Renormalized interactions with a realistic single particle basis
aa r X i v : . [ nu c l - t h ] D ec Renormalized interactions with a realistic single particle basis
Angelo Signoracci , B. Alex Brown , and Morten Hjorth-Jensen Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory,Michigan State University, East Lansing, Michigan 48824-1321, USA and Department of Physics and Center of Mathematics for Applications, University of Oslo, N-0316 Oslo, Norway (Dated: November 12, 2018)Neutron-rich isotopes in the sdpf space with Z ≤
14 require modifications to derived effectiveinteractions to agree with experimental data away from stability. A quantitative justification is givenfor these modifications due to the weakly bound nature of model space orbits via a procedure usingrealistic radial wavefunctions and realistic NN interactions. The long tail of the radial wavefunctionfor loosely bound single particle orbits causes a reduction in the size of matrix elements involvingthose orbits, most notably for pairing matrix elements, resulting in a more condensed level spacingin shell model calculations. Example calculations are shown for Si and Si.
PACS numbers: 21.60.Cs, 21.60.Jz
I. INTRODUCTION
New facilities for rare isotope beams will push the ex-perimental capabilities of nuclear physics with radioac-tive beams to more unstable, shorter-lived nuclei. Prop-erties of these nuclei exhibiting different behavior thanstable nuclei, like the evolution of shell structure, are ofsignificant interest for the next decades of research. Anew theoretical technique and its behavior for stable andexotic nuclei has been studied to examine the importanceof refining theoretical approaches for the production ofmodel space interactions for unstable nuclei.Much research has been done using renormalizationmethods to convert a realistic interaction fit to nucleon-nucleon (
N N ) scattering data into an interaction in thenuclear medium. The goal is to renormalize the interac-tion to valence orbits outside of a stable, semi-magic ordoubly magic nucleus treated as a vacuum in further cal-culations. A typical example would use O as the coreand renormalize the
N N interaction into the sd modelspace. For such an application, the harmonic oscillatorbasis of the form Ψ nlm l ( ~r ) = R HOnl (r) Y lm l ( θ , φ ) is gener-ally used. Additionally, all the valence orbits are boundin the harmonic oscillator basis. For more exotic closed-subshell nuclei, loosely bound orbits often play a role.The harmonic oscillator basis is less applicable furtherfrom stability. Loosely bound orbits particularly deviatefrom the oscillator basis, as they exhibit a “long-tail” be-havior with a larger spread in the radial wavefunctions.However, few calculations have been done with a realis-tic radial basis for unstable nuclei with renormalized N N interactions.Experimental interest in neutron-rich silicon isotopesand the failure of some shell model Hamiltonians to re-produce data in the region have led to modifications inthe SDPF-NR interaction [1], which had been the stan-dard for shell model calculations in the sdpf model space.The new SDPF-U interaction has different neutron-neutron pairing matrix elements for Z ≥
15 and Z ≤
14 to account for the behavior of pf neutron orbits relativeto the number of valence protons. The Z ≤
14 version ofthe interaction treats neutron-rich unstable nuclei thatexhibit different shell behavior than the less exotic nu-clei in the Z ≥
15 nuclei. The interest in silicon isotopesand the nature of the SDPF-U interaction make Si asuitable choice for the renormalization procedure with arealistic basis. A similar effect occurs for the neutron-rich carbon isotopes around the N = 14 closed subshell,requiring a 25% reduction in the neutron-neutron two-body matrix elements from the effective interactions de-rived for the oxygen isotopes [2]. II. RENORMALIZATION PROCEDURE
We begin with the realistic charge-dependent
N N in-teraction N LO derived at fourth order of chiral pertur-bation theory with a 500 MeV cutoff and fit to experi-mental
N N scattering data [3]. The N LO interactionis renormalized using a similarity transformation in mo-mentum space with a sharp cutoff of Λ = 2.2 fm − toobtain the relevant low momentum interaction [4]. Wewill refer to this technique as a v lowk renormalization.Skyrme Hartree-Fock calculations are performed with theSkxtb interaction [5] for a chosen closed sub-shell targetnucleus to determine the binding energy, single particleradial wavefunctions, and single particle energy spectrafor neutrons and protons of the target nucleus. The lowmomentum interaction is then renormalized into a modelspace of interest using Rayleigh-Schr¨odinger perturba-tion theory [6] to second order including excitations upto 6¯ hω , summing over folded diagrams to infinite order.We will compare three options for the renormalizationto produce an effective interaction: harmonic oscillatorsingle particle energies and wavefunctions (HO), SkyrmeHartree-Fock single particle energies and wavefunctions(SHF), and Skyrme Hartree-Fock single particle ener-gies and harmonic oscillator single particle wavefunctions TABLE I: Single-particle energies for Si and Ca using theSkxtb interaction. Values in bold are in the model space.n l j Si Si Ca Caproton neutron proton neutron0 s / -37.73 -32.79 -30.49 -38.180 p / -27.60 -23.10 -22.14 -29.700 p / -22.39 -21.74 -19.03 -26.670 d / -17.29 -13.07 -12.79 -20.200 d / -9.08 -9.03 -7.23 -14.651 s / -13.49 -10.04 -8.31 -15.750 f / -5.97 -2.62 -2.68 -9.89 f / -2.43 p / -1.06 -0.40 -5.48 p / -0.27 -3.66 g / g / (CP).The CP basis and HO basis give identical results tofirst order in perturbation theory since they use identi-cal wavefunctions. The energies, which are different inthe two procedures, come into higher order diagrams viaenergy denominators, as discussed in [6]. Therefore, thelast option is a core-polarization basis (CP basis) sincethe core-polarization diagrams are affected to leading or-der even though the result at first order is unchanged.Skyrme Hartree-Fock radial wavefunctions, oncesolved, are implemented in the renormalization by usingan expansion in terms of the harmonic oscillator basisvia: ψ SHFnlj ( ~r ) = X n a n R HOnl ( r )[ Y l ( θ, φ ) ⊗ χ s ] j , (1)where a n gives the percentage of a specific harmonic os-cillator wavefunction component in the Skyrme Hartree-Fock wavefunction. The Skyrme Hartree-Fock wavefunc-tions and single particle energies can only be determinedfor bound states. For unbound orbits, the harmonic os-cillator basis remains in use, but the Gram-Schmidt pro-cess is used to ensure orthonormality of the single par-ticle wavefunctions. The effective interaction, consistingof the derived two-body matrix elements and the SkyrmeHartree-Fock single particle energies, can then be used ina shell model program directly. III. APPLICATION TO SDPF MODEL SPACE
Neutron-rich silicon isotopes present an interesting ap-plication of the procedure outlined in the last section. Adeeper understanding of the need for multiple interac-tions in the sdpf model space, as seen by the form ofSDPF-U, can be gained by performing the renormaliza-tion for the same model space in multiple ways. The
TABLE II: Single-particle energies for Si and Ca in theharmonic oscillator basis. The energy shift is chosen so thatthe valence energy is identical in both bases. Values in boldare in the model space.n l j Si Si Ca Caproton neutron proton neutron0 s / -36.93 -34.59 -32.22 -39.210 p / -25.42 -23.09 -21.20 -28.190 p / -25.42 -23.09 -21.20 -28.190 d / -13.91 -11.58 -10.18 -17.170 d / -13.91 -11.58 -10.18 -17.171 s / -13.91 -11.58 -10.18 -17.170 f / -2.40 -0.07 -6.15 f / -2.40 -0.07 -6.15 p / -2.40 -0.07 -6.15 p / -2.40 -0.07 -6.15 g / g / model space chosen is the sd proton orbits and pf neu-tron orbits. The renormalization procedure is done usingall three options for two different target nuclei, produc-ing a total of six interactions. The two target nuclei cho-sen are the stable Ca doubly magic nucleus, and theneutron-rich Si semi-magic nucleus. Single particle en-ergies of the SHF basis, using the Skxtb interaction, arepresented in Table I for both target nuclei. For an SHFstate that is unbound, the radial wavefunction is approx-imated by a state bound by 200 keV that is obtained bymultiplying the SHF central potential by a factor largerthan unity. The energy of the unbound state is estimatedby taking the expectation value of this bound state wave-function in the original SHF potential.In the SHF basis, the calculation of single particle en-ergies shows that the proton orbits are shifted down inenergy for Si relative to Ca, while the neutron or-bits are shifted up. For the valence neutrons, this shiftresults in a switch from four orbits for Ca bound by5.4 Mev on average to four orbits for Si centered at0.0 MeV. This change, specifically the loosely bound en-ergies of the p / and p / , has a significant effect onthe wavefunctions, which will be discussed in more detaillater. For comparison, the single particle energies usedin the HO basis are given in Table II. The Blomqvist-Molinari formula [7] ¯ hω = (45 A − / − A − / ) MeVgives 11.508 MeV for A =34 and 11.021 MeV for A =40. The absolute value of the harmonic oscillator basis isirrelevant, as only energy differences come into the dia-grams in Rayleigh-Schr¨odinger perturbation theory. Fora better comparison to the SHF basis, the absolute valueis chosen separately for protons and neutrons such that n val P (2 J +1) ǫ α is identical in the HO and SHF bases, where n val , the number of valence orbits, is three for protons -3(cid:10)-2(cid:10)-1(cid:10)0(cid:10)1(cid:10)-3(cid:10) -2(cid:10) -1(cid:10) 0(cid:10) 1(cid:10) S i H O ba s i s Ca HO basisneutron − neutron orbits FIG. 1: Comparison of pf neutron-neutron matrix elements(in MeV) for the renormalization procedure in the HO basisfor the two target nuclei. The solid line y = x denotes wherethe matrix elements would be identical. -3(cid:10)-2(cid:10)-1(cid:10)0(cid:10)1(cid:10)-3(cid:10) -2(cid:10) -1(cid:10) 0(cid:10) 1(cid:10) S i S H F ba s i s Ca SHF basisneutron − neutron orbits FIG. 2: Comparison of pf neutron-neutron matrix elements(in MeV) for the renormalization procedure in the SHF ba-sis for the two target nuclei. The solid line y = x denoteswhere the matrix elements would be identical. Black dotscorrespond to matrix elements with J >
0, while the J = 0matrix elements are split into three groups: ff − ff (crosses), ff − pp (diamonds), and pp − pp (plus signs). and four for neutrons and ǫ α is the energy of the singleparticle orbit given by the α = n, l, j quantum numbers.In order to avoid divergences from the calculation of en-ergy denominators, all model space orbits are set to thesame valence energy such that the starting energy [6] ofeach diagram is constant. R (r) radius (fm) p radius (fm) p HFHO R (r) f Si neutron radial wavefunctionsf
FIG. 3: Comparison of the single particle radial wavefunctionsfor Si in the HO and SHF bases.
Fig. 1 shows a comparison of the pf matrix elementsin MeV for both target nuclei with the HO basis usedin the renormalization procedure. The values deviateslightly from the line of equality but agree well with eachother. Therefore, the choice of target nucleus, whether Si or Ca, has little effect on the matrix elements inthe HO basis. However, when we look at Fig. 2, wherethe comparison is for both target nuclei in the SHF basis,we see a reduction in the strength of the interaction for Si. This reduction with Si as the target nucleus is dueto the weakly bound nature of the pf neutron orbits.In the SHF basis, the f / orbit is bound by 2.6 MeV,and its radial wavefunction agrees well with the harmonicoscillator wavefunction as seen in Fig. 3. The Skyrmewavefunction is expanded in the harmonic oscillator ba-sis up to n = n max and the a n coefficients are renormal-ized to ensure that n max P n =0 a n = 1. For our renormalizationprocedure, orbits up to (2 n + l ) = 9 are included, whichgives n max = 3 for the f / and f / orbits and n max = 4for the p / and p / orbits. This includes over 99% ofthe strength for the f orbits, but only 93% and 92% forthe p / and p / orbits respectively. A first order calcu-lation can be done to n max = 6 for all orbits, which gives100%, 98%, and 97% for the f , p / , and p / expansionsrespectively.With this procedure, 99% of the f / orbit is repre-sented by the R HO wavefunction. The p / and p / orbits are only bound by 400 and 269 keV, respectively.The expected harmonic oscillator component R HO onlymakes up 80% and 78% of the respective radial wave-functions. Higher n orbits which extend farther awayfrom the center of the nucleus contribute the remaining TABLE III: Core polarization and total matrix elements inMeV of the form h aa V bb i J =0 for different renormalizationprocedures. Si Caa b HO CP SHF HO CP SHFf / f / core pol. -0.449 -0.377 -0.529 -0.637 -0.649 -0.931total -1.855 -1.869 -1.985 -1.957 -1.982 -2.282p / p / core pol. -0.037 0.001 0.010 -0.021 -0.005 -0.015total -1.319 -1.313 -0.944 -1.267 -1.252 -1.270p / p / core pol. 0.068 0.082 0.047 -0.038 -0.069 -0.087total -1.456 -1.462 -0.875 -1.469 -1.488 -1.420 strength. The f / orbit is unbound by three MeV, butthe solution for the Skyrme radial wavefunction is deter-mined by assuming that the orbit is bound by 200 keV.With this method, 97% of the realistic radial wavefunc-tion is given by the R HO wavefunction. Single particleradial wavefunctions of valence space neutron orbits areshown in Fig. 3 in both the HO and SHF basis. The longtail behavior of the loosely bound p orbits can be seenin the SHF basis, as the wavefunctions have significantstrength beyond 8 fm unlike the oscillator wavefunctions.The J = 0 matrix elements in Fig. 2 deviate morefrom the line of equality, i.e. the pairing matrix ele-ments are reduced for Si when the N LO interactionis renormalized in the SHF basis. The SDPF-U inter-action has different neutron-neutron pairing matrix ele-ments for Z ≥
15 and for Z ≤
14 to account for 2p-2h ex-citations of the core correctly, depending on whether Sior Ca should be considered the core [1]. The SDPF-Uneutron-neutron pairing matrix elements are reduced by300 keV for Z ≤
14 in order to produce results in betteragreement with experimental data. The pairing matrixelements in Fig. 2 are reduced for the Si target by 213keV on average, relative to the case with Ca as the tar-get. While the connection here to the Z -dependence inSDPF-U is only suggestive, the change in target mim-ics the change in core for calculations in the sdpf region,cited by Nowacki and Poves as the cause of their 300 keVreduction [1]. The reduction of 213 keV is due solely tothe change in occupation of the d / proton orbit, whichcan affect the single particle energies and radial wave-functions, as well as the available diagrams in the corepolarization. We find that the change in single particleradial wavefunctions plays the most significant role, butare also able to analyze the effect of the core polarization.Table III isolates a few matrix elements and comparesthe total matrix elements and the component due to corepolarization for both target nuclei in all three bases. Thereduction for total matrix elements involving p orbits isdramatic ( ≈ Si is not a halo nucleus,the loosely bound p orbits behave in much the same wayas the valence nucleons in a halo nucleus. The reductionin core polarization is seen going from the Ca targetto the Si target in any basis in Table III, although thesize of the polarization is reduced for nucleons far fromthe core. As noted in [8], the core interacts less withnucleons far away, so the excitations of the core are re-duced. The core polarization for matrix elements solelyinvolving p orbits is under 100 keV. We observe that thecore polarization can be reduced significantly without thetotal matrix element changing in the same proportion.For instance, the h f / f / V f / f / i matrix element isonly reduced by 5% from Ca to Si in the HO ba-sis even though the core polarization is reduced by 30%.In the SHF basis, which takes into account the realis-tic wavefunction, the total matrix element is reduced by13% even though the core polarization is reduced by 43%.We would prefer to compare matrix elements involvingthe p / or p / orbits, but the core polarization becomesvery small for loosely bound orbits, skewing percentagecomparisons. Ogawa et al. [9] produce results whichseem to be consistent with ours, identifying a 10%-30%reduction in nuclear interaction matrix elements involv-ing loosely bound orbits using a realistic Woods-Saxonbasis. However, they were limited to comparisons of ra-tios of matrix elements and did not include core polar-ization. We show that core polarization suppression andreduction due to spread of the wavefunctions are bothimportant effects which should be included, but do nottell the entire story. The f / wavefunction is very similarin the HO and SHF bases, as seen in Fig. 3, and yet the h f / f / V f / f / i matrix element does not follow thesame trend as the core polarization component in TableIII. Other diagrams which are included at second orderare relevant, and the full treatment of the renormaliza-tion in a realistic basis, as developed here, is necessary foraccurate results. Our improvements enable us to performcalculations for neutron-rich silicon isotopes directly. IV. CALCULATIONS FOR SI AND SI The effect of the different interactions on nuclear struc-ture calculations has been studied as neutrons are addedto Si. In order to obtain a consistent starting point,the proton-proton and proton-neutron matrix elementsof SDPF-U have been used, with proton single particleenergies (SPEs) chosen to reproduce those obtained bySDPF-U. Because SDPF-U does not reproduce the bind-ing energy of Si, the SDPF-U neutron SPEs have beenincreased by 660 keV. The six interactions use neutronSPEs that reproduce the values of this modified SDPF-Uinteraction.The only difference in the six interactions used in thecalculations are the neutron-neutron matrix elements.Calculations have been done in the model space discussed -10(cid:10)-9(cid:10)-8(cid:10)-7(cid:10)-6(cid:10)-5(cid:10)-4(cid:10) E ne r g y r e l a t i v e t o S i ( M e v ) SiHO SiCP SiSHF CaHO CaCP CaSHF0 + + + + FIG. 4: Calculations for the lowest energy states for J =0 , , , Si relative to Si from the renormalization pro-cedure for Si and Ca, in the HO, CP, and SHF bases forboth target nuclei. Crosses are used for calculations with Caas the target nucleus. in the last section with the shell model code NuShellx[10]. Fig. 4 shows the lowest J = 0 , , , Si, relative to Si. The HO basis and the CP basisfor the same target nucleus deviate by no more than 20keV. However, the SHF basis noticeably shifts the states,with the largest effect being 170 keV more binding in theground state with Ca as the target nucleus. The bind-ing energy of Si changes by nearly 500 keV dependingon which renormalization procedure is used. Further-more, the level schemes for Si are more spread out forthe crosses where Ca is chosen as the target nucleus.Fig. 5 shows the same states in Si relative to Si,but now the comparison includes the SDPF-U calcula-tions and experimental data. The CP basis results arenot included since they are so similar to the HO basis cal-culations. We see that the level scheme for Si is morespread out for the Z ≥
15 SDPF-U calculation than forthe Z ≤
14 calculation, in agreement with our resultsdiscussed above. Our calculations for each method arein reasonable agreement with the comparable SDPF-Ucalculation, except for the 0 + state which differs by over300 keV in both instances. The experimental bindingenergy relative to Si is taken from a new mass mea-surement of Si which is 140 keV higher in energy thanpreviously measured [11]. The excitation energies of the Z ≤
14 SDPF-U calculation are comparable to experi- -10(cid:10)-9(cid:10)-8(cid:10)-7(cid:10)-6(cid:10)-5(cid:10)-4(cid:10) E ne r g y r e l a t i v e t o S i ( M e v ) SiSHF SiHO CaHO CaSHFZ > Z < SDPF − U Exp. 0 + + + + FIG. 5: Calculations for the lowest energy states for J =0 , , , Si relative to Si using neutron-neutron matrixelements from SDPF-U and the renormalization procedure forboth Si and Ca as target nuclei, using the SHF and HObases. Experimental data is shown for comparison, with anew mass from [11]. Crosses are used for calculations with Ca as the target nucleus. ment, as expected from an interaction fit specifically toneutron-rich silicon isotopes. While no interaction repro-duces the experimental data very well, general trends canbe seen. The calculations with Ca as the target nucleusdepicted by crosses result in level schemes that are toospread out in comparison to the experimental data. Thereduction in the strength of the interaction for Si usingthe SHF basis results in a reduction of the energy of thestates in Si, especially for the ground state (the pairingmatrix elements were most reduced). The rms betweenexperiment and theory with Si as the target nucleus inthe SHF basis is 214 keV for the four states shown. Onereason for this deviation is the lack of three body forcesin the procedure. The inclusion of the
N N N interaction,at least via the effective two-body part, is important fora higher level of accuracy. Additionally, the chosen SPEsmay contribute to the deviation, which would be betterconstrained if all the single particle states in Si wereknown experimentally. For exotic nuclei, the calculatedsingle particle state is often above the neutron separa-tion energy and determination of the experimental singleparticle states may not be possible with current facilities.Thus it is essential to improve energy density functionalssuch that they provide reliable single particle energies. -18(cid:10)-17(cid:10)-16(cid:10)-15(cid:10)-14(cid:10) E ne r g y r e l a t i v e t o S i ( M e v ) SiSHF SiHO CaHO CaSHFZ > Z < SDPF − U Exp. 0 + + FIG. 6: Calculations for the lowest energy states for J = 0 and J = 2 in Si relative to Si using neutron-neutron matrixelements from SDPF-U and the renormalization procedurefor both Si and Ca as target nuclei, using the SHF andHO bases. Crosses are used for calculations with Ca as thetarget nucleus.
The importance of using a realistic basis and an appro-priate target nucleus to renormalize an interaction intothe nuclear medium for calculations of exotic nuclei is ev-ident from the calculations shown here. Otherwise, theinteraction will be too strong and will produce signifi-cant overbinding even for the two particle case, an effect that gets magnified as more particles are added, as seenin Fig. 6 for Si. Only the 0 + and 2 + states are shownsince the 4 + and 6 + states are not known experimentally,but the binding energy is only approximately reproducedfor the calculations with Si in the SHF basis. As notedin the Si case, the excitation energy of the 2 + state istoo high in the SHF basis but is reproduced well by the Z ≤
14 SDPF-U calculation for Si.
V. SUMMARY AND CONCLUSIONS
The microscopic nucleon-nucleon interaction N LOwas renormalized using v lowk and many-body perturba-tion theory in order to produce an effective interaction inthe nuclear medium that could be used in a shell modelcode. The renormalization was performed in three dif-ferent bases: harmonic oscillator, core polarization, andSkyrme Hartree-Fock. The choice of basis can signifi-cantly affect the value of matrix elements, as shown inthe comparisons of pf neutron-neutron matrix elementsfor the stable Ca and the neutron-rich Si nuclei. Thedifference results from valence orbits being bound by onlya few hundred keV, resulting in a long tail in the radialwavefunction relative to the harmonic oscillator wave-function. The loosely bound orbits cause a reductionin the overall strength of the interaction, an effect thatbecomes magnified as full scale shell model calculationsare performed. Accounting for the properties of the or-bits by using a realistic basis is essential for an accuratedescription of the nuclear interaction in exotic nuclei asdetermined by the renormalization of an
N N interaction.
Acknowledgments
Support for this work was providedfrom National Science Foundation Grant PHY-0758099,from the Department of Energy Stewardship ScienceGraduate Fellowship program through grant number DE-FC52-08NA28752, and from the DOE UNEDF-SciDACgrant DE-FC02-09ER41585. [1] F. Nowacki and A. Poves, Phys. Rev. C , 014310(2009).[2] M. Stanoiu et al., Phys. Rev. C , 034315 (2008).[3] D.R. Entem and R. Machleidt, Phys. Rev. C ,041001(R) (2003).[4] S.K. Bogner, T.T.S. Kuo, and A. Schwenk, Phys. Rept.386 (2003) 1.[5] B.A. Brown et al., Phys. Rev. C , 545(1968).[8] T.T.S. Kuo et al., Phys. Rev. Lett. , 2708 (1997).[9] K. Ogawa et al., Phys. Lett. B ∼ brown/resources/resources.html[11] L. Gaudefroy, W. Mittig, et al., private communication. S i H O ba s i s Ca HO basisneutron − neutron orbits S i S H F ba s i s Ca SHF basisneutron − neutron orbits E ne r g y r e l a t i v e t o S i ( M e v ) SiHO SiCP SiSHF CaHO CaCP CaSHF0 + + + + E ne r g y r e l a t i v e t o S i ( M e v ) SiSHF SiHO CaHO CaSHFZ > Z < SDPF − U Exp. 0 + + +6