Analysis of the 3 He(α,γ ) 7 Be and 3 H(α,γ ) 7 Li astrophysical direct capture reactions in a modified potential-model approach
aa r X i v : . [ nu c l - t h ] J u l Analysis of the He( α, γ ) Be and H( α, γ ) Li astrophysical directcapture reactions in a modified potential-model approach E.M. Tursunov, ∗ S.A. Turakulov, † and A.S. Kadyrov ‡ Institute of Nuclear Physics, Academy of Sciences,100214, Ulugbek, Tashkent, Uzbekistan Curtin Institute for Computation and Department of Physics and Astronomy,Curtin University, GPO Box U1987, Perth, WA 6845, Australia
Abstract
Astrophysical S factors and reaction rates of the direct radiative capture processes He( α, γ ) Beand H( α, γ ) Li, as well as the primordial abundance of the Li element, are estimated in theframework of a modified two-body potential model. It is shown that suitable modification ofphase-equivalent α − He potentials in the d waves can improve the description of the astrophysical S factor for the direct He( α, γ ) Be radiative capture reaction at energies above 0.5 MeV. Anestimated Li / H abundance ratio of (4 . ± . × − is in very good agreement with the recentmeasurement of (5 . ± . × − by the LUNA collaboration. PACS numbers: ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] . INTRODUCTION Realistic estimation of the primordial abundances of the lithium isotopes Li and Li, thetwo heaviest elements in Big Bang nucleosynthesis (BBN), is one of the most important andunsolved problems of nuclear astrophysics. The primordial abundances of these elementscan be extracted from an analysis of astronomical observations of old metal-poor halo stars.For the Li abundance astronomical data provide a value of Li / H=(1.58 +0 . − . ) × − [1]which is 2 to 4 times less than an estimate of Li / H=(4.68 ± × − of the BBN model[2]. On the other hand, there is the so-called second lithium problem which is related to theabundance ratios of the lithium isotopes. A recent analysis of the direct measurements dataof the LUNA collaboration yielded a value Li / Li = (1 . ± . × − [3] which is 3 ordersof magnitude lower than the astronomical observation [4]. These problems were subjects ofintense discussions during a recent topical workshop [5]. This demonstrates that they arestill far from being solved.An important question is whether or not the lithium problems originate from astronomyor nuclear physics. From one side a small primordial abundance of the Li element is welldescribed in nuclear physics from both experimental and theoretical perspectives. This ele-ment was mainly produced during the BBN epoch via the direct capture d ( α, γ ) Li process.Until recently the main problem in theoretical studies of this process was connected with aconsistent description of the isospin-forbidden E1 transition. Finally, results of theoreticalcalculations within the most realistic three-body model [6–9] are now in very good agreementwith the direct data of the LUNA collaboration [3, 10]. Good agreement was obtained forall observable of practical interest including astrophysical S factor, reaction rates and theprimordial abundance of the Li element. The absolute values and temperature dependenceof the reaction rates of the LUNA data have been reproduced with a good accuracy, whichwas a consequence of the correct treatment of the isospin-forbidden E1 transition in contrastto two-body models based on so-called exact-mass prescription [9, 11]. The calculated valueof (0 . ± . × − [8, 9] for the Li/H primordial abundance ratio is consistent withthe estimate (0 . ± . × − of the LUNA collaboration [3].The Li isotope was produced mainly through radiative capture reactions He( α, γ ) Beand H( α, γ ) Li during the BBN period [12]. These direct capture reactions play a significantrole also in stellar nucleosynthesis [13], as well as in the pp chain of solar hydrogen burning214]. The primordial abundance of the Li element is evaluated from the reaction rates of thetwo capture processes mentioned above. The Be nucleus is produced in the He( α, γ ) Bedirect capture process and subsequently decays through electron capture resulting in the Lielement. The H( α, γ ) Li process then gives a small additional contribution to the lithiumprimordial abundance.In recent years, the lithium abundance problem was discussed extensively from both ex-perimental and theoretical viewpoints [2, 15]. One has to note that experimental measure-ments of these reactions in low-energy region face serious difficulties due to strong Coulombrepulsion. Nevertheless, direct data for the astrophysical S factor of the He( α, γ ) Be cap-ture process at several energies around 100 keV were obtained by the LUNA collaboration inthe underground facility [16, 17]. Later, this data set was supplemented with a more accu-rate value of the astrophysical S factor at Gamow peak energy region, S (23 +6 − keV)=0.548 ± × − for the Li / H abundance, close to the standard BBN value and morethan three times larger than the astronomical data. Recently, the astrophysical S factor wasreevaluated at the solar Gamov energy peak and its value, S (23 +6 − keV)=0.590 ± He( α, γ ) Be was recently extended up to 4.5 MeV in the center-of-massframe energy [20, 21].Theoretically, the astrophysical capture processes He( α, γ ) Be and H( α, γ ) Li have beenstudied in potential [22–25] and microscopic models [26–29], a microscopic R-matrix ap-proach [30], as well as in a semimicroscopic phenomenological approach [31], a fermionicmolecular dynamics (FMD) method [32] and a no-core shell model with continuum (NC-SMC) [33, 34]. The most realistic microscopic approaches [29, 32–34] still have problemswith simultaneous description of the above mirror capture reactions, including the bothabsolute values and energy dependence of the astrophysical S factor.In Ref. [22] a realistic potential model was developed for the description of the capturereactions mentioned above. It was shown that the potential model is able to describe theastrophysical S factors at low energies, below 0.5 MeV, which include the BBN energyregion of E cm =180-400 keV, leading to good agreement with the experimental data [16–18]. However, the existing data sets at intermediate energies are underestimated and this3iscrepancy increases with the energy. An important question is, whether the potentialmodel can describe the astrophysical S factor of the direct capture processes He( α, γ ) Beand H( α, γ ) Li at low and intermediate energies simultaneously. Answering this questionmay have important implications for both nuclear theory and astrophysical applications.The aim of the present study is to describe the existing data for the astrophysical S factorsof the He( α, γ ) Be and H( α, γ ) Li direct capture reactions at low- and intermediate-energyregions and to estimate the reaction rates of these processes and the primordial abundanceof the Li element in the potential model. As it is known from the literature [22], thedipole E1-transition operator yields the main contribution to the above processes at low andintermediate energies. The E2 transition contributes only in the resonance energy regionnear 3 MeV in the center-of-mass frame. The M1 transition is even more suppressed andthis is the case at all energies.As it was shown in Ref. [22], below 0.5 MeV the main contribution to the E1 S factorcomes from the initial α + He and α + H s -wave scattering states. However, at intermediateenergies the role of the d -wave scattering states increases and their contribution becomesdominant beyond 2 MeV. On this basis it would be very useful to search for optimal d -wave α + He and α + H potentials, which would allow to better describe the astrophysical S factor data for the aforementioned capture reactions. In this way we perform an optimizationprocedure among phase equivalent α + He potentials in the partial d / and d / waves.The two-body Gaussian potentials [23] will be examined. In Ref. [22] the potentialparameters in the s wave were adjusted to reproduce the astrophysical S factor of the α + He direct capture reaction at low energies in addition to the phase shift data. In the p / and p / partial waves the potential parameters were additionally adjusted to reproducethe bound state properties: binding energies and the values of the asymptotic normalizationcoefficients (ANC) for the Be(3 / − ) ground and Be(3 / − ) excited states extracted fromthe analysis of the experimental data within the DWBA method [35].This article is organized as follows. In Section II the theoretical model will be brieflydescribed, Section III is devoted to the analysis of numerical results. Conclusions will bedrawn in the last section. 4 I. THEORETICAL MODEL
Astrophysical S factor of the radiation capture process is expressed in terms of the crosssection as [36] S ( E ) = E σ ( E ) exp(2 πη ) (1)where E is the collision energy in the center-of-mass (cm) frame and η is the Sommerfeldparameter. The cross section reads as [23, 36] σ ( E ) = X J f λ Ω σ J f λ (Ω) , (2)where Ω = E or M (electric or magnetic transition), λ is a multiplicity of the transition, J f is the total angular momentum of the final state. For a particular final state with totalmomentum J f and multiplicity λ we have σ J f λ (Ω) = X J (2 J f + 1)[ S ] [ S ] 32 π ( λ + 1)¯ hλ ([ λ ]!!) k λ +1 γ C ( S ) × X lS k i v i | h Ψ J f l f S k M Ω λ k Ψ JlS i | , (3)where Ψ JlS and Ψ J f l f S are the initial and final state wave functions, respectively, M Ω λ is theelectric or magnetic transition operator, l, l f are the orbital momenta of the initial and finalstates, respectively, k i and v i are the wave number and velocity of the α − He (or α − H)relative motion of the entrance channel, respectively; S , S are spins of the clusters α and He (or H), k γ = E γ / ¯ hc is the wave number of the photon corresponding to energy E γ = E th + E , where E th is the threshold energy. The spectroscopic factor [36] C ( S )within the potential approach is equal to 1, since the potential reproduces the two-bodyexperimental data, energies and phase shifts in partial waves [37]. We also use short-handnotations [ S ] = 2 S + 1 and [ λ ]!! = (2 λ + 1)!!. Further details of the wave functions andmatrix element calculations can be found in Ref. [22].5 II. NUMERICAL RESULTSA. Details of the calculations and phase-shift descriptions
We use simple Gaussian-form potentials for the α − He and α − H two-body interactions[22, 23]: V lSJ ( r ) = V exp( − α r ) + V c ( r ) , (4)where the Coulomb part is given as V c ( r ) = Z Z e /r if r > R c ,Z Z e (3 − r /R c ) / (2 R c ) otherwise , (5)with the Coulomb parameter R c , and charge numbers Z , Z of the first and second clusters,respectively. The parameters α and V of the potential are specified for each partial wave.In Ref. [22] we examined several potential models for the description of the α − He and α − H interactions. As discussed in the introduction, the d-wave potentials can be furtherimproved by modifying the depth ( V ) and width ( α ) parameters for the better descriptionof the astrophysical S factors at intermediate energies.The Schr¨odinger equation in the entrance and exit channels are solved with the α − Heand α − H central potentials as defined in Eq.(4) with the corresponding Coulomb partfrom Eq.(5). The same entry parameter values as in Ref. [22] are used: ¯ h / m N =20.7343MeV fm and R c =3.095 fm (Coulomb parameter), however the nuclear masses are taken as m He = 4 m N and m He = m H = 3 m N , where m N is the nucleon mass.The expressions for the astrophysical S factor and cross section given above are validonly for the radial scattering wave function (the radial component of the initial state wavefunction Ψ JlS ) normalized at large distances as u ( lSJ ) E ( r ) → r →∞ cos δ lSJ ( E ) F l ( η, kr ) + sin δ lSJ ( E ) G l ( η, kr ) , (6)where k is the wave number of the relative motion, F l and G l are regular and irregularCoulomb functions, respectively, and δ lSJ ( E ) is the phase shift in the ( l, S, J )th partial wave.The scattering wave function u E ( r ) of the relative motion is calculated as a solution of theSchr¨odinger equation using the Numerov method with an appropriate potential subject tothe boundary condition specified in Eq. (6). 6he depth and width parameters of the α − He and α − H model potentials V nD and V nM are given in Tables I and II, respectively. In 3th and 4th columns of the tables theenergies of forbidden states are presented. The potentials contain two forbidden states inthe s waves, while a single forbidden states in the each of p / , p / , d / , d / partial waves.These potentials differ from each other only in the s and p waves. At the same time, modelpotentials V nD and V nM are similar to potentials V aD and V aM from Ref. [22], respectively.The only difference is in the d -wave parameter values. The latter have now been fitted tobetter reproduce the astrophysical S factors at larger energies. TABLE I: Values of the depth ( V ) and width ( α ) parameters of the α − He ( H) potential V nD indifferent partial waves (see Eq. (4)). L J V (MeV) α (fm − ) E BeFS (MeV) E LiFS (MeV) s / -78.0 0.186 -40.03; -7.03 -41.34; -8.09 p / -83.8065 0.15747 -27.11 -28.33 p / -82.0237 0.15747 -26.02 -27.24 d / -180.0 0.4173 -11.96 -13.22 d / -190.0 0.4017 -18.13 -19.39 f / -75.9 0.15747 - - f / -85.2 0.15747 - -TABLE II: Values of the depth ( V ) and width ( α ) parameters of the α − He ( H) potential V nM in different partial waves (see Eq. (4)). L J V (MeV) α (fm − ) E BeFS (MeV) E LiFS (MeV) s / -50.0 0.109 -25.70; -5.17 -26.95; -6.11 p / -75.59760 0.13974 -24.58 -25.78 p / -70.75751 0.13308 -22.55 -23.74 d / -180.0 0.4173 -11.96 -13.22 d / -190.0 0.4017 -18.13 -19.39 f / -75.9 0.15747 - - f / -85.2 0.15747 - -
7n Fig. 1 the experimental data [38] for the He + α (panel a) and H + α (panel b) d -wavescattering phase shift are compared with the theoretical calculations using the new modelpotentials V nD and V nM . The phase shift description in the other partial waves were given inRef. [22]. Additionally, the presented models reproduce the energy spectrum of the Be and Li nuclei, as well as the empirical values of the ANC for the ground p / and the first excited p / bound states of the Be nucleus [22]. Indeed, the V nD model yields C (3 / − )=4.34 fm − / and C (1 / − )=3.71 fm − / , while the alternative V nM model reproduces the ANC valuesof C (3 / − )= 4.785 fm − / and C (1 / − )=4.242 fm − / extracted from the analysis of theexperimental data using the DWBA method [35]. Spiger 1967 ( d ) Spiger 1967 ( d ) V nD ( d ) V nD ( d )(a) e l [ d e g ] E c.m. [MeV] Spiger 1967 ( d ) Spiger 1967 ( d ) V nD ( d ) V nD ( d )(b) l [ d e g ] E c.m. [MeV] FIG. 1: d -wave phase shifts for the He + α (panel a) and H + α (panel b) scattering withinpotential models V nD and V nM in comparison with experimental data from Ref. [38]. B. Astrophysical S factor of the He( α, γ ) Be reaction For the study of the He( α, γ ) Be direct radiative capture process we first use the potential V nD . Partial E1 astrophysical S factors, estimated with the V nD potential are presented inFig. 2. Panel a compares the present results for the initial d-wave contribution with thecorresponding ones obtained in Ref. [22] using the potential model V aD . In panel b thecontributions from different initial s and d partial waves are shown. As can be seen fromthe figure, the d -wave contribution increases significantly at larger energies.Contributions from the E1, E2 and M1 astrophysical S factors for the He( α, γ ) Be direct8 .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.510 -3 -2 -1 -3 -2 -1 (a) d p d p [22] d p d p [22] d p d p [22] S f ac t o r [ k e V b ] E c.m. [MeV] (b) s p s p d p d p d p S f ac t o r [ k e V b ] E c.m. [MeV] FIG. 2: Partial E1 astrophysical S factors for the He( α, γ ) Be capture reaction calculated withthe V nD model potential in comparison with the results of Ref. [22]. -3 -2 -1 E1 V nD E1 V aD [22] E2 M1 He( Be S f ac t o r [ k e V b ] E c.m. [MeV] FIG. 3: E1, E2 and M1 components of the astrophysical S factors for the He( α, γ ) Be directcapture reaction calculated with the model potential V nD . The corresponding E1 component fromRef. [22] is also shown. capture reaction calculated with the model potential V nD are presented in Fig. 3. As canbe seen from the figure, modification of the potential in d waves significantly increases theastrophysical S factor in comparison with the results of Ref. [22] at energies above 0.5 MeV.Figure 4 compares the astrophysical S factors calculated with modified potentials V nD .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.50.10.20.30.40.50.6 V aM1 [22] V nM1 V aD [22] V nD He( Be S f ac t o r [ k e V b ] E c.m. [MeV] FIG. 4: Astrophysical S factor for the He( α, γ ) Be direct capture reaction calculated with modifiedpotentials V nD and V nM in comparison with experimental data from Refs. [16–21, 46–49] and theresults of Ref. [22]. and V nM with experimental data from Refs. [16–21, 46–49] and the results of Ref. [22]. Asubstantial improvement is achieved within the new models V nD and V nM at energies aroundand above the resonance energy.In Fig. 5 the final results for the astrophysical S factors of the He( α, γ ) Be direct capturereaction are compared with the available data and results of ab-initio calculations from Refs.[32, 33]. As can be seen from the figure, the potential models V nD and V nM describe bothabsolute values and energy dependence of the experimental data for the astrophysical S factor in a wide energy region from tens of keV to a few MeV. C. Astrophysical S factor of the H( α, γ ) Li As noted in the Introduction, the same model potentials V nD and V nM are applied for thestudy of the mirror capture reaction H( α, γ ) Li. The Coulomb part of these potentials,defined in Eq. (5), is modified according to the charge value of the H cluster, Z =1. Asdemonstrated in Fig. 1 (panel b), the phase shifts in the d / and d / partial waves arewell described. The binding energies E b (3 / − )=2.467 MeV and E b (1 / − )=1.990 MeV of thebound states have been reproduced in Ref. [22].10 .01 0.1 10.20.30.40.50.60.70.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.50.00.10.20.30.40.50.60.70.80.9 (b) FMD [32] NCSM [33] NCSM(ph.) [33] V nM1 V nD Carmona 2012Bordeanu 2013TakÆcs 2015TakÆcs 2018Sz(cid:252)cs 2019 Nara Singh 2004Bemmerer 2006Brown 2007Comfortola 2007Di Leva 2009 He( Be S f ac t o r [ k e V b ] E c.m. [MeV](a) FMD [32] NCSM [33] NCSM(ph.) [33] V nM1 V nD Carmona 2012Bordeanu 2013TakÆcs 2015TakÆcs 2018Sz(cid:252)cs 2019 Nara Singh 2004Bemmerer 2006Brown 2007Comfortola 2007Di Leva 2009 He( Be S f ac t o r [ k e V b ] E c.m. [MeV] FIG. 5: (a) Astrophysical S factor for the He( α, γ ) Be direct capture reaction calculated withmodified potential models V nD and V nM in comparison with experimental data from Refs. [16–21, 46–49] and ab-initio calculations from Refs. [32, 33]. Panel (b) highlights the low-energyregion. In Fig. 6 we compare the contributions of the E E M S factorsfor the H( α, γ ) Li direct capture reaction calculated with the potentials V nD and V aD fromRef. [22]. As in the case of Be, the relative contribution of the E1 transition increases withthe energy in comparison with the results of Ref. [22].Figure 7 presents the astrophysical S factor for the H( α, γ ) Li direct capture reactioncalculated with modified potentials V nD and V nM in comparison with experimental data fromRefs. [39–45] and the results of Ref. [22]. An increase of the astrophysical S factor withinthe models V nD and V nM is seen for energies E > . V nM model.In Fig. 8 the astrophysical S factors for the H( α, γ ) Li direct capture reaction calculatedwith modified potential models V nD and V nM are compared with available experimental dataand ab-initio calculations. As can be seen, the best description of the data for both absolutevalue and energy dependence of the astrophysical S factor is obtained with the new potentialmodels V nD and V nM . As noted above, all the parameters of the model potentials have beenadjusted to the data for the Be nucleus. With that the results for the astrophysical S factorfor the mirror Li nucleus are obtained without any fitting parameters. Additionally, the11 .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.510 -4 -3 -2 -1 E1 V nD E1 V aD [22] E2 M1 H( Li S f ac t o r [ k e V b ] E c.m. [MeV] FIG. 6: Comparison of contributions of the E1, E2 and M1 astrophysical S factors for the H( α, γ ) Li direct capture reaction calculated with modified potential model V nD compared withthe results of Ref. [22]. same potentials describe the binding energies and phase shifts for the mirror Li nucleus [22].
IV. REACTION RATES AND PRIMORDIAL ABUNDANCE OF THE LI ELE-MENTA. Estimation of reaction rates for the He( α, γ ) Be process In Table III estimated values for the reaction rate are given for the He( α, γ ) Be directcapture process in the temperature interval 10 K ≤ T ≤ K (0 . ≤ T ≤ V nD and V nM are in a good agreement with those obtained using the models V aD and V aM [50],respectively.In Fig. 9 we present estimated reaction rates for the direct He( α, γ ) Be capture processwithin the modified potential models V nD and V nM , normalized to the standard NACRE1999 experimental data [36]. For comparison we also display the lines corresponding to theresults of Refs. [18, 51, 52] and more recent NACRE II 2013 data [53]. As can be seen fromthe figure, the potential model results lye between the lines for the microscopic R-matrixapproach from Ref.[52] and the NACRE II data. Other models [18, 51] overestimate the12 .0 0.5 1.0 1.5 2.00.020.040.060.080.100.12 V aM1 [22] V nM1 V aD [22] V nD H( Li S f ac t o r [ k e V b ] E c.m. [MeV] FIG. 7: Astrophysical S factor for the H( α, γ ) Li direct capture reaction calculated with modifiedpotential models V nD and V nM in comparison with available experimental data from Refs. [39–45]and the results of Ref. [22]. NACRE II data.In order to estimate the primordial abundance of the Li element the well knownPArthENoPE [54] public code is employed. It operates with an analytical form of thereaction rate dependence on the temperature T . For this reason the theoretical reactionrates from Table III are approximated (within an uncertainty of 0.971% for the V nD and0.582% for the V nM ) by the analytical form N A ( σv ) = p T − / exp( − C T − / ) × (1 + p T / + p T / + (7)+ p T + p T / + p T / ) + p T − / exp( − C T − ) . The coefficients of the analytical polynomial approximation of the He( α, γ ) Be reactionrates estimated within the potential models V nM and V nD are given in Table IV in the tem-perature interval 0 . ≤ T ≤
1. In addition, for this process the other coefficients are C = 12 .
813 and C = 15 . He( α, γ ) Be direct capture reaction to theprimordial abundance of the Li element. If we adopt the Planck 2015 best fit for the baryondensity parameter Ω b h = 0 . +0 . − . [55] and the neutron life time τ n = 880 . ± . .0 0.5 1.0 1.5 2.00.000.050.100.150.20 V nM1 V nD Griffiths 1961Schr(cid:246)der 1987Burzynski 1987Utsunomiya 1990Brune 1994Tokimoto 2001Bystritsky 2017 FMD [32] NCSM [33] NCSM (ph.) [33] H( Li S f ac t o r [ k e V b ] E c.m. [MeV] FIG. 8: (a) Astrophysical S factor for the H( α, γ ) Li direct capture reaction calculated withmodified potential models V nD and V nM in comparison with available experimental datafrom Refs.[39–45] and ab-initio calculations [32, 33]. Panel (b) highlights the low-energy region. s [56], for the Li/H abundance ratio we have an estimate (4 . ± . × − withinpotential model V nD and the estimate (4 . ± . × − within the model V nM whichagree well, within 2%, to be specific. As discussed below, these numbers barely change the Li/H abundance ratio if the contribution from the H( α, γ ) Li direct capture reaction isincluded.
B. Estimation of reaction rates for the H( α, γ ) Li direct capture process In Table V we give theoretical estimations for the H( α, γ ) Li direct capture reaction ratesin the temperature interval 10 K ≤ T ≤ K (0 . ≤ T ≤
1) calculated with the samemodified potential models V nM and V nD which have been used for the He( α, γ ) Be process.Figure 10 displays these results normalized to the standard NACRE 1999 experimentaldata [36]. For the comparison we also display the lines corresponding to the results of themicroscopic R-matrix method [52] and new NACRE II 2013 data [53].The coefficients of the analytical polynomial approximation of the H( α, γ ) Li reactionrates estimated within the potential models V nM and V nD are given in Table VI in the temper-ature interval 0 . ≤ T ≤
1. The remaining coefficients are C = 8 .
072 and C = 3 . ABLE III: Theoretical estimates of the reaction rates for the direct He( α, γ ) Be capture processin the temperature interval 10 K ≤ T ≤ K (0 . ≤ T ≤ T V nM V nD T V nM V nD . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × . × . × − . × − . × . × . × − . × − . × . × . × − . × − . × . × . × − . × − . × . × In this case, the analytical formula (7) with the parameter values from Table VI reproducesthe theoretical reaction rates from Table V (within an uncertainty 0.599% for V nD and 0.647%for V nM ). 15 ABLE IV: Fitted values of the coefficients of analytical approximation for the direct capturereaction He( α, γ ) BeModel p p p p p p p V nM . × V nD . × Descouvemont et al. (2004) NACRE II 2013 Kontos et al. (2013) TakÆcs et al. (2015) V nM1 V nD He( Be N A ( v ) / N A ( v ) NA CR E T FIG. 9: Reaction rates of the He( α, γ ) Be direct capture process normalized to the NACRE 1999experimental data in comparison with results from Refs.[18, 51, 52] and more recent NACRE II2013 data [53].
Now including the obtained theoretical reaction rates for both He( α, γ ) Be and H( α, γ ) Li capture processes into the nuclear reaction network with the help of thePArthENoPE [54] code, we can evaluate the primordial abundance of the Li element.Adopting the aforementioned values of the baryon density and the neutron life time, forthe Li/H abundance ratio we have an estimate (4 . ± . × − within the model V nD , while the model V nM yields (4 . ± . × − [50]. These numbers are slightlydifferent than the corresponding estimates based exclusively on the He( α, γ ) Be process.16
ABLE V: Theoretical estimates of the reaction rates for the H( α, γ ) Li direct capture process inthe temperature interval 10 K ≤ T ≤ K (0 . ≤ T ≤ T V nM V nD T V nM V nD . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × . × . × − . × − . × . × . × − . × − . × . × . × − . × − . × . × . × − . × − . × . × . × − . × − . × . × . × − . × − . × . × . × − . × − . × . × . × − . × − . × . × . × − . × − . × . × . × − . × − . × . × . × − . × − . × . × . × − . × − . × . × V. CONCLUSIONS
The astrophysical He( α, γ ) Be and H( α, γ ) Li direct capture reactions have been stud-ied in an updated two-body potential model. The parameters of the central potentials of17
ABLE VI: Fitted values of the coefficients of analytical approximation for the H( α, γ ) Li directcapture reaction Model p p p p p p p V nM . × V nD . × Descouvemont et al. (2004) NACRE II 2013 V nM1 V nD H( Li N A ( v ) / N A ( v ) NA CR E T FIG. 10: Reaction rates of the direct H( α, γ ) Li capture process normalized to the NACRE 1999experimental data [36] in comparison with the results of Ref.[52] and new NACRE II 2013 data[53] a simple Gaussian form have been adjusted to reproduce the α + He phase shifts in the s , p , d and f partial waves and the binding energies of the Be ground 3/2 − and first excited1/2 − states. At the same time, properties of the mirror Li nucleus, phase shifts in thepartial waves and the binding energies of the ground 3/2 − and first excited 1/2 − states arereproduced without any additional adjustment parameters.It is found that due to the dominance of the E1 transition in the capture processes,there is a possibility to adjust the parameters of the potential in the initial s - and d -wavesin order to optimize the description of the astrophysical S factor at low and intermediateenergy regions, respectively.In conclusion, the potential models V nM , V nD have been suggested for the description ofthe α + H and α + He interactions. These models reproduce spectroscopic properties and18hase shifts of both Be and Be nuclei. They describe well the experimental data for theastrophysical S factor of the capture process He( α, γ ) Be in a wide energy region, extendingto 4.5 MeV. This includes the new data of the LUNA collaboration around 100 keV and thelatest data at the Gamov peak obtained on the basis of the observed neutrino fluxes fromthe Sun, S (23 +6 − keV)=0.548 ± S factor for the mirror capture reaction H( α, γ ) Li with a good accuracy.The calculated values of the astrophysical S factors and reaction rates for the He( α, γ ) Beand H( α, γ ) Li direct capture reactions are in good agreement with the results of micro-scopic models and ab-initio calculations. For the primordial abundance of the Li elementan estimate (4 . ± . × − have been obtained. This result is within the range of thestandard BBN model estimates. Acknowledgements
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