corresponding triaxial deformation parameters $\beta$ and $\gamma$ are also given. The suitable states for the appearance of the chirality are marked as blue with stars.
Table 1. The total energies $E_{\rm tot}$, triaxial deformation parameters $\beta$ and $\gamma$, and their corresponding valence nucleon configurations of minima states in the configuration-fixed constrained triaxial RMF calculations for $^{72,74,76,78,80,82}$As. The states suitable for the construction of chirality are marked with stars.
State Configuration $E_{\rm tot}$ ($\beta, \gamma$) $E_{x}$(calc)
Valence nucleons Unpaired nucleons (MeV) (MeV)
$^{72}$As A $\pi (f_{5/2},p_{3/2})^{5}\otimes\nu g^{1}_{9/2}p^{-2}_{1/2}$ $\pi (f_{5/2},p_{3/2})^{1}\otimes\nu g^{1}_{9/2}$ $-$620.69 (0.20,60.0$^{\circ}$) 0.00
B $\pi (f_{5/2},p_{3/2})^{5}\otimes\nu g^{2}_{9/2}p^{-2}_{1/2}p^{-1}_{3/2}$ $\pi (f_{5/2},p_{3/2})^{1}\otimes\nu p^{-1}_{3/2}$ $-$620.53 (0.24,48.3$^{\circ}$) 0.16
C $\pi (f_{5/2},p_{3/2})^{5}\otimes\nu g^{3}_{9/2}p^{-2}_{1/2}p^{-2}_{3/2}$ $\pi (f_{5/2},p_{3/2})^{1}\otimes\nu g^{1}_{9/2}$ $-$619.94 (0.28,41.4$^{\circ}$) 0.75
D $\pi g^{1}_{9/2}(f_{5/2},p_{3/2})^{4}\otimes\nu g^{3}_{9/2}p^{-2}_{1/2}p^{-2}_{3/2}$ $\pi g^{1}_{9/2}\otimes\nu g^{1}_{9/2}$ $-$619.55 (0.33,28.8$^{\circ}$) 1.14
E $\pi g^{1}_{9/2}(f_{5/2},p_{3/2})^{4}\otimes\nu g^{4}_{9/2}p^{-2}_{1/2}f^{-2}_{5/2}p^{-1}_{3/2}$ $\pi g^{1}_{9/2}\otimes\nu p^{-1}_{3/2} $ $-$619.09 (0.38,16.7$^{\circ}$) 1.60
F $\pi g^{2}_{9/2}(f_{5/2},p_{3/2})^{3}\otimes\nu g^{4}_{9/2}p^{-2}_{1/2}f^{-2}_{5/2}p^{-1}_{3/2}$ $\pi (f_{5/2},p_{3/2})^{1}\otimes\nu p^{-1}_{3/2} $ $-$618.90 (0.43,11.6$^{\circ}$) 1.79
G $\pi (f_{5/2},p_{3/2})^{5}\otimes\nu g^{4}_{9/2}p^{-2}_{1/2}f^{-2}_{5/2}p^{-1}_{3/2}$ $\pi (f_{5/2},p_{3/2})^{1}\otimes\nu p^{-1}_{3/2}$ $-$618.36 (0.33,13.1$^{\circ}$) 2.33
H$^{\ast}$ $\pi g^{1}_{9/2}(f_{5/2},p_{3/2})^{4}\otimes\nu g^{5}_{9/2}p^{-2}_{1/2}f^{-2}_{5/2}p^{-2}_{3/2}$ $\pi g^{1}_{9/2}\otimes\nu g^{-1}_{9/2}$ $-$618.10 (0.39,31.4$^{\circ}$) 2.59
$^{74}$As A $\pi (f_{5/2},p_{3/2})^{5}\otimes\nu g^{3}_{9/2}p^{-2}_{1/2}$ $\pi (f_{5/2},p_{3/2})^{1}\otimes\nu g^{1}_{9/2}$ $-$640.19 (0.23,55.6$^{\circ}$) 0.00
B$^{\ast}$ $\pi g^{1}_{9/2}(f_{5/2},p_{3/2})^{4}\otimes\nu g^{5}_{9/2}p^{-2}_{1/2}p^{-2}_{3/2}$ $\pi g^{1}_{9/2}\otimes\nu g^{-1}_{9/2}$ $-$639.32 (0.37,23.6$^{\circ}$) 0.87
C $\pi g^{1}_{9/2}(f_{5/2},p_{3/2})^{4}\otimes\nu g^{4}_{9/2}p^{-2}_{1/2}p^{-1}_{3/2}$ $\pi g^{1}_{9/2}\otimes\nu p^{-1}_{3/2}$ $-$638.99 (0.32,24.9$^{\circ}$) 1.20
D $\pi (f_{5/2},p_{3/2})^{5}\otimes\nu g^{4}_{9/2}p^{-2}_{1/2}p^{-1}_{3/2}$ $\pi (f_{5/2},p_{3/2})^{1}\otimes\nu p^{-1}_{3/2}$ $-$638.62 (0.28,22.0$^{\circ}$) 1.57
E$^{\ast}$ $\pi g^{2}_{9/2}(f_{5/2},p_{3/2})^{3}\otimes\nu g^{5}_{9/2}p^{-2}_{1/2}p^{-2}_{3/2}$ $\pi (f_{5/2},p_{3/2})^{1}\otimes\nu g^{-1}_{9/2}$ $-$638.38 (0.41,20.0$^{\circ}$) 1.81
F$^{\ast}$ $\pi (f_{5/2},p_{3/2})^{5}\otimes\nu g^{5}_{9/2}p^{-2}_{1/2}p^{-2}_{3/2}$ $\pi (f_{5/2},p_{3/2})^{1}\otimes\nu g^{-1}_{9/2}$ $-$638.32 (0.32,20.1$^{\circ}$) 1.87
G $\pi (f_{5/2},p_{3/2})^{5}\otimes\nu g^{1}_{9/2}$ $\pi (f_{5/2},p_{3/2})^{1}\otimes\nu g^{1}_{9/2}$ $-$638.21 (0.11,17.5$^{\circ}$) 1.98
$^{76}$As A$^{\ast}$ $\pi g^{1}_{9/2}(f_{5/2},p_{3/2})^{4}\otimes\nu g^{5}_{9/2}p^{-2}_{1/2}$ $\pi g^{1}_{9/2}\otimes\nu g^{-1}_{9/2}$ $-$657.59 (0.30,25.6$^{\circ}$) 0.00
B$^{\ast}$ $\pi (f_{5/2},p_{3/2})^{5}\otimes\nu g^{5}_{9/2}p^{-2}_{1/2}$ $\pi (f_{5/2},p_{3/2})^{1}\otimes\nu g^{-1}_{9/2}$ $-$657.57 (0.26,21.3$^{\circ}$) 0.02
C $\pi g^{1}_{9/2}(f_{5/2},p_{3/2})^{4}\otimes\nu g^{6}_{9/2}p^{-2}_{1/2}p^{-1}_{3/2}$ $\pi g^{1}_{9/2}\otimes\nu p^{-1}_{3/2}$ $-$657.36 (0.35,24.0$^{\circ}$) 0.21
D $\pi (f_{5/2},p_{3/2})^{5}\otimes\nu g^{4}_{9/2}p^{-1}_{1/2}$ $\pi (f_{5/2},p_{3/2})^{1}\otimes\nu p^{-1}_{1/2}$ $-$657.29 (0.21,15.8$^{\circ}$) 0.28
E $\pi (f_{5/2},p_{3/2})^{5}\otimes\nu g^{3}_{9/2}$ $\pi (f_{5/2},p_{3/2})^{1}\otimes\nu g^{1}_{9/2}$ $-$657.03 (0.17,10.4$^{\circ}$) 0.54
F$^{\ast}$ $\pi g^{1}_{9/2}(f_{5/2},p_{3/2})^{4}\otimes\nu g^{7}_{9/2}p^{-2}_{1/2}p^{-2}_{3/2}$ $\pi g^{1}_{9/2}\otimes\nu g^{-1}_{9/2}$ $-$656.00 (0.37,27.0$^{\circ}$) 1.57
$^{78}$As A $\pi (f_{5/2},p_{3/2})^{5}\otimes\nu g^{5}_{9/2}$ $\pi (f_{5/2},p_{3/2})^{1}\otimes\nu g^{-1}_{9/2}$ $-$675.53 (0.20,5.3$^{\circ}$) 0.00
B $\pi (f_{5/2},p_{3/2})^{5}\otimes\nu g^{6}_{9/2}p^{-1}_{1/2}$ $\pi (f_{5/2},p_{3/2})^{1}\otimes\nu p^{-1}_{1/2}$ $-$675.04 (0.23,12.9$^{\circ}$) 0.49
C$^{\ast}$ $\pi g^{1}_{9/2}(f_{5/2},p_{3/2})^{4}\otimes\nu g^{7}_{9/2}p^{-2}_{1/2}$ $\pi g^{1}_{9/2}\otimes\nu g^{-1}_{9/2} $ $-$674.13 (0.31,28.0$^{\circ}$) 1.40
D$^{\ast}$ $\pi (f_{5/2},p_{3/2})^{5}\otimes\nu g^{7}_{9/2}p^{-2}_{1/2}$ $\pi (f_{5/2},p_{3/2})^{1}\otimes\nu g^{-1}_{9/2}$ $-$674.01 (0.26,21.6$^{\circ}$) 1.52
$^{80}$As A $\pi (f_{5/2},p_{3/2})^{5}\otimes\nu g^{7}_{9/2}$ $\pi (f_{5/2},p_{3/2})^{1}\otimes\nu g^{-1}_{9/2}$ $-$692.24 (0.19,0.0$^{\circ}$) 0.00
B$^{\ast}$ $\pi g^{1}_{9/2}(f_{5/2},p_{3/2})^{4}\otimes\nu g^{7}_{9/2}$ $\pi g^{1}_{9/2}\otimes\nu g^{-1}_{9/2}$ $-$690.07 (0.23,15.4$^{\circ}$) 2.17
$^{82}$As A $\pi (f_{5/2},p_{3/2})^{5}\otimes\nu g^{9}_{9/2}$ $\pi (f_{5/2},p_{3/2})^{1}\otimes\nu g^{-1}_{9/2}$ $-$705.60 (0.15,0.0$^{\circ}$) 0.00
B $\pi (f_{5/2},p_{3/2})^{5}\otimes\nu g^{8}_{9/2}s^{1}_{1/2}$ $\pi (f_{5/2},p_{3/2})^{1}\otimes\nu s^{1}_{1/2}$ $-$705.11 (0.22,0.0$^{\circ}$) 0.49
cpl-37-11-112101-fig2.png
Fig. 2. Proton single-particle levels obtained in constrained triaxial RMF calculations as a function of deformation $\beta$ for $^{74,76,78}$As. Positive (negative) parity states are marked by dashed (solid) lines. Occupations corresponding to the minima in Fig. 1 are represented by filled circles (two particles) and stars (one particle).
One can see from Table 1, two triaxial deformed states with the configurations $\pi g^{1}_{9/2}\otimes\nu g^{-1}_{9/2}$ (state C) and $\pi (f_{5/2},p_{3/2})^{1}\otimes\nu g^{-1}_{9/2}$ (state D) in $^{78}$As satisfy the conditions of establishing chiral doublet bands, thereby possibly forming $M\chi$D in $^{78}$As. Moreover, the present calculations demonstrate that the state B in $^{80}$As has the triaxial deformation and the $\pi g^{1}_{9/2}\otimes\nu g^{-1}_{9/2}$ configuration. A pair of chiral doublet bands is expected in $^{80}$As. It should be noted that the negative parity chiral states in $^{74,76,78}$As also involve pseudospin orbits $2p_{3/2}$ and $1f_{5/2}$ in the present calculations. Therefore, the present calculations also allow us to study the possible coexistence of chiral and pseudospin symmetries in the odd-odd As isotopes. To demonstrate the coexistence of chiral and pseudospin symmetries, we present the proton single-particle levels based on the $\pi g^{1}_{9/2}\otimes\nu g^{-1}_{9/2}$ configuration as a function of deformation $\beta$ for $^{74,76,78}$As in Fig. 2. The positive (negative) parity levels are marked by dashed (solid) lines, and the occupations corresponding to the minima of the chiral configurations in Figs. 1(b)–1(d) are represented by filled circles (two particles) and stars (one particle) in Fig. 2. The corresponding quantum numbers for the spherical case are labeled on the left side of the levels. One can see from Fig. 2, the two levels dominated by the ($2p_{3/2}$,$1f_{5/2}$) components (denoted in bold) are nearly degenerate in a large scale of the $\beta$ region for $^{74,76,78}$As, which manifests the characteristic of the pseudospin symmetry in single-particle spectra. For the state F in $^{74}$As, state B in $^{76}$As, and state D in $^{78}$As, it is difficult to distinguish the unpaired protons from occupying the $2p_{3/2}$ or $1f_{5/2}$ orbits due to the strongly mixing of these two levels. Therefore, the negative parity chiral states in $^{74,76,78}$As could also associated with pseudospin states. A novel type of rotational bands originated from the coexistence of chiral and pseudospin symmetries have been introduced as “chirality-pseudospin triplet (or quartet) bands” in Ref. [31]. Such bands were claimed to be observed in $^{131}$Ba recently.[32] The present calculations demonstrate the coexistence of chiral and pseudospin symmetries in the As isotopes. It should be noted that the pseudospin symmetry with the triaxial deformation is still an open problem.[23,24] Therefore, it is highly encouraged to explore the possible chirality-pseudospin bands in the $A \approx 80$ mass region experimentally and gain a better understanding of the pseudospin symmetry with the triaxial deformation. In summary, the deformations and the corresponding configurations of the odd-odd As isotopes have been investigated using the adiabatic and configuration-fixed constrained triaxial relativistic mean field (RMF) theory. Energy minima with triaxial deformations and $\pi g^{1}_{9/2}\otimes\nu g^{-1}_{9/2}$ configurations are obtained in $^{72,74,76,78,80}$As, where the chiral doublet bands are possible to appear. In addition, the triaxial deformed states with $\pi (f_{5/2},p_{3/2})^{1}\otimes\nu g^{-1}_{9/2}$ configuration are also obtained in $^{74,76,78}$As, thereby possibly forming multiple chiral doublet ($M\chi$D) in $^{74,76,78}$As. Based on the calculated single-particle levels, we also find the possible coexistence of chiral and pseudospin symmetries in $^{74,76,78}$As. The numerical calculations in this study were carried out on the supercomputing system in the Supercomputing Center and an HP Proliant DL785G6 server hosted by the Institute of Space Science in Shandong University, Weihai. 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