Pairing and Blocking in High-K Isomers : Variation of the Collective Parameter

Using the principle of additivity, the quasi-particle contribution to magnetism in high-K isomers of Lu – Re has been estimated. Based on these estimates band structure branching ratio data is used to explore the behavior of the collective contribution as the number and neutron/proton nature (Np, Nn), of the quasi-particle excitations, change. A striking systematic variation of the collective g-factor gR with the difference, Np – Nn, is revealed. Basic ideas of pairing, its quenching by quasi-particle excitation and the consequent changes to moment of inertia and collective magnetism are discussed. The new found systematic behaviour of gR opens a fresh window on these effects amenable to detailed theoretical investigation.


Introduction
Nuclei of elements between Z = 71 (Lu) and Z = 75 (Re) show strong, approximately constant, deformation and have been well described by the axially symmetric deformation model.In this model quasi-particle motion is described as giving rise to a quantum number K, the projection of the quasi-particle angular momentum J on the axis of deformation, collective motion being associated with quantum number R which describes rotation about an axis perpendicular to the deformation axis.The spin of each nuclear state is the vector sum, I = J + R. The region typically exhibits rotational bands built upon each quasi-particle state, which forms the band head.The band heads are either single quasi-particles in the deformed potential, or multi-quasi-particle states made up of combinations of single-quasi-particle states.Frequently such multiple states are isomeric, having total K values differing from neighboring states.These states, known as high-K isomers, which have well defined quasi-particle make-up, and the bands built upon them, have been the subject of intense investigation since the first was found in the early 1960's [1].Both quasiparticle and collective components of the state wavefunction contribute to the magnetic moment of the states, through g-factors g K and g R , the total moment being given by the expression (for general state of spin I in a band built upon a band-head of intrinsic spin K) (1) which for the bandhead, I = K, becomes and for band states built on the ground state (K = 0) of even-even nuclei g R I .
(3) Two properties of the band states, the E2/M1 multipole mixing ratio in transitions between states of a band having spins differing by one, and the branching ratio in the decay of a band state of spin I to lower states in the band having spin I -1 and spin I -2 also depend upon both g K and g R , through the expressions where Q 0 is the intrinsic quadrupole moment in eb and E in MeV.It has been a widespread feature of analysis of high-K isomers that their g K factors are used to assist identification of the quasi-particle make-up of the state.
To this end, empirical g Ki values of individual quasiparticles of spin K i are adopted based on magnetic moment measurements on single-quasi-particle band head (I = K i ) states of similar deformation in the same region of nuclei.The adopted empirical g Ki values then give estimates for g K of multi-quasi-particle states of total spin K = K i using the additivity relation There are few measurements of magnetic moments of high-K isomers and relatively few of the required E2/M1 mixing ratios so that usually the estimated g K values are used, with assumed values of g R and estimates of Q 0 , to compare with measured branching ratios as an aid to identification of the quasi-particle configuration.The values taken for g R have been set within a small range, between 0.25 and 0.35.Variation in g R , as the number or identity of the quasi-particles involved in the state under consideration changes within a single isotope, has been little discussed [2].Thus to date, although the valuable physics information contained in the parameter g R has been recognized, in the field of high K-isomers no systematic investigation has been made (see e.g.[3], [4] and, in particular, [5]).
Classically g R relates directly to the degree to which the rotating body carries charge, so that, for a nucleus made up of Z protons and N neutrons rotating as a uniform solid body, the simple expectation is g R = Z/(Z + N) or Z/A.However the fact that nuclear rotation is more complex was recognized from the 1960's when early studies of rotation bands in even-even nuclei yielded moments of inertia (MoI) much reduced from classical body predictions.These reductions were understood as being associated with the phenomenon of pairing, whereby nucleon pairs, treated as bosons, fall into a superfluid state and do not contribute to the MoI.The number of nucleons involved in the superfluid state is a function of the pairing strength.As the rotational frequency increases in higher members of the band, nucleon pairs are broken and the MoI rises towards the classical value [6].
In relation to g R we note that pairing affects protons and neutrons separately.Furthermore the effect of breaking a pair, as in odd-A nuclei, should block that state from participation in the pairing effect and so weaken the effect of pairing for the nucleon type involved.Thus pairing affects the contributions of protons and neutrons to the MoI and to g R .Bohr and Mottelson [7] in their landmark text remark that g R values for single quasiparticle states having a broken proton/neutron pair should be changed compared to the even-even neighbor states.They tabulate evidence, based on data then available, that odd proton single quasi-particle states have higher g R values and odd neutron single particle states lower g R values than their even-even neighbors (see [7] Table 5-14, Vol II).Further direct discussion of the variation in g R has been limited, despite the great activity in study of high-K isomers in the intervening years.Stuchbery et al. [8], focusing on properties of low-spin singlequasiparticle bands, concluded that Coriolis mixing was more important than pairing effects.The present study, by selecting high-spin single and multiple quasiparticle bands, where Coriolis effects are much reduced, allows a systematic, up to date, examination of the importance of pairing.

Data Analysis and Results
In the analysis reported in this contribution a different approach has been adopted.High-K isomers involve states having single, multiple and combinations of quasineutron and quasi-proton excitations, frequently in the same nucleus and certainly in close neighbors with very similar nuclear deformation.Thus, if good estimates can be made of the g K parameters of the band heads in these nuclei, g R values can be extracted from branching ratios and the influence of specific quasi-particle excitations on pairing and the superfluid state explored with a new perspective.
To establish the potential of this approach it is necessary to test the quality of estimates of the g K parameter.For this we have taken the assumption of additivity as expressed in (6).Comparison is made between estimates EPJ Web of Conferences based on the adopted g Ki values and experimental gfactors measured in multi-quasi-particle states in this region.The chosen g Ki values are given in Table 1, with the basis for their choice.All are derived from measured moments of states in the region.Table 2 compares the estimates derived from additivity with experimental measurements.Although making this comparison requires use of an estimate of g R (see Eq. 2), which could give rise to a circular argument, an important feature of the two expressions (2) and ( 5), allows this procedure to be followed without problems.The weight of the g R term in the moment expression (2) is less, by a factor K of the state, than the weight of the g K term, whereas, in (5) the g R and g K terms have equal weight.Since K has values of 6 and above in this study, an approximate value of g R can be used in (2) to estimate the moment and give rise to only a small additional uncertainty, never more than a few %, in the resulting moment prediction.
Table 2 Check for additivity in magnetic moments of multi-quasi-particle states.Kg K are from Eq. 6 and est from Eq. 2 with g R = 0.29 (5).exp are taken from [10] 2 shows that the assumption of additivity produces magnetic moment values which deviate from experiment by an average of less than 5%.For the remainder of this work we have used the adopted g Ki values from Table 1 in all estimates of g K .No previous attempt to use this finding to explore the behavior of g R , as reported here, has been made.
Having established a reliable method for estimating g K for multi-quasi-particle states we can use published values of the parameter |(g K -g R )/Q 0 |, which are available from branching ratio measurements in many bands built upon high-K isomeric states, to obtain values of g R .The necessary choice of sign of this parameter has been made in most cases to eliminate unrealistically high positive or negative values of g R .
The results are plotted in Fig. 1.Details of the data shown are available on request [9].The errors in g R are a compound of uncertainty in g K , and in the branching ratio.For the intrinsic quadrupole moment Q 0 we have adopted the value 7.2 eb.Other Q 0 values in the literature vary by, at most, 5% from this for all states considered, so giving a relatively small contribution to the final uncertainty in g R .The results for g R are grouped and plotted as a function of the variable N p -N n the difference between the numbers of proton and neutron quasiparticles involved in the isomer.This variable was chosen since, to first order, the effects of breaking proton Figure 1 Values of g R for multi-quasi-particle isomers plotted as a function of N p -N n .

01017-p.3
Heavy Ion Accelerator Symposium 2013 and neutron pairs are expected to cancel.Fig. 1 shows a clear systematic behavior in g R , with variation from close to zero to 0.6.As expected the lowest values are found for the greatest excess in numbers of broken neutron pairs and the highest for the most broken proton pairs.The fact that, for example at N p -N n = 0, there is considerable variation in g R , is understood since we do not expect exact cancellation between the effects of breaking different proton and neutron pairs.

Discussion
It is not the objective of this paper to offer a quantitative explanation for the variation of g R that has been revealed by the present approach.Rather we offer the results, and some comparisons with existing empirical and theoretical observations.The variation of the g R parameter is a challenging new and very direct window into the collective behavior of nucleons in these isotopes, in particular how pairing and superfluidity are modified by the blocking effects of specific proton and neutron excitations.
The first comparison utilizes the basic assumption that the relation g R = p /( p + n ), where p,n is the proton, neutron contribution to the MoI, holds good.We examine the change found in g R in states having a single quasi-particle excitation as compared to the quasi-particle vacuum ground state bands of eveneven nuclei.Differentiation of g R with respect to change in p and n yields the results for neutron excitation (8).
Here p and n are the changes produced in the MoI in the single quasi-particle nuclei as compared to their even-even (quasi-particle vacuum) neighbors in which g R(e e) is the 2 + 1 state g-factor.Relevant p,n values are found in Table 5.17 of [7] and the comparison between the predicted changes in g R and the current analysis is given in Table 3.There is good order of magnitude agreement for both single proton and single neutron excitation, with quantitative agreement for specific single proton excitation.
The second point of discussion compares predictions for g R derived from existing analyses of quasi-particle excitation energies with the experimental g R values here extracted from the bands built upon them.Several models have been used to provide values of the pairing gaps for calculation of the MoI for high-K isomers in Hf, Ta and W isotopes, e.g blocked BCS [4] and Lipkin-Nogami [11,12].Using p,n obtained from these analyses and the expression   In each panel of the figure the horizontal line is at the quasi-particle vacuum g value.The round data points are this work and the squares and diamonds are obtained using the pairing gaps.The full and dashed sloping lines are least squares fits to the respective data to guide the eye and emphasize the differences between them.Although the trend of the g R variation with N p -N n is correct in all cases, detailed agreement is seen to be poor, the pairing gap data showing too little slope, that is too small a variation of g R , by about a factor of two.

Summary
This work has used the assumption of additivity in estimation of the intrinsic quasi-particle contribution to magnetism in high-K isomers to extract information regarding the behavior of the collective contribution parameter g R .A striking systematic behaviour of g R has been found, showing wider variation than previously appreciated, for large excess of either proton or neutron quasi-particles.The behavior of g R is closely connected to the question of pairing and the superfluid state of protons and neutrons in nuclei, offering a sensitive new window on this phenomenon previously known in association with reduced moments of inertia.Simple ideas have been shown to give broadly satisfactory qualitative consistency between effects found in the g R parameter as in the moment of inertia for single quasi-particle excitation.Broader comparison with more detailed theoretical analyses based on various pairing models has been found not to reproduce the g R results well.It is hoped that this work will stimulate new activity in several directions.It is clear that in many cases better precision in experimental spectroscopic studies would improve our knowledge of the variation of g R .More and precise direct magnetic moment measurements of high-K isomers would be of great value.However it would appear that existing data should be enough to encourage more theoretical work to reconcile the g R and moment of inertia results.
Valuable discussions with J.R Stone, P.M.Walker and C.R.Bingham are gratefully acknowledged.The research was supported by the US DoE Office of Science.

Figure 2 g 4 EPJ
Figure 2 g R obtained in this work as a function of the number of qps N p , N n , in comparison with model calculations[4],[11],[12].Comparison with experimental g R values produced by the present analysis in the same Hf, Ta and W isomers is

Table 3 .
Comparison between changes in g R , estimated from 177,179Hf ___________________________________________________ depends on choice of sign of [g K -g R ]