Structure of collective bands in deformed nuclei from the microscopic point of view

Nuclear Physics A347(1980)99-122. @North-HollandPublishing Co., Amsterdam Not to be reproduced by photoprintor microfilmwithoutwrittenpermission

from the publisher.

STRUCTURE OF COLLECTIVE BANDS IN DEFORMED NUCLEI FROM THE MICROSCOPIC POINT OF VIEW Raymond A. Sorensen Department of Physics Carnegie-MellonUniversity Pittsburgh, Pennsylvania U.S.A.

Many body calculations with two nucleon residual interactions show that properties of rotational bands up to moderate spins can be explained by a microscopic theory consistent with many low energy nuclear properties. The prospects are good for explaining very high spin bands on the same basis. INTRODUCTION Since the spin 16-18 barrier was broken by the discovery of backbending (1) eight or nine years ago, a vast amount of experimental information has been obtained about energies and transition rates in bands to high spin in both even and odd nuclei. The corresponding theoretical efforts have moved from the original microscopic calculations of the moment of inertia by Griffin and Rich (Z), and by Nilsson and Prior (3), beyond the remarkably successful VMI phenomenologicalmodel (4), to the more complex and intriguing, but still phenomenological IBA models of Arima and Iachello (S), and the microscopic self-consistent cranking calculations being made by a number of people (6). It is difficult to evaluate the comparative success of these calculations. The strictly phenomenologicalmodels tend to fit data rather well within their limited domain (otherwisenobody would pay much attention). The VMI fits energy levels below spin lo-12% of deformed even nuclei with astonishing accuracy for reasons yet unknown. The IBA, which is beginning to be connected to an underlying particle structure, is able to discuss energies and transitions of some nuclei containing a number of interconnected collective bands. A microscopic calculation involving neutrons and protons can, on the other hand, address itself to the whole array of diverse nuclear nroperties (short of those involving meson and other particle degrees of freedom) and then one can learn whether the behavior of a rotational band in a rare earth nucleus can be explained on the same basis as the appearance of a nearby fission isomer or a hexadecapole moment. It is this understanding of a wide variety of nuclear properties on a unified basis for which we should strive. CHOICE OF THE FORCE The ultimate aim of nuclear structure theory is to develop an understanding of nuclear properties in terms of the more fundamental and elementary interactions on which they depend. The quark and gluon constituents have so far added no clarification to nuclear structure, so at best, the "fundamental" interactions are taken to be effective potentials derived in part from an understanding of the strong interactions, and in part phenomenologicallyfitted to nucleonnucleon scattering and properties of the deuteron. Gradual imnrovements over many years in the treatment of the many body problem from the Bethe, Brueckner, Goldstone expansions to those using hypernetted chains convince many experts that the calculation is under control. The failure to fit nuclear matter (5 MeV too little binding at the correct density, or too low a density by about 15% if E/A is fit) is attributed at least in part to inadequacies in the force.

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This level of error is entirely unacceptable for any meaningful discussion of nuclear band structure which is dependent in detail on deformation, single particle energies, and pairing correlations. Thus parameters of the interaction must be empirically adjusted to some features of the many body system. Since the force must be adjusted, two extreme points of view may be taken. In the first, a simple force with few parameters is chosen, whose form is taken to achieve a proper fit to a few selected many body properties. The quadrupole plus pairing force (7) is the prime example. Nilsson deformation calculations with no explicit treatment of a two body field producing force are in the same spirit. In the second, a force compatible as much as possible with two body data, is adjusted in some parameters to fit selected many body information. The various Skyrme interactions are typical of this point of view. With either approach the aim is to develop a model involving nucleons in interact ion, which can make it possible to extranolate from some nuclear properties such as radius, magic level spacings, odd-even mass differences, etc. used to fit the parameters, to other nuclear properties of more complexity such as moments of inertia and transition rates in band structures, excitations of states above the Yrast line, etc. Both of these approaches continue to have validity when handled with care. In the second method the nuclear size and shape is to some degree derived from the two body force and finer details fit to the many body properties. With the first point of view no attempt is made to explain saturation, and the calculator feels free to choose magic gaps and single particle level spacings in agreement with experimental results. The residual effective interaction is then only required to allow extrapolations in which N and Z change by a few units, along with changes in angular shape, angular momentum, etc. required to describe a Extrapolations are more reliable for a more fundamental theory, band structure. but they are also more reliable if only a relatively short extrapolation is A good fit to low energy properties must precede the claim of derequired. tailed understanding of higher spin band members. The Nilsson-Strutinsky (8) method uses no two body force at all. It is based on the idea of a self-consistent field V derived from a two body force, v, with the assumptions that : 1) the self-consistent density matrix, o, can be averaged to produce an average i and corresponding one body potential V = tr &-which is a smooth function of particle number and deformation, and that 6p = p-p is and furthermore that v can be accurately represented by a standard Wood small, Saxon or similar potential; and 2) except for small corrections of order (6 P)~ which are ignored, the energy E consists ofsmoothpart I? and the shell correction N 6E

1

=

i=l

where liquid

the E. are eigenstates drop’energy.

of

N E.

’

-(

T + v,

1

i=l

E.) 1 AVERAGED

and z? can be represented

by a standard

It might seem to be too much to expect that any effective two body interaction But since a good interaction must be adjusted to produce could give all this. and since a good Wood-Saxon does good (experimental) single particle levels, just that, point 1) is likely to be satisfied. Likewise, if the interaction has enough features to saturate and is adjusted e.g. to the size of the Pb it will probably produce liquid dron effects and satisfy point 2). In nucleus, fact direct comparisons (8) between Hartree-Fock and Nilsson-Strutinsky for some Skyrme type forces have shown that for all but light nuclei, the two give very similar results. The differences may still be large enough, however, that details like the prolate-oblate energy differences for transitional nuclei may be significantly affected by thehipherorder terms in Hartree-Fock, not present

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in the Nilsson-Strutinskyapproximation. Thus except for such finer details, any reasonably chosen residual interaction will give similar results for its field producing effects since these depend mainly on the spatial geometry of the nucleus, which a reasonable interaction will have been chosen to fit. Band structure on the other hand would seem to be more dependent on the details of the residual interaction. If there are several close configurations,the details of the interaction may determine which lies lowest, but more important, the moments of inertia depend strongly on the superfluid correlations produced by the force. And so far it has not been possible to derive these correlations either from the two nucleon data or from a parameter tuning based on one body features of the many body system. Most of the band structure calculations have used the conventional J = 0 pairing force with the force strength or the gan parameter adjusted to the emphirical odd-even mass difference. Fine tuning of parameters.is often based on the low band energies themselves. Other nuclear properties particularly sensitive to pairing such as level spacings and single-particleoccupations have been studied in great detail, and confirm the importance of pairing. However, such studies have not been able to specify with precision the pairing force matrix elements important for the band structure. Inclusion of J = 2 pairing has been found to have important effects, but pn pairing which may also be important at high spin has not been considered for heavy nuclei. MOMENT OF INERTIA AT LOW SPIN The first calculation of moments of inertia (2+ state energies of rare earth nuclei) on the basis of the usual pairing force, fit to odd-even mass differences, by Griffin and Rich (2) was a spectacular success agreeing with experiment to about 15%. A more detailed study by Nilsson and prior (3) of all the parameters entering the calculation suggested that the theoretical moment of inertia might be about 20% too low, still within the overall uncertainty of the calculation of about 20%. Both of these calculations treated the pairing correlations by the use of B.C.S. (9). It was recognized from the start by Migdal (10) that the schematic J = 0 pairing force could produce spurious effects on rotation which would not appear for a residual interaction such as a delta force which depends on the coordinate of separation of the two particles. Hamamoto (11) showed that the J = 2 Migdal pairing effect could increase the theoretical moment of inertia to its experimental value. On the other hand Meyer et al. (12) using the Migdal effective interaction (suitable for the treatment of isotope shifts) found cancelling effects which left the moment of inertia at low spin unchanged from the value obtained by Nilsson and Prior. .More recent calculations by Wakai and Faessler (13) of such modified pairing using an improved cranking formalism including number projection still leaves in doubt whether the Migdal effect can unambiguously improve the low spin moment of inertia. With or without such modifications, however, the pairing correlations have a large effect on the moment of inertia reducing it by a factor of three from the large rigid value expected in the absence of correlations. Thus the success of pairing theory is impressive regarding the moment of inertia of even rare earth nuclei, and most nuclear nhysicists believe that the qualitative features are well understood in terms of the pairing superfluid correlations. BAND PROPERTIES

AT LOW SPIN

The spectra of rotational nuclei all deviate from the J(J + 1) rule, corresponding to a smooth increase of the moment of inertia I with increasing angular

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momentum. Up to about spin 10 ti the levels are fit extremely well by VMI, which on the standard I vs. ti2 plot corresponds to a linearly increasing function. None of the cranking calculations succeed very well in reproducing this effect. All the calculations do indeed show a moment of inertia which increases with increasing angular momentum, but most show an I vs. w2 plot with I curving upwards at too low a spin value thus failing to reproduce the observed long linear region. Although VMI gives the best energy fit (up to 8 to 10 ?I), it is incapable of explaining other details without ad hoc additional assumptions. Other phenomenological models such as IBA can discuss transition rates as well as band energies to the extent that the underlying microstructure can be related to the particles making the transitions. But a theory described in terms of neutrons and protons in interaction is capable of explaining (or nredicting) a vast array of properties associated with band structure in even and odd nuclei. Also such a microUnlike a more phenomenological scopic model is capable of more vigorous testing. model for which a failure can be excused by simply saying that the particular nucleus, or level, or property is outside the scope of the model, a microscopic model should fit all properties (involving the degrees of freedom included) and any failure can only reflect an inability to perform a calculation of sufficient accuracy, or be used to say that the particular interaction, configuration space, etc. is wrong or inadequate. For even nuclei measurable quantities in the ground band include a) the change in E2 transition rate and diagonal quadrupole moment of excited states, b) the ~~~~~i~+~2PF~~~~~~~~i~~cifica11y gR, and c) the size change as reflected . All of these quantities reinforce our belief that the superfluid effects as reflected in the cranking model with pairing gives a correct description of low spin states of deformed nuclei. The microscopic calculations based on the HFB (Hartree Fock Bogolyubov) method with cranking agree that the most important cause of the increasing moment in inertia in well deformed nuclei at low spin is a decrease of the pairing correlations due to the Coriolis effects rather than a centrifugal stretching which might also have This is consistent with the obserbeen thought to be a natural explanation. vation that for these nuclei, the B(E2)‘s retain the rotational value corresponding to the O+ to 2+ transition all the way up to spin 10 to 12+ (with 8-10% Only for accuracy at best) even while the moments of inertia are increasing. some soft rotors at N = 90 are the transition rates observed to increase beyond also the rotational value with increasing spin, and for these the calculations show stretching to be of some importance. While the B(E2)‘s rule out centrifugal stretching for well deformed nuclei, the experimental evidence in favor of Coriolis antipairing is less direct and depends on the form of the residual interaction. The evidence concerns the O+ to 2+ isomer shift which shows the RMSradius of the 2+ nearly unchanged from that of the O* ground state (clearly consistent with no stretching) and in a few cases the 2+ state actually has a smaller RMSradius than the ground state. This “antistretching” is explained in a calculation (14) using Migdal’s force as a decreasing pairing correlation with a corresponding decrease in the occupation of higher lying and thus larger single particle orbits with increasing spin. This result depends in detail on the force used as well as the precise positions of the independent particle energies used in the calculation. already noted in the first calculations, and Another detail of the pairing, generally agreed upon, is the fact that the proton pairing is stronger than that of the neutrons for the rare earth nuclei. The proton odd-even mass difference is larger than that of the neutrons so that the gap Ap is larger than of the protons to the nuclear rotation An by 15 to 20%. Thus the contribution is reduced by pairing more than that of the neutrons explaining the observation that the gyromagnetic ratio gR (the ratio of charge to mass flow in the rotation) is about 20% less than Z/A.

STRUCTURE OF COLLECTIVE BANDSIN DEFORMED NUCLEI

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THE BACKBEND REGION For nearly all good rotors the moment of inertia of the yrast states increases rapidly toward the rigid value over the spin range 12+ to about 18+, in many energies actually decases involving “backbending’ in which AJ = 2 transition crease with increasing spin for a few steps. It is reasonable to assume that such a rapid energy change will correspond also to a significant change in the In structure of the yrast wave function, particularly for the backbenders. fact it is widely assumed that at the backbending there is simnly a crossing of two bands of different structure so that the one is yrast below and the other yrast above the backbend. There are numerous cases in both odd and even nuclei in which several bands with different quantum numbers can be followed over a sequence of angular momenta including band crossings. But there are still only a couple of cases in which members of both of the crossing even J positive parity bands are seen on both sides of the assumed crossing point. Thus there is little direct evidence as to whether the backbend is a single band crossing or a compression or merging of many bands. As indicated before there is ample evidence that the yrast band below the sudden rise in moment of inertia is the rotation of a deformed superfluid state. The nature of the yrast band above the backbend is widely believed to be the rotation of a deformed superfluid of A-2 particles, with the extra two neutrons rotating with maximal angular momentumin a j = 13/2 + orbit, the axis of their rotation being the same as that of the collective rotation (15). Such a pair of neutrons is called “decoupled” since the j, = 13/2 and jx = 11/2 neutron states are occupied while the time reverse pairs jx = -13/2 and -11/2 are empty. Thus these neutrons are decoupled from the pair field and make no contribution to the pairing correlations. Likewise since the collective rotation is along the x axis also, there is no coupling of the pair depending on the rate of collective rotation. This simple picture ignors the quantum fluctuations in the angular momentumdirections of the two parts (A-2 core plus 2 neutrons) making up the nucleus. What is the empirical evidence that the backbend leads to a decoupled pair for the yrast states at higher spin? In order to answer that question it is necessary to consider the other possible changes in the y-cast structure which might also be involved. The possibilities taken most seriously are 1) the decoupling of a high spin pair (particularly i 13/2 neutrons) as discussed above, the Rotational Alignment RAL effect, 2) a phase transition of all the nucleons (or at least all the neutrons) from the pairing superfluid state to a normal unpaired state, the so called Coriolis Antipairing (16) phase transition CAP, and 3) a Shape Transition involving a sudden change in the magnitude or geometry of the deformation. Of course it is also possible that the backbend occurs without any such drastic state change, or that some combination of the above changes occurs. It is easiest to distinguish the third possibility from the other two, for a sudden deformation change would be characterized by a change in B(E2) value, i.e. in the two crossing bands nicture, the two bands would have different B(E2) values. In a recent preprint from Lanchow, China by Xu Gong-ou and 7 it is argued on this basis that such a shape change occurs ~~a~~eJ~“,~;~~ f36$e 132,13dCe, and 156Er. It is more difficult to distinguish between 1) RAL and ;) CAP. In both cases a large change of configuration is involved not particularly involving deformation so that a substantial reduction of the interband transitions is exnected in the crossing bands picture, while the B(E2)‘s within the two bands could be rather similar. Likewise, whether only two particles are decoupled from the pairing field and then aligned RAL, or whether the entire neutron pair field collapses with more than two particles contributing to the angular momentum CAP, it is not clear that any measurable property such as B(E2) or Q, or u could uniquely distinguish which was the case.

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Probably the best direct evidence for CAP would be observation of the disappearance of the pairing gap. While RAL may be expected to be a crossing of the paired ground band with a band based on the aligned pair, CAP should be a convergence of many bands into the vicinity of the yrast line with increasing spin as the gap disappears. Since there are still so few cases in which the two (or more) bands are both observed at high spin this distinction is not well tested. The experimental evidence most often cited as distinguishing RAL and CAP is less direct. It is based on the fact that while CAP corresponds to a small change for many particles in many different orbits, RAL requires a pair of particles in a specific high spin orbit, neutron i 13/2 in the light rare earth region. Frank Stehens et al. suggested that if the backbending in even nuclei corresponded to the alignment of ani13/2 neutron pair, then in neighboring odd mass nuclei, those bands based on a single i 13/Z neutron quasiparticle should behave differently from all odd proton nuclei and from odd neutron nuclei with the quasiparticle in a different orbit. A number of cases were then studied in which several odd particle deformed bands were seen to backbend at high spin while those based on the neutron i 1312 orbit did not. The odd neutron is assumed to be blocking the RAL effect in that orbit thus eliminating the backbending (18). Similar arguments suggest that backbending in the neutron rich even nuclei are caused by an aligned decoupled h 9/Z proton pair. These arguments are really rather convincing, but nevertheless the situation is not quite so clear cut as one might hope. In the middle of the rare earth region there are some backbending Yb and Hf even nuclei, Here also there is backbending in a number of neighboring odd mass bands. But in this region the backbending is absent in odd neutron i 13/2 bands of l%'b and 169Hf and also absent in odd proton h Q/2 bands in 16STm and 16'Lu. Thus if the cause of backbending in this region is CAP, it follows that some odd levels can block the backbending in that case as well. If it is RAL of the neutron i 13/2 level then there is not the postulated unique relation between backbending in odd and even nuclei. An interpretation of the backbend in this region involving RAL of both neutrons in i 13/2 and protons in h 9/2 is not reasonable, as such a double RAL would occur at a much higher spin value. In a calculation by Faessler (19), the backbend in this region is attributed to neutron RAL in i 13/Z. In that calculation the failure of the odd proton h 9/2 band to backbend involves a delicate coupling of the protons to the nuclear field, which causes a shift in the neutron levels which in turn are responsible for blocking the backbend. But if this is really the case, how is one to know that the delicate effects don't happen in other parts of the rare earth region? Clearly, the only completely convincing arguments as to the nature of the yrast states above the backbend must (in the absence of data on magnetic g factors and level densities near the yrast line) rely on a comparison of data on odd bands (and excited bands in even nuclei) and theoretical microscopic calculation which fit that data as well as that of the backbending yrast line of the even nuclei. Fully self-consistent calculations such as cranked HFB with any two body force are still prohibitively expensive so that systematic calculations over the rare earth region are not possible. Even more difficult are improved calculations involving projection to good angular momentum and particle number, which have significant effects on such a detail as the critical angular momentum for backbending, band crossing, or pairing collapse. Thus we are left with a relatively small number of calculations for specific nuclei, and all these are deficient in some respects. These calculations do often show the first singular behavior on increasing spin to be a RAL decoupling of a single pair (of neutrons) rather than a total collapse of the neutron pair field.

STRUCTURE OF COLLECTIVE BANDS IN DEFOBMED NUCLEI

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The calculations which show RAL as the cause of backbending in even nuclei invariably start the rapid rise in moment of inertia at spin values which are too low. This defect, already present in the early calculations of Banerjee et al. (20) is not significantly repaired by number projection, or use of forces for the pairing in addition to the monopole forces (13). It persists in the calculations of Diebel, Mantri, and Mosel (21) submitted to this meeting who find for a Strutinsky-HFB calculation, RAL for lS6Dy, an intermediate result for ls8Dy, and only gradual alignment of several pairs with the pairing gap going smoothly to zero for 164Dy and 1648,. They stress that measurement of the high spin magnetic g factors is the clearest way to distinguish the two possibilities. No significant deformation changes occur up to spin 20. SIMPLE MODELS FOR BACKBENDING Realistic calculations, with parameters chosen to fit as well as posssible the known nuclear properties, are needed in order to be sure we understand the changes induced in the nuclear wave function by rotation. Since these calculations are so expensive, it is useful to study simple models in which it is possible to determine in detail, the dependence on various parameters pertinent to the models. A number of such calculations have been made, which are useful in determining under what circumstances a rapid change in the moment of inertia is likely. The most frequently used model involves some particles in a j shell or some other simple configurations coupled to a rotor. The first such model presentation by Szymanski and Krumlinde (22) used a two level model and showed that sharpness of the up or backbend depended on the strength of the pairing force relative to the single particle level splitting (which represented the deformation). This model could not distinguish CAP from RAL since it had only two levels and thus had no way to distinguish a particular decoupled pair--all such pairs are equivalent in the model. A single large j shell plus a rotor is able to display most of the features of a realistic calculation thought to be pertinent to backbending. This is the model used by Bengtsson, Hamamoto, and Mottelson (23) to show that in a model with a large spin two quasi-particleband crossing the quasi-particle vacuum ground state band, the sharpness of the backhending should be an oscillating function of the fermi energy (or particle number). Faessler (24) reproduced this oscillation in a more detailed model in which the interaction allowed for pair exchanges between the j shell and the core. These calculations assume the importance of a high spin decoupled pair. The likelihood of occurance of a decoupled pair, backbending, parallel bands, etc. and their dependence on the type of force and type of approximation can be studied in an even simpler model, a single high spin j shell (or j,jl for protons and neutrons) without any rotor at all. The two body forces we use include the deforming Q*Q force and the Delta and J = 0 pairing forces to produce the superfluid correlations. The two particle spectra are shown in Fig. 1. For either like particles or for an n-p interaction, the Q-Q force is attractive (or least repulsive) for the lowest several spins and the highest spins, and repulsive for intermediate spins for which the two angular momentum vectors are approximately at right angles. The j = 0 pairing force is only active for like particles and only for a J = 0 pair. For like particles the Delta force resembles the pairing force in that the J 5 0 state is by far the most strongly bound, with much less binding for the J = 2,4 etc. The n,p Delta force with the usual singlet triplet ratio is attractive for both low and high spin states. Particularly for the cases where j,.+ Ilp- j, - a, = even, it rather resembles the Q-Q force, while for j, + tp - ln - 9.,= odd it is more like monopole pairing with a large binding for the lowest spin state. It is clear that while the pairing force

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t.J::C 12 4681012ld 1234567802468

2345678

I I'234567t

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Figure 1 The Two Particle Spectrum for Q*Q and Delta Forces may simulate a short range interaction for low spin states of a many particle system, for higher spin states, particularly with both neutrons and protons present it must be highly suspect even though it continues to be widely used. NON-SELF-CONSISTED ~RTREE-FOCK The results to be presented below are for the case of a half filled j shell. The simplest approximation is the non-self-consistentNilsson like treatment in which the shape of the nucleus is set by a one-body force. We choose the usual quadrupole deformation with parameters 0 and y and cranking frequency w. (There are actually only two parameters since only the ratio w/0 and y determine the single particle wave functions and their energy sequence. The many body wave function is obtained by placing particles in the lowest states, and the energy is computed using the two body force alone, and the angular moments is taken as the expectation value for jx. A degree of self-consistency can be obtained by using the free parameter y to minimize the energy at each angular momentum. For all the forces, the system makes use of this degree of freedom moving from the axial shape at low spin to a triaxial shape at intermeidate spin values. There are, of course, no pairing correlations in this approximation. For a j = 15/Z shell (half full.)and the Q*Q force, y shifts from 0 or 60“ (axial shape) to y = 30" over the spin range from J = 6 to J = 10 and then remains at 30° up to the highest spins. For the delta or pairing forces the energy surface is flat at spin zero and moves at low spin to Y = 30'. The early appearance of triaxiality may be an unrealistic feature of this simple model, but it doesn't affect strongly the conclusions concerning the effects of different force forms, of pairing correlations, etc. For a prolate shape, the most bound orbits become highly aligned at low w. For y = 30°, as seen in Fig. 2 it is the orbits in the middle of the shell (the fermi surface for a l/2 filled shell) which are most aligned at low w, but the full alignment of a single orbit (jx = j) only occurs gradually at high spin when many particles are angular momentum aligned as much as possible, consistent with the Pauli principle.

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STRUCTUREOF COLLECTIVEBANDS IN DEFORHRDNUCLEI

-0.8

.

Occupied

Levels

for

- 1 .o Half

Full Shell = 8 Particles

0.05

0.10

w

Figure2 SingleParticleEnergiesand AngularMomenta for a CrankedDeformedj Shell of Gamma = 30" In this non-self-consistent HF treatment,the energyof the many body system is given by the expectationvalue of the two body interactionin the state with the lowestsingleparticlestatesoccupied. We use j = 15/Z half full with 8 particlescapableof producingstateswith J = 0 to J = 32 ii. Note that a single fully alignedpair would have J = 14, two such pairs would have J = 24 and three

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pairs would have J = 30. Fig. 3 shows the surprising result that the J = 0 pairing force in this approximation gives an energy spectrum with a distinct bend at J = 14, another at J = 24

‘24

’

I

I

I

I

I

Nilsson 0

4

8

12

16 20

24

28 J

Figure 3 Energy vs. J and Moment of Inertia vs. w2 for a Cranked Deformed j Shell with Pairing, Delta, and Quadrupole Forces

STRUCTURE OF COLLECTIVE BAWDS IN DEFORMED NUCLEI

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and one at J = 30. The conventional moment of inertia I vs. W* plot generated from this energy spectrum shows backbending not dissimilar from that observed experimentally. In contrast, the Q*Q force or a delta force give similar smoothly rising energy curves. There is no backbending for these forces and the only qualitative distinction between them is a slight hint of a bump at J = 24 for the delta force. The gross difference in behavior with angular momentum between the delta force and the pairing force should make one hesitate to draw firm conclusions on such a delicate matter as backbending on the basis of most of the calculations which have been made to date. These pairing force results look very much as if a band crossing with a decoupled aligned band has occured. This is not a special feature of j = 15/2; but occurs for other j's as well. But this energy vs. J curve is associated at J = 14 with a state in which the angular momentum is shared by many particles as can be seen from Fig. 2. There is nothing in the wave function that looks like an aligned pair. Thus it is all too easy to get backbending in this simplest calculation, but to obtain a true decoupled aligned pair at moderate spin the inclusion of pairing correlations is essential as we will see. SELF-CONSISTENTHARTREE-FOCK In a real Hartree-Fock calculation, the one body potential is not chosen a priori as in a Nilsson treatment, but is determined self-consistentlyfrom the two body force. In our j shell model with Q-Q forces mixed with some delta force component, the quadrupole shapes are nearly self-consistent,so that the fully self-consistentHartree-Fock calculation only produces a very small lowering of the energy compared with the previously discussed results. Likewise it is probably true that the gain in understanding or in numerical precision in going from a good Nilsson or Wood Saxon potential to a self-consistent one is not very great. This is because there is a great deal of experience with these potentials and extra flexibility e.g. hexadecapole shapes have been added as needed to fit data. This confidence in the quality of non-self-consistently determined fields cannot at this time be extended to the correlations of the pairing field particularly with increasing angular momentum, since there is much less experience and very little empirical evidence concerning the details of the pairing field and its relation to space, spin, and isospin degrees of freedom. One of the benchmarks for the moment of inertia I is the expectation that in the absence of pairing, it will have the value for rigid rotation. Griffin (25) has recently called this result into question by calculations involving the cranking of a not fully self-consistentrectangular box of particles. Although I = I rigid on the average, the moment of inertia for a specific number N of particles deviates grossly from that average even for N as big as one thousand. But no self respecting two body nuclear force would yield a rectangular one body potential, and that peculiar non-self-consistencyis probably the cause of the large fluctuations. Thus it is reasonable to consider any large deviation of the moment of inertia from the rigid value as indicative of some type of correlation not simply produced by a deformed spatial field. HARTREE-FOCK BOGOLYUBOV For HFB a half filled (six particles) j = 11/2 shell is used since this system is large enough to illustrate the features of decoupling. For this system the angular momentum goes from J = 0 to J = 18 and an aligned pair has J = 10. For the Q*Q force alone there are pairing correlations in the ground state which is also deformed and axial at J = 0. As it is cranked, the moment of inertia rises to a maximum at J = 8, by which point the pairing has vanished, and at higher spins the solution is the same as the HF solution. With the inclusion of pairing correlations the Q*Q and delta forces are really different,

B.A.

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SORENSEN

in contrast with the HF calculation for which they looked quite similar. As shown in Fig. 4, the addition of small amounts of a delta force to the Q*Q force lowers the energy more at low than at high spin, lowering the moment of sharpening the rise to J = 8 and finally producing a backbend between inertia, J = 6 and J =_ 8. For all except the strongest delta force-shown, the pairing gap vanishes by J = 8, and the J = 10 state is not a fully aligned state. This is the situation for delta force components strong enough to produce backbending which in this case would have to be associated with the CAP effect with the pairing gap going to zero at the backbend. For an even larger proportion of the delta force, the J = 10 state remains paired and becomes a true aligned pair decoupled state. For J, cranking, the density matrix for the J = 10 state has Jx = 11/Z and 9/2 fully occupied i.e. a pairing occupation parameter V = 1.0000 and J, = -11/2 and -9/2 exactly empty. Thus these two particles alone produce all the angular momentum. The remaining four particles form the paired state of maximal spherical symmetry in the remaining space with J, = ?7/2, *S/2, +3/2, *l/2 pair occupied with 50% probability for the occupation of each state, thus making no contribution to the angular momentum. Superficially this state does not look so different from the J = 10 state in the case of weaker pairing since in the later case the J, = 11/2 and 9/2 single particle states are occupied with more than 90% probability. But the decoupled pair state resulting from the stronger delta force is really of a different shape, the full alignment being made possible by the pairing to a symmetrical state of the remaining four particles. Without the residual pairing at high spin, the full pair alignment is not possible (unless the system consists of just one pair). If the short range attractive delta component of the force is strong enough to make the yrast state for J = 10 an aligned pair with the remaining particles paired to zero angular momentum, it is interesting to note that this same force has enough monopole pairing strength that the J = 0 ground state is no longer deformed as preferred by the Q.Q force, but is spherical as preferred by the delta force. This is seen from the fact that for such a strong delta component all the quasi particle energies are degenerate at J = 0, and the system cannot Of course this relation between the decoupled pair state and the be cranked. deformed ground state cannot be expected to hold for a more realistic calculation with many j shells, but it points out that a rather delicate balance is required--a pairing strength weak enough to allow the nucleus to be deformed, yet strong enough at higher spins to pair up all but two of the high j particles to a symmetric form so that the two may be decoupled from the pair field of the rest and fully aligned. Even with the many calculations which have been done,

some of which are submitted to this conference, it is difficult to say whether a two body force compatible with ground state information such as deformations and the odd-even mass difference will lead at higher J values to a decoupled aligned pair except for cases with rather low occupation of the high spin orbit for

prolate

shapes.

In their contribution to this conference Diebel et al. (21) do the deformation using triaxial Nilsson-Strutinskybut do the monopole pairing fully self-

consistently. These calculations show sudden alignment only for the case with one high spin neutron pair and more gradual alignment for the cases with two and three pairs. The Erlangen-Nurenberg group (26) discuss, in a series of contributions, a new method involving the choice of reference frame so as to best separate the internal motion and collective rotation. This new method called Specific Decoupling SDHFB, with HFB for the pairing, is calculated for an i 13/2 subshell plus axial rotor for pairing-plus-Q-Q forces and compared with exact and HFB solutions. Backbending occurs only for small particle numbers. HFB backbends at too low J values as usual. In the SDHFB, the backbend is delayed to the correct J value to agree with the exact results for the model. The use of delta instead of pairing forces also shifts the calculated backbend

111

STBUCTUBE OF COLLECTIVE BANDS IN DEFORMED NUCLEI

60

LQ 40

20 t

I

I

0.1

I

I

0.2 I

I

w2

I

I 0.

1°’

---

-22

,=

v =-CbQ

Deformed Spherical :G= 733 0

2

+ G

l

Delta Force

HFB I

I

I

I

1

t

4

6

8

IO

I2

14

Figure 4 Energy vs. 3 and Moment of Inertia vs. o2 in the HFB Approximation. The Force is the Two Body Q-Q Force Plus Delta Forces of Strength G.

J

112

R.A. SORENSEN

to higher J values, so it may be that we are approaching a time when calculations can agree with experiment in more detail. Calculations with a similar model by Ikeda et al. (27) study the alignment in terms of seniority components. For softer nuclei, bands of excitation still clearly exist, and their description in terms of models like VMI and IBM have had some remarkable success, but a rigorous connection with an underlying microscopic description in terms of protons and neutrons and their orbits is much more difficult to achieve. For well deformed nuclei the 82, 84, and even 86 (28) as well as details of the single particle energies can be measured by several methods and compared in a meaningful way with e.g. Hartree-Fock calculations. For softer nuclei the shape must be treated as a dynamical variable and the best microscopic treatment available is still the Kumar-Baranger method (29). Since it is difficult to be sure enough of the two body force and the approximations to be confident about a calculated residual pairing correlation at spin 16 fi or 26 5 it is important to have measurements of as many different properties as possible at high spin. An important advance is our increasing knowledge of different bands of plus and minus parity and even and odd spin at high spin. We are also beginning to have self-consistent calculations of excited bands based on two quasi-particle or particle hole states in the rotating frame (30). In a number of cases such multiple bands can be followed over a large angular momentumintervalin both even and odd mass nuclei and a successful comparison of theory and experiment aided by the signiture and parity quantum numbers adds to our confidence in the validity of the calculation. The number of cases for which multiple even spin-even parity bands are seen over a significant range of high spins is rather small and more experimental information would be useful. For these states the spin-parity quantum numbers do not specify the configuration which could involve collective vibrations, pairing vibrations or collapse, as well as the usually assumed two quasi-particle decoupled aligned state. Praharaj (31) in a contribution to this conference calculates KT # O+ bands in an unusual way using a self-consistent formulation with J2 cranking instead of the usual cranking with a component of J. A difficulty remaining in the formalism is the lack of a good way to extend the self-consistency into the backbending (or band crossing) region. If the backbend takes place over a significant interval of angular momentum and its increase is due to small contributions from a number of particles, model calculations (32) indicate that the self-consistent cranked HFB may be valid even in the backbending region. In themorelikely event that the rapid rise in J is due to alignment of only two particles, the self-consistent calculations may have large spurious components in the backbending region. An interesting suggestion is made by Horibata (33) to increase the stability of cranking calculations in the backbending region. Instead of one Lagrange multiplier, two are used. The single particle space is divided into a valance part of one large j orbital, and the rest of the space called the core, with the cranking term V

C

added to the Hamiltonian. The w's are adjusted to give the desired -wJ -wJ vx cx value of J = Jl + Jz and minimize the energy. The solutions have wv and wc moving together except in the backbending regions where the stable solution can have oc rather different from w . V

alternative is to assume a two level crossing model in which each band maintains its character through the crossing region. The backbending then arises from a weak interaction between the bands at J critical where the two bands will have rather different w values, perhaps related to w,,and wc above. But in the simple band crossing model as it used, the individual bands cannot be adjusted self-consistently in the backbending region. This is only a problem for crossing of bands with the same quantum numbers. The

STRUCTURE OF COLLECTIVE BANDS IN DEFORMED NUCLEI

113

VERY HIGH SPIN STATES Much of the current interest concerns the study of states of very high spin. From the experimental side this involves gamma spectroscopy with the emission of very many gamma rays. In the high spin region of interest it is not usually possible to identify in a singles spectrum, lines corresponding to discrete transitions; there is an apparent continuum of gammas. A few of these gammas do come from the region of great level density far above the yrast line and form a true high energy gamma continuum, but most of them probably arise from transitions between states not so far above the yrast line and correspond to a large number of discrete lines. The appearance of a continuum from these lines is from lack of resolution. The statistical transitionsdof the true continuum are of some interest to the study of band structure since their strength may be in competition with collective rotational transitions and thus be relevant to a determination of the extent to which band structures exist at high excitation above the yrast line (34). Much more accessible to experimental study is the region within an MeV or so of the yrast line. One of the exciting recent developments is the use of yy coincidence in the continuum region to study the possible band structure near the yrast line. Even if there are so many bands and transitions that the spectrum appears to be continuousI if there is a structure of approximately parallel bands with transitions within a band much stronger than transitions between bands, then transitions in coincidence with a transition at any selected energy will be limited to bands one of whose transitions has the selected energy. Thus only a limited number of paths are available to the coincident gammas which may then show the regularities of a band structure which is invisible in the singles spectrum. In several nuclei this sort of parallel band structure at high spin has been observed. The experiments can also locate regions of irregularity in this parallel structure and places where there are strong paths leading down to the yrast line. If the irregularitiesare so great as to put kinks in the yrast line itself, this is observed more simply by the existence of isomeric states. With the new techniques it should be reasonably possible to determine a number of quantities of interest which may be related to the microscopic underlying structure. These include: 1) the number, or density of parallel bands near the yrast line, 2) the degree to which the bands deviate from being parallel (rate of band crossings), 3) the average slope of the bands compared with the slope of the yrast line. There might, for example, be many high K bands which start at some high J at or near the yrast line and then slope up more steeply. Conversely there may be bands which start at lower spins and then terminate at or near the yrast line at high spin. There may also be states which are not organized into bands at all. Most of the calculations at very high spin have not been based on a two body force, but rather have been of the Nilsson or Nilsson-Strutinskytype. Such calculations can predict irregularitiesof the yrast line which can produce isomers. In particular, if the calculated shape is axial with the symmetry axis in the direction of the rotation axis, there is no collective rotation, and states of increasing J are produced by switching particles one or a few at a time to states of higher spin producing an irregular yrast line. If the nucleus is axial with the symmetry axis perpendicular to the rotation axis, there will be a rotational band based on that intrinsic shape; other parallel bands are not so easy to calculate and may be based on other shapes, or on single particle or collective excitations of the intrinsic state. If the shape is not axial, the single intrinsic state will correspond to a series of parallel bands at high spin whose spacing depends on the three moments of inertia, and of course there may be other bands as well based on excitations or other intrinsic states. It has been shown that under rather general conditions and in an average sense, the slope of the bands should correspond to the rigid value for the moment of

114

R.A. SORENSEN

inertia. This result does not depend on the details of the two body force, and also the shape resulting from a self-consistent Hartree-Fock calculation may not depend so much on the details of the force. But the nature of the approximately parallel bands at high spin, beyond the average character expressed by the moment (or moments) of inertia, does depend on the two body residual interaction and may be difficult to extract from the results of a one body calculation. J SHELL AT HIGH SPIN, EXACT RESULTS The single very large half full j shell is a model which can display all the features of multiple bands at high spin discussed in the preceding paragraphs, and since exact diagonalization at high spin can easily be done it is a useful model for study of the effect of the two body force. The calculation is easy since at the highest spins there are only a limited number of states at each spin no matter how large the j of the shell is. Also the results are nearly independent of the j of the shell (at large j and J). For example we will show results for a i = 23/2 shell half filled with 12 particles. The unioue state of maximum spin is the fully aligned J = 72 Ti state.- Fig. 5 shows all the states from J = J max down to J = 62 5. There are only 12 states which have J = 62. J=

62

64

66

68

70

---

AJ=

72

I E2

.a-....., Q* Q Force in

Figure 5 Energy vs. J for Exact Diagonalization of the Q'Q Force for,the High Spin States. The Solid and Dashed Lines Connect States with Large Stretched B(E2) Values (the Thickness of the Line Indicates the Strength). The Dot-Dashed Lines Show the Strong AJ = 1 B(E2) Transitions. We will also show some results for two j shells, one for protons and one for

115

STRUCTURE OF COLLECTIVE BANDSIN DEFORMED NUCLEI neutrons . In that case the number of states at each spin grows more rapidly away from J max, and we stop at J max - 6 with 29 states. Again we compare delta and Q-Q forces.

Fig. 5 shows that the Q-Q force organizes the states into very regular parallel bands with the strongest quadrupole strength for stretched E2 transitions in the bands nearest the yrast line, which is itself the lowest band. There is a large even J-odd J staggering of the yrast line. The bands above the yrast line do not all go up to the highest possible spin, but terminate at less than the maximum. The stretched cascades are accurately parallel with no suggestion of any band crossings. These are the results expected for a rotating triaxial shape modified by the limitations on angular momentumof the j shell, and are not unexpected on the basis of the relation of the Q-Q force to the group SU(3). The interband transitions are extremely weak, the B(E2)‘s being several orders of magnitude weaker than those within the bands. It is interesting to note that the non-stretched B(E2) strength for AJ = 1 is concentrated rather high in the spectrum. Fig.

J=

6 shows that the band structure

resulting

from diagonalization

of a delta

62

72

---

AJ=

Delta

1 E2

Force

in

$ I

I

I

High Spin Energies

I

I

I

I

y

Full I

Figure 6 for the Delta Force (See Fig.

Shell

I

I

5 Caption)

Again there are bands with strong stretched E2 force is entirely different. transitions connecting levels near the yrast line, but the connected levels move on and off the yrast line with a scalloped shape, and the bands are not The bands in this case must be defined as parallel but have sharp crossings.

I

116

R.A.

SORENSEN

the states connected by large B(E2) matrix elements and not by their energy. here the unmarked transitions between bands are weaker by several orders of nitude. The non-stretched E2’s with AJ = 1 are again concentrated somewhat in the spectrum.

Also maghigh

The scalloped behavior of the lowest E2 band has its cusps at J = J max - 2 = 70 fi and J = J max - 8 = 64 71. This is the same behavior just barely observable in the Hartree Fock-Nilsson calculation shown previously for a delta force. At J max = 72 all the N = 12 particles are aligned. The cusps occur at J max - 2 where all the angular momentum can be furnished by the full alignment of N - 2 = 10 particles and at J max - 8 where all the angular momentum can be furnished by the full alignment of N - 4 = 8 particles. The delta force wave functions for the cusp states show that indeed these are aligned states. It is at these same angular momenta that the sharp backbends occur in the pairing force calculation as seen in Fig. 3, i.e. where all the angular momentum can be produced But with the delta by one aligned pair, then two pairs, three pairs, etc. force, the deviation from a smooth behavior is much greater in the exact calculation than in the self-consistent approximations. This is of course expected since the deformed field approximation averages over angular momenta. Clearly in this situation the deformed field approximation is a poor guide for the existence of isomers. The exact calculation with the delta force would have yrast trap isomers at spins 64 h and 70-K since only states at higher energy have significant E2 matrix elements connecting to these states, while the deformed field approximation with the same force gives results almost indistinguishable from the smooth behavior of the Q-Q force. Figs. 7 and 8 show analogous results for the highest spin states of a two j shell one for protons and one for neutrons with each shell half full. model, The case shown has the j values of 11/2 and 15/2 and the opposite parity for the two levels. In the p,n case the Q.Q force (chosen with pp, pn, and nn components equal) always leads to smooth parallel bands, while with the delta force (chosen with a triplet to singlet ratio of 1.5) the bands are more irregular and not -The largest irregularities occur for cases where Jp + R - J, - en parallel. The reason for this behavior is easily seen from Fin. 1 wh?ch shows that = odd. the pn delta force two body matrix elements, for Jp + II - &, - II, = even, look somewhat like the Q-Q force. The angular momentum at wR.ich the energy in the band is anamolously low is at J = J max - 4. These results show that the pn correlations produced by the pn component of the residual interaction can have important effects at high spin, an effect so far omitted in all calculations. CORRELATIONS AT HIGH SPIN Are the large differences from the Q-Q spectrum caused by special (e.g. pairing) correlations produced by the delta interaction? In order to answer that question One way is simply some way must be found to characterize the w?ve functions. to calculate the overlap between the eigenstate of Q*Q and those of the delta of states on or near the yrast line connected by strong force .’ For the series B(E2)‘s as shown on Fig. 5,6 the overlap is remarkably large. For example the first excited J = J max - 10 state of the delta force has an overlap with the lowest J = J max - 10 state for Q*Q of 0.91. The smallest overlap, .69, is for the lowest J max - 8 state (which, for the delta force, is aligned). Thus for the energies are drastically the strong collective states near the yrast line, with relatively modest change in wave function. Similar overlap facehanged, tors occur between corresponding states in the collective bands in the np calculations of Figs. 7 and 8, for delta vs. Q-Q forces. The collective bands thus look rotational as far as their wave functions are It would be of interest to know if concerned for either delta or Q-Q forces. the wave functions really look as if they were angular momentum projected from a deformed state. Although a deformed state can be projected to good angular

117

STRUCTURE OF COLLECTIVE BANDS IN DEFORMED NUCLEI

J-=44

50

48

46

J=

--- A5=2 1

-A ---

Q*Q

Figure 7. High spin energies for the Q*Q force

(see

Force

I E2 E2 in

Fig. 5 caption).

118

R.A. SORENSEN

J=44

46 6

I 0

0

I

48I

I

50 -I

0

0

Figure 8. High spin energies for the delta force with neutrons and protons (see Fig. 5 caption)

I

STRUCTURB OF COLLECTIVE BANDS IN DEFORMED NUCLEI

119

momentum, there is no way to go backwards from an exact "shell model" state and determine whether it has a large intrinsic "deformed" parantage. In a recent paper Watt et al. (35) study bands in light nuclei in the SD shell with exact shell model calculations. They conclude that the resulting bands usually can be associated with an intrinsic structure characterized by the numbers of neutrons and protons occupying the subshells of the SD shell i.e. these numbers are the same or similar for all members of a band (defined in terms of energy regularity and collective transition rates). On this basis the authors conclude that some of the rotational bands observed in the SD shell should have a high spin cut off significantly lower than the total allowed by the full SD shell. Verification of such behavior is strong evidence of a reasonably good understanding of a collective band in terms of microscopic structure. It is not clear whether the termination of excited bands in heavy nuclei observed in the j shell model above will occur as it does in the SD shell, but it is possible that as the J limits of a large j shell are reached some sort of very high spin backbend might occur. For the large j shell model the intrinsic state cannot be characterized by subshell occupancies as there are no subshells. Another way to determine if a many particle high spin angular momentum eigenstate is deformed, is to study the two particle spatial correlations of the state. Since the J shell model has a single common radial wave function the many body state contains no interesting radial information, but there is angular information. The high spin states we calculate have M = J and are thus axially symmetric about the z axis. Thus the one body density p(e,g) is independent of 4. But the two body density, the probability that one particle be located at 81, $1, and the second at 02, $2 is an interesting and complicated function of these variables. Looking for example at the equator, 91 = 82 = 90", this two body correlation depends only on the difference 0, - el = $. If the nucleus is really deformed and rotating about the z axis, the nucleus can be thought of as being extended (to greater density) in, for example, the f y directions and pulled in (to smaller density) in the + x directions. This does not show up in the one body density since this shape is averaged over the asymuthal angle 4. But the two body density correlation as a function of e, the asymuthal difference between the two particles, should show a maximum at 6 of 0' and 180'. In fact for a simple quadrupole deformation it should look like P(elA2) = P(@) g cos2 0 + PO Fig. 9 shows this two body correlation for the lowest few states of J = J max - 6 for the Q-Q and delta forces. It is seen that the lowest, most collective state in each case has a smooth behavior corresponding to a deformed quadrupole shape. The only difference is that there are small deviations from the smooth behavior which look like some presence of higher multipoles in the delta force case. The higher less collective states of the same spin look less deformed as expected. Thus both in terms of wave function overlap and more detailed two body correlation shapes the Q-Q and delta lowest collective states look similar even though the energies behave very differently. These conclusions hold both for the j shell with one kind of particle, and for the two j shell calculations with protons and neutrons present. The pairing force has been omitted from this discussion since its energy spectrum is highly degenerate depending only on the seniority of.the states. Forces containing both Q*Q and delta components produce collective states at high spin whose wave functions are quite similar to the rotational states produced by either force separately. The mixtures of forces produce spectra which go smoothly from the scalloped shapes for a large delta component to the perfectly parallel bands produced by the Q*Q force. Thus the quadrupole plus short range force

120

R.A. SORENSEN

60

180

Figure 9 The two body spatial correlations for the four J = 66 states of Figs. 5 and 6. The states are listed 1, 2, 3, 4 in order of increasing excitation.

The $1and S means that the correlation pictured is for two particles of opposite spin. model for the residual interaction is probably adequate to study the behavior of the multiplicity of bands at high spin. The calculations would suggest that for a delta component strong enough to produce backbending or more specifically to produce a decoupled, aligned, state above the first backbend, with pairing still remaining for the unaligned particles, it would also be strong enough to produce significant deviations from a parallel band structure. Much work needs to be done, both experimentally, and with more realistic calculations, to determine whether the average spread of the slopes of collective high spin bands in various nuclei can be explained on the same microscopic basis as the odd-even mass difference. In particular more work is needed to determine the importance of short range n,p correlations at high spin.

STRUCTURE OF COLLECTIVE BANDS IN DEFORMED NUCLEI

SUMMARY Calculational methods seem to be sufficiently good that one can say that the properties of rotational bands of well deformed nuclei are reasonably well understood up to and through the first backbend on the basis of a microscopic model whose parameters are chosen to fit properties at low excitation. There is growing evidence that the monopole pairing force is inadequate for some purposes, but that a residual interaction containing a deforming component like Q-Q, together with a short range attractive component is probably adequate. There are still problems, the Coriolis attenuation in odd nuclei still has no clear cut explanation. At higher snins there are no obvious discrepancies with this picture, but specific interpretationsof irregular behavior particularly of even J plus parity states for which the quantum numbers give little identification are rather uncertain. Many other not so rotational bands have been studied and some are described in detail by quasi-microscopicmethods such as IBA. But until the IBA can itself be placed on a more secure microscopic foundation, it is hard to say whether or not it gives a description of the collective states on the basis of a picture which can at the same time be shown to be consistent with the wide variety of nuclear data currently available on the low energy properties of odd and even nuclei. REFERENCES

11 Johnson, A., H. Ryde and S. A. Hjorth, Nucl. Phys. Al79 (1972) 753 21 Griffin, J. J., and M. Rich, Phys. Rev. 118 (1960) 850 32(16) 3) Nilsson, S. G., and 0. Prior, K. Dan. Vidensk. Selsk. Mat.-Fys. Medd. (1961) 4) Mariscotti, M. A. J., G. Scharff-Goldhaber,and B. Buck, Phys. Rev. -178 (1969) 1869 5) Arima, A., and F. Iachello, Ann. Phys. (N.Y.) 99 (1976) 253; -111 (1978) 201 6) Goodman, A. L., Adv. Nucl. Phys. 11 (1979) 2637) Bes, D., and R. A. Sorensen, Adv.%cl. Phys. 2 (1969) 129 8) Brack, M., and P. Quentin, "Nuclear Self Consistent Fields" ed. G. Ripka and M. Parneuf, North Holland (1975) 3.53 9) Bardeen, J., L. N. Cooper, and J. R. Schrieffer, Phys. Rev. -108 (1957) 1175 10) Migdal, A. B., Nucl. Phys. 13 (1959) 655 11) Hamamato, I., Nucl. Phys. A232 (1974) 445 12) Meyer, J., J. Speth, and J. H. Vogler, Nucl. Phys. Al93 (1972) 60 13) Wakai, M., and A. Faessler, Nucl. Phys. A295 (1978) 86 14) Meyer, J. et al., Nucl. Phys. A203 (1973) 17 Nucl. Phys. Al38 (1972) 257 15) Stephens, F. S., and R. S. SimK 16) Mottelson, B. R., and J. G. Valatin, Phys. Rev. Lett. 5 (1960) 511 17) Xu Gong-Ou, and Zhang Jing-Ye, prenrint 18) Lieder, R. M., and H. Ryde, Adv. Nucl. Phys. 10 (1978) 1 19) Faessler, A., K. R. Sandhya Devi, and A. BarrEo, Nucl. Phys. A286 (1977) 101 20) Banerjee, B., H. J. Mang, and P. Ring, Nucl. Phys. A215 (1973) 366 21) Diebel, M., A. N. Mantri, and U. Mosel, "InternationalConference on Band Structure and Nuclear Dynamics", Vol. 1, contributed papers, ed. A. Goodman (1980) p. 89 22) Krumlinde, J., and Z. Syzmanski, Phys. Lett. 40B (1972) 314 23) Bengtsson, R., I. Hamamoto, and B. Mottelson,Phys. Lett. 73B (1978) 259 24) Grummer, F., K. W. Schmid, and A. Faessler, Nucl. Phys. A3K(1979) 1 Physics", 25) Griffin, J. J., "Proceedings of the Latin American School Mayaguez, Puerto Rico (1978) 26) Herald, H., M. Reinecke, H. Ruder, and G. Wunner, "InternationalConference on Band Structure and Nuclear Dynamics", Vol. 1, contributed papers, ed. A. Goodman (1980) p. 76 27) Ikeda, I., ibid., p. 63

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28) Melin, R. C., R. M. Ronningen, J. A. Nolen, Jr., G. M. Crawley, C. H. King, J. E. Finck, C. E. Bemis, Jr., ibid., p. 69 29) Gupta, J. B., ibid., p. 66 30) Mottelson, R. B., "Proceedings-Symposiumon High Spin Phenomina in Nuclei", March (1979) Argonne National Laboratory ANL/PHY-79-4 31) Praharaj, C. R., "International Conference on Band Structure and Nuclear Dynamics", vol. 1, contributed papers, ed. A. Goodman (1980) P. 72. 32) Sorensen, R. A., Nucl. Phys. A269 (1976) 301 33) Horibata,.T., "International Conference on Band Structure and Nuclear Dynamics", Vol. 1, contributed papers, ed. A. Goodman (1980) p. 95. 34) Liotta, R. J., and R. A. Sorensen, Nucl. Phys. A297 (1978) 136 35) Watt, A., D. Kelvin, and R. R. Whitehead, J. Phys. G: Nucl. Phys. 5 (1980) 35