Determination of the {}^{3}{\rm{He}}+\alpha\to\rm{{}^{7}Be} asymptotic normalization coefficients (nuclear vertex constants) and their application for extrapolation of the {}^{3}{\rm{He}}(\alpha,\gamma)^{7}{\rm{Be}} astrophysical S factors to the solar energy region

Determination of the asymptotic normalization coefficients (nuclear vertex constants) and their application for extrapolation of the astrophysical factors to the solar energy region

S.B. Igamov, Q.I. Tursunmahatov and R. Yarmukhamedov Corresponding author, E-mail: rakhim@inp.uz
Abstract

A new analysis of the modern precise measured astrophysical factors for the direct capture reaction [B.S. Nara Singh et al., Phys.Rev.Lett. 93, 262503 (2004); D. Bemmerer et al., Phys.Rev.Lett. 97, 122502 (2006); F.Confortola et al., Phys.Rev.C 75, 065803 (2007), T.A.D.Brown et al., Phys.Rev. C 76, 055801 (2007) and A Di Leva, et al.,Phys.Rev.Lett. 102, 232502 (2009)] populating to the ground and first excited states of is carried out based on the modified two - body potential approach. New estimates are obtained for the indirectly determined”  values of the asymptotic normalization constants (the nuclear vertex constants) for (g.s.) and (0.429 MeV) as well as the astrophysical factors at E 90 keV, including =0. The values of asymptotic normalization constants have been used for getting information about the -particle spectroscopic factors for the mirror ()-pair.

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

PACS: 25.55.-e;26.35.+c;26.65.+t

1 Introduction

The reaction is one of the critical links in the and branches of the –chain of solar hydrogen burning [1–3]. The total capture rate determined by processes of this chain is sensitive to the cross section (or the astrophysical factor ) for the reaction and predicted neutrino rate varies as [2, 3].

Despite the impressive improvements in our understanding of the reaction made in the past decades (see Refs [4–11] and references therein), however, some ambiguities connected with both the extrapolation of the measured cross sections for the aforesaid reaction to the solar energy region and the theoretical predictions for (or ) still exist and they may influence the predictions of the standard solar model [2, 3] .

Experimentally, there are two types of data for the reaction at extremely low energies: i) six measurements based on detecting of -rays capture (see [4] and references therein) from which the astrophysical factor extracted by the authors of those works changes within the range 0.470.58 and ii) six measurements based on detecting of (see [4] references therein as well as [6-11]) from which extracted by the authors of these works changes within the range 0.530.63 . All of these measured data have a similar energy dependence for the astrophysical factors . Nevertheless, the adaptation of the available energy dependencies predicted in [12, 13] for the extrapolation of each of the measured data to low experimentally inaccessible energy regions, including =0, leads to a value of that differs from others and this difference exceeds the experimental uncertainty.

The theoretical calculations of performed within different methods also show considerable spread [12,14–19] and the result depends on a specific model used. For example, the resonating-group method calculations of performed in Ref.[12] show considerable sensitivity to the form of the effective NN interaction used and the estimates have been obtained within the range of 0.312 0.841 .

The estimation of =0.52 0.03 [20] also should be noted. The latter has been obtained from the analysis of the experimental astrophysical factors [21], which were performed within the framework of the standard two-body potential model in the assumption that the dominant contribution to the peripheral reaction comes from the surface and external regions of the nucleus [22]. At this, in [20] the contribution from the nuclear interior (, =4 fm) to the amplitude is ignored. In this case, the astrophysical factor is directly expressed in terms of the nuclear vertex constants (NVC) for the virtual decays (or respective the asymptotic normalization coefficients (ANC) for ) [24, 25]. As a result, in Ref. [20], the NVC-values for the virtual decays and were obtained, which were then used for calculations of the astrophysical factors at 180 keV, including =0. However, the values of the ANCs (or NVCs) and the obtained in [20] may not be enough accurate associated both with the aforesaid assumption in respect to the contribution from the nuclear interior () and with a presence of the spread in the experimental data [21] used for the analysis. As far available values of these ANCs obtained in [13, 16], they depend noticeably on a specific model used. Therefore, determination of precise experimental values of the ANCs for (g.s.) and (0.429 MeV) is highly desirable since it has direct effects in the correct extrapolation of the astrophysical factor at solar energies.

Recently, a modified two-body potential approach (MTBPA) was proposed in [23] for the peripheral direct capture reaction, which is based on the idea proposed in paper [22] that low-energy direct radiative captures of particle by light nuclei proceed mainly in regions well outside the range of the internuclear interactions. One notes that in MTBPA the direct astrophysical factor is expressed in terms of ANC for rather than through the spectroscopic factor for the nucleus in the () configuration as it is made within the standard two-body potential method [26, 27]. In Refs.[23, 28, 29], MTBPA has been successfully applied to the radiative proton and -particle capture by some light nuclei. Therefore, it is of great interest to apply the MTBPA for analysis of the reaction.

In this work new analysis of the modern precise experimental astrophysical factors for the direct capture reaction at extremely low energies ( 90 keV) [6-11] is performed within the MTBPA [23] to obtain indirectly determined”  values both of the ANCs (the NVCs) for (g.s.) and (0.429 MeV), and of at 90 keV, including =0. In this work we quantitatively show that the reaction within the aforesaid energy region is mainly peripheral and one can extract ANCs for directly from the reaction where the ambiguities inherent for the standard two -body potential model calculation of the reaction, which is connected with the choice of the geometric parameters (the radius and the diffuseness ) for the Woods–Saxon potential and the spectroscopic factors [18, 17], can be reduced in the physically acceptable limit, being within the experimental errors for the .

The contents of this paper are as follows. In Section 2 the results of the analysis of the precise measured astrophysical factors for the direct radiative capture reaction is presented (Subsections 2.1–2.3). The conclusion is given in Section 3. In Appendix basic formulae of the modified two-body potential approach to the direct radiative capture reaction are given.

2 Analysis of reaction

Let us write () for the relative orbital (total) angular moment of and -particle in nucleus , () for the orbital (total) angular moment of the relative motion of the colliding particles in the initial state. For the reaction populating to the ground and first excited (=0.429 MeV; =1/2) states of , the values of are taken to be equal to 3/2 and 1/2, respectively, the value of is taken to be equal to 1 as well as =0, 2 for the -transition and =1 for the -transition.

The basic formulae used for the analysis are presented in Appendix.

2.1 The asymptotic normalization coefficients for

To determine the ANC values for (g.s) and (0.429 MeV) the recent experimental astrophysical factors, , for the reaction populating to the ground (=1 and =3/2) and first excited (=0.429 ; ,   =1 and =1/2) states [6–11] are reanalyzed based on the relations (A1)–(A7). The experimental data analyzed by us cover the energy ranges =92.9–168.9 keV [7–9], 420–951 keV [6], 327–1235 keV [10] and 701–1203 keV [11] for which only the external capture is substantially dominant [19, 22]. Also, in [9] the experimental astrophysical factors for the the reaction populating to the first and excited states of the nucleus have been separated only for the energies of =92.9, 105.6 and 147.7 keV . Whereas, in [10] the experimental astrophysical factors have been separated for all experimental points of from the aforesaid energy region by means of detecting the prompt ray (the prompt method) and by counting the activity (the activation).

The Woods–Saxon potential split with a parity (-dependence) for the spin-orbital term proposed by the authors of Refs. [30–32] is used here for the calculations of both bound state radial wave function and scattering wave function . Such the choice is based on the following considerations. Firstly, this potential form is justified from the microscopic point of view because it makes it possible to take into account the Pauli principle between nucleons in - and -clusters in the () bound state by means of inclusion of deeply bound states forbidden by the Pauli exclusion principle. The latter imitates the additional node () arising in the wave functions of relative motion in similarly as the result of the microscopic resonating-group method [12]. Secondly, this potential describes well the phase shifts for -scattering in the wide energy range [31, 32] and reproduces the energies of low-lying states of the nucleus [33].

We vary the geometric parameters (radius and diffuseness ) of the adopted Woods–Saxon potential in the physically acceptable ranges ( in 1.62–1.98 fm and in 0.63–0.77 fm [23]) in respect to the standard values (=1.80 fm and =0.70 fm [31, 32]) using the procedure of the depth adjusted to fit the binding energies. As an illustration, Fig.1 shows plots of the dependence on the single-particle ANC, for = 1 and =3/2 and 1/2 only for the two values of energy . The width of the band for these curves is the result of the weak residual” -dependence of the on the parameters and (up to ) for the [23, 44]. The same dependence is also observed at other energies. For example, for fig.1 plotted for =0.1056 (0.1477) MeV overall uncertainty ( ) of the function in respect to the central values corresponding to the central values of comes to = 4.5(4.5)% for the ground state of and = 3.4(2.9)% for the excited state of . As it is seen from here that the (0.429 MeV) reaction is slightly more peripheral than the (g.s.) reaction since the binging energy for (0.429 MeV) is less than that for (g.s). The similar dependence of on the values is observed for the other aforesaid energies and the value of is no more than 5.0%. It follows from here that the condition (A2) is satisfied for the considered reaction at the energies above mentioned within the uncertainties not exceeding the experimental errors of . It should be noted that values of becomes larger as the energy increases (for more than 1.3 MeV).

Thus, over the energy region 92.9E 1200 keV the -dependence on is exactly sufficiently weak being within the experimental uncertainties for the . Such dependence is apparently associated also with the indirect taking into account the Pauli principle mentioned above within the nuclear interior in the adopted nuclear potential leading as a whole to reduction of the contribution from the interior part of the radial matrix element into the function, which is typical for peripheral reactions.

We also calculated the -elastic scattering phase shifts by variation of the parameters and in the same range for the adopted Woods–Saxon potential. As an illustration, the results of the calculations corresponding to the and waves are only presented in Fig.2 in which the width of the bands corresponds to a change of phase shifts values with respect to variation of values of the and parameters. As it is seen from Fig.2, the experimental phase shifts [34, 35] are well reproduced within uncertainty of about 5%. The same results are also obtained for the and waves.

This circumstance allows us to test the condition (A3) at the energies of = 92.9, 105.6 and 147.7 keV for which the (g.s.) and (0.429 MeV) astrophysical factors were separately measured in [9]. As an illustration, for the same energies as in Fig.1 we present in Fig.3 (the upper panels) the results of -value calculation given by Eq.(A3) (=(1 3/2) and (1 1/2) ) in which instead of the the experimental factors for the reaction populating to the ground and first excited states of were taken. It should be noted that the same dependence occurs for other considered energies. The calculation shows the obtained values also weakly (up to 5.0 %) depend on the -value. Consequently, the reaction within the considered energy ranges is peripheral and a use of parametrization in terms of the ANCs given by Eq.(A1) is adequate to the physics of the reaction under consideration. However, the values of the spectroscopic factors and corresponding to the -configuration for (g.s.) and (0.429 keV), respectively, change strongly about 1.7 times since calculated that vary by 1.75 times (see, the lower panels in Fig.3).

For each energy experimental point (=92.9, 105.6 and 147.7 keV) the values of the ANCs are obtained for and (0.429 MeV) by using the corresponding experimental astrophysical factor ( and , (the activation)) [7, 9] in the ratio of the r.h.s. of the relation (A1) instead of the and the central values of corresponding to the adopted values of the parameters and . The results of the ANCs, and , for these three energy experimental points are displayed in Fig.4 and (filled circle symbols) and the second and third columns of Table 1. The uncertainties pointed in this figure correspond to those found from (A1) (averaged square errors (a.s.e.)), which include the total experimental errors (a.s.e. from the statical and systematic uncertainties) in the corresponding experimental astrophysical factor and the aforesaid uncertainty in the . One should note that the same results for the ANCs are obtained when ( and ) [7, 9] are used in Eq.(A5) (in Eq.(A6) and (A7)) instead of ( and ). Then in Eq.(A6), inserting the averaged means of (=0.666), obtained from the three data, and replacing of the in the l.h.s. of Eq.(A5) with for the others, the two experimental points of energy (=126.5 and 168.9 keV) from [7, 8], four one (=420.0, 506.0, 615.0 and 951.0 keV) from [6], the three one (=93.3, 106.1 and 170.1 keV) from [9] and the ten one (=701–1203 keV) from [11] can also determine values of ANCs, and . The results of the ANCs for (g.s.) and (0.429 MeV) are displayed in Fig.4 in which the open cycle and triangle symbols obtained from the analysis of the data of [6–9] as well as the filled triangles symbols obtained from the analysis of the data of [11](the recoils). Besides, the results obtained from the data of [7–9] are also presented in the second and third columns of Table 1. The same way the values of the ANCs are obtained by using the separated experimental astrophysical factors ( and ) [10]. The results for these ANCs are also presented in Figs.4 and both for the activation (filled star symbols) and for the prompt method (filled square symbols).

As it is seen from Figs.4, for each of the independent measured experimental astrophysical factors the ratio in the r.h.s. of the relation (A4) does not practically depend on the energy within the experimental uncertainties, although absolute values of the corresponding experimental astrophysical factors for the reactions under consideration depend noticeably on the energy and change by up to about 1.7 times when changes from 92.6 keV to 1200 keV. This fact allows us to conclude that the energy dependence of the experimental astrophysical factors [6–11] is well determined by the calculated function and . Hence, the experimental astrophysical factors presented in [6–11] can be used as an independent source of reliable information about the ANCs for (g.s.) and (0.429 MeV). Also, in Fig.4 and Table 2 the weighted means of the ANCs-values and their uncertainties (the solid line and the band width, respectively), derived both separately from each experimental data and from all of the experimental points, are presented.

As it is seen also from the first and second (fifth and sixth) lines of Table 2 the weighted means of the ANCs-values for (g.s.) and (0.429 MeV) obtained by the analysis performed separately for the activation, the prompt method and the recoils of the experimental data from the works [6–9] ([10, 11]) are in a good agreement with one another. These results are the first ones of the present work. Nevertheless, the weighted means [5] of the ANC-values obtained by using separately the experimental data of the works Refs.[6–9] and of the works Refs.[10, 11] noticeably differ from one another (up to 1.13 times for (g.s.) and up to 1.12 times for (0.429 MeV), see the parenthetical figures in Table 2). The main reason of this difference is in the systematical discrepancy observed in absolute values of the experimental astrophysical factors measured by authors of works Refs [6–9](the set I) and of the works Ref. [10, 11] (the set II, see Fig.5). Also, the central values of the weighted means for the ANC-values for (g.s.) and (0.429 MeV) obtained from all of the experimental data [6–11], which is presented in the last line of Table 2, differ up to 10% more (3% less) than those deduced from the data of [6–9]([10, 11]). As, at present, there is no a reasonable argument to prosecute to some of these experimental data measured by two groups ([6-9](the set I) and [10, 11](the set II)), it seems, it is reasonable to obtain the weighted means of the ANCs derived from all these real experimental ANCs with upper and lower limits corresponding to the experimental data of the set II and Set I, respectively. This leads to the asymmetric uncertainty for the weighted means of ANCs and this is caused with the aforesaid systematical discrepancy observed in absolute values of the experimental data of the sets I and II (see the last line of Table 2). In this connection, from our point of view a new precisely measurement of is highly encouraged since it allows one to get an additional information about the ANCs. Nevertheless, below we will use these ANCs for extrapolation of the astrophysical factors at energies down, including =0. The corresponding values of NVCs obtained by using Eq.(A8) are given in Table 2.

A comparison of the present result and that obtained in paper [20] shows that the underestimation of the contribution both of the nuclear interior and of the nuclear exterior indeed occurs in [20] since the contribution of the nuclear interior ( 4.0 fm) to the calculated astrophysical factors and use the experimental data [6-11] more accurate than those in Ref.[20] can influence the extracted values of the ANCs. Besides, one would also like to note that in reality the values of the ANCs, and , should not be equal, as it was assumed in [13] and the values of =14.4 fm were obtained from the analysis of performed in within the -matrix method.

The resulting ANC (NVC) value for (g.s.) obtained by us is in good agreement with the value =25.3 fm (=1.20 fm) and that for (0.429 MeV) differs noticeably from the value =22.0 fm (=1.04 fm), which were obtained in [36] within the ()-channel resonating-group method. Also, the results of the present work differ noticeably from the values =12.6 1.1 fm and fm(=3.55 0.15 fm, =2.90 0.10 fm, =0.596 0.052 fm and =0.397 0.030 fm) [16] as well as those =14.4 fm (=3.79 fm and =0.680 fm) [13]. In this connection one would like to note that in [16] the bound state wave functions, which correspond to the binding energy for (g.s.) in the ()-channel differing noticeably from the experimental ones (see Table I in Ref.[16]), and the initial state wave functions were computed with different potentials and, so, these calculations were not self-consistent. Since the ANCs for are sensitive to the form of the NN potential, it is desirable, firstly, to calculate the wave functions of the bound state using other forms of the NN potential, and, secondly, in order to guarantee the self-consistency, the same forms of the NN potential should be used for such calculation of the initial wave functions.

2.2 -particle spectroscopic factors for the mirror ()–pair

The indirectly determined”  values of the ANCs for presented in the last line of Table 2) and those for deduced in Ref.[23] can be used for obtaining information on the ratio for the virtual decays of the bound mirror ()-pair, where is the spectroscopic factor for () in the ()(())-configuration. For this aim, from ( and ) we form the relation

(1)

where () is the ratio of squares of the ANCs (single-particle ANCs) for the bound mirror ()-pair and =3/2(1/2) for the ground (first excited) state of the mirror nuclei. Besides, it should be noted that the relation (1) allows one to verify a validity of the approximation (, i.e. 1) used in Refs.[37] for the mirror () conjugated decays.

For the bound and first excited state of the mirror ()-pair the values of and change by the factor of 1.3 under the variation of the geometric parameters ( and ) of the adopted Woods–Saxon potential [31, 32] within the aforesaid ranges, while the ratios and for the bound and first excited states of the mirror ()-pair change by only about 1.5% and 6%, respectively. It is seen that the ratios do not depend practically from variation of the free parameters and , which are equal to =1.37 0.02 and =1.40 0.09. They are in good agreement with those calculated in [37] within the microscopic cluster and two-body potential models (see Table I there). The ratios for the ANCs are =1.83 and =1.77. From (1) the values of the ratio are equal to =1.34 and =1.26 for the ground and the first excited states, respectively. Within their uncertainties, these values differ slightly from those of =0.9950.005 and =0.990 calculated in Ref.[37] within the microscopic cluster model. One notes that the values of calculated in [37] are sensitive to the model assumptions (the choice of the oscillation radius and the form of the effective potential) and such model dependence may actually influence the mirror symmetry for the -particle spectroscopic factors. The mirror symmetry breakup for the -particle spectroscopic factors can also be signalled by the results for the ratio of at zero energies for the mirror ()-pair obtained in [12] within the resonating-group method by using the seven forms for the effective potential. As shown in [12], this ratio is sensitive to a form of the effective potential used and changes from 1.0 to 1.18 times in a dependence from the effective potential used. One of the possible reasons of the sensitivity observed in [12] can apparently be associated with a sensitivity of the ratio to a form of the effective potential used. In a contrast of such model dependence observed in [12, 37], the problem of the ambiguity connected with the model -dependence for the values of the ratios found by us from Eq.(1) is reduced to minimum within the experimental uncertainty.

It is seen from here that the empirical values of exceed unity both for the ground state and for the first excited state of the mirror ()-pair. This result for is not accidental and can be explained qualitatively by the following consideration. The fact is that the spectroscopic factor (or ) is determined as a norm of the radial overlap function of the bound state wave functions of the , and (or , and ) nuclei and is given by Eqs.(100) and (101) from Ref. [24]. The interval of integration () in Eq. (101) can be divided in two parts. In the first integral denoted by for and for , the integration over covers the region 0 (the internal region), where nuclear ( or ) interactions are dominate over the Coulomb interactions. In the second integral

(2)

where in the asymptotic region the radial overlap function entering the integrand is replaced by the appropriate Whittaker function (see, for example, Eq.(108) of Ref.[24]), interaction between and -particle ( for or for ) is governed by the Coulomb forces only (the external region). In (2), and is the Whittaker function. One notes that the magnitudes () and () define the probability of finding (or ) in the () configuration (or the () configuration) at distances of and of , respectively. Obviously and .

An information about values of and can be obtained from (2) by using the values of the ANCs for and recommended in [23] and in the present work, respectively. For example, for 4.0 fm (the surface regions for the mirror ()-pair) the calculation shows that the ratio is equal to 1.43 (1.31) for the ground (excited) states of the and nuclei, i.e. the ratio . Owing to the principle of equivalency of nuclear interactions between nucleons of the ()-pair in the nucleus and ()-pair in the nucleus [37], the values of and should not differ appreciably. If one suggests that 1, then the ratio 1.

2.3 The astrophysical factor at solar energies

The equation (A1) and the weighted means of the ANCs obtained for the (g.s) and (0.429 MeV) can be used for extrapolating the astrophysical factor for capture to the ground and first excited states as well as the total astrophysical factor at solar energies ( keV). We tested again the fulfilment of the condition (A2) in the same way as it is done above for E 90 keV and similar results plotted in Fig.1 are also observed at energies of E keV.

The experimental and calculated astrophysical factors for the (g.s.), (0.429 MeV) and (g.s.+0.429 MeV) reactions are presented in Table 1 and displayed in Fig.5a, b and c, respectively. In Figs.5a and b, the open diamond and triangle symbols (the filled triangle symbols) show our results for the (g.s.) and (0.429 MeV) reactions (see Table 1 also), which are obtained from the analysis of the total experimental astrophysical factors of [7–9] and [6]([11]), respectively, by using the corresponding values of the ANCs for each energy experimental point presented Figs.4a and b ( see Table 1 too). There the open circle symbols lying along the smooth solid lines are the results of the extrapolation obtained by us in which each of the quoted uncertainties is the uncertainty associated with that for the ANCs adopted. All these results are the second ones of the present work. In Figs.5a and b the experimental data plotted by the filled circle symbols (filled star and square symbols) are taken from [9] (from [10]). The solid lines present our calculations performed also with the standard values of geometric parameters =1.80 fm and =0.70 fm. In Fig.5c, the symbols are data of all experiments [6–11] and the solid line presents our calculations performed with the standard values of geometric parameters =1.80 fm and =0.70 fm by using the weighted means of the ANCs ( and ) presented in the last line of Table 2. There the dashed (dot-dashed) lines are the results of calculation obtained by using the aforesaid lower (upper) limit values of the ANCs pointed out in the last line of Table 2 and the standard values of the geometric parameters ( and ), and the dotted line is the result of Ref.[18, 19]. As it is seen from these figures, the equations (A1), (A4) and (A5) allow one to perform a correct extrapolation of the corresponding astrophysical factors at solar energies. But, the noticeable systematical underestimation between the results of calculations performed in Ref.[18, 19] in respect to the experimental data occurs.

The weighted means of the total astrophysical factor at solar energies (=0 and 23 ) obtained by us are presented in the the last line of Table 2. As it is seen from Table 2 the weighted means of , deduced by us separately from each the activation and the prompt method of the experimental data from works [6–9] and [10, 11] (the first and second lines as well as the fifth and sixth lines), agree well within their uncertainties with each other and with those recommended in [9–11]. But, these weighted means of obtained by us from the independent analysis of the different data (the set I and the set II) differ also noticeably from one another (about 11%) and this distinction is mainly associated with the aforesaid difference observed in magnitudes of the corresponding ANCs presented in the third and seventh lines of Table 2. Nevertheless, the weighted mean of (0)=0.613 , obtained by us by using the weighted means of the ANCs presented in the last line of Table 2, within the asymmetric uncertainty, which is caused with the asymmetric uncertainty for the ANCs presented in the last line of Table 2, agrees also with that recommended in [9–11,38]. But, it is interesting to note that the central value of it is closer to that given in the third line of Table 2, than to the central value of the weighted mean given in the seventh line of Table 2. Also, the astrophysical factors, calculated by using the values of the ANCs obtained separately from the set I, the set II and both them (see Table 2), are fitted independently using a second-order polynomial within three energy intervals (0 500 keV , 0 1000 keV and 0 1200 keV). The results for the slop (0)/(0) are to be -0.711 MeV, -0.734 MeV and -0.726 MeV in dependence on the aforesaid intervals, respectively, and they do not depend on the values of the ANCs used. One notes that they also are in agreement with -0.73 MeV [19] and -0.920.18 MeV [38]. It is seen from here that the calculated by us (the solid lines in Fig.5a, b and c) and that obtained in [19, 38] have practically the same energy dependence within the aforesaid energy interval but they differ mainly with each other by a normalization.

Our result for differs noticeably on that recommended in Refs.[16] [20] and [13]( 40 keVb, 0.520.03 keVb and 0.510.04 keV b, respectively). This circumstance is apparently connected with the underestimation of the contribution from the external part in the amplitude admitted in these works. Besides, the result of the present work is noticeably larger than the result of =0.516 (0.53) keV b [18]([19]) obtained within the standard two-body () potential by using potential deduced by a double-folding procedure. One of the possible reason of this discrepancy can be apparently associated with the assumption admitted in [18, 19] that a value of the ratio for the bound mirror ()-pair is taken equal to unity ( = =1.17 [39]), which in turn results in the observed systematical underestimation of the calculated in respect to the experimental factors (see the dashed line in Fig.5c). But, as shown in Subsection 2.3 the values of for the ground and first excited states of the mirror ()-pair are large unity. Therefore, the underestimated values of and used in [18] also result in the underestimated value of for the direct capture reaction. Perhaps, the assumption about equal values of the spectroscopic factor ( = =1.17 [18, 39]) is correct only for the spectroscopic factors since the value of obtained in [23] and [18] for the direct capture reaction agree excellently with each other. One notes that in Ref.[23] the analysis of the experimental astrophysical factors [40] has also been performed within MTBPA and the ANC for (g.s.), which has been deduced there from the results of Ref.[18], also is in good agreement with that recommended by authors of Ref.[23].

Nevertheless, we observe that the value of = 0.56 [14] obtained within the microscopical ()-cluster approach is in agreement with our result within the uncertainty. Besides, our result is also in excellent agreement with that of = 0.609 [12] and = 0.621 [36] obtained within the ()-channel of version of the resonating-group method by using the modified Wildermuth-Tang (MWT) and the near-Serber exchange mixture forms for the effective NN potential, respectively. It follows from here that the mutual agreement between the results obtained in the present work and works of [12, 14, 36], which is based on the common approximation about the cluster structure of the , allows one to draw a conclusion about the dominant contribution of the clusterization to the low-energy cross section both in the absolute normalization and in the energy dependence [6–11]. Therefore, single-channel approximation for [12, 14] is quite appropriate for this reaction in the considered energy range.

Also, it is interesting to note that the ratios of the indirectly determined”  astrophysical factors, (0) and , for the reaction populating to the ground and first excited states obtained in the present work to those for the mirror reaction populating to the ground and first excited states deduced in Ref.[23] are equal to =6.87 and =6.11 , respectively. These values are in a good agreement with those of =6.6 and =5.9 deduced in Ref.[37] within the microscopic cluster model. This result also confirms directly our estimation for the ratio obtained above since the ANCs for (g.s) and (0.478 MeV) as well as the ANCs for (g.s) and (0.429 MeV) determine the astrophysical factors for the and reactions at zero energies and, consequently, the ratios and are proportional to and , respectively.

Fig. 5d shows a comparison between the branching ratio obtained in the present work (the open triangle and square symbols) and that recommended in Refs.[41] (the filled square symbols), in [7, 9] (the filled circle symbols) and [10] (the filled triangle symbols). The weighted mean of the recommended by us is equal to =0.43 0.01. As it is seen from Fig.5d, the branching ratio obtained in the present work and in [7, 9, 10] is in a good agreement with that recommended in Ref.[41] although the underestimation occurs for the obtained in Ref.[41]. Such a good agreement between two of the experimental data for the can apparently be explained by the fact that there is a reduction factor in [41], being overall for the (g.s.) and (0.429 MeV) astrophysical factors. The present result for is in an excellent agreement with those of 0.43 0.02 [41] and 0.43 [18, 42] but is noticeably larger than 0.37 [16] and 0.32 0.01 [43].

3 Conclusion

The analysis of the modern experimental astrophysical factors, , for the reaction, which were precisely measured at energies =92.9-1235 keV [6–11], has been performed within the modified two-body potential approach [23]. The performed scrupulous analysis shows quantitatively that the reaction within the considered energy ranges is mainly peripheral and the parametrization of the direct astrophysical factors in terms of ANCs for the is adequate to the physics of the peripheral reaction under consideration.

It is shown that the experimental astrophysical factors of the reaction under consideration [6–11] can be used as an independent source of getting the information about the ANCs (or NVCs) for (or for the virtual decay ), and the found ANCs can predict the experimental astrophysical factors separated for the (g.s.) and (0.429 MeV) reactions at low experimentally acceptable energy regions (126.5 E 1203 keV) obtained from the total experimental astrophysical factors [6–9,11]. The new estimation for the weighted means of the ANCs for and NVCs for the virtual decay are obtained. Also, the values of ANCs were used for getting the information about the -particle spectroscopic factors for the mirror ()-pair.

The obtained values of the ANCs were also used for an extrapolation of astrophysical factors at energies less than 90 keV, including =0. In particular, the weighted mean of the branching ratio (=0.43 0.01) and the total astrophysical factor ((0)=0.613 ) obtained here are in agreement with that deduced in [7–11] from the analysis the same experimental asprophysical factors. Besides, our result for is in an agreement with that =0.56 keV [14] obtained within the microscopical single-channel () cluster model, = 0.609 [12] and = 0.621 [36] obtained within the ()-channel of version of the resonating-group method, but it is noticeably larger than the result of =0.516 (0.53) keV [18]([19]) obtained within the standard two-body () potential by using potential deduced by a double-folding procedure.

Acknowledgments

The authors are deeply grateful to S. V. Artemov, L.D. Blokhintsev and A.M. Mukhamedzhanov for discussions and general encouragement. The authors thank also D.Bemmerer for providing the experimental results of the updated data analysis. The work has been supported by The Academy of Science of The Republic of Uzbekistan (Grant No.FA-F2-F077).

Appendix: Basic formulae

Here we repeat only the idea and the essential formulae of the MTBPA [23] specialized for the astrophysical factor that are important for the following analysis.

According to [23], for fixed and we can write the astrophysical factor, , for the peripheral direct capture reaction in the following form

Here, is the ANC for , which determines the amplitude of the tail of the nucleus bound state wave function in the (