A S Jensen

D V Fedorov
email: fedorov@phys.au.dk

E Garrido
email: 3e.garrido@iem.cfmac.csic.es

Rearrangements in three-body decaying resonances

Keywords: numbers: 21.45.-v, 21.60.Gx, 24.30.Gd

Introduction

The nuclear N-body system has in general a number of bound and excited states, some of which decay by emission of clusters of nucleons. This immediately implies that description of such decays involves continuum structures. Since all bound nuclear clusters except for the neutron are charged the long-range Coulomb interaction is almost always present. Obviously binding is provided by the short-range strong interaction. Thus in decays we are in general dealing with N -body degrees of freedom contracting into much fewer cluster degrees of freedom. The process obeys the laws of quantum mechanics, which, to determine the decisive continuum structures, require simultaneous treatment of both small and large distances for combinations of short and long-range interactions

The structure problem is in general already extremely difficult as evidenced by the fact that ab initio computations of bound nuclear N -body states at the moment only are within reach when N is smaller than 12. Descriptions of decaying structures into few fragments add to the difficulties due to the complications with reduction of degrees of freedom, and simultaneously contributing short and long-range interactions. Two examples of two-body decaying systems, α-emission [1] and binary fission [2], suffice to illustrate the difficulties.

In the early days of quantum mechanics α-emission was an unavoidable phenomenon that convinced physicists of the necessity to embrace the new theory. The tunneling probability through centrifugal and Coulomb barrier from nuclear surface to infinity accounted for variations in the α-decay rate by 30 orders of magnitude [START_REF] Siemens | Elements of nuclei. Many-body physics with the strong interaction[END_REF]. Energy and momentum conservation determines the energy of the α-particle and the daughter nucleus. The problem of contracting the degrees of freedom enters only when details of the decaying short-distance structure are important. This is taken care of by phenomenological spectroscopic or preformation factors [START_REF] Siemens | Elements of nuclei. Many-body physics with the strong interaction[END_REF] which are necessary to account for the remaining discrepancies of one to two orders of magnitude.

Binary fission is another complicated two-body decaying structure [4]. Now the short-distance structure is all decisive for which fragments emerge after the divison in two pieces. Obviously α-emission can be viewed as an extremely asymmetric fission. The mass distribution of the fission fragments emerges after the process of contracting the many degrees of freedom down to two is completed. The decay rate is in contrast to α-decay far from determined by tunneling through Coulomb and centrifugal barriers. This process has to be described by a complicated combination of diffusion, evaporation and dynamic evolution [START_REF] Helmut | The Physics of Warm Nuclei with Analogies to Mesoscopic Systems[END_REF].

The simplest extension is from two to three-body decaying structures [START_REF] Garrido | [END_REF]. All the problems of two-body decays remain and in addition now the energy can be distributed continuously between the three particles without violating energy and momentum conservation. The unstable beam facilities combined with detector development have recently allowed accurate and kinematically complete determination of the fragments arising from three-body decays, see e.g. [7]. These distributions are far from understood.

Several questions are still to be fully answered: How to perform model calculations and compare in details to measurements [8]?, and even more challenging how much can model-independently be inferred from the measured large-distance to the initially decaying short-distance structures?

We shall in this report formulate and discuss the problems illustrated by the decay process of a resonance. We shall assume that this continuum structure is populated in beta-decay or by an unspecified reaction. In this way we simplify the story by omitting everything prior to population of the resonance. This excludes discussions of some reactions but allows focus on decay mechanisms of resonances and analysis of possible model independent connections to measurements. This seems to be a reasonable beginning since resonances are cornerstones in continuum structures. In any case the idea is that detailed formulation of a problem often helps to solve it.

Formulation

We assume a resonance is formed by a system of N particles. We imagine population occurred by a reaction or through beta-decay. However, the formation history does not influence the subsequent decay. Before proceeding we have to define a resonance. Our choice is usually the wavefunction corresponding to the complex energy of a pole in the S-matrix or equivalently the wavefunction with a complex energy and only outgoing flux in all channels. It is also possible to piece a resonance together by continuum wavefunctions with real energies in an appropriate interval where the cross sections are strongly varying and substantially above the background.

Let us focus on three-body decays and three-body computations even though most of the following applies more generally or easily can be extended. We use our resonance definition which implies some analytical extension into the complex plane which can be achieved by complex scaling of the coordinates [9]. The advantage is that the resonance wavefunction is a solution to the (complex scaled) ordinary Schrödinger equation with bound-state boundary conditions. The disadvantage is that the complex scaling method only provides solutions for relatively narrow resonances.

The overlap of the resonance wavefunction with asymptotic large-distance outgoing plane waves provides the momentum distributions of the emerging three particles [10]. This is loosely speaking the Fourier transform at large distance of the resonance wavefunction. The main difficulty of diverging Fourier integrals is solved by the Zeldovic regularization. The momentum distributions are determined by the momentum-space wavefunction multiplied by a Breit-Wigner shape in three-body resonance energy and width.

The procedure can in the simplest version be formulated as computing a threebody resonance by complex scaling, Fourier transforming by Zeldovic regularization, and integrating over all momenta except the observables of interest. The simplifying assumptions are three-body structures and narrow resonances defined by complex scaling. This allows a relatively clean presentation of the problem.

Open problems

Decay of a many-body resonance into three fragments presents two problems. The first also appears in two-body decay, i.e. reduction of the N -body degrees of freedom relevant at small distance to those describing the (two or) three fragments at large distance where the decay products emerge. The second problem arises when the three-body resonance wavefunction is allowed to change character from small to large distance. How to reduce the degrees of freedom, and how to understand and incorporate properly the couplings causing a dynamical evolution.

Contraction of superfluous degrees of freedom

The decay into three particles prescribes use of a three-body model at the (large) distances where the fragments have gained their final identity. On the other hand, at small distances all N particles interact and the final state cluster structures are sometimes far from being realized. Numerous examples exist, e.g. nuclear resonances with a small partial width for decay into three fragments, where the small width can be attributed to a total lack of cluster structure at small distances, see f.ex. the structure and decay of 1 ± states in 12 C, see [11,12]. Thus the many-body structure should be smoothly joined to the three-body cluster structure. One could try two extremes of either full N -body or only three-body models for all distances. The latter case is the usual three-body problem for point-like particles where boundary conditions both at zero and infinity are used. Then the short-distance behavior is crucial for occurrence and properties of resonances. It is from the beginning unclear from which distance it is sufficient to use a description in terms of cluster structure.

One strong indication is from tunneling through the barrier of the dominating adiabatic three-body effective potentials. We illustrate the idea in Fig. 1 by the 12 C resonances [13]. In some cases a minimum is found in the three-body computation but far from always. This means that the short-distance structures of the resonances are far from the cluster structure emerging after the decay, because a minimum inside a barrier is needed to produce a resonance.

If we artificially introduce a strong square-well attraction at short distance the resonance energy can be adjusted to the meassured position and tunneling through the remaining barrier determines the liftetime. In Fig. 2 we show the WKB estimates of these lifetimes. They vary by five orders of magnitude but are within one order of magnitude correlated with the tunneling probability. This result is not trivial as the energy and angular momentum dependence is completely different from those of the well-known two-body decay modes. The conclusion seems to be that the cluster structure is established at rather small distances roughly corresponding to a common nuclear surface. However, closer inspection reveals that the states traditionally denoted as "shell model" states, in contrast to cluster states, are further away from three-body lifetime estimates. In any case to get accuracies within an order of magnitude requires more short-distance details than provided for point-like cluster particles.

It then seems obvious to perform N -body shell-model computations which are designed to account for short-distance correlations, and known to have severe difficulties in describing large-distance and cluster properties, see e.g. [14]. Then combine these shell-model results with three-body computations at small to intermediate distances corresponding to about nuclear surface radii. This procedure seems to be straightforward. However, in practice it has not been consistently implemented due to conceptual, as well as technical, problems. Assume that the results of a large shell model computation is expanded in a basis of three-body relative wavefunctions multiplied by intrinsic cluster wavefunctions. Under the very uncertain assumption that shell-model accuracy is sufficient up to distances where the cluster structures have separated out, we have in principle established new boundary conditions for the three-body computation. At best we have then a six-dimensional surface in relative three-body coordinates but supplemented by the problem related to the center-of-masses of total and individual clusters. The shell-model energies are far from the high accuracy obtained in the threebody calculations. Therefore they cannot be used as guides. Still all clusters may not be established at the same time, one could come first whereas the other two only appears after the first has moves away. The Pauli principle has to be obeyed for identical particles inside and across individual clusters. The change from N to three particles requires a change of interactions, and furthermore a change of Hilbert space which in turn requires a change to correponding effective interactions [15]. Since the effective cluster-cluster interaction cannot be derived from the nucleon-nucleon interaction, this presents a conceptual problem.

It would help to employ a similar basis in the N and three-body computations but that is also very difficult since the three-body continuum is necessary but this is easily in conflict with completeness of the shell-model basis. Thus it is an open problem to contract the degrees of freedom from N to three particles.

Dynamical evolution of the three-body structure

Within the three-body model with point-like particles a crucial change of structure may occur from small to large distance. Small distances are decisive for the resonance properties and supply the largest part of the energy, but large distances reflect the fragment distributions after decay. These structures are linked continuously by quantum mechanics. The change of structure may occur for a variety of reasons. The cluster structure of 9 Be ( 9 B) consists of two α's and a neutron (proton). At small distance the α-nucleon p 3/2 -interaction provides the attraction crudely speaking with the nucleon between the two α-particles. At large distance s-waves are preferred and 8 Be in the (unbound) ground state moves away from the neutron. The dramatic change of partial wave decomposition with distance is shown in Fig. 3. The dominating p-waves (l x = 1) below ρ = 7.5 fm are quickly replaced by s-waves (l x = 0) for larger hyperradii. In general, the lowest energy is favored in adiabatic evolution but if possible the system may prefer to maintain the structure in diabatic evolution [16]. In actual cases a fraction of each can be expected as the resulting compromise between lowest energy or structure change [17]. The amounts of each can vary from 0 to 100% are detemined by the couplings. This can be seen for the two lowest 0 + resonances of 12 C as shown in Fig. 4. The probablilities vary strongly for occupying each of the adiabatic potentials. The Hoyle state at 0.38 MeV consists by far mostly of the lowest component for all distances. In contrast, the next 0 + resonance at 3.95 MeV is at large distance mostly of the same structure as the Hoyle state but leaving about 10% of a different component at large distance. Lowest adiabatic effective potentials for the 3/2 -resonance in 17 Ne at 0.34 MeV. The interactions are described in [18]. The deepest potential (black curve) describes the 16 F d-wave resonance and the last proton in an s-wave.

There may be more than one type of these coupled structures and corresponding attempts to change characteristics as e.g. from symmetry due to identical particles [8]. It could also be that only a virtual change occurs if energy conservation forbids the process [18]. For example, a low-lying excited state of one of the clusters may give an important contribution to the resonance structure at relatively small distance. This channel may still be closed in the resonance decay and rearrangements are necessary. These decays can not occur as purely sequential via a lower-lying resonance in a subsystem. This is the decay analog of Borromean structure. All subsystems have no resonance in the energy interval from zero to the decaying resonance energy scaled by the mass ratios determined by momentum conservation. The energy is measured relative to the energy where all particles are far apart.

This Borromean decay happens for the first excited state, the 3/2 -resonance in the Borromean nucleus 17 Ne, which only can decay into two protons and the 15 O core. We show in Fig. 5 the adiabatic potentials where the deepest, after the barrier, decreases very slowly. The structure corresponds to the d-wave resonance of 16 F with the last proton in an s-wave, and the (almost) antisymmetric configuration of 16 F in an s-wave with the last proton in a d-wave. The decrease matches the Coulomb and centrifugal potentials of one proton relative to the spatially more confined two-body configurations of 16 F. The 17 Ne resonance is dominated by such a structure, even after other potentials have crossed and appear with lower energies. This is then an example of maintaning the structure rather than following the lowest energy. However, this can not continue to infinity because the energy is too small. The 16 F structure must exchange energy with the last proton in order to conserve the total energy in the decay process. Thus, the decay mechanism is virtual sequential decay where the energetically forbidden 16 F structures are exploited as vehicles until the last proton is at the edge of the barrier.

W T=1 (ρ) ( x 2.5 ) W T=0 (ρ) ( x 2.5 ) W T=0 (ρ)/W(ρ) 2 + in 6 Li Figure 6.
The probability of finding isospin 1 and 0 as function of hyperradius (ρ) for the 2 + state in 6 Li.

An important example is symmetries arising from quantum statistics and nearly fulfilled conservation laws [8]. Small violation as for the weakly broken isospin causes a very weak coupling between states related to adiabatic potentials of different isospins [17]. Such states would mix but the accuracy requirement might be overwhelmingly demanding. Imagine population of an almost isospin pure T = 1 resonance in 6 Li which decays by neutron plus proton emission. The remaining α-particle has isospin zero but the isospin 1 neutron-proton system couples weakly to the isospin 0 deuteron with the much lower energy. The coupling is caused by the Coulomb interaction and efficient where it dominates, i.e. outside the range of the strong interaction, which in turn means at intermediate and large distances where the accuracy is especially difficult to maintain. This is illustrated in Fig. 6 where the abrupt change from isospin 1 to 0 at ρ ∼ 20 fm is followed by an equally abrupt change at ρ = 48 fm to a steady level at about 10% for large ρ.

The violation causes the system to change structure at intermediate distances and fall down in energy to a lower-lying state of different sysmmetry. Three-α decays of isospin 1 states in 12 C must change structure before the three α-particles are formed. This means at small distances where the many-body structure is dominant. This is reflected in the potential of 1 + state in 12 C, see Fig. 1.

Another general scenario, a two-body resonance can keep two of the particles close together while the third is moving far away before this resonance decays. This is sequential or virtual sequential two-body decays which can be dealt with separately [19], see Fig. 3 and Fig. 5, respectively. However, if the lifetimes of the initial threebody system and the two-body resonance are comparable this sequential decoupling cannot be made. Then one cluster is formed and emitted (slightly) before the other two clusters emerge from the N particles. The boundary conditions in the contraction from N particles to 3 clusters is now ill-defined since the (small) distance is ill-defined, see previous subsection. When a two-body substructure is prominent and present at intermediate distances, it has to decay or couple to the three-body continuum.

Then the pertinent questions are if the couplings responsible for the structure rearrangments are defined, known, and possible to treat in practice. Obviously couplings to neglected degrees of freedom are absent and most likely related to the problem of contracting the degrees of freedom. However, the resulting wavefunctions in combination with the correspondingly renormalized operators contain all the information. The couplings within the remaining (three-body) Hilbert space are in principle described in the (three-body) model from the initial two-body interactions. However, these (twobody) interactions are as hard to get as the renormalization due to reduction of Hilbert space. Phenomenology is then almost inevitable, but spatial shape is unspecified and the accuracy requirements can be very severe. Further complications are possible since the influence of couplings can appear anywhere from small to large distances.

In particular, it is difficult when more than one substructure coherently contribute and couple perhaps at different distances. It helps to employ Faddeev equations which allow simultaneous treatment of different subsystems on the same footing. For longlived substructures the couplings are tiny and continuously occurring until the two-body decay has taken place or equivalently until no trace remains of the two-body structure. The Coulomb interaction enhances these difficulties as the Coulomb coupling extends to infinity. Combinations with Efimov-like resonance structures of extreme spatial extensions probably cannot be dealt with numerically [20]. Incompatibility between analytic derivations, phenomenological and numerical couplings can be disastrous.

In total, the rearrangements within the possible three-body structures are in principle determined by well defined couplings. Unfortunately, the technical problems are tremendous in many cases although possible to handle with a fair accuracy in other cases. In close connection it is necessary to address the question of how detailed information about the fragment distributions can be used to deduce decay mechanism and initial structure or extract as severe constraint as possible. These are open problems.

Summary, Conclusions and Perspectives

We have presented the open problems related to understanding of three-body decay processes of N -body resonances. The key issues are first that the initial degrees of freedom must be reduced to describe the final state of three clusters, and second that even three-body resonance structures "dynamically" evolve from small to large distances. The first issue involves a many-body structure which has to be smoothly joined to a three-body structure. The second issue is a few-body problem but with the interaction input either fully phenomenologically or derived from properties of the same N -body system as for the first issue.

The thresholds for breakup into different parts play a crucial role. When the nucleon driplines are approached these thresholds tend to approach each other and wash out the differences between dynamical evolution and contraction of degrees of freedom. The reason is that reduction could be in steps as already indicated by three-cluster formation via two subsequent two-cluster processes. In cold atomic gases the two-body interaction can effectively be changed. The closest we get to that in nuclei is by electron screening of the Coulomb repulsion. The thresholds for combination of two and three α-particles into ground state of 8 Be and first 0 + resonance of 12 C then both approach zero [START_REF] Jensen | Three alpha-particle system in a dense helium plasma[END_REF].

We have in this paper indicated how to proceed in special cases but we focussed here more on the problems than on the previous attempts. The formulated problem is only a small piece, perhaps the simplest, of the general issue of decays of N -body resonances, e.g. into more than three pieces. It is also only a small piece of reaction descriptions where a resonance is not an intermediate stepping stone, e.g. for short-lived resonances where the lifetime is comparable to the reaction time or when resonance structures are circumvented all-together.

Treating broad resonances introduces a number of other problems than discussed here. It first of all raises questions about a proper definition of resonances as a quasistationary state. If the total energy is off the center of the resonance, the decay is asymmetric as higher and smaller energies "see" smaller and larger barriers, respectively. This suggests that broad resonances are better treated by staying on the real axis while varying the energy. However, this essentially implies that the resonance concept is not used and at the same time knowledge of the population history is required, and even this simplification disappears.

Figure 1 .

 1 Figure 1. The computed lowest adiabatic potential as function of hyperradius for a number of resonances of 12 C (α+α+α). The horizontal lines mark the resonance energies measured above the three-body threshold. The corresponding excitation energies are given above each of the panels.

Figure 2 .

 2 Figure 2. The computed widths compared to the measured values for different resonances of 12 C with excitation energy E. The WKB approximation is used for tunneling through the lowest adiabatic potential where the inner turning point is fixed to a hyperradius of ρ = 4 fm.

Figure 3 .

 3 Figure 3. The partial wave decomposition of the dominating adiabatic angular wavefunction for 9 Be and 9 B as function of hyperradius ρ. The partial angular momenta l x and l y refer to the α-nucleon and the α-(α-nucleon) Jacobi coordinates.

Figure 4 .

 4 Figure 4. The ratio of probabilities for the four lowest adiababtic potentials as functions of hyperradius, ρ, for each of the two 0 + resonances of 12 C at 0.38 MeV and 3.95 MeV above threshold or at excitation energies of 7.63 MeV and 11.2 MeV. The ratios to the dominating probability |f 1 | 2 are given for both states.

Figure 5 .

 5 Figure 5.Lowest adiabatic effective potentials for the 3/2 -resonance in17 Ne at 0.34 MeV. The interactions are described in[18]. The deepest potential (black curve) describes the 16 F d-wave resonance and the last proton in an s-wave.

Acknowledgment. Useful comments from K. Riisager are highly appreciated.


P D Stevenson
email: p.stevenson@surrey.ac.uk

S Fracasso

Extracting Structure Information from TDHF

This contribution to the "Open Problems in Nuclear Structure Theory" special edition looks at some issues with using Time-Dependent Hartree Fock and related techniques to study structural phenomena in nuclear physics. We limit the discussion to structures like giant resonances and discuss some open questions regarding the interpretation of TDHF calculations.

Introduction

The time-dependent Hartree-Fock (TDHF) approach was written down by Dirac in the 1930s [1] but first applied in anger to nuclear physics problems in the 1970s. As a method able to deal with large amplitude motion, the application of TDHF has been most prominent in reaction dynamics. Though such studies shed much light on nuclear structure as well as reaction mechanisms, we concentrate in this contribution on the use of TDHF in studies of giant resonances in which one is purely concerned with the structure of a single nuclide.

Giant resonances are collective excitation modes, occurring largely above the particle emission threshold and usually characterised by their multipolarity and spin and isospin character. The basic microscopic approach (within the realm of mean field models) to calculating giant resonances is the Random Phase Approximation (RPA). The RPA can be derived as the small-amplitude limit of TDHF, so in principle TDHF should do all RPA does and more. Usually RPA calculations of giant resonances calculate the strength function defined as

S(E) = ν ν| F |0 2 δ(E -E ν ), ( 1
)
where F is the operator describing the excitation mode of interest. In TDHF, this quantity is calculated by exciting the system with the operator F and following the time-dependent expectation value of the same operator F (t). According to linear response theory, the strength function can be calculated in TDHF as [2] S where F (E) = F (hω) is the Fourier transform of the time-dependent response F (t) and f (E) = f (hω) is the Fourier transform of the time-dependence of the external excitation F causing the resonance. Note that the expression (2) holds under the assumption that the response depends linearly on the strength of the external field [START_REF] Fetter | Quantum Theory of Many-Particle Systems[END_REF][START_REF] Blaizot | [END_REF].

(E) = - 1 π Im F (E) f (E) , (2)
Early TDHF calculations for giant resonances concentrated on monopole resonances [5,6,7] since they require only a spherical one-dimensional code, and were also restricted to simplified forms of effective interactions. However, the principles of extracting information from the calculations remain the same as more powerful computers led to the ability to calculate giant resonances with non-spherical instantaneous deformations and those built on nuclei of arbitrary shape [8,9,10,11,12]. In addition to the increase in computer power allowing the relaxation of symmetry assumptions, it is also the combined sophistication and simplicity of the Skyrme-type interactions that allow useful TDHF calculations to take place [13].

Observables, Excitation Operators and Modes

A general question that arises in the study of giant resonances is what are the interesting quantities to calculate? As mentioned, the standard quantity calculated in RPA is the strength function, which can be related to an experimental cross section, though it is not an instrinsic property of the nucleus, but also on the particular experiment and the associated momentum transfer [7,14]. One interesting possibility that TDHF allows is the direct simulation of a realistic time-profile for the excitation and an analysis of the results, or even the direct simulation of a collision. It is an open question whether one will get results more closely related to the experimental situation, but there is scope to try.

A typical RPA analysis of the strength looks also at individual contributions from particular particles in the basis of the uncorrelated ground state. This is less straightforward to do in TDHF. In practical TDHF calculations a large basis of unperturbed single particles states is not needed (or desirable) as the excitations are realised by time-evolved single particle states with a different Slater Determinant at each time. Combining this set of Slater Determinants by means of a Fourier transform then corresponds to the RPA case. By an analysis of the excitation energies, though, it can be possible to identify the dominant single particle state in a structure in the strength function [15], when indeed a single one is dominant. There is no analogue in TDHF with the so-called unperturbed RPA response. However, one can be imaginative in the way that one excites the nucleus. Figure 1 shows the giant monopole resonance in 16 O using the SkM* force in which the total response is shown, alongside the response of the π1s orbital to an excitation of only the π1s orbital, which shows that the π1s orbital is responsible for the structure around 30MeV. The interpretation, however, is not trival. The excitation of the π1s state induces oscillations in the other single particle states. This overall excitation is different to the total GMR excitation, so the peak positions do not line up exactly. Unlike in RPA, we cannot easily disentangle the contributions from each particle state. Comparing with RPA calculations can therefore be difficult. In RPA, moreover, one can isolate the contribution to the response easily from different parts of the residual interaction, which, as well-known is obtained as a second derivative of the Hartree-Fock energy density expressd in terms of the static densities. In TDHF one does not explicitly construt the residual interaction, the only ingredient being the time-dependent mean-field, and the ability to understand which parts of the interaction are contributing is understood in a different way. One cannot simply count up which parts of the underlying interaction in TDHF give rise to the strength as in RPA.

As well as a straightforward multipolar decomposition, there are many possibilities for combining space and time functions of the excitation of the nucleus that are allowed by TDHF -e.g. one can excite only the surface, only protons, only specific orbitals, with angular momentum, with a sudden kick, with a fast or slow external field or a harmonic driving force, all of which can give information about the structure of the nucleus. One can also look for new kinds of excitation that may arise such as the axial spin-twist mode [16], though how to think of a clean signal for its observation is another matter.

TDHF with no spatial constraints permits, and demands, a completely general spatial response to a particular excitation. One cannot (in general) give a nucleus a monopole excitation and expect no quadrupole to be excited in addition. At any given frequency, one might attempt to find the normal modes of excitation, presumably in a multipolarity basis [17]. An example of this mode mixing is easily seen in monopole calculations of the inducement of isovector modes to isoscalar excitations and vice versa [18].

The study of giant resonances and collective modes (by whatever theoretical method) has been of considerable use in understanding the link between effective interactions and the properties of nuclear matter [START_REF] Blaizot | [END_REF] and has been used for understanding how effective interactions fitted to ground state data can be further constrained by excited state properties [19,20,21]. What further role can TDHF play in this area? Constraining those terms in the functional which are made of time-odd densities, including those that arise only from the tensor force [22], is certainly now a possibility given the existing TDHF codes, and a suitable suite of observables needs to be chosen. Nuclear matter properties can also be probed in collisions and TDHF gives an ideal method to combine understanding the properties of nuclear matter simultaneously on structure and reaction observables.

As a final point about the comparison between RPA and TDHF, we bring up the ground state energy correlations. As is well known, particle-hole correlations are explicitly taken into account in RPA, contributing through the Y amplitudes to the response function and the ground state energy [24]. To the best of our knowledge, no attempt has been made to realise an approach to use TDHF to calculte the same correlated ground state as in RPA [25], though some interesting uses are made of time-dependent methods to build in certain classes of ground state correlations [26]. Today's symmetry unrestricted TDHF codes could in principle be used to build in more sophisticated correlations.

Nonlinearities

TDHF is inherently able to deal with nonlinearities in the sense that the nuclear response to an excitation is not, in TDHF, necessarily proportional to the magnitude of the As an example of the kind of size of effect that appears, the nonlinear response of a monpole excitation in 16 O is presented in figure 2. In extracting information from the nonlinear regime, different groups have taken different approaches, including a spectral analysis [15], an exploration of the coupling matrix elements between different modes [17], an information-theoretic approach [27], in terms of quasiperiodic orbits [28],

or the analysis of level-spacing [18]. There has, so far, been little attempt to unify or compare different approaches to exploring nonlinearities -though the informationtheoretic approaches are looking more at the inherent nonlinearities of the TDHF equations than the linearity of the response. As well as a lack of comparison of existing methods for exploring nonlinearities, we are not aware of work coupling particlevibration coupling with time-depdendence.

The Continuum and the Open Quantum System

The two topics of the continuum and open quantum systems are related in that giant resonances can decay by particle emission. TDHF inherently preserves the norm of the wavefunction and so to study particle emission in TDHF one must couple the nucleus to the environment and allow emitted particles to disappear into the continuum. The easiest way of doing this is to use a large coordinate space grid in which the normal TDHF equations are solved, but consider a small region of the grid in which the nucleus is said to exist -and in which region alone nuclear observables are calculated [7]. This is practical only in one-dimension [7,9]. In a full three-dimensional calculation such large boxes become prohibitive. Non-uniform gridding and absorbing boundaries can help [12], though it cannot give an exact solution. One possible route to a proper treatment of the continuum is to use some form of exact boundary condition [29].

It is an open question whether such an approach will be sufficiently efficient in the case of 3D TDHF. Note that it is the reflecting boundaries and the discretisation of the continuum that allows for the kind of nonlinear analysis in [18] and [27]. It also allows for the construction of periodic orbits. If the system is treated as an open quantum systems with nucleus and environment, then dealing with periodic orbits for the nucleus only will require some serious thought.

The process of separating the space into a nucleus and the environment is necessary to treat the continuum. It has its problems, though, in that the point at which to make the cut is chosen with rules of thumb, and results depend slightly on where the choice is made. As an example, Figure 3 shows the time-evolution of the isoscalar giant monopole moment for 16 O with a kick size of 0.0001 fm -2 , in a 240fm box with time evolved to 1024 fm/c (and no flux hitting the edge of the box). The different lines show the result of making different choices for the cut point. In detail, no two lines quite agree with each other. The resulting Fourier transforms are shown, and show that the effect on the strength is not strong, providing the cutoff between nucleus and environment is taken at a large enough radius. Larger radii make nonlinearities more obvious for the same external perturbation size, as indicated by the failure of the Fourier transform to produce a positive function, though the strength in the peak region is remarkably unchanged.

A better understanding of how TDHF can be interpreted as describing an open quantum system is now called for, alongside a comparison to methods which explicitly add an environmental coupling [30].

Beyond TDHF

In a sense, we have already talked about going "beyond TDHF" in the entire nucleus + environment system as obeying the TDHF equations, but analysing a restricted part of the density matrix and identfying it with the nucleus, thus allowing the study of particle emission within a framework that preserves norm. However, there are many ways one would like to go more genuinely beyond TDHF to improve our understanding of nuclear structure. It is well-known that TDHF misses the collisional width in giant resonances associated with 2p-2h (and higher) collisions. Various methods exist, which have TDHF as a base, which can be used to look beyond TDHF [31,32,33], including the inclusion of pairing, which is naturally an important ingredient for structure studies [34]. The time is now ripe for some serious comparisons based on the current generation of sophisticated codes.

Conclusion

We have outlined a few areas in which questions exist in the use of TDHF for nuclear structure calculations. Many of these relate to the comparison of TDHF to other methods. Since the recent advent of symmetry-unrestricted TDHF codes, these questions will be able to be tackled.

Figure 1 .

 1 Figure1. The total giant monopole response ("Full Response") in 16O and the response of the proton 1s state to that state alone being excited. The calculation is a continuum calculation since the edge of the box is not reached by the flux up until the final time of 1024 fm/c. This relatively small final time gives an energy resolution of ∼1.2 MeV.

Figure 2 .

 2 Figure 2. Nonlinearities induced by a strong instantaneous boost with kick size k = 0.01 fm -2 compared with a smaller kick in the linear regime

Figure 3 .

 3 Figure 3. The response of16 O to an isoscalar monopole excitation, with the meansquare radius measured out to a finite cut-off which is less than the total size of the box. The left plot show the time-dependence of the mean-square radius, and the right plot the Fourier transform.

Acknowledgements

This work was supported by the UK Science and Technology Facilities Council under grant number PP/F000596/1.


D Rudolph
email: dirk.rudolph@nuclear.lu.se

I Ragnarsson

W Reviol

C Andreoiu

M A Bentley

M P Carpenter

R J Charity

R M Clark

M Cromaz

J Ekman

C Fahlander

P Fallon

E Ideguchi

A O Macchiavelli

M N Mineva

D G Sarantites

D Seweryniak

S J Williams

Rotational bands in the semi-magic nucleus 57 28 Ni 29

Keywords: numbers: 21.60.Cs, 23.20.En, 23.20.Lv, 27.40.+z

Two rotational bands have been identified and characterized in the proton-magic N = Z + 1 nucleus 57 Ni. These bands complete the systematics of well-and superdeformed rotational bands in the light nickel isotopes starting from doubly-magic 56 Ni to 60 Ni. High-spin states in 57 Ni have been produced in the fusion-evaporation reaction 28 Si( 32 S,2p1n) 57 Ni and studied with the γ-ray detection array Gammasphere operated in conjunction with detectors for evaporated light charged particles and neutrons. The features of the rotational bands in 57 Ni are compared to those of neighbouring isotopes and interpreted by means of configurationdependent cranked Nilsson-Strutinsky calculations. The two observed high-spin bands are considered signature partners and assigned to configurations with one 1g 9/2 proton and one 1g 9/2 neutron, resulting in an unambiguous understanding of the energetically favoured signature α = -1/2 band but a somewhat less satisfactory description of the signature α = +1/2 band.

Introduction

Spectroscopic observables of doubly-magic nuclei and isotopes in their close vicinity represent benchmarks of effective nucleon-nucleon interactions, which remain key ingredients of any contemporary nuclear model. In particular medium-mass nuclei are interesting in that respect, since they are accessible by different microscopic and macroscopic-microscopic theoretical treatments. As prime examples, the doubly-magic and hence near-spherical N = Z = 20 and N = Z = 28 nuclei 40 Ca and 56 Ni have revealed coexisting strongly deformed rotational spectra already at modest excitation energies and angular momenta [1,2]. For instance, extensive large-scale shell-model calculations [2,3,4], Monte-Carlo shell-model calculations [5], complex variational calculations [6], or cranked Hartree-Fock and Hartree-Fock-Bogolyubov calculations [2] as well as the cranked Nilsson-Strutinsky approach [7] have been applied to characterize and describe shape coexistence in 56 Ni.

In addition, neutron-deficient N ∼ Z nuclei in the mass A ∼ 60 regime have been established as a region of well-and superdeformed (SD) nuclear shapes. The region is centered around the N = Z = 30 'doubly-magic superdeformed' core 60 Zn [8], owing to a significant prolate shell gap at particle number 30, which holds for a wide range of high angular momentum [9]. This gap, which is related to the particle number 28 shell gap at the 2:1 axis ratio (ε 2 = 0.6) of the harmonic oscillator [10], is characterized by having four particles in N = 4, 1g 9/2 intruder orbitals and four holes in the N = 3, 1f 7/2 extruder orbitals, which are all strongly shape driving. With this background, bands in this mass region which are built on a similar number of particle-hole excitations are generally referred to as superdeformed even though their deformations are often much smaller than ε 2 = 0.6 [8,11,12].

At intermediate spins and excitation energies, these SD bands often compete with less deformed bands, which contain a smaller number of cross-shell excitations in their configuration. In fact, the average amount of quadrupole deformation of a specific rotational A ∼ 60 band has been found to essentially scale with the sum of number of holes in the 1f 7/2 orbital and number of particles in the 1g 9/2 orbital [13]. Subsequently and somewhat different from these energetically favoured cases at or near N = Z, a plethora of rotational bands has been found in a number of N > Z nuclei, which often are less straight-forward to characterize, namely in 57 27 Co [14], 58-60 Ni [15,16,17,18,19], 61 Cu [21,22], or 61,62 Zn [21,22,23,24].

One missing link is the presence of deformed structures in 57 Ni, which due to its N = Z + 1 nature is expected to provide only a limited number of energetically favored rotational bands; these are reported and discussed here, thus closing the nickel chain from 56 Ni to 60 Ni. Near-spherical medium-spin states of 57 Ni have been studied in detail in [25,26,27]. Indeed, these studies form a proper basis for the present extension towards rotational high-spin states. Preliminary results on such a rotational band in 57 Ni have been presented in [28]. Section 2 provides information on the experimental approach and the data analysis, and the experimental results are the subject of Section 3.

The bands in 57 Ni are discussed and characterized by means of configuration-dependent Nilsson-Strutinsky calculations in Section 4.

Experiments and Data Analysis

The present study relies on data from two almost identical experiments. Details are given in [29,30,31]. A 32 S heavy-ion beam impinged with an energy of 130 MeV on isotopically enriched 0.5 mg/cm 2 28 Si targets inducing fusion-evaporation reactions. The targets were supported with ∼1 mg/cm 2 Ta or Au foils. These faced the beam and thus led to an effective mid-target beam energy of about 122 MeV. The γ-rays were detected in the Gammasphere germanium-detector array [32], which comprised 78 Ge-detectors with associated anti-Compton suppression shields. The Heavimet collimators in front of these suppression shields were removed to allow for event-by-event γ-ray calometric and fold information. This can provide additional reaction channel selectivity by so-called total-energy gating [33]. Both experiments employed the 4π CsI-array Microball [34], which is used to detect evaporated light charged particles. The Neutron Shell [35] replaced about one third of Gammasphere in forward direction to allow for the detection of evaporated neutrons.

The details of the data handling and data analysis of the two experiments have also been described earlier [29,30,31]. In brief, standard pulse-shape discrimination techniques are used to distinguish different light charged particles in Microball, mainly protons and α-particles [34,36], and to discriminate neutrons from γ-rays detected in the liquid-scintillator detectors of the Neutron Shell [35,37].

Since the directions and energies of the evaporated particles are known, an eventby-event kinematic reconstruction can be performed, which leads to a much improved Doppler correction of the emitted γ-rays, provided they are emitted after the recoiling nucleus left the thin target foil. This is not the case for most of the γ-ray transitions originating from well-or superdeformed bands, i.e. the lifetimes of the associated rotational states do not exceed the transit time of the recoils through the target (≤ 200 fs). Therefore, the corresponding peaks in the γ-ray spectra become broader than γ-ray peaks from normally deformed (ND) and usually longer-lived states. However, this additional Doppler shift for fast transitions can be accounted for by assuming an average deformation, hence quadrupole moment, of the rotational bands, which in conjunction with the modelling of the slowing-down process in the thin target foil yields revised higher velocities of the recoils at the time of γ-ray emission. The use of such a scheme sharpens the peaks belonging to rotational bands in the γ-ray spectra, while in turn peaks belonging to ND states become broader. In fact, investigating spectra with and without the additional correction provides useful hints on whether or not a γray transition belongs to a SD band, its decay-out regime, or the lower-lying ND section of the excitation scheme. For the present study, all spectra and γγ correlation matrices and γγγ correlation cubes use the refined Doppler correction. In the work of Reviol et al [28], an average quadrupole moment for one of the bands in 57 Ni has been derived by investigating these additional Doppler shift effects in some more detail. While the suggested value of Q 0 = 2.3 +0.9 -0.7 eb clearly hints towards a very large deformation of the rotational structure, the limited statistics prevented (and prevents) to deduce a more precise value, which would be necessary to distinct between different theoretical descriptions.

Judging from the yields of prominent γ-ray transitions of all residual nuclei observed in the 32 S+ 28 Si reaction and taking into account the detection efficiencies of p = 65(1)% for protons, α = 50(2)% for α particles, and n = 25(1)% for neutrons for the present experiments, the 2p1n reaction channel leading to 57 Ni is found to account for less than 2% of the total fusion-evaporation cross-section. Therefore, the detection of exactly two evaporated protons and in particular one evaporated neutron is needed to discriminate γ-ray transitions associated with 57 Ni residues against the overwhelming 'background' of γ rays originating from more abundant reaction channels, even though the statistics in the γ-ray objects used in the 57 Ni analysis is reduced by about an order of magnitude.

Major remaining contaminations in the analysis originate from the 3p1n channel 56 Co [38], which leaks into the 2p1n γ-ray object in case one proton missed detection. A considerable fraction of such contaminants can be suppressed by applying the selection method described in [33], and by eventually subtracting spectra with an additional, third proton required in coincidence. The Radware software package [39] and the tailored γray spectroscopy analysis code Tv [40], which has been developed at the University of Cologne, were used to construct the level scheme and derive the relative intensities of the γ-ray transitions.

Due to limited statistics, multipolarity assignments of transitions are based on ratios of yields R 150-97 [26]. This is the ratio of efficiency-corrected γ-ray yields measured in the three most backward oriented rings of Gammasphere (average angle θ = 150 • with respect to the beam axis) versus the corresponding yield for the central section of Gammasphere (θ = 97 • ) (cf. [26]). To ensure selectivity, coincidences are required with relevant γ-ray transitions from 57 Ni detected anywhere in the Ge-detector array (θ = 124 • ). Stretched quadrupole transitions should reveal R 150-97 ∼ 1.2-1.3, whereas stretched dipole transitions are predicted to have R 150-97 ∼ 0.8. Smaller or larger values are indicative for mixed ∆I = 1, E2/M 1 transitions. Pure non-stretched (∆I = 0) dipole transitions have values similar to quadrupole transitions.

Experimental Results

The results of the data analysis are illustrated in figure 1 and summarized in table 1. Figure 1 provides the relevant part of the decay scheme of 57 Ni, focusing on the two new high-spin rotational sequences SD1 and SD2 on the left-hand side and their decay-out. Compared with [25,26,27], the I π = (25/2 -) level at 10974 keV excitation energy has been added, and some spin-parity assignments have been possible in the band-like structure B1 in the centre of the figure (cf. The spectra in the lower part of figure 2 document the analysis of SD2. Panel (d) shows transitions in coincidence with the band members at 1746, 2120, 2500, 2959, and 3239 keV. (Note that the 1746 keV transition of SD2 is a doublet with the 1745 keV transition in structure B1.) Next to the members of SD2, including the lower-lying 1386 keV transition, the spectrum in figure 2(d) reveals peaks from decays of all relevant ND states, in particular at 968 and 1818 keV (B1), and the related peak at 3349 keV. The former peaks are members of the B1 structure, namely the decays from the 9856 keV, 23/2 -state. The latter peak connects the 13205 keV state of SD2 with just that ND level. This is proven in figure 2(e), which shows the γ-ray spectrum in coincidence with the 3349 keV link. The transitions in SD2 are seen with decreasing yield (starting at 1746 keV), the 968 and 1818 keV are pronounced, and the flux is collected again in the 1287, 1455, and 2577 keV members of the ground-state sequence. The angular distribution ratio of the 3349 keV line, R 150-97 = 0.90 (12), indicates (pure) dipole character. This affirms a spin assignment I = 25/2 for the state at 13205 keV. However, the parity of SD2 cannot be determined experimentally since the data is unable to distinguish between M1 or E1 character of the 3349-keV linking transition. The angular distribution ratios of the members on top of the 13205 keV level are all consistent with stretched quadrupole character, establishing the band up to the 25769 keV, I = 45/2 state. Between 15 and 20 MeV excitation energy, SD2 carries about 10 % of the yield of the 2577 keV ground-state transition.

Panels (a) and (b) of figure 2 focus on SD1. Panel (b) is created in a similar fashion as panel (d); it is the sum of spectra in coincidence with the band members at 1694, 2087, 2508, 2973, and 3377 keV. Though close in energy, the SD1/SD2 doublets at 2973/2959 and 2508/2500 can be resolved in the coincidence analysis. This is indicated by the vertical dashed lines in figure 2. In addition to the band members mentioned, transitions at 1306, 1614, 3610, 3696, and 4308 keV are enhanced. With an angular distribution ratio of R 150-97 = 1.32 (15), the 1306 keV line most likely continues SD1 to lower spin values. The 1614-keV line, however, is found to be a dipole transition. The subsequent decay pattern following the 1614-keV transition could not be delineated, while its relative yield in combination with the lack of coincidences with, for example, the lines at 1306 and 4308-keV ensure the placement in figure 1.

The three high-energy lines are clearly visible in the insets of panels (b) and (c). While the transitions at 3610 and 4308 link SD1 with the ND states, the peak at 3696 keV is most likely associated with SD1 but could not be placed unambiguously in the decay scheme of figure 1. Figure 2(a) represents the spectrum in coincidence with the 4308 keV line. It is found to feed the 9495 keV 23/2 -yrast state, since the spectrum shows peaks at 890, 906,1151,1924,1939, and 2042 keV next to the SD1 band members and the 1455-1287-2577 keV ground-state sequence. Furthermore, the 3610 keV line fits the energy gap between the lowest state of SD1 at 12497 keV and the 8888 keV 21/2 -level in B1 and concurs with the necessary coincidence relationships. The angular distribution ratios of the 3610 and 4308 keV lines provide values consistent with ∆I = 1, mixed E2/M 1 and E2 character, respectively, which in combination allows for a firm spin and parity assignment of SD1. Finally, SD1 comprises a yield of about 25 % of the 2577 keV transition in the excitation energy range of 15 to 20 MeV. This is a remarkably large number similar to, for example, the favoured deformed band in the neighbouring isotope 58 Cu [12]. The higher yield of SD1 over SD2 is also a clear sign for SD1 lying closer to (or likewise marking) the yrast line at very high spin in 57 Ni. This is consistent with the spin assignments of both bands based on the angular distribution ratios of the linking transitions and the absence of any other hitherto unknown intense peaks in the 57 Ni reference spectrum in figure 2(c). Finally, transitions connecting SD1 and SD2 have not been observed in the present data.

Another note concerns the possibility of prompt, discrete-energy particle decays, which are known in the A ∼ 60 region to connect rotational states in the decay-out regime with near spherical states in the respective daughter nuclei [7,12,15,17]. Taking S p = 7332 keV [41] and assuming a 1g 9/2 , = 4 proton decay of the 13205 keV level in SD2, the yrast 8 + state at 3638 keV in 56 Co [38] could be reached with Q p = 13205 -(7332 + 3638) keV = 2235 keV, a value similar to existing cases. A similar option can be estimated for the 11819 keV state at the bottom of SD2, while the lack of rather low-lying negative-parity states with matching spins in 56 Co prohibit a solid estimate for proton decay energies from states in SD1. Nevertheless, a search for proton decays from both SD1 and SD2 into low-lying excited states of 56 Co has been performed but proved negative. The phase space for α decay of low-lying states of SD1 and SD2 is considered too low compared with the known cases in 58 Ni [15,17].

Discussion

The observed SD bands of 57 Ni are interpreted from comparisons with the bands calculated in the configuration dependent cranked Nilsson-Strutinsky (CNS) model [9,42,43]. This model is based on a modified oscillator potential which is cranked around the x-axis. The total energy of the different bands is obtained from the sum of the single-particle energies of the orbitals, which are occupied in a specific configuration. This sum is calculated at the value of the rotational frequency, which corresponds to a specific spin value obtained from the sum of the expectation values of the j x operator. The configurations are defined by the filling of the different groups of orbitals, which are specified by the N -shells -or rather N rot -shells -and the signature quantum number, α. When possible, i.e. generally at small deformation, a distinction is also made between orbitals of high-j and low-j character, respectively.

The energies of the different configurations are Strutinsky renormalized [44] to a rotating liquid-drop energy based on the Lublin-Strasbourg drop model (LSD) [45]. Rigid moments of inertia are calculated from a nucleus with a diffuse surface having a radius parameter r 0 = 1.16 fm and a diffuseness parameter, a = 0.6 fm [43]. The total energies are then generally shown relative to the rotating liquid drop energy, based on the same LSD model and minimized with respect to deformation at each spin value.

In the CNS calculations the standard modified oscillator single-particle parameters are used [42] but with an increased gap at proton number Z = 28. This modification was introduced in [16] such that the crossing frequency between the bands with two and three 1f 7/2 proton holes comes at the observed spin value in 58 Ni. This is achieved by increasing the distance between the high-j (1f 7/2 ) and low-j (2p 3/2 , 1f 5/2 , and 2p 1/2 ) subshells by 0.0423 ω 0 corresponding to ∼ 0.4 MeV, while keeping their centre of mass unchanged. In a similar way as for 58 Ni, the modification of the single-particle energies has only a minor influence on the calculated bands, but it moves the crossing between the bands with two and three 1f 7/2 proton holes to somewhat higher spin values (∆I ∼ 2 ).

It is instructive to first compare the SD bands in 57 Ni with similar bands in the neighbouring Ni isotopes as presented in figure 3. A representation is chosen where the energies are plotted relative to the rotating liquid drop. The figure suggests a continuous trend from SD2 in 56 Ni via SD1 in 57 Ni to the SD bands in 58 Ni. The bands in 56 Ni and 58 Ni are labelled according to previous interpretations by the number of 1f 7/2 holes, p 1 and n 1 , and 1g 9/2 particles, p 2 and n 2 , respectively, for protons and neutrons, [p 1 p 2 , n 1 n 2 ]. It can be seen in figure 3 that in the low-spin range of the respective bands, the two additional neutrons in 58 Ni compared with 56 Ni are placed in the 1g 9/2 shell. This suggests that the yrast SD1 band in 57 Ni simply lies in between, i.e. it has one additional 1g 9/2 neutron relative to the band SD2 in 56 Ni.

The observed bands SD1 and SD2 of 57 Ni are redrawn in the upper panel of figure 4. They are compared with the calculated bands having one 1g 9/2 proton and one 1g 9/2 neutron shown in the middle panel of figure 4, relative to the same reference rotating liquid drop energy, E rld . The two lowest bands of each signature are calculated with no attempt to distinguish between high-j and low-j orbitals. However, when comparing with the single-particle orbitals shown in figure 5, it becomes possible to determine the configurations of the calculated bands in more detail. Figure 5 is drawn for protons but due to its N ∼ Z nature, the neutron orbitals of 57 Ni behave very similar. In fact, the rather stable neutron configuration is based on the large gap associated with 29 particles, which dominates the approximate frequency range of 0.5-2.0 MeV in figure 5 and which is responsible for the exclusive [21,21] band known in 58 Cu (N = Z = 29) [12]. This configuration implies that the highest 1f 7/2 orbital, which can be seen to be signature degenerate up to ω = 2.0 MeV, is empty, i.e. the most stable neutron configuration for low-lying rotational bands in 57 Ni is also , n 1 n 2 ] =, 21]. The calculated deformation for these configurations -ε 2 ∼ 0.32 at spin I ∼ 21 -is not very large. This can be understood from the fact that while there are four or five holes in the 1f 7/2 orbitals defining the underlying SD gap, there are only two particles in 1g 9/2 orbitals.

Nevertheless, the deformation is consistent with the average quadrupole deformation reported in [28].

For the 28 protons of 57 Ni and with one of them in the lowest 1g 9/2 orbital, there is the necessity for one proton hole in the orbitals of either 2p 3/2 1f 5/2 -labelled (f p) in figure 5 -or 1f 7/2 character below the Z = 29 gap. At low frequencies, the lowest energy band will have a hole in the (f p) orbitals corresponding to a [p 1 p 2 , = [21, configuration. Then, however, the 1f 7/2 and (f p) orbitals come close in energy and exchange character at ω ≈ 1.5 MeV. Consequently, at higher frequencies the lowest energy band is expected to have the hole in an orbital of 1f 7/2 character instead, which thus corresponds to a [p 1 p 2 = [31, configuration. One could also note that two bands with a hole in the signature degenerate 1f 7/2 orbital would be expected to be connected by M1 transitions at low frequencies. The absence of such transitions in 57 Ni supports the present interpretation that the SD1 and SD2 bands have a hole the (f p) orbital in the low-spin range.

The comparison between calculations and experiment in figure 4 shows that the SD1 band is very well described by the calculated lowest energy band with one 1g 9/2 proton and one 1g 9/2 neutron; the difference curve in the lower panel is almost horizontal, which means that the transition energies are nicely reproduced. The absolute value of the difference is off by approximately -1 MeV, i.e. it has a value which is within the range of the typically expected accuracy, ±1 MeV. The straightforward conclusion is that the SD1 band should be associated with the [21,21] configuration and that it gradually changes its configuration towards [31,21] at ω ≈ 1.5 MeV, i.e. close to the highest spin values observed. This configuration change receives support from the differences in the transition energies, ∆E γ . Starting from the 23/2 state at the bottom of the band, the values are ∆E γ = 388, 393, 421, 465, and 404 keV, i.e. this difference suddenly decreases at the top. This corresponds to an increasing J (2) moment of inertia and hence suggests that the highest spin state is pushed down from an interaction with another band. Such a crossing between the [21,21] and [31,21] configurations is also consistent with the general trends seen in figure 3 and discussed in [16].

The SD2 band is somewhat more difficult to describe theoretically. The parity is not determined, and therefore both positive-or negative-parity bands have to be considered. To generate energetically competitive positive-parity bands, configurations with either one or three 1g 9/2 particles need to be considered. Therefore, the lowest calculated band with one 1g 9/2 particle, namely [20,21], is included in figure 4. It is evident that this band has a very different behaviour than both the observed SD1 and SD2 bands with a strong increase relative to the reference in the spin range I = 10 -20. Interestingly, however, the 12189 keV, I = 25/2 level could be interpreted as a member of this [20,21] band. The 1614-keV transtion would then mark an E1 link between the [21,21] and the [20,21] bands in 57 Ni in close resemblance to the 1626-keV (E1) link connecting the corresponding [21,20] and [20,20] bands in 56 Ni [2,7].

Similarly, positive-parity bands having three 1g 9/2 particles show a strong decrease relative to the reference in the spin range I = 20 -30, which once more is very different from the trend of both bands observed in 57 Ni. It is thus safe to interpret also the SD2 band in 57 Ni as a negative-parity band with two 1g 9/2 particles, i.e. it relates to the bands of this type shown in the middle panel of figure 4 as well, but with the opposite signature compared with SD1. Using an assignment to the lowest band with signature α = +1/2, the difference comes out as drawn by the dashed line and filled circles in the corresponding lower panel. It is, however, evident that the calculated band, which should be labelled [21,21] at low spin, interacts with and comes close to the [31,21] band at I = 41/2. Assuming that these two bands cross, the two crossing bands will appear as drawn by dotted lines. The two crossing bands would then be labelled [21,21] and [31,21] over the full spin range. If SD2 were assigned to this [21,21] band, the difference comes out as shown by a dotted line in the lower panel of figure 4. This difference is clearly an improvement compared with the one based on the original band assignment indicated by the dashed line. However, the slope of the difference curve remains unsatisfactorally large.

Summary

In conclusion, two SD bands have been observed in the N = Z + 1 nucleus 57 Ni in the reaction 28 Si( 32 S, 2pn) by means of prompt particle-γγ and -γγγ coincidence investigations using the Gammasphere Ge-detector array in conjunction with the Microball and Neutron Shell ancillary devices. Based on cranked Nilsson-Strutinsky calculations the two bands are interpreted as signature partners having the CNS configuration [21,21] in the spin range I = 10-20. Around spin I = 20, however, the predicted [21,21] bands are expected to interact with [31,21] bands, and in particular the signature α = +1/2 branch. For a complete and thorough understanding it would therefore be of great value if the respective non-yrast sections of the two [21,21] and the two [31,21] negative-parity bands could be discriminated experimentally. This is, however, difficult with present-day instrumentation and beam-target combinations. Nevertheless, the unambigous experimental characterisation and theoretical assignment of the yrast SD band in 57 Ni provides an important link in the nickel chain for a complete assessment of superdeformation in the A ∼ 60 region. [7] (dashed lines, circles), 57 Ni (full lines, squares), and 58 Ni [16] (dashed lines, diamonds), which are interpreted as having one, two, and three 1g 9/2 particles, respectively. They are plotted relative to the rotating liquid drop energy, E rld , as a function of spin. The 58 Ni bands are labelled B1, B2 and B3 in [16] and Q4, Q5 and Q6 in [17], and with standard CNS labels (see text) at low and high spin. The arrows indicate that the configuration of the yrast 58 Ni band has two 1f 7/2 proton holes at low spin and three such holes at high spin. A similar transition occurs for the 57 Ni yrast band according to the interpretation in figure 4. Proton Routhians, i.e. proton single-particle energies in the rotating frame, e ω i , shown as a function of rotational frequency at the approximate deformation calculated for the lowest energy π(g 9/2 ) 1 ν(g 9/2 ) 1 , α = -1/2 band for spin values I = 39/2, 43/2, and 47/2. In this spin range, the band interacts with the next highest band with the same symmetry (cf. figure 4), while the deformation remains essentially constant.

Figure 1 .Figure 2 .

 12 Figure 1. Relevant part of the decay scheme of 57 Ni. Energy labels are in keV. The widths of the arrows correspond to the relative intensities of the γ rays.

Figure 3 .

 3 Figure 3. Observed rotational bands in 56 Ni[7] (dashed lines, circles), 57 Ni (full lines, squares), and 58 Ni[16] (dashed lines, diamonds), which are interpreted as having one, two, and three 1g 9/2 particles, respectively. They are plotted relative to the rotating liquid drop energy, E rld , as a function of spin. The 58 Ni bands are labelled B1, B2 and B3 in[16] and Q4, Q5 and Q6 in[17], and with standard CNS labels (see text) at low and high spin. The arrows indicate that the configuration of the yrast 58 Ni band has two 1f 7/2 proton holes at low spin and three such holes at high spin. A similar transition occurs for the 57 Ni yrast band according to the interpretation in figure4.

Figure 4 .

 4 Figure 4. The upper panel shows the observed SD bands in 57 Ni drawn as in figure 3.The middle panel provides the calculated bands with one and two 1g 9/2 particles relative to the same rotating liquid-drop reference, E rld . The difference between calculations and experiment is shown in the lower panel. See text for details.

Figure 5 .

 5 Figure5. Proton Routhians, i.e. proton single-particle energies in the rotating frame, e ω i , shown as a function of rotational frequency at the approximate deformation calculated for the lowest energy π(g 9/2 ) 1 ν(g 9/2 ) 1 , α = -1/2 band for spin values I = 39/2, 43/2, and 47/2. In this spin range, the band interacts with the next highest band with the same symmetry (cf. figure4), while the deformation remains essentially constant.

table 1

 1 Ni. It is taken in coincidence with either the 1287 or 2577 keV transition of the 11/2 -→ 7/2 -→ 3/2 -ground-state cascade of 57 Ni and the reaction channel selection described in Section 2. Next to the transitions known to belong to 57 Ni, this spectrum reveals rather intense peaks at, for instance, 1694, 2087, 2120, ∼ 2505, and ∼ 2965 keV. These peaks dominate the spectrum in figure2(c) beyond E γ = 2 MeVbesides the 2577 keV ground-state transition. They are found to form two rather intense rotational bands at high spin in 57 Ni, SD1 and SD2.

). The spectrum in panel (c) of figure 2 selects exclusively γ-ray transitions belonging to

57

Table 1 .

 1 Level energies, E x , γ-ray energies, E γ , γ-ray intensities relative to the 2577 keV ground-state transition in 57 Ni, Y γ , angular-distribution ratios, R 150-97 , and multipolarities, as well as spin-parity assignments of initial and final states, I π i and I π f . Parentheses indicate tentative assignments.

	E x	E γ	Y γ	R 150-97	multi-	I π i	I π f
	(keV)	(keV)	(%)		polarity	( )	( )
	B1
	7142(1)	723(1)	3(1)	-	∆I = 0	17/2 -	17/2 -
		1630(1) 3(1)	0.99(10)	E2/M 1	17/2 -	15/2 -
	8038(1)	585(1)	2(1)	-	∆I = 0	19/2 -	19/2 -
		896(1)	2(1)	-	E2/M 1	19/2 -	17/2 -
		1620(1) 4(1)	0.72(15)	E2/M 1	19/2 -	17/2 -
		2526(2) 2(1)	-	E2	19/2 -	15/2 -
	8888(1)	545(1)	1(1)	-	∆I = 0	21/2 -	21/2 -
		1745(2) 5(1)	1.27(9) a	E2	21/2 -	17/2 -
	9856(1)	968(1)	2(1)	1.05(19)	E2/M 1	23/2 -	21/2 -
		1818(2) 3(1)	1.55(20)	E2	23/2 -	19/2 -
	10974(1) 1119(2) 2(1)	-	(E2/M 1) (25/2 -) 23/2 -
		2084(2) 3(1)	1.25(7) a	(E2)	(25/2 -) 21/2 -
	SD1
	12497(2) 3610(3) 2(1)	0.48(11)	E2/M 1	23/2 -	21/2 -
	13803(2) 1306(1) 3(1)	1.32(15)	E2	27/2 -	23/2 -
		1614(2) 4(1)	0.83(10)	∆I = 1	27/2 -	25/2
		3946(4) 1(1)	-	E2	27/2 -	23/2 -
		4308(4) 4(1)	1.35(19)	E2	27/2 -	23/2 -
	15497(2) 1694(1) 26(2)	1.38(8)	E2	31/2 -	27/2 -
	17584(3) 2087(2) 25(2) 1.25(7) a	E2	35/2 -	31/2 -
	20092(4) 2508(3) 21(2) 1.37(7) a	E2	39/2 -	35/2 -
	23065(5) 2973(3) 16(2) 1.28(12) a	E2	43/2 -	39/2 -
	26442(7) 3377(4) 6(1)	1.68(21)	E2	47/2 -	43/2 -
	SD2
	13205(3) 1386(1) 4(1)	-	(E2)	25/2	(21/2)
		3349(3) 4(1)	0.90(12)	∆I = 1	25/2	23/2 -
	14951(4) 1746(2) 10(1) 1.27(9) a	E2	2 9 /2	25/2
	17071(4) 2120(2) 10(1) 1.69(23)	E2	3 3 /2	29/2
	19571(5) 2500(3) 9(1)	1.37(7) a	E2	3 7 /2	33/2
	22530(7) 2959(4) 6(1) 1.28(12) a	E2	4 1 /2	37/2
	25769(8) 3239(4) 2(1)	1.21(16)	E2	4 5 /2	41/2

a Doublet structure.

Acknowledgments

First of all, we would like to thank the accelerator crews and the Gammasphere support staff at Argonne and Berkeley for their supreme efforts. The target maker, Jette Agnete Sørensen, at the Niels Bohr Institute, Copenhagen, Denmark, is also warmly thanked. This work is supported in part by the Swedish Research Council and the U.S. Department of Energy under grants No. DE-AC03-76SF00098 (LBNL), DE-FG05-88ER-40406 (WU), and W-31-109-ENG38 (ANL).