Nuclear Resonances, Scattering and Reactions from First Principles: Progress and Prospects
NNuclear Resonances, Scattering and Reactions from First Principles: Progress and Prospects
Sofia Quaglioni and Petr Navrátil Lawrence Livermore National Laboratory, P.O. Box 808, L-414, Livermore, CA 94551, USA TRIUMF, 4004 Wesbrook Mall, Vancouver, BC V6T 2A3, Canada
Introduction
The study of rare isotopes at radioactive beam facilities has opened new frontiers in nuclear physics. To enable the discovery of model deficiencies and missing physics it is essential that the new insights from these experiments be confronted with predictive theoretical frameworks capable of describing the interplay of many-body correlations and continuum dynamics characteristic of exotic nuclei, as well as the nuclear reactions used to produce and study them. The predictive power garnered, in part, through this process is paramount to the use of atomic nuclei as a doorway to explore some of the most fundamental laws of the Universe through precision experiments. A primary example is neutrinoless double-beta (0 nbb ) decay. If observed, this exceedingly rare decay would provide definite evidence that neutrinos are their own antiparticles (i.e., Majorana particles), and a means to determining the neutrinos’ masses. Essential for this latter goal are accurate predictions of 0 nbb nuclear transitions, which can only be determined theoretically. A predictive theory of nuclear structural and reaction properties is also desirable to aid in precisely determining thermonuclear reaction rates that play an important role in fusion-energy experiments, the predictions of stellar-evolution models, and simulations of nucleosynthetic processes. Very difficult or even impossible to measure at the relevant energies of tens to hundreds of keV due to the hindering effect of Coulomb repulsion, thermonuclear fusion rates are almost always estimated by extrapolation from higher-energy experimental data. Without the help of a predictive theory, such extrapolations can be a major source of uncertainty. In addition to reducing the uncertainty in reaction rates at very low temperatures, a first-principles theory can also help resolving open issues such as the effect of polarization induced by the strong magnetic fields in plasma environments (such as at the international project ITER in France and at the National Ignition Facility in the USA) on the deuterium-tritium and deuterium-deuterium thermonuclear reaction rates.
What is understood under ab initio nuclear theory?
At the low energies relevant for most studies of nuclear structure and nuclear reactions, currently the best path to achieving a predictive theory of nuclear properties combines effective field theories of quantum chromodynamics—that organize the nuclear force into systematically improvable expansions—with ab initio methods—that solve the quantum many-nucleon problem with controlled approximations. In the last few years, the emergence of powerful ab initio approaches to nuclear structure has dramatically accelerated this journey, allowing for the description of nuclei as heavy as
Sn. [1]
Fewer ab initio efforts have been focused on attaining a unified description of bound-state and continuum properties and developing techniques applicable to nuclear reactions. Nevertheless, since the first ab initio calculation of neutron scattering on He in 2007 [2] , progress has been quite remarkable also in this direction. Large scale computations combined with new and sophisticated theoretical approaches have enabled high-fidelity predictions for nucleon- and deuterium-induced scattering and reactions on light targets and some medium-mass nuclei, and the ab initio description of H- He and He- He scattering and radiative capture processes (see Ref. [3] and references therein) as well as He- He scattering [4] . More recently, an avenue for arriving at the ab initio description of scattering and reactions in medium-mass nuclei has been opened by the development of optical nucleon-nucleus potentials [5] . The ab initio no-core shell model with continuum
To arrive at an ab initio description of low-energy nuclear reactions, over the past 13 years we have been developing the no-core shell model with continuum (NCSMC), [6] a unified framework for the treatment of both bound and unbound states in light nuclei. With chiral two- (NN) and three-nucleon (3N) interactions as the only input, we are able to predict structure and dynamics of light nuclei and, by comparing to available experimental data, test the quality of chiral nuclear forces. Describing a reaction–for example the scattering of He with He—requires addressing both the correlated short-range behavior occurring when the reactants are close together, forming a composite nucleus ( Be in our example), and the clustered long-range behavior occurring when the reactants ( He and He in our example) are far apart. The NCSMC accomplishes this by adopting a generalized cluster expansion for the wave function of the reacting system, which in the Be example is given by In the first term, consisting of a expansion over (square-integrable) eigenstates of the composite system ( Be) obtained within the no-core shell model (NCSM) [7] and indexed by 𝜆 , all A nucleons are treated on the same footing. In the second term, corresponding to a resonating-group method [8] expansion over (continuous) antisymmetrized channels the wave function is factorized into products of cluster components ( He and He) and their relative motion, with proper bound-state or scattering boundary conditions. Here, 𝑟⃗ $,& is the separation between the centers-of-mass of He and He and ν is a collective index for the relevant quantum numbers. The discreet expansion coefficients 𝑐 () * and the continuous relative-motion amplitudes 𝛾 ,) * (𝑟) are obtained as Be ) * () * , λ 𝐽 ; < ( + 7 > 𝑑𝑟 𝑟 @ 𝛾 ,) * (𝑟) 𝒜B , CΦ ,,E) * , CΦ ,,E) * $ <8 He & Nucleon elastic scattering on He is the most straightforward process to calculate within the NCSMC. The He nucleus, also known as 𝛼 particle, is tightly bound with the lowest excited state at 20.2 MeV. Consequently, it is typically sufficient to consider only the He ground state in the generalized cluster expansion (1). Recently, 𝑛 - 𝛼 scattering became also treatable within the Faddeev-Yakubovsky formalism. [9] Benchmark calculations using the same chiral NN interaction as input demonstrated agreement between the computed phase shifts, providing validation for these two vastly different methods. [9] Having the capability to also include the chiral three-nucleon (3N) interaction in the calculations, we were able to reproduce the experimental phase shifts in the region of the Y resonance [located at about 0.8 (1.5) MeV above the He+n ( He+p) threshold], corresponding to the ground state of He ( Li). That allowed us to compute the He(p,p) He and H(α,p) He proton elastic scattering and recoil reactions, respectively, at energies of a few MeV per nucleon. These processes are the leading means for determining the concentrations and depth profiles of helium and hydrogen, respectively, at the surface of materials or in thin films. As seen in Fig. 1, our calculated cross sections are in a good agreement with experimental data. [10] Figure 1 Computed (lines) H(α,p) He angular differential cross section at proton recoil angles φ p = 4◦, 15◦, 20◦, and 30◦ as a function of the incident He energy compared with data (symbols). Figure from Ref. [10]. Nuclear scattering and reactions involving deuterium ( 𝑑 ), the deformation and breakup of which cannot be neglected even at low energies due to its weak binding, pose an additional challenge. Deuteron- 𝛼 elastic scattering is the simplest of such processes. We successfully applied the NCSMC to the 𝑑 - He system and gained insight on the structure of Li resonances as well as of its E α [MeV] d σ / d Ω p [ b / s r] Nurmela et al. , 4 o Nurmela et al. , 15 o Kim et al. , 20 o Nagata et al. , 20 o Pusa et al. , 20 o Wang et al. , 20 o Keay et al. , 30 o ϕ p = 4 o H( α , p ) He ϕ p = 15 o ϕ p = 20 o ϕ p = 30 o ound ground state. [11] The deuteron deformation and breakup was taken into account by including proton-neutron excited pseudo states, i.e., a form of approximate discretization of the proton-neutron continuum. These calculations, carried out with chiral NN+3N forces provide a realistic description of Li and highlight the sensitivity of the system to the 3N interaction. Omitting the chiral 3N interaction results in significant overestimation of the excitation energy of the Z state, the lowest excitation of Li, as well as an underprediction of the splitting among the Z - Z - Z resonance triplet, dominated by d- He D -wave of relative motion coupled with the deuteron’s S =1 spin. This demonstrates that the 3N interaction contributes substantially to the nuclear spin-orbit force strength. The calculated elastic differential cross section off the Z resonance region is described very well as seen in Fig. 2. The Li S - and D - wave asymptotic normalization constants, i.e., the normalizations of the wave function tail, are also close to the values inferred from analyses of experimental data. The S-wave asymptotic normalization constant determines the H(α,γ) Li radiative capture cross section, responsible for the Big-Bang nucleosynthesis of Li. Figure 2 Calculated (lines) and measured (symbols) center-of-mass angular distributions at E d =2.93; 6.96; 8.97, and 12 MeV are scaled by a factor of 20, 5, 2, and 1, respectively. For details see Ref. [11]. A well-known reaction important for both Big-Bang nucleosynthesis and the Solar proton-proton chain is the He(α,γ) Be radiative capture. Initial NCSMC calculations of He- He and H- He scattering [12] were carried out starting from a two-nucleon Hamiltonian. The properties of the low-lying resonances as well as those of the two bound states of Be and Li were reproduced rather well. With the obtained scattering and bound state wave functions, we also computed the astrophysical S-factor for the He(α,γ) Be solar fusion cross section (Fig. 3) as well as that of its mirror reaction H(α,γ) Li. [12] At very low energies, the He(α,γ) Be S-factor is in a good agreement with the measurements taken at the underground LUNA facility (Co07). However, its overall shape does not match some of the recent data, likely owing to an overestimation of the non- adopted N force model, particularly concerning thestrength of the spin-orbit interaction.The inclusion of the d þ He states of Eq. (2) results alsoin additional binding for the þ ground state. This stemsfrom a more efficient description of the clusterization of Liinto d þ α at long distances, which is harder to describewithin a finite HO model space, or — more simply — from theincreased size of the many-body model space. Indeed, asshown in Fig. 3 and in Table I for the absolute value of the Lig.s. energy, extrapolating to N max → ∞ [37] brings theNCSM results into good agreement with the NCSMC, forbound states and narrow resonances. However, only with thelatter do the wave functions present the correct asymptotic,which for the g.s. are Whittaker functions. This is essentialfor the extraction of the asymptotic normalization constantsand a future description of the H ð α ; γ Þ Li radiative capture[5]. The obtained asymptotic D - to S -state ratio is notcompatible with the near-zero value of Ref. [38], but rather isin good agreement with the determination of Ref. [39],stemming from an analysis of Li þ He elastic scattering.Next, in Figs. 4(a) and 4(b), respectively, we compare the H ð α ; d Þ He deuteron elastic recoil and He ð d; d Þ Hedeuteron elastic scattering differential cross sections com-puted using the NN þ N Hamiltonian to the measuredenergy distributions of Refs. [7 – – þ resonance, the calculations are in fairagreement with experiment, particularly in the low-energyregion of interest for the big-bang nucleosynthesis of Li,where we reproduce the data of Besenbacher et al. [42] andQuillet et al. [8]. The 500 keV region below the resonancein Fig. 4(a) is also important for material science, where theelastic recoil of deuterium knocked by incident α particlesis used to analyze the presence of H. At higher energies,near the þ and þ resonances, the computed cross sectionat the center-of-mass scattering angle of θ d ¼ ° reproduces the data of Galonsky et al. [44] and Mani et al. [45], while we find slight disagreement with the data ofRef. [9] in the elastic recoil configuration at the laboratoryangle of φ d ¼ °. At even higher energies, the calculatedcross section of Fig. 4(b) lies above the measured one. Thisis likely related to the fact that the þ state is too broad. Theoverall good agreement with experiment is also corrobo-rated by Fig. 4(c), presenting He ð d; d Þ He angular dis-tributions in the . ≤ E d ≤ . MeV interval of incidentenergies. In particular, the theoretical curves reproduce thedata at 2.93 and 6.96 MeV, while some deviations arevisible at the two higher energies, in line with our previousdiscussion. Nevertheless, in general, the present resultswith N forces provide a much more realistic description ofthe scattering process than our earlier study of Ref. [14].Finally, we expect that an N max ¼ ð Þ calculation(currently out of reach) would not significantly changethe present picture, particularly concerning the narrow þ resonance. Indeed, much as in the case of the g.s. energy,here the NCSMC centroid is in good agreement with theNCSM extrapolated value, 0.99(9) MeV. Conclusions. — We presented an application of the ab initio NCSMC formalism to the description of deuteron-nucleusdynamics. We illustrated the role of the chiral N force andcontinuous degrees of freedom in determining the bound-state properties of Li and d - He elastic scattering observ-ables. The computed g.s. energy is in excellent agreementwith experiment, and our d þ α asymptotic normalizationconstants support a nonzero negative ratio of D - to S -statecomponents for Li. We used deuterium backscattering andrecoil cross-section data of interest to ion beam spectroscopyto validate our scattering calculations and found goodagreement at low energy in particular. The overestimationby about 350 keV of the position of the þ resonance is anindication of remaining deficiencies of the Hamiltonian E α [MeV] ( ∂ σ / ∂ Ω ) l a b . [ b / s r] Besenbacher et al. Browning et al. Kellock et al. Nagata et al. Quillet et al. H( α , d ) He ϕ d = 30 o (a)3 + + E d [MeV] ( ∂ σ / ∂ Ω ) c . m . [ b / s r] Galonsky et al. (168 o )Mani et al. (165 o )Mani et al. (163 o ) He( d , d ) He θ d = 164 o (b)3 + + + θ d [deg] ( ∂ σ / ∂ Ω ) c . m . [ b / s r] He( d , d ) He x 20x 5x 2x 1 (c) FIG. 4 (color online). Computed (a) H ð α ; d Þ He laboratory-frame and (b) He ð d; d Þ He center-of-mass frame angular differentialcross sections (lines) using the NN þ N Hamiltonian at the deuteron laboratory and c.m. angles of, respectively, φ d ¼ °and θ d ¼ ° as a function of the laboratory helium ( E α ) and deuteron ( E d ) incident energies, compared with data (symbols) fromRefs. [7 – – E d ¼ . ; . ; . [46], and12 MeV [47] are scaled by a factor of 20,5,2, and 1, respectively. All positive- and negative-parity partial waves up to J ¼ wereincluded in the calculations. PRL week ending29 MAY 2015 esonant S -wave phase shifts. A more quantitative prediction will require the inclusion of chiral 3N forces. One interesting observation is the that no microscopic theoretical approach is currently able to reproduce simultaneously the experimental normalizations of both the He(α,γ) Be and H(α,γ) Li S-factors. Our calculations overpredict the latter. [12] Figure 3 Astrophysical S factor for the He(α,γ) Be radiative-capture processes obtained from the NCSMC approach compared with other theoretical approaches and with experiments. For details see Ref.[12]. Structure of weakly bound exotic nuclei NCSMC calculations have also helped shedding light on the properties of halo nuclei, exotic weakly bound systems with one or two nucleons (typically neutrons) well-separated from the rest of the nucleus. One of the best examples is Be, famous for the “parity-inversion” of its ground and first excited states. With 7 neutrons, the standard shell model expects a Y ground state with the last neutron occupying the 0 p level. However, experimentally Be has a Z ground state bound by only about 500 keV with respect to the Be+ 𝑛 threshold. Understandably, the dynamics of the Be+ 𝑛 system needs to be properly taken into account to realistically describe Be, something the NCSMC is perfectly suited for. Still, the details of the nuclear interaction continue to play a critical role and only a chiral force that describes well the nuclear density is capable to reproduce well the Be properties. [13] In Fig. 4, we show the overlap of the calculated Be Z ground-state wave function with Be+ 𝑛 as a function of Be and neutron separation. The halo S -wave component (solid and dashed black lines) extends beyond 20 fm, i.e., very far beyond the range of the nuclear interaction. Also shown (dotted black line) is the S -wave overlap obtained from describing Be within the NCSM alone, i.e., without Be+n cluster component. While the spectroscopic factors obtained by integrating the square of the two overlap functions are about the same, the NCSM result is close to zero already at about 8 fm. J. Dohet-Eraly et al. / Physics Letters B 757 (2016) 430–436 Fig. 3. (Color online.) 1 / + phase shifts for different values of the SRG parameter: ! = . fm − (dotted lines), ! = . fm − (solid lines), and ! = . fm − (dashed lines). For ! = . fm − , different values of N max are considered; the N max value used for computing the colliding-nuclei wave functions is given. Table 5 / + scattering length for the α + He collision for different values of the SRG parameter ! and different values of N max ; the N max value used for computing the colliding-nuclei wave functions is given. ! [ fm − ] N max a / + [ fm ] − . . . . . Fig. 4. Differential α + He elastic cross sections ( d σ / d $ ) normalized by the dif-ferential Rutherford cross sections ( d σ R / d $ ) as a function of the scattering angle measured in the c.m. frame. Experimental data come from Ref. [56]. For negative-parity partial waves, the discrepancy between the-oretical and experimental resonances seen in Fig. 1 is also visible in the phase shifts. Moreover, the splitting between the / − and / − is underestimated, as it can be seen from the comparison of the phase shifts and of the scattering lengths. Instead of analyz-ing the phase shifts and the scattering lengths, we can compare directly theoretical and experimental cross sections. In Fig. 4, the differential α + He elastic cross sections are displayed for differ-ent angles at two particular colliding energies and compared with experimental data from Ref. [56], for which no phase-shift analysis exists. Our approach reproduces the general trends of the experi-mental data.To evaluate the impact of the discrepancies in the elastic scat-tering on the He( α , γ ) Be and H( α , γ ) Li astrophysical S factors, we adopt a phenomenological model based on the NCSMC results in the largest model space. The basic idea is to consider the en- Fig. 5. (Color online.) Astrophysical S factor for the He( α , γ ) Be and H( α , γ ) Li radiative-capture processes obtained from the NCSMC approach and from its phe-nomenological version and compared with other theoretical approaches [3,20] and with experiments [57–60,6–13,61–63,15]. Recent data are in color (online) and old data are in light grey. Table 6 He( α , γ ) Be and H( α , γ ) Li astrophysical S factors extrapolated at zero collision energy. Experimental data come from Refs. [5,15]. For the He( α , γ ) Be reaction, the numbers in parentheses are the errors in the least significant digits coming from the experiments and from the theoretical extrapolation while for the H( α , γ ) Li reaction, they are the statistical and systematic errors.NCSMC Exp. Refs. S He ( α , γ ) Be ( ) [ keV b ] S H ( α , γ ) Li ( ) [ keV b ] ergies of the square-integrable NCSM basis states E λ , appearing in Eq. (5), as adjustable parameters. These new degrees of free-dom are then used to reproduce the experimental Be and Li bound-state and resonance energies and reducing the gap between theoretical and experimental / + phase shifts.The He( α , γ ) Be and H( α , γ ) Li astrophysical S factors ob-tained with the NCSMC approach and with its phenomenologi-cal version are displayed in Fig. 5 and compared with experi-ment [57–60,6–13,61–63,15]. The astrophysical S factors extrap-olated at zero colliding energy are given in Table 6. The electric E E as well as the magnetic M have been considered. For the energy ranges which are considered, the contribution of the E is dominant while M is essentially negligible and the E play a small but visible role in the He( α , γ ) Be radiative capture, mostly near the / − resonance energy. Qualitatively, the He( α , γ ) Be astro-physical S factors agree rather well with the experimental ones. The results obtained with the phenomenological model are sim-ilar up to approximately the / − resonance energy. Indeed, the peak in the experimental S factor at a relative collision energy of about corresponds to a E from the / − reso-nance to the / − ground state. Since the / − resonance energy Figure 4 Overlap of Be ground-state wavefunction with B+n as a function of Be and neutron separation. The black dashed line corresponds to the original NCSMC calculation for the S-wave, while the black full line is obtained after a phenomenological correction to fit the experimental threshold. The black dotted line shows the NCSM S-wave result that serves as one of the inputs into the NCSMC. For further details see Ref. [13]. With the availability of the first re-accelerated C beam at TRIUMF, we teamed up with the TRIUMF IRIS collaboration and investigated the C(p,p) C elastic scattering and the structure of the unbound N nucleus. [14] As the collision energy was about 4 MeV in center of mass, the cross section was sensitive to the Z and Y resonances in N, analogs of well-known lowest-lying resonances in Be. Perhaps not surprisingly, the chiral NN+3N force that described successfully the parity inversion of the Be ground state also provided the best description of the measured differential cross section, although the overall normalization was overestimated as seen in Fig. 5. Interestingly this investigation demonstrated how an observable that is rather straightforward to measure, such as the elastic scattering cross section, can be a sensitive probe of nuclear force models. Figure 5 Measured differential cross section of C(p,p) C(gs) at E cm = 4.15 MeV. The curves are ab initio theory calculations. For details see Ref.[14]. r [fm] -0.5-0.4-0.3-0.2-0.100.10.20.30.4 c l u s t e r f o r m f ac t o r [f m - / ] Be(0 +1 )+n s=1/2 l=0 Be(2 +1 )+n s=3/2 l=2 Be(2 +1 )+n s=5/2 l=2 Be(2 +2 )+n s=3/2 l=2 Be(2 +2 )+n s=5/2 l=2 Be(1/2 + ) N LOsat 60 90 120 150 180 Θ cm [deg] d σ / d Ω [ m b / s r] chiral NNchiral NN+3N400chiral NN+3N N LO sat chiral NN+3N N LO sat / 2.3Experiment C(p,p) C E cm = 4.15 MeVNCSMC ore complex halo nuclei are Borromean systems with two loosely bound neutrons. The simplest example is He, which presents a He-n-n three-body cluster structure with none of the two-body subsystems being bound. We were able to generalize the NCSMC formalism to include three-body cluster dynamics. [15] The formalism and the computational effort become significantly more challenging. Compared to binary processes, the number of channels ( 𝜈 ) increases dramatically. Further, the wave function has to be described up to very large hyperradial distances (on the order of 100 fm) to reach the asymptotic region. Because of the increased complexity, so far three-cluster NCSMC calculations do not include 3N forces. In spite of this, our calculations reproduce rather well the properties of the bound ground state of He (including the characteristic di-neutron and cigar configurations of the probability distribution, shown in Fig. 6, the charge radius and the two-neutron separation energy) as well as its low-lying resonances. [15] Figure 6 Probability distribution the 𝐽 ; = 0 Z ground state of the He. The r nn and r α,nn are, respectively, the distance between the two neutrons and the distance between the c.m. of He and that of the two neutrons. For further details see Ref. [15]. Transfer reactions Ab initio approaches to nuclear dynamics hold the promise to provide a more profound understanding not only of nuclear scuttering but also complex reactions. The most advanced application of the NCSMC so far is the calculation of polarized deuterium-tritium (DT) thermonuclear fusion [16] . The DT fusion, i.e. the ( 𝑑, 𝑛 ) transfer process H( 𝑑, 𝑛 ) He, is the most promising of the reactions that could power thermonuclear reactors of the future. This reaction, used at facilities such as ITER and NIF in the pursuit of sustained fusion energy production, is characterized by a pronounced Z resonance just above the DT threshold. It may lead to even more efficient energy generation if obtained in a polarized state, that is with the spins of the deuteron ( Z ) and H ( Z ) aligned. While the unpolarized DT fusion has been investigated in many experiments, very few measurements with polarized 𝑑 and/or H nuclei have been THREE-CLUSTER DYNAMICS WITHIN THE AB … PHYSICAL REVIEW C , 034332 (2018)TABLE V. Percentage of the norm of the He g.s. wave functionthat comes directly from the NCSM part of the basis ( ! λ c λ ). N max λ SRG = . − λ SRG = . − comparison in terms of N max provides a better picture of therelevance of each component in the full calculation. We alsonote that the last three columns of Table I in Ref. [18] presenta mismatch with respect to the model space size reported inthe first column, showing results obtained with an N max valuelarger by 2 units. Therefore, we call the reader to consider thepresent tables to be the accurate representation of the results.As seen in Table IV, convergence is not as obviously reachedwhen using the harder potentials with λ SRG = . − . Withinthe NCSMC, there still is a 200-keV difference between the N max = 10 and 12 results. However, the fact that the valueobtained for N max = 12 ( − . 17 MeV) is in agreement withthe NCSM extrapolation from Ref. [39] [ − . ! λ c λ . These percentages are shown in Table V forthe two different potentials used, as well as for different sizesof the model space. We find that, as one would expect, theNCSM component of the basis is able to describe a much largerpercentage of the wave function when using the softer potentialcorresponding to the λ SRG = . − resolution scale, andalso a larger percentage as the HO model space size increases. 1. Spatial distribution In Fig. 5, we show the probability density, as defined inSec. II E, for the ground state of He in terms of the the distancebetween the two halo neutrons ( r nn = √ η nn ) and the distancebetween the He core and the center of mass of the externalneutrons ( r α ,nn = √ / η α ,nn ). This density plot presents twopeaks, which correspond to the two preferred spatial config-urations of the system. The dineutron configuration, whichcorresponds to the two neutrons being close together, clearlypresents a higher probability respect to the cigar configurationin which the two neutrons are far apart and at the opposite sidesof the core. This distribution is in agreement with previousstudies [31,39,50,67–70]. In order to estimate the reliability ofthe approximation of Eq. (49), which uses the projection of theNCSMC wave function into the cluster basis, we integrated theprobability density given by Eq. (49). This integral is equivalentto the square of the norm of the projected wave function. Weobtained 0.971 for the potential with λ SRG = . − and0.967 for the potential with λ SRG = . − . Given that wework with normalized wave functions, the proximity of theseintegrals to the unity indicates that only a small part of thewave functions was lost when performing the projection.When the He ground-state wave function is calculatedwithin the NCSM basis, the probability density can be obtained FIG. 5. Probability distribution the J π = + ground state of the He. Here r nn = √ η nn and r α ,nn = √ / η α ,nn are, respectively, thedistance between the two neutrons and the distance between the c.m.of He and that of the two neutrons. by projecting into a cluster basis in the same way as it is donefor the NCSMC in Eq. (46). The obtained projected wavefunction presents the same distribution observed in the caseof the NCSMC, with the difference that it is less extended.This picture is consistent with the results previously reportedin Ref. [39] and is to be expected given that within this basisthe three-body asymptotic behavior is not well described. Thisis easily appreciated in Fig. 6, where the contour diagramof the probability distribution is shown for the NCSMC inFig. 6(b) and for the NCSM component in Fig. 6(c). In thecontour plots, it is also easier to determine the position onthe probability maxima: Within the dineutron configurationthe highest probability density appears when the neutrons areabout 2 fm apart and the He core about 3 fm from them. Withinthe cigar configuration, the neutrons are about 4 fm apart andthe core is around 1 fm from their center of mass.In Fig. 6(a), the most relevant hyper-radial components˜ u ν K ( ρ ) of the α + n + n relative motion are shown. Thehyper-radial components ˜ u ν K ( ρ ) are analogous to u ν K ( ρ ) fromEq. (37) but defined for the projected wave function fromEq. (46). The solid blue lines are the components from the fullNCSMC wave function while the dashed red lines represent thecontribution to the full NCSMC wave function coming fromthe discrete NCSM eigenstates. This figures also provides agood visualization of how the short range of the NCSM wavefunction is complemented with the cluster basis to reproducethe extended wave function typical of halo nuclei by means ofthe NCSMC. 2. Radii The spatial extension of a particular state can be estimatedby its matter radius as described in Sec. II F. In Table VI, erformed due to experimental challenges. NCSMC calculations with modern chiral NN and 3N interactions as the only input were able to demonstrate the small contributions of partial waves with orbital momentum ℓ > 0 in the vicinity of the Z resonance. They predict the DT reaction rate for realistically polarized reactants and show that the reaction rate increases compared to the unpolarized one and, further, the same reaction rate as the unpolarized one can be achieved at a lower temperature (see Fig. 7). The future New technical developments of the NCSMC approach currently under way will allow calculations of reactions induced by three-nucleon projectiles, two-nucleon transfer processes, the description of 𝛼 -clustering, 𝛼 scattering and capture processes, and ( 𝛼 , 𝑁 ) transfer reactions. Among our long-term goals are studies of systems with three-body clusters, e.g., Borromean two-neutron halo nuclei such as Li, and in general reactions with three-body final states. Overall, ab initio calculations of nuclear structure and reactions made tremendous progress in the past decade and have a bright future. These calculations became feasible beyond the lightest nuclei; they make connections between low-energy quantum chromo-dynamics and many-nucleon systems and find applicability to nuclear astrophysics, nuclear reactions relevant for energy generation, as well as to evaluations of nuclear matrix elements needed in tests of fundamental symmetries and physics beyond the standard model. In synergy with experiments, Figure 7 NCSMC calculated H( 𝑑, 𝑛 ) He reaction rate with and without polarization. A realistic 80% polarization of D and T with their spins aligned is considered. The arrows in the figure show that, with polarization, a reaction rate of equivalent magnitude as the apex of the unpolarized reaction rate is reached at lower temperatures. Shown evaluations of experimental data are for unpolarized nuclei. For further details see Ref. [16]. b initio nuclear theory is the right approach to understand low-energy properties of atomic nuclei. Acknowledgments This work was supported by the NSERC Grant No. SAPIN-2016-00033 and by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under Work Proposals No. SCW0498. TRIUMF receives federal funding via a contribution agreement with the National Research Council of Canada. This work was prepared in part by LLNL under Contract No. DE-AC52-07NA27344. 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