Configurations and hindered decays of K isomers in deformed nuclei with A>100

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Abstract

Spectroscopic information on the decay properties of high-K isomers in deformed and transitional nuclei has been evaluated and collated. Assigned multi-quasiparticle configurations are included. Factors that control the transitions strengths, such as various contributions to K mixing, are outlined. The systematics of K-forbidden transitions for different multipolarities are discussed for selected cases in terms of the hindrances, FW, and of the reduced hindrance factor per degree of K forbiddenness, fν, where ν=|ΔKλ|, ΔK is the K-value difference between the initial and final state and λ is the transition multipole order. With the improved statistics for E1, M1 and E2 transitions, a factorization into the product of the underlying multipolarity-dependent transition strength and a ν-dependence, due to K forbiddenness (f0), is possible. This suggests a weaker dependence on K forbiddenness than is commonly assumed.

Introduction

Isomers have often played a significant role in the understanding of nuclear properties, beginning with the ideas of Soddy [1], [2], the early discoveries of Hahn  [3] and the crucial theoretical insight of von Weizsäcker  [4]. In fact, von Weizsäcker showed that the combination of a large angular momentum change and low transition energy could lead to a long lifetime for electromagnetic decay. The subsequent observation of isomers in many nuclei and the ensuing attempts at interpretations, configuration assignments and quantitative analyses of the lifetimes (see, for example Refs.[5], [6], [7], [8], [9], [10], [11]) stimulated the development of nuclear models. In effect, they provided a window into the important intrinsic states in nuclei, formed by the occupation of single-particle orbits near the proton and neutron Fermi surfaces.

Not the least of these developments was the primary evidence for the existence of deformed nuclei and the understanding that evolved from the seminal paper of Bohr and Mottelson  [12]. These authors interpreted the γ-ray cascade following the decay of a 5.5 hour isomer in 180Hf  [13] as a sequence of transitions within the rotational band of an axially-deformed, even–even nucleus. The fact that the moment of inertia deduced from the state energy spacings was only about a third of that expected for rigid rotation of a nucleus with the deformation implied by the quadrupole moments  [14], [15] led to the concept of pairing-induced superfluidity in nuclei  [16]. At first, the long half-life of the initial state in 180Hf was assumed to arise from a transition of high multipolarity, but the interpretation changed to attribution of the hindrance to the breaking of a selection rule associated with a new quantum number, K, identified as the projection of the nuclear total angular momentum on the deformation axis  [17].

The early definition of an isomeric (metastable) state was that a level with a “measurable” lifetime (already in the 1950s down to a limit of 1010 s). Later this drifted to mean states with lifetimes that were “directly” measurable; i.e., by electronic means, implying lifetimes of a nanosecond or more. The semantics are not important and, perhaps, a more careful definition would focus on whether the lifetimes are unusually long, implying a state of a different structure, and especially, an intrinsic state. Consider, for example, that the 2+ members of rotational bands in well-deformed heavy nuclei have lifetimes of a few nanoseconds by virtue of the low energy of their E2 decays, despite their large enhancement: these are not of interest in the present context. The central point is not whether the lifetimes are arbitrarily long or short, but whether the involved transition strengths that depend in detail on the underlying nuclear structure can be understood, a point emphasized many years ago by Moszkowski  [9]. Of course, from the point of view of possible applications, lifetimes and the feasibility of excitation by non-nuclear means are often of more specific interest  [18].

As well as being a signature of intrinsic states, isomers provide an experimental tool that enables high-sensitivity spectroscopy, exposing not only the isomers themselves, but also the decay path of the states preceding and following them, which otherwise might be difficult to observe. Extensive studies have been carried out in the past few decades, particularly with heavy-ion induced, fusion–evaporation reactions, exploiting time-correlated techniques that allow the unambiguous placement of isomeric states and, importantly, the identification of rotational bands located above them (see Section  3.2). However, such reactions are hampered by their limitations in reaching neutron-rich, deformed nuclei, but developments in using deep-inelastic reactions with heavy beams and relativistic fragmentation have seen a resurgence in isomeric studies. Those new approaches extended considerably the information on more neutron-rich, deformed systems, as reviewed recently  [19].

The focus of the present review is on isomers with seniority (the number of non-paired particles) equal to two and higher in even–even nuclei, three and higher in odd-A nuclei, and greater than two in odd–odd nuclei. Although they are indirectly relevant, K-forbidden, one-quasiparticle decays in odd-A nuclei or two-quasiparticle isomers in odd–odd nuclei, are therefore not included in the present tables.

There are a number of existing compilations with information on intrinsic structures in deformed nuclei that we list here for reference. They include early compilations of transition strengths in odd-A nuclei  [20], odd–odd nuclei  [21] and a range of odd-A, odd–odd and even–even isotopes  [22]. Later studies, that incidentally do not examine the decay properties in detail, include the early work of Jain et al.  [23] on intrinsic states and their configurations in deformed nuclei in the range 60<Z<76 and 90<N< 114; a series of studies  [24], [25], [26], [27] covering configurations, intrinsic states and associated rotational bands in even–even and odd–odd, medium-heavy and heavy nuclei; and a recent compilation by Singh et al.  [28] focused primarily on the properties of three-quasiparticle rotational bands and configurations in the A=153187 region. Note also that, in the recent NUBASE2012 compilation  [29], associated with the 2012 Atomic Mass Evaluation  [30], excited isomeric states were included together with the nuclear ground states, although there are some differences in the coverage.

The identification of high-K isomeric states as a means of probing the position of specific orbitals near the Fermi surface, of testing competing mean-field models of superheavy nuclei, and of investigating correlations, such as pairing, has been the subject of considerable recent activity  [31], [32], [33], [34], [35], [36], [37]. However, challenges remain in characterizing fully such states and their inhibited decays.

In investigations of properties of superheavy nuclei, the population of isomers, rather than the nuclear ground states, can complicate the interpretation of the corresponding α-decay chains. This has been recently emphasized in theoretical calculations  [38], but in fact it was first raised much earlier  [39] (see Section  5).

Given the extensive background and recent development in the study of isomers and, in this particular case, of K isomers, it seemed appropriate to review the numerous nuclear structure physics aspects that govern their formation and properties. These are predominantly, but not exclusively, effects that will shorten the lifetimes by introducing allowed transitions in the context of otherwise nominally forbidden transitions. When using hindrances as a tool in nuclear spectroscopy, it is important to appreciate that a number of factors can contribute, sometimes individually, sometimes in concert, in altering the properties and, thus, in confusing the interpretation.

Section snippets

Deformed single-particle levels

The proton and neutron Nilsson levels of importance for the formation of prolate-deformed K isomers are given in Fig. 1, Fig. 2, Fig. 3. These were calculated using the Woods–Saxon potential with the so-called “universal” parametrization  [40] and β4=β22/6, with β6=0. Essentially three regions can be clearly separated: the lighter deformed nuclei (mass ∼100–150), the mass 170–190 region, which extends at its borders into the transitional nuclei, and the heavy nuclei with mass ∼250.

The important

Nuclear structure and experimental observations

It should be remembered that isomeric states in deformed nuclei, as observed in the laboratory, are not bare intrinsic states. Deformation implies that the wave function will include a rotational component, so that an isomer cannot be viewed separately from the rotational band of which it forms the bandhead. Indeed, the band properties are sensitive to the components making up the configuration and can be used to test the proposed configuration  [41]. Furthermore, rotation inevitably implies

Factors affecting hindrances

Long lifetimes occur when low-energy and/or high-multipolarity transitions are involved  [18], and also when K conservation is challenged. When the γ-ray multipolarity, λ, fails to match the difference in K between the initial and final states, the shortfall, known as the forbiddenness, is given by ν=ΔKλ. However, neither the presence or absence of isomers, nor the magnitude of individual lifetimes by themselves, are necessarily a measure of the conservation of the K quantum number. The

Fission branches in the Fermium region

In principle, high-K isomers should occur in both the first (normal-deformed) and second (superdeformed) wells in the heavy nuclei that exhibit multiple minima in their nuclear potentials. Following the early discovery of two-quasiparticle isomers in the first well of 250Fm (Z=100) and 254No (Z=102)  [116], a number of spectroscopic studies have been carried out in the Fermium region using various techniques, which have differing sensitivities, both in terms of decay branches (α decay, γ decay,

E1, M1 and E2 hindrances: dependencies

Given the historical development of the study of hindered transitions and how they have been used to make decisions on which values are acceptable in terms of spin/parity assignments and structure effects, we show here the E1 strengths in a form that partly matches the original presentation of Löbner  [61]. The important difference here is that, whereas Löbner showed a simple range for each value of ΔK, with an expanded data set it is now possible to show the distribution in each case by

E3 transitions

At this stage, the number of E3 cases is insufficient to warrant a centroid analysis for different values of ΔK. The individual data are illustrated in Fig. 16. The open points correspond to several different regions, e.g. the transitional 190Os and 192Os nuclei and examples from the lighter 130Ba and 132Ce isotopes. A fit only to the filled points presented in Fig. 16 gives F0=3.7(+3619) and f0=41(+119) (compare with Table C).

Coarse statistics

Finally, we include here some observations about the data base and statistics in general. The decay of approximately 370 isomeric states and the strength of their numerous deexcitation decays (∼1050  γ-ray transitions) are listed in the tables. Only ∼65% of all cases have firm spin and parity assignments, the remainder being uncertain, as summarized in Fig. 17. As can be seen from the statistics for the different multipolarities in Fig. 18, there are few cases known for the main multipolarities

Intrinsic states with sub-nanosecond lifetimes: experimental opportunities

Significant progress has been made in recent years in the area of determination of sub-nanosecond lifetimes with the development of LaBr3 detectors. These are characterized by moderate energy resolution, but excellent timing characteristics, reaching into the tens of picosecond range. The use of such detectors in conjunction with high-resolution, multi-detector Ge arrays has opened up the possibility of isolating individual states in complex decay schemes, and determining lifetimes well below

Scope and policies

As indicated above, there is a large body of information on isomeric states in deformed nuclei, and with emerging experimental techniques, this is likely to grow. The focus of the present review, however, is on those states that have been sufficiently well characterized in terms of spin, K value, and parity or at least limited ambiguity in assignment, and with well-defined decay branches. This is sufficient information to evaluate reliably the transition strengths. The tables include

Relevant formulas and definitions

The lifetime of an isomeric state is related to the total decay width, Γ, a linear sum of all partial decay widths (γ ray, conversion electrons, α decay, β decay, fission, etc.), through the uncertainty relationship (in convenient units): Γ×τ=ħ=0.6582×1015[eVs], where τ is the level mean life, which is related to the half-life as T1/2=ln2×τ.

For an isomeric state with N branches, predominantly γ rays and internal conversion in the present cases, the partial γ-ray mean life of an individual

Acknowledgments

This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under contract No. DE-AC02-06CH11357 and by the Australian Research Council Discovery Program grants DP0986725, DP03445844 and DP140102986. The authors are thankful to Dr. R.V.F. Janssens and Dr. D.J. Hartley for the critical reading of the manuscript, and for many useful comments, and suggestions.

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