Microscopic investigation of the 8 Li( n,γ ) 9 Li reaction
Callum McCracken, Petr Navratil, Anna McCoy, Sofia Quaglioni, Guillaume Hupin
AAb initio investigation of the Li( n, γ ) Li reaction
Callum McCracken ∗ TRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia V6T 2A3, Canada andUniversity of Waterloo, 200 University Ave, Waterloo, Ontario N2L 3G1, Canada
Petr Navr´atil † and Anna McCoy ‡ TRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia V6T 2A3, Canada
Sofia Quaglioni § Lawrence Livermore National Laboratory, P.O. Box 808, L-414, Livermore, California 94551, USA
Guillaume Hupin ¶ Universit´e Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France (Dated: September 3, 2020)
Background:
The Li( n, γ ) Li reaction plays an important role in several astrophysics scenarios.It cannot be measured directly and indirect experiments have so far provided only cross sectionlimits. Theoretical predictions differ by an order of magnitude.
Purpose:
In this work we study the properties of Li bound states and low-lying resonances andcalculate the Li( n, γ ) Li cross section within the no-core shell model with continuum (NCSMC)with chiral nucleon-nucleon and three-nucleon interactions as the only input.
Methods:
The NCSMC is an ab initio method applicable to light nuclei that provides a unifieddescription of bound and scattering states well suited to calculate low-energy nuclear scattering andreactions.
Results:
Our calculations reproduce the experimentally known bound states as well as the lowest5 / − resonance of Li. We predict a 3 / − spin-parity assignment for the resonance observed at 5.38MeV. In addition to the a very narrow 7 / − resonance corresponding presumably to the experimental6.43 MeV state, we find several other broad low-lying resonances. Conclusions:
Our calculated Li( n, γ ) Li cross section is within the limits derived from the 1998National Superconducting Cyclotron Laboratory Coulomb-dissociation experiment [Phys. Rev. C , 959 (1998)]. However, it is higher than cross sections obtained in recent phenomenologicalstudies. It is dominated by a direct E1 capture to the ground state with a resonant contribution at ∼ . / − resonance. I. INTRODUCTION
In neutron rich astrophysical environments, reactionsinvolving the short-lived Li nucleus may contributeto the synthesis of heavier nuclei by bridging the sta-bility gap of mass A = 8 elements. In particu-lar, the Li( n, γ ) Li capture reaction plays an impor-tant role in inhomogeneous big bang nucleosynthesisand in the r-process. There, it competes with the Li( α, n ) B reaction and the Li beta decay, affect-ing the reaction path to A> Li( n, γ ) Li( α, n ) B( n, γ ) B( β + ) Cand Li( n, γ ) Li( n, γ ) Li( α, n ) B( β + ) C [1–4]. Inaddition, the reaction chain with two-neutron cap-tures He(2 n, γ ) He(2 n, γ ) He( β − ) Li( n, γ ) Li( β − ) Be,of which the Li( n, γ ) Li is also a component, has beenconsidered as an alternative to the triple-alpha process ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] in overcoming the A = 8 mass gap in the r-process forsupernovae of type II [5, 6].As the Li half-life is 840 ms and a neutron target is notavailable, the Li( n, γ ) Li reaction cannot be measureddirectly. There have been several attempts to determineits cross section by indirect methods. Using a radioactivebeam of Li and the Coulomb-dissociation method withU and Pb targets, only upper limits on the Li( n, γ ) Licross section were determined as it was not possible toestimate the nuclear contribution to the dissociation [7].A follow-up Coulomb-dissociation experiment using a Pbtarget reported a null result and consequently a very lowlimit on the capture cross section [8].In Ref. [9], the direct Li( n, γ ) Li g . s . capture cross sec-tion was computed in the framework of the potentialmodel by deducing the single particle spectroscopic fac-tor for the ground state of Li from a measurement of theangular distribution of the Li( d, p ) Li g . s . transfer reac-tion at E c . m . =7 . Be( Li, Li) Be transfer reactionmeasured with a 27 MeV Li radioactive nuclear beam,was reported in Ref. [10]. The obtained reaction rate was a r X i v : . [ nu c l - t h ] S e p comparable to that from Ref. [9].There were several other studies focused on the struc-ture of Li. Notably, the H( Li, p ) Li reaction with 76MeV radioactive Li beam was studied with the goal toobtain single-neutron spectroscopic factors for states in Li [11]. Spectroscopic factors for the Li ground statehave also been investigated through the d ( Li, t ) Li one-neutron transfer reaction at E/A = 1.68 MeV [12]. Thefirst excited state of Li was studied by the inelastic scat-tering of Li from deuterons [13]. A very recent experi-ment investigated the structure of C, the mirror of Li,using proton resonant scattering [14].The Li( n, γ ) Li cross section and its reaction ratehave also been the focus of several theoretical investi-gations, based on various approaches. In Refs. [4, 15],the reaction rate was estimated based on the existing in-formation for other nuclei. Calculations combining theshell model and the potential model were reported inRefs. [16, 17]. The potential model was also applied toa simultaneous study of the Li( n, γ ) Li reaction and itsmirror, B( p, γ ) C [18]. In Ref. [19], the neutron captureon Li was investigated by means of the microscopic clus-ter model. The Coulomb dissociation of Li on heavy tar-gets was calculated in Refs. [20, 21] and the principle ofdetailed balance was then used to obtain the Li( n, γ ) Lireaction rate. More recently, this reaction was inves-tigated within the framework of the modified potentialcluster model with the state classification of nucleons ac-cording to the Young tableaux [22]. Overall, theoreticalpredictions of the reaction rate span more than an orderof magnitude.In this work, we report the first ab initio calculationof the Li( n, γ ) Li cross section. We apply the no-coreshell model with continuum (NCSMC) [23–25] and usechiral nucleon-nucleon (NN) and three-nucleon (3N) in-teractions as the only input. In particular, we employthe chiral Hamiltonian from Ref. [26] shown to describewell both light and medium mass nuclei. The NCSMCprovides a unified description of bound and scatteringstates and allows us to investigate bound states of Li aswell as its low-lying resonances.The paper is organized as follows: In Sec. II we brieflyreview the NCSMC formalism. In Sec. III, we presentour results for Li, Li, and for the capture cross section.Finally, in Sec. IV we draw our conclusions.
II. THEORETICAL FRAMEWORK
The starting point of our approach is the microscopicHamiltonian H = 1 A A (cid:88) i 2. The coordinate (cid:126)r , in Eq.(3)is the separation vector between the Li target and theneutron.The translationally invariant eigenstates of the ag-gregate ( | Li λJ π T (cid:105) ) and target ( | Li λ J π T (cid:105) ) nucleiare all obtained by means of the no-core shell model(NCSM) [45–47] using a basis of many-body harmonicoscillator (HO) wave functions with the same frequency,Ω, and maximum number of particle excitations N max from the lowest Pauli-allowed many-body configuration.In this work we used the HO frequency of (cid:126) Ω = 20 MeVfound as optimal for p -shell nuclei in Ref. [26].The discrete expansion coefficients c J π Tλ and the con-tinuous relative-motion amplitudes γ J π Tν ( r ) are the solu-tion of the generalized eigenvalue problem derived by rep-resenting the Schr¨odinger equation in the model space ofthe expansions (2) [25]. The resulting NCSMC equationsare solved by means of the coupled-channel R-matrixmethod on a Lagrange mesh [48–50].In general the sum over the index ν in Eq. (2) includesall the mass partitions involved in the formation of thecomposite system Li, i.e., Li+ n , Li+ n + n , He+ Hetc. Here, we limit the present calculations to the Li+ n clusters of Eq. (3), which are by far the most relevant forthe low-energy Li( n, γ ) Li capture. The channel statesfor the other mass partitions are energetically closed andtheir effect is in part accounted for by means of the firstterm in Eq. (2). Applications of the NCSMC with three-body clusters and with coupling between different masspartitions can be found, e.g., in Refs. [51] and [52], re-spectively. III. RESULTSA. NCSM calculations for , Li The present NCSMC calculations require as inputNCSM eigenstates and eigenenergies of Li and Li. For Li, we performed calculations up to N max =10, while for Li up to N max =8 and 9 for the negative- and positive-parity states, respectively. The ground-state energy de-pendence on the basis size for both isotopes is presentedin Fig. 1. The NCSM extrapolated Li ground state en-ergy of -42.1(5) MeV for the interaction used here hasbeen reported in Ref. [26]. Comparing to the experimen-tal value of -45.34 MeV, the calculation underbinds bya few percent. For Li we find the ground-state energy-39.4(3) MeV compared to the experimental -41.28 MeV.The theoretical uncertainty is due to the extrapolationto the infinite basis size.Excitation energies of Li low-lying states are shownin the left panel of Fig. 2. The convergence of the NCSMapproach for the experimentally bound 1 + state and thenarrow 3 + resonance is quite good. The second 1 + state N max -45-40-35-30-25-20-15-10 E g s [ M e V ] Li 2 + NCSM Li 2 + Extrap Li 2 + Expt Li 3/2 - NCSM Li 3/2 - NCSMC Li 3/2 - Extrap Li 3/2 - Expt NN+3N(lnl) Figure 1. Li (circles) and Li (diamonds) ground state en-ergy dependence on the size of the NCSM and for Li also NC-SMC (triangles) basis. Extrapolations to infinite N max withtheir uncertainties are presented on the right. The experi-mental values are shown by dashed-dotted lines. The SRG-evolved NN+3N(lnl) chiral interaction [26] at the resolutionscale of λ SRG = 2 . − and the HO frequency (cid:126) Ω=20 MeVwas used. Experimental data are from Ref. [53]. is a broad resonance in experiment. In the NCSM calcu-lations, this is reflected by rapid changes of the excitationenergy with the size of the model space N max . Comparedto the known levels, we predict additional states close tothe 1 +2 , most notably a 0 + resonance. We note that boththe predicted 0 + and the 2 + resonances have been previ-ously investigated by studying the n + Li continuum [54]working within a predecessor of the NCSMC approach,known as NCSM/RGM. Experimental evidence for theseresonances in B, the isospin mirror of Li, has been re-ported in Ref. [55].NCSM results for the low-lying excitation energies of Li with the interaction used here have been reportedin Ref. [26]. For completeness, we present the negative-parity level energies in the right panel of Fig. 2. Theconvergence of the experimentally bound 1 / − state issatisfactory, though the experimental 1 / − − / − split-ting is underestimated in the calculation. We find the5 / − state quite close to the experimentally established5 / − resonance. In addition, we predict a 3 / − and a7 / − level that might correspond to the experimentallyobserved resonances at 5.38 MeV and 6.43 MeV with un-determined spins and parities.Calculated ground state properties of the two isotopesand the M E x [ M e V ] Li NN+3N(lnl) Ω Ω Ω Ω Ω Expt 2 + + + + + + E x [ M e V ] Li NN+3N(lnl)2h- Ω Ω Ω Ω Expt 3/2 - - - ??7/2 - - - Figure 2. Comparison between the NCSM-calculated and the experimental energy spectra of Li (left panel) and Li (rightpanel). The SRG-evolved NN+3N(lnl) chiral interaction [26] at the resolution scale of λ SRG = 2 . − . The HO basisfrequency was (cid:126) Ω=20 MeV. Experimental data are from Ref. [53]. E g . s . [MeV] Q [ e fm ] µ [ µ N ] B ( M 1) [ µ ] LiNCSM -39.4(3) +2.95(15) +1.48 4.164Expt -41.28 +3.14(2) +1.654 5.0(16) LiNCSM -42.1(5) -2.5(2) +2.91 3.23Expt -45.34 -3.06(2) +3.437 N/ATable I. , Li ground state energies, quadrupole andmagnetic moments, and the M B ( M 1; 1 + → + )and B ( M 1; 1 / − → / − ) for Li and Li, respectively, isshown. NCSM calculations have been performed with theNN+3N(lnl) chiral interaction. Experimental results are fromRefs. [53, 59]. SRG evolution of the transition operators [34, 57, 58].For the microscopic cluster component of the NCSMCexpansion, Eq. (3), we used two NCSM eigenstates cor-responding to the two Li bound states, the 2 + groundstate and the 1 + excited state. In principle, we couldhave included also the experimentally narrow 3 + state.However, since our focus is on the low-energy Li( n, γ ) Liradiative capture, the impact of the 3 + state is expectedto be negligible while the technical complexity of the cal-culations would increase substantially. As for the com-posite Li states entering the expansion (2), we used theeight lowest negative-parity and six lowest positive-parityNCSM eigenstates of Li with total angular momentum J ∈ { / , / , / , / } and isospin T =3 / J π T N max =4 N max =6 N max =8 Expt1 / − / / − / Li bound-state energies, in MeV, with respect to the Li+ n threshold. NCSMC calculations have been performedwith the NN+3N(lnl) chiral interaction [26] at the resolutionscale of λ SRG = 2 . − . The HO basis frequency was (cid:126) Ω=20MeV. Experimental data are from Ref. [53]. B. NCSMC results for Li We performed NCSMC calculations for Li for N max =4 , , Li NCSM positive-parity states entering the expansion (2) were obtainedin N max +1 spaces, i.e., up to N max =9. We found twobound states, the 3 / − ground state and the 1 / − ex-cited state, in agreement with experiment. The NCSMCground-state energies are shown in Fig. 1 and the sepa-ration energies with respect to the Li+ n threshold forboth the 3 / − and 1 / − states are given in Table II.NCSMC calculations increase the binding energies com-pared to the NCSM results at any fixed N max due to theinclusion of the cluster basis component. The separationenergies are quite stable with varying N max . The calcu-lated 1 / − separation energy is quite close to the exper-imental one while the ground state separation energy isunderestimated by about 1.2 MeV. This could be due toa weaker spin-orbit strength and/or missing strength inthe T =3 / Li+ n energy of 4 MeV in the center ofmass, we find three P -wave resonances corresponding totwo 3 / − and a 5 / − state. Corresponding eigenphase d [ d e g ] N max = 8 N max = 6 N max = 4 -60-300306090 d [ d e g ] E kin [MeV] d [ d e g ] - - -6 P P n+ Li P Figure 3. Dependence of the Li+ n eigenphase shifts (toppanel) on the NCSMC basis size characterized by N max forlow-lying − (red) and − (blue and green) resonances of Li. The same for selected − (middle panel) and − (bot-tom panel) P -wave phase shifts. The NN+3N(lnl) chiral in-teraction was used. shifts and selected partial wave phase shifts are shown inFigs. 3. The convergence with respect to N max is quitesatisfactory, especially for the two sharper resonances.We note that the eigenphase shifts are obtained fromthe S -matrix eigenvalues while the partial wave phaseshifts are obtained from diagonal matrix elements of the S -matrix.In the leftmost three panels of Fig. 4, we show thebound-state energies, in addition to the energies andwidths of the three resonances for the N max =4 , , N max =8 space arethen given in Table III. Selected eigenphase shifts and S -wave phase shifts obtained in the N max =8 space arepresented in Fig. 5. It is clear that the calculated 5 / − resonance is a good match to the experimentally knownresonance at 4.296 MeV. We predict that the 5.38 MeVlevel is 3 / − . On the other hand, the experimentallyvery narrow 6.43 MeV level does not correspond to ourcalculated very broad second 3 / − . Rather, it presum-ably corresponds to the calculated 7 / − state shown inthe right panel of Fig. 2 and in the top panel of Fig. 5as an extremely narrow resonance. For a more realisticdescription of this state, we would most likely need toinclude the 3 + state of Li in the NCSMC cluster expan-sion (3) [60]. The Li 3 + state that appears at 2.255 MeV in experiment (see the left panel of Fig. 2) would obvi-ously also impact other higher lying – and in particularhigher spin – resonances, e.g., the 7 / + and the second5 / + , shown in Fig. 5.The decreasing 3 / − and 1 / − eigenphase shifts thatstart at δ =0 o in Fig. 5 correspond to the two boundstates. On the other hand, all calculated S -wave phaseshifts and their associated eigenphase shifts are risingat their respective thresholds, i.e., the correspondingscattering lengths are negative. In particular, for the S / (2 + ) partial wave we find the scattering length of-0.44 fm while for S / (2 + ) -0.13 fm. We note that abroad 5 / + T =3 / Be, an isospin ana-log of a resonance in Li, was very recently reported inRef. [61]. It was found below the T =3 / / − resonance,the isospin analog of the 4.296 MeV resonance in Li.Before proceeding with the calculation of the capturecross section, the NCSMC results were phenomenologi-cally adjusted to reproduce experimental thresholds andpositions of known resonances in an approach known asNCSMC-pheno [62, 63]. This was accomplished first byadjusting the Li excitation energy of the 1 + state to itsexperimental value and, second, by fitting the Li NCSMinput energies to reproduce the experimental Li energiesin the NCSMC calculations. We performed the NCSMC-pheno calculations for the N max =6 and N max =8 modelspaces. As seen in the left panel of Fig. 2, our calcu-lated excitation energy for the Li 1 + state is quite closeto experiment. Consequently, it only needs a − 45 keVadjustment in the N max =8 calculation. Next, we adjustthe lowest NCSM Li eigenenergies in the 3 / − , 1 / − and 5 / − channels (used as input in the NCSMC calcu-lation) to reproduce the experimental separation energiesof the 3 / − and 1 / − bound states and the 5 / − reso-nance centroid energy. As seen in the middle panel ofFig. 4, the NCSMC 1 / − energy is already quite close toexperiment, therefore a shift of -0.3 MeV in the lowest1 / − NCSM eigenvalue is sufficient to reproduce the sep-aration energy. For the 3 / − and 5 / − channels, we needto modify the eigenvalues by about -1 MeV, i.e., 2.5% ofthe calculated ground-state (g.s.) energy.The resulting NCSMC-pheno bound-state energies,centroids and widths of the lowest three calculated res-onances and selected eigenphase shifts for the N max =8model space are presented in the fourth panel of Fig. 4and in Fig. 6, respectively. Due to the negligible ad-justment of the Li 1 + energy, channels other than the1 / − , / − , / − are basically unmodified compared tothe original NCSMC calculation.In Table III, we summarize bound-state energies, aswell as centroid energies and widths of the lowest threecalculated resonances obtained in the N max =8 NCSMCand NCSMC-pheno calculations. Within the table, theseare compared to available experimental data. The res-onance energies and width have been determined fromthe eigenphase shift derivatives as well as from an S -matrix analysis in the complex momentum space. Thetwo methods agree very well for all the resonance ener- Figure 4. Energies of Li bound states and low-lying resonances with respect to the Li+ n threshold. The leftmost threepanels show NCSMC calculations at N max =4 , 6, and 8. The fourth panel shows the NCSMC-pheno N max =8 calculation. TheNN+3N(lnl) chiral interaction was used. Coloured bars represent the widths of resonances. Experimental data in the rightmostpanel are from Ref. [53]. Question marks are used where data is unavailable. -300306090120150 δ [ d e g ] E kin [MeV] -300306090 δ [ d e g ] - - - - - - - - n+ Li + + +2 S (1 + ) S (2 + ) S (2 + ) S (1 + )7/2 + Figure 5. Li+ n eigenphase shifts for selected negative-parity (top panel) and positive-parity (bottom panel) chan-nels. Dashed lines in the bottom panel represent S -wavephase shifts. NCSMC calculations performed in N max =8space with the NN+3N(lnl) chiral interaction. gies and the widths of the two sharper resonances whilethey give a few hundred keV differences for the width ofthe broad 3 / − resonance. This could be interpreted asa theoretical uncertainty, indicated in the table. We re-iterate that only the bound-state energies and the 5 / − resonance energy were fitted in the NCSMC-pheno calcu-lations. The widths of the resonances are predictions as E kin [MeV] -300306090120150 δ [ d e g ] n+ Li - - - + - - - - + + Figure 6. Li+ n eigenphase shifts obtained from the N max = 8 NCSMC-pheno calculation for selected negative-parity (solid) and positive-parity (dashed) channels.NCSMC NCSMC-pheno Expt J π T E Γ E Γ E Γ3 / − / / − / a a / − / / − / / − / a Experimental spin and parity assignment uncertain Table III. Li bound-state and resonance energies with re-spect to the Li+ n threshold with the corresponding reso-nance widths. All values in MeV. NCSMC and NCSMC-pheno calculations have been performed with the NN+3N(lnl)chiral interaction in the N max =8 space. Experimental dataare from Ref. [53]. ANC [fm − / ] SF NCSM SF NCSMC-pheno P / (2 + ) 1.026 0.64 0.59 P / (2 + ) 0.995 0.41 0.41 P / (1 + ) -1.009 0.39 0.37 P / (1 + ) -0.663 0.11 0.11Table IV. Li 3 / − g.s. asymptotic normalization coefficients(ANC) obtained in the NCSMC-pheno calculations and spec-troscopic factors (SF) obtained in the NCSM and NCSMC-pheno. Calculations were performed in the N max =8 space.See Fig. 7 for other details. r [fm] -0.4-0.200.20.4 c l u s t e r f o r m f ac t o r [f m - / ] Li 3/2 - P (2 + ) P (2 + ) P (1 + ) P (1 + ) Figure 7. Li 3 / − g.s. cluster form factors. Only P -wavecomponents are shown. The full lines represent the N max =8NCSMC-pheno calculations, the dashed lines (for the Li 2 + state channels only) are NCSM results. The coupling betweenthe Li and neutron in the cluster state is given in Eq. (3). well as the energies of the two 3 / − states. Our calcula-tions reproduce very well the experimental properties ofthe 5 / − resonance, and the first calculated 3 / − reso-nance matches the energy and width of the experimental5.38 MeV state.A realistic description of the structure of the Liground state is essential for the description of the cap-ture reaction. In Fig. 7, we show the cluster form fac-tors (overlap functions) for the Li 3 / − g.s., defined by r (cid:104) Φ J π Tνr, − | A ν | Ψ J π TA = , − (cid:105) with the states from Eqs. (2)and (3). The dashed lines represent the NCSM clusterform factors that serve as input to the NCSMC equa-tions [24, 25]. While the NCSMC-pheno overlaps ex-tend beyond n - Li separations of 10 fm, the NCSM onesare basically zero starting at 7 fm. By integrating theoverlap functions squared, one obtains spectroscopic fac-tors (SF), which we present in Table IV together withthe asymptotic normalization coefficients (ANCs). Al-though the NCSM and NCSMC-pheno cluster form fac-tors differ, the spectroscopic factors are very similar.Still, we observe some reduction when continuum mi- croscopic cluster states are included. Overall, our SFsare in excellent agreement with those obtained withinthe Variational Monte Carlo (VMC) method [64, 65].The Li(g.s.) ↔ Li(g.s.)+ n NCSMC-pheno total SF, 1.00,is in good agreement with the experimental value of0.90(13) reported in Ref. [11]. Smaller SFs were re-ported in Refs. [9] (0.68(14)), [10] (0.62(13)), and [12](0.65(15)). Our calculated ANC values can be comparedto the experimental determination of (ANC) =1 . − obtained from the angular distribution analysis ofthe Li( d, p ) Li gs transfer reaction [66]. A slightly smaller(ANC) =0 . − was reported in Ref. [67] whichis in excellent agreement with our calculations. C. Li( n, γ ) Li radiative capture Our calculated Li( n, γ ) Li capture cross section is pre-sented in Fig. 8. We compare NCSMC-pheno results ob-tained in the N max =8 and N max =6 spaces. Overall, wefind a good stability of the calculations. By increasingthe model space, the cross section gets reduced slightlyand the difference can serve as an estimate of the uncer-tainty. The capture to the Li ground state dominatesthe total cross section. The excited state contribution issuppressed by more than an order of magnitude. In thelow-energy region displayed in Fig. 8, the non-resonantE1 capture is the leading contribution. The E2/M1 cap-ture enhanced by the 5 / − resonance is visible as a bumparound 0.23 MeV.Our calculated cross section is on the higher sidebut still within the limits derived from the 1998 NSCLCoulomb dissociation experiment [7] shown in Fig. 8 byblack points and vertical lines. These limits should becompared to the E1 contribution to the capture to theground state.The Li( n, γ ) Li reaction rate obtained from our to-tal capture cross section is shown in Fig. 9. In addition,we present the contribution of the capture to the groundstate to the overall reaction rate. Our results are smallerby a factor of 4 and 2 compared to values reported inRefs. [15] and [16], respectively. However, they are higherby a factor of 2 compared to the recent potential clustermodel calculations from Ref. [22]. One of the reasons forthe smaller reaction rate obtained in the latter calcula-tions is the lower value of the spectroscopic factor usedas input for the potential cluster model calculations com-pared to the spectroscopic factor obtained as an outputof our many-body calculations. IV. CONCLUSIONS We applied the ab initio NCSMC to study propertiesof Li bound states and low-lying resonances, and cal-culated the Li( n, γ ) Li cross section. Chiral nucleon-nucleon and three-nucleon interactions from Refs. [33]and [26] served as the only input for our calculations, Figure 8. Li( n, γ ) Li capture cross section obtained inthe NCSMC-pheno calculations. We compare N max =6 (dot-ted lines), N max =8 (dashed lines), the total N max =8 cross-section (solid line), and experimental limits from Ref. [7](black points). Cross-section contributions from the groundstate are shown in blue, contributions from the first excitedstate are in green.Figure 9. Li( n, γ ) Li reaction rate obtained in the N max =8NCSMC-pheno calculations. The upper line shows the to-tal reaction rate, and the lower line shows the ground-statecontribution. though for the purpose of predicting the capture crosssection we adjusted the thresholds and the position ofthe lowest resonance to their experimental values.Our calculations reproduce experimentally known bound states as well as the lowest 5 / − resonance of Li.We predict the 5.38 MeV resonance to be a 3 / − state. Inaddition to the very narrow 7 / − resonance, correspond-ing most likely to the experimental 6.43 MeV state, wefind several other broad low-lying resonances. In partic-ular, at 2.6 MeV above the Li+ n threshold we find abroad 3 / − resonance with the width of 2.5 MeV. Thedescription of the 7 / − resonance and of the higher lying7 / + and 5 / ± resonances can be improved by includ-ing the Li 3 + state in the NCSMC trial wave function(Eqs. (2), (3)). We plan to perform such calculations inthe future.Our calculated Li( n, γ ) Li capture cross section is onthe higher side but within the limits derived from the1998 NSCL Coulomb dissociation experiment. It is dom-inated by the direct E1 capture to the ground state witha resonant contribution around 0.23 MeV due to E2/M1radiation enhanced by the 5 / − resonance.The reaction rate obtained from our calculated capturecross section is lower than early evaluations. However, itis higher by about a factor of two compared to recent po-tential cluster model calculations. Our results indicatethat the Li( n, γ ) Li reaction might play a more impor-tant astrophysical role than recently considered.Results presented in this paper demonstrate currentcapabilities of the NCSMC. With high-precision chi-ral NN+3N interactions as the input, we are able topredict with confidence properties of light nuclei evenwith a large neutron excess. NCSMC calculations ofseveral other radiative capture reactions important forastrophysics including Be( p, γ ) B, C( p, γ ) N, and C( n, γ ) C are under way. ACKNOWLEDGMENTS