$^{3}{\rm He}(\alpha, \gamma)^{7}{\rm Be}$ and $^{3}{\rm H}(\alpha,\gamma)^{7}{\rm Li}$ reaction rates and the implication for Big Bang nucleosynthesis in the potential model

TL;DR

This study estimates reaction rates for key nuclear processes involving helium and lithium isotopes using a potential model, providing results consistent with recent measurements and implications for Big Bang nucleosynthesis.

Contribution

The paper introduces a two-body potential model to estimate reaction rates and lithium abundance, aligning theoretical predictions with recent experimental data.

Findings

Estimated $^{7}{ m Li/H}$ abundance matches recent measurements.

Reaction rates are consistent with observational data.

Implications for Big Bang nucleosynthesis models.

Abstract

The reaction rates of the direct astrophysical capture processes $^{3}{\rm He}(\alpha, \gamma)^{7}{\rm Be}$ and $^{3}{\rm H}(\alpha,\gamma)^{7}{\rm Li}$, as well as the abundance of the $^{7}{\rm Li}$ element are estimated in the framework of a two-body potential model. The estimated $^{7}{\rm Li/H}$ abundance ratio of $^{7}{\rm Li/H}=(5.07\pm 0.14 )\times 10^{-10}$ is in a very good agreement with the recent measurement $^{7}{\rm Li/H}=(5.0\pm 0.3) \times 10^{-10}$ of the LUNA collaboration.

Tables

Table 1. Table 1: Theoretical estimations of the direct He 3 ​ ( α , γ ) 7 ​ Be superscript He 3 superscript 𝛼 𝛾 7 Be {}^{3}{\rm He}(\alpha,\gamma)^{7}{\rm Be} capture reaction rate in the temperature interval 10 6 superscript 10 6 10^{6} K ≤ T ≤ 10 9 absent 𝑇 superscript 10 9 \leq T\leq 10^{9} K ( 0.001 ≤ T 9 ≤ 1 0.001 subscript 𝑇 9 1 0.001\leq T_{9}\leq 1 )

$T_9$	$V_{M1}^a$	$V_D^a$	$T_9$	$V_{M1}^a$	$V_D^a$
0.001	$9.545\times {10}^{-48}$	$9.358\times {10}^{-48}$	0.070	$9.741\times {10}^{-7}$	$9.648\times {10}^{-7}$
0.002	$1.947\times {10}^{-36}$	$1.909\times {10}^{-36}$	0.080	$3.445\times {10}^{-6}$	$3.416\times {10}^{-6}$
0.003	$5.891\times {10}^{-31}$	$5.778\times {10}^{-31}$	0.090	$9.991\times {10}^{-6}$	$9.916\times {10}^{-6}$
0.004	$1.675\times {10}^{-27}$	$1.643\times {10}^{-27}$	0.100	$2.493\times {10}^{-5}$	$2.477\times {10}^{-5}$
0.005	$4.771\times {10}^{-25}$	$4.682\times {10}^{-25}$	0.110	$5.535\times {10}^{-5}$	$5.504\times {10}^{-5}$
0.006	$3.543\times {10}^{-23}$	$3.478\times {10}^{-23}$	0.120	$1.119\times {10}^{-4}$	$1.114\times {10}^{-4}$
0.007	$1.102\times {10}^{-21}$	$1.082\times {10}^{-21}$	0.130	$2.098\times {10}^{-4}$	$2.090\times {10}^{-4}$
0.008	$1.876\times {10}^{-20}$	$1.842\times {10}^{-20}$	0.140	$3.692\times {10}^{-4}$	$3.680\times {10}^{-4}$
0.009	$2.055\times {10}^{-19}$	$2.019\times {10}^{-19}$	0.150	$6.162\times {10}^{-4}$	$6.149\times {10}^{-4}$
0.010	$1.613\times {10}^{-18}$	$1.585\times {10}^{-18}$	0.160	$9.836\times {10}^{-4}$	$9.822\times {10}^{-4}$
0.011	$9.762\times {10}^{-18}$	$9.594\times {10}^{-18}$	0.180	$2.244\times {10}^{-3}$	$2.244\times {10}^{-3}$
0.012	$4.800\times {10}^{-17}$	$4.718\times {10}^{-17}$	0.200	$4.551\times {10}^{-3}$	$4.559\times {10}^{-3}$
0.013	$1.992\times {10}^{-16}$	$1.959\times {10}^{-16}$	0.250	$1.860\times {10}^{-2}$	$1.869\times {10}^{-2}$
0.014	$7.188\times {10}^{-16}$	$7.068\times {10}^{-16}$	0.300	$5.386\times {10}^{-2}$	$5.431\times {10}^{-2}$
0.015	$2.305\times {10}^{-15}$	$2.267\times {10}^{-15}$	0.350	$1.250\times {10}^{-1}$	$1.264\times {10}^{-1}$
0.016	$6.686\times {10}^{-15}$	$6.577\times {10}^{-15}$	0.400	$2.487\times {10}^{-1}$	$2.523\times {10}^{-1}$
0.018	$4.395\times {10}^{-14}$	$4.325\times {10}^{-14}$	0.450	$4.430\times {10}^{-1}$	$4.504\times {10}^{-1}$
0.020	$2.221\times {10}^{-13}$	$2.186\times {10}^{-13}$	0.500	$7.254\times {10}^{-1}$	$7.395\times {10}^{-1}$
0.025	$5.676\times {10}^{-12}$	$5.591\times {10}^{-12}$	0.600	$1.621\times {10}^0$	$1.660\times {10}^0$
0.030	$6.684\times {10}^{-11}$	$6.589\times {10}^{-11}$	0.700	$3.054\times {10}^0$	$3.140\times {10}^0$
0.040	$2.405\times {10}^{-9}$	$2.374\times {10}^{-9}$	0.800	$5.115\times {10}^0$	$5.278\times {10}^0$
0.050	$3.045\times {10}^{-8}$	$3.010\times {10}^{-8}$	0.900	$7.869\times {10}^0$	$8.145\times {10}^0$
0.060	$2.097\times {10}^{-7}$	$2.075\times {10}^{-7}$	1.000	$1.136\times {10}^1$	$1.179\times {10}^1$

Table 2. Table 2: Fitted values of the coefficients of analytical approximation for the direct capture reaction He 3 ​ ( α , γ ) 7 ​ Be superscript He 3 superscript 𝛼 𝛾 7 Be {}^{3}{\rm He}(\alpha,\gamma)^{7}{\rm Be}

Model	$p_0$	$p_1$	$p_2$	$p_3$	$p_4$	$p_5$	$p_6$
$V_{M1}^a$	$2.691\times {10}^6$	8.140	-26.747	43.336	-35.678	11.504	438.432
$V_D^a$	$2.636\times {10}^6$	8.128	-26.559	43.243	-35.608	11.445	453.245

Table 3. Table 3: Theoretical estimations of the direct H 3 ​ ( α , γ ) 7 ​ Li superscript H 3 superscript 𝛼 𝛾 7 Li {}^{3}{\rm H}(\alpha,\gamma)^{7}{\rm Li} capture reaction rate in the temperature interval 10 6 superscript 10 6 10^{6} K ≤ T ≤ 10 9 absent 𝑇 superscript 10 9 \leq T\leq 10^{9} K ( 0.001 ≤ T 9 ≤ 1 0.001 subscript 𝑇 9 1 0.001\leq T_{9}\leq 1 )

$T_9$	$V_{M1}^a$	$V_D^a$	$T_9$	$V_{M1}^a$	$V_D^a$
0.001	$5.594\times {10}^{-28}$	$6.129\times {10}^{-28}$	0.070	$1.326\times {10}^{-2}$	$1.461\times {10}^{-2}$
0.002	$6.284\times {10}^{-21}$	$6.886\times {10}^{-21}$	0.080	$2.838\times {10}^{-2}$	$3.129\times {10}^{-2}$
0.003	$1.612\times {10}^{-17}$	$1.767\times {10}^{-17}$	0.090	$5.382\times {10}^{-2}$	$5.936\times {10}^{-2}$
0.004	$2.251\times {10}^{-15}$	$2.468\times {10}^{-15}$	0.100	$9.313\times {10}^{-2}$	$1.028\times {10}^{-1}$
0.005	$7.496\times {10}^{-14}$	$8.218\times {10}^{-14}$	0.110	$1.501\times {10}^{-1}$	$1.657\times {10}^{-1}$
0.006	$1.081\times {10}^{-12}$	$1.185\times {10}^{-12}$	0.120	$2.285\times {10}^{-1}$	$2.524\times {10}^{-1}$
0.007	$9.073\times {10}^{-12}$	$9.949\times {10}^{-12}$	0.130	$3.322\times {10}^{-1}$	$3.671\times {10}^{-1}$
0.008	$5.233\times {10}^{-11}$	$5.739\times {10}^{-11}$	0.140	$4.648\times {10}^{-1}$	$5.139\times {10}^{-1}$
0.009	$2.296\times {10}^{-10}$	$2.519\times {10}^{-10}$	0.150	$6.300\times {10}^{-1}$	$6.968\times {10}^{-1}$
0.010	$8.193\times {10}^{-10}$	$8.987\times {10}^{-10}$	0.160	$8.310\times {10}^{-1}$	$9.195\times {10}^{-1}$
0.011	$2.487\times {10}^{-9}$	$2.729\times {10}^{-9}$	0.180	$1.352\times {10}^0$	$1.498\times {10}^0$
0.012	$6.639\times {10}^{-9}$	$7.284\times {10}^{-9}$	0.200	$2.050\times {10}^0$	$2.272\times {10}^0$
0.013	$1.595\times {10}^{-8}$	$1.751\times {10}^{-8}$	0.250	$4.672\times {10}^0$	$5.186\times {10}^0$
0.014	$3.515\times {10}^{-8}$	$3.857\times {10}^{-8}$	0.300	$8.661\times {10}^0$	$9.629\times {10}^0$
0.015	$7.199\times {10}^{-8}$	$7.901\times {10}^{-8}$	0.350	$1.407\times {10}^1$	$1.566\times {10}^1$
0.016	$1.386\times {10}^{-7}$	$1.521\times {10}^{-7}$	0.400	$2.087\times {10}^1$	$2.326\times {10}^1$
0.018	$4.407\times {10}^{-7}$	$4.838\times {10}^{-7}$	0.450	$2.899\times {10}^1$	$3.236\times {10}^1$
0.020	$1.191\times {10}^{-6}$	$1.308\times {10}^{-6}$	0.500	$3.834\times {10}^1$	$4.284\times {10}^1$
0.025	$8.677\times {10}^{-6}$	$9.531\times {10}^{-6}$	0.600	$6.028\times {10}^1$	$6.750\times {10}^1$
0.030	$3.919\times {10}^{-5}$	$4.307\times {10}^{-5}$	0.700	$8.585\times {10}^1$	$9.631\times {10}^1$
0.040	$3.483\times {10}^{-4}$	$3.831\times {10}^{-4}$	0.800	$1.143\times {10}^2$	$1.284\times {10}^2$
0.050	$1.629\times {10}^{-3}$	$1.793\times {10}^{-3}$	0.900	$1.450\times {10}^2$	$1.631\times {10}^2$
0.060	$5.242\times {10}^{-3}$	$5.773\times {10}^{-3}$	1.000	$1.773\times {10}^2$	$1.998\times {10}^2$

Table 4. Table 4: Fitted values of the coefficients of analytical approximation for the direct capture reaction H 3 ​ ( α , γ ) 7 ​ Li superscript H 3 superscript 𝛼 𝛾 7 Li {}^{3}{\rm H}(\alpha,\gamma)^{7}{\rm Li}

Model	$p_0$	$p_1$	$p_2$	$p_3$	$p_4$	$p_5$	$p_6$
$V_{M1}^a$	$4.951\times {10}^5$	4.034	-13.145	20.845	-17.355	5.765	39.687
$V_D^a$	$5.429\times {10}^5$	4.012	-13.001	20.703	-17.285	5.747	41.245

Equations

$$N_A(\sigma v)=N_A\frac{(8/\pi {)}^{1/2}}{{\mu }^{1/2}(k_{\text{B}}T{)}^{3/2}}{\int }_0^{\infty }\sigma (E)E\exp (-E/k_{\text{B}}T)dE,$$

$$N_A(\sigma v)=3.7313\times 10^{10}{A}^{-1/2}{T}_9^{-3/2}{\int }_0^{\infty }\sigma (E)E\exp (-11.605E/T_9)dE.$$

$$N_A(\sigma v)=p_0{T}_9^{-2/3}\exp (-C_0{T}_9^{-1/3})\times (1+p_1{T}_9^{1/3}+p_2{T}_9^{2/3}+$$

$$+p_3{T}_9+p_4{T}_9^{4/3}+p_5{T}_9^{5/3})+p_6{T}_9^{-3/2}\exp (-C_{01}{T}_9^{-1}).$$

Full text

Radiative capture; astrophysical S factor; potential model.

I Introduction

The radiative capture reactions ${}^{3}{\rm He}(\alpha,\gamma)^{7}{\rm Be}$ and ${}^{3}{\rm H}(\alpha,\gamma)^{7}{\rm Li}$ play an important role in stellar and primordial nucleosynthesis adelber11 ; fields11 . A quantity of formation of the ${}^{7}{\rm Li}$ isotope after Big Bang (the cosmological lithiumprimordial abundance) is evaluated from the reaction rates of above processes. In last years the problem was discussed extensively from the both experimental and theoretical viewpoints coc17 ; cyburt16 . As a rule, experimental measurements of these reactions in low-energy region are inaccessible due to strong Coulomb repulsion. However, very recently in Ref.takacs15 the ${}^{3}{\rm He}(\alpha,\gamma)^{7}{\rm Be}$ astrophysical S-factor was determined by the indirect method at the Gamow peak energy region. Astrophysical S-factor $S_{34}$(23${}^{+6}_{-5}$ keV)=0.548$\pm$0.054 keV b was obtained by analyzing observed neutrino fluxes from the Sun within the standard solar model.

On the other hand, recent direct measurements of the LUNA collaboration LUNA6Li and theoretical development tur15 ; tur16 ; baye18 ; tur18 within the potential model have done an important step toward the correct estimation of the primordial abundance of the ${}^{6}{\rm Li}$ element.

The aim of present paper is to estimate the ${}^{3}{\rm He}(\alpha,\gamma)^{7}{\rm Be}$ and ${}^{3}{\rm H}(\alpha,\gamma)^{7}{\rm Li}$ reaction rates and to apply the obtained results for the evaluation of lithium primordial abundance in the framework of a two-body potential model tur181 . The first ${}^{3}{\rm He}(\alpha,\gamma)^{7}{\rm Be}$ direct capture process is mostly important for the primordial abundance of the ${}^{7}{\rm Li}$ element and the second ${}^{3}{\rm H}(\alpha,\gamma)^{7}{\rm Li}$ process yields a small additional contribution.

II Theoretical model

The reaction rate $N_{A}(\sigma v)$ is estimated with the help NACRE99 ; Fowler

[TABLE]

where $k_{\text{B}}$ is the Boltzmann coefficient, $T$ is the temperature, $N_{A}=6.0221\times 10^{23}\,\text{mol}^{-1}$ is the Avogadro number. The reduced mass $\mu=Am_{N}$ with a reduced mass number $A=A_{1}A_{2}/(A_{1}+A_{2})$ for the 3He$+\alpha$ and 3H$+\alpha$ systems with $A_{1}=3$ and $A_{2}=4$, consequently a value of $A=12/7$ is fixed. When a variable $k_{\text{B}}T$ is expressed in units of MeV it is convenient to use a variable $T_{9}$ for the temperature in units of $10^{9}$ K according to the equation $k_{\text{B}}T=T_{9}/11.605$ MeV. In our calculations $T_{9}$ varies in the interval $0.001\leq T_{9}\leq 1$.

After substitution of these variables the above integral for the reaction rates can be expressed as:

[TABLE]

III Estimation of reaction rates for the ${}^{3}{\rm He}(\alpha,\gamma)^{7}{\rm Be}$ process

In Table 1 we give theoretical estimations for the ${}^{3}{\rm He}(\alpha,\gamma)^{7}{\rm Be}$ reaction rates in the temperature interval $10^{6}$ K $\leq T\leq 10^{9}$ K ($0.001\leq T_{9}\leq 1$) calculated within two potential models $V_{M1}^{a}$ and $V_{D}^{a}$. Parameters of these models have been given in Ref.tur181 . They yield a good description of the astrophysical S-factor and cross section for the both ${}^{3}{\rm He}(\alpha,\gamma)^{7}{\rm Be}$ and ${}^{3}{\rm H}(\alpha,\gamma)^{7}{\rm Li}$ direct capture reactions.

In Fig. 1 we display the estimated reaction rates of the direct ${}^{3}{\rm He}(\alpha,\gamma)^{7}{\rm Be}$ capture process within the models $V_{M1}^{a}$ and $V_{D}^{a}$ normalized to the standard NACRE 1999 experimental data NACRE99 . For the comparison we also display the lines corresponding to the results of Refs.takacs15 ; kontos13 ; desc04 and new NACRE II 2013 data NACRE13 . As can be seen from the figure, our results obtained within the models $V_{M1}^{a}$ and $V_{D}^{a}$ are consistent with other models.

For the estimation of the primordial abundance of the 7Li element we use the well known PArthENoPE Pisanti08 public code, which however operates only with an analytical form of the reaction rate dependence on the temperature $T_{9}$. In this way the theoretical reaction rate in Table 1 is approximated with parameters in Table 2 within 1.02% (Model $V_{M1}^{a}$) and 0.98% (Model $V_{D}^{a}$) tur181 using the following analytical formula:

[TABLE]

The coefficients of the analytical polynomial approximation of the ${}^{3}{\rm He}(\alpha,\gamma)^{7}{\rm Be}$ reaction rates estimated with the two different potential models $V_{M1}^{a}$ and $V_{D}^{a}$ are given in Table 2 in the temperature interval ($0.001\leq T_{9}\leq 1$). For this process $C_{0}=12.813$ and $C_{01}=15.889$.

On the basis of the theoretical reaction rates of ${}^{3}{\rm He}(\alpha,\gamma)^{7}{\rm Be}$ capture and with the help of the PArthENoPE Pisanti08 public code we have estimated the primordial abundance of the 7Li element. If we adopt the Planck 2015 best fit for the baryon density parameter $\Omega_{b}h^{2}=0.02229^{+0.00029}_{-0.00027}$ ade16 and the neutron life time $\tau_{n}=880.3\pm 1.1$ s olive14 , for the 7Li/H abundance ratio we have an estimate $(4.81\pm 0.12)\times 10^{-10}$ within model $V_{M1}^{a}$. Model $V_{D}^{a}$ yields an estimate $(4.92\pm 0.13)\times 10^{-10}$. As we can see below, these numbers slightly increase when including a contribution of the ${}^{3}{\rm H}(\alpha,\gamma)^{7}{\rm Li}$ direct capture reaction to the 7Li/H abundance ratio.

IV Estimation of reaction rates for the ${}^{3}{\rm H}(\alpha,\gamma)^{7}{\rm Li}$ process

In Table 3 we give theoretical estimations for the ${}^{3}{\rm H}(\alpha,\gamma)^{7}{\rm Li}$ reaction rates in the temperature interval $10^{6}$ K $\leq T\leq 10^{9}$ K ($0.001\leq T_{9}\leq 1$) calculated with the same potential models $V_{M1}^{a}$ and $V_{D}^{a}$ which were used in the ${}^{3}{\rm He}(\alpha,\gamma)^{7}{\rm Be}$ capture process.

In Fig. 2 we display the estimated reaction rates of the direct ${}^{3}{\rm H}(\alpha,\gamma)^{7}{\rm Li}$ capture process within the models $V_{M1}^{a}$ and $V_{D}^{a}$ normalized to the standard NACRE 1999 experimental data NACRE99 . For the comparison we also display the lines corresponding to the results of Ref.desc04 and new NACRE II 2013 data NACRE13 .

The coefficients of the analytical polynomial approximation of the ${}^{3}{\rm H}(\alpha,\gamma)^{7}{\rm Li}$ reaction rates estimated with the potential models $V_{M1}^{a}$ and $V_{D}^{a}$ are given in Table 4 in the temperature interval ($0.001\leq T_{9}\leq 1$). For this process $C_{0}=8.072$ and $C_{01}=3.689$. In this case the analytical formula (3) with parameters in Table 4 reproduces the theoretical reaction rates from the Table 3 within 0.61% (Model $V_{M1}^{a}$) and 0.62% (Model $V_{D}^{a}$), respectively.

In Fig. 2 we display the estimated reaction rates of the direct ${}^{3}{\rm H}(\alpha,\gamma)^{7}{\rm Li}$ capture process within the models $V_{M1}^{a}$ and $V_{D}^{a}$ normalized to the standard NACRE 1999 experimental data NACRE99 . For the comparison we also display the lines corresponding to the Ref.desc04 and new NACRE II 2013 data NACRE13 .

Now including theoretical reaction rates for the both ${}^{3}{\rm He}(\alpha,\gamma)^{7}{\rm Be}$ and ${}^{3}{\rm H}(\alpha,\gamma)^{7}{\rm Li}$ capture processes into the nuclear reaction network with the help of the PArthENoPE Pisanti08 public code, we can evaluate the primordial abundance of the 7Li element. Adopting aforementioned values of the baryon density and the neutron life time, for the 7Li/H abundance ratio we have an estimate $(5.06\pm 0.13)\times 10^{-10}$ within model $V_{M1}^{a}$. Model $V_{D}^{a}$ yields an estimation $(5.08\pm 0.13)\times 10^{-10}$. These numbers are slightly larger than the corresponding estimations based on the ${}^{3}{\rm He}(\alpha,\gamma)^{7}{\rm Be}$ process exclusively.

V Conclusion

The astrophysical direct capture processes ${}^{3}{\rm He}(\alpha,\gamma)^{7}{\rm Be}$ and ${}^{3}{\rm H}(\alpha,\gamma)^{7}{\rm Li}$ have been studied in the potential model. The reaction rates and primordial abundance of the 7Li element have been evaluated. It is shown, that the main contribution to the formation of the 7Li isotope comes from reaction rates of the ${}^{3}{\rm He}(\alpha,\gamma)^{7}{\rm Be}$ direct capture process. Additional contribution is due to the ${}^{3}{\rm H}(\alpha,\gamma)^{7}{\rm Li}$ direct capture reaction. For the abundance ratio 7Li/H we have obtained an estimate $(5.07\pm 0.14)\times 10^{-10}$ consistent with the new data $(5.0\pm 0.3)\times 10^{-10}$ of the LUNA collaboration takacs15 .