New density-independent interactions for nuclear structure calculations

This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail. Author(s): Title: Year: Version: Please cite the original version: All material supplied via JYX is protected by copyright and other intellectual property rights, and duplication or sale of all or part of any of the repository collections is not permitted, except that material may be duplicated by you for your research use or educational purposes in electronic or print form. You must obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not an authorised user. New density-independent interactions for nuclear structure calculations Bennaceur, K.; Dobaczewski, Jacek; Raimondi, F. Bennaceur, K., Dobaczewski, J., & Raimondi, F. (2014). New density-independent interactions for nuclear structure calculations. In S. Lunardi, P. Bizzeti, S. Kabana, C. Bucci, M. Chiari, A. Dainese, P. D. Nezza, R. Menegazzo, A. Nannini, C. Signorini, & J. Valiente-Dobon (Eds.), INPC 2013 – International Nuclear Physics Conference Firenze, Italy, June 2-7, 2013 (Article 02031). EDP Sciences. EPJ Web of Conferences, 66. https://doi.org/10.1051/epjconf/20146602031 2014 New density-independent interactions for nuclear structure calculations K. Bennaceur1, J. Dobaczewski2,3, and F. Raimondi4 1 Université de Lyon, F-69003 Lyon, France; Institut de Physique Nucléaire de Lyon, CNRS/IN2P3, Université Lyon 1, F-69622 Villeurbanne Cedex, France 2 Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, ul. Ho˙za 69, PL-00681 Warsaw, Poland 3 Department of Physics, PO Box 35 (YFL), FI-40014 University of Jyväskylä, Finland 4 TRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia V6T2A3, Canada Abstract. We present a new two-body finite-range and momentum-dependent but density-independent effective interaction, which can be interpreted as a regularized zerorange force. We show that no three-body or density-dependent terms are needed for a correct description of saturation properties in infinite matter, that is, on the level of lowenergy density functional, the physical three-body effects can be efficiently absorbed in effective two-body terms. The new interaction gives a satisfying equation of state of nuclear matter and opens up extremely interesting perspectives for the mean-field and beyond-mean-field descriptions of atomic nuclei. The two most widely used effective forces for nonrelativistic energy-density-functional calculations, the Gogny [1, 2] and the Skyrme interaction [3], have in common a two-body density dependent term. For the case of the Skyrme force, the original three-body contact force of the form t3δ(r1−r2)δ(r1−r3) [3] was in fact replaced by an isoscalar density-dependent two-body contact force, 1 6t3(1+x3ˆ Pσ)ρα 0(r1+r2 2)δ(r1−r2), due to the appearance of spin instabilities [4–6]. Later, it was employed as a convenient way of simulating the density dependence of the effective interaction rather than a three-body force [7]. For Skyrme and Gogny effective interactions, the density dependent term appears to play a determinant role, especially in generating the mechanism of saturation. The non-integer powers of the density seem to be mandatory if one wants to obtain acceptable values for incompressibility and isoscalar effective mass. These considerations concern only the description of nuclei at the mean-field level. However, several years ago it was identified that effective interactions that depend on non-integer powers of density do not allow for beyond-mean-field calculations that use standard techniques of symmetry restoration or Generator Coordinate Method [8, 9]. This observation triggered efforts to define a new generation of effective interactions without density dependence. A promising way is to consider two- and three-body momentum dependent zerorange terms along with a four-body momentum independent one [10]. Another way, discussed in this study, consists in using a form very similar to the two-body part of the Skyrme interaction and replacing the δDirac functions by finite-range form factors. This corresponds to the next-to-leading-order expansion of the effective interaction introduced in Ref. [11]. DOI: 10.1051/ C Owned by the authors, published by EDP Sciences, 2014 , / 02031 (2014) 201 66 epjconf EPJ Web of Conferences 46602031 This is an Open Access article distributed under the terms of the Creative Commons Attribution License 2.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Article available at http://www.epj-conferences.org or http://dx.doi.org/10.1051/epjconf/20146602031 Below we show that the use of a finite-range momentum-dependent, but density-independent, two-body interaction allows us to describe most of the medium effects in a realistic way. We use the notation x≡(r, σ, q) where σand qare the spin and isospin projections. For a general nonlocal two-body effective interaction v, the mean-field average value of the potential energy can be written as ⟨V⟩=1 2∫dx1dx2dx3dx4v(x1,x2;x3,x4)[ρ(x3,x1)ρ(x4,x2)−ρ(x4,x1)ρ(x3,x2)], where ρ is the nonlocal one-body density. Following Ref. [11], we choose the interaction as: v= 2 ∑ k=0[T(1) kˆ δσq+T(2) kˆ δqˆ Pσ−T(3) kˆ δσˆ Pq−T(4) kˆ Pσˆ Pq]ˆ Ok(k∗ 12,k34)δ(r1−r3)δ(r2−r4)ga(r2−r1),(1) where ˆ Pσand ˆ Pqdenote the standard spin and isospin exchange operators, respectively, ˆ δσ,ˆ δq, and ˆ δσqare the identity operators in the spin and/or isospin spaces, and ˆ O0=1, ˆ O1=1 2[k∗2 12 +k2 34],ˆ O2= k∗ 12 ·k34, and ga(r)=e−r2/a2/(a√π)3. Here, ki j represents the relative momentum operator between particles iand jand the regularized δfunction ga(r) defines the profile of the interaction, with arepresenting the regularization scale and range of the interaction. This effective interaction depends on 13 parameters, that is, on 12 parameters T(i) k, for k=0,1,2 and i=1,2,3,4, which control the strengths in each channel, and on the range a. As usual, integration by parts of matrix elements transforms the nonlocal interaction (1) into a local finite-range momentum-dependent pseudopotential. Although a construction of the analogous finite-range spin-orbit force poses no difficulty, in the present implementation we use the standard zero-range form of it, with one strength parameter W0. By using methods of symbolic programming, one can derive the nuclear-matter characteristics of interaction (1). For that, we introduce the following auxiliary functions: F0(ξ)=12 ξ3 1−e−ξ2 ξ3−3−e−ξ2 2ξ+√π 2Erf ξ,(2) F1(ξ)=48 ξ8(1−e−ξ2)+(12 ξ6+6 ξ4)(e−ξ2−5)+(1+5 ξ2)6√π ξ3Erf ξ , (3) F2(ξ)=720 ξ10 (e−ξ2−1)+120 ξ8(e−ξ2+5)+60 ξ6(e−ξ2−5)+60 √π ξ5Erf ξ , (4) G0(ξ)=12 ξ6(e−ξ2−1)+6 ξ4(e−ξ2+1),(5) G1(ξ)=12 ξ6(1−e−ξ2)−12 ξ4−3 ξ2(1+e−ξ2)+6√π ξ3Erf ξ , (6) G2(ξ)=60 ξ8(e−ξ2−1)+30 ξ6(3e−ξ2−1)+15 ξ4(e−ξ2−1)+30 √π ξ5Erf ξ , (7) and H0(ξ)=(1−e−ξ2)/ξ2,H1(ξ)=G0(ξ) . In the following, we use these functions at ξ=kFa, which fixes their dependence on the Fermi momentum kFand range a. With definitions Aρ0 i=1 2T(i) 1+1 4T(i) 2− 1 4T(i) 3−1 8T(i) 4,Bρ0 i=−1 8T(i) 1−1 4T(i) 2+1 4T(i) 3+1 2T(i) 4,Aρ1 i=−1 4T(i) 3−1 8T(i) 4, and Bρ1 i=−1 8T(i) 1−1 4T(i) 2we then have the energy per particle, E A, incompressibility, K∞, effective mass, m∗, symmetry energy, J, slope of the symmetry energy, L, and curvature of the symmetry energy, Ksym, given as: E A=~2 2m τ0 ρ0 +[Aρ0 0+Bρ0 0F0(ξ)]ρ0+1 2(Aρ0 1+Aρ0 2)τ0+1 2(Bρ0 1−Bρ0 2)[F1(ξ)−ξ2 10 F2(ξ)]τ0,(8) K∞=−2~2 2m τ0 ρ0 +Bρ0 0[4ξF′ 0(ξ)+ξ2F′′ 0(ξ)]ρ0+5(Aρ0 1+Aρ0 2)τ0 +1 2(Bρ0 1−Bρ0 2)[10F1(ξ)+8ξF′ 1(ξ)+ξ2F′′ 1(ξ)−14ξ2 5F2(ξ)−6ξ3 5F′ 2(ξ)−ξ4 10 F′′ 2(ξ)]τ0,(9) ~2 2m∗=~2 2m−1 2Bρ0 0a2ρ0G0(ξ)+1 2(Aρ0 1+Aρ0 2)ρ0+1 2(Bρ0 1−Bρ0 2)ρ0[G1(ξ)−ξ2 5G2(ξ)],(10) EPJ Web of Conferences 02031-p.2 J=5 9 ~2 2m τ0 ρ0−Bρ0 0 ξ2 6G0(ξ)ρ0+5 18 (Aρ0 1+Aρ0 2)τ0+5 18 (Bρ0 1−Bρ0 2)[G1(ξ)−ξ2 5G2(ξ)]τ0 +[Aρ1 0+Bρ1 0H0(ξ)]ρ0+5 6(Aρ1 1+Aρ1 2)τ0+5 6(Bρ1 1−Bρ1 2)[H0(ξ)−ξ2 6H1(ξ)]τ0,(11) L=10 9 ~2 2m τ0 ρ0−Bρ0 0[5ξ2 6G0(ξ)+ξ3 6G′ 0(ξ)]ρ0+25 18 (Aρ0 1+Aρ0 2)τ0+3Aρ1 0ρ0 +1 18 (Bρ0 1−Bρ0 2)[25G1(ξ)−7ξ2G2(ξ)+5ξG′ 1(ξ)−ξ3G′ 2(ξ)]τ0+3Bρ1 0[H0(ξ)+ξ 3H′ 0(ξ)]ρ0 +25 6(Aρ1 1+Aρ1 2)τ0+25 6(Bρ1 1−Bρ1 2)[H0(ξ)+ξ 5H′ 0(ξ)−7ξ2 30 H1(ξ)−ξ3 30 H′ 1(ξ)]τ0,(12) Ksym =−10 9 ~2 2m τ0 ρ0−Bρ0 0[5ξ2 3G0(ξ)+4ξ3 3G′ 0(ξ)+ξ4 6G′′ 0(ξ)]ρ0+25 9(Aρ0 1+Aρ0 2)τ0 +1 9(Bρ0 1−Bρ0 2)[25G1(ξ)−14ξ2G2(ξ)+20ξG′ 1(ξ)−6ξ3G′ 2(ξ)+5ξ2 2G′′ 1(ξ)−ξ4 2G′′ 2(ξ)]τ0 +25 3(Bρ1 1−Bρ1 2)[H0(ξ)+4ξ 5H′ 0(ξ)+ξ2 10 H′′ 0(ξ)−7ξ2 15 H1(ξ)−ξ3 5H′ 1(ξ)−ξ4 60 H′′ 1(ξ)]τ0 +Bρ1 0[4ξH′ 0(ξ)+ξ2H′′ 0(ξ)]ρ0+25 3(Aρ1 1+Aρ1 2)τ0.(13) In the present study, we present results for two preliminary sets of a=0.8 fm parameters. First, by using conditions T(i) 2=−T(i) 1, we obtained parameters REG2a.130531 that correspond to a local finiterange momentum-independent potential [11]. Second, by releasing these constraints, we obtained parameters REG2b.130531. On the one hand, both sets correspond to the same values of ρsat = 0.16 fm, E/A=−16, K∞=230, and J=32. On the other hand, they correspond to different values of L=100.2 and 58, Ksym =83.26 and −175, and m∗/m=0.38 and 0.41, respectively (all energies are in MeV). Values of parameters are listed in Table 1. In both cases, saturation in symmetric matter is obtained through the interplay between the T(i) 0attractive and T(i) 1and T(i) 2repulsive terms. In Fig. 1 we show equations of state (EOS) for symmetric, neutron, polarized symmetric and polarized neutron matter for the effective interactions Skyrme SV [7] (zero-range density-independent), Gogny D1N [12] (finite-range density-dependent) and the two regularized δinteractions. The Skyrme SV interaction is the only two-body density-independant interaction commonly used nowadays while D1N is a succesful finite-range density-dependent interaction. The regularized δinteractions have effective masses that are too low at saturation density but a more realistic incompressibility, similar to that obtained with D1N. Different dependences on momenta and densities of the four EOS make their behavior in neutron, polarized symmetric, and polarized neutron matter very different, although symmetric matter remains the ground state upto very high densities. 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.2 0.4 0.6 0.8 1.0 m*/m -20 0 20 40 60 0.0 0.1 0.2 0.3 0.4 0.5 E/A (MeV) 0.0 0.1 0.2 0.3 0.4 0.5 REG2bD1NSV REG2a 0.0 0.1 0.2 0.3 0.4 0.5 ρ (fm -3 ) Figure 1. Equation of state of the symmetric (full circles), neutron (full squares), polarized symmetric (open circles), and polarized neutron (open squares) infinite matter (left scale). The effective mass (right scale) is shown with diamonds. INPC 2013 02031-p.3 Table 1. Values of coupling constants T(i) kthat define the a= 0.8 fm parametrizations REG2a.130531 (top, T(i) 2=−T(i) 1) and REG2b.130531 (bottom) of the regularized δinteraction (1), in units of MeV fm3(k=0) and MeV fm5(k=1,2). Standard zerorange spin-orbit forces with W0=209.30 and 168.35 MeV fm5, respectively, were used. T(i) ki=1i=2i=3i=4 k=0−968.645 1645.515 −1400.680 −451.892 k=1−653.038 1349.673 −2011.063 1692.524 k=0−12250.143 7277.075 −6952.679 10744.723 k=1−1149.333 1594.666 −2342.666 2413.333 k=2 3184.584 −513.696 4681.530 −5127.553 -10 -5 0 5 10 SV REG2a REG2b 40Ca 48Ca 56Ni 78Ni 100Sn 132Sn 208Pb E - Eexp (MeV) Figure 2. Deviations of ground-state energies of doubly magic nuclei from experimental data. To calculate finite nuclei, in the HFODD (v2.64p) solver [13] we implemented self-consistent solutions for the regularized δinteraction (1). Masses of doubly magic nuclei are shown in Fig. 2. We see that REG2a.130531 and, even more, REG2b.130531 represent an improvement compared to the Skyrme SV interaction. The new effective interaction presented in this work seems to be promising. Although it does not contain any density-dependent term, it reproduces all empirical properties of the saturation point but the isoscalar effective mass, which is rather low. To our knowledge, this has never been achieved with any other density independent two-body interaction. Since this interaction has a finite range, it can be used in any calculation at the mean field level or beyond, without the need to introduce any additional momentum cut-off. This work has been supported in part by the Academy of Finland and University of Jyväskylä within the FIDIPRO programme and by the Polish French agreement COPIN-IN2P3 Project no. 11- 143. 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