Decay patterns of multi-quasiparticle bands—a model independent test of chiral symmetry

Chiral symmetry plays an important role in nature. The left- or right-handedness of a molecule determines its chemical interactions and has a crucial impact on its activity in a living cell. The nuclei, however, due to their uniform density, were believed unable to form chiral structures. Thus, the idea of Frauendorf and Meng [1] that nuclear chiral structures may form in angular momentum space drew a lot of attention and inspired a search for such systems. A number of investigations were carried out over the last two decades and many striking features of nuclear chirality have been discovered. These include the existence of more than one chiral formation in a nucleus [2, 3], the presence of a chiral structure which shows an additional octopule degree of freedom [4], etc. However through the two decades of nuclear chirality research and despite the joint efforts of theory and experiment, a very important open question remains: is the identification of chiral symmetry structures reliable and unambiguous?

In the macro world, e.g. in chemistry, the observed chiral structures are identical except for their different handedness. An ideal nuclear chiral system should also produce two identical structures—two rotational bands that become degenerate in the spin region where chiral symmetry forms [1]. It was thus expected that a nuclear chiral system can be easily recognised by the observation of a pair of degenerate rotational bands. These are two bands with the same excitation energies for the levels with the same spin and parity, and with the same transition probabilities for the corresponding in-band and out-of-band transitions [1]. However, as more experiments on chiral symmetry were performed, it became clear that chiral partner bands may not be identical. The real nuclear chiral systems tend to involve non-chiral components, which generate divergence in the properties of the chiral partners, for instance in the excitation energies, in the alignments and in the transition probabilities [5]. It makes the unambiguous identification of chiral symmetry structures difficult, because rotational bands that show some similarity but not degeneracy, can be produced, not only by an intrinsic symmetry, but also in several alternative ways, e.g. by involving different single-particle configurations, or by coupling to vibrational degrees of freedom.

New ideas on how to make the identification of chiral structures more reliable were considered. For instance, it was suggested that since the nuclear chiral system is formed by three mutually orthogonal angular momenta, it should not exhibit Coriolis effects. Therefore two partner bands with a chiral nature should show no energy staggering [6]. In addition theoretical investigations of the transition probabilities suggested that rotational bands with a chiral nature should exhibit a specific staggering in the $B(M1)$ rates with the same phase for both partners [7]. A later work showed that these two features of chiral bands cannot be used for a reliable identification of chiral symmetry, because while they are a characteristic observable for an ideal chiral system, they become considerably distorted in a more realistic description of chiral bands [8]. It was also shown that these two features can be generated by other phenomena, unrelated to chiral symmetry [8]. Therefore ideas on how to make the identification of nuclear chiral symmetry more robust are very necessary.

In this work we examine the decay patterns of a multiplet of rotational bands. It is found that the pattern should be distinctly different for bands with a chiral nature and for bands built on different nucleon configurations. This difference can be used as a model-independent test of the chiral nature of the bands.

Let us consider two rotational bands that have (i) the same parity and similar excitation energies, (ii) similar alignments and (iii) similar reduced transition probabilities. Let us also assume that the nuclear shape is triaxial and that the nucleon configuration involves valence nucleons with both a particle and hole nature. These conditions generate a system with a total angular momentum that has large projections on the three major nuclear axes, with particle angular momentum along the short axis, hole angular momentum along the long axis and rotational angular momentum along the intermediate axis. These three angular momenta can be arranged in a left- or in a right-handed system, therefore a chiral geometry can form. The observed rotational bands would therefore be interpreted as likely chiral symmetry partners. However, is this interpretation unambiguous? While a pair of degenerate bands are undoubtedly produced by a broken symmetry, a pair of similar rotational bands might also be generated by a different mechanism. Therefore, apart from the chiral symmetry interpretation, other scenarios might also be possible. For most chiral candidates it is possible to consider an alternative interpretation in terms of two different nucleon configurations, which involve different orbitals from the active proton and neutron shells.

Two rotational bands with a chiral nature are by default associated with the same nucleon configuration, e.g. a ${h}_{9/2}$ proton and an ${i}_{13/2}$ neutron. One can use additional superscripts to indicate which orbital of the corresponding shell is occupied, e.g. $\pi {h}_{9/2}^{(1)}\otimes \nu {i}_{13/2}^{(7)}$, where the superscript (1) indicates that the proton occupies the 1^rst, i.e. the lowest-energy orbital of the ${h}_{9/2}$ shell (it therefore has a particle nature), while the neutron occupies the 7th, i.e. the highest-energy orbital of the ${i}_{13/2}$ shell (it has a hole nature). We will call these orbitals e and A respectively. A simple alternative to the chiral symmetry interpretation is to consider that the yrast band has a $\pi {h}_{9/2}^{(1)}\otimes \nu {i}_{13/2}^{(7)}$ configuration, while the partner band is assigned the $\pi {h}_{9/2}^{(2)}\otimes \nu {i}_{13/2}^{(7)}$ or $\pi {h}_{9/2}^{(1)}\otimes \nu {i}_{13/2}^{(6)}$ configuration, i.e. either the proton or the neutron occupies a different orbital in their corresponding shells. The second lowest-energy ${h}_{9/2}$ proton orbital will be called f, while the second highest-energy ${i}_{13/2}$ neutron orbital is denoted with B. Since the two bands involve close-lying single-particle orbitals from the same shell they will have similar energies, alignments and reduced transition probabilities. This makes it difficult to rule out this alternative interpretation on the basis of the currently used chirality fingerprints. However the decay patterns of the multi-quasiparticle bands can shed more light on the intrinsic nature of the bands and may also in some cases rule out such an alternative interpretation.

As an example, we will consider the chiral candidates in the neighbouring ${}^{\mathrm{193,194}}$Tl nuclei, see figure 1, [9–13]. All bands shown in the figure are built on $\pi {h}_{9/2}\otimes \nu {i}_{13/2}^{-n}$ configurations, where n = 0, 1, 2, 3 shows the number of ${i}_{13/2}$ neutron holes. The low-spin bands A0 and B0 in ¹⁹³Tl are built on a $\pi {h}_{9/2}$ configuration, while at high spin bands A2, B2, and C2 have $\pi {h}_{9/2}\otimes \nu {i}_{13/2}^{-2}$ configurations. In ¹⁹⁴Tl the bands at lower energy, A1, B1, and C1 are assigned the $\pi {h}_{9/2}\otimes \nu {i}_{13/2}^{-1}$ configuration, while above the corresponding band crossings the A3, B3 and C3 bands are associated with the $\pi {h}_{9/2}\otimes \nu {i}_{13/2}^{-3}$ configuration. The number in the label of the band shows the number of neutron holes in the configuration, e.g. A3 indicates three neutron holes.

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First, the number of the ${h}_{9/2}$ proton and ${i}_{13/2}$ neutron orbitals near the Fermi surfaces will be considered. In ¹⁹³Tl the yrast band (labelled A0, see figure 1) is assigned $\pi {h}_{9/2}^{(1)}$ configuration. There is evidence for negative-parity levels that might belong to a band built on the yrare ${h}_{9/2}$ orbital, with a $\pi {h}_{9/2}^{(2)}$ configuration. This band is labelled B0. Let us denote these two proton orbitals with e and f. There is no experimental evidence for bands that involve two or more ${h}_{9/2}$ protons in the neighbouring nuclei. Thus it will be assumed that the other ${h}_{9/2}$ proton orbitals are not near the Fermi surface.

In ¹⁹³Hg the yrast band is built on the highest-energy $\nu {i}_{13/2}^{(7)}$ neutron orbital. There is evidence of two levels that belong to the yrare $\nu {i}_{13/2}^{(6)}$ band. Although no other single-neutron ${i}_{13/2}$ bands are known in this and in the neighbouring odd-Hg nuclei, excitations involving two, three and four ${i}_{13/2}$ neutrons are well known in the Hg isotopes, thus we will assume that four ${i}_{13/2}$ neutron orbitals lie sufficiently close to the Fermi surface. They are labelled A, B, C and D.

In figure 1, two levels, 21/2⁻ and 23/2⁻, are shown separately from the rotational bands. They belong to the band crossing region of the $\pi {h}_{9/2}$ and the $\pi {h}_{9/2}\otimes \nu {i}_{13/2}^{-2}$ bands. These two levels decay strongly towards band A0. In addition the 21/2⁻ level decays through a M1 transition towards band B0. To deduce whether these two levels have a larger contribution from the e or from the f proton orbitals their E2 decay branches were considered. Both levels have strong E2 decays towards band A0 (with proton configuration e), but no such decays were found towards band B0 (with proton configuration f), [9]. For instance the upper limit on a possible E2 decay branch from the 21/2⁻ level towards band B0 was estimated at about 3% of the observed 21/2⁻ $\to$ 17/2⁻ decay towards band A0. This shows that the levels at the band crossing region have predominant proton configuration e and are therefore linking band A0 with the three-quasiparticle bands.

Thus the possible multiplets of rotational bands built on the e, f, A, B, C and D orbitals in ${}^{\mathrm{193,194}}$Tl will be discussed, in particular the expected decays patterns depending on whether the bands have a chiral nature or are built on different quasiparticle configurations.

In order to build a coherent description of all $\pi {h}_{9/2}\otimes \nu {i}_{13/2}^{-n}$ bands, we examine their decay patterns in an odd–even and in an odd–odd neighbouring nuclei.

Let us first examine the expected decay patterns of a multiplet of rotational bands built on different nucleon configurations from $\pi {h}_{9/2}$ and $\nu {i}_{13/2}$ shells. Let us assume that the e and f proton orbitals actively participate in the nucleon configuration of the multiplet. The expected decay patterns are shown in figure 2. In ¹⁹³Tl two single-particle bands, e and f, would form at low spin. They would undergo neutron pair breaking and produce four bands with eAB, eAC, fAB and fAC configurations. These three-quasiparticle bands would decay to the e or f one-quasiparticle bands depending on the similarity of the corresponding configurations. The eAB and eAC bands would decay to band e, while the fAB and fAC bands would decay towards band f.

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In the neighbouring ¹⁹⁴Tl the two-quasiparticle bands will have configurations eA, eB, fA and fB, as shown in figure 2. After breaking a neutron pair, four-quasiparticle bands would form with configurations eABC, eABD, fABC and fABD. These four-quasiparticle bands decay towards the two-quasiparticle bands with similar configurations. For instance the eABC band can decay to both eA and eB bands. The decay towards the eA band would be stronger because the eA band is yrast. This more intense decay is shown with a solid arrow in figure 2, while the weaker decay, towards the non-yrast, eB band is shown with a dashed arrow.

These decay patterns can be compared with the experimental data, shown in figure 1. It is clear that there is a considerable difference. For instance, the observed decay patterns of the three-quasiparticle bands in ¹⁹³Tl do not show the predicted distinct separation into bands built on e and f proton configurations, respectively. Should some of the multi-quasiparticle bands involve the proton f orbital, for instance in configuration fAB, this band should decay towards the non-yrast B0 band, and furthermore this decay path should be enhanced with respect to any other decays due to the similarity in their configurations. However, none of the three-quasiparticle A2, B2, and C2 bands decays strongly to band B0. These three bands decay towards the 23/2⁻ and 21/2⁻ levels, which have a predominant contribution from the e proton orbital.

The decay pattern of the four-quasiparticle bands in ¹⁹⁴Tl (see figure 2) predicted within this scenario is also inconsistent with the experimental observations. Assume that the proton f orbital is involved in the configurations of bands B1 and B3, while bands A1, A3, C1 and C3 involve the proton e orbital. Then bands B1 and B3 will have configurations fA and fABC. It is then expected that: (i) the fABC band decays strongly towards band fA; (ii) direct decays that require two changes in the configuration, such as the decay from the fABC band to the eA band, or from the eABC band towards the fA band, are not observed. However band C3 (which we assumed involves proton e orbital and three neutrons) decays strongly not only towards bands A1 and C1, but also towards band B1. The last decay would involve two changes in the configuration and should not occur.

Therefore the decay patterns of the observed three- and four-quasiparticle bands in ${}^{\mathrm{193,194}}$Tl rule out a scenario in which the multiplet is produced by different configurations involving e, f, A, B, C and D orbitals.

In another test we examine the predicted decay patterns of a multiplet of rotational bands produced by coupling one proton in the e orbital with neutrons in the A, B, C and D orbitals. The expected multiplet is illustrated in figure 3. In ¹⁹³Tl the one-quasiparticle band is associated with the e orbital. At high energy, where a neutron pair breaks, bands with configurations eAB, eAC, eBC and eAD will form. These three-quasiparticle bands should decay strongly towards the one-quasiparticle band e. This decay pattern is consistent with the experimental observations shown in figure 1, where the A2, B2 and C2 bands decay (passing through a band crossing region) into the levels of band A0.

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However the observed decay patterns in the odd–odd ¹⁹⁴Tl show some inconsistencies with this scenario. At first glance it seems that the bands can be reasonably well explained. For instance, one can form four two-quasiparticle bands by coupling the odd e proton with the four neutrons, i.e. eA, eB, eC and eD bands, see figure 3, three of which were observed experimentally. At high spin, where a pair of neutrons break, a multiplet of four-quasiparticle bands should occur. One can generate four such bands with configurations eABC, eABD, eACD and eBCD. Each one of these bands will decay three-fold due to similarity in the configurations; for instance, band eABC will decay towards bands eA, eB and eC. Among these three decays, the most intense should be the decay towards the yrast eA band. For three out of the four four-quasiparticle bands, the most intense decay should be towards band eA, because they all involve a neutron in the A orbital.

Examining the experimental data on ¹⁹⁴Tl, see figure 1, one notices that band C3 decays towards bands A1, B1 and C1. To make this observation consistent with the predicted decay patterns, band C3 will be assigned eABC configuration, bands A1, B1 and C1 will be assigned eA, eB, and eC configurations respectively. Bands A3 and B3 do not show three-fold decays as expected. Therefore we will assume that the observed decay path corresponds to the most intense decay, while the weaker ones remain unseen. However, the most intense decay of band B3 is not towards the yrast two-quasiparticle band, A1, but towards the yrare B1 band. To explain this we have to assume that the configuration of B3 does not involve a neutron in the A orbital, and therefore band B3 is only consistent with the eBCD configuration. This assignment, however, creates inconsistencies.

For instance, band A3 in this case would be assigned one of the two remaining configurations, i.e. eABD or eACD. Let us assume that the configuration of A3 is eABD, while the band with configuration eACD remains unseen. But then there is a problem explaining why the band with configuration eBCD, which is the band with highest energy among the multiplet eABC, eABD, eACD, eBCD, is observed, while one of the bands with lower excitation energy is not. Furthermore it is not clear how to explain the observed linking transitions among the A3, B3 and C3 bands within this scenario. Since these bands have similar configurations, decays among all three of them are expected, with strongest links feeding the yrast band, C3. This is in disagreement with the experimental observations [11, 12]. There is no evidence for any links between band A3 and band C3. Furthermore only one weak ${\rm{\Delta }}I$ = 0 linking transition is found between band B3 and band C3, contrary to the observed strong M1 links from this band towards band A3. In order to have such a distinct difference in the linking transitions going out of band B3, which shows strong preference towards feeding the non-yrast band A3, one needs to assume an additional similarity between bands A3 and B3, something that this scenario cannot offer.

Let us now consider the decay patterns of the multiplets assuming that the bands have a chiral nature. Chiral systems built on the $\pi {h}_{9/2}\otimes \nu {i}_{13/2}^{-n}$ configurations are illustrated in figure 4. In ¹⁹³Tl a chiral system cannot form for the one-quasiparticle configuration, thus band e has no chiral partner. At high energy a neutron pair breaks. The figure shows two chiral systems, built on the eAB and eAC configurations, each of them consisting of two chiral partner bands. All four bands are expected to decay directly to the one-quasiparticle e band. Such a decay is a characteristic feature of chiral symmetry systems, since it reflects the transition from chiral systems at high energy to a non-chiral band at low energy. This predicted decay pattern is in agreement with the experimental observations, where several three-quasiparticle bands decay towards the yrast one-quasiparticle band.

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In ¹⁹⁴Tl one can construct two chiral systems at low energy, associated with the eA and eB configurations, and built of two partner bands each. These bands will go through a neutron pair alignment, forming two chiral systems with eABC and eABD configurations. Each of these chiral systems consists of two chiral partner bands. Within this scenario the bands of a two-quasiparticle chiral nature develop into bands with a four-quasiparticle chiral nature. One could thus consider the pair (A1 and B1), and (A3 and B3) as chiral partner bands, corresponding to configurations eA and eABC respectively. The decay of band C3 towards A1, B1 and C1 is understood as a decay of the eABD band towards bands with configurations eA and eB. In addition, the links between the partners of a chiral system, i.e. bands A3 and B3, would be stronger than those between two different chiral systems, because the partner bands belong to the same nucleon configuration, while the configurations of two different chiral systems are different. This is in agreement with the experimental data, which shows linking transitions between bands B3 and A3, but no such decays between bands B3 and C3.

One can note that the chiral interpretation was carried out for two chiral systems and for two-, three- and four-quasiparticle configurations. This suggests a multiplet of four rotational bands. The experimentally observed multiplet consists of three bands only. Since no fourth band has yet been found, the presence of a second chiral system in these nuclei is not established. However, whether there is a second chiral system or not is of no importance for the present discussion. The aim of this work is to show that the decay pattern of a multiplet of rotational bands is different for bands with a chiral nature and for bands built on different single-particle configurations. It can therefore provide additional information on the nature of the bands, distinguishing between these competing interpretations.

It should be noted that the proposed decay pattern analysis is most suitable for rotational bands built on pure high-j proton and neutron configurations, as for instance the bands involving intruder $\pi {h}_{9/2}$ and $\nu {i}_{13/2}$ orbitals in the Tl isotopes. Should the proton and/or the neutron configurations include additional low-j orbitals, (which tend to mix and thus affect the decay probability patterns), the proposed analysis may become inconclusive.

It is suggested that an examination of the decay patterns of a multiplet of chiral candidates built on multi-quasiparticle configurations can reveal, in a model-independent way, information on the nature of the rotational bands. Distinct differences in the decay patterns of chiral structures and of rotational bands built on different quasiparticle configurations exist, thus an examination of the decay patterns may rule out one of these scenarios. This analysis is best suited for rotational bands built on pure high-j configurations. It was applied for the multiplet of quasiparticle bands in ${}^{\mathrm{193,194}}$Tl.

This work is partially supported by the National Research Foundation, South Africa, under grants GUN 76632, 88646, 93531 and 109134.