Abstract
The fusion of deuterium (D) with tritium (T) is the most promising of the reactions that could power thermonuclear reactors of the future. It may lead to even more efficient energy generation if obtained in a polarized state, that is with the spin of the reactants aligned. Here, we report firstprinciples predictions of the polarized DT fusion using nuclear forces from effective field theory. By employing the ab initio nocore shell model with continuum reaction method to solve the quantum mechanical fivenucleon problem, we accurately determine the enhanced fusion rate and angular distribution of the emitted neutron and ^{4}He. Our calculations demonstrate in detail the small contribution of anisotropies, placing on a firmer footing the understanding of the rate of DT fusion in a polarized plasma. In the future, analogous calculations could be used to obtain accurate values for other, more uncertain thermonuclear reaction data critical to nuclear science applications.
Introduction
Thermonuclear reaction rates of light nuclei are critical to nuclear science applications ranging from the modeling of bigbang nucleosynthesis and the early phases of stellar burning to the exploration of nuclear fusion as a terrestrial source of energy. The lowenergy regime (tens to hundreds of keV) typical of nucleosynthesis and fusion plasmas is challenging to probe due to low counting rates and the screening effect of electrons, which in a laboratory are bound to the reacting nuclei. A predictive understanding of thermonuclear reactions is therefore needed alongside experiments to achieve the accuracy and/or provide part of the nuclear data required by these applications. A salient example is the fusion of deuterium (D) with tritium (^{3}H or T) to generate a ^{4}He nucleus (αparticle), a neutron, and 17.6 MeV of energy released in the form of kinetic energy of the products. This reaction, used at facilities such as ITER^{1} and NIF^{2} in the pursuit of sustained fusion energy production, is characterized by a pronounced resonance at the centerofmass (c.m.) energy of 65 keV above the free D and T nuclei due to the formation of the J^{π} = 3/2^{+} resonance of the unbound ^{5}He nucleus. Fifty years ago, it was estimated^{3} that, in the ideal scenario in which the spins of the reactants are perfectly aligned in a totalspin 3/2 configuration and assuming that the reaction is isotropic, one could achieve an enhancement of the cross section by a factor of δ = 1.5, thus improving the economics of fusion energy generation^{4}. However, while the unpolarized cross section and some analyzingpower data exist, no correlation coefficients have been measured yet to confirm this prediction^{5}. More generally, what little is known about the properties of the polarized DT fusion was inferred from measurements of the D^{3}He reaction^{6}.
The DT fusion is a primary example of a thermonuclear reaction in which the conversion of two lighter elements to a heavier one occurs through the transfer of a nucleon from the projectile (D) to the target (T). Despite the fairly small number of nucleons involved in this process, arriving at a comprehensive understanding—in terms of the laws of quantum mechanics and the underlying theory of the strong force (quantum chromodynamics)—of the interweaving of nuclear shell structure and reaction dynamics giving rise to the DT fusion already represents a formidable challenge for nuclear theory.
Towards this goal, a pioneering ab initio study of the DT fusion was carried out in ref. ^{7}, using a nucleonnucleon (NN) interaction that accurately describes twonucleon data and representing the wave function on a basis of continuous “microscopiccluster” states^{8} made of D+T and n+^{4}He pairs in relative motion with respect to each other. However, this approach was unable to yield results of adequate fidelity, due to the omission of the threenucleon (3N) force—disregarded for technical reasons. Numerous studies have shown that this component of the nuclear interaction is essential for the reproduction of singleparticle properties^{8,9,10,11,12}, masses^{13,14,15}, and spin properties^{10,16}, all impactful in the present case. Besides the 3N force, the approach of ref. ^{7} also lacked a complete treatment of shortrange fivenucleon correlations, which are crucial to arrive at the accurate description of the 3/2^{+} resonance. The formation of this rather longlived resonance as a correlated, localized system of five nucleons built up during the fusion process is integral to the reaction mechanism. Finally, no polarization observables were calculated in the study of ref. ^{7}.
In the following, we report on ab initio predictions for the polarized DT fusion using validated NN and 3N forces derived in the framework of chiral effective field theory (EFT)^{17,18}, a powerful tool that enables the organization of the interactions among protons and neutrons in a systematically improvable expansion linked to the fundamental theory of quantum chromodynamics. The quantummechanical fivenucleon problem is solved using the nocore shell model with continuum (NCSMC)^{10,19}, where the model space includes D+T and n+^{4}He microscopiccluster states, plus conventional static solutions for the aggregate ^{5}He system^{20}. This enables a fully integrated description of the reaction in the incoming (outgoing) channel, where the reactants (products) are far apart, as well as when all five nucleons are close together. We show that this approach yields an accurate reproduction of the DT cross section for unpolarized reactants, discriminating among reaction rates from phenomenological evaluations and demonstrating in detail the small contribution of anisotropies in the vicinity of the 3/2^{+} resonance. The maximum enhancement of the polarized cross section varies as a function of the deuterium incident energy, dropping significantly above 0.8 MeV. However, such variation is slow in the narrow range of optimal energies for the reaction, resulting in a rather constant enhancement of the rate of fusion, compatible with the historic approximate estimate.
Results
Validation of model for unpolarized reaction observables
We begin our study with a validation of our ab initio reaction method on existing experimental data for the unpolarized DT reaction^{21,22,23,24,25}. In Fig. 1a, we review the agreement of our computed astrophysical Sfactor with established measurements. The Sfactor isolates the nuclear dynamics by factoring out the Coulomb component of the total reaction cross section. The experimental peak at the c.m. energy of E_{c.m.} = 49.7 keV corresponds to the enhancement from the 3/2^{+} resonance of ^{5}He (see Supplementary Note 2). We underpredict by 15% the experiment (green dashed line versus red circles), an outcome that can be traced back to the overestimation of the 3/2^{+} resonance centroid by a few keV, stemming from residual inaccuracies of the nuclear interaction^{10}. To overcome this issue and arrive at an accurate evaluation of polarized DT reaction observables, we apply a phenomenological correction of −5 keV to the position of the resonance centroid, achieving a remarkable agreement with the experimental Sfactor over a wide range of energies (blue line). A detailed explanation of how such correction was obtained can be found in the Methods section. As a further validation of our calculations in Fig. 1b we present the differential cross section in the centerofmass frame at the scattering angle of θ_{c.m.} = 0^{°} over a range of energies up to the deuterium breakup threshold. Our results (blue solid and green dashed lines) match the differential cross section of ref. ^{26} (red circles), obtained from a Legendre coefficient fit of measurements.
Polarization enhancement and reaction anisotropy
Having validated our calculation on precision measurements of unpolarized DT fusion, we now turn to the fusion of polarized DT fuel. The tritium has a spin of 1/2, consequently its initial spin state is fully characterized by the Cartesian spin projection onto the axis of quantization (zaxis), q_{z}. On the other hand, the deuterium is a spin1 particle. Therefore, besides the equivalent p_{z} projection, an extra tensor value (p_{zz}) is required to fully specify the spin state of the D beam. For the special case considered here, in which both reactants are aligned along the zaxis, the polarized differential cross section assumes a fairly simple form and is given by
where \(A_{{\mathrm{zz}}}^{(b)}\) and \(C_{{\mathrm{z,z}}}\) are the beam tensor analyzing power and spin correlation coefficient, respectively. The general expression for arbitrary orientation of the spins is more complicated and can be found in refs. ^{27,28,29}. The main assumption used to estimate the 50% enhancement of the cross section for polarized DT fuel is that the reaction proceeds entirely through the \(J^\pi = 3/2^ +\) partial wave with an orbital relative angular momentum of the D+T pair ℓ = 0 (that is, in an swave of relative motion). Under such an assumption, the unpolarized differential cross section is isotropic (that is, independent from the scattering angle). Furthermore, the integrals of \(A_{{\mathrm{zz}}}^{(b)}\) and \(C_{{\mathrm{z,z}}}\) over the scattering angle can be computed analytically and are 0 and 1/3, respectively. This yields the estimate for the polarized reaction cross section \(\sigma _{{\mathrm{pol}}} \approx \sigma _{{\mathrm{unpol}}}\left( {1 + \frac{1}{2}p_{\mathrm{z}}q_{\mathrm{z}}} \right)\), that is an enhancement factor of δ = 1.5 when \(p_{\mathrm{z}} = q_{\mathrm{z}} = 1\).
The study of the anisotropy in the unpolarized differential cross section stands as a first stringent test of this estimate. When investigating the angular differential cross section divided by the reaction cross section (its integral over the scattering angle), as done before in the experiment of ref. ^{30}, these appear as deviations from unity. As shown in Fig. 2, a departure from a pure swave behavior is apparent. In particular, pwaves \(\left( {\ell = 1} \right)\) are responsible for the oblique slope, and dwaves \(\left( {\ell = 2} \right)\) for the making of a bump at 90°. Overall, we find good agreement with experiment once the centroid of the 3/2^{+} resonance is correctly located. It is worth noting that the degree of anisotropy does not exceed the 1.6% level, leading to an absolute variation of the differential cross section of about 6.6 mb between 0° and 180°. The overall good reproduction of the data gives once again evidence of the highquality of the computed collision matrix. Thereafter we present our ab initio results including the phenomenological adjustment of the 3/2^{+} resonance centroid, and comment when appropriate, on its effect.
Validation of model for polarized reaction observables
As a further test, we computed \(A_{{\mathrm{zz}}}^{({\mathrm{b}})}\) and \(C_{{\mathrm{z}},{\mathrm{z}}}\) from the components of the Smatrix using the formalism of the density matrix. As benchmark, we verified that (under the condition of an unpolarized target) we could reproduce the beam analyzing powers derived and computed independently. In principle both these observables can be measured in a laboratory by analyzing the differences with respect to the unpolarized cross section when the deuteron beam is vector and tensorpolarized, the tritium is vectorpolarized, and beam and target polarizations are aligned along the zaxis. In practice, however, only the tensor analyzing power at \(\theta _{{\mathrm{c}}.{\mathrm{m}}.} = 0^ \circ\) has been measured in the energy region relevant for thermonuclear fusion \(\left( {0.24_{ + 0.18}^{  0.9} \, {\mathrm{MeV}}} \right)\)^{31}. Our computed result (−0.975) agrees well with experiment (−0.929 ± 0.014) (see also Supplementary Note 3 and Supplementary Figs. 67). The only available experimental data to test the angular distribution of the differential cross section, and hence the contribution of partial waves other than the \(J^\pi = 3/2^ + ,\ell = 0\) component at the relevant energies are measurements of the mirror D^{3}He fusion process. Such contribution of additional partial waves is exemplified in Fig. 3a, where we compare theoretical and experimental results for the D^{3}He tensor analyzing power at the deuteron incident energy of 0.424 MeV after subtraction of the swave contribution, which is simply given by the Legendre polynomial \( P_2(\cos \theta _{{\mathrm{c}}.{\mathrm{m}}.})\). Our results are in fair agreement with the experimental data^{32}, particularly for what concerns the shape of the distribution. At the same time, we find notable differences with respect to the predicted DT \(A_{{\mathrm{zz}}}^{({\mathrm{b}})}\) at the corresponding energy of E_{D} = 0.1 MeV (where we take into account the difference in Qvalues), highlighting a somewhat different partialwave content in the two mirror reactions. This indicates that some caution has to be taken when using D^{3}He as a proxy for the study of polarization in the DT fusion process. All in all, Fig. 3a gives added confidence in the polarization observables predicted for the DT fusion. More details on the calculation of the D^{3}He reaction observables can be found in Supplementary Note 4 (see also Supplementary Figs. 811).
Reaction cross section enhancement
The differential cross section for all angles is required for the computation of the polarized reaction cross section σ_{pol} and the enhancement factor δ, which we obtain (for any initial spin configuration) as the ratio of the latter to σ_{unpol}. As shown in Fig. 4a, at the deuteron incident energy of 100 keV the ab initio calculation recovers and confirms the ideal enhancement factor for \(p_{\mathrm{z}},q_{\mathrm{z}} = 1.0\), which is a result independent of the model space size and the phenomenological correction. Indeed, while our ab initio calculations show that the reaction is not exactly isotropic, at this energy the deviations of \(A_{{\mathrm{zz}}}^{({\mathrm{b}})}\) and C_{z,z} from a pure \(J^\pi = 3{\mathrm{/}}2^ + ,\ell = 0\) contribution are substantial only in the proximity of \(\theta _{{\mathrm{c}}.{\mathrm{m}}.} = 180^\circ\) (see, for example, Fig. 3b), and hence have only a minor effect on angleaveraged observables, such as the reaction cross section. We note that δ is nearly independent of the value of p_{zz}, indicating that the analyzing power of the deuterium does not play any role in the enhancement of the cross section (a consequence of the nearlyzero value of the integral of \(A_{{\mathrm{zz}}}^{({\mathrm{b}})}\)). However, the factor δ varies as a function of the energy and drops significantly above the deuteron incident energy of 0.8 MeV. This is shown in Fig. 4b for the maximum enhancement (which is found for \(p_{\mathrm{z}}q_{\mathrm{z}},p_{{\mathrm{zz}}} = 1\)). Interestingly, the peak value of the maximum enhancement (located around E_{D} = 0.4 MeV) is somewhat larger than the estimated 1.5 value. This is mainly an effect of \(3{\mathrm{/}}2^ + ,\ell = 2\) contributions. For comparison we also show the maximum enhancement obtained when we only include the \(J^\pi = 3{\mathrm{/}}2^ +\), and \(1/2^ +\) partial waves with an orbital relative angular momentum of the D+T pair of \(\ell = 0\). This shows the influence of \(1/2^ +\)components of the wave function below and above the \(3/2^ +\)resonance even in a purely swave picture of the reaction. When also the \(1/2^ +\) partial wave is removed, we recover the (energy independent) 1.5 estimate.
Reaction rate enhancement
In Fig. 5 we show the polarized fusion reaction rate for typical values of vector and tensor polarization of the deuterium \(\left( {p_{\mathrm{z}},p_{{\mathrm{zz}}}} \right)\) and tritium (q_{z}) that can be readily obtained in the laboratory, that is \(p_{\mathrm{z}},p_{{\mathrm{zz}}} = 0.8\) and \(q_{\mathrm{z}} = 0.8\), respectively. This quantity, obtained from averaging the reaction cross section over the distribution of the reactants’ speeds (assumed to be Maxwellian)^{33},
is a measure of how rapidly the reaction occurs and is an important input in astrophysics and plasma simulations. The constant μ is the reduced mass of the reacting nuclei (D and T), k_{B} and T are respectively the Boltzmann constant and the temperature, S(E) stands for the (computed) Sfactor and E_{G} is the Gamow energy given by \(2\mu \left( {{\mathrm{\pi }}e^2} \right)^2{\mathrm{/}}\hbar ^2\). In the figure we also compare our unpolarized reaction rate to those obtained from the parameterization of DT fusion data of Bosch and Hale^{34}, the phenomenological Rmatrix fit of Descouvemont et al.^{35}, and the potential model calculation adopted in the NACRE compilation^{36}, which is intended for applications in astrophysics simulations. Overall, we find that they agree well even at energies above the resonance. In more detail, our calculation agrees best with the phenomenological Rmatrix evaluation, particularly at higher energies where data are typically scarcer. In our case, the uncertainties due to the finiteness of the model space are indistinguishable from the line width. The convergence of our ab initio model is discussed in Supplementary Note 1 (see also Supplementary Figs. 15 and Supplementary Table 2). A further analysis of the systematic and statistical uncertainties associated with the adopted nuclear interaction model, such as those stemming from the order of the chiral expansion or the uncertainty in constraining its parameters, is presently computationally prohibitive (see also Supplementary Discussion). The phenomenological correction induces a global shift towards the reaction threshold, commensurate with that of the resonance centroid. In practice, this fine tuning is tightly constrained by the requirement to match Sfactor data in the energy range below the resonant peak. The polarized reaction rate shows the same shape, albeit globally enhanced by a factor of ~1.32, in agreement with the approximate estimate for the chosen polarization. This result follows from the rather slow variation of the enhancement factor of the reaction cross section as a function of the energy in the narrow Gamow window (deuteron incident energies below a few hundred keV) where the product of the Maxwell–Boltzmann distribution with the tunneling probability of the nuclei through their Coulomb barrier is significantly different from zero. It is interesting to note that with polarization a reaction rate of equivalent magnitude as the apex of the unpolarized reaction rate is reached at lower temperatures, that is less than 30 keV compared to 65 keV (where both rates peak), as highlighted in Fig. 5 by the arrows. As a naive illustration, this means that by using polarized DT fuel the output of a standard fusion reactor could either be enhanced by 32% or its operational temperature decreased by as much as 45%. A more comprehensive discussion of the economics of using polarized fuel in the case of inertial confinement fusion can be found in ref. ^{4}.
Angular distribution of the polarized reaction products
While the deviations from a pure \(J^\pi = 3{\mathrm{/}}2^ + ,\ell = 0\) contribution are small and have only a minor effect, particularly on angleaveraged observables such as the reaction cross section or the reaction rate, they play a somewhat larger role on the angular distribution of the reaction products, especially when the reactants’ spins are not in a parallel setting. In particular, while the tensor analyzing power \(A_{{\mathrm{zz}}}^{({\mathrm{b}})}\) has virtually no impact on the enhancement factor, it is the main driver of the shape of the angular distribution of the polarized cross section, shown in Fig. 6. To better visualize the situation in the laboratory, in addition to the differential cross section in the c.m. frame we also plot the differential cross section in the laboratory frame as a function of the laboratory neutron and αparticle angles in yellow shortdashed and green dotted lines, respectively. The anisotropy of the angular differential cross section is highly sought after because it can be used to force the emitted neutrons and α to be two to five times more focused towards the reactor blanket (Fig. 6a, c), which collects the energy released, than along the polarization axis, or twice the exact opposite (see Fig. 6b). That is, the reaction products are more focused along the magnetic field. The former conditions can be achieved using only polarized deuterium or fully polarized DT fuel. The latter is obtained in the situation where the D and T spins are antialigned, leading to a reduction of the cross section of up to a factor of 0.5, as illustrated in Fig. 4a.
Discussion
In conclusion, we have performed ab initio NCSMC calculations with modern chiral EFT NN and 3N interactions for the DT fusion and its mirror D^{3}He reaction. We were able to reproduce the cross sections of these reactions with unpolarized reactants. Our calculations discriminate among DT reaction rates from phenomenological evaluations and demonstrate in detail the small contribution of \(\ell > 0\) partial waves in the vicinity of the 3/2^{+} resonance. We predict the DT reaction rate for realistically polarized reactants \(\left( {p_{\mathrm{z}},q_{\mathrm{z}} \sim 0.8} \right)\) and show that the reaction rate increases by about 32% compared to the unpolarized one and, further, the same reaction rate as the unpolarized one can be achieved at ~45% lower temperature. These results also endorse the application of the present approach to the evaluation of the polarized DD fusion, where the nonresonant character of the reaction prevents even a simple estimate of the enhancement factor in the ideal scenario of perfect spinalignment of the reactants.
Methods
Nocore shell model with continuum
Our approach to the description of the DT fusion reaction is the ab initio NCSMC introduced in ref. ^{19} and applied to nucleon^{9,11,12}, deuterium^{37}, tritium, and ^{3}He induced reactions^{38} and the threecluster continuum dynamics of the Borromean ^{6}He nucleus^{39,40}. Presently, it is the only ab initio reaction method capable to efficiently describe complex lightnuclei reactions and in particular transfer reactions, though a complementary approach based on lattice EFT offers a more efficient avenue to the calculation of scattering and reactions induced by α particles^{41}.
The approach starts from the wave functions of each of the colliding nuclei and of the aggregate system, obtained within the ab initio nocore shell model (NCSM)^{20} by working in a manybody harmonic oscillator (HO) basis. This is a configuration interaction method in which all nucleons are treated as active degrees of freedom and the model space includes all possible excitations of the system up to a maximum of N_{max} quanta above the minimumenergy configuration. It then uses the NCSM static solutions for the aggregate system and continuous “microscopiccluster” states, made of pairs of nuclei in relative motion with respect to each other, as an overcomplete basis to describe the full dynamical solution of the system. That is, the ansatz for the fivenucleon (A = 5) wave function takes the form of a generalized cluster expansion (here specifically shown for the present case of a ^{5}He aggregate system):
where J, π and I denote respectively total angular momentum, parity, and isospin quantum numbers. The first term on the righthand side of the equation is an expansion over the discrete energy eigenstates of the ^{5}He nucleus (indexed by λ) obtained within the NCSM method to incorporate the physics of the five nucleons in close contact. The second term is tailored to tackle the scattering and longrange clustering of the system. The index \(\nu = \{ \nu _{{\mathrm{TD}}},\nu _{{\mathrm{\alpha n}}}\}\) runs over the reaction channels, defined by the mass partition (D+T and n+^{4}He, respectively) and the quantum numbers characterizing the reacting bodies and their relative motion. The continuum basis states \(\Phi _{\nu \;r}^{J^\pi I}\) are antisymmetrized by the operator \(\hat A_\nu\), and, in the case of the present binary collision, read:
and
The first set of continuum states describes the incoming T and D nuclei in relative motion, with \(\vec r_{{\mathrm{TD}}}\) the separation between their centers of mass, while the second set represents the outgoing wave of relative motion between the ejected α and neutron particles with separation \(\vec r_{{\mathrm{\alpha n}}}\). (Expressions in squared brackets denote angular momentum coupling.) The discrete coefficients \(c_\lambda ^{J^\pi I}\) and continuous amplitudes of relative motion \(\gamma _\nu ^{J^\pi I}(r)\) are obtained by solving the generalized eigenvalue problem derived from representing the nonrelativistic Bloch–Schrödinger equation in the model space spanned by the discrete and continuum basis states of the NCSMC. The scattering matrix—and from it all reaction observable—are finally obtained by matching these solutions with the known asymptotic behavior of the wave function at r = 18 fm, using the coupledchannel Rmatrix method on a Lagrange mesh^{42,43}_{.}
Details of the calculation
We start from a fivenucleon Hamiltonian including NN^{44} and 3N^{45,46} interactions at the fourth and third order of chiral EFT, respectively, with a 500 MeV cutoff (also adopted in the studies of refs. ^{9} and ^{37}). This interaction is then softened by the means of the similarity renormalization group (SRG) technique to a resolution scale of \({\mathrm{\Lambda }}_{{\mathrm{SRG}}} = 1.7\;{\mathrm{fm}}^{  1}\), enabling good convergence properties within the currently largest HO basis size achievable. The computational challenges of the present work limited such a basis size to a maximum number of HO excitations of N_{max} = 11. For the HO frequency, we chose the value of \(\hbar \omega = 16\,{\mathrm {MeV}}\), which was found to speed up the convergence rate with respect to N_{max} (see Supplementary Note 1, Supplementary Figs. 4 and Supplementary Table 2).
Besides the size of the HO model space, the convergence properties of the present calculations are also affected by the number of discrete eigenstates of the 2,3,4 and 5nucleon systems used to construct the NCSMC trial wave function. We included the first 14 discrete energy eigenstates of the ^{5}He system (two J^{π} = 1/2^{−}, three 3/2^{−}, 5/2^{−}, 7/2^{−}, three 1/2^{+}, two 3/2^{+}, 5/2^{+}, 7/2^{+}), the ground state and up to 8 positiveenergy eigenstates (5 in the \(\;^3S_1\)\(\;^3D_1\) and 3 in the \(\;^3D_2\) channels) of the deuterium, and the ground states of the ^{3}H and ^{4}He nuclei. The close vicinity of the energy continuum of the deuterium, only bound by 2.224 MeV, leads to distortion effects the description of which necessitates the inclusion of positiveenergy eigenstates^{7,37}. Analogous distortion effects are less pronounced in the more bound triton and α particles, and are efficiently addressed indirectly through the inclusion of the eigenstates of the aggregate ^{5}He system^{9,10}.
A particular challenge in the presence of 3N forces is the dependence on the parameter E_{3max}. This embodies the size of the threenucleon singleparticle HO basis used to represent the 3N interaction. For technical reasons, the largest E_{3max} value computationally achievable is currently of 17 HO quanta. High energy 3N force components of the NCSMC Hamiltonian can be slowly converging as a function of this parameter. Since they represent a small perturbation with respect to the NN contribution, we omit them for basis states at the boundary of the model space.
Phenomenological correction
Remaining inaccuracies in the adopted chiral Hamiltonian prevent an accurate (of the order of less than a few keV) reproduction of the sub pshell levels. This was already observed for the ^{5}He system in, e.g., Figure 16 of ref. ^{10}, which illustrates the residual imprecisions for the reproduction of pshell spectroscopy. It is then not surprising that the DT fusion Sfactor is not perfectly reproduced (see Fig. 1a). To address this difficulty, we treated the eigenvalue of the second 3/2^{+} NCSM energyeigenstate (one of the static basis states that serve as input to represent our solution) as an adjustable parameter and constrained it to the value that yielded the best fit of the experimental Sfactor data for energy below the resonance. In practice, this resulted in a shift of −86 keV of the N_{max} = 11 ^{5}He 3/2^{+} eigenenergy computed within the NCSM, which was initially −8.186 MeV, while the microscopic n+^{4}He and D+T cluster potentials and all other characteristics of the scattering matrix continued to be predicted within the ab initio method. The amplitude of the correction is less substantial than it appears. In the NCSMC Hamiltonian, the coupling matrix elements between the aggregate system and microscopiccluster states are given by the NCSM eigenvalues multiplied by the cluster form factor (the overlap between the two type of basis states). As a consequence, the effect of this adjustment is a considerably smaller shift of −5 keV of the resonance centroid E_{r} extracted from the 3/2^{+} eigenphase shifts computed within the NCSMC, shown in Supplementary Table 1. Because the 3/2^{+}resonance is close to the D+T threshold, the Sfactor is very sensitive to its centroid. Assuming a Breit–Wigner formula for the reaction cross section one can estimate the Sfactor to be proportional to \(1/E_r^2\) close to threshold and to follow a \(1/E_{{\mathrm{c}}{\mathrm{.m}}{\mathrm{.}}}^2\) slope after the resonance. This explains our results, and why our phenomenological adjustment is tightly constrained by reproducing the Sfactor close to threshold. We refer to the modified calculation as NCSMCpheno.
Data availability
The data acquired in this study are available from the corresponding author upon reasonable request.
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Acknowledgements
Computing support for this work came from the Lawrence Livermore National Laboratory (LLNL) institutional Computing Grand Challenge program. This article was prepared by LLNL under Contract DEAC5207NA27344. This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under Work Proposals No. SCW1158 and SCW0498, and by the Natural Sciences and Engineering Research Council of Canada (NSERC) Grants No. SAPIN201600033. TRIUMF receives funding via a contribution through the Canadian National Research Council of Canada.
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G.H. carried out the calculations in consultation with S.Q. and P.N. All authors discussed the results and contributed to the manuscript at all stages.
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Hupin, G., Quaglioni, S. & Navrátil, P. Ab initio predictions for polarized deuteriumtritium thermonuclear fusion. Nat Commun 10, 351 (2019). https://doi.org/10.1038/s41467018080526
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DOI: https://doi.org/10.1038/s41467018080526
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