NEW DETERMINATION OF THE ²H(d,p)³H AND ²H(d,n)³He REACTION RATES AT ASTROPHYSICAL ENERGIES

Channel selection was achieved by gating on the ΔE−E two-dimensional (2D) plots to select coincident proton and ³He/³H loci. A typical ΔE−E 2D plot is shown in Figure 4, where the proton, triton, and ³He loci are clearly visible. The kinematics were reconstructed under the assumption of either a proton (for the p+³H channel) or a neutron (for the n+³He channel) as undetected particle. The Q-value spectra for the selected events are reported in Figures 5 (p+³H channel) and 6 (n+³He channel).

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A prominent peak is observed in each figure around E = −1.5 MeV (Figure 5) and E = −2.2 MeV (Figure 6) in very good agreement with the theoretical Q values of −1.46 MeV and −2.22 MeV for the p+³H+p and n+³He+p three-body channels of interest, respectively. No other reactions contribute, except a small background not larger than 4%. This confirms the correct identification of the reaction channel, as well as the accuracy of the calibration. Only events inside these peaks were selected for the further analysis.

After selecting events belonging to the ²H(³He,p³H)¹H and ²H(³He,n³He)¹H reactions, the next step is to determine the contribution of the QF process to the total coincidence yield. This is necessary because the equations exploited are valid only upon the assumption that particle s, here the proton, acts as a spectator to the A − x interaction. Indeed, the analysis of the experimental results can reveal the presence of other reaction mechanisms, such as sequential decay (SD) and direct break-up, filling up the same final state. A study of the reaction dynamics can be carried out looking into the relative energies for any two of the three final particles, deduced from the measured energies and emission angles, for possible excitations of their respective intermediate systems. In the present case, SD mechanisms may only be due to the excitation of ⁴He levels above 20 MeV of excitation energy that feed the two p+³H and n+³He exit channels. However, none of these levels can be seen as resonances, their widths being of the order of 5–10 MeV. Thus, no useful information can be gathered from the investigation of relative energy spectra.

A physical quantity very sensitive to the reaction mechanism is the shape of the experimental momentum distribution of the spectator. Only in the case of ³He QF break-up does the proton momentum distribution keep the same shape as inside ³He. Thus, the agreement between the shape of the theoretical ³He momentum distribution and the experimental one is a clear signature of the occurrence of the QF mechanism (Spitaleri et al. 2004; Tumino et al. 2008). To establish the momentum dependence of the coincidence yield, the modulation due to possible feedings of ⁴He states has to be removed. This is achieved by deducing the proton momentum distribution within p+³H/n+³He relative energy bins of 100 keV. To correct for the phase-space effects, a Monte Carlo simulation was performed considering the angular ranges in the experiment and the real detection thresholds. The experimental momentum distribution obtained for both ²H(³He,p³H)¹H and ²H(³He,n³He)¹H reactions experience the same shape. As an example, the momentum distribution from the ²H(³He,p³H)¹H reaction is shown in Figure 7 as black full circles.

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Data are projected in 10 MeV/c bins over the momentum axis of the detected proton, p_s, with error bars only including statistical errors. The theoretical distribution, shown as a solid line, refers to a Wood–Saxon p − d bound state potential with standard geometrical parameters r₀ = 1.586 fm, a = 0.50 fm, and V₀ = 53.858 MeV (Pizzone et al. 2009). The shape agreement within 3% between the experimental momentum distribution and the theoretical one is a clear signature of the existence and dominance of the QF mechanism. This 3% enters in the further extraction of the astrophysical S(E) factor as the systematic uncertainty coming from theory (Pizzone et al. 2005a, 2009; Mukhamedzhanov et al. 2006, 2008; La Cognata et al. 2010). Thus, the factorization of Equation (6) is reliable for the present investigation and the QF mechanism is dominant in the phase space here spanned. In further analysis, only coincidence events within 40 MeV/c were considered.

The functional form of the ²H(d,p)³H and ²H(d,n)³He two-body cross sections was determined starting from previous R-matrix calculations by Descouvemont et al. (2004). If no resonance is present in the energy range investigated, the R-matrix can be parameterized by a constant value. This is the case in the present study and two non-resonant components for the l = 0 and l = 1 partial waves are needed in the whole relative energy E_dd region investigated. In this way, we account for the several ⁴He levels feeding the ³H+p and ³He+n decay channels, which, as mentioned, do not contribute as promiment resonances because of their huge widths from 5 to 10 MeV. Nonetheless, they are also responsible for the anisotropies in the angular distributions, as shown later. With orbital angular momentum l = 0, possible J^π values are 0⁺ and 2⁺, whereas l = 1 yields J^π = 0⁻, 1⁻, 2⁻.

Given that the energy dependence of the s- and p-wave contributions is essentially provided by their penetrability factors (Descouvemont et al. 2004), the functional form of the binary cross section was assumed as

$$\begin{equation} \frac{d\sigma }{{dEd}\Omega } (d+d \rightarrow C+c) = \frac{1}{E_{dd}} \sum _{l=0,1} C_{l} P_{l}^{2} k_{dd}R T_{l}(k_{dd}R),\quad \end{equation} \tag{ 11 }$$

with C + c = t+p or ³He+n in the present case. The relative energy E_dd and the momentum k_dd are connected in QF kinematics by Equation (6), which in the present case becomes

$$\begin{equation} E_{dd}= \hbar k_{dd}^{2}/(2 \mu _{dd}) - B_{dp}, \end{equation} \tag{ 12 }$$

with the reduced mass μ_dd and the binding energy B_dp of ³He with respect to the d+p threshold. E_dd is determined from the measured p³H/n³He relative energies by means of the energy conservation law and it is referred to as E_cm, as given by (Spitaleri et al. 2011)

$$\begin{equation} E_{{\rm cm}}=E_{p^3{\rm H}/n^3{\rm He}}-Q_{2-{\rm body}}, \end{equation} \tag{ 13 }$$

with $E_{p^3{\rm H}/n^3{\rm He}}$ and Q_{2 − body} relative energy and Q value, respectively, of any two of the exit channels.

To fix the contributions of the two partial waves l = 0 and l = 1 to the three-body coincidence yield, Equation (11) enters the factorization of Equation (6) in a Monte Carlo simulation (Tumino et al. 2003), based on the theoretical approach reported in (Typel & Wolter 2000; Typel & Baur 2003; Tumino et al. 2003), under the assumption that the reaction proceeds through a pure QF mechanism. The theoretical distribution so obtained needs to fit the experimental three-body coincidence yield using the scaling factors C_l and the channel radius R as free parameters. As mentioned above, for the 18 MeV run each fusion channel has triggered two couples of coincidence detectors and thus two coincidence yields have to be fitted at the same time. This constraint helped to unequivocally fix how to distribute the measured THM cross section to the l = 0 and l = 1 components for the further extraction of the S(E) factor. For the 17 MeV run, this procedure was applied only to the working pair of coincidence detectors, providing scaling factors that are consistent within few percent with those from the 18 MeV run. An example of the TH coincidence yield, projected onto the E_dd relative energy axis, is shown as solid dots in Figure 8. The solid line superimposed onto the distribution represents the corresponding fit from E_dd = 1.5 MeV down to zero. The best description of the THM yields were obtained for relative weight C₀/C₁ = 1.3 ± 0.14 (1.2 ± 0.13) and R = 6.15 ± 0.05 fm (5.25 ± 0.04 fm) for the ³H+p (³He+n) case with an overall $\tilde{\chi }^2$ of 0.24 (0.54). Values for R are similar to the channel radii in the R-matrix analysis reported in Descouvemont et al. (2004).

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Owing to the virtual character of cluster x (²H in the present case), the THM provides the half-of-energy-shell (HOES) binary cross section (Tumino et al. 2007, 2008). Thus, an angular distribution analysis of the fragments is needed to check if the different partial waves enter in the same way in both the on-shell and HOES cross sections. A positive result would mean that no distortions are induced by the emitted deuteron. The invariant scattering angle in direct measurements is determined as the angle between the relative momenta of the final and initial particles. In the c.m. system of the subreaction, such an angle is the one between the momentum of any of the two fragments (p/n or ³H/³He) and the beam direction. The corresponding invariant TH angle was calculated as reported in Slaus et al. (1977), for the case of projectile QF break-up, and is given by

$$\begin{eqnarray} &&\theta _{{\rm cm}} = arccos \frac{\boldsymbol {q} \cdot (\boldsymbol {v_C}-\boldsymbol {v_c})}{|\boldsymbol (q)||\boldsymbol {v_C}-\boldsymbol {v_c}|}, \end{eqnarray} \tag{ 14 }$$

with c+C = p+³H or n+³He and $\boldsymbol {q}$ the momentum of the transferred particle. Angular distributions for both ²H(d,p)³H and ²H(d,n)³He reactions were extracted for several energies. In the 18 MeV experiment, θ_cm ranges from 40° to 120°, with the most forward couple of coincidence telescopes, and from 140° to 170° with the second one, consisting of the less and the most forward detectors placed at opposite sides. For the 17 MeV run, θ_cm goes from 80° to 100°. A comprehensive systematics is shown in Figures 9 (³H+p channel) and 10 (³He+n channel) for E_dd = 20, 50, 130, 270, 980 keV, and 1.25 MeV (±30 keV) as full dots. The solid lines represent the results of fits on the OES distributions measured at those energies (Krauss et al. 1987; Schulte et al. 1972). Due to the identical boson character in the entrance channel, those fits have been performed using only even Legendre polynomials with a maximum order of four. A good agreement is observed, as supported by the following $\tilde{\chi }^2$ values for ³H+p/³He+n: 0.29/0.25 at 20 keV, 0.33/0.36 at 50 keV, 0.28/0.37 at 130 keV, 0.36/0.49 at 270 keV, 0.37/0.28 at 980 keV, and 0.35/0.33 at 1.25 MeV. The positive results confirm our hypothesis about the partial wave relative contribution and gives further confidence on the validity of the theoretical approach to extract the THM S(E) factor.

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The S(E) factor from both the ²H(d,p)³H and ²H(d,n)³H reactions was determined from the definition in Equation (4), using the combination of the two l = 0 and l = 1 components as given in Equation (11) as σ(E), with the scaling ratio C₀/C₁ fixed from the fit to the three-body coincidence yields as explained before. For the ²H(d,p)³H reaction, results from the 17 and 18 MeV runs were combined accounting for their 10% and 4% statistical errors, respectively. An error calculation for E_dd was performed, leading to a value of about 20 keV. The S(E) factors thus obtained must be normalized to the direct data in a region with E_dd > 15 keV, where electron screening effects are still negligible. However, available sets of data below 200 keV have different accuracies and evaluations of past work on both ²H(d,p)³H and ²H(d,n)³H reactions indicated the possibility of large systematic errors in some of these data sets (Angulo et al. 1999). A classification of direct data available in the literature is reported as follows.

• 1.  
  McNeill & Keyser 1951. p+³H channel: E_cm = 0.06–0.15 MeV, statistical error: 3%–6%; systematic error: up to 20%.n+³H channel: E_cm = 0.06–0.125 MeV, statistical error: 3%–6%; systematic error: up to 20%;
• 2.  
  Schulte et al. 1972. E_cm = 0.98–3.1 MeV, total error: 0.25%–0.28%.
• 3.  
  The First Research Group 1985. E_cm = 15.3–150 keV, total error: 3.6%–4.8%.
• 4.  
  Krauss et al. 1987. p+³H channel: E_cm = 2.98–49.67 keV, statistical error < 1% above 6 keV and 4%–15% below 6 keV;³He+n channel: E_cm = 2.63–49.67 keV, statistical error from 1.3% to 10%; for both channels two quoted systematic errors from independent sources: 6.4% and 5% (due to variations in the alignment of beam and jet target profiles).
• 5.  
  Brown & Jarmie 1990. p+³H channel: E_cm = 9.97–58.45 keV, statistical error: 0.42%–1.3% except in the lowest point (2.8%),³He+n channel: E_cm = 9.97–58.45 keV, statistical error: 0.46%–1.8% except in the lowest point (5%), for both channels systematic error: 1.3%.
• 6.  
  Greife et al. 1995. p+³H channel: E_cm = 1.62–14.95 keV in close geometry and 14.85–128 keV in far geometry, total error: 0.7%–1.5% in close geometry except in the lowest point (15%) (to be more precise, from 1.5% up to 15% below 3 keV, <1.5% between 3 and 10 keV and 1% between 10 keV and 15 keV) and 3%–4% in far geometry;³He+n channel: E_cm = 2.4–13.1 keV in close geometry (no absolute cross section, only relative ratio) and 14.85–128 keV in far geometry, total error: 0.6%–2.9% in close geometry and 3%–4% in far geometry.
• 7.  
  Leonard et al. 2006. E_cm = 55–325 keV, total error: about 4%.

Seeking to benefit from the information given for the individual data sets in the references, the THM S(E) factor was normalized to all cited experimental data in the range E_dd = 15 keV–1.5 MeV. However, it is practically impossible to separate the statistical and systematic errors for each direct data set, because in several cases not enough information is provided. This indicates that the total errors should be used to find the most probable normalization, assuming that the systematic errors are independent for the different data sets. It is reasonable to use a weighted normalization, where data sets with larger errors constrain the normalization less. The number of THM data points overlapping with direct data chosen for the normalization is 20 for the 17 MeV run and 43 for the 18 MeV one. As a result, the 4% (10%) THM statistical error entering the normalization is divided by $\sqrt{43}$ ($\sqrt{20}$), thus supplying a 0.61%(2.2%) uncertainty. This uncertainty, combined in quadrature with that of direct data in the weighted normalization, provides an overall normalization error of about 1% (2.3%). This small error is essentially provided by those direct points with total error of less than 1%, which drive the normalization procedure in terms of absolute value as well as normalization uncertainty.

The robustness of this procedure was checked for the 18 MeV experiment, with a normalization for each individual data set below 300 keV combined with the higher energy data by Schulte et al. (1972) for both ³H+p and ³He+n channels. This provided very consistent results with a total normalization error of about 0.9% for ³H+p and 0.94% for ³He+n (Tumino et al. 2011a, 2011b). From these, one can comment that data sets from Greife et al. (1995) (in "far geometry"), The First Research Group (1985), Brown & Jarmie (1990), Leonard et al. (2006), and Schulte et al. (1972) are quite consistent, in particular for the ³H+p channel, whereas those by Krauss et al. (1987) and McNeill & Keyser (1951) reflect a normalization problem due to the quoted systematic errors, despite a very small statistical uncertainty (⩽1% and 3%, respectively).

The THM S(E) factors of the ²H(d,p)³H and ²H(d,n)³He reactions are shown in Figures 11 and 12, respectively, as black symbols (full squares and full circles for the 17 and 18 MeV run, respectively), with uncertainties accounting for statistical and normalization errors. TH data for the p+³H run at 17 MeV experience a larger scatter with respect to those at 18 MeV, due to the lower statistics in that experiment. Nonetheless, a fair agreement between them is apparent and this is the proof of consistency of the two independent ²H(d,p)³H experiments. Direct data from the different sources are shown as colored symbols, as reported in the legend. Their overall trend shows a large scatter, with deviations of more than 15%, in particular in the ³He+n channel, where a new direct study of the cross section over a larger energy region would be desirable. In contrast, the THM S(E) factors exhibit a much smoother behavior, with deviations from the direct data at ultra-low energies (see Figure 11), because of the screening, and in the SBBN region (see Figure 12). For a complete investigation, the THM S(E) factors for both the ²H(d,p)³H and ²H(d,n)³He reactions were also compared to previous polynomial/R-matrix fits to direct data (Cyburt 2004; Descouvemont et al. 2004), usually exploited in the calculation of the reaction rates, and to the results of recent ab initio calculations using realistic NN interactions (Arai et al. 2011). The comparison is shown in Figures 13 and 14, respectively, using black symbols (full squares and full circles for the 17 and 18 MeV run, respectively) to represent THM data, green (Angulo et al. 1999), blue (Cyburt 2004), and yellow (Descouvemont et al. 2004) dashed lines for earlier fits and a red dashed line for the ab initio calculations. For the ³H+p channel, the green line seems to describe the low-energy trend of TH data, whereas for E_cm > 200 keV the blue line is more appropriate. For the ³He+n channel, the green line matches the full trend, although the slope is not the same. However, the polynomials of Angulo et al. (1999) resulting in the green line were fitted only to the low-energy direct data and it cannot be expected that they could reproduce the full energy covered in the THM experiments. The blue lines at low energy fall below the THM data. The origin of this behavior, already observed for most of the direct data, is discussed in Cyburt (2004). The yellow lines resulting from the R-matrix fits of Descouvemont et al. (2004) resemble the green line for the ²H(d,p)³H reaction and the blue line for the ²H(d,n)³He one. The red lines from ab initio calculations (Arai et al. 2011) are systematically higher than THM data for E_cm > 30 keV. Similar behavior is shown in the comparison to direct data reported in Figure 3 of their paper (Arai et al. 2011), although it is not very obvious, due to the chosen scaling of the y axis. Considering that the ab initio approach gives absolute values of the cross section, it is remarkable that the deviation from experimental data is less than 10% in most cases. For the ³H+p channel, the red lines are comparable to the THM data at the lowest energies, which is not the case for the ³He+n channel, where red lines experience a much steeper energy dependence. This might be caused by not reproducing the exact binding energies of the nuclei and, as a consequence, the Q value of the reaction in the calculation. An updated theoretical calculation based on improved realistic microscopic potentials is required.

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A new parameterization of the THM S(E) factor, based on the present data, is given here for both channels as a sum of the theoretical l = 0 and l = 1 components. Bringing Equation (11) for the two-body cross section in the definition of the S(E) factor (Equation (4)) with the relative weight C₀/C₁ = 1.3 ± 0.14 (1.2 ± 0.13) for the ³H+p (³He+n), obtained from the fit to the measured three-body coincidence yields, one is left with the total S(E) factors shown as black solid lines in Figures 15 (³H+p) and 16 (³He+n). The l = 0 and l = 1 contributions are also shown as red and blue solid lines, respectively. THM data from both the 17 (red open squares only in Figure 15) and 18 MeV (red open circles) experiments are also reported. The solid lines for the ³H+p channel are the result of a weighted average of the parameterizations from the 17 and 18 MeV runs, with the total uncertainty of THM data from each run (10% and 5%, respectively) as relative weight. The lines essentially resemble those reported in Tumino et al. (2011a, 2011b), based only on the 18 MeV data. This is due to the close agreement between THM data from the two experiments and to the procedure that favors the most accurate 18 MeV data. To better compare the THM results with the direct data, the residuals to the theoretical THM curves were derived following the prescriptions reported in Cyburt (2004). The residual scatter in the direct data about the TH curves was divided by the weighted dispersion σ (1.82 keVb for the p+³H and 4.24 keVb for the n+³He channel), as given by

$$\begin{equation} \sigma ^{2}=\frac{\sum _{i}\left(\frac{y_{i}-\mu }{\sigma _{i}}\right)^{2}}{\sum _{i}\frac{1}{\sigma _{i}^{2}}}, \end{equation} \tag{ 15 }$$

with y_i and σ_i direct data points and its total uncertainty, respectively, and μ the corresponding S(E) value on the THM theoretical curve (black solid line) at the same energy. Residuals are shown in Figures 17 and 18; the dashed horizontal lines represent the 1σ error bars. These kinds of scatter diagrams provide an easier visualization of the dispersion of each direct data set about the THM S(E) factor and one may further comment on the quality and reliability of the direct experimental data. One can observe that the group of data by Brown & Jarmie (1990), The First Research Group (1985), and Leonard et al. (2006; only for the ³H+p channel) converge to practically the same normalization, with the first two being affected by a smaller uncertainty. For the ³He+n channel, data by Leonard et al. (2006) turn out systematically above the THM S(E) factor. Data by Greife et al. (1995) in far geometry, which are less precise than their data in close geometry, appear to have a higher independent normalization. Data by Krauss et al. (1987) and McNeill & Keyser (1951) are affected by systematic uncertainties from independent sources that are probably responsible for their steeper slope with respect to that of the other data sets.

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From the present work, the S(0) values are 57.7 ± 1.8 keVb for ³H+p and 60.1 ± 1.9 keVb for ³He+n, with uncertainties including the normalization error and the one coming from theory, combined in quadrature. These values differ by 15%–20% from those usually taken as reference (Angulo et al. 1999; Cyburt 2004; Descouvemont et al. 2004), and the present ratio (³H+p/³He+n) of 0.96 ± 0.04 contrasts with their estimates, which are larger than one. Our ratio agrees with that of Greife et al. (1995) using screened data, thus providing an additional test of the isotopic invariance of the electron screening. Our ratio is also consistent with the value predicted by Kimura & Bonasera (2007), based on the difference in Q values of the two mirror d+d fusion processes.

The lack of screening effects in the THM S(E) factors gives the possibility to return the screening potential U_e from comparison with direct data. The only data that allows for an extraction of U_e are those of Greife et al. (1995) in "close geometry," whose screened behavior below 15 keV is clearly visible in Figure 11. These data are not in absolute units and first require a renormalization to the THM S(E) factor. This was done by normalizing the point at 14.95 keV to the THM S_bare(E) (the black solid line of Figure 15), and then performing a weighted fit to the whole data set with the screening function (Equation (1)), with U_e as free parameter. This brings it to a value of U_e = 13.4 ± 0.6 eV, which is below the adiabatic limit (14 eV) for a molecular deuteron gas target, but consistent with it if the uncertainty is considered. The uncertainty accounts for the 5% from THM data, 1.5% from data by Greife et al. (1995) in "close geometry," and 1.8% errors from the fit procedure, combined with the standard error propagation. The result is shown in Figure 19 as a black dashed line, with only the low-energy data by Greife et al. (1995) used for the fitting procedure (red solid stars) and the THM S_bare(E) (black solid line).

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A tentative estimate of U_e can also be derived for the n+³He channel, using the cross section ratio ³H+p/³He+n reported in Greife et al. (1995). Several technical problems prevented a precise analysis of the ³He+n channel to be carried out in close geometry in the ultra-low energy region. Nonetheless, as already mentioned, the authors were able to provide the cross section ratio between the two channels from 13.1 down to 2.4 keV in close geometry. Multiplying this ratio by the corresponding low-energy ³H+p direct data already normalized to the THM S_bare(E), this gives a normalized ³He+n data set systematically underestimated by a factor 1.02 with respect to the higher energy data in far geometry by the same authors (Greife et al. 1995). If the normalization of these low-energy data is corrected by this factor, the absolute units in the crossing region between data sets in close and far geometry become consistent.

This is shown in Figure 20, where red stars represent direct data by Greife et al. (1995) in "close geometry" with the additional 1.02 factor in the normalization, and the blue star corresponds to the lowest energy point obtained in "far geometry." Solid lines have the same meaning as in Figure 19. When fitting these low-energy data for the ³He+n channel with the same screening function as before (dashed line in Figure 20), this provides a U_e value of 11.7 ± 1.6, in agreement within the experimental errors with those extracted from the ³H+p channel. These results strongly disagree with U_e = 25 ± 5 eV by Greife et al. (1995), possibly because this value was obtained using different bare nucleus S(E) factors for the normalization and/or because of an overestimate of the lowest energy point affected by a larger uncertainty, with respect to that of the other data in "close geometry." A future upgrade in the accuracy of direct low-energy data would be helpful to fix the electron screening potential.

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The reaction rates for the ²H(d,p)³H and ²H(d,n)³H processes were calculated assuming the bare nucleus S(E) factors from the THM measurement. The reaction rate between particles 1 and 2, R = N_A〈σv〉 (measured in cm³/(s mol)), can be written following Iliadis (2007) as

$$\begin{eqnarray} N_{A} \langle \sigma v \rangle &=&\frac{3.7318\times 10^{10}}{T_{9}^{{3}/{2}}} \nonumber\\ && \times \sqrt{\frac{m_{1}+m_{2}}{m_{1}m_{2}}} \int E\sigma (E)e^{-11.605E/T_{9}}{dE}, \end{eqnarray} \tag{ 16 }$$

with T₉ = T/10⁹ K, where T is the temperature of the astrophysical site; m₁ and m₂ are the particle masses in a.m.u.; and the cross section σ is measured in barns b. The numerical values of the reaction rates for the ²H(d,p)³H and ²H(d,n)³He processes are reported in Table 1.

The rate uncertainty is 4.5% for the ³H+p channel (from the S(E) factor combination of the 18 and 17 MeV experiments using the standard error propagation on the average) and 5% for the ³He+n. Upper and lower limits are also shown in the figures, however they cannot be distinguished from the thick solid line. The region of astrophysical relevance for the SBBN goes from 0.1T₉ to T₉, whereas for the PMS, as well as for inertial fusion, we are interested essentially in the reaction rate corresponding to the lowest temperature shown in the figure, 0.001T₉. We recommend an analytical expression for the reaction rates, exploiting the same formulas as reported in the REACLIB library (Thielemann et al. 1987). This expression is reliable in the whole range of temperatures reported in Table 1 within an accuracy better than 0.5%. The expression is

$$\begin{eqnarray} N_{A} \langle \sigma v \rangle \; &=& \exp \,\left(a_{1}+a_{2}T^{-1}_{9}+a_{3}T^{-1/3}_{9} +a_{4}T^{1/3}_{9}+a_{5}T_{9}\right. \nonumber\\ &&\left. +a_{6}T^{5/3}_{9}+a_{7}lnT_{9}\right), \end{eqnarray} \tag{ 17 }$$

with parameters a_i (1 ⩽ i ⩽ 7) as given in Table 2, resulting from a fit performed using the reaction rate parameterizer from NUCASTRODATA toolkit (CINA 2004).

The results are compared with the calculated reaction rates available in the literature, taken from Angulo et al. (1999), Cyburt (2004) for p+³H, R. H. Cyburt (2004, private communication) for n+³He (the ²H(d,n)³He reaction rate from Cyburt 2004 not consistent with his S(E) factor), and Descouvemont et al. (2004). The ratio of the THM reaction rate to those quoted above is reported in Figures 21(a), (b), and (c), and 22(a), (b), and (c) for ²H(d,p)³H and ²H(d,n)³H, respectively, with the colored solid lines referring to: blue line = rate ratio THM–NACRE (Angulo et al. 1999) (a); red line = rate ratio THM–(R. H. Cyburt 2004, and private communication) (b); green line = rate ratio THM–(Descouvemont et al. 2004) (c). Each colored region refers to the range of variation due to the experimental uncertainty in the THM reaction rates. The dashed black line in each figure represents the reference rate (Nacre in (a), R. H. Cyburt (2004, private communication) in (b), and Descouvemont et al. (2004) in (c)). In the lower temperature region (below 0.02T₉), the reaction rate for the ²H(d,p)³H process can be up to 20% larger than the lowest estimate, given by (R. H. Cyburt 2004, and private communication), while in the higher energy region relevant for the SBBN, an increase by up to 10% is observed. By comparing the reaction rate of Descouvemont et al. (2004; Figure 21(c)), one can deduce that the temperature dependence is closer to the THM, with a maximum deviation of about 10% at the BBN temperatures. As for the ²H(d,n)³He reaction, a similar variation of up to 20% is shown in the lowest temperature region, while in the SBBN region an opposite behavior shows up, with a decrease in the reaction rate by more than 10%. None of the temperature slopes resemble the one from THM rate.

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These deviations call for a new run of the BBN codes. This is also suggested by a recent sensitivity study by (Coc & Vangioni 2010), which puts the importance of the ²H(d,p)³He and ²H(d,n)³H reaction rates for the SBBN model on more solid ground. In this paper, the η value is fixed at the most recent Wilkinson Microwave Anisotropy Probe result, and used to calculate the sensitivity of a primordial abundance Y to a change of ±15% in reaction rates 〈σv〉, as given by: (∂log Y)/(∂log 〈σv〉). According to the quoted abundance sensitivities, a change in the reaction rates of both ²H(d,p)³He and ²H(d,n)³H processes may imply significant variations in the ²H and ⁷Li abundances. If we consider our lower and upper limits in the reaction rates from Figures 21 and 22, we can estimate the following ranges for the abundances reported in Table 3.

From this work, the ²H(d,n)³He reaction is quite influential on ⁷Li and the present variation in the ⁷Li abundance is in favor for solving the unexplained discrepancy between the calculated and observed lithium primordial abundances. A consistent result is obtained using the abundance sensitivity formulas reported in Cyburt (2004). The present rate variations in the T₉ relevant region for the SBBN imply an increase in deuterium abundance of about 2%, but a decrease of 10% in the lithium abundance. This moves the model toward solving the lithium depletion problem. However, more quantitative results for both ²H(d,p)³He and ²H(d,n)³H processes can be obtained from BBN codes once the new THM rates enter the full T₉ range of interest.

Interesting results can also be derived from the strong rate variations in the region of T₉ relevant for PMS and for plasma physics. Very little information was available, mainly due to the difficulty of providing experimental data at ultra-low E_cm energies from direct experiments, which are screened anyway. To determine the influence of these new rates on PMS evolution, they were included in an updated version of the FRANEC evolutionary code (Degl'Innocenti et al. 2008, Tognelli et al. 2011) and stellar models were computed exploiting five different stellar masses (0.1 M_☉, 0.2 M_☉, 0.4 M_☉, 1 M_☉, 2 M_☉). Usually in stellar evolutionary codes only the most efficient deuterium burning channel ²H(p,γ)³He is included. Thus, THM rates for the other two deuterium burning d+d channels were included in the FRANEC stellar evolutionary code. As expected, the inclusion of the two additional destruction channels (²H(d,p)³H and ²H(d,n)³He) did not affect the global properties of a star (i.e., luminosity, radius, age, etc.). The analysis has been focused on the evolutionary parameter most affected by the change of the burning rate, (i.e., the deuterium abundance inside the stellar structure).

The deuterium "total" destruction rate in terms of the three relevant ²H(p,γ)³He, ²H(²H,p)³H, and ²H(²H,n)³He channels is given by

$$\begin{eqnarray} \frac{dn_d}{dt} |_{{\rm dis}} &=& {-}\left [ n_p n_d \langle \sigma v\rangle _{{{{pd}},}^3\mathrm{He}}\right. \nonumber \\ & &\left. +\frac{n_d^2}{2}\langle \sigma v\rangle _{{{{dd}},}^3\mathrm{He\,{{n}}}}+ \frac{n_d^2}{2}\langle \sigma v\rangle _{{{{dd}},}^3\mathrm{H\,{{p}}}}\right ]\\ \nonumber & \equiv & {-} n_p n_d \langle \sigma v\rangle _{{{{pd}},}^3\mathrm{He}} (1 + \mathcal {F}), \end{eqnarray} \tag{ 18 }$$

with n_p and n_d the numerical hydrogen and deuterium densities, respectively, and F defined as

$$\begin{equation} \mathcal {F} \equiv \frac{n_d}{2n_p}\left [ \frac{\langle \sigma v\rangle _{{{{dd}},}^3\mathrm{He\,{{n}}}}}{\langle \sigma v\rangle _{{{{pd}},}^3\mathrm{He}}} + \frac{\langle \sigma v\rangle _{{{{dd}},}^3\mathrm{H\,{{p}}}}}{\langle \sigma v\rangle _{{{{pd}},}^3\mathrm{He}}}\right ]. \end{equation} \tag{ 19 }$$

For the present calculations, the initial deuterium mass fraction abundance has been set to X_d = 2 × 10⁻⁵ ($X_d^0$) as a representative value of the initial deuterium abundance present in population I stars (see, e.g., Geiss & Gloeckler 1998; Linsky et al. 2006; Steigman et al. 2007). However, as soon as deuterium is burnt inside a star, its abundance rapidly drops (X_d ≈ 10⁻¹⁸–10⁻¹⁷, see, e.g., Rolfs & Rodney 1988). The d-burning rate is different in each region of the star, due to the temperature stratification inside the stellar structure, however, for all the calculated models the burning occurs in a fully convective star, and, consequently, the deuterium abundance is kept homogeneous by the continuous mixing.

The left panel of Figure 23 shows the calculated time evolution of the central deuterium mass fraction abundance, X_d, (which is the same abundance inside the whole structure) normalized to the initial one, for stars of different masses. Results from model calculations with (solid lines) and without (dashed lines) the inclusion of the ²H(d,p)³H and ²H(d,n)³He channels are reported. For a better assessment, the relative difference between solid and dashed lines of the same color (for a given stellar mass) is shown in the right panel as a function of the star age (black dashed lines). Colored dashed lines in this panel connect models of different masses for the same relative time reduction of central deuterium (calculated for the case without the inclusion of the two d+d channels). It turns out that the difference between deuterium abundance with and without the inclusion of the two additional burning channels is usually below 10%, except for larger stellar masses and when deuterium is already nearly exhausted. However, such young evolutionary stages might be affected by other uncertainties, essentially depending on the initial abundances for the computations (see Baraffe et al. 2002). Recently Baraffe et al. (2009) and Hosokawa et al. (2011) have shown that the inclusion of an accretion phase at young ages strongly affects the stellar structure, particularly for what concerns the chemical evolution (see also Baraffe & Chabrier 2010). Indeed, as light elements (i.e., deuterium and lithium) are destroyed inside the star due to their relatively low nuclear burning temperature, the presence of a continuous accretion of matter (i.e., from an accretion disk), which supplies non-nuclear-processed material, partially compensates for the destruction of such elements. Here we do not discuss how accretion processes could modify the present results, because the accretion phase is extremely sensitive to several parameters, such as accretion rate, accretion history, and/or accretion energy, which at the moment are far from being well constrained (Baraffe et al. 2012).

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