aa r X i v : . [ nu c l - e x ] S e p Recent progress in multiple chiral doublet bands *Shou-Yu Wang
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
Abstract:
The recent progress of multiple chiral doublet bands (M χ D) is reviewed for both experimental andtheoretical sides. In particular, the experimental findings, theoretical predictions, selection rule for electromagnetictransitions, M χ D with octupole correlations and some related topics are highlighted. Based on these discussions,it is of highly scientific interest to search for the more M χ D as well as the possible chiral wobblers, chirality-parityquartet and chirality-pseudospin triplet (or quartet) bands in nuclear system.
Key words: chiral doublet bands, M χ D, octupole correlations, chirality-parity quartet bands
PACS:
Handedness or chirality is a well-known phenomenonin chemistry, biology and particle physics. Many bio-logical molecules occur in identical but left- and right-handed modes. In particle physics chirality is a dynami-cal feature of massless particles, which distinguishes theparallel or antiparallel orientation of spin and momen-tum. However, scientists have long believed that theatomic nucleus are too symmetrical to exist as left- andright-handed versions. The question about the existenceof chiral nuclei is thus of great interest.The first prediction of chirality in atomic nuclei wasmade by Frauendorf and Meng in 1997 [1]. They pointedout the existence of this phenomenon in triaxial odd-oddnuclei where three angular momentum vectors may cou-ple to each other in either a left- or right-handed mode.Such a chiral geometry may give rise to pairs of nearlydegenerate ∆ I = 1 bands with the same parity, i.e., thechiral doublet bands. To test the theoretical predictionof chirality in atomic nuclei, much effort has been de-voted to further exploring this interesting phenomenon.So far, such chiral doublet bands have been reported inthe A ∼
80 [2–4], 100 [5–12], 130 [13–29], and 190 [30–36]mass regions of the nuclear chart. For details, see recentreviews on nuclear chirality and related topics [37–45] ordata tables [46].In 2006, based on the adiabatic and configuration-fixed constrained triaxial relativistic mean field (RMF)calculations, triaxial shape coexistence with high- j proton-hole and neutron-particle configurations is found in Rh, which demonstrates the possible existence ofmultiple chiral doublet (acronym M χ D) bands in thisnucleus [47]. In the last decade, the theoretical predic-tion of M χ D has stimulated a lot of experimental ef-forts [3, 12, 28, 48–52]. In this review, we will presentthe recent progress of M χ D. χ D In this section, our attempt is to provide an overviewof experimental findings of M χ D. In 2013, two distinctsets of chiral doublet bands based on the πh / ⊗ νh − / and πh / ( g / ) − ⊗ νh − / configurations were identi-fied in the odd- A nucleus Ce, which was regardedas the strong experimental evidence for the existence ofM χ D [28]. The experimental observations of M χ D rep-resent an important confirmation of triaxial shape coex-istence and its geometrical interpretation. Later, a noveltype of M χ D bands with the identical configuration wasfound in
Rh [12], which shows that chiral geometrycan be robust against the increase of the intrinsic exci-tation energy. The M χ D with octupole correlations wasidentified in the odd-odd Br, which provides the firstexample of chiral geometry in octupole soft nuclei, andindicates that nuclear chirality can be robust against theoctupole correlations [3]. It also indicates that a simulta-neous breaking of chiral and space-reflection symmetriesmay exist in nuclei. In 2018, the M χ D involving 3 and 5quasiparticle configurations has been observed in odd-A
Tl [52], which is the first observation of such bandsin A ∼
190 mass region. Five pairs of nearly degenerate
Received xxxx 201x ∗ This work was supported by the Shandong Natural Science Foundation, China (Grant No. JQ201701), and the National NaturalScience Foundation of China (Grant No. 11622540).1) E-mail: [email protected] c (cid:13) oublet bands were reported in the even-even Nd [51].Very recently, a new pair of chiral doublet bands withthe πg / h / ⊗ νh / configuration was identified in Nd [53], which is the isotone of
Ce. The new ob- served chiral doublet bands together with the previouslyknown chiral bands with the πh / ⊗ νh / configura-tion [20] constituted a new example of M χ D bands.
Table 1. The observed M χ D nucleus candidate, reaction used to produce M χ D, number of pairs for the observedchiral doublet bands, and single-particle configuration of M χ D Nucleus Reaction Number Single-particle configuration Br [3] Zn( C, p3n) 2 πg / ⊗ νg / , πf / ⊗ νg / Rh [12] Zr( B, 4n) 3 πg / ⊗ νh / , πg / ⊗ νh / g / Rh [7, 8]
Mo( B, α Zr( C, p3n) 2 πg / ⊗ νh / , πg / ⊗ νh / g / Ag [56–58]
Mo( B, 4n), Zr( O, p3n) 2 πg − / ⊗ νh / , πg − / ⊗ νh / ( d / /g / ) Ce [28]
Cd( Ne, 5n)
Ce 2 πh / ⊗ νh / , πg / h / ⊗ νh / Nd [20, 53]
Pd( Si, 5n),
Mo( Ar, 5n) 2 πh / ⊗ νh / , πg / h / ⊗ νh / Nd [51]
Mo( Ar, 5n) 5 πh / ( d / , g / ) − ⊗ νh − / ( s / , d / ) − , πh / ( d / , g / ) − ⊗ νh − / ( s / , d / ) − , πh / ( d / , g / ) − ⊗ νh − / ( f / , h / ) , πh / ( d / , g / ) − ⊗ νh − / ( s / , d / ) − πh / ⊗ νh − / ( s / , d / ) − Tl [52] , Re( C, xn) 2 πh / ⊗ νi − / , πi / ⊗ νi − / ( p / f / ) − In fact, the existence of more than one chiral configu-ration in one nucleus had been noticed in 2004. the can-didate chiral doublet bands in
Rh with πg − / ⊗ νh / configuration [7], and another ones with tentatively sug-gested πg − / ⊗ νh / ( d / /g / ) configuration [8] were re-spectively reported by the two different groups. Thetriaxial RMF approaches have been applied to investi-gate their triaxial deformations with the correspondingconfigurations in Rh. Two pairs of doublet bands in
Rh were suggested as the candidate M χ D bands [54].A similar discussion was also applied to
Ag [55]. Twopairs of doublet bands with the πg − / ⊗ νh / and πg − / ⊗ νh / ( d / /g / ) configurations in Ag have been ob-served in the experiments [56–58]. The RMF calculationsshowed the πg − / ⊗ νh / and πg − / ⊗ νh / ( d / /g / )bands in Ag have obvious triaxial deformation, γ =27.2 ◦ and γ = 28.1 ◦ , respectively. These are favorabledeformation parameters for chirality. Using these defor-mation parameters as input, the multiparticle plus rotormodel (MPRM) calculations well reproduced the avail-able data for the two pairs of doublet bands. The chiralgeometry of the aplanar rotation for two pairs of doubletbands was further confirmed by analyzing the angularmomentum components [55].The observed M χ D nucleus candidate, reaction usedto produce M χ D, number of pairs for the observed chiraldoublet bands in each nucleus, and single-particle con-figuration of M χ D are summarized in Tab. 1. As shown in Tab. 1, all M χ D nucleus candidates were discoveredin the fusion evaporation reactions using in-beam γ -rayspectroscopy. For the single-particle configurations ofM χ D, the high- j intruder orbitals (for instance, g / , h / and i / ) are involved in all observed M χ D nucleuscandidates. Some low- j orbitals also appeared in themultiparticle configurations, and usually acted as a spec-tator in the formation of chiral geometry. Furthermore,the observed M χ D nucleus candidates can be roughlydivided into two categories. One is M χ D bands withthe distinct configurations that differ from each other intheir triaxial deformations and configurations. For ex-ample, two distinct chiral doublet bands based on theconfigurations πh / ⊗ νh / and πg / h / ⊗ νh / in Ce [28]. The second is M χ D bands with the identicalconfiguration. A unique example is M χ D bands with the πg / ⊗ νh / g / configuration in Rh [12]. χ D Theory-wise, M χ D has been investigated with thetriaxial PRM [59–63], the combination of triaxial PRMand RMF approaches [3, 12, 28, 51, 53, 64], the tiltedaxis cranking model (TAC) with the collective Hamilto-nian [65, 66] and the projected shell model [67], etc. Inthis section I shall only focus on the theoretical predic-tions of M χ D. β and γ , corresponding valence nucleon and unpairednucleon configurations of minima, as well as excitation energies E x in the predicted M χ D nuclei.
Configuration ( β, γ ) E x Nuclei Valence nucleons Unpaired nucleons (MeV) Co [68] πf − / ⊗ ν ( g / f − / ) πf − / ⊗ νg / (0.26,18.2 ◦ ) 8.39 π ( g / f − / ) ⊗ νf − / πg / ⊗ νf − / (0.26,17.3 ◦ ) 8.1 Co [68] πf − / ⊗ νg / ( fp ) πf − / ⊗ νg / ( fp ) (0.20,24.0 ◦ ) 5.79 πf − / ⊗ ν ( g / ) πf − / ⊗ νg / (0.25,36.0 ◦ ) 11.82 Co [68] πf − / ⊗ νg / ( fp ) πf − / ⊗ νg / (0.28,27.0 ◦ ) 2.07 πf − / ⊗ νg / ( fp ) πf − / ⊗ νg / ( fp ) (0.30,15.1 ◦ ) 6.75 Br [69] π ( g / f − / p − / p − / ) ⊗ ν ( g / f − / p − / p − / ) πg / ⊗ νf − / (0.43,23.2 ◦ ) 0.44 π ( g / f − / p − / p − / ) ⊗ ν ( g / f − / p − / p − / ) πg / ⊗ νg − / (0.45,27.5 ◦ ) 0.47 Br [69] π ( g / f − / p − / p − / ) ⊗ ν ( g / p − / p − / ) πg / ⊗ νg − / (0.41,20.8 ◦ ) 0.08 π ( g / f − / p − / ) ⊗ ν ( g / p − / p − / ) πg / ⊗ νp − / (0.36,32.0 ◦ ) 0.42 π ( f − / p − / p − / ) ⊗ ν ( g / p − / p − / ) πf − / ⊗ νg / (0.28,40.3 ◦ ) 2.52 Br [69] π ( g / f − / p − / p − / ) ⊗ ν ( g / p − / ) πf / ⊗ νg − / (0.31,23.7 ◦ ) 0.96 π ( g / f − / p − / p − / ) ⊗ ν ( g / p − / ) πg / ⊗ νg − / ∗ (0.34,25.2 ◦ ) 1.54 Br [69] π ( g / f − / p − / p − / ) ⊗ ν ( g / p / p − / ) πg / ⊗ νg − / (0.41,17.5 ◦ ) 6.62 π ( g / p − / p − / ) ⊗ ν ( g / ) πg / ⊗ νg − / (0.15,33.6 ◦ ) 2.77 π ( g / f − / p − / p − / ) ⊗ ν ( g / ) πg / ⊗ νg − / ∗ (0.27,10.2 ◦ ) 3.78 Rb [70] π ( g / f − / p − / ) ⊗ ν ( g / f − / p − / ) πg / ⊗ νg − / (0.37,34.0 ◦ ) 0.26 π ( g / f − / p − / ) ⊗ ν ( g / f − / p − / ) πf / ⊗ νg − / (0.32,38.8 ◦ ) 0.51 π ( g / f − / p − / ) ⊗ ν ( g / f − / p − / ) πg / ⊗ νg − / (0.40,44.5 ◦ ) 1.46 Rb [70] π ( g / p − / p − / ) ⊗ ν ( g / p − / ) πg / ⊗ νg − / (0.22,45.1 ◦ ) 0.31 π ( g / f − / p − / ) ⊗ ν ( g / p − / p − / ) πg / ⊗ νp − / (0.33,37.3 ◦ ) 1.19 π ( g / p − / p − / ) ⊗ ν ( g / p − / p − / ) πg / ⊗ νg − / (0.35,39.1 ◦ ) 1.98 Rb [70] π ( g / p − / p − / ) ⊗ ν ( g / p − / ) πp / ⊗ νg − / (0.26,42.7 ◦ ) 0.07 π ( g / p − / p − / ) ⊗ ν ( g / p − / ) πg / ⊗ νg − / (0.22,36.9 ◦ ) 0.24 π ( g / f − / p − / ) ⊗ ν ( g / p − / ) πg / ⊗ νg − / (0.29,45.5 ◦ ) 0.56 Rh [47, 71–73] πg − / ⊗ νh / ( d / ord / ) πg − / ⊗ νh / ∗ (0.25,23.3 ◦ ) 0.636 πg − / ⊗ νh / πg − / ⊗ νh / (0.30,22.9 ◦ ) 1.219 Rh [71] πg − / ⊗ νh / πg − / ⊗ νh / (0.26,40.6 ◦ ) 0 πg − / ⊗ νh / πg − / ⊗ νh / (0.31,18.7 ◦ ) 0.51 Ag [74] π ( g − / p − / ) ⊗ ν ( h / d / g / ) πg − / ⊗ νh / g − / ∗ (0.23,34.9 ◦ ) 2.46( g − / p − / ) ⊗ ν ( h / d / g / ) πg − / ⊗ νh / d / (0.23,26.5 ◦ ) 3.10( g − / p − / ) ⊗ ν ( h / g / ) πg − / ⊗ νh / (0.25,38.5 ◦ ) 3.94 Cs [75] π ( g / h / ) ⊗ ν ( sd ) h / πh / ⊗ νh − / ( sd ) (0.25,26.3 ◦ ) 2.75 π ( g / h / ) ⊗ ν ( sd ) h / πh / ⊗ νh − / (0.26,24.3 ◦ ) 4.78 Cs [75] π ( g / h / ) ⊗ ν ( sd ) h / πh / ⊗ νh − / ( sd ) (0.21,13.4 ◦ ) 2.78 π ( g / h / ) ⊗ ν ( sd ) h / πh / ⊗ νh − / (0.21,22.7 ◦ ) 2.31 Cs [75] π ( g / h / ) ⊗ ν ( sd ) h / πh / ⊗ νh − / ( sd ) (0.18,22.0 ◦ ) 2.11 π ( g / h / ) ⊗ ν ( sd ) h / πh / ⊗ νh − / (0.17,24.8 ◦ ) 3.00 ∗ : the configurations have been experimentally observed. The adiabatic and configuration-fixed constrainedtriaxial RMF approaches were developed for the firsttime to investigate the triaxial shape coexistence andpossible chiral doublet bands in 2006 [47]. The existenceof multiple chiral doublets (M χ D) was suggested in
Rhfrom the examination of the deformation and the cor-responding configurations. Similar investigations have also been performed for several isotope chains. Thesecalculations predicted that the M χ D phenomenon mightexist in , , Co [68], , , , Br [69], , , Rb [70], , Rh [71–73],
Ag [74] and , , Cs [75] basedon the triaxial deformations of the local minima andthe corresponding high- j particle(s) and hole(s) config-urations. The predicted multi-chiral nuclei are listed in ab. 2, together with the calculated triaxial deformationparameters β and γ , corresponding valence nucleon andunpaired nucleon configurations of minima, as well asexcitation energies. Thereinto, the configurations of ex-perimentally observed chiral doublet bands are markedwith an asterisk. From Tab. 2, the excitation energiesof most chiral configurations are less than 3 MeV. It iseasy to be populated in experiment. The further experi-mental explorations are highly expected to search for the M χ D in these nuclei.It is worthwhile to mention that a three-dimensionaltilted axis cranking (3DTAC) method based on covariantdensity functional theory has been recently establishedand used to investigate the M χ D for the first time ina fully self-consistent and microscopic way [73]. Thismodel reproduced well the available experimental spec-tra and B ( M /B ( E
2) ratios in
Rh, which exhibiteda high predictive power. + + + + + + + + + + + + + + + + + + + + + Chiral Doublet Bands A Chiral Doublet Bands B + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + Fig. 1. Calculated level scheme for M χ D based on the configuration πh / ⊗ νh − / coupled with γ = 90 ◦ rotor fromRef. [63]. Red and black arrows represent M I → I −
1) and E I → I −
2) transitions, respectively. χ D Besides the experimental explorations and theoreticalpredictions of M χ D, it is also interesting to study the fin-gerprints of M χ D. For a single chiral doublet bands, theknown fingerprints for the ideal chirality are as follows:(i) the nearly degeneracy of doublet bands, (ii) the spinindependence of S ( I ), (iii) the similar spin alignments,(iv) the similar B ( M
1) and B ( E
2) values, (v) the stag-gering of B ( M E χ D withthe distinct configurations, the chiral fingerprints are stilleffective for every pair of chiral doublet bands.As mentioned above, in comparison with the M χ Dthat differ from each other in their triaxial deforma-tions and configurations, M χ D may also exist in a sin-gle nucleus with the identical configuration. The chiralbands, including the yrast and yrare bands as well as the higher excited bands, have been studied by the tri-axial PRM [59–62], which has been extensively used instudies of chiral doublet bands and yielded lots of suc-cesses [64, 77, 80–89]. The PRM calculations [59–62]showed that the properties of the two higher excitedbands, including the excitation energies and selectionrule for electromagnetic transitions, were very similar tothose of the yrast and yrare bands, which indicated thatexcited doublet bands could be a pair of chiral partnersas well. However, we noted the existence of a number oflinking transitions between the lowest-lying chiral bandsand higher excited bands. It is necessary to study theproperties of these linking transitions.In Ref. [63], the selection rule of electromagnetictransitions for these linking transitions between the low-est and excited chiral doublet bands in M χ D was alsostudied based on the triaxial PRM. The calculated levelscheme for two pairs of chiral doublet bands based on theconfiguration πh / ⊗ νh − / coupled with γ =90 ◦ rotorwas shown in Fig. 1. These bands are labeled as 1, 2, 3 nd 4. Bands 1 & 2 and bands 3 & 4 form the chiral dou-blet bands A and B, respectively. In the calculations, thedeformation parameters β = 0 . γ = 90 ◦ , moment ofinertia J = 30 M eV − ~ , intrinsic quadrupole moment Q = (3 / √ π ) R Zβ =3.5 and g p ( g n ) − g R = 0 . − . M M M B ( M
1) and B ( E
2) values in the excitedchiral doublet bands have the same order of magnitudeas those in lowest chiral doublet bands. However, the B ( M
1) and B ( E
2) values of transitions which link theexcited to the lowest chiral doublet bands are two or-ders of magnitude smaller than those in the lowest (orexcited) chiral doublet bands. The selection rules andthe quantitative relations of electromagnetic transitionsprobabilities would be helpful for confirming the exis-tence of M χ D bands with the identical configuration inthe real nuclei.In Fig. 1, the interband E E E V can be expressed as [79] V = √ R ∆ E I ∆ E I − p ( R + 1) (∆ E I + ∆ E I − ) + 2∆ E I ∆ E I − ( R − , (1)where R = B [ E ,I yrare → ( I − yrast ] B [ E ,I yrare → ( I − yrare ] . According to theEq. (1), we extracted the interaction strength betweenchiral doublet bands A and B. The values of B ( E
2) and∆ E come from the calculated results of PRM [63]. Thecalculations show that the average interaction strength V between chiral doublet bands A and B is approximatelyequal to 200 keV in the chiral range, which implies that the chiral geometry is mixed by the vibrational compo-nent [79, 90]. It should be noted that the Eq. (1) arededuced from the two-band mixing picture. Thus, thepresent calculated V is an approximate solution to M χ Dbands.In order to show the picture more clearly, the mixingratio and the percentage of E I =1linking transitions are calculated in the present workby the PRM with the ideal case i.e. the configuration πh / ⊗ νh − / with γ =90 ◦ . The calculations show ∼ E I =1 transitions in thechiral doublet bands A and B, respectively, suggestingthese transitions as being essentially of a pure M E ∼ I =1 linking transitions betweenthe chiral doublet bands A and B. The enhance E p h R i of the core rotation along the quantizationaxis, and found that components of the core rotationalong the long and short axis for the excited doubletbands are larger than those for the lowest chiral bands.It implied that the excited pair exhibited a chiral geome-try which is realized with a wobbling motion of the core.Hence, the excited pair was claimed as the chiral wob-blers [92]. Further detailed studies beyond the scope ofthis paper are needed to study whether the excited dou-blet bands are associated with the chiral wobblers. Thepossible coexistence of chirality and the other rotationalmodes will be discussed in the following section. χ D with octupole correlations andsome related topics
If another spontaneous breaking of discrete symme-try takes place in addition to chirality, the degree of en-ergy degeneracy will accordingly be increased resultingin multiple degenerate ∆ I =1 rotational bands. For in-stance, in the case of the chiral and reflection symme-try breakings, parity and chiral doubling bring about aset of four degenerate ∆ I =1 bands. The first such ex-ample, though rather soft breaking of reflection symme-try, has been found in Br [3]. In Ref. [3], two pairsof positive- and negative-parity doublet bands togetherwith eight strong electric dipole transitions linking theiryrast positive- and negative-parity bands in Br havebeen found. It provided the evidence for M χ D bandswith octupole correlations, reported the first example ofchiral geometry in octupole soft nuclei, and indicatedthat nuclear chirality can be robust against the octupolecorrelations. This observation also pointed to the ex- iting possibility of observing the chirality-parity quar-tet (CPQ) bands i.e., four ∆ I =1 alternating-parity ro-tational bands with the same configuration in a singlenucleus with both stable triaxial and octupole deforma-tions. So far, CPQ bands have not been experimentallyobserved.Note that, a pair of positive-parity doublet bandsand several E1 transitions linking yrast positive- andnegative-parity bands in Cs have been reported byRef. [93] and Refs. [94, 95], respectively. Based on theTPRM calculations, Ref. [86] suggested that the positive-parity doublet bands in
Cs might correspond to a typ-ical chiral vibration pattern. Recently, lifetime measure-ments have been carried out using the Doppler shift at-tenuation method (DSAM) for the yrast positive- andnegative-parity bands in
Cs [96]. The measured re-sults [96] show that the B ( E
1) rates are of the order of10 − W.u., thereby indicating coexistence of a pair ofchiral doublet bands and octupole correlations in
Cs.In order to search for the possibly candidate coresto construct CPQ bands, the potential energy surfaces(PES) of the even-even Se, Ba, and Ra isotopes were cal-culated by using the macroscopic-microscopic method in a multidimensional space { α λ,µ } including quadrupole( λ =2, µ =0, 2) and octupole ( λ =3, µ =0, 1, 2, 3) degreesof freedom [97]. The calculated results showed that theeven-even isotopes Se, , , − Ba and − Racan exhibit the coexistence of triaxial and octupole de-formations. It is therefore expected that CPQ bands canbe observed experimentally in these even-even nuclei andtheir neighboring odd-A/odd-odd nuclei. As an example,the calculated PES of
Ra in the β − γ and β − β planes using the macroscopic-microscopic method [98–100] are shown in Fig. 2(a) and 2(b), respectively. Onecan see from Fig. 2 that Ra has the obviously triaxialand octupole deformations.It is necessary to note here that in Tab. 1, the config-urations of some M χ D involve the orbits of pseudospindoublet states (e.g. 1g / , 2d / ). Thus, a compet-ing interpretation of these doublet bands would includethe pseudospin doublet bands. Pseudospin symmetry inatomic nuclei was introduced in 1969 [101, 102]. A pair ofnearly degenerate doublet bands with the configurationinvolving pseudospin doublet states have been observedand suggested as the pseudospin doublet bands in severalnuclei [103–111]. (b) Ra (a) -0.3 -0.2 -0.1 0.0 0.1 0.2 0.30.00.10.2 Fig. 2. The calculated PES of
Ra in the β − γ (a) and β − β (b) planes. The energies are normalized withrespect to the ground state. The contour separation is 0.2 MeV. A specific calculation [74] for the nearly degeneratetriplet bands with the πg − / ⊗ νh / ( d / /g / ) configu-ration in Ag was performed by using the RMF theoryand the MPRM. The configuration-fixed constrained tri-axial RMF calculations exhibited the pseudospin sym-metry in single particle spectra and triaxial shape coex-istence. The experimental excitation energies and elec-tromagnetic transition probabilities for the triplet bands were well reproduced by the MPRM calculations. Thus,the first & second lowest energy bands and the second& third bands were interpreted as the pseudospin dou-blet bands and chiral doublet bands, respectively. Thiswork also motivated the investigation to search for thechirality-pseudospin triplet (or quartet) bands in the nu-clei.
Summary and perspectives
The recent progress in M χ D is reviewed for bothexperimental and theoretical sides. In particular, theexperimental findings, theoretical predictions, selectionrule of electromagnetic transitions, M χ D with octupolecorrelations and some related topics are highlighted.Based on the above discussion, it is of highly scientificinterest to search for the more M χ D as well as the possi-ble chiral wobblers, chirality-parity quartet and chirality-pseudospin triplet (or quartet) bands in nuclear system.On the other hand, these exotic nuclear phenomena havebrought severe challenges to current nuclear models and, thus, require the development of new approaches. Veryrecently, to study the M χ D with octupole correlations in Br, a reflection-asymmetric triaxial PRM with a quasi-proton and a quasi-neutron coupled with a reflection-asymmetric triaxial rotor has been developed [112]. Ac-cording to the present review, we would also like to at-tract more experimental and theoretical efforts on the in-vestigation of chirality or multiple chirality in the atomicnucleus.
The author is grateful to B. Qi, C. Liu, H. Jia, andN. B. Zhang for helpful discussions and careful readingsof the manuscript.
References , 131 (1997).2 S. Y. Wang, B. Qi, L. Liu, S. Q. Zhang, H. Hua, X. Q. Li, Y.Y. Chen, L. H. Zhu, J. Meng, S. M. Wyngaardt, P. Papka, T.T. Ibrahim, R. A. Bark, P. Datta, E. A. Lawrie, J.J. Lawrie,S. N. T. Majola, P. L. Masiteng, S. M. Mullins, J. G´al, G.Kalinka, J. Moln´ar, B. M. Nyak´o, J. Tim´ar, K. Juh´asz, and R.Schwengner, Phys. Lett. B , 40 (2011).3 C. Liu, S. Y.Wang, R. A. Bark, S. Q. Zhang, J. Meng, B. Qi,P. Jones, S. M. Wyngaardt, J. Zhao, C. Xu, S.-G. Zhou, S.Wang, D. P. Sun, L. Liu, Z. Q. Li, N. B. Zhang, H. Jia, X.Q. Li, H. Hua, Q. B. Chen, Z. G. Xiao, H. J. Li, L. H. Zhu,T. D. Bucher, T. Dinoko, J. Easton, K. Juh´asz, A. Kamblawe,E. Khaleel, N. Khumalo, E. A. Lawrie, J. J. Lawrie, S. N. T.Majola, S. M. Mullins, S. Murray, J. Ndayishimye, D. Negi, S.P. Noncolela, S. S. Ntshangase, B. M. Nyak´o, J. N. Orce, P.Papka, J. F. Sharpey-Schafer, O. Shirinda, P. Sithole, M. A.Stankiewicz, and M. Wiedeking, Phys. Rev. Lett. , 112501(2016).4 C. Liu et al. , Phys. Rev. C , 054309 (2019).5 C. Vaman, D. B. Fossan, T. Koike, K. Starosta, I. Y. Lee, andA. O. Macchiavelli, Phys. Rev. Lett. , 032501 (2004).6 P. Joshi, D. G. Jenkins, P. M. Raddon, A. J. Simons, R.Wadsworth, A. R. Wilkinson, D. B. Fossan, T. Koike, K.Starosta, C. Vaman, J. Tim´ar, Zs. Dombr´adi, A. Kraszna-horkay, J. Moln´ar, D. Sohler, L. Zolnai, A. Algora, E. S. Paul,G. Rainovski, A. Gizon, J. Gizon, P. Bednarczyk, D. Curien,G. Duchˆene, and J. N. Scheurer, Phys. Lett. B , 135 (2004).7 J. Tim´ar, P. Joshi, K. Starosta, V. I. Dimitrov, D. B. Fossan, J.Moln´ar, D. Sohler, R. Wadsworth, A. Algora, P. Bednarczyk,D. Curien, Zs. Dombr´adi, G. Duchene, A. Gizon, J. Gizon,D.G. Jenkins, T. Koike, A. Krasznahorkay, E. S. Paul, P. M.Raddon, G. Rainovski, J. N. Scheurer, A. J. Simons, C. Vaman,A. R. Wilkinson, L. Zolnai, and S. Frauendorf, Phys. Lett. B , 178 (2004).8 J. A. Alc´antara-N´u˜nez, J. R. B. Oliveira, E. W. Cybulska, N.H. Medina, M. N. Rao, R. V. Ribas, M. A. Rizzutto, W. A.Seale, F. Falla-Sotelo, K. T. Wiedemann, V. I. Dimitrov, andS. Frauendorf, Phys. Rev. C , 024317 (2004).9 J. Tim´ar, C. Vaman, K. Starosta, D. B. Fossan, T. Koike, D.Sohler, I. Y. Lee, and A. O. Macchiavelli, Phys. Rev. C ,011301(R) (2006).10 P. Joshi, M. P. Carpenter, D. B. Fossan, T. Koike, E. S. Paul,G. Rainovski, K. Starosta, C. Vaman, and R. Wadsworth,Phys. Rev. Lett. , 102501 (2007).11 T. Suzuki, G. Rainovski, T. Koike, T. Ahn, M. P. Carpen-ter, A. Costin, M. Danchev, A. Dewald, R. V. F. Janssens, P. Joshi, C. J. Lister, O. M¨oller, N. Pietralla, T. Shinozuka, J.Tim´ar, R. Wadsworth, C. Vaman, and S. Zhu, Phys. Rev. C , 031302(R) (2008).12 I. Kuti, Q. B. Chen, J. Tim´ar, D. Sohler, S. Q. Zhang, Z. H.Zhang, P. W. Zhao, J. Meng, K. Starosta, T. Koike, E. S. Paul,D. B. Fossan, and C. Vaman, Phys. Rev. Lett. , 032501(2014).13 K. Starosta, T. Koike, C. J. Chiara, D. B. Fossan, D. R.LaFosse, A. A. Hecht, C. W. Beausang, M. A. Caprio, J. R.Cooper, R. Kr ü cken, J. R. Novak, N. V. Zamfir, K. E. Zyrom-ski, D. J. Hartley, D. L. Balabanski, Jing-ye Zhang, S. Frauen-dorf, and V. I. Dimitrov, Phys. Rev. Lett. , 971 (2001).14 T. Koike, K. Starosta, C. J. Chiara, D. B. Fossan, and D. R.LaFosse, Phys. Rev. C , 061304(R) (2001).15 R. A. Bark, A. M. Baxter, A. P. Byrne, G. D. Dracoulis, T.Kib´edi, T. R. McGoram, S. M. Mullins, Nucl. Phys. A691 , 577(2001).16 A. A. Hecht, C. W. Beausang, K. E. Zyromski, D. L. Balaban-ski, C. J. Barton, M. A. Caprio, R. F. Casten, J. R. Cooper, D.J. Hartley, R. Kr¨ucken, D. Meyer, H. Newman, J. R. Novak,E. S. Paul, N. Pietralla, A. Wolf, N. V. Zamfir, Jing-ye Zhang,and F. D¨onau, Phys. Rev. C , 051302(R) (2001).17 D. J. Hartley, L. L. Riedinger, M. A. Riley, D. L. Balaban-ski, F. G. Kondev, R. W. Laird, J. Pfohl, D. E. Archer, T. B.Brown, R. M. Clark, M. Devlin, P. Fallon, I. M. Hibbert, D. T.Joss, D. R. LaFosse, P. J. Nolan, N. J. O ’ Brien, E. S. Paul,D. G. Sarantites, R. K. Sheline, S. L. Shepherd, J. Simpson,R. Wadsworth, Jing-ye Zhang, P. B. Semmes, and F. D¨onau,Phys. Rev. C , 031304(R) (2001).18 E. Mergel, C. M. Petrache, G. Lo Bianco, H. H¨ubel, J. Dom-scheit, D. Roßbach, G. Sch¨onwaßer, N. Nenoff, A. Neußer, A.G¨orgen, F. Becker, E. Bouchez, M. Houry, A. H¨urstel, Y. LeCoz, R. Lucas, Ch. Theisen, W. Korten, A. Bracco, N. Blasi,F. Camera, S. Leoni, F. Hannachi, A. Lopez-Martens, M. Re-jmund, D. Gassmann, P. Reiter, P.G. Thirolf, A. Astier, N.Buforn, M. Meyer, N. Redon, and O. Stezowski, Eur. Phys. J.A , 417 (2002).19 T. Koike, K. Starosta, C. J. Chiara, D. B. Fossan, and D. R.LaFosse, Phys. Rev. C , 044319 (2003).20 S. Zhu, U. Garg, B. K. Nayak, S. S. Ghugre, N. S. Pattabira-man, D. B. Fossan, T. Koike, K. Starosta, C.Vaman, R. V. F.Janssens, R. S. Chakrawarthy, M. Whitehead, A. O. Macchi-avelli, and S. Frauendorf, Phys. Rev. Lett. , 132501 (2003).21 S. Y. Wang, Y. Z. Liu, T. Komatsubara, Y. J. Ma, and Y. H.Zhang, Phys. Rev. C , 017302 (2006).22 E. Grodner, J. Srebrny, A. A. Pasternak, I. Zalewska, T. Morek,Ch. Droste, J. Mierzejewski, M. Kowalczyk, J. Kownacki, M.Kisieli´nski, S. G. Rohozi´nski, T. Koike, K. Starosta, A. Ko- dyasz, P. J. Napiorkowski, M. Woli´nska-Cichocka, E. Ru-chowska, W. P l´ociennik, and J. Perkowski, Phys. Rev. Lett. , 172501 (2006).23 D. Tonev, G. de Angelis, P. Petkov, A. Dewald, S. Brant, S.Frauendorf, D. L. Balabanski, P. Pejovic, D. Bazzacco, P. Bed-narczyk, F. Camera, A. Fitzler, A. Gadea, S. Lenzi, S. Lunardi,N. Marginean, O. M¨oller, D. R. Napoli, A. Paleni, C. M. Pe-trache, G. Prete, K. O. Zell, Y. H. Zhang, Jing-ye Zhang, Q.Zhong, and D. Curien, Phys. Rev. Lett. , 052501 (2006).24 S. Mukhopadhyay, D. Almehed, U. Garg, S. Frauendorf, T. Li,P. V. Madhusudhana Rao, X. Wang, S. S. Ghugre, M. P. Car-penter, S. Gros, A. Hecht, R. V. F. Janssens, F. G. Kondev,T. Lauritsen, D. Seweryniak, and S. Zhu, Phys. Rev. Lett. ,172501 (2007).25 Y. X. Zhao, T. Komatsubara, Y. J. Ma, Y. H. Zhang, S. Y.Wang, Y. Z. Liu, and K. Furumo, Chin. Phys. Lett. , 082301(2009).26 E. Grodner, I. Sankowska, T. Morek, S.G. Rohozi´nski, Ch.Droste, J. Srebrny, A. A. Pasternak, M. Kisieli´nski, M. Kowal-czyk, J. Kownacki, J. Mierzejewski, A. Kr´ol, and K. Wrzosek,Phys. Lett. B , 46 (2011).27 K. Y. Ma, J. B. Lu, D. Yang, W. H. Dong, Y. Z. Liu, X. G.Wu, Y. Zheng, and C. Y. He, Phys. Rev. C , 037301 (2012).28 A. D. Ayangeakaa, U. Garg, M. D. Anthony, S. Frauendorf, J.T. Matta, B. K. Nayak, D. Patel, Q. B. Chen, S. Q. Zhang, P.W. Zhao, B. Qi, J. Meng, R. V. F. Janssens, M. P. Carpenter,C. J. Chiara, F. G. Kondev, T. Lauritsen, D. Seweryniak, S.Zhu, S. S. Ghugre, and R. Palit, Phys. Rev. Lett. 110, 172504(2013).29 K. Y. Ma, J. B. Lu, Z. Zhang, J. Q. Liu, D. Yang, Y. M. Liu,X. Xu, X. Y. Li, Y. Z. Liu, X. G. Wu, Y. Zheng, and C. B. Li,Phys. Rev. C , 014305 (2018).30 D. L. Balabanski, M. Danchev, D. J. Hartley, L. L. Riedinger,O. Zeidan, Jing-ye Zhang, C. J. Barton, C. W. Beausang, M. A.Caprio, R. F. Casten, J. R. Cooper, A. A. Hecht, R. Kr ü cken,J. R. Novak, N. V. Zamfir, and K. E. Zyromski, Phys. Rev. C , 044305 (2004).31 E. A. Lawrie, P. A. Vymers, J. J. Lawrie, Ch. Vieu, R. A. Bark,R. Lindsay, G. K. Mabala, S. M. Maliage, P. L. Masiteng, S. M.Mullins, S. H. T. Murray, I. Ragnarsson, T. M. Ramashidzha,C. Schuck, J. F. Sharpey-Schafer, and O. Shirinda, Phys. Rev.C , 021305(R) (2008).32 E. A. Lawrie, P. A. Vymers, Ch. Vieu, J. J. Lawrie, C. Sch¨uck,R. A. Bark, R. Lindsay, G. K. Mabala, S. M. Maliage, P. L.Masiteng, S. M. Mullins, S. H. T. Murray, I. Ragnarsson, T.M. Ramashidzha, J. F. Sharpey-Schafer, and O. Shirinda, Eur.Phys. J. A , 39 (2010).33 P. L. Masiteng, E.A. Lawrie, T. M. Ramashidzha, R. A. Bark,B. G. Carlsson, J. J. Lawrie, R. Lindsay, F. Komati, J. Kau,P. Maine, S. M. Maliage, I. Matamba, S. M. Mullins, S. H. T.Murray, K. P. Mutshena, A. A. Pasternak, I. Ragnarsson, D.G. Roux, J. F. Sharpey-Schafer, O. Shirinda, P. A. Vymers,Phys. Lett. B , 83 (2013).34 P. L. Masiteng, E. A. Lawrie, T. M. Ramashidzha, J. J. Lawrie,R. A. Bark, R. Lindsay, F. Komati, J. Kau, P. Maine, S. M.Maliage, I. Matamba, S. M. Mullins, S. H. T. Murray, K. P.Mutshena, A. A. Pasternak, D. G. Roux, J. F. Sharpey-Schafer,O. Shirinda, and P.A. Vymers, Eur. Phys. J. A , 119 (2014).35 P. L. Masiteng, A. A. Pasternak, E. A. Lawrie, O. Shirinda, J.J. Lawrie, R. A. Bark, S. P. Bvumbi, N. Y. Kheswa, R. Lind-say, E. O. Lieder, R. M. Lieder, T. E. Madiba, S. M. Mullins,S. H. T. Murray, J. Ndayishimye, S. S. Ntshangase, P. Papka,and J. F. Sharpey-Schafer, Eur. Phys. J. A , 28 (2016).36 J. Ndayishimye, E. A. Lawrie, O. Shirinda, J. L. Easton, S.M. Wyngaardt, R. A. Bark, S. P. Bvumbi, T. R. S. Dinoko,P. Jones, N. Y. Kheswa, J. J. Lawrie, S. N. T. Majola, P. L.Masiteng, D. Negi, J. N. Orce, P. Papka, J. F. Sharpey-Schafer,M. Stankiewicz, M. Wiedeking, Acta Physica Polonica B , 343 (2017).37 S. Frauendorf, Rev. Mod. Phys.
463 (2001).38 J. Meng, B. Qi, S. Q. Zhang, and S. Y. Wang, Modern PhysicsLetters A , 2560 (2008).39 J. Meng and S. Q. Zhang, J. Phys. G: Nucl. Part. Phys. ,064025 (2010).40 R. A. Bark et al. , Int. J. Mod. Phys. E , 1461001 (2014).41 J. Meng and P. W. Zhao, Phys. Scr. , 053008 (2016).42 A. A. Raduta, Prog. Part. Nucl. Phys. , 241 (2016).43 K. Starosta, T. Koike, Phys. Scr. , 093002 (2017).44 S. Frauendorf, Phys. Scr. , 043003 (2018).45 Q. B. Chen and J. Meng, Nuclear Physics News. , 11-15(2020).46 B. W. Xiong and Y. Y. Wang, At. Data Nucl. Data Tables 125,193 (2019).47 J. Meng, J. Peng, S. Q. Zhang, and S. G. Zhou, Phys. Rev. C73, 037303 (2006).48 D. Tonev et al. , Phys. Rev. Lett. , 052501 (2014).49 E. O. Lieder et al. , Phys. Rev. Lett. , 202502 (2014).50 N. Rather et al. , Phys. Rev. Lett. , 202503 (2014).51 C. M. Petrache et al. , Phys. Rev. C , 041304(R) (2018).52 T. Roy, G. Mukherjeea, M. Asgara, S. Bhattacharyyaa,S. Bhattacharyaa, C. Bhat-tacharyaa, S. Bhattacharyaa, T.Ghosha, K. Banerjeea, S. Kundua, T. Rana, P. Roya, R.Pandeya, J. Meena, A. Dhal, R. Palitd, S. Saha, J. Sethi, S.Thakur, B. Naidu, S. Jadav, R. Dhonti, H. Pai, A. Goswami,Phys. Lett. B , 768 (2018).53 B. F. Lv et al. , Phys. Rev. C , 024314 (2019).54 J. Li, S. Q. Zhang, and J. Meng, Phys. Rev. C , 037301(2011).55 B. Qi, H. Jia, N. B. Zhang, C. Liu, and S. Y. Wang, Phys. Rev.C , 027302 (2013).56 D. Jerrestam et al. , Nucl. Phys. A , 786 (1994).57 F. R. Espinoza-Qui˜ n ones et al. , Phys. Rev. C , 1548 (1997).58 B. Zhang et al. , Chin. Phys. C, , 1009 (2011).59 Ch. Droste, S. G. Rohozinski, K. Starosta, L. Pr´ o chniak, andE. Grodner, Eur. Phys. J. A , 79 (2009).60 Q. B. Chen, J. M. Yao, S. Q. Zhang, and B. Qi, Phys. Rev. C , 067302 (2010).61 I. Hamamoto, Phys. Rev. C , 024327 (2013).62 H. Zhang, Q. B. Chen, Chin. Phys. C 40, 024102 (2016).63 H. Jia, B. Qi, S. Y. Wang, S. Wang, and C. Liu, Chin. Phys.C 40, 124103 (2016).64 Q. B. Chen, B. F. Lv, C. M. Petrache, and J. Meng, Phys.Lett. B 782, 744 (2018).65 Q. B. Chen, S. Q. Zhang, P. W. Zhao, R. V. Jolos, and J.Meng, Phys. Rev. C , 044301 (2016).66 X. H. Wu, Q. B. Chen, P. W. Zhao, S. Q. Zhang, and J. Meng,Phys. Rev. C , 064302 (2018).67 Y. K. Wang, F. Q. Chen, P. W. Zhao, S. Q. Zhang, and J.Meng, Phys. Rev. C , 054303 (2019).68 J. Peng, and Q. B. Chen, Phys. Rev. C , 024320 (2018).69 B. Qi, H. Jia, C. Liu, and S. Y. Wang, Sci. Chin. Phys., Mech.& Astr. 62, 012012 (2019).70 B. Qi, H. Jia, C. Liu, and S. Y. Wang, Phys. Rev. C , 014305(2018).71 J. Peng, H. Sagawa, S. Q. Zhang, J. M. Yao, Y. Zhang, and J.Meng, Phys. Rev. C , 024309 (2008).72 J. M. Yao, B. Qi, S. Q. Zhang, J. Peng, S. Y. Wang, and J.Meng, Phys. Rev. C , 067302 (2009).73 P. W. Zhao, Phys. Lett. B , 1 (2017).74 H. Jia, B. Qi, C. Liu, and S. Y. Wang, J. Phys. G: Nucl. Part.Phys. 46, 035102 (2019).75 J. Li, Phys. Rev. C , 034306 (2018).76 T. Koike, K. Starosta, C. Vaman, T. Ahn, D. B. Fossan, R. M.Clark, M. Cromaz, I. Y. Lee, and A. O. Macchiavelli, in Fron-tiers of Nuclear Structure, edited by P. Fallon andR.Clark, AIPConf. Proc. No. 656 (AIP, Melville, New York, 2003), p. 160. , 664 (2007).79 L. Liu, S. Y. Wang, B. Qi, and C. Liu, Int. J. Mod. Phys. E , 1350060 (2013).80 J. Peng, J. Meng, and S. Q. Zhang, Phys. Rev. C 68, 044324(2003).81 S. Q. Zhang, B. Qi, S. Y. Wang, and J. Meng, Phys. Rev. C75, 044307 (2007).82 S. Y. Wang, S. Q. Zhang, B. Qi, and J. Meng, Phys. Rev. C75, 024309 (2007).83 S. Y. Wang, S. Q. Zhang, B. Qi, J. Peng, J. M. Yao, and J.Meng, Phys. Rev. C 77, 034314 (2008).84 B. Qi, S. Q. Zhang, J. Meng, and S. Frauendorf, Phys. Lett. B675, 175 (2009).85 B. Qi, S. Q. Zhang, S. Y. Wang, J. M. Yao, and J. Meng, Phys.Rev. C 79, 041302(R) (2009).86 S. Y. Wang, B. Qi, and D. P. Sun, Phys. Rev. C 82, 027303(2010).87 E. A. Lawrie and O. Shirinda, Phys. Lett. B , 66 (2010).88 B. Qi, S. Q. Zhang, S. Y. Wang, J. Meng, and T. Koike, Phys.Rev. C , 034303 (2011).89 O. Shirinda and E. A. Lawrie, Eur. Phys. J. A 48, 118 (2012).90 B. Qi, S. Q. Zhang, S. Y. Wang, and J. Meng, Chin. Phys.Lett. , 112101 (2010).91 S. W. Ødeg˚ard et al. , Phys. Rev. Lett. 86, 5866 (2001).92 T. Koike, K. Starosta, I. Hamamoto, D. B. Fossan and C. Va- man, AIP Conf. Proc. , 87 (2005).93 A. Gizon et al. , Nucl. Phys. A , 63 (2001).94 D. Yang et al. , Chin. Phys. Lett. 26, 082101 (2009).95 J. Lu et al. , Phys. Rev. C , 057304 (2000).96 K. Selvakumar et al. , Phys. Rev. C , 064307 (2015).97 C. Liu, S. Y. Wang, B. Qi, and H. Jia, Chin. Phys. C ,074105 (2018).98 W. Nazarewicz, P. Olanders, I. Ragnarsson, J. Dudek, G. A.Leander, P. M¨oller, and E. Ruchowsa, Nucl. Phys. A429 , 269(1984).99 J. Dudek, K. Mazurek, and B. Nerlo-pomorska, Int. J. Mod.Phys. E , 117 (2004).100 K. Mazurek, and J. Dudek, AIP Conf. Proc. , 153 (2005).101 A. Arima, M. MHarvey, K. Shimizu, Phys. Lett. B , 517(1969).102 K. T. Hecht, A. Adler, Nucl. Phys. A , 129 (1969).103 Q. Xu et al. , Phys. Rev. C , 064301 (2008).104 C. Liu et al. , Phys. Rev. C , 037301 (2013).105 C. Liu et al. , Int. J. Mod. Phys. E , 2351 (2011).106 S. Y. Wang et al. , Phys. Rev. C , 057303 (2010).107 S. Y. Wang et al. , Phys. Rev. C , 037302 (2007).108 W. Hua et al. , Phys. Rev. C , 034303 (2009).109 S. Y. Wang et al. , Phys. Rev. C , 064302 (2012).110 J. Sun et al. , Phys. Rev. C , 064301 (2016).111 S. Y. Wang, Sci Sin-Phys Mech Astron, 46, 012011 (2016) (inChinese).112 Y. Y. Wang, S. Q. Zhang, P. W. Zhao and J. Meng, Phys.Lett. B , 454 (2019)., 454 (2019).