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(Uzbekistan, 2018)

International Journal of Modern Physics: Conference Series Vol. 49 (2019) 1960017 (14 pages)

c

The Author(s)

DOI: 10.1142/S2010194519600176

Determination of the asymptotic normalization coefficient (nuclear vertex constant) for α + d →6_{Li from the new direct}

measured d(α, γ)6Li data and its implication for extrapolating the d(α, γ)6_{Li astrophysical S factor at Big Bang energies}

K. I. Tursunmakhatov

Physical and Mathematical Department of Gulistan State University, Gulistan 120100, Uzbekistan

R. Yarmukhamedov∗

Institute of Nuclear Physics, Uzbekistan Academy of Sciences, Tashkent 100214, Uzbekistan

[email protected]

Published 25 July 2019

The results of the analysis of the new experimental astrophysical S factors Sexp₂₄ (E) [D. Trezzi et al., Astropart. Phys. 89, 57 (2017)] and those measured earlier [R. G. Robertson et al., Phys. Rev. Lett. 47, 1867 (1981)] for the nuclear-astrophysical d(α, γ)6_{Li reaction directly measured at extremely low energies E, which is derived} within the modified two-body potential method, are presented. New estimates and their uncertainties have been obtained for values of the asymptotic normalization coefficient for α + d →6_{Li and for the direct astrophysical S factors at Big Bang energies.} Keywords: Astrophysical S factor; asymptotic normalization coefficient; Big Bang energy.

1. Introduction

At present, the nuclear-astrophysical radiative capture reaction

d + α →6Li + γ (1)

is of great interest due to the so-called second lithium puzzle, which is associated with an existence of three order of the discrepancy between the observational and

∗_{Corresponding author.}

This is an Open Access article published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 4.0 (CC-BY) License. Further distribution of this work is permitted, provided the original work is properly cited.

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calculated ratios6Li/7Li.1, 2Besides, the reaction (1) is considered as the only source of the 6_{Li production in the standard Big Bang model.}3 _{The amount of the} 6_Li

production in the Big Bang via the reaction (1) depends in turn on the nuclear cross sections (or respective astrophysical S factors S24(E)) at Big Bang energies

(30 . E . 400 keV).

Despite the impressive improvements in our understanding of the reaction (1) made in the past decades (see Refs. 1–3 and references therein), however, some ambiguities connected with both the extrapolation of the measured astrophysical S factors S₂₄exp_{(E) to the energy region (E . 100 keV) and the theoretical predictions} for S24(E) still exist and they may influence the predictions of the Big Bang model.3

Experimentally, there are two types of data for S₂₄exp(E) at astrophysically rel-evant energies: i) four direct experimental data of S₂₄exp(E) presented in Refs. 4–7, where the experimental errors in the astrophysically relevant energy region (80 ≤ E ≤ 1316 keV) change from 7.3% at E = 1316 keV to more than 100% at E = 80 keV (but, 60% at E = 93 keV),4, 6, 7 whereas, the data measured in Ref. 4 cover energies of 1–3.5 MeV; ii) two indirect data obtained from the Coulomb breakup

208_Pb(6_{Li, αd)}208

Pb reaction in the energy regions of 70 . E . 410 keV in Ref. 8 and 107 ≤ E ≤ 250 keV in Ref. 9. The data obtained in Refs. 4, 6, 7, 9 have a similar energy dependence for the astrophysical S factors S24(E), which differ from

that of the data in Ref. 8. Moreover, the S₂₄exp(E) data of Ref. 8 have been obtained by using the interpolated formula S24(E) = (0.91 ± 0.18) ± (2.92 ± 0.66E) in the 208_Pb(6_{Li, αd)}208_{Pb triple-differential cross section (TDCS) in the energy region}

mentioned with taking into account only the E2 contribution. Besides, in Ref. 8, the experimental TDCSs in the energy region above have rather large errors. As is seen from here, the presence of the sufficiently large experimental errors impedes extrapolating of these measured data to low experimentally inaccessible energies in a correct manner.

The most late theoretical calculations of S24(E) have been performed within

different two-body10–15 _{and three-body}16–19 _{potential models as well as}

semi-microscopic20 _{and microscopic}21, 22 _{models. These methods show considerable}

spread in the calculated S24(E) at Big Bang energies and the result depends

notice-ably on a specific model used. Nevertheless, in most cases theoretical calculations show practically the same energy dependence for the calculated S24(E), but, they

have different normalization. These ambiguities are mainly associated both with a choice of the methods and with the used input free parameters, which must really be adequate to physics of the considered reaction.

Therefore, it is of great interest to apply a modified two-body potential method (MTBPM) proposed in Ref. 23 for reanalysis of the direct measured astrophysical S factors4, 6, 7of the reaction (1). In the MTBPM, ambiguities, which arise due to the used input free parameters, can be reduced within the experimental errors for the astrophysical S factors.

Below, we use the system of units c = ~ = 1.

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2. Basic Formulae of the MTBPM

Here we present only the idea and the essential formulae of the MTBPM23 special-ized for the direct astrophysical S factor for the reaction (1) that are important for the following analysis.

Within the two-body (α + d) potential method the matrix element of the reac-tion (1) in the long-wavelength approximareac-tion can be presented to the form23, 24

M = hIαd6Li(ρ)|O(ρ)|Ψ (+)

k (ρ)i. (2)

Herein:

M = I_αd6Li(ρ) = N_αd1/2hΨα(ζα)Ψd(ζd)|Ψ6_Li(ζ_α, ζ_d; ρ)i (3)

is the overlap function of the antisymmetric internal wave functions of the α particle (Ψα(ζα)), the deuteron (Ψd(ζd)) and the6Li nucleus (Ψ6_Li(ζ_α, ζ_d; ρ))25; Ψ(+)_k (ρ) is

the wave function of a relative motion of the α particle and the deuteron in the initial state with the relative momentum k and O(ρ) is the electromagnetic-transition operator, where Nαdis the factor taking into account the nucleons identity (Nαd= 9

for the6_{Li nucleus in the (α + d) configuration),}26_ζ

Ais a set of the internal relative

coordinates of the bound A system and ρ is the radius vector connecting the centers of mass of the α particle and the deuteron.

In the matrix element (2), the contribution of the (α + d) configuration for the residual 6_{Li nucleus to the amplitude of the reaction (1) can assume to be}

dominant. Besides, only the s wave for the overlap function for the six-nucleonic6_Li

wave function in the (α + d) channel is taken into account in our calculations. This approximation is associated by the fact that the absolute value of the asymptotic normalization coefficient (ANC)26of the radial overlap function (denoted by α+d →

6_{Li everywhere below) corresponding to the d wave is very small.}20, 27, 28_Therefore,

the contribution of the d wave component of the overlap function to the calculated astrophysical S factors is strongly suppressed and, so, it can be ignored.20, 27 _The

asymptotic behaviour of the s wave component of the overlap integral (I0(ρ) in

which the N_αd1/2 factor is absorbed) outside the range of the nuclear interaction ρ > ρ(N) _{is given by the relation:}

I0(ρ) ' C0W−η_{6 Li};1/2(2καdρ)/ρ, (4)

where C0 is the s wave ANC, which is proportional up to the known multiplier

to the nuclear vertex constant (NVC) (G0) for the virtual decay 6Li → α + d26;

Wα;β(ρ) is the Whittaker function; η6_Li = 2e2µ_αd/κ_αd is the Coulomb parameter

for the 6_{Li[= (α + d)] bound state in which κ} αd =

√

2µαdαd; αd is the binding

energy with respect to the virtual decay6_{Li → α + d; µ}

αd is the reduced mass for

the α particle and the deuteron and ρ(N) is the nuclear interaction radius between the α particle and the deuteron in the (α + d) bound state.

In the standard two-body potential model calculation, the unknown radial over-lap function I0(ρ), which indeed satisfies a set of the integro-differential equations,25

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is approximated by a s wave model function as following I0(ρ) ≈ Z

1/2

0 ϕn(ρ). (5)

Herein, ϕn(ρ) is the s wave single-particle wave function of the bound 6Li(α + d)

state, which satisfies the radial Schr¨odinger equation and is calculated usually with the phenomenological Woods-Saxon potential containing the geometric parameters (the radius r0 and the diffuseness a), and Z0 is the s wave spectroscopic factor

(SF).26 _{The SF Z}

0 is the norm of the I0(ρ) overlap function and determines the

probability of the (α+d) configuration in the6_{Li nucleus. The asymptotic behaviour}

of the ϕn(ρ) wave function for ρ > ρ(N) has the form as

ϕn(ρ) ' b0W−η_{6 Li};1/2(2καdρ)/ρ, (6)

where b0 is the s component of the single-particle ANC and n is the number of

nodes of ϕn(ρ).

According to Ref. 23, within the framework of the MTBPM, the astrophysical S factor for the peripheral direct capture reaction (1) can be presented in the form S24(E) = C02R24(E; b0), (7) where R24(E; b0) = ˜ S24(E; b0) b2 0 , S˜24(E; b0) = X λ [ ˜S_24;λ(Eλ)(E; b0) + ˜S (M λ) 24;λ (E; b0)]. (8)

Herein ˜S_24;λ(Eλ)(E; b0) and ˜S (M λ)

24;λ (E; b0) are the electric and magnetic components of

the single-particle astrophysical S factors ( ˜S24(E; b0)),29 respectively, and λ is the

multipole order of the electromagnetic transition. The explicit forms of the ˜S_24;λ(Eλ) and ˜S(M λ)_24;λ components are rather cumbersome. Nevertheless, one notes only that they contain the radial integral

IlijiL(E) =

Z ∞

0

dρϕn(ρ)ρL+2ψliji(ρ; E), (9)

where L = λ (λ − 1) for the electric (magnetic) transition, ψlijiis the radial relative

wave function for the dα-scattering in the initial state and li(jj) is the orbital (total)

angular momentum of the relative motion of the α particle and the deuteron in the initial state. In Eqs. (7) and (8), the parameters b0 and C0are not known.

To make the dependence of the calculated ˜S24(E; b0) and R24(E; b0)

func-tions on the (r0, a) pair more explicit, in Eq. (9), we split the space of

inter-action of the colliding particles into two parts separated by the channel radius ρ(N): 1) the interior part (0 ≤ ρ ≤ ρ(N)) denoted by I_l<

ijiL(E) below, where

nuclear forces between the α particle and the deuteron are important; 2) the exterior part (ρ(N) ≤ ρ < ∞) denoted by I>

lijiL(E) below, where the

interac-tion between the α particle and the deuteron is governed by Coulomb force only. The I_l>

ijiL(E) function contains explicitly the free model parameter b0 since the

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wave function ϕn(ρ) for ρ > ρ(N) can be approximated by its asymptotic

behav-ior (6). In this case, in Eq. (9), the dependence of the model wave functions (ϕn(ρ)

and ψliji(ρ; E)) on the geometric r0 and a parameters of the adopted

Woods-Saxon potential is determined by the single-particle ANC b0[= b0(r0, a)30] for

the ϕn(ρ) wave function and by the free r0 and a parameters for the ψliji(ρ; E)

wave function. This means indeed that ϕn(ρ) = ϕn(ρ; b0) ≡ ϕn(ρ; b0(r0, a))30 and

ψliji(ρ; E) = ψliji(ρ; E; r0, a). Hence, the I

<

lijiL(E) function depends both on the

tree r0 and a parameters and on the model single-particle ANC b0(r0, a), i.e.,

IlijiL(E) = IlijiL(E; b0(r0, a); r0, a) = I

<

lijiL(E; b0(r0, a); r0, a) + I

>

lijiL(E; b0(r0, a)).

From here and Eq. (9), the expression (8) can be reduced to the form as

R0(E; b0) ∼X λ      C_λ(Eλ)X jili δliλA (Eλ) lijiλ[I <

lijiλ(E; b0(r0, a); r0, a)/b0(r0, a) + ˜I

> lijiλ(E)] 2 + C_λ(M λ)X jili δliλ−1A˜ (M λ) lijiλ[I <

lijiλ−1(E; b0(r0, a); r0, a)/b0(r0, a) + ˜I

> lijiλ−1(E)] 2     . (10) Herein C_λ(Eλ)= µλ_αd[2/mλα+ (−1)λ/mλd], C (M λ) λ = mpµλαd(2/m λ+1 α + 1/m λ+1 d ), ˜ I_l> ijiL(E) = Z ∞ ρ(N) dρW_−η 6 Li;1/2(2καdρ)ρ L+1_ψ(as) liji(ρ; E), (11) where ψ(as)_l

iji(ρ; E) is the asymptotics of the radial dα-scattering wave function and

C0= Z 1/2 0 b0(r0, a). (12) In (10), A(Eλ)_l ijiλand ˜A (M λ)

lijiλ are the known spin multiplier for the electric and magnetic

transitions, respectively, |li− Jd| ≤ ji ≤ li+ Jd in which Jd = 1 is the spin of the

deuteron, and δij is the Kronecker-Copelli symbol.

As is seem from Eqs. (7), (8) and (10), the contribution of the exterior part to the matrix element (2) (or S24(E)) is determined only by the ANC (C0). But, the

contribution of the interior part to the matrix element depends both on the SF (Z0)

and on the model free parameters (b0(r0, a), r0and a) of the bound and continuum

states. Therefore, the simultaneous change of the values of b0(r0, a) for the bound

state and the r0 and a parameters for the dα-scattering may strongly change the

contribution of the nuclear interior region to the radial integral (9) (or the matrix element (2)). It may become a cause of arising of the additional uncertainty in the S24(E) values calculated at Big Bang energies. One notes that the expression

for S24(E) derived within the framework of the standard two-body method can

be obtained by inserting Eq. (12) in the right-hand side of Eq. (7) in which the astrophysical S factor is expressed in the term of the SF Z0.

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In Eqs. (10) and (11), the exterior part of the radial matrix element does not contain explicitly the free model parameter b0. Consequently, the parametrization

of the astrophysical S factor in the form (7) makes it possible to fix a contribution from the exterior region (ρ(N) _{≤ ρ < ∞), which is dominant for the peripheral}

reaction (1),31 _{by a model independent way if the squared ANC C}2

0 are known. In

Eq. (7), the contribution from the interior part into the R0(E; b0) function must

exactly determine the dependence of the R0(E; b0) function on b0. For the

periph-eral reaction considered at Big Bang energies, this contribution into the R0(E; b0)

function must strongly be suppressed. Therefore, the equation (7) can be used for determination of the squared ANC C2

0.

Thereto, the following additional requirements23

R0(E; b0) = f (E) (13)

and

C₀2= S24(E) R0(E; b0)

= const. (14)

must be simultaneously fulfilled as a function of the free parameter b0 for each

energy E = Ei(i = 1, 2, . . . , N , and N is a number of the experimental point) from

the range E1≤ E ≤ ENand values of the function of R0(E; b0) from (13).

The fulfilment of the relations (13) and (14) (or their violation within the exper-imental uncertainty for S₂₄exp(E) and the experimental phase shifts for the α + d scattering) enables one, firstly, to determine an interval for energies E where the dominance of extra-nuclear capture occurs and, secondly, to obtain the “indirectly determined” (“experimental”) value (C₀exp)2 _{for α + d →} 6_{Li using the directly}

measured astrophysical S factors S₂₄exp(E)4, 6, 7 _{instead of S}

24(E), i.e. (C₀exp)2= S exp 24 (E) R0(E; b0) . (15)

Then, the “experimental” ANC (C₀exp)2 can be used for extrapolation of the astrophysical S factor S24(E) to the region of experimental inaccessible energies

(0 ≤ E < E1), including Big Bang ones, by using the obtained value (C exp 0 )

2_{in (7).}

3. Asymptotic Normalization Coefficient (Respective Nuclear Vertex Constant) for α + d →6Li and the Astrophysical S Factors for the d(α, γ)6_{Li at Big Bang Energies}

To determine the ANC (NVC) value for α + d →6_{Li, the recent and earlier directly}

measured experimental astrophysical S factors, S₂₄exp(E), for the radiative capture reaction (1) are reanalyzed based on the relations Eqs. (7) and (8) as well as Eqs. (13)–(15). The experimental data analyzed by us cover the energies E = 93, 120 and 133 keV7 as well as E = 993 and 1315 keV4 for which only the external capture is substantially dominant.31_{One notes that the S}exp

24 (E) data measured in

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Ref. 6 for E = 94 and 134 keV coincide practically with that of Ref. 7 considered only below.

For this reaction, the li= 1 and 2 for the E1 and E2 transitions, whereas, li= 0

for the M 1 transition. In Eqs. (7) and (8) (see Eq. (10) also), the experimental values for the masses of the deuteron and the α particle are used, which are equal to 2.013553212724 and 4.001506179127 amu, respectively. In this case, as is shown in Ref. 10 for the first time, the reaction (1) at low energies can be considered within the two-body cluster model not using the isospin formalizm in which the isoscalar E1 transition (Ti = 0 → Tf = 0) is forbidden, where Ti and Tf are the

total isospins of the initial and final state wave functions, respectively. The problem of the forbidden isoscalar E1 transition, arising within the total microscopic isospin formalizm considered in Ref. 24 for the first time, is discussed in detail in Ref. 18, where the microscopic three-body (α + n + p) model is used. Though, a use of the exact experimental masses mentioned in the calculations of S24(E), performed

in Ref. 10 in the two-body (α + d) cluster model, resulted in the fact that the contribution of the E1 transition to the S24(E) calculated at low energies may be

the same order (even large) as that of the E2 one.

The real potential in the Woods-Saxon form with spin-orbital term used in Ref. 4 is taken both for the continuum state and for the bound one. We vary the geomet-ric parameters (r0 and a) of the adopted Woods-Saxon potential in the physically

acceptable ranges (r0 in 1.13 ÷ 1.37 fm and a in 0.58 ÷ 0.72 fm) with respect to the

standard (r0 = 1.25 fm and a = 0.65 fm) values using the procedure of the depth

adjusted to fit the binding energies for the bound state and to describe simultane-ously the experimental phase shifts within their errors for the continuum state. The contribution of the M 1 transition to the calculated astrophysical S factor is negli-gible small (∼1–2%), whereas, that of the E1- and E2-components are important. A choice of the limit of variation of the parameters r0 and a above allows us

to supply fulfilment of the conditions (13) and (14) for the energies above with the high accuracy as a comparison with the experimental errors for the S₂₄exp(E). Below, as an example, the limits of the change of the ˜S24(E; b0) and R0(E; b0) functions

as well as the SF Z0 and the squared ANC C02 in a dependence on variation of the

single-particle ANC b0are given only at E = 93 keV, where b0is changed within the

ranges 2.369 ≤ b0≤ 2.858 fm−1/2. They are 2.82 × 10−6≤ ˜S24≤ 4.08 × 10−6keV·b,

0.66 ≤ Z0 ≤ 0.96, 4.85 × 10−7 ≤ R0 ≤ 5.15 × 10−7keV·b·fm−1 and 5.24 ≤ C02 ≤

5.57 fm−1. Here the experimental astrophysical S factor (S₂₄exp(E)) at E = 93 keV is taken instead of the S24(E). It is seen that the change of the ˜S24(E; b0) and

Z0 on a variation of b0 within the interval above is rather noticeably (about 1.45

times), whereas the same change for the R0(E; b0) function and C02 is rather weak

(1.06 times). The same dependence is also observed at the other energies above. It follows from here that the conditions (13) and (14) is fulfilled for the considered reaction with a high accuracy not exceeding the experimental errors of Sexp₂₄ (E).

To test self-consistency, we also calculated phase shifts of the dα-elastic scatter-ing by variation of the parameters r0 and a in the range mentioned above for the

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adopted Woods-Saxon potential. The results of the calculations of the phase shifts for the 2_S

1, 3Pj (j = 0, 1 and 2) and 3Dj (j = 2 and 3) waves show rather a well

agreement with the experimental data.32–35 Besides, the calculated values of phase shifts with respect to variation of the r0and a parameters within the intervals above

are changed within the uncertainty of about ∼2–3%.

As is seen from the analysis performed above, if the SF, which enters the expres-sion for S24(E) calculated within the framework of the standard two-body method,

should be set to Z0= 1, as it is assumed in Refs. 12, 14, 29, then, in reality, three

are infinite number of the phase-equivalent Woods-Saxon potentials resulting in the calculated S24(E) values with the theoretical uncertainty about 45%. This is a result

of the fact that the ANC C0 (C0= b0(r0, a)) becomes strongly a model-dependent.

Though, all these potentials lead to the calculated phase shits, which are in a well agreement with the experimental data within the uncertainty up to ∼3%. It fol-lows from here that the results for S24(E) derived in Refs. 12, 14, 29 are strongly

a model-dependent. In this case, the SF Z0 cannot be determined unambiguously.

That is the main reason why the SF value should not be set to Z0= 1, a priori.

Thus, it is seen that the reaction (1) at sufficiently low energies is strongly peripheral and the contribution of the nuclear interior dα interaction region to the astrophysical S factors is up to 3%. Consequently, for S24(E) given by Eq. (7), a use

of parametrization in terms of the squared ANC is adequate to the reaction physics. Taking into account this fact, the values of the squared ANC ((C₀exp)2_{) can be}

obtained by using the corresponding S₂₄exp(E) in Eq. (15) for each experimental point

Fig. 1. The squared ANC for α + d → 6_{Li and the astrophysical S factors for the d(α, γ)}6_Li reaction. In (a): the values of the squared ANC, ((C₀exp)2_{), obtained for each the experimental} points of the energy E, where the solid line present our results for the weighted mean and the width of the band is the weighed uncertainty; In (b): the experimental and calculated astrophysical S factors; the experimental data are taken from Ref. 7 (open triangle symbols) and Ref. 8(square symbols); the solid, dashed and dashed-dotted lines in (b) are our result for the total, E2 and E1 compotents of the S24(E), respectively; star points correspond to our data at Big Bang energies; all the calculated data are the result of calculation performed with the standard values of geometric parameters (r0= 1.25 fm and a = 0.65 fm) and the weighted mean and uncertainty for (Cexp0 )2. The width of each of the band corresponds to the weighted uncertainty for the squared ANC.

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of E (E = 93, 120, 133, 993 and 1315 keV). The results are displayed in Fig. 1(a). The uncertainty plotted for each the experimental point of the energy E is the averaged squared error found from Eq. (15), which involves both the experimental errors in the corresponding S₂₄exp(E) and the aforesaid uncertainty in the R0(E; b0).

As is seen from Fig. 1(a), the ratio in the right-hand side of (15) practically not depend on the energy within the experimental errors for S₂₄exp(E), although absolute values of the S₂₄exp(E) data for the energies above depend noticeably on the energy E and are changed by up to 22.6 times. The result of the weighed mean and its weighed uncertainty for the squared ANC ((C₀exp)2_{) for α+d →}6_{Li and the squared}

modulus of the respective NVC (|Gexp₀ |2_{) for the virtual decay}6_{Li → α + d as well}

as those obtained by other authors are presented in Table 1.

As is seen from Table 1, the resulting ANC (NVC) value obtained by us is in a good agreement with the values recommended in Refs. 9, 20, 36–40, which were derived within the quite independent methods. In this connection, one notes

Table 1. The ANC (C2

0) for α + d →6Li and the respective NVC (|G0|2). Figures in square bracket are experimental and theoretical uncertainty, respectively, whereas, those in bracket are the weighed means and their total uncertainties.

The method and the reaction C2 0, (fm −1₎ _|G 0|2, (fm) Refs. TBPM the d(α, γ)6_{Li analysis}(1) _{5.41 [0.18; 0.12]} (5.41 ± 0.21) 0.423 [0.014; 0.009] (0.423 ± 0.017) The present work TBCBM208_Pb(6_{Li, αd)}208_{Pb with}

the E1- and E2-multipoles(2)

5.50 [0.46; 0.45] (5.50 ± 0.64)

0.43 [0.04; 0.04] (0.43 ± 0.05)

9

The analytical continuation for the phase shifts of the dα-scattering using the Pad´e-approximation

5.37 ± 0.26 0.42 ± 0.02 37

The dispersion peripheral model with the exchange

d6_{Li-scattering}

5.24 ± 0.77 0.41 ± 0.06 36

The analytical continuation for the DCS of the exchange

d6Li-scattering using the Pad´e-approximation(3)

4.22 ± 1.28 0.33 ± 0.10 38

The three body (α − n − p) wave function for6_Li

5.24 0.41 39, 40

The microscopic three body (α − n − p) model(4)

6.66 (7.73) 0.52 (0.60) 22

The six-body wave function calculation(5)

5.10 0.40 20

The three body (α − n − p) hyperspherical method(6)

4.20 0.33 18

The three body (α − n − p) 4.48 0.35 19

Faddeev’s method(7) _{2.69 ÷ 6.66} _{0.21 ÷ 0.59} ₄₂

Notes: (1)Two-body potential method (TBPM); (2)three-body Coulomb breakup method (TBCBM);(3)_{the differential cross section (DCS);}(4)_{for the MN(V2) form of the N N potential;} (5)_for6_{Li with the Argonne v}

18and Utbana IX forms of the N N potential;(6)for the MN and Voronchev et al. forms of the N N and N α potentials, respectively;(7)_{for the six and four forms} of the N N and N α potentials, respectively.

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that, in Ref. 38, the squared ANC (NVC) was extracted from the analysis of the experimental differential cross section (EDCS) of the exchange d6_{Li scattering at}

the back center-of-mass scattering angles. The analysis was performed within the analytical continuation (extrapolation) method26_{in which the Pad´}_{e-approximation}

for the extraplated EDCS is used, similar to that used in Ref. 37 for the experimental phase shifts of the dα-scattering. At this, the overall uncertainty for the squared ANC (∆C2

0) derived in Ref. 38 has been obtained within the posed-ill problem41

in a correct manner, which provides the stability of the analytical extrapolation (∆C2

0 → 0 at ∆σexp→ 0, where ∆σexp is the error of the EDCS). One notes that

the uncertainty involves both the errors of the EDCS (∆σexp_{) and that arising}

due to a choice of the acceptable orders of the Pad´e-approximation. Therefore, the uncertainty for the squared ANC C2

0derived in Ref. 38 is correct from the standpoint

of the posed-ill problem. Whereas, the uncertainty of the ANC (NVC) obtained in Ref. 37 involves only the uncertainty associated with the acceptable orders of the Pad´e-approximation. As is seem from here, due to the fact that the same analytical continuation method is used in Refs. 37, 38, the uncertainty derived in Ref. 37 would also be obtained within framework of the posed-ill problem. Apparently, this is one of the reason why the ∆C2

0 value in Ref. 37 is underestimated against to that in

Ref. 38. Our result is in reasonable with that of Ref. 20 obtained the six-nuclonic calculations for6Li using the Argonne v18and Utbana IX forms of the N N potential

as well as with that of Refs. 39, 40 obtained in the three-body (α+n+p) calculations for the6Li wave function using the Reid and exchange Majorana component forms for the N N and N α potentials, respectively. This means that the potentials used in Refs. 20, 39, 40 reproduce well the asymptotic behavior of the radial overlap function of the6_{Li nucleus in the (α + d) channel given by Eq. (4).}

Nevertheless, the result of the present work differs noticeably from the calculated values obtained in Refs. 18, 19, 22, 42 within the framework of the different body methods. Moreover, the results of Refs. 22 and 42, obtained within the three-body (α + n + p) cluster microscopic generator coordinate method and the strict Faddeev’s one, respectively, show a noticeable sensitivity to the forms of the used N N and N α potentials. One of them that of C2

0 = 5.56 fm−1 (|G0|2 = 0.45 fm),

calculated in Ref. 42 using the Malfiet-Tjon and Sack-Biedenharm-Breit forms the N N and N α potentials, respectively, is the closest to that obtained in the present work.

Besides, one should note the results of Ref. 43 obtained for the squared ANC from the analysis of the Coulomb breakup208Pb(6Li, αd)208Pb reaction. The anal-ysis is performed by using the three-body method proposed in Ref. 9. At this, it is assumed that the projectile 6_{Li moves on the Rutherford trajectory with the}

impact parameter b. As a result, the values of the squared ANCs(NVCs) obtained in Ref. 43 become noticeably overestimated [C2

0 = 30.51 ± 1.70 and 6.72 ± 0.73 fm−1

(|G0|2 = 2.75 ± 0.15 and 0.61 ± 0.07 fm)] as a comparison with that in Ref. 9

(see Table 1). In this connection, one notes that, in Ref. 9, the contribution of the Coulomb208Pb−6Li interaction region (R0≤ R ≤ b), which is larger than the radius

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of the nuclear208Pb −6Li interaction, but less than the Rutherford impact distance, to the matrix element is also taken into account. Here, R is the relative distance between the centers of mass of the colliding nuclei, R0= RN + πzLizPbe2/4Ei< b

in which zβe is a charge of the particle β, RN is the radius of the nuclear208Pb −6Li

interaction and Ei is a relative kinetic energy of the colliding nuclei. Due to the fact

that the mutual good agreement between the ANC obtained by the independent method in the present work and that recommended in Ref. 9 occurs, the ANC of Ref. 43, obtained without taking into account the contribution of the R0≤ R ≤ b

region to the matrix element, is not reliable. It is related to the S24(E) values

derived in Ref. 43 by using the overestimated C2

0 values of Ref. 43.

Thus, it is seen from above that the mutual agreement between the result obtained here and other authors within the aforesaid different methods9, 20, 36–40

confirms a correctness of the assumption made by us above of the dominant contri-bution of the (α + d) clusterization to the sufficiently low-energy d(α, γ)6_{Li S factor}

both in the absolute normalization through the squared ANC and in the energy dependence.

Below, the weighted mean of the squared ANC (C₀exp)2 _{derived in the present}

work is applied for calculation of the astrophysical S factors for the d(α, γ)6Li at Big Bang energies using Eq. (7). The results of extrapolation of the astrophysical S factor at Bing Bang energies E (E = 30 ÷ 400 keV) are displayed in Fig. 1(b) by the solid line and stars points. The overall uncertainty for the calculated S24(E)

is about 4% on the average, which involves both the experimental errors and the theoretical uncertainty. It is seen that our result is in an excellent agreement with the experimental data.4, 7, 9_{Whereas, the noticeable discrepancy occurs between our}

result and that of Ref. 8. In this connection, one should note that the latter was obtained by using the interpolate formula mentioned above. Its energy dependence reproduces only that of the E2 component of the total S24(E) but not the E1

component of it. One notes that the calculations of S24(E) performed in Refs. 20,

44 at Big Bang energies show higher than that presented here. For example, at the most effective Big Bang energy (E = 70 keV), our result is S24(70 keV) = 2.424 ±

0.081(exp) ± 0.054(theor)[2.424 ± 0.097(total)] MeV nb, whereas that is 4.0 MeV nb in Ref. 20 and 3.16 MeV nb in Ref. 44. Besides, our result for S24(70 keV) is in a good

agreement with that of 2.58 eV nb obtained in Ref. 15. Though, the uncertainty of the latter can be larger than that obtained the present work. It is associated with the model assumption used in Ref. 15.

4. Conclusion

The scrupulous analysis of the earlier and recent directly measured experimental astrophysical S factors, S₂₄exp(E), for the reaction (1) at energies E = 93– 1315 keV,4, 7 _{beyond the resonance energy region, has been performed within the}

MTBPM.23It shows quantitatively that the reaction within the considered energy ranges is strongly peripheral and a use of the parametrization of the direct

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astrophysical S factors in terms of ANC for α + d →6Li is adequate to physics of the peripheral reaction (1).

The new estimation for the weighted mean of the ANC(NVC) are obtained to be (C₀exp)2 _{= 5.41 ± 0.21 fm}−1 _(|G

0|2 = 0.423 ± 0.017 fm), which has the overall

uncertainty about 4% on the average. They are in a good agreement with those obtained in Refs. 9, 20, 36–40. Though, they differ noticeably on those obtained in Refs. 18, 19, 22.

The found ANC was also used for an extrapolation of astrophysical S factors at energies less than 93 keV, including up to E = 0. In particular, at the most effective Big Bang energy(E = 70 keV), our result is S24(70 keV) = 2.42 ± 0.10 MeV nb with

the overall uncertainty about 4% on the average. The result is in an agreement with S24(70 keV) = 2.58 MeV nb15and noticeably lesser than that in Ref. 20, 44.

Acknowledgments

This work has been supported in part by the Ministry of Innovations and Technolo-gies of the Republic of Uzbekistan (grant No. HE F2-14).

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