Nuclear Physics A 845 (2010) 33–57 www.elsevier.com/locate/nuclphysa A particle-number conserving description of rotational correlated states H Laftchiev a,∗ , J Libert b , P Quentin c , Ha Thuy Long d a INRNE, Bulgarian Academy of Sciences, 72 Tzarigradsko chaussee blvd., 1784 Sofia, Bulgaria b IPN-Orsay, Université Paris XI, CNRS-IN2P3, 15 rue Georges Clémenceau, F-91406 Orsay, France c Université Bordeaux I, CNRS-IN2P3, CENBG, F-33175 Gradignan Cedex, France d Department of Nuclear Physics, Hanoï Vietnam National University, Hanoï University of Sciences, 334 Nguyen Trai, Hanoï, Viet Nam Received 24 February 2010; received in revised form 23 April 2010; accepted 23 April 2010 Available online May 2010 Abstract The so-called Higher Tamm–Dancoff Approximation (HTDA) has been designed to describe microscopically 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 by using the Skyrme mean field associated with the single-particle density matrix of the correlated wavefunction 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 Moments of inertia predicted by cranked HTDA in the Yrast superdeformed (SD) bands of some A ∼ 190 nuclei are compared with those deduced from experimental SD sequences as well as those produced by current cranked Hartree–Fock–Bogoliubov approaches under similar hypotheses © 2010 Published by Elsevier B.V Keywords: Microscopic mean field; Collective nuclear rotation; Pairing correlations; Nuclear shell model; Tamm–Dancoff approximation Introduction The theoretical study of nuclear structure using effective phenomenological nucleon–nucleon interactions has met during the past decades years with numerous successes Within this micro* Corresponding author Tel.: +359 878 743 489; fax: +359 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 34 H Laftchiev et al / Nuclear Physics A 845 (2010) 33–57 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 concerning superdeformed (SD) bands, on which huge efforts (both experimental and theoretical) have been focused – see e.g a review in Ref [1] As well known, the spectroscopic properties of these level sequences provide indeed stringent tests of dynamical approaches in such cases where rotational modes are reasonably well decoupled from other degrees of freedom The most refined independent quasiparticle variational approaches – the HFB (Hartree–Fock–Bogoliubov) and the RHB (Relativistic Hartree–Bogoliubov) approximations, combined with approximate projection methods (to restore the broken symmetries of particle number, angular momentum, etc.) – constitute the state of the art in the study of rotational bands in heavy nuclei They were shown in many calculations 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 momentum is widely influenced by the evolution of the pairing correlations Therefore any microscopic approach able to reproduce this function relies upon three essential conditions: The first one is to make sure that it gives reasonable values of the moments of inertia at low spins In that respect, one may single out approaches as in Ref [6] where the global rotation and quadrupole vibrations are treated on the same footing within an adiabatic approach valid at low spins By reasonable in the present context – i.e for SD in Hg–Pb nuclei – we mean values that are 10–15% greater than the experimental values as it will be discussed later Within our microscopic 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 to study the behavior of moments of inertia associated with a theoretical treatment of the rotational motion We will compare below two different treatments of these modes When doing so, we will adjust their respective pairing strengths so as to start, in each of them, the rotational sequence from 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 various approaches the same amount of correlations (having, of course, defined a relevant measuring stick for that) The second prerequisite is to allow a correct handling of the so-called Coriolis anti-pairing mechanism It corresponds to the purely collective mechanism governing the decrease of pairing 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 region under study here, it should be noted that the spin-dependence of the moments of inertia is generally not associated with a change in deformation It is rather due to a variation of another intrinsic property, namely the amount of pairing correlations in the nuclear wavefunctions (see in this context the model approach of Ref [8]) Of course another mechanism which is not collective but related with the disalignment of two paired nucleons might a priori be also at work It appears however that this is not the dominant mechanism here 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 occupied and the first unoccupied single particle level increases over some critical threshold Of course such transitions induce large effects on collective kinetic energies and therefore on the deduced inertia parameters (moments of inertia for what this paper is concerned with) This third point H Laftchiev et al / Nuclear Physics A 845 (2010) 33–57 35 constitutes clearly the basic motivation to develop a Routhian approach in which pairing correlations are present, but where the quasiparticle approximation is avoided together with its most undesirable effects related to the spurious spread in the particle number To that effect we have made use of the Higher Tamm–Dancoff Approximation (HTDA) of Refs [9,10,17–19] which explicitly conserves the particle number As already mentioned, the SD bands in the Hg–Pb region (particularly the Yrast ones) have been widely used as testing grounds for many theoretical microscopic approaches In the framework 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 simultaneously, upon using the D1S Gogny force [3] As well known, the Gaussian form of the residual interaction (as e.g when using a Gogny force) yields a more robust behavior of the pairing field when lowering the single particle (sp) level density near the Fermi surface than in the seniority or delta cases As a consequence, strong variations in the moment of inertia have sometimes been 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 behavior have 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, in more 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 implemented in cranked HFB calculations using the Gogny force [5,23] as well as within the cranked Relativistic Hartree–Bogoliubov framework (Cr.RHB – see Ref [24]) However, being highly approximate in a poorly controllable fashion, LN solutions not necessarily improve the results 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 and surface-favored zero-range delta pairing interaction have been performed for 192 Hg in Ref [4] There, similarly to what has been obtained with the Gogny force, the calculated moments of inertia reproduce the data rather well qualitatively, but not at all quantitatively The same is true also for the RHB calculations of Ref [24] For instance, in all calculations using an LN approximate projection technique, the moment of inertia of the first SD band in 192 Hg 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 SD band of 192 Hg still awaits for a correct theoretical description Our HTDA calculations of Refs [9,10], being in essence similar to the traditional shell model approaches, 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 constitutes the 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 r Y20 ) and of a single particle space which is limited to a single subshell Some other more recent approaches, devoted also to the study of superdeformed bands in the same region as ours [15,16], have included a Nilsson model single particle basis, the monopole and quadrupole pairing interactions and a rather limited many-body basis size (typically 1000) Our physical purpose and technical difficulties are quite different 36 H Laftchiev et al / Nuclear Physics A 845 (2010) 33–57 here, since we use self-consistent single particle orbitals issuing from a state of the art Skyrme effective interaction, using a full fledged delta residual interaction with a many-body basis which could 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 approach, whereas Section is devoted to the description of the residual interaction, the study of the symmetry properties of the corresponding cranked Hamiltonian and the discussion of the results of some numerical tests It also includes an assessment of the ability of such calculations to convincingly describe the considered phenomenon In Section 4, we present and discuss the results of the cranked HTDA calculations of the Yrast SD bands in the 190,192,194 Hg and 192,194,196 Pb nuclei by comparing them with experimental data as well as with the corresponding cranked HFB results using similar forces Finally, Section is devoted to a summary of our results, together with some conclusions and perspectives offered by this new approach The cranked HTDA formalism We will not describe here the HTDA method which has been described in the usual static case 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 Jx of the angular momentum is merely added, writing therefore the Routhian in the general form: R = K + V − Ω Jx (1) where Ω is the angular velocity or the Lagrange multiplier associated with the dynamical constraint Jx , the x-axis component of the angular moment vector, while K is 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 U0 the one body reduction of V associated with |0 We rewrite now R as R = R0 + Vres (2) where the one-body operator R0 is defined as R0 = K + U0 − Ω Jx − 0|V |0 (3) so that 0|R0 |0 = 0|R|0 (4) and where the residual interaction Vres is defined as Vres = V − U0 + 0|V |0 (5) so that 0|Vres |0 = (6) It is to be noted that the above presentation of Vres overlooks the density dependence of the Skyrme interaction in use Going beyond such a simplification (made here only for the purpose of a clear presentation) would not have any practical consequences in view of the approximate treatment of Vres discussed below Considering formally a multipole–multipole expansion of Vres we will in the following ignore its low multipole part and mock up the rest, as usual, by a two-body delta interaction While the H Laftchiev et al / Nuclear Physics A 845 (2010) 33–57 37 latter is responsible mostly for pairing-type correlations, the former generates vibrational (RPAlike) correlations Their inclusion in the HTDA framework has been performed elsewhere [18] The HTDA method consists in solving the static Schrödinger equation for R within a highly 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 ρˆ 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, we will take for |0 the ground state solution of the static Schrödinger equation obtained for the onebody operator R(ρ) ˆ defined with the one-body reduction of V (called U (ρ)) ˆ associated with ρ ˆ With this choice, one incorporates in |0 some information on the one-body effects of the correlations which are present in |Ψ It should be noted, en passant, that U (ρ) ˆ 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–h channels For the former, i.e to define the mean fields U (ρ) ˆ 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 the actinide mass region As already mentioned, a (zero-range) delta pairing interaction (sometimes dubbed as “a volume delta interaction”) has been used for the residual part of the Hamiltonian, namely Vres = W − W0 + 0|W |0 , (7) with V0 ˆ − r2 ) (1 + x Pσ )δ(r (8) Such a density-independent version of the zero-range delta force has been proven to give in many cases similar results as a surface-dependent version (see for instance Ref [27] in the 254 No case) but at the expense, in the latter case, of introducing more parameters One may note also that since we are only dealing here with neutron–neutron and proton–proton pairing correlations (|Tz | = 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 these truncations will be discussed in Section below To close this section, let us schematize the calculations as they have been performed Our method is iterative Each iteration consists of the following steps: a) Given a correlated solution obtained in the previous step, one determines the corresponding density matrix ρ ˆ b) The ground state eigensolution of the one body operator R(ρ) ˆ 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 + Vres from |0 with the replacement of V by a zero-range δ force to define Vres as discussed previously W (1, 2) = 38 H Laftchiev et al / Nuclear Physics A 845 (2010) 33–57 e) One gets finally the ground state solution |Ψ of R as: 1p1h |Ψ = χ|0p0h + χi 2p2h |1p1hi + i χi |2p2hi + · · · (9) i In order to get this solution one relies on a standard Lanczös algorithm In practice, we have made use of the numerical code written by B.N Parlett and D.S Scott [28] available as an open source The energy matrix is diagonal by blocks defined by the retained symmetries (see below) 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 ρ ˆ A convergence test is performed with respect to the corresponding eigensolution of R, in order to stop the iteration process 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 to the 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 continuous parameter like the deformation or the angular velocity), we have introduced, as it is customary, a smoothing factor f (ei ) defined by a cutoff parameter X (here X = MeV) and a smoothing parameter μ (here, μ = 0.5 MeV) given by: f (ei ) = + exp − X μ + exp (ei −λ)−X μ (10) The matrix elements of the residual interaction (for the charge state q ) take thus the form [26]: pair Vij kl = V (q) ij ||(1 − σ1 σ2 )δ(r1 − r2 )|kl f (ei )f (ej )f (ek )f (el ), (11) where V (q) is the pairing strength for the charge state q, and σi are 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 Vij kl , the sp-Routhian energies and the total HF-energy, by means of the Wick theorem (see Appendix B or Refs [10,17]) 3.2 Self-consistent symmetries It is well known (see for instance Refs [2,29,30]) that the HF cranking Skyrme Hamiltonian (with the angular velocity vector aligned on the Ox axis) preserves the parity and x-signature symmetries The corresponding sp spectra are also parity–signature symmetrical The selection rules for the zero-range volume pairing interaction are thus: pair Vij kl = 0, when si sj sk sl = or πi πj πk πl = 1, (12) H Laftchiev et al / Nuclear Physics A 845 (2010) 33–57 39 where si , sj , sk , sl and πi , πj , πk , πl 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 operator basis are real It is to be noted that np–nh states are also eigenstates of the x-signature and parity operators (with obvious notations, σx = i=1,A σx (i) and π = i=1,A π(i)) Therefore, the HTDA Hamiltonian matrix becomes block-diagonal with respect to four types of basis Slater determinants: those which have the same parity and signature as the quasivacuum |φ0 , those where only the parity or the signature has been changed and those where both the parity and signature have been changed This allows us to perform a separate diagonalization in each block for each charge state Of course, the final result for the lowest-energy state, belongs to the block which gives the lowest total Routhian energy R = Ψ |R|Ψ It defines a ground state Ψ , for which an expansion is made over Slater determinants (Eq (9)) having all the same symmetry properties The deduced one-body ρˆ corr density matrix is of course block-diagonal with respect to the same symmetries and hence, the diagonalization of the operator R(ρˆ 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 (Jx = Lx + Sx ) and the quadrupole deformation moments (Q20 and Q22 ) operators are evaluated in terms of the one body reduced density matrix ρˆ (dropping here the mention of “corr” in ρˆ corr ) of the correlated solution |Ψ as (with a straightforward and usual notation): F = fij ρj i (13) ij The expectation value Jx is related, as usual, to the total angular momentum I, through the standard semiclassical assumption of a global rotation around the x-axis by Jx = I (I + 1) (14) 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: As in shell model calculations, it is worth noting that a large amount of coefficients in the many body state are known a priori (in particular for symmetry reasons) to be vanishing This leads to the introduction of numerical recipes to deal with non-zero terms only Prescriptions eliminating the storage of very weak matrix elements are also in use to minimize calculation time and storage 40 H Laftchiev et al / Nuclear Physics A 845 (2010) 33–57 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 upon using the density matrix ρˆ associated with the correlated wavefunction to define the Skyrme one body Routhian yielding the HTDA reference quasiparticle vacuum The importance of such an optimized character of the HTDA reference quasiparticle vacuum state has been clearly illustrated in the static calculations of Refs [9,10,17–19] The cutoff parameter (see Eq (10)) involved in the actual calculation of the pairing interaction matrix elements Eq (11) leads automatically to a cutoff in the many-body states taken into account in the eigenvalue problem In practice, the exponential cut-off factor is arbitrarily set to be exactly zero for |λ − X| > 52 μ 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 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 that beyond the vacuum state, merely the particular 2p–2h states corresponding to pair transfers (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 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 significant yet very small contribution to the correlated solution This, as a matter of fact, may play a very important role when fitting a pairing residual interaction on properties involving the Fermi 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, one may 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 computational price In this context, the present calculation should be considered as a first attempt yielding probably most of the physics, yet being not completely satisfactory in terms of a reasonable convergence 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 190 Hg, 192 Hg, and 196 Pb nuclei Our primary goal is to validate our approach and compare its predictions with those of a Cranking HFB under comparable conditions and using similar interactions However, for that purpose, we have to find, for the two different approaches in use, protons , V0neutrons ) It should be stressed that the simplistic idea of realistic pairing strengths (V0 comparing 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 substantial differences (due to the different simplified treatments of V res in both cases) rendering irrelevant any direct comparison Our present objective, thus, will not consist in pining down some definitive values for these parameters in both approaches but rather to make sure that we can provide a sensible comparison of their impact on the behavior of SD bands upon increasing angular momentum To so, we have taken the following steps: p i) We have used in Cranking HFB calculations the values V0 = V0n = 300 MeV fm3 which yield good values for the moment of inertia of 192 Hg at low spins given our present sp space cutoff 194 Hg, 192 Pb, 194 Pb H Laftchiev et al / Nuclear Physics A 845 (2010) 33–57 41 conditions As alluded to earlier, the resulting values obtained for the moment of inertia (kinetic and dynamical) at low spins are 10–15% greater than the corresponding data This leaves some room for further correction due to effects of the vibrational degrees of freedom (as discussed in the GCM + GOA calculations using HFB wavefunctions of Ref [6]) p ii) We have chosen our cranking HTDA strengths V0n = V0 = 1400 MeV fm3 (for a space which includes 0p0h, 1p1h and 2p2h excitations and with the above discussed truncation conditions) in order to get roughly the same low spin moments of inertia as the cranking HFB approach, for one of nuclei under study The choice of equal values for the strengths in the proton and neutron channels is merely motivated here by practical simplicity It is of course not physically 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 the validation of the cranking HTDA approach and the comparison between the cranking HTDA approach 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 There, for a given value of the angular velocity ω, the kinetic moment of inertia and the quadrupole moment are plotted for 192 Pb as functions of the number of iterations The starting point of the iterative process corresponds to a converged cranking HF solution at the same angular velocity One may remark that both quantities are rather quickly converging even though the convergence of the axial quadrupole deformation Q20 requests a somewhat larger number of iterations (∼ 50) This number might have to be increased upon increasing the number of particles and holes 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 to define some relevant measuring stick As soon as the considered wavefunction is no longer a Slater determinant the operator X = ρˆ − ρˆ (15) 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 X is allowed due to the non-negative definite character of the ρˆ and 1ˆ − ρˆ operators The consideration of α = 1/2 is particularly interesting since upon defining in the canonical basis {|i } non-negative, quantities ui and vi by ui = − vi2 , vi = √ ρii (16) (where ρii is the eigenvalue of the one body reduced density matrix ρ corresponding to its eigenstate |i ) one gets C = Tr X = ui vi i (17) 42 H Laftchiev et al / Nuclear Physics A 845 (2010) 33–57 Fig Some example of numerical convergence in HTDA calculations: For the 192 Pb nucleus, the kinetic moment of inertia J (1) (lower plot) and the mass quadrupole moment Q20 (upper plot) are displayed as functions of the iteration number As well known the product ui vi is associated in the BCS theory to non-vanishing (quasidiagonal, i.e proportional to δi j¯ ) matrix element of the abnormal density matrix κ Let us mention in particular that in the seniority force model of BCS pairing correlations, Tr X 1/2 after multiplication by an energy scale (namely the common value G of an “average” pairing type matrix element ¯ v|j i i| ˜ 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 Fermisurface diffusivity but a priori not as a direct measure of pair transfers Indeed, if the residual interaction 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 ∼ δˆ − δˆHF (ρ) is no longer even under time-reversal, since δˆHF (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 One cannot define thus, a pair of two 1p1h states which would be connected through the time-reversal H Laftchiev et al / Nuclear Physics A 845 (2010) 33–57 43 Fig Variation of the kinetic moment of inertia J (1) (upper plots) and of the correlation measure C = ui vi (lower plots) as a functions of the interaction strength Vk for two different values of the angular velocity ω (the small bump observed in the curves around Vk = 1400 MeV fm3 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 ui vi 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 inertia J on the residual interaction strength Vk 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 194 Pb As clearly seen on this figure, the kinetic moments of inertia reach a plateau upon increasing the strength of the residual interaction It is noticeable that the common retained value for the neutron and proton residual interaction strengths (Vk = 1400 MeV fm3 ) corresponds to cases where the values of J reach 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 Vk , it is more and more difficult to implement particle–hole excitations 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 44 H Laftchiev et al / Nuclear Physics A 845 (2010) 33–57 Fig Results of HTDA calculations at no spin (ω = 0) for the 194 Pb nucleus The neutron (resp proton) correlation measure C = ui vi is 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 (resp configurations with n together with most important n = configurations – see text for details) or exceeds the plateau value This points out a clean-cut limitation of our calculations in their present (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 basis, we have performed in the non-rotating case (ω = 0) HTDA calculations including also some 4p–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 per cent of all the 4p–4h excitations The results for the diffusivity parameter C are plotted in Fig along with the corresponding 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 property (like the value J IB of the Inglis–Belyaev moment of inertia for instance) one may encounter situations where the 2p–2h calculated J values have a much poorer sensitivity to the residual interaction strength V0 than 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 V0 is about twice larger in the 4p–4h than in the 2p–2h case, and thus smaller values for V0 could be chosen when 4p–4h Slater determinants are included in the configurational space The above described behavior underlines the phenomenological importance of having a sufficiently large many-body basis size While limiting it as we will in this preliminary study only to 2p–2h excitations seems to be somewhat too limited from that point of view, it was necessitated and justified by the exploratory character of our present work Clearly this should, and will be improved soon Results and discussion 4.1 Definitions of the moments of inertia The theoretical kinetic J and dynamic J moments of inertia have been determined here according to their usual definitions (see for instance Refs [2] or [31]) Namely, they are completely H Laftchiev et al / Nuclear Physics A 845 (2010) 33–57 45 determined from the calculated function Jx of the angular velocity ω through the standard formulae : J1 = Jx ω and J = d Jx dω (18) In practice, the J moment of inertia is obtained directly for each calculation corresponding to a given ωi value of the angular velocity through the evaluation of Jx i When the whole set of calculations is completed, a three point derivation over the calculational mesh is performed on the function Jx considered as a function of ω in order to get the J moment On the other hand, experimental data to be compared with these two quantities are extracted from the SD energy sequences available in particular in the systematic compilation of Ref [1], through the following relations (with a usual notation): ωexp = Eγ+ (I ) + Eγ− (I ) (MeV), √ I (I + 1) Jexp (I ) = + (I ) = + and Jexp − Eγ (I ) + Eγ (I ) Eγ (I ) − Eγ− (I ) (19) (20) (in units of h¯ /MeV) They relate the angular velocity and moment of inertia at each angular momentum I with the transition energies above and below the considered level which is observed at the energy E(I ) Namely, one has Eγ+ (I ) = E(I + 2) − E(I ) and Eγ− (I ) = E(I ) − E(I − 2) A remark about our calculations of moments of inertia is worth mentioning The set of formulae (18) does not involve at all the total energy This choice turns out to be very important It is not motivated by any concern about the accuracy or convergence of the energy of our calculations As a matter of fact, it results from a particular aspect of our current HTDA Hamiltonian The replacement of the residual interaction as deduced from the Skyrme effective Hamiltonian in use in the particle–hole channel, by a delta force introduces some spurious dependence of the HTDA Hamiltonian in terms of the chosen quasiparticle vacuum (see e.g the discussion of the isomeric state energy in 178 Hf in Ref [9]) In our case, the Hamiltonian varies slightly e.g with the considered angular momentum As a result, evaluating the moments of inertia from transition energies would request a difficult and somewhat arbitrary correction from the spurious energy shift generated by the above mentioned lack of consistency Thus, the choice made to evaluate the moments of inertia is such that this current weak point of our HTDA approach does not affect directly our results on the J and J moments 4.2 Presentation and discussion of our results We display in Fig for the Yrast SD bands of the six isotopes considered in this study (190,192,194 Hg and 192,194,196 Pb), the kinetic moments of inertia J , as functions of the angular velocity ω for the two HFB and HTDA cranking approaches These calculated values are compared with experimental results The HFB results yield a rather smooth behavior in reasonable agreement with the data at low angular velocities At higher angular velocities however, they are at variance with the experimental ones These discrepancies, which are due to sudden variations of the pairing properties as discussed below, not appear in the cranked HTDA results which exhibit a regular behavior in good qualitative agreement with the data up to the higher experimentally known spin values One may note, in this context, that the jumps in the HTDA kinetic moment for the 192 Pb values will be explained later in terms of sudden switches in the 46 H Laftchiev et al / Nuclear Physics A 845 (2010) 33–57 Fig For 190,192,194 Hg (upper plots) and 192,194,196 Pb (lower plots) experimental (dots), HFB (full gray line and open circles) and HTDA (full black line and triangles) are displayed as functions of the angular velocity ω quadrupole deformation Moreover, it is worth recalling at this point that, as above justified, the residual interaction strength has been voluntarily chosen such that calculated values of J (and similarly J ) moments of inertia, in the vanishing angular velocity limit are overestimated by about 10–15 per cent The dynamical moments of inertia for the first SD bands of the studied Hg (and Pb) isotopes are displayed in Fig (and Fig respectively) as functions of the angular velocity ω It is found, as expected, that upon varying ω the J moments are strongly affected by the amount of correlations This is apparent from the display of the quantity C of Eq (17) This is in particular the case for the HFB results since there, whenever the sp level density at the Fermi surface diminishes below some critical values, the pairing correlations break down This happens for the protons in all three studied isotopes of Hg for ω 0.2 MeV This results in a sudden rise of the moments of inertia The same phenomenon is observed for the 194,196 Pb isotopes (we reserve the discussion of the more complicated 192 Pb case for a further consideration) Interestingly enough, even though in these two isotopes, the proton correlations not vanish, one finds that the jumps in the moments of inertia are even more abrupt than what has been observed in the Hg isotopes when the C values were vanishing However, contrarily to the latter case, in the considered Pb isotopes one experiences a sudden drop of C This underlines the fact, that the key factor for a jump in the HFB rotational mass parameter is the derivative of the pairing gaps, as it has been clearly evidenced both analytically and numerically in the adiabatic limit discussed in Ref [32] The HTDA results, exhibiting merely a smooth decrease of C, not yield such accidents in the moments of inertia This pattern is quite in line with available experimental data However, one should note that the strong experimental increase of J for 190 Hg, at large ω, is not reproduced by our HTDA calculations A very important remark is appropriate at this point It concerns the low angular velocity part of the variation of J (and also of J ) moments Their HTDA calculated values appear to yield a slope slightly too weak as compared with the experimental trend, especially in the case of the Hg isotopes This is most probably related with the above mentioned deficiency of our calculations to reproduce, within our currently too restricted H Laftchiev et al / Nuclear Physics A 845 (2010) 33–57 47 Fig HFB (full lines and open symbols), HTDA (full lines and full symbols) and experimental (full dots) results in the 190,192,194 Hg SD band In the upper plots (resp medium) (resp lower plots), the moment of inertia J (2) (resp the neutron and proton correlation measures C = ui vi – see central plots for connecting symbols and functions) (resp the quadrupole moment variation for each distribution k, Qk (ω) = Qk (ω) − Qk (ω = 0)) is displayed as a function of the angular velocity ω many body wavefunction space, the correct diffusivity of the Fermi surface in these isotopes at vanishing values of ω This is taken care only in an approximate fashion by enlarging artificially the residual interaction strength Of course, other reasons for this discrepancy could be advocated as a not so good reproduction of the sp properties with the Skyrme force in use However, it could be arguably asserted that in the particular considered states of the even–even nuclei under study the spectra not seem to be so sensitive to the details of particular sp orbitals (which are not so badly reproduced otherwise anyway with the interaction in use) but to a collective response to the centrifugal forces of the Coriolis anti-pairing type, as amply discussed above In all nuclei but the 192 Pb (discussed below), the quadrupole deformation of the HTDA solutions are varying very weakly with ω The related HFB pattern is more complicated reflecting to some extent the variations of the amount of pairing correlations In more general terms, the Q20 quadrupole moment as well as the diffusivity parameter C may be used as good markers for band crossing or pair alignment whenever one observes sudden changes in their otherwise smooth behavior As can be seen in Fig 6, their abrupt variations for the SD bands calculated using HFB for the 194,196 Pb nuclei at ω ≈ 300 keV could correspond to a proton pair breaking The sharing into various types of many body configurations in the correlated cranked HTDA solution is shown as a function of ω in Figs and for the calculated Hg and Pb SD bands 48 H Laftchiev et al / Nuclear Physics A 845 (2010) 33–57 Fig Same as Fig for 192,194,196 Pb nuclei To interpret these results, one may first recall that for vanishing angular velocities, previous HTDA results, Refs [9,10,18,19], have shown that almost all the correlations are built through a particular class of 2p–2h states, namely those corresponding to transfers of pairs of Kramers degenerate sp states, thus retrieving in a particle number conserving approach the basic assumption of the Cooper pair approach When the rotation sets in, these pairs will gradually disappear (as explained within the so-called Mottelson–Valatin (Ref [7]) Coriolis anti-pairing mechanism analogous to what is observed with a magnetic field in type-1 supra-conductivity) The corresponding strength (corresponding to 2p–2h states thus) will then decrease to be distributed, within our limited many-body wavefunction space, partly into 1p–1h states, partly into the vacuum state For large enough angular velocities now, the disappearance of counter-rotating intrinsic vortical modes (see e.g Refs [8,34,35] ) will lead to a classical mode of global rotation which may be adequately described by the Routhian approximation, yielding thus an almost pure vacuum state The latter is proposed to offer an explanation for the observed maximum of the occupancies of 1p–1h states, going from zero to zero while increasing at small ω values Needless to say, this does not provide any clue as to why these maxima occur at the particular ω values (around 0.3 MeV) obtained in our calculations Let us describe briefly now the patterns obtained for the 192 Pb nucleus At the transition spin value of 8h¯ the Yrast states change their deformation towards a higher values, thus reducing accordingly the angular velocity (to obtain similar angular momenta) Consistently with what has been above discussed, this yields an increase in the 2p–2h occupancies and correlatively a decrease in the vacuum probabilities The reverse phenomenon is observed at the highest spin H Laftchiev et al / Nuclear Physics A 845 (2010) 33–57 49 Fig For 190,192,194 Hg nuclei, the HTDA abundances for neutrons (resp protons) are displayed in full black lines (resp full gray lines) as functions of the angular velocity ω Upper plots refer to 2p2h (triangles) and 0p0h (squares) abundances whereas lower plots refer to 1p1h ones (circles) Fig Same as Fig for 192,194,196 Pb values where the opposite switch of bands is obtained Such behavior is described when finding the equilibrium states for each spin by the minimization process of the self-consistent CHTDA approach, as it is demonstrated by the HTDA(a&b) curves in Fig By using axial deformation constraints on Qˆ 20 one could follow these composing two bands (HTDA(a) and HTDA(b)) beyond their Yrast values The quadrupole deformations obtained here are compatible with Wilson et al [33] A similar behavior is not found by the CHFB calculations due to configuration mixing between states with different particle number 50 H Laftchiev et al / Nuclear Physics A 845 (2010) 33–57 Fig Band mixing in 192 Pb SD1 sequences In the upper (resp lower) plots, the HTDA(a), HTDA(b) and HTDA(a&b) values (see text for definitions) of the moment of inertia J (1) (resp J (2) ) are compared with experimental ones as functions of the angular velocity ω The weakness of 1p–1h occupancies over all, is a direct consequence of the fact that our residual interaction is of a zero range type As well known, it is therefore not suited to produce RPA correlations which are mostly produced by low multipole components of the residual interaction Note also that to correctly assess the collective 1p–1h excitation of the RPA type one should of course couple the neutron and proton excitations This is not done here Assessing the amount of vibrational correlations in SD states would be very useful, in particular, as already mentioned, to describe adequately moments of inertia which are partly contingent on them This has been done in a Bohr collective Hamiltonian approach in Ref [6] It could be done within the HTDA framework according to the methods discussed in the seminal work of Ref [18] Conclusions and perspectives This paper has presented fully microscopic cranking calculations which explicitly conserves from the onset the particle numbers The HTDA model has been shown to provide a tractable approach to that However, as pointed out all along this paper, some approximations have been found to be necessary They concern the limited analytical form of the residual interaction and the restriction of the many-body wavefunction space up to 2p2h excitations Nevertheless, the current approach has proven to be rather effective in curing some important deficiencies of cur- H Laftchiev et al / Nuclear Physics A 845 (2010) 33–57 51 rent quasiparticle description in particular at spins leading to a weak pairing regime Moreover, it has allowed for a precise treatment of some spectroscopic properties specific to a given isotope (as e.g in the 192 Pb case) which are not, thus, spuriously smeared out among many neighboring nuclei Some of the deficiencies of HFB calculations could be cured by projection (approximate or exact) on exact particle number states Our method while obviously better to the projection after variation approach, is equivalent in principle to an exact projection before variation The latter would also need very costly computational efforts In practice, the superiority of one approach over the other should depend on the quality of the many body basis on which the solution of the secular equation is to be solved, in our case, and on the relevance of the variational ensemble of projected wavefunctions, in the other case In our opinion, a perspective way for theoretical study of rotational collectivity of heavy nuclei is also the projected shell model described in Ref [36] The new approach discussed here opens a very large area of investigations In particular, it offers a theoretical framework in which even, odd–odd and odd–even nuclei could be described in a consistent manner As already mentioned vibrational correlations (of the RPA type) around an equilibrium deformation could also be considered within the same framework This would require however, to improve the residual interaction in use to include its low multipole part In our opinion, the most important task to be achieved would be to enlarge the many body wavefunction space in order to improve the description of pairing correlations Such a work is currently undertaken Acknowledgements Part of this work has been funded through the agreement #12533 between the Bulgarian Academy of Sciences (BAS-Bulgaria) on one hand and the Centre National de la Recherche Scientifique (CNRS-France) on the other hand They are gratefully acknowledged Also the contract #F-1502/05 of the Bulgarian Scientific Fund has provided support for this research work We appreciate the technical help and fruitful discussions with A Minkova, A Korichi, T Kutsarova, D Tonev, P Petkov and E Stefanova about experimental data and results Appendix A Calculation of the residual interaction matrix-elements in the case of parity–signature symmetry A.1 The zero range delta parity interaction matrix elements The operator of the two body zero range delta parity interaction is: − σ1 σ2 δ(r1 − r2 ), (A.1) where r1 , r2 are the coordinates of the particles, and σ1 , σ2 are the spin projections on the x-axis, and V0τ is a scalar parameter for the strength of the interaction for the given isospin τ In order to express it in its second quantization form one must find the matrix elements Vij kl , where in our case the indices correspond to single particle states |ϕi , |ϕj , |ϕk , |ϕl with parity–signature symmetry In the most general case these elements could be written as: V pair = V0τ d r1 d r2 ϕi∗ (r1 σ1 )ϕj∗ (r2 σ2 ) Vij kl = V0τ σ1 σ2 =± 12 × − σ1 σ2 δ(r1 − r2 )ϕk (r1 σ1 )ϕl (r2 σ2 ) (A.2) 52 H Laftchiev et al / Nuclear Physics A 845 (2010) 33–57 In the antisymmetrized matrix elements the spin part ( 1−σ41 σ2 ) of the operator disappears because {ϕk (r1 σ1 )ϕl (r2 σ2 ) − ϕl (r1 σ1 )ϕk (r2 σ2 )} is zero for triplets, hence d r1 d r2 ϕi∗ (r1 σ1 )ϕj∗ (r2 σ2 ) δ(r1 − r2 ) Vij kl = V0τ σ1 σ2 =± 12 × ϕk (r1 σ1 )ϕl (r2 σ2 ) − ϕl (r1 σ1 )ϕk (r2 σ2 ) , (A.3) where the sp-functions are composed of two parts with opposite spins: ϕi (rσ ) = ϕi+ (r)χ + + Si ϕi− (r)χ − , (A.4) where χ + and χ − being the spin-functions: χ + = σ |+ 12 , χ − = σ |− 12 and where Si is the eigenvalue (±1) associated with the signature operator Π23 , namely: Π23 ϕi (rσ ) = Si ϕi (rσ ) (A.5) For the matrix element (Eq (A.3)) and using Eq (A.4) one gets: σ1 =+ 12 σ2 =− 12 Vij kl = Vij kl σ1 =+ 12 σ2 =− 12 Vij kl σ1 =− 12 σ2 =+ 12 + Vij kl = V0τ with d r1 d r2 Sj ϕi+∗ (r1 )ϕj−∗ (r2 )χi+∗ χj−∗ δ(r1 − r2 ) × Sl ϕk+ (r1 )ϕl− (r2 ) − Sk ϕl+ (r1 )ϕk− (r2 ) χk+ χl− (A.6) and σ1 =− 12 σ2 =+ 12 Vij kl = V0τ d r1 d r2 Si ϕi−∗ (r1 )ϕj+∗ (r2 )χi−∗ χj+∗ δ(r1 − r2 ) × Sk ϕk− (r1 )ϕl+ (r2 ) − Sl ϕl− (r1 )ϕk+ (r2 ) χk− χl+ (A.7) or, in a more compact form: Vij kl = V0τ d 3r Sj ϕi+∗ (r)ϕj−∗ (r) − Si ϕi−∗ (r)ϕj+∗ (r) × Sl ϕk+ (r)ϕl− (r) − Sk ϕl+ (r)ϕk− (r) (A.8) Under the present hypotheses, the ϕi states are also eigenstates of the parity operator P : P ϕi (r σ ) = πi ϕi (r σ ), (A.9) with πi = ±1 Moreover, if ϕi refers to the |nr nz Λ σ ket – eigenstate of the (axially symmetric harmonic oscillator) ASHO basis – one has P ϕi (r σ ) = πi ϕi (r σ ) = (−1)nz +Λ ϕi (rσ ) (A.10) In what follows, one will also use the notations: ϕi (r, + 12 ) = ϕi+ (r)χ + and ϕi (r, − 12 ) = Si ϕi− (r)χ − The properties and the notations of the sp-particle wavefunctions with parity–signature symmetry used here are taken from Ref [29], where they are described in details One could obtain some selection rules for the matrix elements Vij kl , when one use the symmetry properties of the sp-functions |ϕi , |ϕj , |ϕk , |ϕl This greatly simplifies the calculating efforts for the cranking HTDA approach H Laftchiev et al / Nuclear Physics A 845 (2010) 33–57 53 A.2 Selection rules and parity symmetry It is easy to verify that V pair = δ(r1 − r2 )( 1−σ41 σ2 ) commutes with the parity operator P Due to the P = property, one has ij |V pair |kl = ij |P V pair |kl = ij |P V pair P |kl = πi πj πk πl ij |V pair |kl (A.11) The parity is self-consistent symmetry regarding the Skyrme and Coulomb mean field operators and the kinetic energy term in the Routhian This implies the selection rule: πi πj πk πl = (A.12) for the matrix elements of the residual and the mean field parts of the interaction A.3 Selection rules and signature symmetry The x-signature operator Π23 acting on any space-spin function Ψ in cylindric coordinates is defined by Π23 Ψ (r, θ, z, σ ) = Ψ (r, −θ, −z, −σ ) which implies that Π23 is a projector: Π23 Π23 = functions χ ± are such that (A.13) Π23 = Space functions ϕi± (r) and spin Π23 ϕi+ (r)χ + = ϕi− (r)χ − , Π23 ϕi− (r)χ − = ϕi+ (r)χ + (A.14) It is well known (see for example Ref [29]) that the x-signature is preserved by the mean field Skyrme and Coulomb interaction operators and its action does not change this symmetry of the wavefunctions The same is true for the volume delta pairing residual interaction term in our case as follows The spin part of the volume delta pairing interaction ( 1−σ41 σ2 ) discriminates the triplet states and allows the singlet states That permits to write ⎧ {σ1 = σ2 }, ⎪0 − σ1 σ2 1⎨ − + + − Sl ϕk ϕl χ χ {σ1,2 = + 12 , − 12 }, Π23 ϕk (r1 σ1 )ϕl (r2 σ2 ) = (A.15) 4⎪ ⎩ Sk ϕk+ ϕl− χ + χ − {σ1,2 = − 12 , + 12 }, and ⎧ ⎪0 − σ1 σ 1⎨ Sl ϕk− ϕl+ χ + χ − Π23 ϕk (r1 σ1 )ϕl (r2 σ2 ) = ⎪ 4⎩ Sk ϕk+ ϕl− χ + χ − {σ1 = σ2 }, {σ1,2 = + 12 , − 12 }, (A.16) {σ1,2 = − 12 , + 12 } Consequently, the operator ( 1−σ41 σ2 ) commutes with the x-signature operator in a wavefunction space made by products of parity–signature symmetric functions as ϕk (r1 σ1 ) and ϕl (r2 σ2 ): Π23 − σ1 σ2 ϕk ϕl = It is easy to verify also that − σ1 σ2 Π23 ϕk ϕl (A.17) 54 H Laftchiev et al / Nuclear Physics A 845 (2010) 33–57 Π23 δ(r1 − r2 ) = δ(r1 − r2 )Π23 , (A.18) thus when V = δ(r1 − r2 )( 1−σ41 σ2 ) ij |V |kl = ij |Π23 V |kl = ij |Π23 V Π23 |kl = Si Sj Sk Sl ij |V |kl (A.19) The non-zero matrix elements correspond thus to the necessary condition Si Sj Sk Sl = 1, (A.20) which is the selection rule for the signature quantum number A.4 Two body matrix elements in the harmonic oscillator basis Here we use notations and developments of Ref [29] In this reference, the advantage in terms of computational time of using a deformed axial oscillator basis for studying deformed rotating nuclei, is demonstrated Using the eigen-wavefunctions of the ASHO one obtains (Ref [29]): ϕi− (r) = βz β⊥ e−(ξ 2π +η) (−1)nz Cαi e−iΛθ η |Λ| Hnz (ξ )L|Λ| nr (η) (A.21) α r The index α corresponds to a set of the ASHO for the stretched variables ξ = βz z and η = β⊥ basis quantum numbers (nz , nr , Λ) and the index α correspond to another values of these quantum numbers:(nz , nr , Λ ) The C matrix is the Hartree–Fock transformation matrix where the element Cαi corresponds to the k-th Hartree–Fock state and the α harmonic oscillator state One has also: ϕi+ (r) = βz β⊥ e−(ξ 2π +η) Cαi e+iΛθ η |Λ| Hnz (ξ )L|Λ| nr (η) (A.22) α Therefore: Sj ϕi+ ϕj− − Si ϕi− ϕj+ ∗ βz β⊥ e−(ξ 2π = +η) j × Cαi Cα η |Λ|+|Λ | |Λ | Hnz Hnz L|Λ| nr Ln r αα × Sj (−1)nz ei(Λ−Λ )θ − Si (−1)nz ei(Λ −Λ)θ (A.23) and, in the same way: Sl ϕk+ ϕl− − Sk ϕl+ ϕk− = βz β⊥ e−(ξ 2π × +η) Cβk Cβl η |Λ|+|Λ | |Λ | Hnz Hnz L|Λ| nr Ln r ββ × Sl (−1)nz ei(Λ−Λ )θ − Sk (−1)nz ei(Λ −Λ)θ (A.24) Substituting Eq (A.24) and Eq (A.23) in Eq (A.8) and switching to the stretched cylindrical r and θ , one obtains: coordinates ξ = βz z, η = β⊥ H Laftchiev et al / Nuclear Physics A 845 (2010) 33–57 Vij kl = V0τ βz β⊥ e−(ξ 2π +η) dη j Cαi Cα η |Λ|+|Λ | dξ ξ =−∞ η=0 × ξ =+∞ η=∞ 55 |Λ | Hnz Hnz L|Λ| nr Ln Cβk Cβl η r αα |Λ|+|Λ | |Λβ | Hnz Hnz Ln|Λr | Ln β r ββ θ=2π × Sj (−1)nz ei(Λα −Λα )θ − Si (−1)nz ei(Λα −Λα )θ α dθ α θ=0 β × Sl (−1)nz ei(Λβ −Λβ )θ β − Sk (−1)nz ei(Λβ −Λβ )θ (A.25) , which is a fairly complicated 3-fold integral and must be simplified When using the parity– signature properties (Eqs (A.12) and (A.20)) and also the property P |nz , nr , Λ, Σ = β β (−1)nz +Λ |nz , nr , Λ, Σ (i.e πi = (−1)nz +Λ , see Ref [29]) it follows that (−1)±nz ±nz ±nz ±nz α α β β = (−1)±Λ ±Λ ±Λ ±Λ Using this property, the angular part of the integral in Eq (A.25) could be developed as follows: α α θ=2π Sj (−1)nz ei(Λα −Λα )θ − Si (−1)nz ei(Λα −Λα )θ α α θ=0 β × Sl (−1)nz ei(Λβ −Λβ )θ β − Sk (−1)nz ei(Λβ β nz α −Λβ )θ dθ β β nz −nz = 4πSj (−1)nz (−1) Sl δ Λ− =0 − Sk (−1) δ Λ+ =0 ⎧ 1, when Λα = Λβ = 0, ⎨ β Λ nαz +nz α −Sk Sl πk πl (−1) , when Λα = − Λβ = 0, = 4π(−1) ⎩ (1 − Sk Sl πk πl ), when Λα = = Λβ (A.26) Here Λα = Λα − Λα , Λβ = Λβ − Λβ , Λ− = Λα − Λα − (Λβ − Λβ ) and Λ+ = Λα − Λα + Λβ − Λβ This leads to the idea to use the Λ quantities for further simplification of V pair If one defines a new function: e−(ξ βz β⊥ f ( Λ, ξ, η) = Sj π ij +η) j Cαi Cα η |Λα |+|Λα | |Λα | Hnz Hnz Ln|Λr α | Ln r (A.27) αα for all α, where Λ = Λα − Λα is fulfilled And if one takes into account this equation and also Eq (A.26) then in Eq (A.25) the matrix elements Vij kl could be developed as +∞ +∞ Vij kl = 2πV0τ dη −∞ f ij ( Λα , ξ, η) dξ Λα ⎧ 1, ⎨ × −Sk Sl πk πl (−1) ⎩ (1 − Sk Sl πk πl ), Λα , f kl ( Λβ , ξ, η) Λβ when when when Λα = Λβ = 0, Λα = − Λβ = 0, Λα = = Λβ , (A.28) 56 H Laftchiev et al / Nuclear Physics A 845 (2010) 33–57 and after straightforward manipulations one finally obtains: +∞ Vij kl = 2πV0τ +∞ dη kl f ij ( Λα , ξ, η) − f j i ( Λα , ξ, η) dξ −∞ Λα >0 × f ( Λα , ξ, η) − f lk ( Λα , ξ, η) + f ij ( Λα , ξ, η) f kl ( Λα , ξ, η) − f lk ( Λα , ξ, η) Λα =0 , (A.29) a result which can easily be used for computations Therefore, one could calculate the Vij kl values in two steps: first, one tabulates the function f for the needed index values and then second, one obtains the values of Vij kl by calculating numerically the two-fold integral in Eq (A.29) The latter procedure simplifies a lot the numerical efforts to compute the 6-fold integral in Eq (A.3) or the 3-fold integral in Eq (A.25) Appendix B Wick theorem for one- and two-body operators Here we sketch the calculations of the nP nH |ρ|mP ˆ mH kl matrix elements needed to obtain the single-particle density matrix from a HTDA many-body state Let us demonstrate how they could be calculated in their n = 2, m = cases When n and m differ from these values one could apply similar reasonings Let us take the i-th and the j -th 2-particle 2-hole excitation as an example: nP nH |V |mP mH kl = 2P 2Hi |ρ|2P ˆ 2Hj kl = 2P 2Hi |al+ ak |2P 2Hj = HF|ai+1 ai+2 ai3 ai4 al+ ak aj+4 aj+3 aj2 aj1 |HF (B.1) The non-zero variants of this matrix element could be found (depending on the values of the indices i1−4 , j1−4 and k, l) by using directly the Wick theorem, but for computational convenience, one could simply apply the following reasoning: One could divide the braket in two parts – left hand one and right hand one For instance: HF|ai+1 ai+2 ai3 ai4 al+ = ϕleft | and ak aj+4 aj+3 aj2 aj1 |HF = |ϕright The scalar product ϕleft |ϕright between two such normalized states will obviously vanish if they are not linearly dependent A necessary and sufficient condition for this is to a have a global equality between the indices of the creation operators from the left side part and the annihilation operators of the right side part and the equality between the indices of the annihilation operators from the left hand part and the creation operators of the right hand part After requiring this equality the braket will have a value of or −1, depending on the signature of the permutation transforming one set of the indices into the other We have used the same type of reasoning to evaluate the matrix elements 1P 1H |al+ ak |2P 2Hj , 1P 1Hi |al+ ak |1P 1Hj 2P 2Hi |al+ ak |1P 1Hj and similar where calculated Additionally, such a procedure is also very well adapted for fast computing of the matrix elements of ρ Also, following the logic described here, the many body Hamiltonian matrix elements were calculated References [1] B Singh, R Zywina, R.B Firestone, Table of 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HTDA calculations A very important remark is appropriate at this point It concerns the low angular velocity part of the variation of J (and also of J ) moments Their HTDA calculated values appear... calculated moments of inertia reproduce the data rather well qualitatively, but not at all quantitatively The same is true also for the RHB calculations of Ref [24] For instance, in all calculations... sp-Routhian operator basis are real It is to be noted that np–nh states are also eigenstates of the x-signature and parity operators (with obvious notations, σx = i=1 ,A σx (i) and π = i=1 ,A π(i))