key: cord-0109833-85dxfqe4
authors: Dubovichenko, S. B.; Tkachenko, A. S.; Kezerashvili, R. Ya.; Burkova, N. A.; Dzhazairov-Kakhramanov, A. V.
title: The $^6$Li$(p, gamma)^7$Be reaction rate in the light of the new LUNA data
date: 2022-01-20
journal: nan
DOI: nan
sha: d141131ee0061b55528dcb16701c8f13edd4b935
doc_id: 109833
cord_uid: 85dxfqe4

We present new calculations of the astrophysical $S-$factor and reaction rate for the $^{6}$Li$(p,gamma )^{7}$Be reaction at energies of 10 keV to 5 MeV in the framework of a modified potential cluster model with forbidden states, including low lying resonances. The astrophysical $S(E)-$factor is compared with the available experimental data and calculations done within different models. The results for the $S-$factor are in good agreement with the data set (for $E<0.3$ MeV) and calculations (for $E<0.6$ MeV) of LUNA collaboration (Phys. Rev. C 102$,$ 052802, 2020). The recommended extrapolated zero value $S(0)$ turned out to be 101 eV $cdot $ b. Using the theoretical total cross-sections$,$ the $^{6}$Li$(p,gamma)^{7} $Be capture reaction rate is calculated at temperatures ranging from 0.01 to 10 $T_{9}$ and compared with NACRE and NACRE II. Analytical expressions for the $S-$factor and reaction rate are given, and the effect of low-lying resonances on the reaction rate is estimated. We suggest to update the NACRE and NACRE II databases in light of the new LUNA data and present calculations.

1. The MPCM is a two-particle model that accounts for the internal characteristics of clusters: their sizes, charges, masses, quadrupole and magnetic momenta, which are used to calculate the reaction total cross-sections or other characteristics of the final nucleus.

2. The classification of cluster states is performed according to Young's orbital diagrams, leading to the concept of forbidden states in some partial waves [28] . The Pauli principle is implemented via exclusion of the forbidden states (FSs), manifesting in proper node behavior of the radial wave function (WF). Forbidden states that lead to low-lying bound states are not physically realized due to the orthogonality of corresponding functions and allowed state functions.

3. The Gaussian type inter-cluster interaction potentials are constructed, taking into account these forbidden states in certain partial waves. For each partial wave with specified quantum numbers, the potential is constructed with two parameters, assuming it depends explicitly on Young's orbital diagrams. 4 . Potentials of the bound states (BSs) are constructed based on asymptotic constants (AC) and binding energies. Potentials of the scattering processes are constructed based on the spectra of the final nucleus or the scattering phase shifts of the particles of the input channel. Parameters of the potentials are fixed or variable within the AC error intervals and vary within the energy or width errors of resonant or excited states. 5 . The radial WFs of the allowed states of the continuous and discrete spectra are tailored appropriately using correct asymptotics.

In light of the new experimental LUNA data [9] , we reexamine the 6 Li(p, γ) 7 Be reaction S −factor and reaction rate within the framework of the MPCM.

The total wave functions (WF) have the form of an antisymmetrized product of completely antisymmetric internal wave functions of clusters Ψ(1, . . . , A 1 ) = Ψ(R 1 ) and Ψ(A 1 + 1, . . . , A) = Ψ( R 2 ), multiplied by the corresponding wave function Φ(R) of relative motion [22] [23] [24] Ψ =Â{Ψ(R 1 )Ψ(R 2 )Φ(R)}.

(1)

In Eq. (1)Â is the antisymmetrization operator that permutes nucleons from the clusters A 1 and A 2 , R 1 and R 2 are the centerof-mass radius-vectors of the clusters, and R = R 1 −R 2 is the relative motion coordinate. The wave functions (1) are characterized by specific quantum numbers, including JLS -total momentum, orbital quantum momentum, and spin, respectively -and Young's diagrams { f }, which determine the orbital part of WF permutation symmetry of the relative motion of the clusters.

In the general case, the possible Young's orbital diagram { f } L of some nucleus A({ f }), consisting of two parts

and is determined by Littlewood's theorem [23, 24] . According to Elliot's theorem, each Young's diagram is associated with a certain orbital angular momentum or their combination.

Spin-isospin diagrams are the direct inner product of the spin and isospin Young diagrams of a nucleus consisting of A nucleons { f } S T = { f } S ⊗ { f } T . For a system with no more than eight particles such diagrams are provided in Table C of Ref. [25] . A detailed procedure for defining the corresponding momenta can be found in the classical monograph [26] . Let us note that in Ref. [26] the definition for inner and outer products is reverses.

The total Young's diagram of the nucleus is defined as the direct inner product of the orbital and spin-isospin

The total wave function of the system under antisymmetrization does not vanish identically, only if it contains an antisymmetric component {1 N }, where N is the number of nucleons. In this case the conjugates { f } L and { f } S T are multiplied. Therefore, the diagrams { f } L conjugated to { f } S T are allowed in this channel. All other orbital symmetries are forbidden since they lead to zero total wave function of the particle system after antisymmetrization.

Let us describe in more detail the procedure for constructing the intercluster partial potentials. Below we define the criteria and outline the sequence for finding parameters for the potentials and indicating their errors as well as ambiguities.

For the bound states of two clusters, the interaction potentials within the framework of the MPCM are constructed based on the requirement imposed to describe the main observable characteristics of such a nucleus. In this case, the potential parameters are fixed. It should be noted that this requirement is an idealized scenario that exists in the nucleus since it assumes that the ground state (GS) is a two-body single channel with probability closed to unity. First, we find the parameters of the bound state potentials. For the GS with a given number of bound allowed and forbidden states in the partial wave, these parameters are fixed unambiguously in terms of the binding energy, the radius of the nucleus, and the AC. When constructing the partial interaction potentials in the MPCM, it is assumed that interactions depend not only on the orbital angular momentum L but also on the total spin S and the total angular momentum J of the system and also depend on Young's orbital diagrams. As in earlier work [28] , we use Gaussian interaction potentials, which depend on the quantum numbers JLS , and Young's diagrams { f } L . Therefore, for different JLS , we have different values of the parameters of the partial potentials.

The accuracy of determining the parameters of the BS potential is connected directly with the accuracy of the AC. The potential does not contain any other ambiguities, since according to Young's diagrams, the classification of states makes it possible to unambiguously fix the number of bound forbidden and allowed states in a given partial wave. The number of bound states ultimately determines the depth of the potential, while the width depends entirely on the value of the AC. If one fixes two parameters of the potential using two particular quantities -the binding energy and the AC -the error of the binding energy is seen to be much less than that of the AC.

It should be noted that any calculations of the charge radius reflect the errors of the underlying model. In any model, the magnitude of such a radius depends on the integral of the model wave functions, thereby compounding sources of error. At the same time, the values of AC are determined from the asymptotic behavior of the model WFs at one point and contain significantly less error. The potentials of the BSs are constructed to obtain the best agreement with the values of the AC extracted independently from the experimental data. For more details, see Ref. [27] .

For the potentials of the continuous spectrum, the intercluster potential of the nonresonant scattering process for a given number of allowed and forbidden BSs in the considered partial wave is also constructed quite unambiguously based on the scattering phase shifts. The accuracy of the potential parameters sometimes as high as 20-30%, is associated with the precision of the extracted scattering phase shifts from experimental data. For the 6 Li(p,γ) 7 Be reaction, the potential is unambiguous since the classification, according to Young's diagrams, makes it possible to fix the number of bound states. This completely determines the potential depth, and its width is determined by the shape of the scattering phase shifts.

When constructing the nonresonant scattering potential based on the data for the nuclear spectra, it is difficult to estimate the accuracy of the parameters even for a given number of BSs. However, one can expect that it will not exceed the error discussed above. This potential should lead to a scattering phase shift close to zero or rise to a smoothly decreasing phase shift at low energies, since there are no resonance levels in the spectra of the nucleus.

In resonance scattering, when a relatively narrow resonance is present in the partial wave at low energies for a given number of BSs, the potential is constructed completely unambiguously. The accuracy of determining the parameters of the interaction potentials is determined by the following factors. The depth of the potential depends on the resonance energy E x and the number of BSs. The width is determined by the accuracy of the experimental values of the level width Γ.

The error of the parameters, approximately 5-10%, usually does not exceed the error of the energy level width. This also applies to the construction of the partial potential from the resonant scattering phase shifts and the determination of its parameters from the spectral resonance of the nucleus [19, 28] .

To calculate the total cross-sections of radiative capture processes, we use the well-known formula for the transitions of N J multipolarity [19, 28] 

where the matrix elements of orbital EJ(L)-transitions have the following form (S = S i = S f )

and the matrix elements of the magnetic M1(S )-transition are written as follows

In Eqs. (2)-(6) K = E γ / ̵ hc is the wave number of the emitted photon with energy E γ , m 1 , m 2 and µ 1 , µ 2 are the masses and magnetic momenta of the clusters, respectively, and µ is the reduced mass of the system. Namely, in the present calculations for the reaction 6 Li(p,γ) 7 Be, m 1 ≡ m p = 1.00727646677 amu, m 2 ≡ m6 Li = 6.0151232 amu [29] and µ 1 ≡ µ p = 2.792847µ 0 , µ 2 ≡ µ6 Li = 0.822µ 0 [30, 31] , where µ 0 is nuclear magneton, ̵ h 2 /m 0 = 41.4686 MeV⋅fm 2 , where m 0 = 931.494 MeV is the atomic mass unit (amu).

The point-like Coulomb potential is of the form V Coul (MeV) = 1.439975Z 1 Z 2 /r, where r is the relative distance between the particles of the channel in fm, and Z 1 and Z 2 are the charges in units of the elementary charge. The Coulomb parameter η = µZ 1 Z 2 e 2 /k ̵ h 2 is represented in the form η = 3.44476 ⋅ 10 −2 Z 1 Z 2 µ/k, where k is the wavenumber in fm −1 and is determined by the energy E c.m. of the interacting particles, k 2 = 2µE c.m. / ̵ h 2 . In our calculations, we use the dimensionless AC denoted as C w [32] χ L (r) = 2k 0 C w W −ηL+1/2 (2k 0 r).

A dimensional asymptotic constant C is related to the asymptotic normalization coefficient (ANC) A NC by the expression [27] A NC = S F C,

where S F is the spectroscopic factor and C is the dimensional AC that can be represented using the asymptotics of the WF

In Eq. (9) R is the large distance where the nuclear potential vanishes and χ L (r) is the wave function of the bound state obtained from the solution of the radial Schrödinger equation and normalized to unity. The Whittaker function W −ηL+1/2 of the bound state determines the asymptotic behaviour of the WF. The wave number k 0 is related to the channel binding energy E b . For a continuous spectrum, the function χ i found numerically is matched to asymptotics u L (R) of the form

Here F L and G L are Coulomb regular and irregular functions [33] . They are the solutions of the Schrödinger equation with the Coulomb potential. δ J S ,L are the scattering phase shifts depending on the JLS momenta of the system and N L is the normalizing constant of the numerical radial function u L (R) for the continuum.

Consider the classification of the BSs of the p 6 Li system according to Young's diagrams. In this case there is only one Young's orbital diagram for the 6 Li nucleus {42}. It is believed that the system's potentials are dependent on the diagrams or combinations of these diagrams in various states. Thus, if the orbital diagram {42} allowed in the 2 H 4 He-cluster channel is accepted for the 6 Li nucleus, then the p 6 Li system with spin S = 1/2 contains a forbidden level with diagram {52} and orbital momenta of L = 0, 2, and the allowed states with configurations {43} for L = 1, 3 and {421} for L = 1, 2. Hence, the p 6 Li potentials must have a forbidden state related to {52} in the S wave. The allowed bound state corresponds to the P wave with the two Young's diagrams {43} and {421}. In the quartet spin channel S = 3/2 of the system, only one diagram {421} is allowed for L = 1, 2 [28] . Since there are two allowed diagrams {43} and {421} in the doublet spin state of the p 6 Li system, the scattering states turn out to be mixed in orbital symmetries. At the same time, only one allowed diagram {43} usually corresponds to the doublet ground state of the 7 Be nucleus in the p 6 Li channel with J π = 3/2 − and L = 1.

Here, the p 6 Li system is completely analogous to the p 2 H channel in the 3 He nucleus. In the latter case the doublet state is also mixed according to Young's diagrams {3} and {21} [24] . Therefore, the potentials constructed based on the elastic scattering phase shifts of the p 6 Li system or the level spectra of the 7 Be nucleus cannot be used to describe the GS of the 7 Be nucleus in the p 6 Li channel. Pure in orbital symmetry with Young's diagram {43}, the 2 P 3/2 potential of the ground state of 7 Be reproduces the binding energy of the GS of the nucleus consistent with the p 6 Li system and its asymptotic constant.

The scattering potentials are constructed based on the spectra of the 7 Be nucleus from Ref. [30] ; it has no major difference from the newer compilation [34] .

The orbital state's classification of the p 6 Li system is shown in Table I . [14, 19] . The following notations are used: S and L are spin and orbital angular momentum of the system, respectively,

, and { f } L are spin, isospin and spin-isospin, and possible orbital Young's diagrams, and { f } AS , { f } FS are Young's diagrams of allowed and forbidden orbital states.

We use a Gaussian potential that depends on the momenta of the system [28] and Young's diagrams

In Eq. (11) V 0 is the potential depth and α is related to the potential width. The choice of parameters for the bound and scattering states is discussed in detail in Sec. VI. A compilation of the AC data for the ground and first excited state in the p 6 Li channel of 7 Be is presented in Table II , with

Summarizing the data in Table II , we conclude that all cited data on the AC are overlapped. While constructing the corresponding GS and FES potentials, we used the average values indicated in bold font in Table II . Just at the end of our calculation story, a publication by Kiss et al. [11] appeared, and it happened that these latest experimental results turned to be within the defined intervals for ANC given above.

We use GS and FES in the form of only doublet 2 P 3/2 and 2 P 1/2 states, but we take the experimental data on ANC from [37] since it is assumed that these states result in the observed ANC values. We do not consider these states as a mix of doublet and quartet states, for instance, 2+4 P 3/2 , is a prime example, as the quartet channel is not allowed for the orbital Young's diagram {43} in Table I .

All AC values are used here as a framework to obtain the parameters of the p 6 Li interaction BSs potentials. These potentials correspond to the lower, upper, and average values of AC and accurately reproduce the binding energies [30] of the bound states. The parameters of the potentials are presented in Table III . We usually assume that the scattering potentials can lead to the FSs [28] , and if the FSs are absent, the potential depth can be set to zero. The present case refers to the scattering P potentials without the FS, while the S and D potentials have the bound forbidden state and, even at zero phase shifts, must have a nonzero depth.

In the presence of one BS, the phase shift starts at 180 • [24] , as shown in Fig. 1 for the S scattering phase shifts. Furthermore, we consider E2 transitions from resonant F scattering states with phase shifts shown in Fig. 2 . The parameters of potentials for all scattering processes for transitions to GS and FES are given in Tables IV and V, respectively. (3) and (5). E x , E res , Γ c.m. and V 0 are provided in MeV.

No res. In addition, we consider the resonance at an excitation energy of 9.9 MeV [30] according to Fig. 3 (3) and (5). E x , E res , Γ c.m. and V 0 are provided in MeV. Table V . The resonance energies and widths are obtained with the corresponding parameters of the scattering potentials. For the P 1/2 scattering wave, zero-depth potentials are used since the scattering P waves do not contain forbidden BSs. For the 2 D wave potentials, 2 S wave parameters are used for L = 2.

Resonant phase shifts of elastic scattering p+ 6 Li are shown in Fig. 2 . The above-threshold resonance at 6.73 MeV with a width of 1.2 MeV indicated in Fig. 3 refers to the 4 He 3 He channel [30] and is not considered in our previous works [14, 15] . Note again that in the MPCM we used, Young's orbital diagram {43} is forbidden in the quartet state, as shown in Table I , and this particular diagram corresponds to the GS of the 7 Li nucleus. Therefore, in the GS there is only a doublet 2 P 3/2 state (without impurity of 4 P 3/2 ), which is allowed for the diagram {43}. Thus, our model [14, 15] predicted the absence of resonance at 6.73

MeV with J π = 5/2 in the nucleon channel, or, in other words, the impossibility of the M1 transition from this resonance to the GS. This has been confirmed by the new LUNA results [9] and, indirectly, by the data of [30] . The width of the resonance peak at 9.29 MeV is taken from Table 7 .10 of [30] , although another state, 2 P 1/2 , is indicated therein. At 9.27 MeV, the given moment is 7/2, as per Table 7 .7 in Ref. [30] , so we infer the presence of an F state. However, this resonance leads to a minimal increase in cross-sections at the E2 transition. It is negligible against the background of the E1 resonance at 4.3 MeV with a transition from the 2 P 3/2 scattering state for the potential parameters #7 (Table IV) or #5 (Table V) , respectively.

In Fig. 1 are shown the doublet and quartet S phase shifts of elastic p 6 Li scattering at low energies. The 2 S potential from Fig. 1 by solid circles. Moreover, it gives phase shifts below 2 MeV that coincide with the phase shifts obtained with the potential from Ref. [14] , with a depth of 126 MeV and a width of 0.15 fm −2 . This early potential has two FSs and does not agree with our new classification from Table I . To compare the results, we construct a new potential (depth 58 MeV) that gives the most overlap in phase shifts from prior work [14] . The phase shift of the new potential is given in Fig. 1 by a solid curve, while dash-dotted curves refers to results from Ref. [14] .

A special feature of cross-sections of nuclear reactions with charged particles at low and ultra-low energies is an extreme reduction by several orders of a cross-section magnitude due to the decrease in transmission probability through the Coulomb barrier. For practical purposes, the astrophysical S-factor is introduced as

where the factor P(E) reflects the permeability of the Coulomb barrier. The advantage of the astrophysical S −factor is that it shows a smooth energy dependence at low energies. Following the excellent manuscript by Christian Iliadis [39] , we would like to provide a brief discussion of the use of the S −factor in conventional calculation schemes in order to clarify the current approach. The definition of the S −factor (12) allows to write the following expression for the reaction rate:

In Eq.(13) κ is the Boltzmann constant and N A is Avogadro's number. A notable effort has been expended to bring the integral in (13) to analytical form. This is possible only if the expansion of the S (E)−factor in the E series, given by

is valid at low energies (c.f. Ref. [39] , Section 3.2). Expression (14) explains the active interest to determine the value S (0) as a key one for the calculation of the reaction rate in the form of (13) with its further analytical parameterizations. Table VI presents the available experimental data of the astrophysical S −factor for the 6 Li(p,γ) 7 Be reaction, as well as the extrapolated S (0) values. Results of previous theoretical calculations and the present work are presented in Table VII . Our calculations are analyzed and interpreted based on the experimental data [9] as one of the newest and most accurate.

In the present work we introduce some corrections to [14, 15] that allow us to extend the energy interval for the cross-sections and corresponding S −factors. It is also worth mentioning that in Ref. [18] , the authors used the calculation scheme based on [15] . The value S (0) = 95.0 eV⋅b is obtained which is consistent with the LUNA experimental data [9] . However, the astrophysical reaction rate is missing in Ref. [18] . Xu et al. [16] , 2013

Direct-capture potential model 88, 34 Dong et al. [17] , 2017 Gamow shell model 103, 9 Gnech and Marcucci [18] , 2019 Potential cluster model 96,5 ± 5,7

Kiss et al. [11] , 2021 Modified two-body potential method 92 ± 12

Kiss et al. [11] , 2021 Modified two-body potential method 98, 3 Present work Modified potential cluster model

The results of the present calculations of S −factors along with available experimental data are shown in Figs. 4, 5, and 6. Fig.  4 shows the astrophysical S (E)−factor of 6 Li(p,γ 0 ) 7 Be capture to the GS of the 7 Be nucleus in an energy range up to 5 MeV. The solid red curve 2 and the two dashed curves, blue 1 and green 3, show the calculation for all transitions to the GS given in Table IV . Parameters V 0 (only integer values) and α of the corresponding potentials as well as C w from Table III are indicated in  the figures, and the parameters of the GS potentials are taken from Table IV. The solid red curve 2 is the result for the potential  with the set of parameters #2 from Table III, Table IV . A band shows the sensitivity to changes in C w .

In Fig. 5 Be capture to FES. Experimental data ○ are taken from [6] . Parameters of the continuum potential are given in Table V . A band shows the sensitivity to changes in C w .

In Fig. 6 , similar curves show the astrophysical S −factor for the total cross-sections corresponding to the transition to GS and FES. The two dashed curves 1 and 3 show the range of S −factor values due to ambiguities in the AC of the GS and FES. For the scattering potentials, the parameters from Tables IV and V are used.

The best agreement of the S −factor with experimental data is achieved for the values of C w = 2.74 for the GS and C w = 1.68 for the FES. We recommend these values as the most reliable benchmarks for future experimental studies. Fig. 6 shows that almost all experimental data lie between the solid red curve 2 and the green dashed curve 3. If we use the GS and FES potentials set of parameters #3 and #4 from Table III , the result is shown in Fig. 6 by the black curve 4. In this case the experimental data [9] are reproduced entirely, and S −factor at 10 keV is found to be 101 eV⋅b. For the scattering potentials, the data from Table IV and Table V are used. Due to the uncertainty of the S −factor that arises from the uncertainty of the AC, it is desirable to select other options for the potentials of the GS and FES to correctly describe the LUNA data [9] . This can be the subject of future work if more accurate data is compiled for the AC of the ground and first excited states of the 7 Be nucleus. The approximation of the S −factor shown by the black curve 4 in Fig. 6 has an analytical form New experimental data from LUNA [9] can be approximated to the first order

with parameters S 0 = 91.952 eV⋅b and S 1 = −75.471 MeV −1 ⋅eV⋅b, leading to χ 2 = 0.6 and S (0) = 92 eV⋅b. To compare the calculated S −factor at zero energy (10 keV), we present the known results for the total S (0): 79(18) eV⋅b [10] , 105 eV⋅b (at 10 keV) [12] and 106 eV⋅b [41] .The S −factor for transitions to the ground state in [40] , 39 eV⋅b is specified, and for the transition to the first excited state, the S −factor value is equal to 26 eV⋅b, the total S −factor is 65 eV⋅b. In our previous works [14, 15] , a value of 114 eV⋅b was obtained. The summary for S −factor experimental and theoretical values are presented in Tables VI and VII. To sum up, our astrophysical S −factor is given in Fig. 7 with a solid red curve, together with experimental data and theoretical calculations. The R−matrix fit of the data from LUNA collaboration [9] and Switkowski et al. [4] is represented with the solid blue curve. A solid green curve was obtained by Kiss et al. [11] using the weighted means of the ANCs from the analysis of the 6 Li( 3 He,d) 7 Be transfer reaction within the modified two-body potential method (MTBPM). In addition, [11] contains the results for the S −factor of the 6 Li(p,γ) 7 Be reaction calculated within the MTBPM, using the values of ANCs obtained from the analysis of the experimental astrophysical S −factors of the 6 Li(p,γ) 7 Be reaction [9] . These results are given in Fig. 7 with the solid black curve.

To calculate the 6 Li(p, γ) 7 Be capture reaction rate in units of cm 3 mol −1 s −1 , we used the expression [43] analogous to Eq. [42] . Results of calculations: red curve -present work; blue curve -Ref. [9] ; black and green curves -Ref. [11] .

In Eq. (17a) E is given in MeV, the total cross-section σ(E) is taken in µb, and µ is the reduced mass in amu and T 9 = 10 9 K [43] . Using real integration limits E min and E max Eq. (17a) becomes Fig. 6 . The yellow band is taken from Ref. [16] .

As noted above, curve 4 in Fig. 6 is in best agreement with all experimental data of the S −factor. Therefore, the corresponding reaction rate, also marked as curve 4 in Fig. 7 is most recommended description of the reaction rate. Curve 4 can be approximated by a function of the form [44] The parameters of approximation (18) with an average value of χ 2 = 0.014 and the error of 5% are given in Table VIII . Comparing our results with the reaction rate presented in NACRE [43] and NACRE II [16] , we follow the format of Fig. 4 from [9] . We added our results for the reaction rate, normalized to the reaction rate from NACRE [43] . The comparison is shown in Figs. 9 and 10.

FIG. 9. Comparison of the astrophysical reaction rates in the range 0.001 to 1 T 9 from [9, 16, 43] to present work, normalized to the NACRE rate [43] . Dotted curves represent the uncertainty of the NACRE [43] rate, while shaded areas represent the uncertainties from LUNA [9] (pale red) and NACRE II [16] (yellow).

FIG. 10. Comparison of the astrophysical reaction rates in the range 0.001 to 10 T 9 from [9, 16, 43] to present work, normalized to the NACRE rate [43] . Dotted curves represent the uncertainty of the NACRE [43] rate, while shaded areas represent the uncertainties from LUNA [9] (pale red) and NACRE II [16] (yellow).

As stated in Ref. [9] , the LUNA "thermonuclear reaction rate is 9% lower than NACRE [43] and 33% higher than reported in NACRE II [16] at 2 MK, and the reaction rate uncertainty has been significantly reduced". Fig. 9 shows that the deviation between the adopted reaction rate obtained in [9] and the present calculations in the range of 0.01 to 1 T 9 does not exceed 5%. Therefore, the present calculations confirm the above conclusion by Piatti et al. [9] . Fig. 10 shows two of our results -the blue curve 1 is the reaction rate calculated on the basis of the total cross sections in the energy range from 1 keV to 5 MeV, and the green one 2 shows the rate at the upper range of the integration limit E max = 0.6 MeV, and this value corresponded to the upper limit of energy in work [9] . The difference between these curves illustrates the importance of the resonance region contribution for the cross sections in the range of 0.6 MeV to 5 MeV at temperatures above 1 T 9 .

We present the results of calculations and analyses of the S −factor and astrophysical reaction rate for the 6 Li(p,γ) 7 Be reaction in the framework of MPCM. It is demonstrated that the MPCM approach has only one ambiguity arising from the accuracy of the experimentally determined asymptotic constants. This effect manifests as bands in Figs. 4 -6 for the astrophysical S −factor. Precise LUNA experimental data played a role of the criterion, in reducing the ANC ambiguity with theoretical simulations.

Comparing the R−matrix method which is constrained by the parameterization of the experimental cross-sections data, MPCM enables to implement calculations in wider energy ranges. We extended the energy interval for the total cross-sections and S −factors up to 5 MeV, including resonances in the continuum. The numerical signature of this extension is seen in Fig. 10 for the reaction rate.

It was also shown in the present work that MPCM had predicted the absence of resonance at 6.73 MeV in the nucleon channel [14, 15] , which was confirmed by the LUNA results [9] , as well as, indirectly, by the data from [30] .

We suggest that the NACRE [43] and NACRE II [16] databases should be updated in light of LUNA data [9] and present calculations. DEDICATION We dedicate this paper to the memory of our colleague, Dr. Albert Dzhazairov-Kakhramanov, who recently passed away from COVID19. 

A solution of a two-body problem for a discrete energy spectrum with a given potential requires finding the binding energy of the system and the wave function of the state. This problem can be solved using the Variational Method and the Finite Difference Method (FDM) [45] . If both methods are used for the same system of particles, it is possible to control the correctness of the search for the binding energy and WF of the state. We already have used such an approach for p 2 H and p 3 H systems in [28, 46] and demonstrated that the FDM provides more precise description of the systems. Below we present the FDM approach.

The calculation of the binding energy of a two-cluster system by the FDM relies on the representation of the Schrödinger equation in finite differences [47] . The radial equation for the central potential [45] u ′′ L (r) + k 2 − V(r) u L (r) = 0 (A1) with some boundary condition for k 2 < 0, (k 2 = 2µE/ ̵ h) takes the form of a Sturm-Liouville type boundary value problem. Recasting the second derivative in finite difference form, we obtain

and (A1) becomes a closed system of linear algebraic equations. Thus, for a certain k 0 , D N (k) = 0

Eq. (A3) allows one to determine the binding energy E b of a system of two particles. The elements of the tridiagonal determinant (A3) are defined as follows:

Here η is the Coulomb parameter, k = | k 2 | is the wave number expressed in fm −1 and determined by the energy of interacting particles in the input channel, and V n = V(r n ) is the interaction potential of clusters at the point r n = nh from the interval of zero to R. The number of equations N or the dimension of the determinant, which usually turns out to be in the range 100 000 − 1 000 000 [45] , h = ∆r/N is the step of the finite difference grid and ∆r is the solution interval of the system (usually from zero to r N = R).

By writing f (k,η,L,Z n ) in the form given in Eq. (A4) it is possible to take the Coulomb interaction into account [33] . The form of the logarithmic derivative of the WF in the external region can be obtained from the integral representation of the Whittaker function [33] f (k, η, L,

where

Calculations show that the value S (η,L,Z) does not exceed 1.05, and its effect on the binding energy of a two-particle system is negligible [45] . When f (k,0,0,Z) = −k in Eq. (A5), the binding energy search process is noticeably accelerated. The calculation of the band determinant D N (k) for a given k is carried out using recurrent formulas of the form [47] D −1 = 0, D n = θ n D n−1 − α n D n−2 , D 0 = 1, n = 1, . . . , N.

Any energy E or wave number k that leads to zero determinant

is an eigenenergy of the system E b or k 0 , and the wave function at this energy, determined by recurrent process below, is an eigenfunction of the problem. Methods for determining the zero of some functional of one variable k are well known [48] . The number N D of determinant values is determined automatically from the accuracy condition of the binding energy value. The latter one is usually set to the level ≈ 10 −5 -10 −9 MeV, and r N = R is fixed on the range 20-30 fm [45] . After determining the eigenenergy E b , the WF of this state is sought. To find the shape of the eigenfunctions of bound states, the recurrent procedure u 0 = 0, u n = θ n−1 u n−1 + u n−2 , u 1 = const, n = 2, . . . , N.

(A9)

is carried out, where u 1 is an arbitrary number, usually fixed on the range 0.01-0.1 [48] . For bound states, the determined WF is normalized to unity. Comparing it to Whittaker asymptotics, one can find an asymptotic constant denoted by C w (see Sec. IV).

The WF search area R is usually of 20 to 30 fm, and the number of steps N WF for the desired WF is fixed between 10 000 and 50 000. Only in the case of a very low binding energy (0.1-0.2 MeV) the WF search area increased to 100-200 fm or more.

The recurrence relation (A9) is also used to search for WFs in the case of a continuous spectrum of eigenvalues at predetermined positive energy (k 2 > 0) of interacting particles [45] . However, the WF must now be matched with asymptotics of the form 

Matching the numerical solution u L (R) of Eq. (A1) for two points at large distances (R on the order of 10-20 fm) with asymptotics (A10), it is possible to calculate the scattering phase shifts for each value of the momenta JLS for a given energy of interacting particles, as well as the normalization of the WF for scattering processes [45] . To calculate the WF, one can also use the Numerov method [49] . When the number of steps exceeds 10 000, both methods yield the same results within the typical required accuracy. Such results can be compared by calculating the values of AC or charge radii for the BS or the matrix elements for the scattering processes [45] .

Essays in nuclear astrophysics: presented to William A. Fowler, on the occasion of his seventieth birthday

Nuclei in the Cosmos: Proceedings of the Second International Symposium on Nuclear Astrophysics

Atomic Data and Nuclear Data Tables

A Unified Theory of the Nucleus

Clusters as Subsystems in Light Nuclei

Nuclear Structure

Thermonuclear processes in Stars and Universe

Energy levels of light nuclei A = 5

Handbook of mathematical function

Supplement to I/25 A-F

Nuclear physics of stars

Thermonuclear reaction rates V

Methods for calculating nuclear characteristics. Nuclear and thermonuclear processes

Application of Numerical Methods to Neutron Cross-Section Calculations

Mathematical Handbook for Scientists and Engineers

Solving Ordinary Differential Equations I

Appendix B