CCold neutrons trapped in external fields
S. Gandolfi, J. Carlson, and Steven C. Pieper Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545 Physics Division, Argonne National Laboratory, Argonne, IL 61801
The properties of inhomogeneous neutron matter are crucial to the physics of neutron-rich nucleiand the crust of neutron stars. Advances in computational techniques now allow us to accuratelydetermine the binding energies and densities of many neutrons interacting via realistic microscopicinteractions and confined in external fields. We perform calculations for different external fields andacross several shells to place important constraints on inhomogeneous neutron matter, and hencethe large isospin limit of the nuclear energy density functionals that are used to predict properties ofheavy nuclei and neutron star crusts. We find important differences between microscopic calculationsand current density functionals; in particular the isovector gradient terms are significantly morerepulsive than in traditional models, and the spin-orbit and pairing forces are comparatively weaker.
PACS numbers: 21.30.-x, 21.60.-n, 21.60.Jz
The properties of inhomogeneous neutron-rich mat-ter are important in both astrophysical and terrestrialregimes. While the equation of state (EOS) and thepairing gap for homogeneous neutron matter have beenstudied extensively in microscopic theories [1–4], inho-mogeneous neutron matter has received comparably lit-tle attention. Understanding the inner crust of neutronstars, which affects transient stellar cooling and deter-mine oscillation modes requires knowledge of inhomoge-neous neutron-rich matter [5–7]. Neutron-rich nuclei arealso the subject of intense theoretical and experimentalinvestigations, driven by their relevance for r-process nu-cleosynthesis as well as the intrinsic interest in the prop-erties of nuclei at large isospin; [8, 9] they are the princi-pal thrust of rare-isotope accelerators [10].Simulations of both the crust of neutron stars and oflarge neutron-rich nuclei employ nuclear energy densityfunctionals fit to nuclei. These density functionals haveproved to be extremely successful in describing many nu-clei, but involve large extrapolations to reach inhomoge-neous neutron matter. To test these extrapolations, weperform calculations of neutron drops – neutrons con-fined by artificial external fields and interacting via real-istic two- and three-nucleon forces. We vary substantiallythe number of neutrons as well as the strength and shapeof the external fields to test the density functional.The EOS of homogeneous neutron matter has oftenbeen included as a constraint to density functional the-ories (eg. [11]); our objective is to allow inhomogeneousneutron matter to be employed in a similar manner. Wefind, for example, that once the bulk terms are fixed fromthe neutron matter EOS, the closed shells of neutrons areprimarily sensitive to the gradient terms in the densityfunctional. These pure neutron matter gradient termshave modest effects on nuclei, and hence they are notwell constrained in fits to nuclear masses [12, 13]. Theclosed-shell systems are nearly independent of spin-orbitand pairing terms, but ground and excited states of asingle neutron outside a closed shell, or of a single neu- tron hole, are a sensitive probe of the spin-orbit inter-action. Mid-shell results are sensitive to both spin-orbitand pairing terms. We compare our calculated results toseveral “standard” Skyrme models, and also to a model inwhich the isovector terms are adjusted to reproduce theab-initio calculations; these changes are expected to haveonly a small effect on the nuclear energies used to fit theoriginal parameters. The goal of these studies is to deter-mine which terms in the density functional can be probedthrough microscopic calculations, and how the adjustedvalues compare to traditional models. A realistic im-proved density functional will require a complete refittingof nuclear properties along with the properties of homo-geneous and inhomogeneous neutron matter [14, 15].
Interaction and Methods:
We report calculations ofneutrons in harmonic oscillators (HO) of two frequenciesand a Woods-Saxon (WS) well. The full Hamiltonian is: H = − (cid:126) m (cid:88) i ∇ i + (cid:88) i V i + (cid:88) i 44 MeV-fm . The neutron-neutron potential V ij is AV8 (cid:48) [16], a slightly simplifiedversion of the AV18 potential [17]; we find less than 0.25%differences in neutron-drop energies for these two poten-tials. We also add the Urbana IX model (UIX) [16] three-nucleon interaction (TNI), including the p-wave two-pionexchange (Fujita-Miyazawa) TNI and a short-range phe-nomenological repulsion. We use this combination of two-and three-nucleon interactions because it produces anEOS consistent with known neutron star masses [1], andbecause several present-day Skyrme models have usedthis EOS to constrain the properties of homogeneousneutron matter. Further studies with different interac-tion models will be valuable, in particular to look at thespin-orbit interactions which might be increased with athree-pion exchange TNI as in Illinois-7 [18]. a r X i v : . [ nu c l - t h ] J a n E / h _ ω N / N E / h _ ω N / AFDMCGFMCSLY4SLY4-adjSKM*SKPBSK17 FIG. 1. (color online) Energies divided by (cid:126) ωN / for neu-trons in HO fields with (cid:126) ω = 10 MeV (top) and 5 MeV(bottom). Filled symbols indicate ab initio calculations; thedashed lines are Thomas-Fermi results (see text); the lowercurves are from the SLY4 interaction and the upper curvesshow the modified SLY4 interaction described in the text. N -14-12-10-8 E / N ( M e V ) AFDMCGFMCSLY4SLY4-adj FIG. 2. (color online) Energies per particle for neutrons inthe Woods-Saxon field, symbols as in Fig. 1. Calculations are performed using Green’s FunctionMonte Carlo (GFMC) [19] and Auxiliary Field Diffu-sion Monte Carlo (AFDMC) [20] quantum Monte Carlo(QMC) methods. These algorithms evolve an initial trialstate, Ψ T , in imaginary time to yield the ground-state.GFMC sums explicitly over spin and isospin states, andcan use very sophisticated Ψ T [16]. However it is lim-ited to small systems, up to 16 neutrons. In additionto sampling the spatial integrals as in GFMC, AFDMCalso samples the spin and isospin degrees of freedom, andhence it can treat larger systems [3]. Both methods usea constraint involving the overlap with Ψ T to eliminatethe Fermion sign problem, and hence are approximate.Studies of light nuclei and neutron matter show they giveresults within 1% of the exact ground-state energy.We use external fields yielding low or moderate den-sities. However, even at small densities neutrons arestrongly interacting and pairing can be important. Re-cent microscopic calculations of neutron matter give s-wave pairing gaps of several MeV [4, 21]. One- and two-nucleon properties including pairing gaps and spin-orbitsplittings can be more sensitive to models of the three-nucleon interaction. Calculations of very small neutrondrops (N=6,7,8) have been performed previously [22–24].Even these calculations indicated a substantial differ-ence with traditional Skyrme models, which overbind thedrops and give too-large spin-orbit splitting. N J π (cid:126) ω = 5 MeV (cid:126) ω = 10 MeVGFMC AFDMC GFMC AFDMC7 1 / − . . . . / − . . . . + . . . . / + . . . . / + . . + . . . . + . . . . / + . . . . / + . + . . . . + . . + . . 3) 1114 . (cid:48) +UIX with HO external fields. Results: The ground-state energies versus neutronnumber N for the HO potentials are given in Fig. 1 andfor the WS potential in Fig. 2. Up to N=16 both GFMCand AFDMC results are included. They agree very wellfor the 10-MeV HO interaction, while for (cid:126) ω = 5 MeV,the AFDMC results are slightly higher than the GFMC;the maximum difference is 3%, and more typically resultsare within 1%. The bigger difference for the lower density5-MeV drops presumably arises because the AFDMC Ψ T does not yet include pairing, while the GFMC does.In addition to the microscopic calculations, re-sults for several different Skyrme models are shownin Fig. 1. We also show results for Thomas-Fermilocal density approximations [25] using E ( ρ n ) /N = ξ (3 / (cid:126) / m )(3 π ρ n ) / ; the upper horizontal line is forfree particles, ξ = 1, and the lower has ξ = 0 . 5, a reason-able approximation to the EOS of low-density neutronmatter. For the 10-MeV well, the density functionalsgive energies significantly below the Monte Carlo resultsfor all N. The energies are also lower for the 5-MeV well,but less so. This overbinding is a general feature of allthe Skyrme models considered. It is intriguing that thesesame Skyrme models underbind the properties of very di-lute neutron systems, typically they are fit to the neutronmatter EOS at ρ = 0 . 04 fm − and above.Since the Skyrme homogeneous neutron matter EOShave been fit to various microscopic calculations, thisoverbinding suggests that the gradient terms in inhomo-geneous neutron matter should be more repulsive. Theobserved differences between ab-initio results and theSkyrme functionals are much larger than the differencesbetween experiments and Skyrme models in nuclei, asexpected, because of the large extrapolations to inhomo-geneous neutron matter. Isovector Gradient Contributions: As is apparent inFig. 1, for harmonic oscillators there are closed shellsat N= 8 , , and 40 neutrons. These closed-shell statesare almost exclusively sensitive to the neutron matterEOS and the isovector gradient terms; pairing and spin-orbit play nearly no role. Hence they are direct probes ofthe gradient terms; to examine them we have altered theisovector gradient terms in the SLY4 interaction [11] toapproximately reproduce the QMC results using a mod-ified version of the ev8 code [26], The gradient terms areadjusted without changing any isoscalar (T=0) parame-ters or the homogeneous neutron matter EOS.The lowest-order gradient contribution to the energydensity for inhomogeneous matter is G d [ ∇ ρ n ] . The con-stants G d are small and often negative, for example, G d = − , − , , − , − for the SLY4, SLY7,BSK17, SkM ∗ , and SkP interactions. Repulsive gradientterms for neutron matter are to be expected on rathergeneral grounds, and are required for the absolute sta-bility of uniform matter in the absence of a backgroundfield. The adjusted interaction SLY4-adj gives G d = 26 . G d = 64. A single adjustment of G d markedly improvesthe agreement with QMC results for both the HO andWS fields. A precise fit to both neutron matter and theseresults would require a more general form of the densityfunctional. Isovector Spin-Orbit: By examining neutron numbersslightly away closed shells, we can constrain the spin-orbit interaction for neutron drops. For example, N =7 , T = 0) and isovec-tor ( T = 1) spin-orbit couplings. We find an even smallerisovector coupling, approximately 1/6 of the isoscalarcoupling, reproduces the ab initio calculations. The com-bined factor of 1 / / Isovector Pairing: The mid-shell results (eg. N=14,30) and odd-even staggerings are sensitive to the pairinginteractions as well as the spin-orbit force. Fixing thespin-orbit strength from near closed-shells, we adjust thepairing strength to fit the calculated spectra. There is asignificant interplay between the pairing and spin-orbitforces required to reproduce microscopic calculations. Asmall spin-orbit force results in many quasi-degeneratelevels which enhances pairing in mid-shell systems. < r > / (f m ) GFMCSLY4SLY4-adj N < r > / (f m ) 10 MeV5 MeVHOWS FIG. 3. (color online) Calculated radii of neutrons confinedin HO (upper) and WS (lower), fields compared to originaland adjusted Skyrme models (see text). Several models of pairing are used in density-functionaltheories. We employ a simple volume parametrizationwith a delta-function spatial dependence, a density cut-off that restricts pairing to ρ n < ρ , and limit the pairingto single-particle orbitals less than 5 MeV from the Fermienergy. We find a reduction from a typical 1 GeV-fm strength to half that value significantly improves agree-ment with microscopic results. A reduction of pairingin neutron-rich nuclei has recently been found to give abetter fit to experimental energy differences of 156 nucleiof mass A=118 to 196 [31].Adjusting these three parameters (gradient term G d = 26.5, spin-orbit coupling = 123 MeV-fm and pairingstrength = 500 MeV) in the density functional increasesthe agreement across all external fields and all particlenumbers. This is shown by the upper solid curves (SLY4-adj) in Fig. 1. Radii and Mass Distributions: Our calculations yieldprecise estimates for RMS radii and the density distribu-tions of the smaller drops. The average densities, definedas (cid:82) d rρ n ( r ) /N , of the drops in the 5-MeV HO well areapproximately 0.02 fm − , or about 1/8 nuclear mattersaturation density, while for the 10-MeV HO and theWS wells they are ∼ − , or almost 1/3 nuclearmatter saturation density.The RMS radii obtained in microscopic calculationsare compared with the original and adjusted Skyrme den-sity functional results in Fig. 3. The density distribu-tions for N=8 and 14 are compared in Fig. 4. Since weare comparing gross properties of inhomogeneous mat-ter, we plot the densities weighted with the phase space: r ρ n ( r ), which gives a better picture of the density distri-butions near the average density of the system. In everycase the adjusted Skyrme interaction produces a betterdescription of the radii and density distributions. The N = 8 calculations depend primarily upon the gradientterms, the reduction in pairing and spin-orbit are also im-portant for N = 14. The improvement in the mid-shellN=14 case is particularly dramatic, as a significant shiftin the density occurs with the modified isovector Skyrme r ρ ( f m - ) ω = 5 MeV ω = 10 MeV SLY4-adjSLY4 r (fm) r ρ ( f m - ) N=8N=14 FIG. 4. (color online) Calculated densities of neutrons in HOpotentials, compared to Skyrme models (see text). parameters, bringing the results into much better agree-ment with microscopic calculations. Conclusions: We have examined the properties of neu-trons confined in external fields to study the propertiesof inhomogeneous neutron matter. These ab-initio calcu-lations place significant constraints on the nuclear energydensity functional in a regime far from that probed by fit-ting to available nuclei. They indicate the need for morerepulsive gradient terms in pure neutron matter, and a re-duced isovector spin-orbit and pairing strength comparedto standard functionals. With a combined fit of densityfunctionals to both nuclei and neutron matter, more reli-able predictions should be possible for very neutron-richnuclei including those participating in r-process nucle-osynthesis. These improved functionals would also be ex-tremely valuable in examining the neutron skin thicknessof lead [32], as can be probed in parity-violating electronscattering. Much more reliable predictions for extremelyneutron-rich astrophysical environments can also be ex-pected. 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