Triaxial Angular Momentum Projection and Configuration Mixing calculations with the Gogny force
TTriaxial Angular Momentum Projection and ConfigurationMixing calculations with the Gogny force
Tom´as R. Rodr´ıguez
GSI Helmholtzzentrum f¨ur Schwerionenforschung ∗ , D-64259 Darmstadt, Germany andDepartamento de F´ısica Te´orica, Universidad Aut´onoma de Madrid, E-28049 Madrid, Spain J. Luis Egido
Departamento de F´ısica Te´orica, Universidad Aut´onoma de Madrid, E-28049 Madrid, Spain (Dated: November 2, 2018)
Abstract
We present the first implementation in the ( β, γ ) plane of the generator coordinate method withfull triaxial angular momentum and particle number projected wave functions using the Gognyforce. Technical details about the performance of the method and the convergence of the resultsboth in the symmetry restoration and the configuration mixing parts are discussed in detail. Weapply the method to the study of Mg, the calculated energies of excited states as well as thetransition probabilities are compared to the available experimental data showing a good overallagreement. In addition, we present the RVAMPIR approach which provides a good description ofthe ground and gamma bands in the absence of strong mixing.
PACS numbers: 21.60.Jz, 21.10.Re, 21.60.Ev, 27.60.+j ∗ Present address a r X i v : . [ nu c l - t h ] A p r . INTRODUCTION Self-consistent mean field methods with effective phenomenological interactions and theirextensions beyond mean field provide the appropriate theoretical tools for describing manyphenomena along the whole chart of nuclides, from light to medium, heavy and superheavynuclei in or far away from the stability valley [1]. On the one hand, the success of thesemethods is related to the high quality of the phenomenological effective interactions used-Skyrme, Gogny or Relativistic Mean Field (RMF). On the other hand, the mean fieldmethod allows the inclusion of many correlations within a very simple intrinsic productwave function. Hence, bulk properties such as masses and radii are very well described atthe mean field level. However, in some cases this picture fails to take into account importantcorrelations and methods beyond the mean field approach have to be applied. Furthermore,because the mean field is defined in the intrinsic frame it is mandatory to go beyond thisapproximation to evaluate excitation energies or transition probabilities in the laboratorysystem.There are several methods to incorporate the correlations missing at the mean field level.Normally, the intrinsic wave functions are allowed to break relevant symmetries of the sys-tem, for example, particle number, rotational and translational invariance, parity, time-reversal, etc. to enlarge the variational space and incorporate, for instance, deformation orsuperfluidity in the mean field picture. This leads to a degeneracy of the wave functions rotated in the gauge space associated to the broken symmetry. An appropriated superposi-tion of these wave functions provides a symmetry conserving many-body wave function andan additional lowering of the energy of the system. In this way, using projection techniques[2], many correlations are obtained by restoring some or all of these symmetries . Further-more, the mixing of different mean field configurations within the general framework of theGenerator Coordinate Method (GCM) [2] allows the inclusion of quantum fluctuations alongsome relevant collective variables such as the multipole moments.Most of the currently used beyond mean field calculations with effective forces include twosymmetry restoration, i.e., particle number (PN) and angular momentum projection (AMP)and configuration mixing along the axial quadrupole deformation [1, 3, 4]. This approach(axial GCM-PNAMP) has been successfully applied to study many phenomena like, for ex-ample, the appearance or degradation of shell closures in neutron rich nuclei [3, 5–7], shape2oexistence in proton rich Kr [8] or Pb [9, 10] isotopes or shape transitions in the A ∼ K = 0, because this assumption simplifies considerably the angularmomentum projection and lightens the computational burden significantly. This restrictionis one of the major drawbacks of the method because it limits its applicability to systemswhere triaxiality does not play an important role. However, many exciting experimental andtheoretical phenomena are closely related to the triaxial degree of freedom, for instance: thepresence of γ -bands in the low lying energy spectra and γ -softness, shape coexistence andshape transitions in transitional regions [13, 15–20]; the lowering of fission barriers along thetriaxial path [21–23]; the influence of triaxial deformation in the ground state for the massmodels [24, 25]; triaxiality at high spin [26–28]; the observation of K -bands and isomericstates in the Os region [29–31]; or some other exotic excitation modes such as wobblingmotion and chiral bands [32–34].From the theoretical point of view some approaches beyond mean field have been proposedto study the triaxial effects. In particular, one of the most widely used is the collectiveHamiltonian [2] given in different versions depending on the underlying nucleon-nucleoninteraction used to define the collective potential, namely Pairing-plus-Quadrupole [35], In-teracting Boson Model [36], Nilsson Woods-Saxon [26], Gogny [37–39] or RMF [40]. Thismodel has been applied successfully to describe some of the experimental features men-tioned above. However, the collective Hamiltonian can be understood as a gaussian overlapapproach (GOA) of the triaxial GCM and this description should be improved includingproperly the effects of the symmetry restoration and the full configuration mixing withoutany GOA approximation.In the past, exact angular momentum projection with triaxial intrinsic wave functionswithout GCM have been carried out only for schematic forces and/or reduced configura-tion spaces. Examples are the projection of BCS [41] or Cranked Hartree-Fock-Bogoliubov(CHFB) states [42] with the Pairing-plus-Quadrupole interaction, the projection of CrankedHartree-Fock (CHF) states (no pairing) with schematic [43] and full Skyrme interactions [44]or angular momentum projection before variation with particle number and parity restora-tion in limited shell model spaces [45, 46] .However, recent improvement of the computational capabilities enabled the first implemen-tations of the angular momentum projection of triaxial intrinsic wave functions in the whole3 β, γ ) plane with effective forces. In particular, Bender and Heenen reported GCM cal-culations with particle number and triaxial angular momentum projection (PNAMP) withthe Skyrme SLy4 interaction [47]. In this work, the intrinsic wave functions were found bysolving the Lipkin-Nogami (LN) equations. On the other hand, Yao et al presented theimplementation of the triaxial angular momentum projection [48] and the extension to theGCM [49] for the Relativistic Mean Field (RMF) framework. In the latter work no particlenumber projection has been performed and the mean field states are found by solving theRMF+BCS instead of the full HFB or LN equations. These approximations could lead toa poor description of important pairing correlations, especially in the weak pairing regimewhere even spurious phase transitions appear [3, 50].In this paper we present the first implementation of the Generator Coordinate Method withParticle Number and Angular Momentum Projected (GCM-PNAMP) triaxial HFB wavefunctions with the finite range density dependent Gogny force [51]. The finite range of theGogny force provides excellent pairing properties and is often used as a benchmark in thisrespect. Furthermore it is able to provide at the same time both good global as well as spec-troscopic properties [52, 53]. The intrinsic HFB states are found by solving the VariationAfter Particle Number Projection (VAP-PN) equations [54]. This fact constitutes the mainmethodological difference with respect to the calculations reported in Ref. [47]. This is avery important difference because VAP-PN allows the inclusion of the pairing correlationsin a very efficient way yielding a significant improvement of the final results with respect toother approaches [3, 54].In nuclei without strong mixing the so called Variation After Mean-field Projection In Real-istic model spaces (VAMPIR) [45] approach has been very successful. In this approach onlyone HFB wave function is considered which is determined by minimization of the projectedenergy, i.e. the VAP approach for both the AM and the PN projections. A full VAMPIRapproach with the Gogny force and large configuration spaces is not feasible yet. Instead weuse an approximation to it, which we call RVAMPIR and which as we shall see, in the casenucleus studied in this article, provides very reasonable results for the ground and γ bandswith much less effort than in the GCM approach.The paper is organized as follows. In Sec. II we will give an overview of the theoreticalframework. Then, we will focus our analysis on the nucleus Mg which has been studiedas a test case in earlier implementations of the GCM-PNAMP method with Skyrme and4elativistic interactions. In particular, in Sec. III we will show a standard axially symmetriccalculation which allows to make an educated guess for some relevant parameters needed inthe full calculation such as the number of major oscillator shells or the relevant deforma-tions ranges. In Sec. IV we will analyze in detail the simpler PNAMP method, studying theconvergence of the integrals in the Euler angles, giving some consistency requirements andshowing the role of having an adequate mesh in the ( β, γ ) plane. In Sec. V we will showthe final results for the calculated spectrum and B(E2) transitions strengths of Mg and acomparison with experimental data. Finally, a brief summary and outlook on future workwill be addressed in Sec. VI.
II. THEORETICAL FRAMEWORKA. Generator Coordinate Method with Particle Number and Angular MomentumProjected states (GCM-PNAMP)
In the present approach, the final many-body wave functions that describe the differentstates of an even-even nucleus with Z ( N ) number of protons (neutrons) are written as: | IM ; N Zσ (cid:105) = (cid:88) Kβγ f I ; NZ,σKβγ | IM K ; N Z ; βγ (cid:105) (1)where ( β, γ ) are quadrupole deformation parameters (see below), σ = 1 , , ... labels thelevels for a given value of the angular momentum I and M, K are the projections of (cid:126)I on the laboratory and intrinsic z − axes respectively. The coefficients f I ; NZ,σKβγ of the linearcombination are found by minimizing the energy within the non-orthogonal set of wavefunctions {| IM K ; N Z ; βγ (cid:105)} . These states are obtained by projecting the intrinsic mean-field states | Φ( β, γ ) (cid:105) onto good particle number and angular momentum: | IM K ; N Z ; βγ (cid:105) = 2 I + 18 π (cid:90) D I ∗ MK (Ω) ˆ R (Ω) ˆ P N ˆ P Z | Φ( β, γ ) (cid:105) d Ω (2)with ˆ P N = π (cid:82) π e iϕ ( ˆ N − N ) dϕ the neutron number projector ( ϕ the associated gauge angleand ˆ P Z protons the proton number projector), ˆ R (Ω) and D I ∗ MK (Ω) are the rotation operatorand the Wigner matrices [55] in the Euler angles Ω = ( a, b, c ) [69], respectively. In principle,the ranges for these angles are (0 ≤ a ≤ π, ≤ b ≤ π, ≤ c ≤ π ). However, for intrinsicHartree-Fock-Bogoliubov (HFB) states ( | Φ( β, γ ) (cid:105) ) which are symmetric under time-reversal5nd simplex symmetries, the intervals for both gauge and Euler angles can be reduced to(0 ≤ ϕ ≤ π/
2) and (0 ≤ a ≤ π/ , ≤ b ≤ π/ , ≤ c ≤ π ), respectively [47].The wave functions (Eq. 2) are eigenstates of the particle number and angular momentumoperators: ˆ N | IM K ; N Z ; βγ (cid:105) = N | IM K ; N Z ; βγ (cid:105) , (3)ˆ Z | IM K ; N Z ; βγ (cid:105) = Z | IM K ; N Z ; βγ (cid:105) , (4)ˆ I | IM K ; N Z ; βγ (cid:105) = ¯ h I ( I + 1) | IM K ; N Z ; βγ (cid:105) (5)ˆ I z | IM K ; N Z ; βγ (cid:105) = ¯ hM | IM K ; N Z ; βγ (cid:105) , (6)ˆ I | IM K ; N Z ; βγ (cid:105) = ¯ hK | IM K ; N Z ; βγ (cid:105) (7)The intrinsic HFB states ( | Φ( β, γ ) (cid:105) ) are obtained by minimizing the particle-number pro-jected energy functional E N,Z (cid:104) ¯Φ( β, γ ) (cid:105) (variation after projection, VAP)[54]. This is one ofthe most relevant parts in the calculation because the quality of the result largely dependson the structure of the intrinsic HFB-type wave functions used. In contrast to other methodslike plain HFB or Projected Lipkin-Nogami (PLN), the VAP-PN performs the restorationof the particle number symmetry in an optimal way, including pairing correlations both inthe weak and strong pairing regimes [3]. This is especially relevant in GCM-like theorieswhere a large grid of ( β, γ ) points is needed. The strength of the pairing correlations hasa strong dependence on the single particle level density and the latter one itself with thedeformation parameters. This implies that a strongly ( β, γ ) dependent oscillating pairingregime appears in the calculations and consequently theories like plain HFB (BCS) or PLN(LN) are unable to cope with this challenge providing wave functions of oscillating goodness.Only a VAP-PN approach warrants high quality solutions independently of the ( β, γ ) values.Dealing with effective forces like Skyrme, Relativistic and Gogny, a natural separation of theinteraction into the two-body Hamiltonian ˆ H on the one hand and the density-dependentpart, ε N,ZDD [Φ] on the other emerges. In our case, we are using the Gogny D1S interaction[51] and ˆ H corresponds to the kinetic energy (the two-body part from the center of masscorrection included) plus the spin-orbit, Coulomb and the finite range central potentials.In the calculations, all direct, exchange and pairing terms are included [56]. The VAP-PN6rinciple, provides δE N,Z (cid:104) ¯Φ( β, γ ) (cid:105)(cid:12)(cid:12)(cid:12)(cid:12) ¯Φ=Φ = 0 (8)where: E N,Z [Φ] = (cid:104) Φ | ˆ H ˆ P N ˆ P Z | Φ (cid:105)(cid:104) Φ | ˆ P N ˆ P Z | Φ (cid:105) + ε N,ZDD [Φ] − λ q (cid:104) Φ | ˆ Q | Φ (cid:105) − λ q (cid:104) Φ | ˆ Q | Φ (cid:105) (9)In a beyond mean field method, and in particular for the particle number projection, weneed a reasonable prescription for the spatial density, which we shall call ρ int ( (cid:126)r ), that entersin ε N,ZDD [Φ], the density dependent term of the interaction. In this work, assuming thephenomenological nature of these interactions and considering that the restoration of theparticle number symmetry is performed not in the coordinate but in the gauge space, wehave chosen the number projected spatial density prescription that has proven to be freeof divergences [54] and to give very good results for describing many phenomena along thenuclear chart: ρ NZ int ( (cid:126)r ) ≡ (cid:104) Φ | ˆ ρ ( (cid:126)r ) P N P Z | Φ (cid:105)(cid:104) Φ | P N P Z | Φ (cid:105) (10)with ˆ ρ ( (cid:126)r ) ≡ (cid:82) d(cid:126)r (cid:48) δ ( (cid:126)r − (cid:126)r (cid:48) ). As shown in [54] for the PNP and in [57] for the Lipkin-Nogamiapproach, the use of the projected density or the so-called mixed prescription (in the casewhen the latter is free of potential divergences) provide very similar results. Furthermore,we see in Eq. 9 that the minimization is performed under constraints on the quadrupole de-formation operators ˆ Q µ . The Lagrange multipliers λ q µ ensure that the following conditionsare fulfilled in the intrinsic state: λ q → (cid:104) Φ | ˆ Q | Φ (cid:105) = q λ q → (cid:104) Φ | ˆ Q | Φ (cid:105) = q (11)In addition, the deformation parameters ( β, γ ) are directly related to ( q , q ) by: q = β cos γC ; q = β sin γ √ C ; C = (cid:115) π π r A / (12)being r = 1 . A the mass number. These constraints allow to explore the ( β, γ )plane to generate the wave functions to be used in the configuration mixing calculations.We now describe the Generator Coordinate Method (GCM) to obtain the final spectrum( E I ; NZ ; σ ) and the coefficients f I ; NZ,σKβγ given in Eq. 1. Minimization of the energy with respectto the coefficients f I ; NZ,σKβγ leads to the Hill-Wheeler-Griffin (HWG) equation (cid:88) K (cid:48) β (cid:48) γ (cid:48) (cid:16) H I ; NZKβγK (cid:48) β (cid:48) γ (cid:48) − E I ; NZ ; σ N I ; NZKβγK (cid:48) β (cid:48) γ (cid:48) (cid:17) f I ; NZ ; σK (cid:48) β (cid:48) γ (cid:48) = 0 , (13)7hich has to be solved for each value of the angular momentum. The GCM norm- andenergy-overlaps have been defined as: N I ; NZKβγK (cid:48) β (cid:48) γ (cid:48) ≡ (cid:104) IM K ; N Z ; βγ | IM K (cid:48) ; N Z ; β (cid:48) γ (cid:48) (cid:105)H I ; NZKβγK (cid:48) β (cid:48) γ (cid:48) ≡ (cid:104) IM K ; N Z ; βγ | ˆ H | IM K (cid:48) ; N Z ; β (cid:48) γ (cid:48) (cid:105) + ε IKK (cid:48) ; NZDD [Φ( β, γ ) , Φ (cid:48) ( β (cid:48) , γ (cid:48) )] (14)In the last expression, we have separated again the energy overlap in the contribution ofthe pure Hamiltonian part of the interaction and the density-dependent term. In the lat-ter, we have used the particle number projected spatial density combined with the mixedprescription for the angular momentum projection and GCM part, namely: ρ NZ int (Ω , (cid:126)r ) ≡ (cid:104) Φ | ˆ ρ ( (cid:126)r ) ˆ R (Ω) P N P Z | Φ (cid:48) (cid:105)(cid:104) Φ | ˆ R (Ω) P N P Z | Φ (cid:48) (cid:105) . (15)This prescription is suitable for dealing with the restoration of broken symmetries in thecoordinate space such as the rotational invariance or the spatial parity.Once we have calculated the corresponding GCM overlaps, the next step consists in solvingthe HWG equations (Eq. 13). To cope with the problem of the linear dependence one firstintroduces a orthonormal basis defined by the eigenvalues n I ; NZ Λ and eigenvectors u I ; NZKβγ ;Λ ofthe norm overlap: (cid:88) K (cid:48) β (cid:48) γ (cid:48) N I ; NZKβγK (cid:48) β (cid:48) γ (cid:48) u I ; NZK (cid:48) β (cid:48) γ (cid:48) ;Λ = n I ; NZ Λ u I ; NZKβγ ;Λ . (16)This orthonormal basis is known as the natural basis and for n I ; NZ Λ values such that n I ; NZ Λ /n I,NZmax > ζ , the natural states are defined by: | Λ IM ; NZ (cid:105) = (cid:88) Kβγ u I ; NZKβγ ;Λ (cid:113) n I ; NZ Λ | IM K ; N Z ; βγ (cid:105) . (17)Obviously, a cutoff ζ in the value of the norm eigenvalues has to be introduced in orderto avoid linear dependences [48]. Then, the HWG equation is transformed into a normaleigenvalue problem: (cid:88) Λ (cid:48) (cid:104) Λ I ; NZ | ˆ H | Λ (cid:48) I ; NZ (cid:105) G I ; NZ ; σ Λ (cid:48) = E I ; NZ ; σ G I ; NZ ; σ Λ . (18)From the coefficients G I ; NZ ; σ Λ we can define the so-called collective wave functions F I ; NZ ; σ ( β, γ ) that account for the probability density, normalized to 1, of finding the state( I, σ ) with given deformation parameters ( β, γ ): F I ; NZ ; σ ( β, γ ) = (cid:88) Λ ,K G I ; NZ ; σ Λ u I ; NZKβγ ;Λ = (cid:88) K F I ; NZ ; σK ( β, γ ) . (19)8e have also introduced F I ; NZ ; σK ( β, γ ) that account for the probability density of finding thestate ( I, σ ) with given values of K and deformation parameters ( β, γ ).Furthermore, the expectation value of a generic operator ˆ O is given by o I ; NZ ; σ = (cid:88) Λ;Λ (cid:48) (cid:88)
Kβγ ; K (cid:48) β (cid:48) γ (cid:48) G I ; NZ ; σ ∗ Λ u I ; NZ ∗ Kβγ ;Λ (cid:113) n I ; NZ Λ (cid:104) (cid:36) | ˆ O | (cid:36) (cid:48) (cid:105) u I ; NZK (cid:48) β (cid:48) γ (cid:48) ;Λ (cid:48) (cid:113) n I ; NZ Λ (cid:48) G I ; NZ ; σ Λ (cid:48) , (20)with (cid:104) (cid:36) | ˆ O | (cid:36) (cid:48) (cid:105) = (cid:104) IM K ; N Z ; βγ | ˆ O | IM K (cid:48) ; N Z ; β (cid:48) γ (cid:48) (cid:105) . This expression can be generalizedto account for transitions associated to the tensorial operator ˆ T : t ( I σ → I σ ) = (cid:88) Λ;Λ (cid:48) (cid:88)
Kβγ ; K (cid:48) β (cid:48) γ (cid:48) G I ; NZ ; σ ∗ Λ u I ; NZ ∗ Kβγ ;Λ (cid:113) n I ; NZ Λ (cid:104) (cid:36) || ˆ T || (cid:36) (cid:48) (cid:105) u I ; NZK (cid:48) β (cid:48) γ (cid:48) ;Λ (cid:48) (cid:113) n I ; NZ Λ (cid:48) G I ; NZ ; σ Λ (cid:48) , (21)where (cid:104) (cid:36) || ˆ T || (cid:36) (cid:48) (cid:105) = (cid:104) I K ; N Z ; βγ || ˆ T || I K (cid:48) ; N Z ; β (cid:48) γ (cid:48) (cid:105) stands for the reduced matrix ele-ment calculated according to the Wigner-Eckart theorem [2, 55]. Detailed expressions for cal-culating these reduced matrix elements for B(E2) transitions and spectroscopic quadrupolemoments within this framework can be found elsewhere [5, 47, 49]. B. Simpler approaches: Particle Number and Angular Momentum Projection(PNAMP) and the RVAMPIR approximation
The expressions given above constitute the most general framework that we are using forsolving the nuclear many body problem. Nevertheless, there are some limiting cases with arelevant physical meaning that can be deduced in a straightforward manner from them. Thefirst one is the particle number projection (PNP) that has been discussed above (Eq. 9).The second approach is the particle number and angular momentum projection (PNAMP)of a single point in the ( β, γ ) plane. Here, the wave function is of the form of Eq. 1 butwithout the mixing in the deformation parameters: | IM ; N Z ; ν ; β, γ (cid:105) = (cid:88) K h I ; NZ,νK ( βγ ) | IM K ; N Z ; βγ (cid:105) , (22)where the label ν stands for the (2 I + 1) different states that can be obtained with theangular momentum projection. However, due to the time reversal and simplex symmetriesimposed on the intrinsic wave functions, this number is reduced to ( I/ I − / I , respectively. Moreover, if we furthermore have axialsymmetry, only one state can be obtained and only for even values of I .9he coefficients h I ; NZ,νK ( βγ ) and the PNAMP energies E I ; NZ ; ν ( β, γ ) are found by solving thesimplified version of the HWG equation (see Eq. 13): (cid:88) K (cid:16) H I ; NZKβγK (cid:48) βγ − E I ; NZ ; ν ( β, γ ) N I ; NZKβγK (cid:48) βγ (cid:17) h I ; NZ ; νK (cid:48) ( β, γ ) = 0 . (23)The remaining expressions used to solve the HWG equations are simplified in the samemanner, i.e., removing the sum over ( β, γ ) from the equations and evaluating only thediagonal part. In addition, the collective wave functions F I ; NZ ; νK ( β, γ ) (Eq. 19), which inanalogy we shall call H I ; NZ ; νK ( β, γ ), now give the spectral distribution in the K space of thecorresponding PNAMP state.A full variation of the HFB wave function in the VAP approach, in the spirit of VAMPIR,for the PN and the AM with large configuration spaces and the Gogny interaction is not yetfeasible. However we can use an approximation to VAMPIR, which we shall call from nowon RVAMPIR, in which the PN is handled in the VAP approach and the AM in a RestrictedVAP (RVAP) one. The RVAP approximation has been thoroughly studied in [58]-[59].In the VAP method the whole Hilbert space associated with the HFB transformation isscanned in the variational procedure. In the RVAP method, however, only a restrictedvariational space of highly correlated wave-functions is allowed in the minimization process.Monopole (pairing) and quadrupole ( β and γ ) correlations are believed to be the mostrelevant degrees of freedom of atomic nuclei and are related to the particle number and theangular momentum symmetries, respectively. Since we are considering the PN symmetry inthe VAP theory it seems reasonable in our case to consider the restricted Hilbert space tocontain a whole set of quadrupole deformed wave-functions | Φ( β, γ ) (cid:105) which parametricallydepend on ( β, γ ). This procedure is justified by theoretical arguments [2] which establishthat a VAP approach is needed for systems with weakly broken symmetries, like in thePN case where only a few Cooper pairs participate, but it can be approximated in caseof strongly broken symmetries, such as deformation, where a large number of nucleonsparticipate. Concerning the differences of this approximation as compared to VAMPIR it isclear [58] that if, besides of considering the quadrupole moments ˆ Q and ˆ Q in Eq. 9, wewill include higher multipole moments ˆ Q LM to increase the variational space, our solutionwould get very close to the one of the genuine VAMPIR. With respect of the quality of ourapproach (again with respect to the full VAMPIR) we expect that in general it will be verysimilar and only in very soft nuclei, where higher modes (hexadecupole for example) are10ery relevant, differences may arise. But for very soft nuclei we have to question also the fullVAMPIR since a GCM-like approach will be more appropriate. That means, RVAMPIR isnot as “restricted” as its name might imply.Specifically the basic RVAMPIR approach consist of the following steps:A.- At each ( β i , γ i ) value of a given set of points in the ( β, γ ) plane the following itemsare performed:A1.- Solve the VAP-PN equations, Eqs.8-9, to determine the β − γ constrained HFBwave function | Φ( β, γ ) (cid:105) .A2.- Carry out simultaneous particle number and angular momentum projection onthe wave function | Φ( β, γ ) (cid:105) , what we have called | IM K ; N Z ; βγ (cid:105) , see Eq. 2, toform the linear combination of the state | IM ; N Z ; ν ; βγ (cid:105) in Eq. 22.A3.- Solve the HWG equation, Eq.23, for different angular momenta.B.- For each value of the angular momentum sort out the energies E I ; NZ ; ν ( β, γ ) of Eq. 23and find out the point ( β I min , γ I min ) providing the energy minimum E I ; NZ ; ν min ( β I min , γ I min )C.- The solutions of the HWG equation at the points ( β I min , γ I min ) provide I/ I − / I values, which allow to build a partial spectrum and to calculatethe transition probabilities among the different states or any other observable.One has to notice that all RVAMPIR states are orthogonal, those with different AMin an obvious way and those with the same AM because they are solution of the sameeigenvalue equation.In the following sections we will give some examples of the convergence, consistency andperformance of the methods described above. All the many body intrinsic wave functionsand operators have been expanded in a cartesian harmonic oscillator single particle basisclosed under rotations [60]. In particular, the rotation operator ˆ R (Ω) has been evaluatedfollowing the expressions given in Ref. [61] and the Neergard method [62] has been usedin the calculation of the norm overlaps in order to determine the correct sign of theOnishi formula [63–65]. The overlaps of a generic operator have been calculated using thegeneralized Wick theorem [64]. 11 II. AXIAL CALCULATIONS FOR MG Due to the huge computational cost of the full triaxial calculation, it is important tostudy first the axial case (with K = 0) in order to fix some relevant quantities. The mostimportant ones are the region of β deformation to be included in the calculation and thenumber of major oscillator shells in which the mean field wave functions are expanded. Thecomputational effort depends critically on these quantities and it is important to ensure theconvergence of the results, at least in the axial case, to have a reasonable choice which thenlater allows to perform the full triaxial calculation.The main advantage of considering only axial symmetric ( K = 0) intrinsic wave functions | Φ( β, γ = 0 ◦ , ◦ ) (cid:105) ≡ | Φ( β ) (cid:105) is that the integration over the Euler angles ( a, c ) can bedone analytically and this fact reduces drastically the computational time. The simplifiedexpressions of the axial GCM-PNAMP method can be found in detail in Ref. [5]. We firstanalyze the results obtained for the nucleus Mg using N shells = 7 oscillator shells and N points = 31 intrinsic wave functions distributed in the interval ( − . ≤ β ≤ .
5) with posi-tive and negative values of β corresponding to prolate γ = 0 ◦ and oblate γ = 180 ◦ shapes.The integration over the gauge angle ϕ for the particle number projection part has beenperformed using the Fomenko expansion [66] while for the integration over the Euler angle b a Gaussian-Legendre quadrature has been used. We have chosen N F om = 9 and N b = 16 asthe number of integration points for the particle number and the angular momentum partsof the projection, respectively. With these assumptions the expectation values for the ˆ N , ˆ Z ,ˆ N , ˆ Z and ˆ I operators differ by less than 10 − from the corresponding eigenvalues. In Fig.1(a) we plot the potential energy surfaces (PES) along the β direction for the VAP-PN andPNAMP approaches. The VAP-PN curve shows two differentiated minima separated by abarrier of ∼ . β = 0 .
5) and the other one inthe oblate part ( β = − .
2) . These minima are shifted towards larger deformations when theangular momentum projection is performed. In particular, a well defined prolate minimumappears at β = 0 . I = 0 , , , , ∼ . β = − . I = 0 , I = 0 oblate minima is12 E ( M e V ) PN-VAPI=0I=2I=4I=6I=8 0 +1 +1 +1 +1 +1 +2 +2 +2 (a)-1.5 -1 -0.5 0 0.5 1 1.5 ! | F | +2 +2 +2 +3 (c) 00.10.20.30.4 | F | +1 +1 +1 +1 +1 (b) FIG. 1: (a) Potential Energy Surfaces (PES) along the β deformation for particle number projectionand particle number and angular momentum projection ( Mg). The bullets correspond to theexcitation energies for the different GCM levels (
I, σ ) with their positions at ¯ β Iσ = (cid:80) β β | F Iσ ( β ) | .The energy is normalized to the GCM ground state energy (0 +1 ). (b) Collective wave functionsfor the ( σ = 1) GCM levels. The values of the ordinate axis is displaced by 0.05 with increasingangular momentum. (c) Same as (b) but for the ( σ = 2) GCM levels and 2 +3 state. Positive andnegative values of β correspond to prolate ( γ = 0 ◦ ) and oblate ( γ = 180 ◦ ) shapes, respectively. ∼ . plateau in the energy as a function of the number of states in the natural basis (Eq. 17) andfulfill the orthonormality condition. To avoid duplications a detailed discussion on theseissues is postponed to the triaxial case. The resulting GCM-PNAMP energies are also rep-resented in Fig. 1(a), while the corresponding collective wave functions (Eq. 19) are plotted13 E x ( M e V ) + + + + + + + + + + + + + + + + Axial 7 shells Axial 11 shells
FIG. 2: Axial PNAMP-GCM excitation spectra of Mg obtained considering 7 shells (left) and11 shells (right) in the calculations. in Fig. 1(b) for σ = 1 and in Fig. 1(c) for σ = 2 and 2 +3 . In these figures we can seethat the σ = 1 states are members of a rotational band, with most of the intensity of thecollective wave functions concentrated around β = 0 .
6. This deformation corresponds tothe location of the prolate minima of the different potential wells. The ground state 0 +1 alsohas a small mixing with the oblate minimum at β = − .
5. The situation is rather differentfor the σ = 2 states. The second 0 + state is a mixing of oblate and prolate configurations,while wave function of the 2 +2 state peaks in the oblate minimum of the corresponding PESand the 4 +2 state could be considered as a vibration built on the I = 4 prolate well witha small contribution of slightly oblate states. In the 2 +3 state the prolate deformations areagain favored. Remembering that the purpose of this axial calculation is to determine therange of values of β needed to obtain converged results in the low-lying energy spectrum,we observe in Fig. 1(b)-(c) that all the collective wave functions studied here drop to zeroat the boundaries. A smaller interval, however, could not be sufficient for describing cor-rectly the collective states. In addition, we have checked the convergence of the results as afunction of the number of the points included in the GCM-PNAMP. Increasing this numberto N points = 61 still yields very similar results for the PES, GCM-PNAMP energies and thecollective wave functions as compared to the ones obtained for N points = 31.Finally, in order to test the convergence with the number of oscillator shells, we have per-formed a calculation with N shells = 11 and N points = 31. It is noteworthy that for a triaxialcalculation, the computational time for N shells = 11 is ∼
30 times larger than the one usedfor N shells = 7. Although this fact complicates the applicability of this method for heavynuclei, for lighter systems the calculation with a smaller number of oscillator shells could14till be sufficient. This is the case for Mg, where the PES and the collective wave functionscalculated with N shells = 11 (not shown) are very similar to the N shells = 7 results. In Fig. 2we compare the spectra obtained in the two calculations and observe a relative error of lessthan 10% for all the levels. While the members of the σ = 1 bands almost match each other,small differences are found in the σ = 2 band. This comparison justifies that all furthercalculations are performed with N shells = 7. IV. CONVERGENCE AND CONSISTENCY OF THE TRIAXIAL PNAMP
In this section we will study some aspects of the simultaneous particle number and an-gular momentum projection with triaxial shapes. Firstly, it is important to note that theparametrization of the quadrupole deformation in terms of ( β, γ ) variables gives a triple de-generacy in the range 0 ◦ ≤ γ ≤ ◦ if we consider time-reversal conserving wave functions[2]. This degeneracy corresponds to the three possible orientations of the intrinsic axis I with respect to the z − axis (see Fig. 3). Therefore, the interval 0 ◦ ≤ γ ≤ ◦ covers all thepossible quadrupole deformations. However, we can take advantage of this symmetry firstto improve the convergence of the integral in the Euler angles that must be carried out inthe PNAMP calculation (Eq. 2) and second to perform consistency checks of the results.We now study the convergence of the integral in the Euler angles with respect to the numberof integration points in Ω = ( a, b, c ). We have considered the symmetries of the intrinsicwave function reducing the integration interval to (0 ≤ a ≤ π/ , ≤ b ≤ π/ , ≤ c ≤ π )(see Refs. [42, 47, 48]) and we have used Gaussian-Legendre quadratures for the numericalintegration. As in the axial case, the number of integration points for the particle numberprojection is kept to N F om = 9, which is sufficient to get eigenstates of the particle numberoperators. Naturally, the best candidate to check the convergence of the angular momen-tum projection is the expectation value of the total angular momentum operator ˆ I that,considering Eq. 5, must be: (cid:104) ˆ I (cid:105) IK = (cid:82) D I ∗ KK (Ω) (cid:104) Φ | ˆ I ˆ R (Ω) P N P Z | Φ (cid:105) d Ω (cid:82) D I ∗ KK (Ω) (cid:104) Φ | ˆ R (Ω) P N P Z | Φ (cid:105) d Ω = ¯ h I ( I + 1) . (24)The convergence in the number of integration points depends on three factors, namely theorientation of the intrinsic axes, the values of ( I, K ) and the deformation β . Let us startwith the two latter factors. In Fig. 4 we plot the mean value of the total angular momentum15 IG. 3: (Color online) Orientations of the intrinsic deformation as a function of the γ parameter. γ = 0 ◦ , ◦ , ◦ and γ = 60 ◦ , ◦ , ◦ correspond to axial symmetric prolate and oblate shapes,respectively. operator as a function of β for projected wave functions with I = 2 , γ = 50 ◦ . The integration has been performed with two sets of integration points in ( a, b, c ), S = (6 , ,
12) and S = (16 , , S the correctresult of the eigenvalue is obtained for all β and I, K . However, the set S fails both forlarge values of β for all I, K and also for smaller deformations with high K = 4 ,
6. Thepoor performance of this choice is clearly seen in the latter case where substantial deviationsfrom the correct number are observed. Therefore, as a rule of thumb, the larger the valuesof (
I, K ) and β the more integration points are needed to have good results. The finalchoice will be the one that is able to provide converged results for all ( I, K, β, γ ) values.Taking into account that the symmetry axis corresponds to pure K = 0 states, one mayassume that close to the symmetry axis only small K -components are present. We thereforeexamine the role of the orientation of the intrinsic axes in the PNAMP method. First, weexplore the convergence of the angular momentum projection using the property given in16 < I > / h _ ! =50 : S S I =2; K =0 I =2; K =2 4141.54242.54343.544 < I > / h _ I =6; K =0 I =6; K =20 0.2 0.4 0.6 0.8 1 " < I > / h _ I =6; K =4 0 0.2 0.4 0.6 0.8 1 " I =6; K =6 FIG. 4: (Color online) Expectation values of the total angular momentum operator calculated withangular momentum projected states | IK (cid:105) as a function of the β deformation ( γ = 50 ◦ ) and fordifferent sets of integration points in the Euler angles ( a, b, c ) (red circles S = (16 , , S = (6 , , I = 2 and I = 6, and theircorresponding K values, respectively. Fig. 3 and projecting symmetric states with the same value of β but with γ (cid:48) = 120 ◦ + γ . Ifour assumption is right, we could reduce the number of integration points using instead agiven wave function an equivalent intrinsic wave function with an orientation closer to the K = 0 case. In Fig. 5 we plot as a function of β the expectation values of the angularmomentum operator for intrinsic states with γ = 50 ◦ and also with γ (cid:48) = 170 ◦ . The sets ofintegration points are the same as in Fig. 4. For the set S with γ = 50 ◦ we observe againthe loss of convergence whenever β and I increase. However, very much improved results areobtained for the same set of integration points, S , but projecting the wave functions withthe γ = 170 ◦ orientation. In addition, the calculation with the set S reveals the numericalorigin of the lack of convergence for the set S with γ = 50 ◦ . Therefore, we will use thisproperty to define the mesh in the ( β, γ ) plane for performing GCM-PNAMP calculations17 < I > / h _ ! =50 : S S ! =170 : S I =0; K =0 66.577.58 I =2; K =0 1212.51313.514 < I > / h _ I =3; K =2 2020.52121.522 I =4; K =00 0.2 0.4 0.6 0.8 1 " < I > / h _ I =6; K =0 0 0.2 0.4 0.6 0.8 1 " I =8; K =0 FIG. 5: (Color online) Expectation values of the total angular momentum operator between angularmomentum projected states I = 0 , , , , K = 0 and I = 3; K = 2 as a function of the β deformation and for different sets of integration points in the Euler angles and orientation of theintrinsic axes, red circles S = (16 , , S = (6 , ,
12) with γ = 50 ◦ and bluefilled diamonds with γ = 170 ◦ . as we will see below.The analysis shown in Figs. 4 and 5 has been performed with diagonal matrix elements.Since in the GCM calculations we have to consider also non-diagonal matrix elements, wehave extended our study to this case. We find that in order to ensure a good convergencein all cases, the final set of integration points in the Euler angles has to be chosen as( N a = 8 , N b = 16 , N c = 16). We can also exploit the degeneracy illustrated in Fig. 3to perform a consistency test of the implementation of the PNAMP method [47]. Usingsymmetry properties of the point group D it can be shown, in the notation of eq. 2, that (cid:104) IM K ; N Z ; βγ = 60 ◦ | ˆ H | IM K ; N Z ; βγ = 60 ◦ (cid:105)(cid:104) IM K ; N Z ; βγ = 60 ◦ | IM K ; N Z ; βγ = 60 ◦ (cid:105) = (cid:104) I N Z ; βγ = 180 ◦ | ˆ H | I N Z ; βγ = 180 ◦ (cid:105)(cid:104) I N Z ; βγ = 180 ◦ | I N Z ; βγ = 180 ◦ (cid:105) , (25)i.e., the projected energy calculated with a HFB wave function with γ = 60 ◦ is K -independent and equal to the projected energy calculated with the HFB wave function18 E ( M e V ) + + + + + + + + + + K =0 K =2 K =4 K =6 K =8 mixed K =0
503 503281 281 281 281278 278 278 278 278 278297 297 297 297 297296 296 296 503281278296 503281278296 ! =60 ! ! =180 ! " =0.625 FIG. 6: (Left panel) excitation energies and B(E2) (in e fm units) values for the states beforeand after K mixing - | IM K (cid:105) and | IM (cid:105) respectively- with β = 0 .
625 and γ = 60 ◦ . (Right Panel)excitation energies and BE(2) values for the state | IM K = 0 (cid:105) with β = 0 .
625 and γ = 180 ◦ with γ = 180 ◦ . A similar relation applies for the transition probabilities. In Fig. 6 we showthe excitation energies and reduced transition probabilities B(E2) calculated with the sameoblate axially symmetric wave function ( β = 0 . γ = 60 ◦ (left panel) and γ = 180 ◦ (right panel). As expected, we find that the γ = 60 ◦ excitation spectrum and transition probabilities are K -independent and therefore identicalto the mixed ones. A look to the right panel corroborates also that these quantities co-incide with the ones generated with the γ = 180 ◦ intrinsic wave function. Once we haveanalyzed the convergence and consistency of the PNAMP method for a given point in the( β, γ ) plane we can study the potential energy surfaces (PES) for the different approaches(VAP-PN, PNAMP with and without K -mixing). We explore first the role of the mesh ofpoints needed to cover all different triaxial shapes. Given the better convergence propertiesfor wave functions with a large K = 0 component (compare Fig. 5), we divide the calculationinto two regions, γ ∈ [0 ◦ , ◦ ] and γ ∈ [150 ◦ , ◦ ] (see Fig.7). The last interval is equivalentto γ ∈ [30 ◦ , ◦ ] and we will transform the results to it whenever we plot the different PESthroughout this paper. Furthermore, the resolution of the PES is affected by the way weperform the discretization of the plane. In the lower panels of Fig. 7 we show the VAP-PNenergy surfaces for a constant step division both in β and γ directions (left part) and for19 !" . . . . . . !" . . . . . . !" . . . . . . !" . . . . . . ! ! ! ! -1.2 -0.8 -0.4 0 0.4 0.8 1.2 ! ! ! ! ! -1.2 -0.8 -0.4 0 0.4 0.8 1.2 ! PN-VAP PN-VAP
VAP-PN VAP-PN
FIG. 7: (Color online) Mesh of points using constant step in β and γ (top left) or triangle division(top right) and the corresponding calculated VAP-PN potential energy surfaces (lower panels)transformed to the interval γ (cid:15) [0 ◦ , ◦ ]. The energy is normalized to the minimum of the PES( − .
01 MeV) and the contour lines are divided in 1 MeV (black dashed lines) and 2 MeV steps(continuous magenta lines). a division based on equilateral triangles (right part). The number of points is N points = 99in both cases. We observe that the distribution of the points in constant steps is not thebest choice neither for small β , where for many points almost degenerated states are ob-tained, nor for large β , where a loss of resolution in γ is observed for increasing values of β .It is precisely in this region where the interpolation between distant points produces arti-facts or wrong results in the PES such as spurious oscillations, as for example in the region( β ∈ [1 . , . , γ ∈ [20 ◦ , ◦ ]) or softening of the contour plots ( β ∈ [0 . , . , γ ∈ [50 ◦ , ◦ ]).This is rectified with a discretization based on triangles and the results presented hereafterare calculated with this mesh. Nevertheless, although only small differences around the min-imum of the PES are obtained in the case of Mg, these effects will be enhanced for rather γ -soft and moderate β deformed nuclei. In those cases, the division based on triangles willgive much better results for the same number of total points included in the calculation andwill save computing time with respect to the other mesh.20 . TRIAXIAL CALCULATIONS FOR MG In the previous sections we have studied several aspects needed to ensure a good perfor-mance of the full generator coordinate method with the particle number and triaxial angularmomentum projected wave functions. This previous research is important because the fullGCM-PNAMP calculation is very demanding in CPU-time and both convergence tests andthe choice of the relevant parameters should be performed in advance, but nonetheless alsochecked afterwards. In this section the final results for Mg are presented, their calculationas mentioned above, have been done with the set of integration points in the Euler angles( N a = 8 , N b = 16 , N c = 16). We choose the triangular mesh with N points = 99 shown inFig. 7 to solve the constrained particle number projection before the variation (VAP-PN)equations. The intrinsic many body wave functions | Φ( β, γ ) (cid:105) are expanded in a cartesian har-monic oscillator basis and the number of spherical shells included in this basis is N shells = 7with an oscillator length of b = 1 . A / . In Fig. 7 the VAP-PN energy landscape is plottedshowing a single and well defined minimum at β = 0 . , γ = 0 ◦ separated by ∼ . ∼ . β = 0 .
25. Theseresults are consistent with the ones obtained in the axial calculation (see Fig. 1) with thedifference of having a saddle point in the ( β, γ ) plane instead of a minimum on the oblateside. Similar PES are obtained for Skyrme (HFB with particle number projection aftervariation (PN-PAV) included) [47] and Relativistic (BCS without PNP) [48] interactionsalthough a softer surface between the spherical point and the minimum is obtained for theSkyrme interaction.
A. Triaxial PNAMP potential energy surfaces and the RVAMPIR approach for Mg The solution of the triaxial HWG equation, Eq. 13, does not require to perform a separateangular momentum projection in the laboratory system for each component of the GCMbasis states in the sense of Eq. 22. However, as in the axial case, we expect the PNAMPpotential energy surfaces to provide insight and a better interpretation of the configurationmixing calculations. We can also separate the energy gain due to the triaxial AMP from the21 !" . . . . . . I =7 +1 !" . . . . . . I =8 +1 !" . . . . . . I =5 +1 !" . . . . . . I =6 +1 !" . . . . . . I =3 +1 !" . . . . . . I =4 +1 !" . . . . . . I =0 +1 !" . . . . . . I =2 +1 MeV
FIG. 8: (Color online) PNAMP potential energy surfaces including K -mixing in the ( β, γ ) planefor I = 0 − K -space. The PES are normalized to the minimumof the surfaces (-200.74, -199.43, -194.04, -196.61, -190.86, -192.27, -186.09, -185.33 MeV for I =0 , , , , , , ,
8, respectively). The contour lines are divided in 1 MeV (black dashed lines) and 2MeV steps (continuous magenta lines). States with projected norm less than 10 − are removed one due to the ( β, γ ) configuration mixing. Furthermore, they are very important becausethe minima of these PES determine the associated RVAMPIR solution. The PNAMP is aninvolved approach that requieres the solution of the HWG equation, Eq. 23, to include the K mixing. The HWG eigenstates, Eq. 22, provide real eigenstates of the symmetry operators22hat can be used, as we shall see below, to generate energy spectra and to calculate transitionprobabilities.In Fig. 8 we plot the normalized PNAMP energy landscapes in the ( β, γ ) plane for thelowest eigenvalue in the K -space for each angular momentum I = 0 +1 − +1 (see Eq.23).In addition, all the points close to the spherical one, and those close to axiality for oddvalues of I , have been removed for I (cid:54) = 0 because their norm is very small. The firstnoticeable aspect is that the VAP-PN axial minimum of Fig. 7 becomes a saddle point, theminimum being displaced towards larger β values and γ > ◦ for all values of the angularmomentum, although the barriers between the new minima and the axial prolate saddlepoints are less than 1 MeV. For I = 0 +1 , +1 the minima are located in ( β ∼ . , γ ∼ ◦ )while with increasing value of the angular momentum we observe a softening of the PES anda displacement of the minimum to larger γ and smaller β deformation, ( β ∼ . , γ ∼ ◦ )for I = 4 +1 , +1 and ( β ∼ . , γ ∼ ◦ ) for I = 6 +1 , +1 , +1 . We also note that in the case ofodd- I values the softening of the PES is in the γ direction towards the oblate saddle point.The energy difference between the VAP-PN and the I = 0 +1 minima is ∼ . ∼ . I = 0 , IK -projected energy E I ; NZK ( βγ ) = H I ; NZKβγKβγ N I ; NZKβγKβγ , (26)in the ( β, γ ) plane for the different K -values. Since the states | IM K ; N Z ; βγ (cid:105) are noteigenstates of the angular momentum in the laboratory frame their energies do not have aphysical meaning. Furthermore the IK -projected energy PES and wave functions dependon the orientation of the axis in Fig. 3. To illustrate this point we present in Fig. 9 PES’scalculated in three approaches for different orientations of the nucleus according to Fig. 3.We observe that, as expected, one sixth of the circle (for instance γ = 0 ◦ − ◦ ) is enoughto describe the PES corresponding to the VAP-PN and the PNAMP ones (corresponding to23 ! -1 0 1 060120180 240 300 ! -1 0 1060120180 240 300 ! -1 0 1 060120180 240 300 ! -1 0 1 060120180 240 300 ! -1 0 1 ! -1 0 1 VAPJ=0 J=2 K=0J=2 K=2 J=2 J=2 MeV MeV I =2 K =0 I =2 K =2 I =2 I =2 I =0 VAP-PN
FIG. 9: (Color online) Potential energy contour plots for Mg in the ( β, γ ) plane for γ = 0 ◦ − ◦ in different approaches and angular momenta normalized to the corresponding minima. Thecontour lines are divided in 1 MeV (black dashed lines) and 2 MeV steps (continuous magenta lines).( Left panels ) Top: Particle number projection (VAP), E min = − .
02 MeV; bottom: PNAMPapproach for I = 0¯ h , E min = − .
74 MeV. (
Middle panels ) IK -projected energies according toEq. 26. Top: I = 2 , K = 0, E min = − .
42 MeV; bottom: I = 2 , K = 2, E min = − .
78 MeV.(
Right panels ) Lowest eigenvalues of the PNAMP approach. Top: I = 2 , E min = − .
43 MeV;bottom: I = 2 , E min = − .
18 MeV. I = 0 and I = 2 , ). However for the K -projected PES’s ( I = 2 , K = 0 and I = 2 , K = 2)a semicircle (for instance γ = 0 ◦ − ◦ ) is needed. Since in the laboratory system all the sixsectors are equivalent we explicitly see that it is the same to use the region of γ = 150 ◦ − ◦ than γ = 30 ◦ − ◦ , as we have done in the GCM calculations. The contour plots in the IK projection can be easily understood looking at Fig. 3. For I = 2 , K = 0, the collective AMis perpendicular to the z-axis and since semi-classically a rotor will prefer to rotate aroundthe axis with the largest moment of inertia it is obvious that the energy minima are around24 + σ ( β, γ ◦ ) min E min ( M eV ) K = 0 K = ± K = ± +1 (0.696, 8.95) 0.000 — —2 +1 (0.696, 8.95) 1.311 +2 (0.696, 8.95) 5.556 0.000 —3 +1 (0.696, 8.95) 6.695 0.000 —4 +1 (0.661, 19.1) 4.129 +2 (0.661, 19.1) 8.116 0.000 +1 (0.661, 19.1) 9.883 0.0000 +1 (0.545, 23.4) 8.471 +2 (0.545, 23.4) 12.139 0.002 +1 (0.545, 23.4) 14.645 0.000 +1 (0.545, 23.4) 15.401 β and γ coordinates of the triaxial PNAMP minima after K -mixing as well as excitationenergies and distribution of K components (i.e., | H I ; NZ ; σK ( β, γ ) | see Eq. 22 and below) as a functionof I πσ . The values of ( β, γ ) min may not coincide exactly with those of Fig. 8 because of the finite sizeof the grid used in the calculations. The quoted values are the actual ones used in the K -mixingcalculation. The K = ± , ± γ = 0 ◦ . For I = 2 , K = 2, the collective AM is parallel to the z-axis and in this case theminima will be around γ = 120 ◦ and γ = 240 ◦ . Specially for the latter case we see thatit can be dangerous to make interpretations based on the γ = 0 ◦ − ◦ sector. For nucleiwith more mixing one should also care about the interpretation of the I = 2 , K = 0 surface.In any case, it is important to note that the K value is not a good quantum number inthe laboratory frame and therefore it is not an observable. In addition, the distributionof K and the corresponding PES can change depending on the orientation of the intrinsicwave function (see Fig. 3). Nevertheless, in cases where the K -mixing is not very large thisquantum number can be useful to give an interpretation of the different bands that couldappear in the spectrum. As we will see below, Mg is a very good example of rather pure | K | bands. One should be aware, however, that even with rather pure | K | = 2 bands, amixing of K = 2 and K = − E x ( M e V ) + + + + + + + + + + + Mg FIG. 10: RVAMPIR spectrum and transition probabilities.
As discussed in Sect. II B, the minima of the PNAMP potential energy surfaces provide anapproximation to an angular momentum projection in a variation after projection approach,which we have called RVAMPIR. In Table I we present the ( β, γ ) of the minima of the twolowest eigenstates together with the K -distribution of the corresponding wave functions.As we observe there is almost no mixing, the 0 +1 , +1 , +1 , +1 , +1 states are K = 0 and the2 +2 , +1 , +2 , +1 , +1 , K = 2. Only at the highest angular momentum we observe very small K -mixing. This is not the general rule. The amount of K-mixing depends strongly on thenucleus and on the ( β, γ ) point. As mentioned Mg seems to be a nucleus with rather small K -mixing. The solution of the HWG equation, Eq. 23, at the point ( β I min , γ I min ) providesthe RVAMPIR energies and wave functions of the corresponding states. In Fig.10 we presentthe energy spectrum and the calculated transition probabilities. Though we will discuss thisfigure in relation with the full GCM results we can compare with the Axial PNAMP-GCMexcitation spectrum of Fig.2. The clear difference is the presence of a well developed gammaband in the RVAMPIR calculations. B. The configuration mixing calculations for Mg The final step in the calculation of the spectrum is the GCM-PNAMP method, in whichsimultaneous mixing of the different deformations ( β, γ ) and K components is performed26 - - - - - - ! -205-200-195-190-185 E ( M e V ) +1 +2 +3 FIG. 11: (Color online) GCM-PNAMP energies ( I = 2) as a function of the corresponding normeigenvalue, normalized to the highest value, used as cutoff in the definition of the natural basis(Eq. 17). (see Eq. 1). As we mentioned in Sec. II A, we have to solve the HWG equations separatelyfor each value of the angular momentum. These generalized eigenvalue problems are solvedremoving the linear dependence of the states with the definition of the orthonormal naturalbasis (Eq. 17). In order to avoid spurious states in this basis, we have to define a cutoffparameter, ζ , to determine the states in the natural basis, see Eq. 17 and text below. Theconvergence of the triaxial PNAMP-GCM method is showed in Fig. 11 where the lowestthree energy values obtained for I = 2 are represented as a function of the parameter ζ .Here we distinguish a region of large ζ in which the energies are decreasing followed by arange of values where the energies are nearly constant. The appearance of these plateaus isthe signature of the convergence of the GCM method [67]. We observe that this plateau isbetter defined for the 2 +1 and 2 +2 states as compared to the 2 +3 . Finally, for small values of ζ the linear dependence shows up and we obtain meaningless values for the energy. The finalchoice for ζ is a value around which we observe a wide plateau for all the levels of interest.This value must be kept constant for a given angular momentum in order to guarantee theorthogonality of the corresponding wave functions. This analysis has been performed fordifferent values of the angular momentum showing in all cases a behavior similar to the onepresented in Fig. 11. Eventually, we have chosen ζ = 10 − as the final value, similar to theone found in Ref. [49]. This procedure can be complemented by an inspection of the shapeof the collective wave function as a function of ζ .In the central panel of Fig. 12 we now plot the spectrum of Mg extracted from the27 E x ( M e V ) + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + Axial Triaxial Experiment Mg FIG. 12: Calculated excitation energies and reduced transition probabilities B(E2) (in e f m )in Mg obtained using axially symmetric (left) and triaxial (middle) GCM-PNAMP approachescompared to the experimental values (right). The widths of the arrows are proportional to thecorresponding B values . The experimental values are taken from [68] triaxial GCM calculations. We classify the different levels in three bands according to thecorresponding B(E2) values. The ground state band is formed by a sequence of even angularmomentum states with a level spacing very similar to that of a rotational band. The firstexcited band consists of states with I = 2 , , , , ... as expected for a γ band. The third bandis built of even- I states on top of the second 0 +2 state. We can also compare the absolutevalue of the ground state energy calculated with different approaches. Evidently, the lowestvalue is obtained with the triaxial GCM-PNAMP method (-201.36 MeV) while -200.74 MeVand -200.67 MeV are the results for RVAMPIR and axial GCM-PNAMP approximationsrespectively. Comparing the first two values we observe that the energy gained by mixingdifferent shapes is ≈ ≈ + σ E (MeV) K = 0 K = ± K = ± K = ± K = ± +1 — — — —2 +1 +1 +1 +1 +2 — — —3 +1 — — —4 +2 +1 +2 +1 +2 — — — —2 +3 +3 K compo-nents ( (cid:80) β,γ | F I ; NZ ; σK ( β, γ ) | ) for the first, second and third bands. The highest values are printedin boldface. The excitation energies of the corresponding states are also provided. This fact indicates different underlying structures of the bands and the absence of mixingbetween them. We can study the nature of these bands decomposing the collective wavefunctions | F I ; NZ ; σ ( β, γ ) | (Eq. 19) into their K components, | F I ; NZ ; σK ( β, γ ) | , summing thecontribution of all deformations ( β, γ ) for each K . The result is shown in Table II , wherewe clearly observe that the first and third bands are rather pure K = 0 while the secondband corresponds mainly to | K | = 2 states. Furthermore, we see that for each level the ± K components have the same values, as a direct consequence of the time-reversal conservationof the intrinsic wave functions.The distribution of the states within these bands is supported by the values of the spectro-29 I h _ -1.5-1-0.500.511.5 Q ( I ) / Q ( ) FIG. 13: (Color online) Spectroscopic quadrupole moments as a function of the angular momentum I calculated for the states in the first (black bullets), second (blue diamonds) and third (magentatriangles) bands and for the vibrational-rotational collective model with K = 0 (red boxes) and K = 2 (black circles). scopic quadrupole moments: Q ( Iσ ) = (cid:115) π I I II − I (cid:104) I ; N Zσ || ˆ M elec2 || I ; N Zσ (cid:105) (27)where ˆ M elec2 µ = er Y µ ( θ, φ ) are the electrical quadrupole moment operators. In the collectiverotational model the spectroscopic quadrupole moments for a given | K | -band take the simpleform [2]: Q coll ( I, K ) = Q K − I ( I + 1)( I + 1)(2 I + 3) (28)with Q a constant deformation of the intrinsic macroscopic state. In Fig. 13 we comparethe triaxial results with the values given for the collective rotational model with K = 0and K = 2 -normalized to I = 2. Here we can clearly observe that the ground state bandcorresponds to a rotational band ( K = 0) while the second band matches to a γ -band( K = 2) and the third band cannot be described in this simple picture.We also plot the probability distribution | F I ; NZ ; σ ( β, γ ) | of each GCM state in the ( β, γ )surface (Fig. 14) summing all the possible K components. The most noticeable aspect isthat all the states belonging to the same band have a very similar probability distribution in30he ( β, γ ) plane and that the overlap between states of different bands is small. One couldassume that these facts will lead to the intraband and interband B(E2) values shown in Fig.12. However, as we shall see below, the reason for the small interband transitions seems tobe more related to a K -hindrance aspect based on the fact that the ground band is a pure K = 0 and the γ -band a rather pure | K | = 2 band. In particular, all the states in the firstband have a well defined maximum at ( β ∼ . , γ = 0 ◦ ) and the probability drops rathersymmetrically in the β and γ directions. For the second band, the probability distributionis concentrated in a region of the plane with ( β ∈ [0 . − . , γ ∈ [0 ◦ , ◦ ]) with maximaaround ( β ∼ . , γ ∼ ◦ ). Finally, the states belonging to the third band show a highprobability of having spherical shape (0 +2 ) or slightly prolate (2 +3 , +3 , +3 ) combined with anon-negligible mixing of strongly deformed states in the range of β ∈ [0 . , . , γ ∈ [0 ◦ , ◦ ].The PNAMP-PES of Fig. 8 can help to understand the probability distribution of the HWGequation. Looking at the ground band panels (0 +1 , +1 , +1 , +1 , +1 ) of Fig. 8 we find that allshow soft triaxial minima close to the axial axis, the contour lines being elongated along theradial direction and rather steep along the γ angle. These states will mix with the mirroredones at γ = 0 ◦ , see Fig. 9, and as a result distributions with a peak at γ = 0 ◦ similar tothe ones in the left panels of Fig. 14 are expected. If we now concentrate on the panels(3 +1 , +1 , +1 ), representative of the γ band, we observe contour lines centered around a softslightly triaxial minimum. These contours, at variance with the ones of the ground band,are softer in the γ angle. We found that the members of the γ band are rather pure K = 2states, that means, the norms of the states along the symmetry axis ( γ = 0) are zero. Thisaxis acts as a barrier between the states above γ = 0 ◦ and the mirrored ones hindering themechanism described for the ground band. As a result distributions similar to the ones inthe middle panels of Fig. 14 will be obtained.In Fig. 12 we have also compared the triaxial results with axial calculations. In orderto better understand the results of this comparison, we investigate first the relationshipbetween the axial and triaxial collective wave functions. The axial states emerge from the γ = 0 ◦ − ◦ path of the K = 0 component of the corresponding triaxial states. In particular,we can relate the ground state bands in both approaches and also the axial 0 +2 , +3 , +2 withthe triaxial 0 +2 , +3 , +3 states (see Figs. 1 and 14). Hence, the comparison between the triaxialand axial calculations reveals that both the energies and reduced transition probabilities ofthe ground state band are very similar in both cases, as expected. Nevertheless, the small K -31 !" . . . I =2 +2 !" . . . I =0 +2 !" . . . I =0 +1 ! . . . I =2 +1 ! . . . I =3 +1 ! . . . I =4 +1 ! . . . I =4 +2 ! . . . I =4 +3 ! . . . I =5 +1 ! . . . I =6 +1 ! . . . I =6 +3 ! . . . I =2 +3 FIG. 14: (Color online) GCM-PNAMP collective wave functions | F I ; NZ ; σ ( β, γ ) | for the groundstate (left), second (middle) and third (right) bands, respectively. Contour lines are separated in0.01 units. mixing for I (cid:54) = 0 lowers the excitation energies of higher angular momentum and therefore,the triaxial ground state band is slightly compressed with respect to the axial band. Thiseffect, although small, helps to improve the description of the moments of inertia within theGCM-PNAMP framework. Larger differences between the axial and triaxial calculationsappear for the second and third bands. Obviously, the axial calculations are unable todescribe the γ -band but also the energies and B(E2) for the third triaxial band ( K = 0mainly) are modified with respect to the corresponding ones in the axial case. This differenceis due to both the small K -mixing and the triaxial configuration around β ∼ . I = 0 (see Fig. 14).In Table III we present the average intrinsic deformation parameters and the spectroscopicquadrupole moments obtained for the axial and triaxial calculations. In general the average β deformation is larger in the triaxial than in the axial calculations. The largest differencescorrespond, obviously, to the states that compose the γ band in the triaxial case and to the32 +2 state due to the fake minimum on the oblate side of Fig. 1. Also interesting to noticeis that though the most probable γ -value in the first band is zero, the average γ values arearound 15 ◦ . Also the average γ values for the γ -band are larger than the most probableones. For completeness we also include the values of the spectroscopic quadrupole moments.At this point we would like to discuss the bands obtained in the RVAMPIR approach andplotted in Fig. 10. At first glance both bands look similar to the corresponding ones of thefull GCM calculations. A more careful analysis shows that the GCM bands are slightly morecompressed than the RVAMPIR ones in a better agreement with the experimental results.The transition probabilities are also very similar in both approaches. It is really surprisingthat the RVAMPIR is able to provide spectra and transition probabilities comparable tothe full GCM approach. There are several reasons which explain this behavior. A look toFig. 14 shows that all states of the first and second band, respectively, do have a similarprobability distribution consisting of one maximum -practically at the same ( β, γ ) pointfor all I -values- and a homogeneous spread around this point. Such distributions can bevery well approximated by a delta function at the given point. Furthermore since there isno K -mixing neither in the GCM nor in the RVAMPIR there is no chance that the twoapproaches can differ in this respect. The maxima of band 2 are located practically at thesame ( β, γ ) point in both approaches while for band 1 the GCM maximum appears closer tothe symmetry axis. However since the surfaces are rather flat around these points this doesnot matter too much. With respect to the B ( E
2) transition probabilities we observe thatthey are rather similar to the ones of the GCM calculations, i.e., they are large for intrabandand small for interband transitions. Interestingly the 2 +1 , the 2 +2 and the 3 +1 RVAMPIR statesdo have the same deformation parameters, see Table I, i.e., we cannot argue, as in the GCMcase, that the small interband transition probabilities are due to the poor overlap of thecorresponding wave functions. The reason is that the ground band is a pure K = 0 andthe γ -band a pure | K | = 2 band. As a matter of fact, in this case, if we look at Eq. 21,we observe that if the factors sandwiched between the collective wave functions do not mixstrongly the K quantum number, then the transition probabilities are very small.Finally we compare the triaxial results with the available experimental data for Mg(see Fig. 12). We find a remarkable qualitative agreement between theory and experimentboth in energies and reduced transition probabilities. In both cases we observe a rotationalground state band, a second band associated to a γ -band and a third band with ∆ I = 2. In33act, the theoretical description of the experimentally observed γ -band is one of the majorachievements of the present model compared to previous implementations. Furthermore,in the particular case of Mg, the excitation energies within the ground band are verywell described even quantitatively with the present calculations as we see in Fig. 12. Inaddition, it is important to emphasize the quality of the theoretical results for the intrabandand interband reduced transition probabilities which reflect the small K mixing between thecorresponding bands. Although the improvement of the results with respect to the axial caseis evident, the band heads of the γ - and, especially, the third band are still calculated too highin excitation energy. This is probably due to the lack of the correlations associated to theangular momentum restoration before the variation and time-reversal symmetry breakingthat are not included in this calculation. Additionally, the inclusion of two-quasiparticlestates would further lower the excitation energies for these band heads. These effects couldalso be present in the ground state bands. However, all these potential improvements arebeyond the scope of the present work. They would probably lead to a better quantitativedescription of the experimental results although we do not expect qualitative changes in thegeneral picture. Research in this direction is in progress. VI. SUMMARY
In summary, we have presented the first implementation of the GCM-PNAMP methodwith fully triaxial intrinsic wave functions found by solving the VAP-PN equations with theGogny interaction. Furthermore, due to the huge computational effort demanded by thistype of calculations, we have established a protocol for a good performance of the method,namely:1. Perform first a GCM-PNAMP with only axial K = 0 wave functions in order to choosethe number of oscillator shells, the relevant interval of β deformation and the densityof points in the collective variable.2. Study the convergence of the triaxial angular momentum projection with the numberof integration points in the Euler angles by looking at the expectation value of ˆ I inthe ( β, γ ) plane and exploit the symmetries of the intrinsic states.34 σ ¯ β tr ¯ γ ◦ tr Q spectr ¯ β ax γ ◦ ax Q specax
3. Choose a triangular mesh in the ( β, γ ) plane in order to avoid both redundancy nearthe spherical shape and spurious effects due to a loss of resolution for increasing β .4. Select the converged states as the ones whose energy belongs to a plateau and ensurethe orthogonality with the other states with the same angular momentum.5. Check the convergence of the results in the full triaxial calculation.The method has been applied to the study of Mg which has been chosen as a test casein previous studies with different interactions. The comparison between axial and triaxialresults shows minor changes in the ground state band which is predicted to be an axialrotational band with K = 0. Only for angular momentum I ≥ K -mixing is observedgiving rise to a small level compression. This result supports the use of axial calculations35n these cases. However, the triaxial calculation is also able to reproduce the second bandassociated to a γ -band ( K = 2) observed experimentally.We have also introduced the RVAMPIR method which provides a more affordable al-ternative to the full GCM procedure for the calculation of ground and γ bands. We findthat this approach provides a good description of the energy levels and the intraband andinterband transition probabilities for the nucleus Mg.Furthermore, the agreement between the theoretical and experimental results is in generalgood although some improvements beyond the scope of this work must be performed inorder to give a better quantitative description. Some work is in progress in order to takeinto account these effects. In any case, Mg is not a very good example for studying strongtriaxial effects like the ones mentioned in the introduction and the method will be appliedin the near future to other systems where both triaxiality and K -mixing play a crucial rolein describing the experimental data. Acknowledgments
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