научная статья по теме ASYMPTOTIC NORMALIZATION COEFFICIENTS (NUCLEAR VERTEX CONSTANTS) FOR AND THE DIRECT ASTROPHYSICAL FACTORS AT SOLAR ENERGIES Физика

Текст научной статьи на тему «ASYMPTOTIC NORMALIZATION COEFFICIENTS (NUCLEAR VERTEX CONSTANTS) FOR AND THE DIRECT ASTROPHYSICAL FACTORS AT SOLAR ENERGIES»

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ASYMPTOTIC NORMALIZATION COEFFICIENTS (NUCLEAR VERTEX CONSTANTS) FOR p + 7Be ^ 8B AND THE DIRECT 7Be(p, y)8B ASTROPHYSICAL S FACTORS AT SOLAR ENERGIES

© 2008 S. B. Igamov, R. Yarmukhamedov*

Institute of Nuclear Physics, Uzbekistan Academy of Sciences, Tashkent Received September26, 2007; infinalform, January 18, 2008

A new analysis of the precise experimental astrophysical S factors for the direct-capture 7Be(p, y)8B reaction [A.J. Junghans et al, Phys. Rev. C 68, 065803 (2003) and L.T. Baby et al., Phys. Rev. C 67, 065805 (2003)] is carried out, basing on the modified two-body potential approach in which the direct astrophysical S factor, S17(E), is expressed in terms of the asymptotic normalization constants for p + 7Be ^ 8B and two additional conditions are involved to verify the peripheral character of the reaction under consideration. The Woods—Saxon-potential form is used for the bound-(p + 7Be)-state wave function and for the p7Be-scattering wave function. New estimates are obtained for the "indirectly measured" values of the asymptotic normalization constants (the nuclear vertex constants) for the p + + 7Be ^ 8B and S17(E) at E < 115 keV, including E = 0. These values of S17(E) and asymptotic normalization constants have been used for obtaining the "indirectly measured" values of the s-wave average scattering length and the p-wave effective-range parameters for p7Be scattering.

PACS:25.40.Lw, 26.35.+c

1. INTRODUCTION

The 7Be(p, y)8B reaction rate given in terms of the zero-energy astrophysical S factor S17(0) is one of the main input data in the solar-neutrino problem because high-energy neutrinos are emitted via the decay 8B ^ 7Be + e+ + ve [1—3]. This quantity is determined both by extrapolating the measured absolute cross sections aexp(E) (or equivalently its experimental S factors S^7p(E)) to the astrophysically relevant energies (^20 keV) [2—4] and by theoretical predictions (see, e.g., [5—7] for review and [8—11]).

Despite the steady and impressive progress in understanding of this reaction made in recent years in measurements of Se7p(E) at extremely low energies [7] and the theoretical predictions for Si7(E) at solar energies (0 < E < 25 keV) [11], some ambiguities associated with the prediction for Si7(0), however, still exist and may considerably influence the predictions of the standard solar model [1, 2].

Experimentally, there are two types of data for the 7Be(p, y)8B cross sections at extremely low energies: (i) Eight direct measured data using radioactive 7Be targets with uncertainties up to 20% [7, 12—19]. All of these measured data have a similar energy

E-mail: rakhim@inp.uz

dependence for the astrophysical S factors but the extrapolation of each of the measured data from the observed energy ranges to low experimentally inaccessible energy regions, including E = 0, gives a value of S17(0) with an uncertainty exceeding noticeably the experimental one. Nevertheless, the recent values of S17(0) recommended in [7] and [19] are 21.4 ± 0.5(exp.) ± 0.6(theor.) and 21.2 ± ± 0.7 eV b, respectively, which have been obtained from the analysis of the precisely measured data for S17 (E) by means of an artificial renormalization of the energy dependence predicted within the microscopic cluster model [8]. (ii) Indirectly measured data [20— 27] obtained from the Coulomb-breakup experiments in which a radioactive beam of8B nuclei is dissociated into two fragments (proton (p) and 7Be) in the field of multicharged heavy nuclei. The 7Be(p, y)8B astrophysical S factors extracted by the authors of those works change within the range 16.7 < S17(0) < < 20.6 eV b. It is seen that there is a discrepancy between the results for S17(0) obtained from two types of the experimental data, the direct and indirect ones, the main reasons of which is not known yet. Besides, as noted in [18] the astrophysical S factor S17(0) must be known to ±5%, so as its uncertainty would not be a dominant error in prediction of the solar-neutrino flux [2].

The theoretical calculations of £17(0) performed within different methods also show considerable spread [4, 8, 9, 11, 28]. However, the microscopic cluster-model calculations performed in [8, 9, 11] show, firstly, considerable sensitivity of £17(0) to the used effective nucleon—nucleon potential and, secondly, a correlation has been revealed between the calculated asymptotic normalization coefficients (ANCs) (or the respective nuclear vertex constants (NVCs) [29]) for p + 7Be ^ 8B (for the virtual decay 8B ^ p + 7Be) and the calculated £17(0). Such correlation does happen since at extremely low energies, due to the strong Coulomb repulsion and rather low binding energy (0.137 MeV) of 8B in the p + 7Be channel, the direct-radiative-capture 7Be(p, 7)8B reaction proceeds mainly in the region well outside the range of the internuclear p7Be interaction [30]. In this case, £17(0) is expressed in terms of the ANCs for p + 7Be ^ 8B [9, 31, 28]. In this connection, one should also note the value of £17(0) = 18.2 ± 1.8 eV b [32, 33] inferred in the ANC method by using the values of the ANCs for p + + 7Be ^ 8B which have been obtained from analysis of the results for the peripheral proton-transfer reactions performed within the modified DWBA approach [33—35]. But the authors of [33] also noted that the reason of the discrepancy between the £17(0) value obtained in [33] and that recommended in [7] is not understood yet. Moreover, as it is noted also by authors of [18, 7, 19], the ANC method is still subject to uncertainties related to the model dependence of the ANC and the extracted £17(0) values (see [36, 37] and below also). From our point of view one of the possible reasons of the observed discrepancy between the results of [7] and [33] is apparently connected with the fact that the available values of the ANCs for p + 7Be ^ 8B obtained in [33—35] may not have enough accuracy [37]. Therefore, determination of precise experimental values of the ANCs for p + + 7Be ^ 8B is highly desirable since it has direct effects in the correct extrapolation of the 7Be(p, 7)8B astrophysical £ factor at solar energies. For this aim all possible applications of the two-body potential model are not exhausted yet.

In this work a new analysis of the highly precise experimental astrophysical £ factors for the direct-capture 7Be(p, 7)8B reaction at the energy regions 116 < E < 400 keV and 1000 < E < 1200 keV [7, 18, 19] is performed within the modified two-body potential approach [36] to obtain "indirectly measured" values both of the ANCs (the NVCs) for p + 7Be ^ ^ 8B and of £17(E) at E < 115 keV, including E = 0. We show the possibility of extraction of ANCs for p + 7Be ^ 8B directly from the 7Be(p, 7)8B reaction,

where the ambiguities inherent for the standard two-body-potential-model calculation of the 7Be(p, y)8B reaction, connected with the choice of the geometric parameters (the radius ro and the diffuseness a) for the Woods—Saxon potential and the spectroscopic factors (for example, see [15, 38—43] and below), can be reduced in the physically acceptable limit, being within the experimental errors for the Sl7P(E) reached in [7, 18, 19].

The contents of this paper are as follows. In Section 2 basic formulae of the modified two-body potential approach to the direct-radiative-capture p + + 7Be ^ 8B + y reaction are given. There the precisely measured astrophysical S factors for the direct-radiative-capture 7Be(p, y)8B reaction is analyzed (Subsections 2.2—2.4). The conclusion is given in Section 3.

2. ANALYSIS OF 7Be(p,7)8B REACTION.

RESULT AND DISCUSSION

2.1. Basic Formulae

Here we give the formulae specialized for the astrophysical S factor to the case of the 7Be(p, 7)8B reaction. Let us write lf (jf) for the orbital (total) angular momentum of the proton in the nucleus 8B(7Be + p), li(ji) for the orbital (total) angular momentum of the relative motion of the colliding particles in the initial state, A for multipole order of the electromagnetic transition, nf for the Coulomb parameter for the 8B(=7Be + p) bound state, and f for the reduced mass of the (p7Be) pair. For the 7Be(p, 7)8B reaction, the value of lf is taken to be equal to 1 and the values of jf are taken to be equal to 1/2 and 3/2, while li = 0 and 2 for the E1 transition and li = 1 for the E2 transition.

According to [44, 36], we can write the astrophys-ical S factor in the form

S17(E)= ^E j Rlf (E,Cff). (1)

Here, Cf jf is the ANC for p + 7Be ^ 8B, which determines the amplitude of the tail of the nucleus-8B bound-state wave function in the 7Be + p channel and is related to the NVC Gf jf for the virtual decay

8B ^ 7Be + p and to the spectroscopic factor Zf jf for the (7Be + p) configuration with the quantum numbers lf ,jf in the nucleus 8B as [29]

Gi

f jf

= _ jlf +nf

Ci

f jf

(2)

and

of Eq. (6):

Ci

_ Z1/2 C(sp) f jf _ Zlf 3f Clf jf '

respectively, and

Rif (E,Cfjf

(sp) ^ _ Slfjf(E)

(3)

(4)

where Sifjf (E) = ^л Sif jfл is the single-particle astrophysical S factor and cjj) is the single-particle

ANC, which determines the amplitude of the tail of the single-particle wave function of the bound 8B(7Be + p) state. In (2) the factor taking into account the nucleons' identity [29] is absorbed in the

dj The single-particle bound-state wave function plfpf (r) is determined by the solution of the radial Schrodinger equation with the phenomenological Woods—Saxon potential for the given quantum numbers n (n is a number of the nodes of pifpf (r)), lf, and jf as well as geometric parameters of ro and a and with depth adjusted to the experimental binding energy ep of the 8B ground state with respect to the 7Be + p channel. Note that in Eq. (4) a dependence of the function Rif (E, dj) on the free parameter dj enters also through the single-particle wave function pif jf (r; djfsjf)(=pif jf (r)) [45], and the single-

particle ANC dlj itself is a function of the geometric parameters of ro and a.

According to [36], the

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