Abstract
An R-matrix model for three-body final states is presented and applied to a recent measurement of the neutron energy spectrum from the T + T → 2n + α reaction. The calculation includes the nα and nn interactions in the final state, angular momentum conservation, antisymmetrization, and the interference between different channels. A good fit to the measured spectrum is obtained, where clear evidence for the 5 He ground state is observed. The model is also used to predict the α-particle spectrum from T + T as well as particle spectra from 3 He + 3 He. The R-matrix approach presented here is very general, and can be adapted to a wide variety of problems with three-body final states.
I. INTRODUCTION
Due to the presence of three particles in the final state, the T + T → 2n + α reaction produces distributions of neutron and α-particle energies. The neutron energy spectrum at an effective E c.m. of 16 keV has recently been measured in an inertial confinement fusion experiment at the National Ignition Facility (NIF) [1] . This paper also presented a sequential-decay R-matrix model for the three-body state. The primary purpose of the present paper is to fully describe this model and to explore a broader range of assumptions for the fitting of the neutron spectrum. We also present a prediction for the α-particle spectrum, for which limited data exists. Finally, we calculate the final-state energy spectra of the mirror reaction 3 He + 3 He and discuss some features of our approach when one of the nuclei in the final state is much heavier than the others.
Our model includes interactions between all pairs of nuclei in the final state. For the T + T case, this implies the nα interaction, including the unbound 3/2 − ground and 1/2 − first excited state of 5 He, and the nn interaction. The calculation also incorporates angular momentum conservation and fermion symmetry. The latter is a particular example of an order-of-emission effect, which give rise to various interference phenomena. In addition, kinematic effects present in the three-body final state are tightly integrated into the model. These details of the model, predictions for particle spectra, and comparisons to available experimental data are discussed below.
The reactions 3 He + 3 He → 2p + α and T + 3 He → n + p + α, which are related by mirror or isospin symmetry to T + T → 2n + α, are also presently under study at inertial confinement fusion facilities. The model presented here can be adapted to these reactions, and a prediction for the 3 He + 3 He → 2p + α case is given in this paper. This R-matrix approach is very general, and additional areas where it could be applied are discussed in the conclusion.
II. THREE-BODY KINEMATICS
The three-body final state from the T + T → 2n + α reaction will be described using non-relativistic kinematics [2] . With the center of mass assumed to be at rest, the kinetic energy available in the final state is given by
where Q = 11.332 MeV and E c.m. is the center-of-mass (c.m.) kinetic energy of the initial state. Here, we will assume E c.m. = 16 keV, unless otherwise indicated. The masses of the final-state particles are m i , where i = 1, 2, or 3, and M = m 1 + m 2 + m 3 . Indices 1 and 2 are used for the neutrons, with index 3 used for the α particle. The momentum and kinetic energies of the final-state particles are given in the three-body c.m. system by p i and E i . The relative momentum and kinetic energy of particles i and j are
Assuming the indices i, j, and k are all distinct, the relative kinetic energy between particle i and the j − k system is given byẼ i and we also have
E tot = E 1 + E 2 + E 3 =Ẽ i + E jk .
III. R-MATRIX MODEL
The energy distribution of particles emitted by reactions proceeding to unbound states can be described using R-matrix methods, as presented by Baker [3] . This approach in essence describes the particle emissions as sequential two-body decays. Due to the low energy in the initial T + T state, we assume it has orbital angular momentum of zero and thus a total spin and parity of 0 + . We consider here two types of sequential decays: neutron emission to unbound 5 He intermediate states and α-particle emission to unbound neutronneutron states. This latter type decay may also be referred to as di-neutron emission [4] . For both of these decay types, a further complication is presented by the fact that the amplitudes must be constructed to be antisymmetric under the exchange of neutrons, which give rise to direct and exchange terms [see Eq. (22) and Eq. (23), respectively]. The R-matrix formalism has been applied to other cases of three-body final states with identical particles in Refs. [5] [6] [7] [8] .
We emphasize that this approach treats the intermediate state rather carefully. The phase shifts between the particles that make up this state are rather well known for the cases considered here -either experimentally (nα scattering) or theoretically (nn scattering). The R-matrix model described below accurately incorporates these phase shifts. The interaction between the first particle emitted and the intermediate state is, however, not well known, as it generally cannot be studied independently in the laboratory. This part of the matrix element is treated in a minimalist R-matrix approach, with just a hard-sphere interaction, which characterizes a non-resonant phase shift. (12) which can be written
The quantities in angled brackets are the Clebsch-Gordan coefficients and Y lm are the spherical harmonics with Ω 1 representing the angles (θ 1 , φ 1 ) that describe the emission of neutron 1 with momentum p 1 = p 1p1 in the three-body c.m. system and Ω 23 representing (θ 23 , φ 23 ) that describe the the emission of the of neutron 2 in with momentum p 23 = p 23p23 the rest frame of the 2 − 3 system. Finally, our amplitude can be made antisymmetric under the exchange of neutrons by adopting
with Ω 2 , Ω 13 , and related quantities defined analogously to the above. Our approach only considers n + α configurations for the description of 5 He states; this approximation is well justified in this case because the thresholds for other configurations, such as d + T, are located much higher in excitation energy. Below neutron energies of 20 MeV, it is found that considering l ≤ 3 is sufficient and that one level plus a constant R ∞ for each channel allows for a good fit to be obtained. We consider here l = 0 and 1 transitions involving the 1/2 + , 1/2 − , and 3/2 − n + α partial waves.
The scattering of neutrons by α particles is very well studied and R-matrix parameters are available [9] . The quality of fit to n + α scattering observables is comparable to modern analyses (e.g., Ref. [10] ) that take into account more multichannel data. We utilize the R-matrix parameters given in Table 2 of Ref. [9] , with R ∞ replaced by a background level at very high (1000 MeV) excitation energy, which we refer to hereafter to as the R ∞ state. Both l = 1 partial waves (1/2 − and 3/2 − ) have a resonant states at low excitation energy, in the range which can be populated by low-energy T+T reactions. Consequently, these partial waves are expected to contribute significantly. Feedings of both the important resonance state and the R ∞ state are considered. We choose the boundary condition constant for these parameters so that the level shift vanishes for the lowest-energy state in each partial wave. The l = 0 1/2 + partial wave is non-resonant, but is included due to its low angular momentum. We consider feeding of the 1/2 + 50-MeV level (which is in fact a background state), but not the R ∞ state. It should be pointed out that the R ∞ state for this partial wave is unphysical, as it has γ where the sum is over three nα channels and one nn channel. The nature of the energy dependence of u c varies with the channel type (nα versus nn); the 12 and 21 notation is used to indicate direct (12) and exchange (21) terms. In principle, all observables can now be calculated. Our primary interest, however, is in calculating the particle energy spectra. The particle distribution in the three-body c.m. system is given by
where the product of factors p i p jk J ijk is the three-body phase space [2, 12] , and J ijk is the Jacobian for the transformation from the (Ẽ i , Ω i , Ω jk ) system to the (E i , Ω i , Ω j ) system. In order to extract the particle energy distributions, it is necessary to integrate out the angular variables. This task can be accomplished most easily by transforming to the (Ẽ i , Ω i , Ω jk ) system:
D. Evaluation of spin and angle-dependent functions
In order to proceed further, it is necessary to evaluate the square of the matrix element, summed over spin projections. In doing so, two types of sums arise:
where the factors of (4π) 2 have been inserted for later convenience. The first type of term can be evaluated using standard techniques (see e.g. Ref. [13] ):
where the W without subscripts is the Racah coefficient, P k is the Legendre polynomial of order k, and cos γ jk =p i ·p jk .
The second type of term arises from antisymmetrization and is more complicated to evaluate, due to the fact that, as written, it depends on two sets of angular variables that are not independent. The angular variables are represented by the unit vectors of the momenta. By using Eqs. (2) (27) so that the expression only depends on the angular variablesp 1 andp 2 . The spherical harmonics harmonics can then be evaluated for these substitutions using the following addition theorem [14] 
where c = a + b with a = aâ, b = bb, and c = cĉ. The second type of term is then found to be:
where {} indicates the Wigner 9-J symbol and
The functions W (i) lJl J are real and are invariant under the interchange of (l, J) and (l , J ), while W (12) lJl J are also real and are invariant under the interchange of (l, J) and (l , J ) and particle labels 1 and 2. These functions are tabulated in Table I for the partial wave combinations considered here.
Due to our assumption of a J = 0 initial state, the particles are emitted isotropically; these functions thus describe the angular correlations between the particles. Note that the particle distribution is a function of two variables, which can be taken to be E i and cos γ jk . From these two quantities, all other needed energies, momentum magnitudes, and angles can be calculated from the kinematics relationships. 
E. Calculation of particle energy spectra
We can now write
cc + g (12) cc ,
where
If the neutrons were distinguishable, the g (1) contribution would be the neutron 1 distribution and g (2) would be the neutron 2 distribution. The g (12) term arises from treating the neutron as indistinguishable fermions. In the case of neutron emission via intermediate 5 He states, we take neutron 1 to the first neutron emitted and neutron 2 to be the second. The neutron energy distribution can be calculated:
The first term will be called the primary distribution, the second the secondary distribution, and the third the exchange distribution. If only nα channels are present, the calculation of the primary contribution can be simplified, because u c then only depends onẼ 1 , andẼ 1 , p 1 , and p 23 are independent of cos γ 23 . The result is: dN
which is free from any angular correlation or interference effects. Note also that it is in this situation that the R-matrix energy distribution formula given in Ref. [3] is recovered. For the α-particle energy distribution, all of the contributions must be calculated by numerical integration, but this task is simplified by noting that
If only the nn (di-neutron) channel is present, the calculation also simplifies. In this case we have dN
For the neutron energy distribution, these must be evaluated by numerical integration, but for the α-particle energy distribution we have
which is also in the form given by Ref. [3] . In the general case, all three contributions to the distribution must be calculated using numerical integration. We do note that the overall contributions of the primary and secondary distributions are equal, i.e., that
Spectra for channels in isolation
We will next investigate the nature of the particle energy distributions resulting for each channel in isolation. Note that we make no effort in this section to adjust the feeding factor parameters of the model to fit experimental data; this is done below in Sec. IV. For the l = 1 nα channels, we have taken the background feeding to be zero. Each channel thus has only a single feeding factor, which has been adjusted so that
The results are shown in Fig. 1 for the neutron energy distributions and in Fig. 2 for the α-particle energy distributions. For the nα channels, the primary, secondary, exchange, and total contributions are shown for neutron energy distributions, and the primary plus secondary, exchange, and total contributions are shown for the α-particle distributions. For the nn channel, only the total is shown, since, as shown by Eq. (39), the sub-contributions are all proportional. It is interesting to note that the interference introduced by antisymmetrization has a general tendency to be constructive in all cases investigated. This point is discussed further below in Sec. VI.
1/2 + nα: The particle spectra for this channel are rather featureless. Both the neutron and α-particle spectra closely approximate elliptical energy distributions, characteristic of uniform phase space population.
1/2 − nα: The first excited state of 5 He gives rise to a broad peak in the primary neutron spectrum, while the secondary neutron spectrum is also broad, but peaks at a lower neutron energy. The effect of antisymmetrization is to make the overall spectrum narrower, with relatively little strength near the endpoints of the spectrum. The α-particle spectrum for this channel is relatively flat, except near the endpoints.
3/2 − nα: The ground state of 5 He gives rise to a narrow peak in the primary neutron spectrum near the maximum neutron energy. The secondary neutron spectrum shows a double-peaked feature below 2 MeV. This structure results from the W (i) = 1 + P 2 (cos γ jk ) angular correlation between the primary and secondary neutrons, which implies a strong tendency for the neutrons to be emitted in the same or opposite directions, but not perpendicular to each other. Due to the recoil of the the secondary neutron energy distribution. This angular correlation also gives rise to a double peak in the α-particle energy spectrum. These effects on the particle energy spectra due to angular correlations were understood over 50 years ago [15, 16] , and were observed for the the α-particle energy spectrum at higher T + T energies [16] . Due to the relatively small energy overlap between primary and secondary spectra, the effect of antisymmetrization on the overall spectra is less important for this channel. nn: In this case, the neutron energy spectrum peaks just below 4 MeV and has considerably less strength near the endpoints compared to the 1/2 + nα channel which has the same quantum numbers. The α-particle spectrum has a very distinctive peak near the maximum energy that is associated with the two neutrons being emitted in nearly the same direction with a low relative energy. Similar results for the effect of the nn interaction on the α-particle spectrum were found in the calculations of Lacina, Ingley, and Dorn [4] .
Interference between channels
Another way to decompose Eq. (35) that is useful when considering multiple channels is
where the second sum is due to interference effects between channels and
The channel interference contributions to the particle spectra are shown in Figs. 3 and 4 , for the partial wave combinations under consideration. The same feeding factors were used as for the calculations shown in Figs. 1 and 2 . Note that the signs of the interference contributions are determined by the relative signs of the feeding factors. The effects are seen to be substantial, comparable in magnitude to the single-channel contributions. In addition, note that the contribution of these effects, integrated over E i , does not vanish.
IV. FITS AND COMPARISONS TO EXPERIMENTAL DATA
In this section, fits and comparisons to experimental neutron and α-particle spectra from low-energy T + T reactions (below E c.m. = 100 keV) are presented.
A. Neutron spectrum
The T(t, 2n) neutron spectrum at an effective E c.m. of 16 keV has recently been measured at the NIF [1] . The neutrons were detected in two liquid scintillators along separate lines of sight located 20.1 and 22.2 m from the source, respectively. The experiment provides raw data in the form of digitized currents from the detectors versus time. The data from the 22.2-m detector has been presented in Fig. 2 of Ref. [1] , where the time of flight has been converted to a nominal neutron energy and the points have been rebinned. Here we will present additional fits to these data. The fits to the raw data utilize the R-matrix description of the neutron spectrum for E c.m. = 16 keV, and take into account the following effects [1] : thermal broadening due to the Maxwell-Boltzmann distribution of particle velocities in the plasma, neutron attenuation and scattering between the T(t, 2n) reaction source and the detector, the light output response of each detector, and the time response of each detector. Finally, the background from the T (d, n) reaction, which was measured separately, was added to the model spectra. The data fitted here are identical to those reported in Ref. [1] , except that some additional points at longer times of flight have been included (giving 812 data points in total), and the errors on the data have been increased by assuming that the attenuation correction has a 20% uncertainty and that the scattering correction has a 50% uncertainty. The raw time-of-flight data that are fitted are shown in Fig. 5 ; note the narrow peak near 500 ns is the peak in the neutron spectrum from the 5 He ground state.
The spectra have been fitted using the R-matrix formalism described above, assuming various combinations of the four channels discussed in Sec. III E 1. In addition, we have considered non-zero background feeding factors for the 1/2 − and 3/2 − nα channels, leading to a total of up 6 variable parameters. Note that the feeding of the low-lying 1/2 − and/or 3/2 − states in nα channels were always fitted and that if a feeding factor was not fitted, its value has been assumed to be zero. The χ 2 min values obtained for the various feeding factor assumptions are presented in Table II .
Not surprisingly, the χ 2 min decreases steadily as the number of free parameters is increased. In Ref. [1] , only the 1/2 − and 3/2 − nα channels were considered. The fit presented there is nearly identical to fit number 9 presented here, with the very small changes arising from the changes in the data set discussed above. The additional channels are seen to make a substantial increase in the quality of the fit. However, we are cautious about placing a large emphasis on this improvement, as the neutron spectrum data contain neutron-energydependent systematic errors from the scattering and attenuation corrections that may be comparable to this improvement in fit (i.e., from χ 2 min = 1085 to 632). Two of the fits, numbers 9 and 16, are shown if Fig. 6 , where the data have been re-binned and plotted versus the nominal neutron energy.
The R-matrix parameters for fit 16 are shown in Table III . Note that the uncertainties on the feeding factors are computed assuming that uncertainties on the data are random and normally distributed; as explained above this is not strictly the case and the true uncertainties are larger (this is also why some fits are able achieve a χ 2 value which is less than the number of data points). The decomposition of this fit into its various channel components and the net interference contribution, according to Eq. (42), is shown in Fig. 7 . It is seen that the fitted 1/2 − and 3/2 − nα channels, and di-neutron channel, are substantial, with the 1/2 + nα channel contributing to a lesser degree. The net interference between channels is also non-negligible. In Ref. [1] , a branching ratio for the the 1/2 − and 3/2 − nα channels was given, which is possible to do if only these two channels are considered, since their interference term is very small. In the general case, it is not possible to determine branching ratios, due to the substantial interference contribution.
In order to facilitate future comparisons with other experiments and calculations, it is desirable to present the neutron spectra in a deconvoluted form, i.e., with the various efficiency, resolution, and background corrections removed. General methods and considerations for the deconvolution of nuclear science measurements are discussed in Ref. [17] . Any approach to these corrections necessarily involve some model dependence. We have deconvoluted the present measurements by assigning a mean energy to each point (analogous to Eq. (8) of Ref. [17] ) and then applying a correction factor to the measured yield for each point (analo- gous to Eq. (6) of Ref. [17] ). This procedure requires that the underlying neutron spectrum be known in advance -for this we use fit 16. In practice, the fit used makes very little difference, as long as it gives a reasonable description of the measured data. The deconvoluted neutron spectrum data is shown in Fig. 8 , where the data from the two detectors are combined and re-binned in energy.
B. α-particle spectrum
Some information about the α-particle spectrum from the low-energy T + T reaction is available from a 1985 conference paper by Jarmie and Brown [18] , where a measured spectrum and a background spectrum are given for an incident triton energy of 115 keV and a laboratory angle of 45
• . We have extracted the α-particle spectrum from their Fig. 8 as follows. The spectrum was first corrected for the background shown along with the spectrum in their figure. Next the spectrum was energy calibrated, using the peak from the T(d, α) reaction to fix the calibration at the high-energy end of the spectrum. The calibration assumed the channel number in the Si detector was linear with α-particle energy with zero offset, with the energy loss in the 30-µg/cm 2 CH 2 foil in front of the detector [19] being taken into account. Finally, the spectrum was converted to the c.m. system assuming the spectrum is isotropic in the c.m. system. It should be noted that this spectrum should be most reliable for the higher energies, where the background is small and the energy calibration is well established. We finally note that this spectrum was measured for E c.m. = 57.5 keV, with the beam-energy loss correction being less than 0.1 keV.
The resulting α-particle spectrum, rebinned such that each point represents 5 channels in the raw spectrum, is shown in Fig. 9 . Also shown are the predictions from R-matrix fits 9 and 16, where the normalizations of the fits has been adjusted to optimize the agreement with data. It is seen that fit 16 supplies a much better description of the spectrum than fit 9 (χ 2 = 46 versus 140 for the 35 data points). The decomposition of the fit 16 α-particle spectrum into its various channel components and the net interference contribution is shown in Fig. 10 . It is seen that the inclusion of the di-neutron channel, which supplies spectral strength near the maximum energy, is crucial for reproducing the spectrum.
C. Dalitz plots
A useful tool for visualizing the the particle energy correlations in a three-body final state is the Dalitz plot. As already noted, the particle distribution given by Eq. (18) is a function of two variables. Taking these to be any pair of particle energies E i and E j , we can write
where the kinematically-allowed region in E i − E j space is an ellipse. The particle distribution resulting from fit 16, plotted as a function of neutron and α-particle energies, is shown in Fig. 11 . The vertical band at E n ≈ 8.7 MeV and the diagonal band in the lower left part of the ellipse are due to the 5 He ground state. The concentration of strength at the top of the ellipse, where E α ≈ 3.8 MeV, is due to the di-neutron. The decomposition of the α-particle spectrum from the fit 16 prediction into its various channel components and the net interference contribution.
The same particle distribution, plotted as a function of the two neutron energies, is shown in Fig. 12 . In this case, the horizontal and vertical bands at E n ≈ 8.7 MeV are due to the 5 He ground state and the di-neutron strength appears at E n1 ≈ E n2 ≈ 3.8 MeV.
V. PARTICLE SPECTRA FROM 3 He + 3 He
It is straightforward to adopt this approach for describing the proton and α-particle spectra from 3 He + 3 He, which is the mirror reaction to T + T → 2n + α. For the pα channels, we utilize again the final R-matrix parameters given by Ref. [9] , which are defined using the same channel radius and boundary condition conventions as their nα parameters used above. For the pp channel, adopting a channel radius of 2.0 fm, E c = 4.865 MeV, and γ 2 c = 34.61 MeV reproduces the scattering length and effective range of the Argonne V18 potential [11] which are −7.8064 fm and 2.788 fm, respectively.
The calculated results for considering each channel in isolation are shown in Figs. 13 and 14. We have assumed E c.m. = 165 keV and the same normalization convention was used as for the T + T calculations shown in Figs. 1 and 2 . The results are very similar to those found for T + T. The main difference is that the 5 Li ground-state peak in 3/2 − pα channel is broader than the 5 He peak in the corresponding nα channel, which simply reflects the different intrinsic widths of the states. In addition, the α-particle energy distribution is somewhat broader for the pp channel compared to the corresponding nn channel.
In Fig. 15 , we show a prediction for the proton spectrum, where we have assumed the feeding factors from fit 16 to T + T neutron spectrum. The corresponding prediction for the α-particle spectrum is shown in Fig. 16 . The main differences from the T + T case are (1) the 5 Li ground state is less prominent than the 5 He ground state peak in the proton spectrum, due to the former being broader, and (2) there is less strength in the α-particle spectrum near the endpoint (di-proton region), due to the Coulomb barrier between the two protons suppressing the amplitude as their relative energy approaches zero.
It should be noted that we have ignored certain complications introduced by long-ranged Coulomb force to the three-body final state. In particular, our factorized form of the amplitude does not correspond to an asymptotic solution to the Schrödinger equation for three charged particles [20, 21] . One can see that our amplitude does not include the effect of the Coulomb barrier as the relative energy goes to zero for all particle pairs. For some energy spectra and channels, this deficiency is exposed at the highest energies in the particle spectra, as the endpoint corresponds to the case where the other two particles recoil in the opposite direction with zero relative energy.
An ad hoc modification to our amplitudes can be made which restores physicallyreasonable behavior near the endpoints. Such an approach may be necessary for describing experimental data in these regions. A simple procedure is to multiply the pα amplitude given by Eq. (9) by C 12 C 13 and the pp amplitude given by Eq. (16) by C 13 C 23 , where
and k ij , a ij , and η ij are the wavenumber, channel radius, and Coulomb parameter for particle pair ij. We have assumed l = 0 for the penetration factor and used the same radii for the pα and pp channels as discussed above. This modification introduces some additional angular dependence to the matrix element that prevents some of the simplifications based The primary plus secondary, exchange, and total are given by the dotted, dot-dashed, and solid curves, respectively. Only the total is shown for the pp case. on integrating over Legendre polynomials discussed in Subsec. III E from being applicable. Otherwise, the computations are unchanged. We find that the shapes of the calculated spectra are little changed except near the endpoint while the normalization (area) of the spectra are reduced by 12-17%, depending up on the particular channel. The results with and without this modification are shown in Fig. 17 for the 1/2 + pα channel near the endpoint, where the normalization of the modified spectrum was adjusted to match the area of the original spectrum.
VI. HEAVY NUCLEI
It is instructive to consider how this formalism behaves in heavier nuclei. In the limit that m 3 m 1 , m 2 , the kinematic relations simplify considerably. In particular, we have E 1 + E 2 = E tot , p 23 = p 2 , p 13 = p 1 , and cos γ 23 = cos γ 13 = cos δ 12 . The is also substantial simplification of the angular functions:
If only a single nα channel is present, the neutron energy spectrum is given by
which is symmetric around the center of the spectrum (E 1 = E tot /2). Note also that the interference due to antisymmetrization is maximally constructive at the center of the spectrum. This result explains the general tendency observed in Fig. 1 for the antisymmetrization interference contribution to the neutron spectrum to be constructive in the T + T case. This formula was determined using the following special result for the Racah coefficient [13] when
This result also implies that the interference between channels with distinct l and J values vanishes in this limit.
The interference between channels with distinct l and J values can thus be interpreted to arise via the recoil of the intermediate state. This recoil is substantial in processes involving light nuclei such as T + T → 2n + α. In heavier nuclei, the interference between channels is much reduced and is found to scale ∝ 1/m 3 .
VII. CONCLUSIONS
A phenomenological R-matrix model for the three-body final state of the T + T → 2n + α reaction has been presented. This approach includes a detailed treatment of the nα and nn interactions in the final state, angular momentum conservation, antisymmetrization, and the interference between different channels. This model is able to supply an excellent fit to the T + T neutron spectrum for E c.m. = 16 keV recently measured at the NIF [1] . The most prominent feature in the spectrum is a peak at E n ≈ 8.7 MeV, which arises for the 3/2 − 5 He ground state. The strength in the spectrum at lower neutron energies arises from the 1/2 − first excited state of 5 He, 1/2 + nα emission, and the nn (di-neutron) emission channel. The best fit to the spectrum includes significant strength in the di-neutron channel, but it should be noted that the distribution of strengths in these additional channels is not well constrained by the data (see Table II ). This best fit provides a prediction for the α-particle spectrum, which is in reasonable agreement with an experimental spectrum that is available in the literature [18] . The agreement of the prediction with the data near the endpoint of the α-particle spectrum provides support for the significant di-neutron channel strength present in the best fit.
Several issues could be clarified by improved experimental data. It would be very useful to extend the T + T neutron spectrum measurements to lower neutron energies, in order to better constrain the fits and to possibly observe the double-humped structured predicted below 2 MeV neutron energy that is associated with the 5 He ground state (see the 3/2 − nα panel in Fig. 1) . A fully documented measurement of the α-particle spectrum from T + T would also be valuable, particularly if the spectrum could be measured up to the endpoint, where the di-neutron contribution is maximal. It would also be interesting to study the dependence of the spectrum on the energy in the entrance channel, as there is some indication the 5 He ground state peak is more prominent at higher entrance channel energies [22] .
It is also interesting to consider the reactions 3 He + 3 He → 2p + α and T + 3 He → n + p + α, which are related by mirror or isospin symmetry to T + T → 2n + α. A prediction for the proton and α spectra resulting from 3 He + 3 He has been given above in Sec. V. Measurements of proton spectra from 3 He + 3 He and T + 3 He are currently being pursued with the inertial confinement fusion technique using the OMEGA facility at the Laboratory for Laser Energetics of the University of Rochester [23] .
On the theoretical side, it would be interesting and useful to extend the formalism presented here to include the energy dependence in the initial state. We expect that the methods presented here can be applied to additional reactions or spectra with three-body final states. For example, our approach could be applied to the decay 16 Be → 2n + 14 Be, where evidence for the di-neutron has been reported [24] . Another area where these methods could be used is the calculation of coherent interference effects between different decay pathways to three-body final states, which has been noted as an important issue for understanding the total widths of states which decay by the emission of three particles [25, 26] . It must also be acknowledged that the phenomenological R-matrix approach presented here includes many approximations. In the future, it is thus hoped that ab-initio techniques based on nucleon-nucleon interactions may be applied to these reactions.

{'paper_id': '119115590', 'title': 'R-Matrix description of particle energy spectra produced by low-energy T+T reactions', 's2_url': 'https://api.semanticscholar.org/CorpusID:119115590', 'source': 'S2ORC'}