We present recent developments obtained in the so-called N2LO extension of the usual Skyrme pseudo-potential. In particular, we discuss the isovector splitting mass in infinite nuclear matter and the pairing gaps of selected semi-magic even-even nuclei

Becker, Pierre

davesne, dany

Navarro, Jesus

Pastore, Alessandro

White Rose Research Online

This is a repository copy of Skyrme N2LO pseudo-potential for calculations of properties ofatomic nuclei.White Rose Research Online URL for this paper:https://eprints.whiterose.ac.uk/125108/Version: Accepted VersionArticle:Becker, Pierre, davesne, dany, Navarro, Jesus et al. (1 more author) (Accepted: 2017) Skyrme N2LO pseudo-potential for calculations of properties of atomic nuclei. Acta Physica Polonica B. ISSN 0587-4254 (In Press) eprints@whiterose.ac.ukhttps://eprints.whiterose.ac.uk/Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. Skyrme N2LO pseudo-potential for calculations ofproperties of atomic nucleiP. Beckera,b, D. Davesnea,b, J. Meyera,b, J. Navarroc, A.Pastoreda Université de Lyon, Université Lyon 1, 43 Bd. du 11 Novembre 1918, F-69622Villeurbanne cedex, Franceb CNRS-IN2P3, UMR 5822, Institut de Physique Nucléaire de Lyonc IFIC (CSIC-Universidad de Valencia), Apartado Postal 22085,E-46.071-Valencia, Spaind Department of Physics, University of York, Heslington, York, Y010 5DD,United KingdomWe present recent developments obtained in the so-called N2LO exten-sion of the usual Skyrme pseudo-potential. In particular, we discuss theisovector splitting mass in infinite nuclear matter and the pairing gaps ofselected semi-magic even-even nuclei.PACS numbers: 21.30.Fe 21.60.Jz 21.65.-f 21.65.Mn1. IntroductionSkyrme’s original idea [1] was to build an effective zero-range pseudo-potential as a momentum expansion of a given finite-range form factor. InRef. [2], we have explicitly discussed how it is possible to derive (startingfrom a finite-range interaction such as Gogny [3] or M3Y [4]), all the termsof the Skyrme interaction up to any order. At order three, the results arein agreement with the previous calculations done in Refs. [5, 6] when oneimposes that the N3LO1 pseudo-potential is invariant under Galilean andlocal gauge transformations.The main motivations behind exploring such extended versions of theSkyrme pseudo-potential have been discussed in Ref. [7]: the current dis-crepancies observed between predicted values with the standard Skyrme1 NℓLO with ℓ = 1, 2, 3, . . . is the name given to the pseudo-potential or a functionalwhen the highest power of the momentum operator kept before truncating the ex-pansion is k2ℓ.(1)2 mazurian printed on December 8, 2017pseudo-potential and the measured observables can’t be further reducedby using improved fitting procedures. It is thus time to explore richerfunctionals that may help us getting theoretical predictions closer to theexperimental measurements.The first decisive step can be found in Ref. [8] where the authors haveused for the very first time the extended N3LO functional [5] to test therole of higher order terms. By means of density matrix expansion (DME),they have shown that the next-to-next-to-leading order (N2LO) plays animportant role in reducing by roughly one order of magnitude the discrep-ancy between the exact result and the DME expansion. By extending theformalism to N3LO, the DME results further improve and get closer andcloser to the exact values, thus showing that the expansion converges.Then, in Ref. [9], we have performed the very first study of the extendedSkyrme N2LO pseudo-potential in the case of spherical even-even nuclei. Inthat article, we have obtained the first parametrisation of such a pseudo-potential using properties of some selected double-magic nuclei. In thepresent article, we continue our investigation, by extending our analysis toopen-shell nuclei and in particular exploring the behaviour of pairing gapsalong some isotopic chains [10] using the N2LO pseudo-potential.The article is organised as follows: in Sec. 2 we briefly summarise thekey-concepts of the N2LO Skyrme pseudo-potential. In Sec.3, we discuss theproperties of the effective mass and in Sec. 4 we discuss pairing propertiesof the SN2LO1 interaction. We finally provide our conclusions in Sec. 5.2. Skyrme N2LOThe N2LO Skyrme pseudo-potential, as described in Refs. [5, 6, 11], isa generalisation of the standard Skyrme interaction, corresponding to theexpansion of the momentum space matrix elements of a generic interactionin powers of the relative momenta k,k′ up to the fourth order. It is writtenas the sum of three terms [12]VN2LO = VCN2LO + VLSN1LO + VDDN1LO . (1)The central term readsV CN2LO = t0(1 + x0Pσ) +12t1(1 + x1Pσ)(k2 + k′2) + t2(1 + x2Pσ)(k · k′)+14t(4)1 (1 + x(4)1 Pσ)[(k2 + k′2)2 + 4(k′ · k)2]+t(4)2 (1 + x(4)2 Pσ)(k′· k)(k2 + k′2). (2)In the above expression, a Dirac function δ(r1−r2) is to be understood [13].The density-dependent term V DDN1LO and the spin-orbit term VLSN1LO have themazurian printed on December 8, 2017 3same structure as in the standard Skyrme interaction [14]. An alternativeto the use of a density-dependent term would be the inclusion of an explicitthree-body term. This possibility has been discussed in details in Ref. [15].From the interaction given in Eq. (1), we are in position to derive thecorresponding functional form by averaging over Hartree-Fock (HF) states.Since the focus of the article is the study of semi-magic even-even nuclei, welimit ourselves to the time-even spherically-symmetric case and we obtainE =∑t=0,1Cρt ρ2t + C∆ρt ρt∆ρt + Cτt ρtτt −12 CTt J2t + C∇Jt ρt∇ · Jt+ C(4)∆ρt (∆ρt)2 + C(4)Mρt(ρtQt + τ2t)+ C(4)Mρt[Re(τt,µν)Re(τt,µν) − Re(τt,µν)∇µ∇νρt]− C(4)Mst[(∇µJt,µν)2 + 4Jt,µνVt,µν − Im(Kt,µνκ)Im(Kt,µνκ)]. (3)We refer to Ref. [9] for a more detailed discussion on the properties of theN2LO functional. By comparing Eq. (3) with the standard form of theSkyrme functional (N1LO), see for example Ref. [16], we observe the ap-pearance of new densities: Vt,µν ,Kt,µνκ, Qt, τt,µν whose complete expressionscan be found in Ref. [9]. In this reference, we have also given some otherexpressions of scalar quantities related to these new densities. 0 0.04 0.08 0.12 0.16 0.2 0.24 0.28 0.32 0  2  4  6  8  10  12Densitiesr [fm]ρ0[fm-3]τ0[fm-5]J0[fm-4]Q0[fm-7]V0[fm-6]K10[fm-5]K20[fm-5]Fig. 1. (Colors online) Isoscalar densities in 132Sn obtained with SN2LO1 func-tional. See text for details.In order to have a physical insight and give some order of magnitude, weshow in Fig. 1, the radial profile of all these local densities for 132Sn using4 mazurian printed on December 8, 2017the SN2LO1 interaction. We observe that the density Q has similar orderof magnitude and shape as the kinetic density τ , while the other additionaldensities are peaked at the surface of the nucleus and almost zero in thebulk of the nucleus. It is presently difficult to estimate the respective roleof these new densities since only one parametrisation is available, but weplan to investigate in the near future their properties in a more systematicway along the nuclear chart.3. Effective massThe value of the effective mass at saturation density [17] has a strongimpact on the density of states around the Fermi energy and thus on the un-derlying pairing properties. In the fitting protocol used to adjust SN2LO1,we have imposed an explicit constraint on the value of the effective massm∗/m at saturation density (ρ0=0.16 fm−3) in infinite symmetric nuclearmatter (SNM). From our fit, we have obtained m∗/m = 0.71; such a valueshould be compared with the one derived for SLy5* m∗/m = 0.7 and fittedwith a similar protocol [18]. In Fig. 2 a, we show the evolution of the effec-tive neutron mass, m∗n/m, as a function of the density ρ for SNM and pureneutron matter (PNM) for both interactions.0 0.08 0.16 0.24 0.32ρ [fm-3]0.50.60.70.80.91m*n/mρ0SN2LO1 (SNM)SN2LO1 (PNM)SLy5* (SNM)SLy5* (PNM)a)0 0.2 0.4 0.6 0.8 1Y0.60.81m*/mSN2LO1 (n)SN2LO1 (p)SLy5* (n)SLy5* (p)ρ0=0.16 fm-3b)Fig. 2. (Colors online) Panel a: neutron effective mass calculated in SNM andPNM for the two interactions as a function of the density of the infinite system.The vertical dashed line represents the saturation density ρ0 of SNM. See text fordetails. In Panel b we show the evolution of neutron and proton effective massesat saturation density and in function of the asymmetry parameter Y.We observe that in both cases (SLy5* and SN2LO1), the behaviour of neu-tron effective mass is very similar. In both cases, we observe the absence ofpoles in the effective mass up to very high densities. In Fig. 2 b, we showmazurian printed on December 8, 2017 5the evolution of proton and neutron effective masses as a function of theasymmetry parameter Y [19], from SNM (Y=0) to PNM (Y=1). We noticethat mass splitting is essentially zero at saturation density. The splitting isslightly bigger in SN2LO1 than in SLy5*, but they are very close to zero.We refer the reader to Ref. [17] for a devoted study of the isovector masssplitting in Skyrme functionals.4. Pairing gapsIn this section, we perform a first systematic study of the pairing gapsusing the SN2LO1 functional. As for the SLy5* functional [18], it hasbeen fitted using properties of infinite nuclear matter and doubly-magicnuclei with the additional stability constraint coming from Linear Responsetheory [20, 21, 22]. This particular choice leaves us complete freedom indetermining the parameters entering the pairing channel. Therefore, forthe current analysis, we have decided to use the numerical code WHISKY [9]to solve the Hartree-Fock-Bogoliubov equations (HFB) [23] for a simpledensity dependent pairing interaction of the form [24]Vpair(r1, r2) = V0[1− ηρ(R)ρ0]δ(r1 − r2) , (4)where R = (r1 + r2)/2 is the center of mass of the interacting particlesand V0 is the strength of the interaction. To avoid an ultraviolet divergencyassociated with such a contact interaction [25], we adopted a sharp-cut off inthe quasi-particle space Ecut = 60 MeV. It is then possible to determine thepairing field ∆(r). As it is well-known, its shape can be modified by settingthe value of the parameter η to 0, 1/2 and 1, thus producing a volume,mixed and surface pairing field [26], respectively. To quantify the amountof pairing correlations, we calculated the average pairing gap defined as [27]∆̄ =∫∆(r)ρ̃(r)dr∫ρ̃(r)dr, (5)where ρ̃q is the pairing density. The results are depicted in Fig. 3. Theaverage pairing gaps have been obtained using SN2LO1 for the three valuesof η = 0, 1/2, 1 mentioned above as a function of the neutron number N .The experimental values have been extracted from experimental bindingenergies [28] using the standard the three-point formula [29]. Since ournumerical code works in spherical symmetry only, we limited ourselves tothe four isotopic chains of semi-magic nuclei.For these calculations, we have used a pairing strength of V0 = −200 MeVfm3 (volume), V0 = −300.2 MeV fm3 (mixed) and V0 = −496.6 MeV fm36 mazurian printed on December 8, 201720 22 24 26 28 30 32 34N0.511.52∆ [MeV]ExpVolumeMixedSurface Ca30 32 34 36 38 40 42 44 46N0.511.52∆ [MeV]Ni60 64 68 72 76 80 84N0.511.52∆ [MeV]Sn104 108 112 116 120 124 128 132N0.511.52∆ [MeV]PbFig. 3. (Colors online) Evolution of average pairing gaps for Ca, Ni, Sn and Pbisotopic chains in function of the neutron number N . Full symbols correspond tothe SN2LO1 interaction, while the empty ones to SLy5*.The theoretical pairinggaps have been calculated according to Eq.5. See text for details.(surface). Since the pairing interaction is not adjusted during the fittingprocedure, we adjusted in a completely arbitrary way the pairing strengthto the same value of pairing gap in 120Sn. From this figure we can concludethat the underlying single-particle spectrum is reasonable and the resultingpairing gaps give a reasonable description of available experimental data.For completeness, we have also repeated the same calculations using SLy5*for the mean field and the same pairing interaction. In the figure, we havereported only (for clarity) the case of volume-type pairing interaction (samestrength). The two other not reported cases give similar results comparedto the SN2LO1 ones. The small differences observed among the two interac-tions are due to small differences in the underlying single-particle structure.mazurian printed on December 8, 2017 75. ConclusionsIn this article, we have discussed some important properties of the ex-tended Skyrme interaction N2LO. By using the set of parameters deter-mined in Ref. [9], we have first studied the behaviour of the effective massas a function of the density and isospin asymmetry of the infinite nuclearmedium. The isovector mass splitting is still not compatible with ab−initiofindings [30], but the current results go in the right direction. Therefore,one may expect, adding such explicit constraint into the fitting protocol,that we should be able to obtain a positive splitting [19].We have discussed the isotopic evolution of pairing gaps for some rel-evant isotopic chains of semi-magic nuclei using the SN2LO1 functionalplus density-dependent contact pairing interaction. At present the extendedfunctional provides us with results that are qualitatively of the same levelof accuracy as SLy5*, both being fitted with a very similar fitting protocolincluding the explicit constraint on the linear response of infinite nuclearmatter [22]. This is an improvement compared to the vast majority ofexisting functionals that manifest instabilities in the spin-channels. Unfor-tunately at present, we have not been able to find a quantitative indicatorthat proves the necessity of using the extended pseudo-potential comparedto the simpler SLy5*. A more rigorous statistical analysis of the propertiesof the new functional is now mandatory [31] as well as a rethinking of thepenalty function used during the fit to identify the most relevant observ-ables that may help better constraining higher order parameters and avoidsloppiness [32] in some directions of parameter space.AcknowledgmentsThe work of J.N. has been supported by grant FIS2014-51948-C2-1-P,Mineco (Spain). The work of A.P. is supported by the UK Science and Tech-nology Facilities Council under Grants No. ST/L005727 and ST/M006433.REFERENCES[1] T.H.R. Skyrme, Nucl. Phys. 9, 615 (1959).[2] D. Davesne, P. Becker, A. Pastore, J. Navarro, Ann. Phys. 375, 288 (2016).[3] J. Dechargé, D. Gogny, Phys. Rev. C 21, 1568 (1980).[4] H. Nakada, ibid. 68, 014316 (2003).[5] B.G. Carlsson, J. Dobaczewski, M. Kortelainen, ibid. 78, 044326 (2008).[6] F. Raimondi, B.G. Carlsson, J. Dobaczewski, ibid. 83, 054311 (2011).8 mazurian printed on December 8, 2017[7] M. Kortelainen, J. McDonnell, W. Nazarewicz, E. Olsen, P-G Reinhard,J. Sarich, N. Schunck, S.M. Wild, D. Davesne, J Erler, et al., ibid. 89,054314 (2014).[8] B.G. Carlsson, J. Dobaczewski, Phys. Rev. Lett. 105, 122501 (2010).[9] P. Becker, D. Davesne, J. Meyer, J. Navarro, A. Pastore, Phys. Rev. C 96,044330 (2017).[10] D.M. Brink, R.A. Broglia, Nuclear Superfluidity: pairing in finite systems,Cambridge University Press 2005.[11] D. Davesne, A. 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Skyrme N2LO pseudo-potential for calculations of properties of atomic nuclei

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Skyrme N2LO pseudo-potential for calculations of properties of atomic nuclei

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