New Thermonuclear Rate of 7Li(d,n)24He Relevant to the Cosmological Lithium Problem

S. Q. Hou; T. Kajino; T. C. L. Trueman; M. Pignatari; Y. D. Luo; C. A. Bertulani

doi:10.3847/1538-4357/ac1a11

1. Introduction

Regarded as a key pillar of modern cosmology, big bang nucleosynthesis (BBN) describes the conditions in which nuclear reactions built the first complex nuclei as the universe expanded and cooled down from an incredibly dense and hot primordial fireball. About half an hour after the start of the Big Bang, the entire process of primordial nucleosynthesis ended, leaving behind as main relics ²H, ³He, ⁴He, and ⁷Li. In the standard BBN model, the prediction of primordial abundances depend on only one free parameter: the baryon-to-photon ratio η, equivalently the present baryon density ω_b = Ω_b h² via the relation η = ω_b/(3.65 × 10⁷), which has been determined quite accurately from observations of the anisotropies of the cosmic microwave background (Planck Collaboration 2020). Thus, provided that no uncertainties exist in the relevant reaction rates, the BBN-predicted light nuclide abundances should be reliable. Current BBN predictions for abundances of D, ³He, and ⁴He are consistent with values inferred from astronomical observations. However, only the ⁷Li abundance is overpredicted by about a factor of 3 (Cyburt et al. 2003; Coc et al. 2004; Asplund et al. 2006; Sbordone et al. 2010). This is called the cosmological lithium problem.

Over the past decade, many attempts to address this issue have been carried out, such as from the perspective of conventional nuclear physics and even exotic physics beyond the standard BBN framework (Angulo et al. 2005; Cyburt et al. 2008, 2016; Boyd et al. 2010; Pospelov & Pradler 2010; Fields 2011; Kirsebom & Davids 2011; Wang et al. 2011; Broggini et al. 2012; Coc et al. 2012, 2013, 2014; Cyburt & Pospelov 2012; Kang et al. 2012; Voronchev et al. 2012; Bertulani et al. 2013; Hammache et al. 2013; He et al. 2013; Kusakabe et al. 2014; Pizzone et al. 2014; Yamazaki et al. 2014; Hou et al. 2015, 2017; Famiano et al. 2016; Damone et al. 2018; Hartos et al. 2018; Luo et al. 2019; Rijal et al. 2019; Clara & Martins 2020). However, despite the fact some solutions using exotic physics have succeeded in resolving this issue, it appears there is still no universally accepted solution in the academic community since validations of these mysterious exotic physics are beyond the capabilities of current science. Conversely, it seems more worthwhile to exclude any potential possibility of resolving the ⁷Li discrepancy from the perspective of nuclear physics. It is known that the majority of the primordial ⁷Li production arises from the decay of ⁷Be by electron capture during the 2 months after BBN stops. Thus, for the solution of the Li problem, reactions involving ⁷Be could be more significant than those involving ⁷Li. Therefore, many reactions that potentially destroy ⁷Be were investigated to solve this discrepancy over past 10 yr (Kirsebom & Davids 2011; Broggini et al. 2012; Hammache et al. 2013; Hou et al. 2015; Hartos et al. 2018). Meanwhile, enormous efforts have been made to refine the reaction rates of key BBN reactions in the past 20 yr (Smith et al. 1993; Descouvemont et al. 2004; Serpico et al. 2004; Cyburt & Davids 2008; Neff 2011; Pizzone et al. 2014; Tumino et al. 2014; Hou et al. 2015; Barbagallo et al. 2016; Iliadis et al. 2016; Kawabata et al. 2017; Lamia et al. 2017, 2019; Damone et al. 2018; Rijal et al. 2019; Mossa et al. 2020), but the probability of solving or alleviating the ⁷Li problem by improving our knowledge of relevant nuclear reaction rates still cannot be eliminated. Recent experiments for key nuclear reactions like ⁷Be(n,p)⁷Li and ⁷Be(d,p)2⁴He allow for a reduction of the ⁷Li production by about 12% (Damone et al. 2018; Rijal et al. 2019) compared to previous calculations. At present, nuclear uncertainties cannot rule out that some of the reactions destroying ⁷Li are indeed more efficient than those currently used (Boyd et al. 2010; Chakraborty et al. 2011).

Despite the fact it is an important ⁷Li destruction reaction, before 2018 the ⁷Li(d,n)2⁴He reaction could not have been well studied, due to limited information on energy levels close to the threshold in ⁹Be. With mounting experimental results concerning the properties of relevant excited states emerging recently, it is the right time to reinvestigate the ⁷Li(d,n)2⁴He reaction rate. The necessity of carrying out this work can be summarized as follows: First, the maximum reduction of uncertainties of ⁷Li(d,n)2⁴He reaction rates can remove the most significant ambiguity in the calculated ⁷Li abundance due to this reaction, and promises substantial improvements in the ⁷Li BBN prediction. Second, more accurate abundance predictions of primordial isotopes are also crucial to probe exotic physics beyond the standard model as well as to constrain cosmological parameters (Pospelov & Pradler 2010; Fields 2011; Coc et al. 2013, 2014; Kusakabe et al. 2014; Yamazaki et al. 2014; Hou et al. 2017; Luo et al. 2019; Mossa et al. 2020). For these reasons, this work is important for continued developments in other interdisciplinary fields such as astronomy, cosmology, and particle physics.

The current ⁷Li(d,n)2⁴He reaction rate most widely used in BBN simulations is taken from Boyd et al. (1993, hereafter referred to as BM93). Differing from the rate compiled by Caughlan & Fowler (1988, hereafter CF88), which only considered the direct component, the BM93 rate not only updated the direct reaction rate, but also took the contributions from the 280 and 600 keV resonances into account. Theoretically speaking, the BM93 rate should be more reliable compared with the evaluation from CF88. Nevertheless, we found that a significant overestimation exists in their assessment, which could potentially impact BBN. In this work, the ⁷Li(d,n)2⁴He reaction rate is investigated systematically and comprehensively, and the separate contributions from the direct components and resonances near the deuteron threshold are studied individually. It is well known that the uncertainties of every individual component come from the uncertainties from its own input parameters. For the purpose of getting more reasonable uncertainties of the total reaction rates, a Monte Carlo approach is used to obtain the total ⁷Li(d,n)2⁴He reaction rate and its corresponding error. In order to study the impact of the new reaction rate on the abundances of primordial nuclei, we perform detailed BBN calculations using two types of models: a uniform density distribution model and a nonuniform density model.

The paper is organized as follows. In Section 2, we introduce the basic formalism for the resonant reaction cross section and its relation to the astrophysical reaction rate. In Section 3, based on an elaborate investigation of each individual term, which contributes to reaction rates, we derive the cross section of the ⁷Li(d,n)2⁴He reaction and its corresponding uncertainties, further obtaining the new astrophysical reaction rate of ⁷Li(d,n)2⁴He. In Section 4, we perform the BBN simulations with a uniform and a nonuniform baryon distribution to investigate the impact of our new ⁷Li(d,n)2⁴He reaction rate on primordial yields. Our conclusions are summarized in the last section.

2. Astrophysical Reaction Rate

The thermonuclear rate is calculated from the reaction cross section σ(E) by integration over the Maxwell–Boltzmann (MB) distribution of the interacting particles in a stellar environment with a temperature T (Rolfs & Rodney 1988; Iliadis 2007)

$\begin{eqnarray}&&\left\langle \sigma \ v\right\rangle =\sqrt{\displaystyle \frac{8}{\pi \mu {\left({kT}\right)}^{3}}}{\int }_{0}^{\infty }\sigma (E)E\exp \left(-\displaystyle \frac{E}{{kT}}\right){dE},\end{eqnarray} \tag{ 1 }$

where μ is the reduced mass, N_A is Avogadro's number, and k is the Boltzmann constant. Obviously, the reaction cross section σ(E) and its energy dependence are the key parameters for determining the reaction rates. σ(E) comprises a resonant and direct reaction cross section.

2.1. Resonant Cross Section

Differing from a one-step process without the formation of an intermediate compound nucleus (direct reaction), a resonant reaction proceeds through the formation of a compound nucleus in the entrance channel, which subsequently decays to the exit channel. The resonant cross section is described by a Breit–Wigner single level formula

$\begin{eqnarray}&&\sigma =\displaystyle \frac{\pi }{2\mu E}\displaystyle \frac{\omega {{\rm{\Gamma }}}_{\mathrm{in}}{{\rm{\Gamma }}}_{\mathrm{out}}}{{\left(E-{E}_{r}\right)}^{2}+{{\rm{\Gamma }}}_{\mathrm{tot}}^{2}/4},\end{eqnarray} \tag{ 2 }$

where the first term is the upper limit for the cross section (i.e., the geometrical cross section), E is the energy in the center of mass frame (CM), E_r is the resonance energy, and Γ_in and Γ_out are the widths of the entrance channel and exit channel, respectively. The total resonance width of the state is defined as Γ_tot = Γ_in + Γ_out+ ⋯. The statistic factor ω is defined as

$\begin{eqnarray}&&\omega =\displaystyle \frac{(2{J}_{C}+1)(1+{\delta }_{01})}{(2{J}_{0}+1)(2{J}_{1}+1)},\end{eqnarray} \tag{ 3 }$

which takes into account the angular momenta J₀ and J₁ of the colliding nuclei and the angular momentum J_C of the excited state in the compound nucleus. The factor (1 + δ₀₁) is included since the cross section for identical particles in the entrance channel increases by two times.

The resonance width Γ_in(Γ_out) can be parameterized by the dimensionless reduced width θ², which incorporates all the unknown properties of the nuclear interior,

$\begin{eqnarray}&&{{\rm{\Gamma }}}_{\mathrm{in}}=\displaystyle \frac{3{\hslash }v}{R}{P}_{l}(E,R){\theta }_{\mathrm{in}}^{2},\end{eqnarray} \tag{ 4 }$

where v is the relative velocity in the CM frame and R is the interaction radius. The function P_l(E, R) refers to the Coulomb and centrifugal barrier penetrability given by

$\begin{eqnarray}&&{P}_{l}(E,R)=\displaystyle \frac{1}{[{F}_{l}{\left(E;R\right)}^{2}+{G}_{l}{\left(E;R\right)}^{2}]}.\end{eqnarray} \tag{ 5 }$

Here, F_l(E; R) and G_l(E; R) are the regular and irregular Coulomb wave functions, respectively.

2.2. Resonant Reaction Rate

2.2.1. Narrow Resonant Reaction Rate

Broadly speaking, the astrophysical reaction rate should be obtained by performing strict numerical integration following Equation (1). However, in the case of a narrow resonance for which the width of the resonance is much smaller than resonance energy, the expression for the rate can be rewritten as

$\begin{eqnarray}\begin{array}{rcl}\left\langle \sigma \,v\right\rangle & = & \sqrt{\tfrac{2\pi }{{\left(\mu {kT}\right)}^{3}}}\exp \left(-\displaystyle \frac{{E}_{r}}{{kT}}\right)\omega \displaystyle \frac{{{\rm{\Gamma }}}_{\mathrm{in}}{{\rm{\Gamma }}}_{\mathrm{out}}}{{{\rm{\Gamma }}}_{\mathrm{tot}}}2\\ & & \times \ {\int }_{0}^{\infty }\displaystyle \frac{{{\rm{\Gamma }}}_{\mathrm{tot}}/2}{{\left({E}_{r}-E\right)}^{2}+{{\rm{\Gamma }}}_{\mathrm{tot}}^{2}/4}\,{dE},\end{array}\end{eqnarray} \tag{ 6 }$

since the partial width and the energy factor from the MB distribution are approximately constant over the total width of the resonance.

By introducing the concept of resonance strength with the definition of ωγ = ωΓ_inΓ_out/Γ_tot and the integral in Equation (6) being calculated analytically, the narrow resonance reaction rate can be simplified as

$\begin{eqnarray}&&\left\langle \sigma \,v\right\rangle ={\left(\displaystyle \frac{2\pi }{\mu {kT}}\right)}^{3/2}{{\hslash }}^{2}\omega \gamma \exp \left(-\displaystyle \frac{{E}_{r}}{{kT}}\right).\end{eqnarray} \tag{ 7 }$

2.2.2. Reaction Rate of Broad Resonance and Subthreshold Resonance

For the case of a broad resonance where the resonance width is not much smaller than the width of the Gamow peak for a given temperature, it can no longer be assumed that partial widths and the MB distribution factor can be pulled out front from the integration as constants. In such a case, the reaction rates must be calculated by numerical integration using Equation (1). Similar cases occur to subthreshold resonances where the compound level lies below the particle threshold and the reaction can proceed via the high energy wing of the resonance extending over the particle threshold. Likewise, the energy dependence of the partial and total widths is required as well. Therefore, we just need to follow the same procedure used for broad resonance to calculate the contribution from a subthreshold resonance to the reaction rate.

3. Derivation of the Reaction rate for ⁷Li(d,n)2⁴He

It is well known that the astrophysical reaction rate is mainly determined by the reaction cross section in the energy region close to the threshold. For the reaction ⁷Li(d,n)2⁴He, it is known that there are four near-threshold resonances: the subthreshold state at −24.9 keV and three above-threshold resonances at 0.28, 0.6, and 0.8 MeV, respectively. We know that primordial nucleosynthesis occurs at temperature around 10⁹ K (that is, T₉ ∼ 1), corresponding to thermal energies of ∼100 keV. Theoretically speaking, the resonances at 280 and −24.9 keV should have a more significant contribution to the ⁷Li(d,n)2⁴He rate than the remaining two resonances. However, an accurate contribution from each separate term is still unclear since the properties of these resonances are still ambiguous and the partial widths of some resonances remain unknown. In this section, we will study each of them in detail utilizing the results from recent experiments. One thing we must clarify in advance is that the 280 keV resonance will be neglected from the analysis in the next dedicated subsection as its width is too narrow to be of interest.

3.1. The Cross Section Breakdown

3.1.1. Consideration of the Subthreshold Resonance

As noted in the introduction, the newest ⁷Li(d,n)2⁴He astrophysical reaction rate widely used in BBN simulations is from the estimation of BM93, which includes the contributions from both direct components and resonances at 280 and 600 keV. Compared with the rates from CF88, wherein only the contribution from the direct term is considered, the BM93 rates integrate the contributions from resonances near the threshold for the first time, which means, at least in principle, it should be reliable. However, neither rate includes the contribution from the ⁹Be resonance state at 16.671 MeV (Tilley et al. 2004), which is only 24.9 keV below the energetic threshold of the ⁷Li + d reaction; this is most likely due to the inaccurate information regarding the energy levels of the 9Be nucleus at that time. Fortunately, fruitful follow-up experimental studies on the energy level of ⁹Be made it possible to assess its contribution to the ⁷Li + d reaction rates.

Owing to the absence of a Coulomb barrier for neutron emission from the ⁹Be compound system, it is conventionally thought the d+⁷Li reaction proceeds mainly through intermediate states in ⁸Be by the ⁷Li(d,n)⁸Be(α)⁴He reaction sequence, and not through intermediate states in ⁵He by the ⁷Li(d,α)⁵He(n)⁴He sequence. Nevertheless, completely differing from our conventional understanding, the recent experimental result from Rijal et al. (2019) strongly indicated that α decay dominates for the 16.849 MeV, 5/2⁺ state in ⁹B (actually corresponding to the 16.71 MeV state in the energy level diagram of ⁹B in the National Nuclear Data Center (NNDC)), which is regarded as the mirror state of the 5/2⁺ state at 16.671 MeV in ⁹Be. According to the mirror symmetry principle, the J^π = 5/2⁺ subthreshold state in ⁹Be should primarily decay by α emission as well, and the counterpart ⁵He then subsequently splits into neutron and α. Therefore, nucleosynthesis calculations should also take into account the ⁷Li(d,α)⁵He reaction, although the final products are the same as the ⁷Li(d,n)⁸Be reaction.

For the purpose of calculating the separate contribution from the subthreshold resonance to the ⁷Li(d,n)2⁴He reaction rate, the energy level information of this state including spin, parity, and partial width for d, n, and α decay of this resonance are required. It is known from NNDC that the spin and parity of the 16.67 MeV resonant state of ⁹Be are determined as J^π = 5/2⁺, while the relevant knowledge of partial decay widths of this state remains unknown. Thus, we have to derive the widths of these particle decays using resonance theory in combination with relevant width information of its mirror state in ⁹B.

From Section 2, we know that the partial decay width Γ_i (i = α, d, n) essentially depends on three aspects: relative velocity v, penetration factor P_l(E, R), and reduced width θ². Among them, relative velocity v can be obtained easily and P_l(E, R) can be calculated analytically for neutron emission, but for the case of charged nuclei, we have to resort to a numerical calculation instead of analytical approximation in order to obtain the P_l(E, R) with relatively high accuracy. In our calculation, a code following the formalism depicted in Iliadis (1997) is used to calculate P_l(E, R). Regarding the reduced width θ², it reflects a measure of the degree to which an actual quasi-stationary state can be described by the motion of particle a and the residual nucleus X in a potential. In principle, it can be estimated on the basis of a nuclear potential approximated as a square well and assuming an average level distance (Blatt & Weisskopf 2010). Nevertheless, the value derived from experiments would be of high priority for its use in calculations.

It is well known that a mirror state is referred to as an analog state at nearly the same excitation energy in mirror nuclei pairs, which can be inter-transformed by exchanging the role of protons with neutrons. According to mirror symmetry, the properties and configuration of the mirror states in the mirror pair of ⁹Be and ⁹B should be identical apart from the Coulomb effects. Thus, it is expected that the θ² value holds constant for identical particle decay from the mirror states pair of ⁹B at 16.849 MeV and ⁹Be at 16.67 MeV. Therefore, the reduced width θ², which will be used to calculate the partial width of the subthreshold level (16.67 MeV) of ⁹Be, can be extracted directly from the relevant widths information of its mirror state in ⁹B (16.849 MeV, 5/2⁺). Fortunately, the partial widths for the mirror state in ⁹B are available thanks to recent cross section measurements of the ⁷Be + d reaction (Rijal et al. 2019). In their analysis, it is shown that the (d,α) channel dominates relative to the (d,p) channel and the values of Γ_α and Γ_d are suggested to be 50 and 3.3 keV, respectively. Γ_P is only 1 keV, implying the contributions from the (d,p) channel are negligible. Using Equation (4), the reduced width ${\theta }_{i}^{2}$ (i = α, d, p) values for the mirror state in ⁹B of 16.849 MeV are obtained and listed in Table 1. Here, the uncertainty of ${\theta }_{d}^{2}$ is mainly caused by the influence of the Coulomb penetration factor P_l from the 5 keV uncertainty of the resonance energy. However, this hardly makes a visible impact on ${\theta }_{a}^{2}$ and ${\theta }_{p}^{2}$ because several kiloelectronvolt uncertainties in the energy level can be totally neglected with respect to the huge energy release for α and p decay. In the present evaluation, we mainly consider the ⁷Li(d,α)⁵He channel since the neutron decay is negligible relative to α decay. Using the above reduced widths of d and α from the mirror state in ⁹B, as shown in the first row of Table 2, the cross section for the ⁷Li(d,α)⁵He reaction proceeding through the subthreshold compound nucleus ⁹Be can be obtained using Equation (1).

Table 1. The Reduced Width of Different Particle Decays for the 16.849 MeV Level in ⁹B

E_r (MeV)	J^π	${{\rm{\Gamma }}}_{{p}_{1}}$	Γ_d	Γ_α	${\theta }_{{p}_{1}}^{2}$	${\theta }_{d}^{2}$	${\theta }_{\alpha }^{2}$
0.361(5)	5/2⁺	1	3.3	50	3.64 × 10⁻⁵	0.119 ± 0.008	3.22 × 10⁻³

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Table 2. Resonance Properties (Energies in Megaelectronvolts, Widths in Kiloelectronvolts) Considered in the Present Calculation

E_r (MeV)	J^π	Γ_P	Γ_d	Γ_α	${\theta }_{p}^{2}$	${\theta }_{d}^{2}$	${\theta }_{\alpha }^{2}$
−0.0249 ± 0.008	5/2⁺					0.119 ± 0.008	3.22 × 10⁻³ ± 1.9 × 10⁻⁶
0.6 ± 0.005	5/2⁻	30	143	27^⋆	0.186 ± 0.006	0.099 ± 0.0014	1.872 × 10⁻³ ± 4.0 × 10⁻⁷ ^⋆
0.8 ± 0.005	7/2⁺	1 ± 0.2	7 ± 3	39 ± 4

Note. Energies are given with respect to the ⁷Li+D threshold. The value with an ^⋆ denotes the upper limit for the given quantity and specific particle decay.

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3.1.2. The 600 keV Resonance

The resonance, 600 keV above the deuterium threshold (16.6959 MeV), corresponding to the 5/2⁻ exited state of ⁹Be at 19.298 MeV, is thought to be very significant in the evaluation by BM93 since they thought its contribution to the ⁷Li+d reaction rates dominates within the temperature range of BBN interest. In their evaluation, the total cross section at resonance energy E_r = 600 keV is determined to be 420 mb, which was obtained by multiplying the value of the cross section measured at 0° by 4π on the assumption that the differential cross section is isotropic (Slattery et al. 1957). According to the Breit–Wigner single resonance formula Equation (2), we know that the reaction cross section σ_r will reach a maximum when the condition of Γ_in = Γ_out = Γ/2 and Γ = Γ_in + Γ_out is satisfied. We also know that the total width of this state is determined as 200 keV from Tilley et al. (2004), so if one sets Γ_in = Γ_out = 100 keV, the obtained maximum limit of the cross section at E_r = 600 keV should be 357 mb, which is still smaller than the value of 420 mb adopted in BM93. If all of these considerations are correct, the cross section for the resonance at 600 keV is overestimated in the evaluation by BM93. The reason for this overestimation is not obvious. A possible reason could be attributed to the assumption of isotropy for the angular distribution.

We now use an indirect method to reevaluate the contribution from the 600 keV resonance on the cross section of ⁷Li(d,n). In contrast to the situation of the ⁷Li(d,n) cross section at E_r = 600 keV, which has a lack of sufficient experimental data, the cross section for ⁷Li(d,p) from this resonance has been measured extensively. The currently existing values for the measured ⁷Li(d,p) cross section range from a maximum value of 211 ± 15 mb to a minimum of 110 ± 22 mb (Adelberger et al. 1998), and the cross section for this resonance recommended by Adelberger et al. (1998) is 147 ± 11 mb based on the comprehensive consideration of previous measurements. Another new cross section measurement of this 600 keV resonance, which is free from the effect of backscattering, presented a slightly bigger value of 155 ± 8 mb (Weissman et al. 1998). Here, the two above proposed cross sections lead to an average value of σ _d,p = 151 ± 10 mb, which will be used in our next calculation.

The recent measurements of resonances in ⁹Be around the proton threshold present the specific partial widths of (p,d), (p,α), and (p,p) of the 19.298 MeV excited state (E_r = 600 keV for deutron threshold) via multichannel R-matrix analysis of the experimental data (Leistenschneider et al. 2018). The final values of Γ_p , Γ_d , and Γ_α for this state and the associated uncertainties are determined to be 40 ± 10, 150 ± 7, and 20 ± 3 keV, respectively. Using the values of Γ_d and Γ_p given above, we attempted to reproduce the value of 151 ± 10 mb by adjusting the values of Γ_d and Γ_p within their own uncertainties and finally Γ_d = 143 keV and Γ_p = 30 keV are proposed.

Recalling the specific derivation of the resonant cross section of ⁷Li(d,n) from the 17.298 MeV resonance in the BM93 estimation, one point we need to highlight is that Γ_α was thought to be negligible compared to Γ_n. However, we found there is no definite evidence to support their conclusion in the literature (e.g., Heggie & Martin 1973). On the contrary, the experimental results from Heggie & Martin (1973) show that the α emission accounts for a significant fraction relative to neutron emission. Our conclusions are not affected by which channel dominates between ⁷Li(d,n) and ⁷Li(d,α) since the final reaction products in a three-body form will be identical for both channels. The sum of the contributions from these two channels can be taken as the resonant cross section for ⁷Li(d,n) or ⁷Li(d,α) at 17.298 MeV. As indicated in Tilley et al. (2004), the γ decay width is only at the level of several electronvolts so the total width (Γ = 200 keV) of the 600 keV resonance consists almost entirely of Γ_p , Γ_n , Γ_d , and Γ_α. Note that the partial widths Γ_p an Γ_d have been set as introduced above, and all of the remaining fraction of the total width Γ (subtracting Γ_p and Γ_d ) can be taken as the value of Γ_n or Γ_α. For this reason, we can set the upper limit of Γ_α as Γ − (Γ_d + Γ_p ). Then the reduced widths ${\theta }_{i}^{2}$ (i = p, d) and the upper limit of ${\theta }_{\alpha }^{2}$ can be obtained via the partial width formula Equation (4), as listed in the second row of Table 2.

Unlike the reaction rate from a single narrow resonance, which can be directly obtained via Equation (7), the calculation of the reaction rate for the 17.298 MeV resonance relies on Equation (1) since this resonance is confirmed to be broad. Similar to the case of subthreshold resonance, the knowledge of the dimensionless reduced widths θ² for different decay channels are required. Using our deduced values of Γ_d and Γ_p for the resonance at E_r = 600 keV in combination with the partial width formula Equation (4), the reduced widths ${\theta }_{i}^{2}$ (i = p, d) can be obtained, as listed in Table 2. If we take the remaining width Γ − (Γ_d + Γ_p) as the upper limit of Γ_α or Γ_n, we can then obtain the maximum value of the cross section for the ⁷Li(d,n)2⁴He from the 600 keV resonance.

3.1.3. The 800 keV Resonance

The 800 keV resonance corresponds to the excited state of ⁹Be at 17.493 MeV, with recommended J^π = 7/2⁺ assignment in Tilley et al. (2004). The nature of this state, like partial decay widths, is still not well understood, despite the fact the total width is confirmed to be 47 keV. Fortunately, information regarding the partial width of this state is given by Leistenschneider et al. (2018) and will be used directly in our evaluation, as shown in third row of Table 2. We emphasize again that the contribution of ⁷Li(d,α)⁵He is regarded as equivalent to the ⁷Li(d,n)2⁴He channel since the final reaction products of both reactions are identical. Through a similar process to that in the previous section, the cross section of the 17.493 MeV resonance can be obtained.

3.1.4. The Direct Contribution

For the sake of comparison with previous results and convenience of discussion below, here we write the cross section in the form of σ(E) = S(E)E⁻¹ e^{−2π
η} σ(E), where S(E) is the astrophysical S-factor and η is the Sommerfield parameter. Examining the previous evaluations of direct contributions from ⁷Li(d,n) in the literature (CF88; BM93), it is clear that the source data used to determine the direct S-factor in BM93 originates from Slattery et al. (1957), which is exactly the same one used to derive the cross section of the 600 keV resonance, while that of CF88 remains unclear. We investigate the relevant literature and assume that the data originates from Baggett & Bame (1952). This assumption is also confirmed by comparing the CF88 rate to the results from a numerical integration over the cross section data in the low energy regions of Baggett & Bame (1952). The direct S-factor S(0) determined in CF88 is about 33.9 MeV barn, which is about two times that derived in BM93 based on data between 1.6 and 2.0 MeV of deuteron energy. It is difficult to conclude which is correct since both of them have their own intrinsic drawbacks.

The uncertainties of the CF88 direct S-factor mainly stem from the constraints of measurements at a very limited solid angle (90° ± 20°) and the assumption of isotropic angular distribution, which has proven to result in an S-factor overestimation (see Section 3.1.2 on the 600 keV resonance). In addition, at low energies, extra contributions from possible resonances near the threshold, such as yet to be identified subthreshold resonances and effects from electron screening, will both lead to overestimation of the S-factor from the direct term. Differing from the case of CF88, the S-factor of the direct term in BM93 is determined as 17 MeV-b based on the fact that the derived S-factor values almost appear to be a constant in the energy range from 1.6−2.0 MeV. Theoretically speaking, this value might be more reliable compared with that from CF88 since the interference from subthreshold resonances and electron screening can be excluded. However, it is likely still overestimated. The reasons are threefold: First, this value of 17 MeV-b from BM93 actually refers to the sum of the ⁷Li(d,n) and ⁷Li(d,p) channels, while the actual contribution from ⁷Li(d,n) is only about 9 MeV-b. The remaining part accounts for the endoergic ⁷Li(d,p)⁸Li reaction, which does not proceed efficiently in BBN as a result of the dominance of its reverse reaction. Therefore, they should be separately treated as two different reactions whenever performing BBN network calculations, but this is not mentioned in all of the previous BBN simulations where the BM93 rate is used directly (Serpico et al. 2004; Pisanti et al. 2008; Arbey 2012; Coc et al. 2012; Consiglio et al. 2018; Pitrou et al. 2018). Second, it is well known that the cross section value from experimental measurements for a fixed reaction can only reflect the total contributions from direct components and resonant components, not the separate contribution. So the value from BM93 still includes the resonant contribution in or near the energy zone of 1.6–2.0 MeV. Third, the same as for the case of the 600 keV resonance where the isotropic angular distribution is assumed for the same source data of cross section, it is inevitable to produce an overestimation of the direct S-factor. In other words, this S-factor of about 9 MeV-b is still larger than the true value taking into account only the direct contribution.

In the present work, instead of adopting the data from Baggett & Bame (1952) and Slattery et al. (1957), we chose data from the recent work where the direct S-factor is determined as (Sabourov et al. 2006)

$\begin{eqnarray}&&S(E)=5400(\pm 1500)-37(\pm 21)E,\end{eqnarray} \tag{ 8 }$

where S(E) and E are in units of keV·b and keV, respectively. As clarified by Sabourov et al. (2006), the negative slope term is probably attributed to the effect of electron screening, and the S(0) should not include the negative slope. Thus, we choose the constant S-factor of S(0) = 5400(±1500) keV·b in our evaluation, which is in good concordance with the value that is evaluated based on the data from Baggett & Bame (1952), as introduced below. In the experiment of Sabourov et al. (2006), the cross section of ⁷Li(d,n) was measured for energies below 70 keV and the emitted neutrons were detected at eight different angles from 0°−150°, so the obtained S-factor value could exclude the influence from the assumption of the isotropic angular distribution in Baggett & Bame (1952). Another reason we choose this value for S(0) is due to its consistency with the value estimated by using the extracted factor of overestimation to scale the S(0) of the ⁷Li(d,n) channel in Baggett & Bame (1952). Specifically, it can be seen that both of the cross section of ⁷Li(d,p) and ⁷Li(d,n) were measured in Baggett & Bame (1952). The cross section of ⁷Li(d,p) at the peak of 600 keV resonance is up to 230 mb in their results, but the proceeding measurements of the ⁷Li(d,p) cross section for this resonance support the value of 147 mb recommended in Adelberger et al. (1998), which is about 64% of the value from Baggett & Bame (1952). Then the factor of overestimation extracted out from these two sets of ⁷Li(d,p) cross section data can be used to scale the direct S-factor of ⁷Li(d,n) derived from the low energy cross section in Baggett & Bame (1952). The obtained value is approximately 5700 keV·b, basically in accordance with the value from Sabourov et al. (2006).

3.2. Astrophysical S-factor and Reaction Rate of ⁷Li(d,n)2⁴He

Using the information of Γ_i and ${\theta }_{i}^{2}$ listed in Table 2 and the newly determined direct S-factor from the previous section, the total S(E) (or cross section σ(E)) of ⁷Li(d,n)2⁴He and its corresponding uncertainties can be obtained via a Monte Carlo simulation considering all the uncertainties of the resonance parameters, as plotted in Figure 1. Here, for the subthreshold resonance, we not only include the uncertainty listed in the first row of Table 2, but we also consider uncertainties of 100% arising from the assumption of mirror symmetry for the ${\theta }_{d}^{2}$ and ${\theta }_{\alpha }^{2}$ in our simulation, as suggested by Nesaraja et al. (2007). The solid blue line refers to the total S-factor for ⁷Li(d,n)2⁴He, and the dashed dark red and dark blue lines correspond to upper and lower limits, respectively. The bigger uncertainties of the S-factor at low energy is mainly caused by the assumption of 100% uncertainties from mirror symmetry. For comparison, we plot five sets of experimental ⁷Li(d,n)⁸Be cross section data from NNDC in Figure 1 (Slattery et al. 1957; Nussbaum 1969; Bochkarev et al. 1994; Hofstee et al. 2001; Sabourov et al. 2006). Among them, the data from Slattery et al. (1957) and Bochkarev et al. (1994) use differential cross sections, so we need to convert them into regular cross sections by assuming isotropic angular distribution. It can be seen from Figure 1 that only two sets of data are located in the energy region of BBN interest and they show huge differences. The existing three sets of data at E > 400 keV show large discrepancies, even the positions of the resonance peaks at 600 and 800 keV are inconsistent for different data. Therefore, it seems difficult to obtain a reliable S-factor and corresponding uncertainties by fitting or analyzing the five sets of measured cross section data mentioned above. In Figure 2, we break down the contributions from the direct component and various resonances for our calculated astrophysical S-factor for the ⁷Li(d,n)2⁴He reaction. The S-factor gets contributions from the direct term and various resonances, which are displayed separately. The dashed–dotted blue line is due to the central subthreshold component. The dashed purple line is for the direct contribution. The solid red and dotted orange curves are the 5/2⁻ resonances at E_x = 17.298 MeV (E_r = 600 keV) and the 7/2⁻ resonance at E_x = 17.493 MeV (E_r = 800 keV), respectively. Here, we did not consider the 1/2⁻ resonance at E_x = 16.9752 MeV (E_r = 280 keV) since its width is too narrow to make a noticeable contribution to the final results.

**Figure 1.** The obtained astrophysical S-factor of the ⁷Li(d,n)2⁴He reaction as a function of energy corresponds to the solid blue line, while the dashed dark red and dark blue lines signify the upper limits and lower limits, respectively. The different data sets from NNDC marked by solid circles, triangles points up, triangles points down, diamonds, squares correspond to the data from the following sources (in order): Slattery et al. (1957), Hofstee et al. (2001), Sabourov et al. (2006), Bochkarev et al. (1994), and Nussbaum (1969). An enlarged view of the energy range from 0.75−0.85 MeV is shown in the inserted panel.
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**Figure 2.** The obtained astrophysical S-factor of the ⁷Li(d,n)2⁴He reaction is shown by the solid green line. The S-factor gets contributions from nonresonant and various resonances displayed separately. The dashed–dotted line is due to the central subthreshold component. The dashed purple line is for the direct component, while the solid red and dotted orange curves are for the 600 and 800 keV resonances, respectively.
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The S-factor obtained above is then inserted into the reaction rate expression to calculate the total reaction rate of ⁷Li(d,n)2⁴He as shown by the green solid line in the upper panel of Figure 3. We also break down our newly obtained total reaction rate in terms of the separate contributions from direct and various resonances. It can clearly be seen in the lower panel of Figure 3 that the reaction rate is dominated by the nonresonant contributions from the direct reaction and subthreshold resonances, rather than by the 600 keV resonance asserted in the previous evaluation from BM93. In particular, the contribution from the −24.9 keV subthreshold resonance dominates for temperatures lower than 0.07 GK. For temperatures T > 0.07 GK, it is mainly contributed by direct reaction. Around the temperature of 1.6 GK, the 600 keV (5/2⁻) broad resonance contributions is comparable with those from direct components. The contribution from the 800 keV resonance can basically be neglected. Here, the narrow resonance at E_r = 280 keV is neglected in our calculation since its contribution is even smaller than that from the 800 keV resonance.

**Figure 3.** Upper panel: The new reaction rate of ⁷Li(d,n)2⁴He as a function of temperature in units of GK is shown by the green solid line. The rate contributions from direct reaction and various resonances are displayed separately. The dashed–dotted line shows the subthreshold resonance (E_r = −25 keV). The solid red and dotted orange lines correspond to the resonances at E_r = 600 and 800 keV, respectively. The dashed purple line indicates the direct component. Lower panel: This figure shows the contribution ratio from separate components to the total reaction rate.
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The uncertainty of the S-factor shown in Figure 1 is also used to calculate the uncertainty for the ⁷Li(d,n)2⁴He reaction rates, as shown in Figure 4 by the green shaded band. For convenience of comparison, the old rates from previous works are also included in Figure 4, where the blue solid line is for the BM93 rate and the red line is for the CF88 rate. It can clearly be seen from Figure 4 that our new rates, including its upper and lower limits, are overall smaller than the previous two evaluations. In particular, for temperatures in the range of BBN importance (up to about 1 GK), our new rate is about 60 times smaller than the rate widely used in current BBN simulations given by BM93, which strongly motivates us to explore its impact on the production of ⁷Li during BBN.

For the sake of convenience for its use by others, the present rate in units of cm³ s⁻¹ mol⁻¹ can be well fitted (less than 0.11% error in 0.01–10 GK) by the following analytic expression in the standard seven parameter format of REACLIB:

$\begin{eqnarray}&&\begin{array}{l}{N}_{{\rm{A}}}\left\langle \sigma \,v\right\rangle =\exp (45.3213+0.180629{T}_{9}^{-1}-16.8231{T}_{9}^{-1/3}\\ \quad -14.9337{T}_{9}^{1/3}+1.22317{T}_{9}-0.0685717{T}_{9}^{5/3}\\ \quad +1.80904\mathrm{ln}({T}_{9})+\exp (0.410313+0.0129319{T}_{9}^{-1}\\ \quad -14.3153{T}_{9}^{-1/3}+36.1545{T}_{9}^{1/3}-9.83075{T}_{9}\\ \quad -0.445434{T}_{9}^{5/3}-8.38412\mathrm{ln}({T}_{9})).\end{array}\end{eqnarray} \tag{ 9 }$

Here, T₉ indicates temperature in units of 10⁹ K. The corresponding uncertainties in the new thermonuclear reaction rate are less than 30% in the whole temperature range from 0.01−10 GK.

4. BBN Calculation

Using the new ⁷Li(d,n)2⁴He reaction rate, we perform a BBN simulation to investigate its effect on the primordial ⁷Li abundance by using a modified Wagoner code (Wagoner 1969) with updated reaction rates. In our calculation, we choose the most up-to-date baryon-to-photon ratio η₁₀ = 6.104(η₁₀ = η × 10¹⁰) (Planck collaboration 2020) and the newest world average for the neutron lifetime (τ = 879.4 ± 0.6 s) from Fields et al. (2020) based on the recent recommendation of Particle Data Group (2018). The predicted light element abundances are shown in Table 3. It is found that the adoption of the new rate hardly results in any remarkable change in the final ⁷Li yield. In particular, the abundance of ⁷Li only increases by about 0.002%. The present nuclear uncertainties associated with the ⁷Li+d channel have no impact on ⁷Li nucleosynthesis. The reason for such a low effect is that the direct production of ⁷Li dominates at low baryon abundances (η₁₀ ≤ 3), whereas the direct production of ⁷Be dominates at higher baryon abundances (η₁₀ ≥ 3), as is remarked in Steigman (2007). In other words, the alteration of the direct production of ⁷Li caused by the new ⁷Li(d,n)2⁴He rate can be completely neglected in regions of high baryon density.

Table 3. The Predicted Abundances of Primordial Light Elements from the Standard BBN Model for the Old and Our New ⁷Li(d,n)2⁴He Reaction Rate

	D/H	³He/H	⁴He	⁶Li/H	⁷Li/H	⁷Be/H
Old	2.567 × 10⁻⁵	1.019 × 10⁻⁵	0.247	1.114 × 10⁻¹⁴	3.119 × 10⁻¹¹	4.539 × 10⁻¹⁰
New	2.567 × 10⁻⁵	1.019 × 10⁻⁵	0.247	1.114 × 10⁻¹⁴	3.120 × 10⁻¹¹	4.539 × 10⁻¹⁰

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With this in mind, it is worthwhile to consider alternative BBN models that can produce low baryon density regions (Rauscher et al. 1994; Orito et al. 1997). One appropriate candidate where we would expect the ⁷Li+d reaction to have an impact is that of an inhomogeneous density distribution at the time of big bang nucleosynthesis (IBBN), in which ⁷Li(d,n)2⁴He probably plays a pronounced role. We adopt the same IBBN model as that in Orito et al. (1997), where the model is parameterized by the density contrast between the high and low-density regions R; the volume fraction of high-density region f_v; the distance scale of the inhomogeneity r, in addition to the average baryon-to-photon ratio η and fluctuation geometry. The present calculation is performed in a cylindrical shell geometry, where the same set of model parameters (R, f_v, r) as in Orito et al. (1997) are used to characterize the density nonuniformity, where R = 10⁶, f_v = 0.15 and r = 10⁶ m, respectively, as an illustrative example. These parameter values are chosen such that the observed constraints on light elemental abundances, except for A = 7, are maximally satisfied. However, for the parameters η, τ_n, and N_ν, the same values are adopted as those used in homogeneous big bang nucleosynthesis (HBBN). The fluctuations are divided into 16 zones of variable width as described in Mathews et al. (1990), where the baryon density of the zone (z_i) increases with the zone number i from 1−16. The relevant thermonuclear reaction rates are the same as those in the HBBN simulation.

First, we investigate how it affects ⁷Li production in local regions, and two extreme cases are chosen: a high-density zone and a low-density zone. Table 4 shows the predicted abundances of primordial nuclides for the low-density region in the columns labeled zone1 and those for the high-density region (zone16), respectively. We only show the results for ⁷Li, ⁷Be, ⁹Be, and ¹⁰B, nuclides A = 11 and A ≥ 12 (marked as 12 m in Table 4) since our new ⁷Li(d,n)2⁴He rate has no impact on the production of other primordial isotopes. It can be seen from zone1 in Table 4 that the adoption of our new ⁷Li(d,n)2⁴He rate increases the abundances of ⁷Li, ⁹Be, and ¹⁰B, nuclides A = 11 and A ≥ 12 by about a factor of 2, compared to the abundances obtained using the old rate. The reason for the growth of ⁷Li can be attributed to a smaller ⁷Li(d,n)2⁴He rate with a weaker capability of ⁷Li destruction, resulting in bigger ⁷Li production. Likewise, more ⁷Li will regulate the reactions flows moving toward the direction of synthesizing more nuclei with A > 7. This explains why the abundances of almost all of the light nuclides heavier than ⁷Li are increased. In comparison to the other extreme case, these features exhibited in the low-density zone will disappear in the high-density zone (zone16), as shown clearly in Table 4. This is owing to the fact that the production of ⁷Be dominates that of ⁷Li in the high-density regions, and therefore the production of A > 7 nuclei is mainly through ⁷Be involved reactions rather than ⁷Li reactions. Thus, the impact from the variation of the ⁷Li(d,n)2⁴He rate can be totally neglected in a high-density region. In order to assess its net impact on the final yields of primordial nuclei, a weighted average of abundances is calculated over the 16 zones for the entire fluctuation region, shown in the column labeled "Average" in Table 4. The results show that the final ⁷Li abundance (⁷Li/H+⁷Be/H) increases by 10% and the abundances of light nuclides with A > 7 also increase by about 40% if we assume an inhomogeneous density distribution during the epoch of BBN.

Table 4. The Predicted Abundances of the Primordial Light Elements for Our IBBN Model with R = 10⁶, f_v = 0.15, and r = 10⁶ m

Nuclide	zone1				zone16				Average
	Y_o	Y_n	m	$\tfrac{{Y}_{n}-{Y}_{o}}{{Y}_{o}}$	Y_o	Y_n	m	$\tfrac{{Y}_{n}-{Y}_{o}}{{Y}_{o}}$	Y_o	Y_n	m	$\tfrac{{Y}_{n}-{Y}_{o}}{{Y}_{o}}$
⁷Li/H(×10^−m)	1.82	2.82	9	55%	3.95	3.96	11	0.25%	1.26	1.70	10	35%
⁷Be/H(×10^−m)	9.38	9.38	12	0%	7.38	7.38	10	0.0%	2.77	2.77	10	0%
⁹Be/H(×10^−m)	1.46	2.28	14	56%	5.30	5.40	18	1.9%	0.71	1.02	15	44%
¹⁰B/H(×10^−m)	2.59	4.09	17	58%	4.46	4.52	21	1.3%	1.34	1.94	18	45%
¹¹X/H(×10^−m)	1.36	2.13	16	57%	4.09	4.09	17	0.0%	2.67	2.96	17	11%
^12mX/H(×10^−m)	4.84	13.0	16	168%	3.48	4.34	15	24%	6.89	9.38	15	36%

Note. Y_o refers to the abundance of the specific nuclide obtained using the ⁷Li(d,n)2⁴He rate from BM93, while Y_n is the nuclide's abundance using our new ⁷Li(d,n)2⁴He rate. m denotes the exponent in the power of 10.

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5. Conclusion

Starting from recent experimental measurements on low energy excited states of the mirror nuclei pair ⁹Be and ⁹B, we make a comprehensive analysis of the nature of near deuteron-threshold resonant states of ⁹Be. For this reason, it is important to reevaluate the ⁷Li(d,n)2⁴He reaction rate since it plays a pivotal role in the destruction of ⁷Li during BBN. For the first time, we present the experimentally constrained uncertainties associated with this important reaction rate. It is shown that both our newly obtained reaction rate and corresponding uncertainties show a remarkable departure from earlier evaluations. In particular, our rate is a factor of 60 times smaller than the most widely used rate (BM93) in current BBN simulations. The cross section breakdown shows that the subthreshold resonance omitted from previous evaluations dominates in the temperature range below T₉ = 0.07, while the rate for T₉ > 0.07 is mainly determined by direct reaction, not by the 600 keV resonance previously thought in BM93. In order to figure out the impact of this new reaction rate, we perform simulations using two different BBN models: a uniform density model and a nonuniform density model. The results obtained can be summarized by the following points: the adoption of the new ⁷Li(d,n)2⁴He rate increases by 0.002% in the final ⁷Li yield for standard BBN models. However, for models of inhomogeneous density distribution, it can lead to about a 10% increase in ⁷Li production and a 40% increase in the final primordial abundances of light nuclei with A > 7 compared to calculations using old reaction rates. Such an increase is due to the impact in the low-density zones, where the ⁷Li yields increases by a factor of 1.55. Therefore, our results confirm the existence of the cosmological lithium problem.

We are grateful to R.N. Boyd for helpful discussions and advice. We also thank C.X. Yuan for his shell model calculation on the properties of ⁹Be excited states. This work was financially supported by the Strategic Priority Research Program of Chinese Academy of Sciences under grant No. XDB34020204, and the Youth Innovation Promotion Association of Chinese Academy of Sciences under grant No. 2019406, and the National Natural Science Foundation of China under grant Nos. 11705244, 11490562, and 11961141004, and in part by the National Science Foundation under grant No. OISE-1927130 (IReNA) and Grants-in-Aid for Scientific Research of The Japan Society for the Promotion of Science (20K03958, 17K05459). M.P. acknowledges the support of NuGrid from STFC (through the University of Hull's Consolidated Grant ST/R000840/1), and access to viper, the University of Hull High Performance Computing Facility. M.P. acknowledges the support from the "Lendulet-2014" Program of the Hungarian Academy of Sciences (Hungary), from the ERC Consolidator Grant (Hungary) funding scheme (Project RADIOSTAR, G.A. No. 724560), by the National Science Foundation (NSF, USA) under grant No. PHY-1430152 (JINA Center for the Evolution of the Elements). M.P. also thanks the UK network BRIDGCE, the ChETEC COST Action (CA16117) supported by COST (European Cooperation in Science and Technology), and the ChETEC-INFRA project funded from the European Union's Horizon 2020 research and innovation programme under grant agreement No 101008324. C.A.B. acknowledges support from the U.S. DOE grant No. DE-FG02-08ER41533 and funding contributed by the LANL Collaborative Research Program by the Texas A&M System National Laboratory Office. Y.L. acknowledges the support from JSPS KAKENHI grant No. 19J22167.

New Thermonuclear Rate of ⁷Li(d,n)2⁴He Relevant to the Cosmological Lithium Problem

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Abstract

1. Introduction