DSpace at VNU: A particle-number conserving description of rotational correlated states

As seen hereafter, our objective in this paper is not to fit pairing strength values but rather tostudy the behavior of moments of inertia associated with a theoretical treatment of the

A particle-number conserving description

of rotational correlated states

H Laftchieva,∗, J Libertb, P Quentinc, Ha Thuy Longd

aINRNE, Bulgarian Academy of Sciences, 72 Tzarigradsko chaussee blvd., 1784 Sofia, Bulgaria

bIPN-Orsay, Université Paris XI, CNRS-IN2P3, 15 rue Georges Clémenceau, F-91406 Orsay, France

cUniversité Bordeaux I, CNRS-IN2P3, CENBG, F-33175 Gradignan Cedex, France

dDepartment of Nuclear Physics, Hanọ Vietnam National University, Hanọ University of Sciences,

334 Nguyen Trai, Hanọ, Viet Nam

Received 24 February 2010; received in revised form 23 April 2010; accepted 23 April 2010

Available online 1 May 2010

Abstract

The so-called Higher Tamm–Dancoff Approximation (HTDA) has been designed to describe

microscopi-cally correlations within a particle number conserving approach It relies upon a truncated n particle–n hole

expansion of the nuclear wavefunction, where the single particle basis is optimized self-consistently byusing the Skyrme mean field associated with the single-particle density matrix of the correlated wavefunc-tion It is applied here for the first time in a rotating frame, i.e within a self-consistent cranking approach(cranked HTDA or CHTDA) aimed at describing the collective rotational motion in well-deformed nuclei

nuclei are compared with those deduced from experimental SD sequences as well as those produced bycurrent cranked Hartree–Fock–Bogoliubov approaches under similar hypotheses

* Corresponding author Tel.: +359 878 743 489; fax: +359 2 9753619.

E-mail address: lafchiev@inrne.bas.bg (H Laftchiev).

0375-9474/$ – see front matter © 2010 Published by Elsevier B.V.

doi:10.1016/j.nuclphysa.2010.04.014

scopic framework, various descriptions of nuclear phenomena became precise enough to reach

a predictive character, and to demonstrate their ability to model fairly well the nuclear behavior.Such studies include rotational collective modes, among them, in particular, those concerningsuperdeformed (SD) bands, on which huge efforts (both experimental and theoretical) have beenfocused – see e.g a review in Ref [1] As well known, the spectroscopic properties of these levelsequences provide indeed stringent tests of dynamical approaches in such cases where rotationalmodes are reasonably well decoupled from other degrees of freedom The most refined indepen-dent quasiparticle variational approaches – the HFB (Hartree–Fock–Bogoliubov) and the RHB(Relativistic Hartree–Bogoliubov) approximations, combined with approximate projection meth-ods (to restore the broken symmetries of particle number, angular momentum, etc.) – constitutethe state of the art in the study of rotational bands in heavy nuclei They were shown in manycalculations to reproduce quantitatively the properties of SD bands, in particular in the vicinity

of A= 190 [2–5] In this region, where the SD phenomenon is observed from very low spins up

to rather high spins, the behavior of the moment of inertia as a function of the angular tum is widely influenced by the evolution of the pairing correlations Therefore any microscopicapproach able to reproduce this function relies upon three essential conditions:

momen-The first one is to make sure that it gives reasonable values of the moments of inertia at lowspins In that respect, one may single out approaches as in Ref [6] where the global rotationand quadrupole vibrations are treated on the same footing within an adiabatic approach valid atlow spins By reasonable in the present context – i.e for SD in Hg–Pb nuclei – we mean valuesthat are 10–15% greater than the experimental values as it will be discussed later Within ourmicroscopic approach, adjusting the pairing strength is the key to successfully meet that goal

As seen hereafter, our objective in this paper is not to fit pairing strength values but rather tostudy the behavior of moments of inertia associated with a theoretical treatment of the rotationalmotion We will compare below two different treatments of these modes When doing so, we willadjust their respective pairing strengths so as to start, in each of them, the rotational sequencefrom a reasonably similar starting point in terms of rotational properties As it will be shown,the adopted pairing strengths will lead us at low spins to wavefunctions having in the variousapproaches the same amount of correlations (having, of course, defined a relevant measuringstick for that)

The second prerequisite is to allow a correct handling of the so-called Coriolis anti-pairingmechanism It corresponds to the purely collective mechanism governing the decrease of pair-ing correlations with angular momentum as first discovered by Mottelson and Valatin [7] As a

common result of microscopical theories for the Yrast SD bands belonging to the A∼ 190 gion under study here, it should be noted that the spin-dependence of the moments of inertia isgenerally not associated with a change in deformation It is rather due to a variation of anotherintrinsic property, namely the amount of pairing correlations in the nuclear wavefunctions (see

re-in this context the model approach of Ref [8]) Of course another mechanism which is not lective but related with the disalignment of two paired nucleons might a priori be also at work Itappears however that this is not the dominant mechanism here

col-Finally, one should make sure that the considered theory is still valid in the low pairing regime

The latter is to be encountered at medium or high spins within a SD band in the A∼ 190 region

In that respect, the BCS or Bogoliubov quasiparticle approximations are known to be at fault,giving rise to spurious normal/superfluid transitions whenever the gap between the last occupiedand the first unoccupied single particle level increases over some critical threshold Of coursesuch transitions induce large effects on collective kinetic energies and therefore on the deducedinertia parameters (moments of inertia for what this paper is concerned with) This third point

constitutes clearly the basic motivation to develop a Routhian approach in which pairing lations are present, but where the quasiparticle approximation is avoided together with its mostundesirable effects related to the spurious spread in the particle number To that effect we havemade use of the Higher Tamm–Dancoff Approximation (HTDA) of Refs [9,10,17–19] whichexplicitly conserves the particle number.

corre-As already mentioned, the SD bands in the Hg–Pb region (particularly the Yrast ones) havebeen widely used as testing grounds for many theoretical microscopic approaches In the frame-work of the cranked HFB (CHFB) approach, calculations in this region have been performed

with the SkM* Skyrme force [20] in the particle–hole channel and seniority or δ forces in the

particle–particle hole–hole channels [21] Similar approaches have been developed ously, upon using the D1S Gogny force [3] As well known, the Gaussian form of the residualinteraction (as e.g when using a Gogny force) yields a more robust behavior of the pairing fieldwhen lowering the single particle (sp) level density near the Fermi surface than in the seniority

simultane-or delta cases As a consequence, strong variations in the moment of inertia have sometimesbeen found in calculations using calculations using the latter forces which have not been yielded

in calculations using Gaussian interactions Various attempts to cure for this unwanted behaviorhave been implemented in this context One should mention in particular the Lipkin–Nogami(LN) approximate restoration of the particle numbers [22] Its use has resulted, as expected, inmore correlated solutions (see Ref [2]) and has produced a displacement of this anomalous trend

at higher spins Later, LN or similar approximate projection techniques have been also mented in cranked HFB calculations using the Gogny force [5,23] as well as within the crankedRelativistic Hartree–Bogoliubov framework (Cr.RHB – see Ref [24]) However, being highly ap-proximate in a poorly controllable fashion, LN solutions do not necessarily improve the results

imple-of the non-projected ones comparing them with data (see for instance Fig 10.9 in Ref [23]).This feature underlines a genuine and severe limitation of this approximate projection technique

as it has been shown in quite general terms when the low-pairing regime is reached [25].Some cranked HFB+ LN calculations using a Skyrme force in the particle–hole channel andsurface-favored zero-range delta pairing interaction have been performed for192Hg in Ref [4].There, similarly to what has been obtained with the Gogny force, the calculated moments ofinertia reproduce the data rather well qualitatively, but not at all quantitatively The same istrue also for the RHB calculations of Ref [24] For instance, in all calculations using an LNapproximate projection technique, the2moment of inertia of the first SD band in192Hg exhibits

a large deviation from the data when the angular velocity ω becomes greater than 0.3 MeV

(reaching there a low pairing regime) Thus, it is fair to consider, for instance, that the Yrast SDband of192Hg still awaits for a correct theoretical description

Our HTDA calculations of Refs [9,10], being in essence similar to the traditional shell modelapproaches, give solutions which are obviously eigenstates of the number of particles They are,therefore, very well suited for the study of high angular velocity (low pairing) regime Applying

them to the Yrast SD bands of some A∼ 190 nuclei is the aim of this paper which thus constitutesthe first test of the cranked version of the HTDA approach

In previous studies one has already diagonalized a microscopic Routhian in a basis consisting

of a vacuum and particle–hole excitations over it Some of these studies [11–14] have made use

of a rather simple one body potential (proportional to r2Y20) and of a single particle space which

is limited to a single subshell Some other more recent approaches, devoted also to the study ofsuperdeformed bands in the same region as ours [15,16], have included a Nilsson model singleparticle basis, the monopole and quadrupole pairing interactions and a rather limited many-bodybasis size (typically 1000) Our physical purpose and technical difficulties are quite different

here, since we use self-consistent single particle orbitals issuing from a state of the art Skyrmeeffective interaction, using a full fledged delta residual interaction with a many-body basis whichcould reach to two orders of magnitude larger and even more.

The paper is organized as follows In Section 2, we briefly describe the cranked HTDA proach, whereas Section 3 is devoted to the description of the residual interaction, the study of thesymmetry properties of the corresponding cranked Hamiltonian and the discussion of the results

ap-of some numerical tests It also includes an assessment ap-of the ability ap-of such calculations to vincingly describe the considered phenomenon In Section 4, we present and discuss the results

con-of the cranked HTDA calculations con-of the Yrast SD bands in the190,192,194Hg and192,194,196Pbnuclei by comparing them with experimental data as well as with the corresponding cranked HFBresults using similar forces Finally, Section 5 is devoted to a summary of our results, togetherwith some conclusions and perspectives offered by this new approach

2 The cranked HTDA formalism

We will not describe here the HTDA method which has been described in the usual staticcase e.g in Ref [9] to which we refer for an extended presentation Its basic principles are to be

kept in the present Routhian approach, in which a linear constraint on the component J xof theangular momentum is merely added, writing therefore the Routhian in the general form:



where Ω is the angular velocity or the Lagrange multiplier associated with the dynamical

con-straint J x , the x-axis component of the angular moment vector, while  Kis the kinetic energy and



V a two-body term composed of the nucleon–nucleon and Coulomb interactions

Let us consider a Slater determinant|0 so far unspecified Let us call U0the one body tion of V associated with|0 We rewrite now Ras

latter is responsible mostly for pairing-type correlations, the former generates vibrational like) correlations Their inclusion in the HTDA framework has been performed elsewhere [18].The HTDA method consists in solving the static Schrödinger equation for Rwithin a highly

(RPA-truncated n-particle n-hole basis Clearly, the convergence of the solution as a function of n

will be highly contingent upon the physical relevance of the “reference” Slater determinant|0.Actually if ˆρ is the one-body density matrix associated with the correlated solution |Ψ  (in a

somewhat consistent version of HTDA as explained below) or a good approximation to it, wewill take for|0 the ground state solution of the static Schrödinger equation obtained for the one-body operator R( ˆρ) defined with the one-body reduction of  V (called U ( ˆρ)) associated with ˆρ.

With this choice, one incorporates in|0 some information on the one-body effects of thecorrelations which are present in|Ψ  It should be noted, en passant, that  U ( ˆρ) is thus different

from U0

Let us discuss now the choice made for the interactions in the p–h channel and in the p–p, h–hchannels For the former, i.e to define the mean fields U ( ˆρ) in the studied region, we have chosen

to use the SkM∗interaction [20] Indeed, during the last decades, it has been amply shown (see

for example Refs [2,4,26]) that this Skyrme force was able to reasonably describe the mean field

properties at all deformations in the A∼ 190 mass region as well as the fission properties in theactinide mass region

As already mentioned, a (zero-range) delta pairing interaction (sometimes dubbed as “a ume delta interaction”) has been used for the residual part of the Hamiltonian, namely

vol-

V res= W− W0+ 0| W |0, (7)with



W ( 1, 2)=V0

2 (1+ x  P σ ) ˆδ( r1− r2). (8)Such a density-independent version of the zero-range delta force has been proven to give inmany cases similar results as a surface-dependent version (see for instance Ref [27] in the254Nocase) but at the expense, in the latter case, of introducing more parameters One may note alsothat since we are only dealing here with neutron–neutron and proton–proton pairing correlations

(|T z | = 1) we are bound to limit ourselves here to spin singlet states so that x = −1.

In practice, the numerical task is simplified by making truncations at two levels: the order n of

the particle–hole expansion of the correlated wavefunction, and a limitation of the single particle(sp) configuration space in which one considers particle and hole states The validity of thesetruncations will be discussed in Section 3 below

To close this section, let us schematize the calculations as they have been performed Ourmethod is iterative Each iteration consists of the following steps:

a) Given a correlated solution obtained in the previous step, one determines the correspondingdensity matrix ˆρ.

b) The ground state eigensolution of the one body operator R( ˆρ) is used as a quasiparticle

vacuum|0 (vacuum for the simple particle–hole quasiparticle transformation)

c) One builds a many-body basis consisting of|0, |1p1h, |2p2h, , |npnh states where

particle and hole states are built with respect to|0 within the restricted sp-configuration space

One stops at some reasonable order n.

d) One computes within such a truncated many-body basis the matrix elements of R= R0+



V res from |0 with the replacement of V by a zero-range δ force to define  V res as discussedpreviously

e) One gets finally the ground state solution|Ψ  of  Ras:

In each of these blocks the typical size of the matrix to be diagonalized is around 20 000 (for

n up to 2) and 700 000 (for additionally restricted calculations with n up to 4, as described in

Section 3.8)

This process is initiated by some reasonable assumption for ˆρ A convergence test is

per-formed with respect to the corresponding eigensolution of R, in order to stop the iterationprocess

3 Some calculational details

3.1 Matrix elements of the residual interaction

The matrix elements of the residual interaction are only calculated for sp states belonging tothe configuration space The latter includes those sp states whose energies are in the vicinity of

the Fermi energy λ To avoid any artificial sharp cutoff energy dependence (due to the appearance

or disappearance of some single particle state into the window upon varying any continuousparameter like the deformation or the angular velocity), we have introduced, as it is customary,

a smoothing factor f (e i ) defined by a cutoff parameter X (here X= 4 MeV) and a smoothing

parameter μ (here, μ = 0.5 MeV) given by:

where V (q) is the pairing strength for the charge state q, and σ iare the usual Pauli matrices The

particular V (q)values in use here will be discussed hereafter in Section 3.5

The calculations of the matrix elements (Eq (11)) for sp-wavefunctions possessing the parity–signature symmetry, are performed using Eq (A.29) of Appendix A Then, the matrix elements

of the Routhian are calculated by using V ij kl, the sp-Routhian energies and the total HF-energy,

by means of the Wick theorem (see Appendix B or Refs [10,17])

where s i , s j , s k , s l and π i , π j , π k , π l are the signatures and parities of the sp states labeled i, j ,

k , and l respectively A discussion of these selection rules is performed in Appendix A We note,

en passant, that the choice of phases of the x-signature basis states is made (as in Refs [30,2,29])

in such a way that the expansion coefficients of the sp eigenstates of the sp-Routhian operatorbasis are real

It is to be noted that np–nh states are also eigenstates of the x-signature and parity tors (with obvious notations, σ x=i =1,A σ x (i) and π=i =1,A π(i)) Therefore, the HTDAHamiltonian matrix becomes block-diagonal with respect to four types of basis Slater determi-nants: those which have the same parity and signature as the quasivacuum|φ0, those whereonly the parity or the signature has been changed and those where both the parity and signaturehave been changed This allows us to perform a separate diagonalization in each block for eachcharge state Of course, the final result for the lowest-energy state, belongs to the block whichgives the lowest total Routhian energyR  = Ψ | R |Ψ  It defines a ground state Ψ , for which an

opera-expansion is made over Slater determinants (Eq (9)) having all the same symmetry properties.The deduced one-body ˆρ corrdensity matrix is of course block-diagonal with respect to the samesymmetries and hence, the diagonalization of the operator R( ˆρ corr )gives a new quasivacuum

|φ0 possessing the same symmetries

3.3 Characterization of the HTDA solutions

The expectation values of one body operators F such as the angular momentum projection

on the rotational axis ( J x= L x+ S x) and the quadrupole deformation moments ( Q20and Q22)operators are evaluated in terms of the one body reduced density matrix ˆρ (dropping here the

mention of “corr” in ˆρ corr) of the correlated solution|Ψ  as (with a straightforward and usual

which allows to determine, through a specific iterative process, the value of the angular velocity

ω necessary to yield a given I

3.4 Truncation schemes for practical calculations

In such a highly truncated shell-model problem, it is of paramount importance to determine

the minimal size of the np–nh basis yielding a reasonable level of convergence Four remarks are

in order in this context:

1 As in shell model calculations, it is worth noting that a large amount of coefficients in themany body state are known a priori (in particular for symmetry reasons) to be vanishing Thisleads to the introduction of numerical recipes to deal with non-zero terms only Prescriptionseliminating the storage of very weak matrix elements are also in use to minimize calculationtime and storage

2 As shown previously, the HTDA reference quasiparticle vacuum is built iteratively in such

a way that it contains the effects of the correlation on the mean field This is achieved uponusing the density matrix ˆρ associated with the correlated wavefunction to define the Skyrme

one body Routhian yielding the HTDA reference quasiparticle vacuum The importance ofsuch an optimized character of the HTDA reference quasiparticle vacuum state has beenclearly illustrated in the static calculations of Refs [9,10,17–19]

3 The cutoff parameter (see Eq (10)) involved in the actual calculation of the pairing tion matrix elements Eq (11) leads automatically to a cutoff in the many-body states takeninto account in the eigenvalue problem In practice, the exponential cut-off factor is arbitrar-ily set to be exactly zero for|λ − X| >5

interac-2μ Therefore, excitation energies of the considered

1p–1h states are limited to 2X + 5μ Similar remarks should hold for the excitation energies

of the 2p–2h excitation However, one have also cut off here the 2p–2h energies to 2X + 5μ

in order to allow the same maximum excitation energies for both type of configurations

4 In the present calculation, we have limited ourselves to 1p–1h and 2p–2h many body states.Previous HTDA calculations for time-even many body solutions [17,19] have shown thatbeyond the vacuum state, merely the particular 2p–2h states corresponding to pair trans-fers (where both the particle and hole states belong each to a some Kramers degenerate

pairs) play a significant role when the maximum value of n (defining the np–nh excitation)

is 2 However, if one raises the maximal n-value to 4, the probability of one-pair transfer

is raised at the expense of the vacuum state, the two-pair transfers, yielding a significantyet very small contribution to the correlated solution This, as a matter of fact, may play avery important role when fitting a pairing residual interaction on properties involving theFermi surface diffusivity (as odd–even mass differences or moments of inertia for instance).The question remains thus wide open to know the real importance in our present approach

of including 3p–3h or 4p–4h components in the calculations As a minimal approach, onemay try to evaluate the influence of states such as one-pair transfer plus a 1p–1h excitation,and by two-pair transfer states This is currently undertaken at a much higher computationalprice In this context, the present calculation should be considered as a first attempt yieldingprobably most of the physics, yet being not completely satisfactory in terms of a reasonableconvergence of the many body basis set

3.5 Pairing strengths and spreading around the Fermi surface

We will present in this paper the results obtained for the Yrast SD band in 190Hg,192Hg,

194Hg,192Pb,194Pb and196Pb nuclei Our primary goal is to validate our approach and compareits predictions with those of a Cranking HFB under comparable conditions and using similarinteractions However, for that purpose, we have to find, for the two different approaches in use,

realistic pairing strengths (V0protons , V0neutrons) It should be stressed that the simplistic idea ofcomparing correlation energies in HFB and HTDA is meaningless As a matter of fact, while

it is possible to define, in both cases, correlation energies, their specific definitions imply stantial differences (due to the different simplified treatments of V res in both cases) renderingirrelevant any direct comparison Our present objective, thus, will not consist in pining downsome definitive values for these parameters in both approaches but rather to make sure that wecan provide a sensible comparison of their impact on the behavior of SD bands upon increasingangular momentum To do so, we have taken the following steps:

sub-i) We have used in Cranking HFB calculations the values V0p = V n

0 = 300 MeV fm3whichyield good values for the moment of inertia of192Hg at low spins given our present sp space cutoff

conditions As alluded to earlier, the resulting values obtained for the moment of inertia (kineticand dynamical) at low spins are 10–15% greater than the corresponding data This leaves someroom for further correction due to effects of the vibrational degrees of freedom (as discussed inthe GCM+ GOA calculations using HFB wavefunctions of Ref [6]).

ii) We have chosen our cranking HTDA strengths V0n = V p

0 = 1400 MeV fm3(for a spacewhich includes 0p0h, 1p1h and 2p2h excitations and with the above discussed truncation con-ditions) in order to get roughly the same low spin moments of inertia as the cranking HFBapproach, for one of nuclei under study The choice of equal values for the strengths in the pro-ton and neutron channels is merely motivated here by practical simplicity It is of course notphysically grounded since these interactions are “effective” in the sense that they are only acting

in restricted model spaces for each charge states whose relative extensions are to be considered

as somewhat arbitrary Moreover as already said, we are concentrating this paper mainly on thevalidation of the cranking HTDA approach and the comparison between the cranking HTDA ap-proach and the cranking HFB approximation, and not on a fine tuning of the residual interactions

3.6 Stability of the solutions

The convergence of our iterative (self-consistent) HTDA process is exemplified in Fig 1

There, for a given value of the angular velocity ω, the kinetic moment of inertia and the

quadrupole moment are plotted for 192Pb as functions of the number of iterations The ing point of the iterative process corresponds to a converged cranking HF solution at the sameangular velocity

start-One may remark that both quantities are rather quickly converging even though the

conver-gence of the axial quadrupole deformation Q20requests a somewhat larger number of iterations(∼ 50) This number might have to be increased upon increasing the number of particles andholes allowed in the many body basis

3.7 Measuring the correlations

In order to compare the amount of correlations present in different approaches one has todefine some relevant measuring stick As soon as the considered wavefunction is no longer aSlater determinant the operator



is not vanishing A global measure of its departure from zero could be provided by the trace of

any positive α power of  X We note, en passant, that a non-integer power of Xis allowed due tothe non-negative definite character of the ˆρ and ˆ1 − ˆρ operators The consideration of α = 1/2 is

particularly interesting since upon defining in the canonical basis{|i} non-negative, quantities

u i and v iby

u i= 1− v2

(where ρ ii is the eigenvalue of the one body reduced density matrix ρ corresponding to its

eigen-state|i) one gets

Fig 1 Some example of numerical convergence in HTDA calculations: For the192Pb nucleus, the kinetic moment of inertiaJ (1) (lower plot) and the mass quadrupole moment Q20(upper plot) are displayed as functions of the iteration number.

As well known the product u i v iis associated in the BCS theory to non-vanishing (quasidiagonal,

i.e proportional to δ i ¯ j ) matrix element of the abnormal density matrix κ Let us mention in ticular that in the seniority force model of BCS pairing correlations, Tr X 1/2after multiplication

par-by an energy scale (namely the common value G of an “average” pairing type matrix element

i¯i|˜v|j ¯j) one gets the so-called pair condensation energy.

It is worth noting that in the HTDA case, C is to be considered as an index of the

Fermi-surface diffusivity but a priori not as a direct measure of pair transfers Indeed, if the residualinteraction and the vacuum state are even under time-reversal for instance, one cannot distinguish

the contribution to C of a given 1p–1h excitation supplemented by its time-reversed partner with

the one resulting from the corresponding pair transfer It has however been found that the latter

are vastly dominating over the former when describing HTDA correlations generated by a δ-force

in non-rotating even–even nuclei (see e.g Ref [9])

In cranked HTDA calculations the residual interaction V res ∼ ˆδ − ˆδ HF (ρ)is no longer even

under time-reversal, since ˆδ HF (ρ)is a priori defined from either a correlated wavefunction|Ψ  or

a vacuum Slater determinant|Ψ0 which breaks this symmetry in our “routhian” approach Onecannot define thus, a pair of two 1p1h states which would be connected through the time-reversal

Fig 2 Variation of the kinetic moment of inertiaJ (1) (upper plots) and of the correlation measure C= u i v i(lower

plots) as a functions of the interaction strength V k for two different values of the angular velocity ω (the small bump observed in the curves around V k= 1400 MeV fm 3 curve is due to a sudden variation of the proton sp configuration space not totally smoothed out in this case by the corrective term of Eq (10)).

operator Consequently, it is clearly meaningless to try to disentangle 1p1h and pair transfer

contributions to C This could be, at best, possible when one deals with rather low ω-values This is why the comparison of the HTDA value of C with the BCS quantity i u i v i is only

meaningful and will only be made here, in the adiabatic (low ω) regime.

3.8 On the consequences of our present limitation of the many-body wavefunction space

In Fig 2, the dependence of the kinetic moment of inertiaJ1 on the residual interaction

strength V k for a very low value of the angular velocity (Ω= 50 keV) and at a much larger

one (Ω= 300 keV) has been displayed for the SD-1 band of 194Pb As clearly seen on thisfigure, the kinetic moments of inertia reach a plateau upon increasing the strength of the residualinteraction It is noticeable that the common retained value for the neutron and proton residual

interaction strengths (V k= 1400 MeV fm3) corresponds to cases where the values ofJ1reach

almost the above mentioned plateau (closer to it at low ω values than at higher angular velocities).

As shown on the same figure, this behavior of the moments of inertia is correlated with the

corresponding variation of the Fermi surface diffusivity assessed by the quantity C This means that beyond some critical value of V k, it is more and more difficult to implement particle–holeexcitations within the limited many-body space in use This feature may lead to a somewhat

delicate phenomenological description of cases where the realistic value of the C index nears

Fig 3 Results of HTDA calculations at no spin (ω= 0) for the 194 Pb nucleus The neutron (resp proton) correlation

measure C= u i v iis displayed as function of the pairing strength in full dark lines (resp full gray lines) The full and

open triangles (resp the full and open diamonds) correspond to calculations including npnh configurations with n 2

(resp configurations with n  2 together with most important n = 4 configurations – see text for details).

or exceeds the plateau value This points out a clean-cut limitation of our calculations in theirpresent (preliminary) stage which we will further discuss now

In order to illustrate the link between this drawback and the limited size of our many-body

ba-sis, we have performed in the non-rotating case (ω= 0) HTDA calculations including also some4p–4h excitations In order to use only small amount of them, they are limited to energies up to

2X + 5μ and correspond to the 200 most probable 2p–2h excitations on top of which we have

added two more particle and holes In doing so, one retains only about 5 per cent of all the 4p–4h

excitations The results for the diffusivity parameter C are plotted in Fig 3 along with the

corre-sponding results for calculations limited as before to all 2p–2h configurations It clearly appears,

as expected, that including more many-body configurations raises the Fermi surface diffusivity

This implies that for a given value of C necessitated by some ad hoc phenomenological

prop-erty (like the valueJ IBof the Inglis–Belyaev moment of inertia for instance) one may encountersituations where the 2p–2h calculatedJ1values have a much poorer sensitivity to the residual in-

teraction strength V0than the 4p–4h corresponding results In the given example, one notices that

a reasonable value of C for the neutrons, should lie in the 8–10 range, where the slope of C with respect to V0is about twice larger in the 4p–4h than in the 2p–2h case, and thus smaller values for

V0could be chosen when 4p–4h Slater determinants are included in the configurational space.The above described behavior underlines the phenomenological importance of having a suffi-ciently large many-body basis size While limiting it as we will do in this preliminary study only

to 2p–2h excitations seems to be somewhat too limited from that point of view, it was tated and justified by the exploratory character of our present work Clearly this should, and will

necessi-be improved soon

4 Results and discussion

4.1 Definitions of the moments of inertia

The theoretical kinetic J1and dynamic J2moments of inertia have been determined here cording to their usual definitions (see for instance Refs [2] or [31]) Namely, they are completely