# 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

S.B. Igamov, Q.I. Tursunmahatov and R. Yarmukhamedov Corresponding author,
E-mail<EMAIL_ADDRESS>
###### Abstract

A new analysis of the modern precise measured astrophysical $S$ factors for
the direct capture ${}^{3}He(\alpha,\gamma)^{7}{\rm{Be}}$ 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 ${}^{7}{\rm Be}$ 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 ${}^{3}{\rm{He}}+\alpha\to{\rm{{}^{7}Be}}$(g.s.) and
${}^{3}{\rm{He}}+\alpha\to{\rm{{}^{7}Be}}$(0.429 MeV) as well as the
astrophysical $S$ factors $S_{34}(E)$ at E$\leq$ 90 keV, including $E$=0. The
values of asymptotic normalization constants have been used for getting
information about the $\alpha$-particle spectroscopic factors for the mirror
(${\rm{{}^{7}Li}}{\rm{{}^{7}Be}}$)-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 ${}^{3}{\rm{He}}(\alpha,\gamma)^{7}{\rm{Be}}$ reaction is one of the
critical links in the ${}^{7}{\rm{Be}}$ and ${}^{8}B$ branches of the
$pp$–chain of solar hydrogen burning [1–3]. The total capture rate determined
by processes of this chain is sensitive to the cross section $\sigma_{34}(E)$
(or the astrophysical $S$ factor $S_{34}(E)$ ) for the
${}^{3}{\rm{He}}(\alpha,\gamma)^{7}{\rm{Be}}$ reaction and predicted neutrino
rate varies as $[S_{34}(0)]^{0.8}$ [2, 3].

Despite the impressive improvements in our understanding of the
${}^{3}{\rm{He}}(\alpha,\gamma)^{7}{\rm{Be}}$ 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 $\sigma_{34}(E)$ (or $S_{34}(E)$) still exist and they may influence the
predictions of the standard solar model [2, 3] .

Experimentally, there are two types of data for the
${}^{3}{\rm{He}}(\alpha,\gamma)^{7}{\rm{Be}}$ reaction at extremely low
energies: i) six measurements based on detecting of $\gamma$-rays capture (see
[4] and references therein) from which the astrophysical $S$ factor
$S_{34}(0)$ extracted by the authors of those works changes within the range
0.47$\leq S_{34}(0)\leq$0.58 ${\rm{keV\,\,\,b}}$ and ii) six measurements
based on detecting of ${}^{7}{\rm{Be}}$ (see [4] references therein as well as
[6-11]) from which $S_{34}(0)$ extracted by the authors of these works changes
within the range 0.53$\leq S_{34}\leq$0.63 ${\rm{keV\,\,\,b}}$. All of these
measured data have a similar energy dependence for the astrophysical $S$
factors $S_{34}(E)$. 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
$E$=0, leads to a value of $S_{34}(0)$ that differs from others and this
difference exceeds the experimental uncertainty.

The theoretical calculations of $S_{34}(0)$ 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
$S_{34}(0)$ 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$\leq S_{34}(0)\leq$ 0.841 ${\rm{keV\,\,\,b}}$.

The estimation of $S_{34}(0)$=0.52$\pm$ 0.03 ${\rm{keV\,\,\,b}}$ [20] also
should be noted. The latter has been obtained from the analysis of the
experimental astrophysical $S$ 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 ${}^{7}{\rm{Be}}$ [22]. At this, in [20] the
contribution from the nuclear interior ($r<r_{cut}$, $r_{cut}$=4 fm) to the
amplitude is ignored. In this case, the astrophysical $S$ factor is directly
expressed in terms of the nuclear vertex constants (NVC) for the virtual
decays ${\rm{{}^{7}Be}}\to\alpha+^{3}{\rm{He}}$ (or respective the asymptotic
normalization coefficients (ANC) for
${\rm{{}^{3}He}}+\alpha\to{\rm{{}^{7}Be}}$) [24, 25]. As a result, in Ref.
[20], the NVC-values for the virtual decays
${\rm{{}^{7}Be}}{\rm{(g.s.)}}\to\alpha+{\rm{{}^{3}He}}$ and
${\rm{{}^{7}Be}}{\rm{(0.429\,\,MeV)}}\to\alpha+{\rm{{}^{3}He}}$ were obtained,
which were then used for calculations of the astrophysical $S$ factors at
$E<$180 keV, including $E$=0. However, the values of the ANCs (or NVCs)
${\rm{{}^{3}He}}+\alpha\to{\rm{{}^{7}Be}}$ and the $S_{34}(0)$ obtained in
[20] may not be enough accurate associated both with the aforesaid assumption
in respect to the contribution from the nuclear interior ($r<r_{cut}$) 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
${\rm{{}^{3}He}}+\alpha\to{\rm{{}^{7}Be}}$(g.s.) and
${\rm{{}^{3}He}}+\alpha\to{\rm{{}^{7}Be}}$(0.429 MeV) is highly desirable
since it has direct effects in the correct extrapolation of the
${\rm{{}^{3}He}}(\alpha,\gamma){\rm{{}^{7}Be}}$ astrophysical $S$ factor at
solar energies.

Recently, a modified two-body potential approach (MTBPA) was proposed in [23]
for the peripheral direct capture ${\rm A}(a,\gamma){\rm B}$ reaction, which
is based on the idea proposed in paper [22] that low-energy direct radiative
captures of particle $a$ by light nuclei ${\rm A}$ proceed mainly in regions
well outside the range of the internuclear $a{\rm A}$ interactions. One notes
that in MTBPA the direct astrophysical $S$ factor is expressed in terms of ANC
for ${\rm A}+a\to{\rm B}$ rather than through the spectroscopic factor for the
nucleus ${\rm B}$ in the (${\rm A}+a$) 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 $\alpha$-particle
capture by some light nuclei. Therefore, it is of great interest to apply the
MTBPA for analysis of the ${\rm{{}^{3}He}}(\alpha,\gamma){\rm{{}^{7}Be}}$
reaction.

In this work new analysis of the modern precise experimental astrophysical $S$
factors for the direct capture ${}^{3}{\rm{He}}(\alpha,\gamma)^{7}{\rm{Be}}$
reaction at extremely low energies ($\gtrsim$ 90 keV) [6-11] is performed
within the MTBPA [23] to obtain „indirectly determined“ values both of the
ANCs (the NVCs) for ${}^{3}{\rm{He}}+\alpha\to{\rm{{}^{7}Be}}$(g.s.) and
${}^{3}{\rm{He}}+\alpha\to{\rm{{}^{7}Be}}$(0.429 MeV), and of $S_{34}(E)$ at
$E\leq$ 90 keV, including $E$=0. In this work we quantitatively show that the
${}^{3}{\rm{He}}(\alpha,\gamma)^{7}{\rm{Be}}$ reaction within the aforesaid
energy region is mainly peripheral and one can extract ANCs for
${}^{3}{\rm{He}}+\alpha\to{\rm{{}^{7}Be}}$ directly from the
${}^{3}{\rm{He}}(\alpha,\gamma){\rm{{}^{7}Be}}$ reaction where the ambiguities
inherent for the standard two -body potential model calculation of the
${}^{3}{\rm{He}}(\alpha,\gamma){\rm{{}^{7}Be}}$ reaction, which is connected
with the choice of the geometric parameters (the radius $R$ and the
diffuseness $a$) 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 $S_{34}(E)$.

The contents of this paper are as follows. In Section 2 the results of the
analysis of the precise measured astrophysical $S$ factors for the direct
radiative capture ${}^{3}{\rm{He}}(\alpha,\gamma)^{7}{\rm{Be}}$ 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 ${}^{3}{\rm{He}}(\alpha,\gamma)^{7}{\rm{Be}}$
reaction are given.

##  2 Analysis of ${}^{3}{\rm{He}}(\alpha,\gamma)^{7}{\rm{Be}}$ reaction

Let us write $l_{f}$ ($j_{f}$) for the relative orbital (total) angular moment
of ${}^{3}{\rm{He}}$ and $\alpha$-particle in nucleus
${}^{7}{\rm{Be}}\,\,(\alpha+{\rm{{}^{3}He}})$, $l_{i}$ ($j_{i}$) for the
orbital (total) angular moment of the relative motion of the colliding
particles in the initial state. For the
${\rm{{}^{3}He}}(\alpha,\gamma){\rm{{}^{7}Be}}$ reaction populating to the
ground and first excited ($E^{*}$=0.429 MeV; $J^{\pi}$=1/2-) states of
${\rm{{}^{7}Be}}$, the values of $j_{f}$ are taken to be equal to 3/2 and 1/2,
respectively, the value of $l_{f}$ is taken to be equal to 1 as well as
$l_{i}$=0, 2 for the $E1$-transition and $l_{i}$=1 for the $E2$-transition.

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

###  2.1 The asymptotic normalization coefficients for
${}^{3}{\rm{He}}+\alpha\to^{7}{\rm{Be}}$

To determine the ANC values for
${\rm{{}^{3}He}}+\alpha\to{\rm{{}^{7}Be}}$(g.s) and
${\rm{{}^{3}He}}+\alpha\to{\rm{{}^{7}Be}}$(0.429 MeV) the recent experimental
astrophysical $S$ factors, $S^{exp}_{l_{f}j_{f}}(E)$, for the
${\rm{{}^{3}He}}(\alpha,\gamma){\rm{{}^{7}Be}}$ reaction populating to the
ground ($l_{f}$=1 and $j_{f}$=3/2) and first excited ($E^{*}$=0.429 $MeV$;
$J^{\pi}=1/2^{-}$, $l_{f}$=1 and $j_{f}$=1/2) states [6–11] are reanalyzed
based on the relations (A1)–(A7). The experimental data analyzed by us cover
the energy ranges $E$=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 $S$ factors for
the the ${\rm{{}^{3}He}}(\alpha,\gamma){\rm{{}^{7}Be}}$ reaction populating to
the first and excited states of the ${\rm{{}^{7}Be}}$ nucleus have been
separated only for the energies of $E$=92.9, 105.6 and 147.7 keV . Whereas, in
[10] the experimental astrophysical $S$ factors have been separated for all
experimental points of $E$ from the aforesaid energy region by means of
detecting the prompt $\gamma$ ray (the prompt method) and by counting the
${\rm{{}^{7}Be}}$ activity (the activation).

The Woods–Saxon potential split with a parity ($l$-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
$\varphi_{l_{f}j_{f}}(r)$ and scattering wave function $\psi_{l_{i}j_{i}}(r)$.
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
${}^{3}{\rm{He}}$\- and $\alpha$-clusters in the ($\alpha+{\rm{{}^{3}He}}$)
bound state by means of inclusion of deeply bound states forbidden by the
Pauli exclusion principle. The latter imitates the additional node ($n$)
arising in the wave functions of $\alpha-{\rm{{}^{3}He}}$ relative motion in
${}^{7}{\rm{Be}}$ similarly as the result of the microscopic resonating-group
method [12]. Secondly, this potential describes well the phase shifts for
$\alpha{\rm{{}^{3}He}}$-scattering in the wide energy range [31, 32] and
reproduces the energies of low-lying states of the ${}^{7}Be$ nucleus [33].

We vary the geometric parameters (radius $R$ and diffuseness $a$) of the
adopted Woods–Saxon potential in the physically acceptable ranges ($R$ in
1.62–1.98 fm and $a$ in 0.63–0.77 fm [23]) in respect to the standard values
($R$=1.80 fm and $a$=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 ${\cal{R}}_{l_{f}j_{f}}(E,C^{(sp)}_{l_{f}j_{f}})$ dependence on the
single-particle ANC, $C^{(sp)}_{l_{f}j_{f}}$ for $l_{f}$= 1 and $j_{f}$=3/2
and 1/2 only for the two values of energy $E$. The width of the band for these
curves is the result of the weak „residual“ $(R,a)$-dependence of the
${\cal{R}}_{l_{f}j_{f}}(E,C^{(sp)}_{l_{f}j_{f}})$ on the parameters $R$ and
$a$ (up to $\pm 2\%$) for the
$C^{(sp)}_{l_{f}j_{f}}=C^{(sp)}_{l_{f}j_{f}}(R,a)=const$ [23, 44]. The same
dependence is also observed at other energies. For example, for fig.1 plotted
for $E$=0.1056 (0.1477) MeV overall uncertainty ($\Delta_{{\cal{R}}}$ ) of the
function ${\cal{R}}_{l_{f}j_{f}}(E,C^{(sp)}_{l_{f}j_{f}})$ in respect to the
central values ${\cal{R}}_{l_{f}j_{f}}(E,C^{(sp)}_{l_{f}j_{f}})$ corresponding
to the central values of $C^{(sp)}_{l_{f}j_{f}}(1.80,0.70)$ comes to
$\Delta_{{\cal{R}}}$=$\pm$ 4.5($\pm$4.5)% for the ground state of
${}^{7}{\rm{Be}}$ and $\Delta_{{\cal{R}}}$ =$\pm$ 3.4($\pm$2.9)% for the
excited state of ${}^{7}{\rm{Be}}$. As it is seen from here that the
${}^{3}{\rm{He}}(\alpha,\gamma)^{7}{\rm{Be}}$(0.429 MeV) reaction is slightly
more peripheral than the ${}^{3}{\rm{He}}(\alpha,\gamma)^{7}{\rm{Be}}$(g.s.)
reaction since the binging energy for ${}^{7}{\rm{Be}}$(0.429 MeV) is less
than that for ${}^{7}{\rm{Be}}$(g.s). The similar dependence of
${\cal{R}}_{l_{f}j_{f}}(E,C^{(sp)}_{l_{f}j_{f}})$ on the
$C^{(sp)}_{l_{f}j_{f}}$ values is observed for the other aforesaid energies
$E$ and the value of $\Delta_{{\cal{R}}}$ is no more than $\sim\pm$ 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 $S^{exp}_{l_{f}j_{f}}(E)$. It should be
noted that values of $\Delta_{{\cal{R}}}$ becomes larger as the energy $E$
increases (for $E$ more than 1.3 MeV).

Thus, over the energy region 92.9$\leq$E$\leq$ 1200 keV the
${\cal{R}}_{l_{f}j_{f}}(E,C_{l_{f}j_{f}}^{(sp)})$-dependence on
$C_{l_{f}j_{f}}^{(sp)}$ is exactly sufficiently weak being within the
experimental uncertainties for the $S^{exp}_{l_{f}j_{f}}(E)$. 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
$\alpha{\rm{{}^{3}He}}$ potential leading as a whole to reduction of the
contribution from the interior part of the radial matrix element into the
${\cal{R}}_{l_{f}j_{f}}(E,C_{l_{f}j_{f}}^{(sp)})$ function, which is typical
for peripheral reactions.

We also calculated the $\alpha^{3}{\rm{He}}$-elastic scattering phase shifts
by variation of the parameters $R$ and $a$ in the same range for the adopted
Woods–Saxon potential. As an illustration, the results of the calculations
corresponding to the $s_{1/2}$ and $p_{3/2}$ 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 $R$ and $a$ parameters. As it is
seen from Fig.2, the experimental phase shifts [34, 35] are well reproduced
within uncertainty of about $\pm$ 5%. The same results are also obtained for
the $p_{1/2}$ and $d_{5/2}$ waves.

This circumstance allows us to test the condition (A3) at the energies of $E$=
92.9, 105.6 and 147.7 keV for which the
${}^{3}{\rm{He}}(\alpha,\gamma)^{7}{\rm{Be}}$(g.s.) and
${}^{3}{\rm{He}}(\alpha,\gamma)^{7}{\rm{Be}}$(0.429 MeV) astrophysical $S$
factors were separately measured in [9]. As an illustration, for the same
energies $E$ as in Fig.1 we present in Fig.3 (the upper panels) the results of
$C_{l_{f}j_{f}}^{2}$-value calculation given by Eq.(A3) ($(l_{f}\,\,j_{f})$=(1
3/2) and (1 1/2) ) in which instead of the $S_{l_{f}j_{f}}(E)$ the
experimental $S$ factors for the ${}^{3}{\rm{He}}(\alpha,\gamma)^{7}{\rm{Be}}$
reaction populating to the ground and first excited states of
${}^{7}{\rm{Be}}$ were taken. It should be noted that the same dependence
occurs for other considered energies. The calculation shows the obtained
$C_{l_{f}j_{f}}^{2}$ values also weakly (up to 5.0 %) depend on the
$C^{(sp)}_{l_{f}j_{f}}$-value. Consequently, the
${}^{3}{\rm{He}}(\alpha,\gamma)^{7}{\rm{Be}}$ 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 $Z_{1\,\,3/2}$
and $Z_{1\,\,1/2}$ corresponding to the $(\alpha+^{3}{\rm{He}})$-configuration
for ${}^{7}{\rm{Be}}$(g.s.) and ${}^{7}{\rm{Be}}$(0.429 keV), respectively,
change strongly about 1.7 times since calculated $\tilde{S}_{l_{f}j_{f}}(E)$
that vary by 1.75 times (see, the lower panels in Fig.3).

For each energy $E$ experimental point ($E$=92.9, 105.6 and 147.7 keV) the
values of the ANCs are obtained for
$\alpha+{\rm{{}^{3}He}}\rightarrow{\rm{{}^{7}Be}}(g.s.)$ and
$\alpha+{\rm{{}^{3}He}}\rightarrow{\rm{{}^{7}Be}}$(0.429 MeV) by using the
corresponding experimental astrophysical $S$ factor ($S_{1\,\,3/2}^{exp}(E)$
and $S_{1\,\,1/2}^{exp}(E)$, (the activation)) [7, 9] in the ratio of the
r.h.s. of the relation (A1) instead of the $S_{l_{f}j_{f}}(E)$ and the central
values of ${\cal{R}}_{l_{f}j_{f}}(E,C^{(sp)}_{l_{f}j_{f}})$ corresponding to
the adopted values of the parameters $R$ and $a$. The results of the ANCs,
$(C_{1\,\,3/2}^{exp})^{2}$ and $(C_{1\,\,1/2}^{exp})^{2}$, for these three
energy $E$ experimental points are displayed in Fig.4 $a$ and $c$ (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 $S$ factor and the aforesaid uncertainty in the
${\cal{R}}_{l_{f}j_{f}}(E,C^{(sp)}_{l_{f}j_{f}})$. One should note that the
same results for the ANCs are obtained when $S^{exp}_{34}(E)$
($S^{exp}_{1\,3/2}(E)$ and $R^{exp}(E)$) [7, 9] are used in Eq.(A5) (in
Eq.(A6) and (A7)) instead of $S_{34}(E)$ ($S_{1\,3/2}(E)$ and $R(E)$). Then in
Eq.(A6), inserting the averaged means of $\lambda_{C}$ ($\lambda_{C}$=0.666),
obtained from the three data, and replacing of the $S_{34}(E)$ in the l.h.s.
of Eq.(A5) with $S^{exp}_{34}(E)$ for the others, the two experimental points
of energy $E$ ($E$=126.5 and 168.9 keV) from [7, 8], four one $E$ ($E$=420.0,
506.0, 615.0 and 951.0 keV) from [6], the three one $E$ ($E$=93.3, 106.1 and
170.1 keV) from [9] and the ten one $E$ ($E$=701–1203 keV) from [11] can also
determine values of ANCs, $C_{1\,\,3/2}^{2}$ and $C_{1\,\,1/2}^{2}$. The
results of the ANCs $(C_{1\,j_{f}}^{exp})^{2}$ for
$\alpha+^{3}{\rm{He}}\rightarrow{\rm{{}^{7}Be}}$(g.s.) and
$\alpha+^{3}{\rm{He}}\rightarrow{\rm{{}^{7}Be}}$(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 ${\rm{{}^{7}Be}}$ 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 $S$ factors
($S_{1\,\,3/2}^{exp}$ and $S_{1\,\,1/2}^{exp}$) [10]. The results for these
ANCs are also presented in Figs.4$b$ and $d$ 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 $S$ factors the ratio in the r.h.s. of the relation (A4) does
not practically depend on the energy $E$ within the experimental
uncertainties, although absolute values of the corresponding experimental
astrophysical $S$ factors for the reactions under consideration depend
noticeably on the energy and change by up to about 1.7 times when $E$ changes
from 92.6 keV to 1200 keV. This fact allows us to conclude that the energy
dependence of the experimental astrophysical $S$ factors [6–11] is well
determined by the calculated function
${\cal{R}}_{l_{f}j_{f}}(E,C^{(sp)}_{l_{f}j_{f}})$ and ${{\cal
R}}_{13/2}(E,C_{13/2}^{(sp)})+\lambda_{C}{{\cal
R}}_{11/2}(E,C_{11/2}^{(sp)})$. Hence, the experimental astrophysical $S$
factors presented in [6–11] can be used as an independent source of reliable
information about the ANCs for
$\alpha+^{3}{\rm{He}}\rightarrow{\rm{{}^{7}Be}}$(g.s.) and
$\alpha+^{3}{\rm{He}}\rightarrow{\rm{{}^{7}Be}}$(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
$\alpha+^{3}{\rm{He}}\rightarrow{\rm{{}^{7}Be}}$(g.s.) and
$\alpha+^{3}{\rm{He}}\rightarrow{\rm{{}^{7}Be}}$(0.429 MeV) obtained by the
analysis performed separately for the activation, the prompt method and the
${\rm{{}^{7}Be}}$ 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 $\alpha+^{3}{\rm{He}}\rightarrow{\rm{{}^{7}Be}}$(g.s.)
and up to 1.12 times for
$\alpha+^{3}{\rm{He}}\rightarrow{\rm{{}^{7}Be}}$(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 $S$ 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
${}^{3}{\rm{He}}+\alpha\rightarrow{\rm{{}^{7}Be}}$(g.s.) and
${}^{3}{\rm{He}}+\alpha\rightarrow{\rm{{}^{7}Be}}$(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 $S_{34}^{exp}(E)$ 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 $S$ factors at energies down, including $E$=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 ($r<$ 4.0 fm) to the calculated astrophysical $S$ 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, $C_{1\,\,3/2}$ and
$C_{1\,\,1/2}$, should not be equal, as it was assumed in [13] and the values
of $C_{1\,\,3/2}^{2}=C_{1\,\,1/2}^{2}$=14.4 fm-1 were obtained from the
analysis of $S_{34}^{exp}(E)$ performed in within the $R$-matrix method.

The resulting ANC (NVC) value for
$\alpha+{\rm{{}^{3}He}}\to{\rm{{}^{7}Be}}$(g.s.) obtained by us is in good
agreement with the value $C_{1\,\,3/2}^{2}$=25.3 fm-1
($|G_{1\,\,3/2}|^{2}$=1.20 fm) and that for
$\alpha+{\rm{{}^{3}He}}\to{\rm{{}^{7}Be}}$(0.429 MeV) differs noticeably from
the value $C_{1\,\,1/2}^{2}$=22.0 fm-1 ($|G_{1\,\,1/2}|^{2}$=1.04 fm), which
were obtained in [36] within the ($\alpha+^{3}{\rm{He}}$)-channel resonating-
group method. Also, the results of the present work differ noticeably from the
values $C_{1\,\,3/2}^{2}$=12.6$\pm$ 1.1 fm-1 and $C_{1\,\,1/2}^{2}=8.41\pm
0.58$ fm-1($C_{1\,\,3/2}$=3.55$\pm$ 0.15 fm-1/2, $C_{1\,\,1/2}$=2.90$\pm$ 0.10
fm-1/2, $|G_{1\,\,3/2}|^{2}$=0.596$\pm$ 0.052 fm and
$|G_{1\,\,1/2}|^{2}$=0.397$\pm$ 0.030 fm) [16] as well as those
$C_{1\,\,3/2}^{2}=C_{1\,\,1/2}^{2}$=14.4 fm-1
($C_{1\,\,3/2}=C_{1\,\,1/2}$=3.79 fm-1/2 and
$|G_{1\,\,3/2}|^{2}=|G_{1\,\,1/2}|^{2}$=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 ${\rm{{}^{7}Be}}$(g.s.) in the
($\alpha+{\rm{{}^{3}He}}$)-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
${\rm{{}^{3}He}}+\alpha\rightarrow{\rm{{}^{7}Be}}$ 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 $\alpha$-particle spectroscopic factors for the mirror
(${}^{7}{\rm{Li}}^{7}{\rm{Be}}$)–pair

The „indirectly determined“ values of the ANCs for
${}^{3}{\rm{He}}+\alpha\rightarrow{\rm{{}^{7}Be}}$ presented in the last line
of Table 2) and those for $\alpha+t\rightarrow{\rm{{}^{7}Li}}$ deduced in
Ref.[23] can be used for obtaining information on the ratio
$R_{Z;j_{f}}=Z_{1j_{f}}(^{7}{\rm{Be}})/Z_{1j_{f}}({\rm{{}^{7}Li}})$ for the
virtual $\alpha$ decays of the bound mirror
(${\rm{{}^{7}Li}}^{7}{\rm{Be}}$)-pair, where
$Z_{1j_{f}}({\rm{{}^{7}Be}})(Z_{1j_{f}}({\rm{{}^{7}Li}}))$ is the
spectroscopic factor for ${\rm{{}^{7}Be}}$ (${\rm{{}^{7}Li}}$) in the
($\alpha+{\rm{{}^{3}He}}$)(($\alpha+t$))-configuration. For this aim, from
$C_{1\,j_{f}}({\rm{B}})=Z_{1\,j_{f}}^{1/2}({\rm{B}})C^{(sp)}_{1\,j_{f}}({\rm{B}})$
(${\rm{B}}={\rm{{}^{7}Li}}$ and ${\rm{{}^{7}Be}}$) we form the relation

$R_{Z;\,j_{f}}=\frac{R_{C;\,j_{f}}}{R_{C^{(sp)};\,j_{f}}},$ (1)

where
$R_{C;\,j_{f}}=\Big{(}C_{1\,j_{f}}(^{7}{\rm{Be}})/C_{1\,j_{f}}({\rm{{}^{7}Li}})\Big{)}^{2}$($R_{C^{(sp)};\,j_{f}}=\Big{(}C_{1\,j_{f}}^{(sp)}(^{7}{\rm{Be}})/C_{1\,j_{f}}^{(sp)}({\rm{{}^{7}Li}})\Big{)}^{2}$)
is the ratio of squares of the ANCs (single-particle ANCs) for the bound
mirror (${\rm{{}^{7}Li}}^{7}{\rm{Be}}$)-pair and $j_{f}$=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
($R_{C;j_{f}}\approx R_{C^{(sp)};\,j_{f}}$, i.e. $R_{Z;\,j_{f}}\approx$ 1)
used in Refs.[37] for the mirror (${\rm{{}^{7}Li}}^{7}{\rm{Be}}$) conjugated
$\alpha$ decays.

For the bound and first excited state of the mirror
(${\rm{{}^{7}Li}}^{7}{\rm{Be}}$)-pair the values of
$C_{1\,j_{f}}^{(sp)}(^{7}{\rm{Be}})$ and
$C_{1\,j_{f}}^{(sp)}({\rm{{}^{7}Li}})$ change by the factor of 1.3 under the
variation of the geometric parameters ($R$ and $a$) of the adopted Woods–Saxon
potential [31, 32] within the aforesaid ranges, while the ratios
$R_{C^{(sp)};\,3/2}$ and $R_{C^{(sp)};\,1/2}$ for the bound and first excited
states of the mirror (${\rm{{}^{7}Li}}^{7}{\rm{Be}}$)-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 $R$ and $a$, which are equal
to $R_{C^{(sp)};\,3/2}$=1.37$\pm$ 0.02 and $R_{C^{(sp)};\,1/2}$=1.40$\pm$
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 $R_{C;\,3/2}$=1.83${}^{{\rm{+0.18}}}_{{\rm{-0.25}}}\,$
and $R_{C;\,1/2}$=1.77${}^{{\rm{+0.19}}}_{{\rm{-0.24}}}\,$. From (1) the
values of the ratio $R_{Z;\,j_{f}}$ are equal to
$R_{Z;\,3/2}$=1.34${}^{{\rm{+0.13}}}_{{\rm{-0.18}}}\,$ and
$R_{Z;\,1/2}$=1.26${}^{{\rm{+0.16}}}_{{\rm{-0.19}}}\,$ for the ground and the
first excited states, respectively. Within their uncertainties, these values
differ slightly from those of $R_{Z;\,3/2}$=0.995$\pm$0.005 and
$R_{Z;\,1/2}$=0.990 calculated in Ref.[37] within the microscopic cluster
model. One notes that the values of $R_{Z;\,j_{f}}$ calculated in [37] are
sensitive to the model assumptions (the choice of the oscillation radius $b$
and the form of the effective $NN$ potential) and such model dependence may
actually influence the mirror symmetry for the $\alpha$-particle spectroscopic
factors. The mirror symmetry breakup for the $\alpha$-particle spectroscopic
factors can also be signalled by the results for the ratio of
$S_{34}({\rm{{}^{7}Be}})/S_{34}({\rm{{}^{7}Li}})$ at zero energies for the
mirror (${\rm{{}^{7}Li}}^{7}{\rm{Be}}$)-pair obtained in [12] within the
resonating-group method by using the seven forms for the effective $NN$
potential. As shown in [12], this ratio is sensitive to a form of the
effective $NN$ potential used and changes from 1.0 to 1.18 times in a
dependence from the effective $NN$ potential used. One of the possible reasons
of the sensitivity observed in [12] can apparently be associated with a
sensitivity of the ratio $R_{Z;\,j_{f}}$ to a form of the effective $NN$
potential used. In a contrast of such model dependence observed in [12, 37],
the problem of the ambiguity connected with the model $(R,a)$-dependence for
the values of the ratios $R_{Z;\,j_{j}}$ found by us from Eq.(1) is reduced to
minimum within the experimental uncertainty.

It is seen from here that the empirical values of $R_{Z;\,j_{f}}$ exceed unity
both for the ground state and for the first excited state of the mirror
(${\rm{{}^{7}Li}}^{7}{\rm{Be}}$)-pair. This result for $R_{Z;\,j_{f}}$ is not
accidental and can be explained qualitatively by the following consideration.
The fact is that the spectroscopic factor $Z_{1\,j_{f}}({\rm{{}^{7}Li}})$ (or
$Z_{1\,j_{f}}(^{7}{\rm{Be}})$) is determined as a norm of the radial overlap
function of the bound state wave functions of the $t$, $\alpha$ and
${\rm{{}^{7}Li}}$ (or ${\rm{{}^{3}He}}$, $\alpha$ and ${\rm{{}^{7}Be}}$)
nuclei and is given by Eqs.(100) and (101) from Ref. [24]. The interval of
integration ($0\leq r<\infty$) in Eq. (101) can be divided in two parts. In
the first integral denoted by $Z_{1\,j_{f}}^{(1)}({\rm{{}^{7}Li}})$ for
${\rm{{}^{7}Li}}$ and $Z_{1\,j_{f}}^{(1)}({\rm{{}^{7}Be}})$ for
${\rm{{}^{7}Be}}$, the integration over $r$ covers the region 0$\leq r\leq
r_{c}$ (the internal region), where nuclear ($\alpha t$ or
$\alpha^{3}{\rm{He}}$) interactions are dominate over the Coulomb
interactions. In the second integral

$Z_{1\,j_{f}}^{(2)}({\rm{B}})=C_{1\,j_{f}}^{2}({\rm{B}})\int_{r_{c}}^{\infty}drW_{\eta_{{\rm
B}};\,3/2}^{2}(2\kappa_{\alpha a}r),$ (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 $a$ and $\alpha$-particle ( $a=t$
for ${\rm B}={\rm{{}^{7}Li}}$ or $a={\rm{{}^{3}He}}$ for ${\rm
B}={\rm{{}^{7}Be}}$) is governed by the Coulomb forces only (the external
region). In (2), $\kappa_{\alpha a}=\sqrt{2\mu_{\alpha a}\varepsilon_{\alpha
a}}$ and $W_{\eta_{{\rm{B}}};\,3/2}(x)$ is the Whittaker function. One notes
that the magnitudes $Z_{1\,j_{f}}^{(1)}({\rm{{}^{7}Li}})$
($Z_{1\,j_{f}}^{(1)}({\rm{{}^{7}Be}})$) and
$Z_{1\,j_{f}}^{(2)}({\rm{{}^{7}Li}})$ ($Z_{1\,j_{f}}^{(2)}({\rm{{}^{7}Be}})$)
define the probability of finding $t$ (or ${\rm{{}^{3}He}}$) in the
($\alpha+t$) configuration (or the ($\alpha+{\rm{{}^{3}He}}$) configuration)
at distances of $r\leq r_{c}$ and of $r>r_{c}$, respectively. Obviously
$Z_{1\,j_{f}}({\rm{{}^{7}Li}})=Z_{1\,j_{f}}^{(1)}({\rm{{}^{7}Li}})+Z_{1\,j_{f}}^{(2)}({\rm{{}^{7}Li}})$
and
$Z_{1\,j_{f}}({\rm{{}^{7}Be}})=Z_{1\,j_{f}}^{(1)}({\rm{{}^{7}Be}})+Z_{1\,j_{f}}^{(2)}({\rm{{}^{7}Be}})$.

An information about values of $Z_{1\,j_{f}}^{(2)}({\rm{{}^{7}Li}})$ and
$Z_{1\,j_{f}}^{(2)}({\rm{{}^{7}Be}})$ can be obtained from (2) by using the
values of the ANCs for $\alpha+t\to{\rm{{}^{7}Li}}$ and
$\alpha+{\rm{{}^{3}He}}\to{\rm{{}^{7}Be}}$ recommended in [23] and in the
present work, respectively. For example, for $r_{c}\approx$4.0 fm (the surface
regions for the mirror (${}^{7}{\rm{Li}}^{7}{\rm{Be}}$)-pair) the calculation
shows that the ratio
$R_{Z;\,j_{f}}^{(2)}=Z_{1\,j_{f}}^{(2)}({\rm{{}^{7}Be}})/Z_{1\,j_{f}}^{(2)}({\rm{{}^{7}Li}})$
is equal to 1.43${}^{{\rm{+0.13}}}_{{\rm{-0.18}}}$
(1.31${}^{{\rm{+0.14}}}_{{\rm{-0.18}}}$) for the ground (excited) states of
the ${\rm{{}^{7}Li}}$ and ${\rm{{}^{7}Be}}$ nuclei, i.e. the ratio
$R_{Z;\,j_{f}}^{(2)}>1$. Owing to the principle of equivalency of nuclear
interactions between nucleons of the ($\alpha t$)-pair in the nucleus
${\rm{{}^{7}Li}}$ and ($\alpha^{3}{\rm{He}}$)-pair in the nucleus
${\rm{{}^{7}Be}}$ [37], the values of $Z_{1\,j_{f}}^{(1)}({\rm{{}^{7}Li}})$
and $Z_{1\,j_{f}}^{(1)}({\rm{{}^{7}Be}})$ should not differ appreciably. If
one suggests that $R_{Z;\,j_{f}}^{(1)}\approx$1, then the ratio
$R_{Z;\,j_{f}}>$1.

###  2.3  The ${}^{3}{\rm{He}}(\alpha,\gamma)^{7}{\rm{Be}}$ astrophysical $S$
factor at solar energies

The equation (A1) and the weighted means of the ANCs obtained for the
${}^{3}{\rm{He}}+\alpha\to{\rm{{}^{7}Be}}$(g.s) and
${}^{3}{\rm{He}}+\alpha\to{\rm{{}^{7}Be}}$(0.429 MeV) can be used for
extrapolating the ${}^{3}{\rm{He}}(\alpha,\gamma)^{7}{\rm{Be}}$ astrophysical
$S$ factor for capture to the ground and first excited states as well as the
total astrophysical $S$ factor at solar energies ($E\leq 25$ keV). We tested
again the fulfilment of the condition (A2) in the same way as it is done above
for E$\geq$ 90 keV and similar results plotted in Fig.1 are also observed at
energies of E$<90$ keV.

The experimental and calculated astrophysical $S$ factors for the
${}^{3}{\rm{He}}(\alpha,\gamma)^{7}{\rm{Be}}$(g.s.),
${}^{3}{\rm{He}}(\alpha,\gamma)^{7}{\rm{Be}}$ (0.429 MeV) and
${}^{3}{\rm{He}}(\alpha,\gamma)^{7}{\rm{Be}}$ (g.s.+0.429 MeV) reactions are
presented in Table 1 and displayed in Fig.5 _a_ , _b_ and _c_ , respectively.
In Figs.5 _a_ and _b_ , the open diamond and triangle symbols (the filled
triangle symbols) show our results for the
${}^{3}{\rm{He}}(\alpha,\gamma)^{7}{\rm{Be}}$(g.s.) and
${}^{3}{\rm{He}}(\alpha,\gamma)^{7}{\rm{Be}}$(0.429 MeV) reactions (see Table
1 also), which are obtained from the analysis of the total experimental
astrophysical $S$ factors of [7–9] and [6]([11]), respectively, by using the
corresponding values of the ANCs for each energy $E$ experimental point
presented Figs.4 _a_ 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.5 _a_ 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 $R$=1.80 fm and $a$=0.70 fm. In
Fig.5 _c_ , the symbols are data of all experiments [6–11] and the solid line
presents our calculations performed with the standard values of geometric
parameters $R$=1.80 fm and $a$=0.70 fm by using the weighted means of the ANCs
($C_{1\,\,3/2}^{2}$ and $C_{1\,\,1/2}^{2}$) 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 ($R$ and $a$), 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 $S$
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 $S$ factor $S_{34}(E)$ at solar
energies ($E$=0 and 23 ${\rm{keV\,\,b}}$) obtained by us are presented in the
the last line of Table 2. As it is seen from Table 2 the weighted means of
$S_{34}(0)$, 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 $S_{34}(0)$ 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 $S_{34}$(0)=0.613${}^{{\rm{+0.\,026}}}_{{\rm{-0.\,063}}}$
${\rm{keV\,\,b}}$, 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 $S$ 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$\leq E\leq$ 500 keV ,
0$\leq E\leq$ 1000 keV and 0$\leq E\leq$ 1200 keV). The results for the slop
$S_{34}^{\prime}$(0)/$S_{34}$(0) are to be -0.711 MeV-1, -0.734 MeV-1 and
-0.726 MeV-1 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-1 [19] and -0.92$\pm$0.18 MeV-1 [38]. It is seen from
here that the $S_{34}(E)$ calculated by us (the solid lines in Fig.5 _a_ , _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 $S_{34}(0)$ differs noticeably on that recommended in Refs.[16]
[20] and [13]($\approx$ 40 keVb, 0.52$\pm$0.03 keVb and 0.51$\pm$0.04 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 $S_{34}(0)$=0.516 (0.53) keV b [18]([19]) obtained
within the standard two-body ($\alpha+{\rm{{}^{3}He}}$) potential by using
$\alpha{\rm{{}^{3}He}}$ 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 $R_{Z;\,j_{f}}$
for the bound mirror (${\rm{{}^{7}Li}}{\rm{{}^{7}Be}}$)-pair is taken equal to
unity ($Z_{1\,3/2}({\rm{{}^{7}Be}})$ = $Z_{1\,1/2}({\rm{{}^{7}Be}})$=1.17
[39]), which in turn results in the observed systematical underestimation of
the calculated $S_{34}(E)$ in respect to the experimental $S$ factors (see the
dashed line in Fig.5 _c_). But, as shown in Subsection 2.3 the values of
$R_{Z;\,j_{f}}$ for the ground and first excited states of the mirror
(${\rm{{}^{7}Li}}{\rm{{}^{7}Be}}$)-pair are large unity. Therefore, the
underestimated values of $Z_{1\,3/2}({\rm{{}^{7}Be}})$ and
$Z_{1\,1/2}({\rm{{}^{7}Be}})$ used in [18] also result in the underestimated
value of $S_{34}(0)$ for the direct capture
${\rm{{}^{3}He}}(\alpha,\gamma){\rm{{}^{7}Be}}$ reaction. Perhaps, the
assumption about equal values of the spectroscopic factor
($Z_{1\,3/2}({\rm{{}^{7}Li}})$ = $Z_{1\,1/2}({\rm{{}^{7}Li}})$=1.17 [18, 39])
is correct only for the spectroscopic factors $Z_{1\,j}({\rm{{}^{7}Li}})$
since the value of $S_{34}(0)$ obtained in [23] and [18] for the direct
capture $t(\alpha,\gamma){\rm{{}^{7}Li}}$ reaction agree excellently with each
other. One notes that in Ref.[23] the analysis of the
$t(\alpha,\gamma){\rm{{}^{7}Li}}$ experimental astrophysical $S$ factors [40]
has also been performed within MTBPA and the ANC for
$\alpha+t\to{\rm{{}^{7}Li}}$(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 $S_{34}(0)$= 0.56 ${\rm{keV\,\,b}}$
[14] obtained within the microscopical ($\alpha+{\rm{{}^{3}He}}$)-cluster
approach is in agreement with our result within the uncertainty. Besides, our
result is also in excellent agreement with that of $S_{34}(0)$= 0.609
${\rm{keV\,\,b}}$ [12] and $S_{34}(0)$= 0.621 ${\rm{keV\,\,b}}$ [36] obtained
within the ($\alpha+{\rm{{}^{3}He}}$)-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 $(\alpha+{\rm{{}^{3}He}})$ structure of the
${\rm{{}^{7}Be}}$, allows one to draw a conclusion about the dominant
contribution of the $(\alpha+{\rm{{}^{3}He}})$ clusterization to the low-
energy ${\rm{{}^{3}He}}(\alpha,\gamma){\rm{{}^{7}Be}}$ cross section both in
the absolute normalization and in the energy dependence [6–11]. Therefore,
single-channel $(\alpha+{\rm{{}^{3}He}})$ approximation for ${\rm{{}^{7}Be}}$
[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 $S$ factors, $S_{1\,\,3/2}$(0) and $S_{1\,\,1/2}(0)$, for the
${}^{3}{\rm{He}}(\alpha,\gamma)^{7}{\rm{Be}}$ reaction populating to the
ground and first excited states obtained in the present work to those for the
mirror $t(\alpha,\gamma)^{7}{\rm{Li}}$ reaction populating to the ground and
first excited states deduced in Ref.[23] are equal to
$R_{S}^{(g.s.)}$=6.87${}^{+0.70}_{-0.87}$ and
$R_{S}^{(exc)}$=6.11${}^{+0.67}_{-0.86}$ , respectively. These values are in a
good agreement with those of $R_{S}^{(g.s.)}$=6.6 and $R_{S}^{(exc)}$=5.9
deduced in Ref.[37] within the microscopic cluster model. This result also
confirms directly our estimation for the ratio $R_{C;\,j_{f}}$ obtained above
since the ANCs for $t+\alpha\to{\rm{{}^{7}Li}}$(g.s) and
$t+\alpha\to{\rm{{}^{7}Li}}$(0.478 MeV) as well as the ANCs for
${\rm{{}^{3}He}}+\alpha\to{\rm{{}^{7}Be}}$(g.s) and
${\rm{{}^{3}He}}+\alpha\to{\rm{{}^{7}Be}}$(0.429 MeV) determine the
astrophysical $S$ factors for the $t(\alpha,\gamma)^{7}{\rm{Li}}$ and
${}^{3}{\rm{He}}(\alpha,\gamma)^{7}{\rm{Be}}$ reactions at zero energies and,
consequently, the ratios $R_{S}^{(g.s.)}$ and $R_{S}^{(exc)}$ are proportional
to $R_{C;\,3/2}$ and $R_{C;\,1/2}$, respectively.

Fig. 5 _d_ shows a comparison between the branching ratio $R^{exp}(E)$
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
$\bar{R}^{exp}$ of the $R^{exp}(E)$ recommended by us is equal to
$\bar{R}^{exp}$=0.43$\pm$ 0.01. As it is seen from Fig.5 _d_ , 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
$S_{34}^{exp}(E)$ obtained in Ref.[41]. Such a good agreement between two of
the experimental data for the $R^{exp}(E)$ can apparently be explained by the
fact that there is a reduction factor in [41], being overall for the
${}^{3}{\rm{He}}(\alpha,\gamma){\rm{{}^{7}Be}}$(g.s.) and
${}^{3}{\rm{He}}(\alpha,\gamma){\rm{{}^{7}Be}}$(0.429 MeV) astrophysical $S$
factors. The present result for $\bar{R}^{exp}$ is in an excellent agreement
with those of 0.43$\pm$ 0.02 [41] and 0.43 [18, 42] but is noticeably larger
than 0.37 [16] and 0.32$\pm$ 0.01 [43].

##  3 Conclusion

The analysis of the modern experimental astrophysical $S$ factors,
$S^{exp}_{34}(E)$, for the ${}^{3}{\rm{He}}(\alpha,\gamma)^{7}{\rm{Be}}$
reaction, which were precisely measured at energies $E$=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
${}^{3}{\rm{He}}(\alpha,\gamma)^{7}{\rm{Be}}$ reaction within the considered
energy ranges is mainly peripheral and the parametrization of the direct
astrophysical $S$ factors in terms of ANCs for the
${}^{3}{\rm{He}}+\alpha\rightarrow{\rm{{}^{7}Be}}$ is adequate to the physics
of the peripheral reaction under consideration.

It is shown that the experimental astrophysical $S$ 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
${}^{3}{\rm{He}}+\alpha\rightarrow\rm{{}^{7}Be}$ (or for the virtual decay
$\rm{{}^{7}Be}\rightarrow\alpha+^{3}{\rm{He}}$), and the found ANCs can
predict the experimental astrophysical $S$ factors separated for the
${\rm{{}^{3}He}}(\alpha,\gamma){\rm{{}^{7}Be}}$(g.s.) and
${\rm{{}^{3}He}}(\alpha,\gamma){\rm{{}^{7}Be}}$(0.429 MeV) reactions at low
experimentally acceptable energy regions (126.5$\leq$ E$\leq$ 1203 keV)
obtained from the total experimental astrophysical $S$ factors [6–9,11]. The
new estimation for the weighted means of the ANCs for
${}^{3}{\rm{He}}+\alpha\rightarrow{\rm{{}^{7}Be}}$ and NVCs for the virtual
decay $\rm{{}^{7}Be}\rightarrow\alpha+^{3}{\rm{He}}$ are obtained. Also, the
values of ANCs were used for getting the information about the
$\alpha$-particle spectroscopic factors for the mirror
(${}^{7}Li{\rm{{}^{7}Be}}$)-pair.

The obtained values of the ANCs were also used for an extrapolation of
astrophysical $S$ factors at energies less than 90 keV, including $E$=0. In
particular, the weighted mean of the branching ratio $\bar{R}^{exp}$
($\bar{R}^{exp}$=0.43$\pm$ 0.01) and the total astrophysical $S$ factor
$S_{34}(0)$ ($S_{34}$(0)=0.613${}^{{\rm{+0.026}}}_{{\rm{-0.063}}}$
${\rm{keV\,\,b}}$) obtained here are in agreement with that deduced in [7–11]
from the analysis the same experimental asprophysical $S$ factors. Besides,
our result for $S_{34}(0)$ is in an agreement with that $S_{34}(0)$=0.56 keV
[14] obtained within the microscopical single-channel
($\alpha+{\rm{{}^{3}He}}$) cluster model, $S_{34}(0)$= 0.609 ${\rm{keV\,\,b}}$
[12] and $S_{34}(0)$= 0.621 ${\rm{keV\,\,b}}$ [36] obtained within the
($\alpha+{\rm{{}^{3}He}}$)-channel of version of the resonating-group method,
but it is noticeably larger than the result of $S_{34}(0)$=0.516 (0.53) keV
[18]([19]) obtained within the standard two-body ($\alpha+{\rm{{}^{3}He}}$)
potential by using $\alpha{\rm{{}^{3}He}}$ 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 ${}^{3}{\rm{He}}(\alpha,\gamma){\rm{{}^{7}Be}}$
astrophysical $S$ factor that are important for the following analysis.

According to [23], for fixed $l_{f}$ and $j_{f}$ we can write the
astrophysical $S$ factor, $S_{l_{f}j_{f}}(E)$, for the peripheral direct
capture ${}^{3}{\rm{He}}(\alpha,\gamma)^{7}{\rm{Be}}$ reaction in the
following form

$S_{l_{f}j_{f}}(E)=C^{2}_{l_{f}j_{f}}{\cal{R}}_{l_{f}j_{f}}(E,C_{l_{f}j_{f}}^{(sp)}).$
$None$

Here, $C_{l_{f}j_{f}}$ is the ANC for
${\rm{{}^{3}He}}+\alpha\to{\rm{{}^{7}Be}}$, which determines the amplitude of
the tail of the ${\rm{{}^{7}Be}}$ nucleus bound state wave function in the
($\alpha+{\rm{{}^{3}He}}$)-channel and is related to the spectroscopic factor
$Z_{l_{f}j_{f}}$ for the ($\alpha+{\rm{{}^{3}He}}$)-configuration with the
quantum numbers $l_{f}$ and $j_{f}$ in the ${\rm{{}^{7}Be}}$ nucleus by the
equation $C_{l_{f}j_{f}}=Z_{l_{f}j_{f}}^{1/2}C^{(sp)}_{l_{f}j_{f}}$ [24], and
${\cal{R}}_{l_{f}j_{f}}(E,C^{(sp)}_{l_{f}j_{f}})=S_{l_{f}j_{f}}^{(sp)}(E)/(C^{(sp)}_{l_{f}j_{f}})^{2}$,
where $S_{l_{f}j_{f}}^{(sp)}(E)$ is the single-particle astrophysical $S$
factor [5] and $C^{(sp)}_{l_{f}j_{f}}$ ($\equiv C^{(sp)}_{l_{f}j_{f}}(R,a)$
[44]) is the single-particle ANC, which determines the amplitude of the tail
of the single-particle wave function of the bound ($\alpha+^{3}{\rm{He}}$)
state
$\varphi_{l_{f}j_{f}}(r)$($\equiv\varphi_{l_{f}j_{f}}(r;C^{(sp)}_{l_{f}j_{f}}$)
[44]) and in turn is itself a function of the geometric parameters (radius of
$R$ and diffuseness $a$) of the Woods-Saxon potential, i.e.
$C^{(sp)}_{l_{f}j_{f}}\equiv C^{(sp)}_{l_{f}j_{f}}(R,a)$[44].

In order to make the dependence of the
${\cal{R}}_{l_{f}j_{f}}(E,C_{l_{f}j_{f}}^{(sp)})$ function on
$C_{l_{f}j_{f}}^{(sp)}$ more explicit, in the radial matrix element [23, 26]
entering in the ${\cal{R}}_{l_{f}j_{f}}(E,C_{l_{f}j_{f}}^{(sp)})$ function, we
split the space of interaction in two parts separated by the channel radius
$r_{c}$: the interior part ($0\leq r\leq r_{c})$, where nuclear forces between
the $\alpha{\rm{{}^{3}He}}$-pair are important, and the exterior part
($r_{c}\leq r<\infty$), where the interaction between the $\alpha$-particle
and ${\rm{{}^{3}He}}$ is governed by Coulomb force only. The contribution from
the exterior part of the radial matrix element into the
${\cal{R}}_{l_{f}j_{f}}(E,C_{l_{f}j_{f}}^{(sp)})$ function does not depend on
$C_{l_{f}j_{f}}^{(sp)}$ since for $r>r_{c}$ the wave function
$\varphi_{l_{f}j_{f}}(r;C_{l_{f}j_{f}}^{(sp)})$ can be approximated by its
asymptotic behavior [24]. Consequently, the parametrization of the
astrophysical $S$ factor in the form (A1) makes one it possible to fix a
contribution from the exterior region ($r_{c}\leq r<\infty$), which is
dominant for the peripheral reaction, by a model independent way if the ANCs
$C_{l_{f}j_{f}}^{2}$ are known. It follows from here that the contribution
from the interior part of the radial matrix element into the
${\cal{R}}_{l_{f}j_{f}}(E,C_{l_{f}j_{f}}^{(sp)})$ function, which depends on
$C_{l_{f}j_{f}}^{(sp)}$ through the fraction
$\varphi_{l_{f}j_{f}}(r;C_{l_{f}j_{f}}^{(sp)})/C_{l_{f}j_{f}}^{(sp)}$ [44,
45], must exactly determine the dependence of the
${\cal{R}}_{l_{f}j_{f}}(E,C_{l_{f}j_{f}}^{(sp)})$ function on
$C_{l_{f}j_{f}}^{(sp)}$.

In Eq. (A1) the ANCs $C_{l_{f}j_{f}}^{2}$ and the free parameter
$C_{l_{f}j_{f}}^{(sp)}$ are unknown. But, for the peripheral
${}^{3}{\rm{He}}(\alpha,\gamma){\rm{{}^{7}Be}}$ reaction the equation (A1) can
be used for determination of the ANCs. For this aim, obviously the following
additional requirements [23]

${\cal{R}}_{l_{f}j_{f}}(E,C^{(sp)}_{l_{f}j_{f}})=f(E)$ $None$

and

$C_{l_{f}j_{f}}^{2}=\frac{S_{l_{f}j_{f}}(E)}{{\cal{R}}_{l_{f}j_{f}}(E,C_{l_{f}j_{f}}^{(sp)})}=const$
$None$

must be fulfilled as a function of the free parameter $C_{l_{f}j_{f}}^{(sp)}$
for each energy $E$ experimental point from the range $E_{min}\leq E\leq
E_{max}$ and values of the function of
${\cal{R}}_{l_{f}j_{f}}(E,C_{l_{f}j_{f}}^{(sp)})$ from (A2).

The fulfillment of the relations (A2) and (A3) or their violation within the
experimental uncertainty for $S_{l_{f}j_{f}}^{exp}(E)$ enables one, firstly,
to determine an interval for energies E where the dominance of extra-nuclear
capture occurs and, secondly, to obtain the value $(C_{l_{f}j_{f}}^{exp})^{2}$
for ${}^{3}{\rm{He}}+\alpha\to{\rm{{}^{7}Be}}$ using the experimental
astrophysical $S$ factors $S^{exp}_{l_{f}j_{f}}(E)$, precisely measured by
authors of Refs. [6–11], instead of $S_{l_{f}j_{f}}(E)$, i.e.

$(C_{{l_{f}j_{f}}}^{exp})^{2}=\frac{S^{exp}_{l_{f}j_{f}}(E)}{{\cal{R}}_{l_{f}j_{f}}(E,C_{l_{f}j_{f}}^{(sp)})}.$
$None$

Then, the value $(C_{l_{f}j_{f}}^{exp})^{2}$ can be used for extrapolation of
the astrophysical $S$ factor $S_{l_{f}j_{f}}(E)$ to the region of experimental
inaccessible energies 0$\leq E<E_{min}$ by using the obtained value
$(C_{l_{f}j_{f}}^{exp})^{2}$ in (A1).

The total astrophysical $S$ factor for the
${}^{3}{\rm{He}}(\alpha,\gamma)^{7}{\rm{Be}}$(g.s.+0.429 MeV) reaction is
given by

$S_{34}(E)=\sum_{l_{f}=1;\,j_{f}=1/2,\,3/2}S_{l_{f}j_{f}}(E)=$ $None$
$=C_{1\,\,3/2}^{2}[{{\cal
R}}_{1\,\,3/2}(E,C_{1\,\,3/2}^{(sp)})+\lambda_{C}{{\cal
R}}_{1\,\,1/2}(E,C_{1\,\,1/2}^{(sp)})]$ $None$ $=C_{1\,\,3/2}^{2}{{\cal
R}}_{1\,\,3/2}(E,C_{1\,\,3/2}^{(sp)})[1+R(E)]$ $None$

in which $\lambda_{C}=(C_{1\,\,1/2}/C_{1\,\,3/2})^{2}$ and $R(E)$ is a
branching ratio.

One notes that in the two-body potential model the ANC $C_{l_{f}j_{f}}$ is
related to the NVC $G_{l_{f}j_{f}}$ for the virtual decay
${\rm{{}^{7}Be}}\to\alpha+{\rm{{}^{3}He}}$ by the equation [24]

$G_{l_{f}j_{f}}=-i^{l_{f}+\eta_{{\rm{\,{}^{7}Be}}}}\frac{\sqrt{\pi}}{\mu}C_{l_{f}j_{f}},$
$None$

where $\eta_{{\rm{\,{}^{7}Be}}}$ is the Coulomb parameter for the
${\rm{{}^{7}Be}}\,(\alpha+{\rm{{}^{3}He}})$ bound state. In (A8) the
combinatorial factor taking into account the nucleon’s identity is absorbed in
$C_{l_{f}j_{f}}$ and its numerical value depends on a specific model used to
describe wave functions of the ${\rm{{}^{3}He}}$, $\alpha$ and
${\rm{{}^{7}Be}}$ nuclei [25]. Hence, the proportionality factor in (A8),
which relates NVC’s and ANCs, depends on the choice of nuclear model [25].
But, as it is noted in [25], the NVC $G_{l_{f}j_{f}}$ is a more fundamental
quantity than the ANC $C_{l_{f}j_{f}}$ since the NVC is determined in a model-
independent way as the residue of the partial $S$-matrix of the elastic
$\alpha{\rm{{}^{3}He}}$-scattering at the pole
$E=-\varepsilon_{\alpha{\rm{{}^{3}He}}}$
($\varepsilon_{\alpha{\rm{{}^{3}He}}}$ is the binding energy of the bound
($\alpha+{\rm{{}^{3}He}}$) state of ${\rm{{}^{7}Be}}$) [24, 25]. Therefore, it
is also of interest to get an information about values of the NVCs from Eqs.
(A4) and (A8).

## References

  * [1] J. N. Bahcall, A. M. Serenelli, and S. Basu, Astrophys. J. 621, L85 (2005).
  * [2] J. N. Bahcall, W.F. Huebner, S. H. Lubow, P. D. Parker, and R. K. Ulrich, Rev.Mod.Phys. 54, 767 (1982).
  * [3] J. N. Bahcall, and M. H. Pinsonneault, Rev.Mod.Phys. 64, 781 (1992).
  * [4] E.G. Adelberger, S.M.Austin, J.N. Bahcall, A.B.Balantekin, G.Bogaert, L.S.Brown, L.Buchmann, F.E.Cecil, A.E.Champagne, L. de Braeckeleer, Ch.A. Duba, S.R. Elliott, S.J. Freedman, M. Gai, G.Goldring, Ch. R. Gould, A. Gruzinov, W.C. Haxton, K.M. Heeger, and E.Henley, Rev. Mod. Phys. 70, 1265 (1998).
  * [5] C.Angulo, M.Arnould, M.Rayet, P.Descouvemont, D.Baye, C.Leclercq-Willain, A.Coc, S.Barhoumi, P.Aguer, C.Rolfs, R.Kunz, J.W. Hammer, A.Mayer, T. Paradellis, S.Kossionides, C.Chronidou, K.Spyrou, S.Degl’Innocenti, G. Fiorentini, B.Ricci, S.Zavatarelli, C.Providencia, H.Wolters, J.Soares, C.Grama, J.Rahighi, A.Shotter, M.L.Rachti, Nucl. Phys. A 656, 3 (1999).
  * [6] B. S. Nara Singh, M. Hass, Y. Nir-El, and G. Haquin, Phys.Rev.Lett. 93, 262503 (2004).
  * [7] D. Bemmerer, F. Confortola, H. Costantini, A. Formicola, Gy. Gyürky, R. Bonetti, C. Broggini, P. Corvisiero, Z. Elekes, Zs. Fülöp, G. Gervino, A. Guglielmetti, C. Gustavino, G. Imbriani, M. Junker, M. Laubenstein, A. Lemut, B. Limata, V.Lozza, M. Marta, R. Menegazzo, P. Prati, V. Roca, C. Rolfs, C. Rossi Alvarez, E. Somorjai, O. Straniero, F. Strieder, F. Terrasi, and H.P. Trautvetter (The LUNA Collaboration), Phys.Rev.Lett. 97, 122502 (2006); private communication.
  * [8] Gy. Gyürky, F. Confortola, H. Costantini, A. Formicola, D. Bemmerer, R. Bonetti, C. Broggini, P. Corvisiero, Z. Elekes, Zs. Fülöp, G. Gervino, A. Guglielmetti, C. Gustavino, G. Imbriani, M. Junker, M. Laubenstein, A. Lemut, B. Limata, V.Lozza, M. Marta, R. Menegazzo, P. Prati, V. Roca, C. Rolfs, C. Rossi Alvarez, E. Somorjai, O. Straniero, F. Strieder, F. Terrasi, and H.P. Trautvetter (The LUNA Collaboration), Phys.Rev. C 75, 035805 (2007) .
  * [9] F. Confortola, D. Bemmerer, H. Costantini, A. Formicola, Gy. Gyürky, P. Bezzon, R. Bonetti, C. Broggini, P. Corvisiero, Z. Elekes, Zs. Fülöp, G. Gervino, A. Guglielmetti, C. Gustavino, G. Imbriani, M. Junker, M. Laubenstein, A. Lemut, B. Limata, V.Lozza, M. Marta, R. Menegazzo, P. Prati, V. Roca, C. Rolfs, C. Rossi Alvarez, E. Somorjai, O. Straniero, F. Strieder, F. Terrasi, and H.P. Trautvetter (The LUNA Collaboration), Phys.Rev. C 75, 065803 (2007).
  * [10] T.A.D. Brown, C. Bordeanu, R.F. Snover, D.W. Storm, D. Melconian, A.L. Sallaska, S.K.L. Sjue, S. Triambak, Phys.Rev. C 76, 055801 (2007).
  * [11] A Di Leva, L. Gialanella, R. Kunz, D. Rogalla, D. Schürmann, F. Strieder, M. De Cesare, N. De Cesare, A. D’Onofrio, Z. Fülöp, G. Gyürky, G. Imbariani, G. Mangano, A. Ordine, V. Roca, C. Rolfs, M. Romano, E. Somorjai, and F. Terrasi, Phys.Rev.Lett. 102, 232502 (2009).
  * [12] T. Kajino, Nucl.Phys. A 460, 559 (1986).
  * [13] P.Descouvement, A. Adahchour, C. Angulo, A. Coc, E. Vangioni-Flam. Atomic data and Nuclear data Tables. 88, 203 (2004).
  * [14] K. Langanke, Nucl. Phys. A 457, 351 (1986).
  * [15] A. Csótó and K. Langanke, Few-body Systems. 29, 121 (2000).
  * [16] K.M. Nollett, Phys. Rev. C 63, 054002 (2001).
  * [17] S.B. Dubovitchenko, A.V. Dzhazairov - Kakhramanov, Yad. Fiz. 58, 635 (1995) [Phys. At. Nucl. 58, 579 (1995)]. 48, 1664 (1982); Nucl.Phys. A 419, 115 (1984).
  * [18] P. Mohr, H. Abele, R. Zwiebel, G. Staudt, H. Krauss, H. Oberhummer, A. Denker, J.W. Hammer and F. Wolf, Phys. Rev. C 48, 1420 (1993).
  * [19] P. Mohr, Phys. Rev. C 79, 065804 (2009).
  * [20] S.B. Igamov, T.M. Tursunmuratov, and R. Yarmukhamedov,Yad.Fiz. 60, 1252 (1997) [Phys. At. Nucl. 60, 1126 (1997)].
  * [21] J.L. Osborne, C. A. Barnes, R.W. Ravanagh, R.M. Kremer, G.J. Mathews, J.L. Zyskind, P.D. Parker, and A.J. Howard, Nucl.Phys.A 419, 115(1984).
  * [22] R.F. Christy and I. Duck, Nucl.Phys. 24, 89 (1961).
  * [23] S.B. Igamov, and R. Yarmukhamedov, Nucl.Phys.A 781, 247 (2007);Nucl.Phys.A 832, 346 (2010).
  * [24] L.D. Blokhintsev, I.Borbely, E.I. Dolinskii, Fiz.Elem.Chastits At. Yadra. 8, 1189 (1977)[Sov. J. Part. Nucl. 8, 485 (1977)].
  * [25] L.D. Blokhintsev and V.O. Yeromenko, Yad.Fiz. 71, 1219 (2008) [Phys. At. Nucl. 71, 1126 (2008)].
  * [26] R.G.H. Robertson, P. Dyer, R.A. Warner, R.C. Melin, T.J. Bowles, A.B. McDonald, G.C. Ball, W.G. Davies, E.D. Earle, Phys.Rev.Lett. 47, 1867 (1981).
  * [27] K.H. Kim, M.H. Park, B.T. Kim, Phys.Rev. C 35 363 (1987).
  * [28] S.B. Igamov and R. Yarmukhamedov, Phys. At. Nucl. 71, 1740 (2008)].
  * [29] S.V. Artemov, S.B. Igamov, K.I. Tursunmakhatov, and R. Yarmukhamedov, Izv. RAN: Seriya Fizicheskaya, 73, 176 (2009)[Bull.RAS:Physics,73, 165 (2009)].
  * [30] V.G. Neudatchin, V.I. Kukulin, A.N. Boyarkina and V.D. Korennoy, Lett.Nuovo Cim.5, 834 (1972).
  * [31] V.I. Kukulin, V.G. Neudatchin, and Yu. F. Smirnov, Nucl.Phys. A 245, 429 (1975).
  * [32] V.I. Kukulin, V.G. Neudatchin, I.T. Obukhovsky and Yu.F. Smirnov. "Clusters as Subsystems in Light Nuclei". In Clustering Phenomena in nuclei, Vol.3, eds. K. Wildermuth and P. Kramer ( Vieweg , Braunschweig, 1983)p.1.
  * [33] S.B. Dubovichenko and M.A. Zhusupov, Ivz. Akad. Nauk Kaz.SSR, Ser. Fiz.-Mat. 4, 44 (1983); Yad. Fiz. 39, 1378 (1984) [Sov. J. Nucl. Phys.39, 870 (1984).]
  * [34] A.L.C. Barnard, C. M. Jones, and G.C. Phillips, Nucl.Phys. 50, 629 (1964).
  * [35] R. J. Spiger, and T. A. Tombrello, Phys.Rev. 163 (1967) 964.
  * [36] H. Walliser, H. Kanada, and Y.C. Tang, Nucl.Phys. A 419, 133 (1984).
  * [37] N.K. Timofeyuk, P. Descouvemont, R.C. Johnson, Phys.Rev. C 75, 034302 (2007).
  * [38] R. H. Cyburt and B. Davids, Phys.Rev.C 78, 064614(2008).
  * [39] D. Kurath and D. J. Millener, Nucl.Phys.A 238, 269 (1975).
  * [40] C.R. Brune, W.H. Geist, R.W. Kavanagh, K.D. Veal, Phys.Rev.Lett. 83, (1999) 4025.
  * [41] H.Kräwinkel, H.W. Becker, L. Buchmann, J. Görres, K.U. Kettner, W.E. Kieser, R. Santo, P. Schmalbrock, H.P. Trautvetter, A. Vlieks, C. Rolfs, J.W. Hammer, R.E. Azuma, W.S. Rodney, Z.Phys. A 304, 307 (1982).
  * [42] T. Altmeyer, E. Kolbe, T. Warmann, K. Langanke, H.J. Assenbaum, Z.Phys. A 330, 277 (1988).
  * [43] U. Schröder, A. Redder, C. Rolfs, R.E. Azuma, L. Buchmann, C. Campbell, J.D. King, T.R. Donoghue, Phys.Lett. B 192, 55 (1987).
  * [44] S. A. Goncharov, J. Dobesh, E. I. Dolinskii, A. M. Mukhamedzhanov and J. Cejpek, Yad. Fiz.35, 662 (1982)[Sov. J. Nucl. Phys. 35, 383 (1982)].
  * [45] A. M. Mukhamedzanov, F. M. Nunes, Phys.Rev. C 72, 017602 (2005).

Figure 1:  The dependence of ${\cal{R}}_{l_{f}j_{f}}(E,C^{(sp)}_{l_{f}j_{f}})$
as a function of the single-particle ANC, $C^{(sp)}_{l_{f}j_{f}}$, for the
${}^{3}{\rm{He}}(\alpha,\gamma)^{7}{\rm{Be}}$(g.s.) ($(l_{f},j_{f})$=(1,3/2))
and ${}^{3}{\rm{He}}(\alpha,\gamma)^{7}{\rm{Be}}$(0.429 MeV
($(l_{f},j_{f})$=(1,1/2)) reactions at different energies $E$.

Figure 2: The energy dependence of the $\alpha^{3}{\rm{He}}$-elastic
scattering phase shifts for the $s_{1/2}$ and $p_{3/2}$ partial waves. The
experimental data are from [34, 35]. The bands are our calculated data. The
width of the bands for fixed energies corresponds to the variation of the
parameters $R$ and $a$ of the adopted Woods–Saxon potential within the
intervals of $R$=1.62 to 1.98 fm and $a$=0.63 to 0.77 fm.

Figure 3: The dependence of the ANCs $C_{l_{f}j_{f}}$ (upper band) and the
spectroscopic factors $Z_{l_{f}j_{f}}$ (lower band) on the single-particle ANC
$C^{(sp)}_{l_{f}j_{f}}$ for the
${}^{3}{\rm{He}}(\alpha,\gamma)^{7}{\rm{Be}}$(g.s.) (the left column,
$(l_{f},j_{f})$=(1,3/2)) and
${}^{3}{\rm{He}}(\alpha,\gamma)^{7}{\rm{Be}}$(0.429 MeV) (the right column,
$(l_{f},j_{f})$=(1,1/2)) reactions at different energies $E$.

Figure 4:  The values of the ANCs, $C^{2}_{1\,\,3/2}$ and $C^{2}_{1\,\,3/2}$,
for $\alpha+{\rm{{}^{3}He}}\to{\rm{{}^{7}Be}}$(g.s.) ((_a_) and (_b_)) and
$\alpha+^{3}{\rm{He}}\to^{7}{\rm{Be}}$(0.429 MeV) ((_c_) and (_d_)) for each
energy $E$ experimental point. The open triangle and cycle symbols and filled
triangles (filled star(the activation), square (the prompt method) symbols)
are data obtained by using the total (separated) experimental astrophysical
$S$ factors from [6](the activation), [7, 9] (the activation and the prompt
method) and [11](the ${\rm{{}^{7}Be}}$ recoils) (from [10]), respectively,
while filled circle symbols are data obtained from the separated experimental
astrophysical $S$ factors from Refs.[7–9]. The solid lines present our results
for the weighted means. Everywhere the width of each of the band is the
weighted uncertainty.

Figure 5: The astrophysical $S$ factors for the
${}^{3}{\rm{He}}(\alpha,\gamma)^{7}{\rm{Be}}$(g.s.) (_a_),
${}^{3}{\rm{He}}(\alpha,\gamma)^{7}{\rm{Be}}$ (0.429 MeV) (_b_) and
${}^{3}{\rm{He}}(\alpha,\gamma)^{7}{\rm{Be}}$ (0.429 MeV) (_c_) reactions as
well as the branching ratio (_d_). In (_a_) and (_b_): the open diamond and
triangle symbols (the filled triangle symbols) are our result separated from
the total experimental astrophysical $S$ factors of Refs.[7-9] and [6],
respectively, ([11]); the filled circle symbols (filled star and square
symbols) are experimental data of Ref.[9] (Ref.[10], the activation and the
prompt method, respectively); the open circle symbols are our results of the
extrapolation; in (_c_), the symbols are data of all experiments [6]–[10] and
the present work(the open cycle symbols). The solid lines present our
calculations performed with the standard values of geometric parameters
$R$=1.80 fm and $a$=0.70 fm. The dashed line is the result of [18, 19]. In
(_d_):the filled circle, triangle and square symbols are experimental data
taken from Refs.[7, 9], [10] and [41], respectively, and the open triangle and
square symbols are our results. The straight line (width of band ) is our
result for the weighted mean(its uncertainty).

Table 1: The „indirectly determined“ values of the asymptotic normalization
constants ($(C_{13/2}^{exp})^{2}$ and ($C_{11/2}^{exp})^{2}$) for
$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{}^{3}{\rm{He}}+\alpha\to^{7}{\rm{Be}}$,
the experimental astrophysical $S$ factors ($S^{exp}_{1j_{f}}$ and
$S^{exp}_{34}(E)$) and branching ratio ($R^{exp}(E)$) at different energies
$E$.

$E$ | $(C_{1\,\,j_{f}}^{exp})^{2}$ | $S^{exp}_{1\,\,j_{f}}$ | $S^{exp}_{34}(E)$ | $R^{exp}(E)$
---|---|---|---|---
(keV) | (fm${}^{-1})$ | (${\rm{keV\,\,\,b}}$) | (${\rm{keV\,\,\,b}}$) |
|  $j_{f}$=3/2 |  $j_{f}$=1/2 |  $j_{f}$=3/2 |  $j_{f}$=1/2 |  |
92.9∗) | 22.0$\pm$1.8 | 14.0$\pm$1.2 | 0.387$\pm$0.031[7, 9] | 0.147$\pm$0.012[7, 9] | 0.534$\pm$0.023[7, 9] | 0.380$\pm$0.030[7, 9]
93.3∗∗) | 21.4$\pm$1.4 | 14.7$\pm$0.9 | 0.374$\pm$0.020 | 0.153$\pm$0.009 | 0.527$\pm$0.027[9] | 0.409$\pm$0.03
105.6∗) | 21.0$\pm$1.8 | 14.6$\pm$1.2 | 0.365$\pm$0.030[7, 8] | 0.151$\pm$0.012[7, 8] | 0.516$\pm$0.031[7, 8] | 0.415$\pm$0.029[7, 9]
106.1∗∗) | 21.2$\pm$1.3 | 14.6$\pm$0.9 | 0.368$\pm$0.020 | 0.150$\pm$0.009 | 0.518$\pm$0.024[9] | 0.408$\pm$0.020
126.5∗) | 21.2$\pm$0.9 | 14.6$\pm$0.6 | 0.366$\pm$0.023 | 0.148$\pm$0.009 | 0.514$\pm$0.019[7, 8] | 0.404$\pm$0.020
147.7∗) | 20.8$\pm$1.1 | 14.6$\pm$0.7 | 0.352$\pm$0.017[7] | 0.147$\pm$0.007[7] | 0.499$\pm$0.017[7, 8] | 0.417$\pm$0.020[7]
168.9∗) | 20.6$\pm$0.8 | 14.1$\pm$0.6 | 0.343$\pm$0.010 | 0.139$\pm$0.006 | 0.482$\pm$0.017[7, 8] | 0.405$\pm$0.020
170.1∗∗) | 21.9$\pm$1.2 | 15.0$\pm$0.8 | 0.362$\pm$0.020 | 0.148$\pm$0.008 | 0.510$\pm$0.021[9] | 0.409$\pm$0.030
420.0∗) | 21.4$\pm$1.7 | 14.7$\pm$1.1 | 0.297$\pm$0.020 | 0.123$\pm$0.009 | 0.420$\pm$0.030[6] | 0.414$\pm$0.050
506.0∗) | 20.9$\pm$1.9 | 14.3$\pm$1.3 | 0.266$\pm$0.020 | 0.113$\pm$0.010 | 0.379$\pm$0.030[6] | 0.424$\pm$0.050
615.0∗) | 21.5$\pm$1.4 | 14.7$\pm$0.9 | 0.254$\pm$0.020 | 0.108$\pm$0.006 | 0.362$\pm$0.020[6] | 0.425$\pm$0.040
951.0∗) | 22.7$\pm$1.2 | 15.6$\pm$0.8 | 0.220$\pm$0.010 | 0.096$\pm$0.005 | 0.316$\pm$0.010[6] | 0.436$\pm$0.030

∗) the activation
∗∗) the prompt method

Table 2: The weighted means of the ANC-values ($C^{{\rm exp}})^{2}$ for ${}^{3}He+\alpha\rightarrow^{7}{\rm Be}$, NVC’s $\mid G\mid^{2}_{{\rm exp}}$ and the calculated values of $S_{3\,4}(E)$ at energies $E$=0 and 23 keV. The the second and third lines (the fifth, sixth and ninth lines) correspond to the results obtained by means of the analysis of data from works pointed out in the first column, and the penultimate line corresponds to that obtained by using data from all experiments [6–11]. The figures parenthetical are the weighted means obtained from corresponding ones given for the activation and the prompt method. Experimental data |  $(C^{{\rm exp}}_{1\,3/2})^{2}$, |  $\mid G_{1\,3/2}\mid^{2}_{{\rm exp}}$, |  $(C^{{\rm exp}}_{1\,1/2})^{2}$, |  $\mid G_{1\,1/2}\mid^{2}_{{\rm exp}}$, |  $S_{3\,4}$(0), |  $S_{3\,4}$(23 keV),
---|---|---|---|---|---|---
| fm-1 | fm | fm-1 | fm | keV b | keV b
[6–9](the activation) | 21.2$\pm$0.4 | 1.00$\pm$0.02 | 14.5$\pm$0.3 | 0.688$\pm$0.013 | 0.560$\pm$0.003 | 0.551$\pm$0.003
[9](the prompt | 21.5$\pm$0.7 | 1.02$\pm$0.04 | 14.8$\pm$0.5 | 0.697$\pm$0.024 | 0.568$\pm$0.002 | 0.558$\pm$0.002
method), the set I | (21.3$\pm$0.4) | (1.01$\pm$0.02) | (14.6$\pm$0.2) | (0.690$\pm$0.011) | (0.566$\pm$ 0.004) | (0.556$\pm$0.003)
|  |  |  |  | 0.560$\pm$0.017 [9] |
[10](the activation) | 24.0$\pm$0.4 | 1.13$\pm$0.02 | 16.2$\pm$0.2 | 0.768$\pm$0.011 | 0.630$\pm$0.008 | 0.619$\pm$0.008
[10](the prompt me- | 24.1$\pm$0.3 | 1.14$\pm$0.02 | 16.4$\pm$0.2 | 0.773$\pm$0.011 | 0.624$\pm$0.010 |  $0.612\pm$0.010
thod) and [11] (the | (24.1$\pm$0.2) | (1.14$\pm$0.01) | (16.3$\pm$0.2) | (0.771$\pm$0.008) | (0.628$\pm$ 0.006) | (0.616$\pm$0.006)
${\rm{{}^{7}Be}}$ recoils),the set II |  |  |  |  | 0.596$\pm$ 0.021 [10] |
|  |  |  |  | 0.57$\pm$ 0.04 [11] |
${\rm{[6-10]}}$ |  |  |  |  | 0.580$\pm$ 0.043 [38] |
[6–11] | 23.3${}^{{\rm{+1.0}}}_{{\rm{-2.4}}}$ | 1.10${}^{{\rm{+0.05}}}_{{\rm{-0.11}}}$ | 15.9${}^{{\rm{+0.6}}}_{{\rm{-1.5}}}$ | 0.751${}^{{\rm{+0.028}}}_{{\rm{-0.072}}}$ | 0.613${}^{{\rm{+0.026}}}_{{\rm{-0.063}}}$ | 0.601${}^{{\rm{+0.030}}}_{{\rm{-0.072}}}$