Triple-differential cross section of the 208Pb(6Li,αd)208Pb Coulomb breakup and astrophysical S -factor of the d(α,γ)6Li reaction at extremely low energies

Nuclear Physics A 673 (2000) 509–525 www.elsevier.nl/locate/npe

Triple-differential cross section of the Pb(6Li, αd)208Pb Coulomb breakup and astrophysical S-factor of the d(α, γ )6Li reaction at extremely low energies 208

S.B. Igamov a , R. Yarmukhamedov a,b a Institute of Nuclear Physics, Uzbekistan Academy of Sciences, Tashkent, 702132, Uzbekistan b The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy

Received 19 November 1999; revised 21 January 2000; accepted 28 January 2000

Abstract A method of calculation of the triple-differential cross section of the 208 Pb(6 Li, αd)208 Pb Coulomb breakup at astrophysically relevant energies E of the relative motion of the breakup fragments, taking into account the three-body (α–d–208 Pb) Coulomb effects and the contributions from the E1- and E2-multipoles, including their interference, has been proposed. The new results for the astrophysical S-factor of the direct radiative capture d(α, γ )6 Li reaction at E 6 250 keV have been obtained. It is shown that the experimental triple-differential cross section of the 208 Pb(6 Li, αd)208 Pb Coulomb breakup can also be used to give information about the value of the modulus squared of the nuclear vertex constant for the virtual decay 6 Li → α + d.  2000 Published by Elsevier Science B.V. All rights reserved. PACS: 24.10-i; 25.35.+c; 25.70.Mn Keywords: Triple-differential cross section; 208 Pb(6 Li, αd)208 Pb Coulomb breakup; Three-body Coulomb effects; Nuclear vertex constant; Astrophysical S-factor; d(α, γ )6 Li reaction

1. Introduction It is well-known that the radiative capture d(α, γ )6 Li reaction is of great interest as one of sources of the 6 Li creation in the early Universe [1–4]. However, up to now in nuclear astrophysics the problem of obtaining reliable experimental astrophysical Sexp factors, Sαd (E), for the d(α, γ )6 Li reaction at astrophysically relevant energies E (E 6 300 keV) is still unsolved. exp A direct measurement of the Sαd (E) for the d(α, γ )6 Li reaction covers the energy exp region E > 700 keV [4]. The only available experimental data for the Sαd (E) in the energy region 100 6 E 6 600 keV are those obtained from the triple-differential cross 0375-9474/00/$ – see front matter  2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 4 7 4 ( 0 0 ) 0 0 1 3 2 - 9

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section (TDCS) analysis of the 208 Pb(6 Li, αd)208 Pb Coulomb breakup taking into account only a contribution from the E2-multipole [5]. However, from the TDCS analysis of the 208 Pb(6 Li, αd)208 Pb Coulomb breakup a reliability of the given information about the exp Sαd (E) depends on taking into account the contributions of both the nuclear breakup amplitude and the interference between the E1- and E2-multipoles in correct manner. But the pure Coulomb breakup can be experimentally approached by the appropriate choice of the reaction kinematics, for instance, for scattering at energies well above the Coulomb barrier but with small scattering angles and a sufficiently large cutoff parameter R0 being beyond the range of the radius RN of the 208 Pb–6 Li nuclear interaction (R0 > RN ). However it is necessary to take into account a contribution of the E1-multipole into the TDCS for the 208 Pb(6 Li, αd)208 Pb Coulomb breakup at astrophysically relevant energies E (E 6 300 keV) of the relative motion of the breakup fragments. This is due to with the following fact. Though the E1-contribution is suppressed compared to the E2-contribution in the amplitude of the 208 Pb(6 Li, αd)208 Pb Coulomb breakup [6–8] nevertheless the noticeable E1-contribution to the TDCS can be expected due to its interference with the E2-contribution. Therefore taking into account the importance of the E1-contribution in the astrophysical S-factor of the direct radiative capture d(α, γ )6 Li reaction at astrophysically relevant energies [9–11], the accurate calculation of the E1- and E2-contributions in the TDCS of the 208Pb(6 Li, αd)208 Pb Coulomb breakup is required for obtaining reliable data exp exp of the Sαd (E) from the TDCS-analysis. Despite this in a paper [5] the data for the Sαd (E) have been obtained by means of the TDCS analysis of the 208 Pb(6 Li, αd)208 Pb Coulomb breakup without taking into account the E1-contribution though it was noted by pass the importance of the E1-contribution at extremely low energies. Perhaps that is one of the possible reasons of the observed discrepancy between the results of the calculation [10,11] and the experimental data [5] for the astrophysical S-factor of the d(α, γ )6 Li reaction in the energy range E 6 300 keV. In additional, at extremely low energies E the three-body Coulomb effects in the final state of the 208Pb(6 Li, αd)208 Pb Coulomb breakup must be taken into account in correct way. In the present work, a method of the TDCS analysis for the 208 Pb(6 Li, αd)208 Pb Coulomb breakup is developed within the framework of the three-body approach [8,12], taking into account the contributions of the E1- and E2-multipoles, including their interference, for obtaining the information of the astrophysical S-factor for the d(α, γ )6 Li reaction at extremely low energies. The contents of the paper is as follows. In Section 2 the expression for the TDCS of the 208 Pb(6 Li, αd)208 Pb Coulomb breakup is derived taking into account the contributions from the E1- and E2-multipoles, including their interference. The consideration is carried out within the framework of the three-body (α–d–A) approach using a new kind of the three-body Coulomb wave function in the continuum, where A = 208Pb. In Section 3 a new method for the extraction of the astrophysical S-factor, exp Sαd (E), for the d(α, γ )6 Li reaction in the energy range E 6 250 keV is proposed using the results of Section 2 for the TDCS of the Coulomb breakup considered. The conclusion is given in Section 4.

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2. The triple-differential cross section for the 208 Pb(6 Li, αd)208 Pb Coulomb breakup In this section we consider the 208 Pb(6 Li, αd)208 Pb Coulomb breakup reaction within the framework of the three-body (α–d–A) approach. We assume that a breakup of 6 Li into the α-particle and the deuteron takes place at relative distances of the colliding nuclei R exceeding a value of the radius RN , i.e., R > R0 > RN , where R0 is the minimum distance of approach of the colliding nuclei beginning with that an influence of nuclear interaction vanishes. If the reaction kinematics is also restricted by rather low energies E then within the framework of distorted waves approximation with the optical Coulomb potential of (A–6Li)-interaction, the amplitude of the 208Pb(6 Li, αd)208 Pb Coulomb breakup can be represented as [8,12] Z (−)∗ (+) c dR dr Ψkαd kf (r, R)W c (r, R)Iαd (r)Φki (R). (1) M (ki , kf , kαd ) = R>R0

Herein Iαd (r) is the overlap integral for the wave function of the 6 Li nucleus in the (α + d)(+) configuration, Φki (R) is the Coulomb distorted wave function of the relative motion of the A–6Li-nuclei with the relative momentum ki , which satisfies the Schrödinger equation for the “optical” Coulomb potential VAc 6 Li (R), and c c (rAα ) + VdA (rdA ) − VAc 6 Li (R), W c (r, R) = VAα

(2)

c (r ) is the Coulomb potential of the interaction between the centers of mass of where Vβγ βγ the particles β and γ and the vectors rAα and rdA are defined by md mα r, rdA = R − r, (3) rAα = −R − m6 Li m6 Li

where mβ is a mass of particle β. (r, R) from the integrand in Eq. (1) describes the relative The wave function Ψk(−) αd kf motion of the three-body (α–d–A) in the final state of the 208Pb(6 Li, αd)208 Pb Coulomb breakup with the relative momentum kαd of the (α–d)-pair and the relative momentum kf of the centers of mass of the (α–d)-pair and the 208 Pb nucleus. We use a procedure of choice of a value of the cutoff parameter proposed in papers [13–15], i.e., the adopted value R0 is chosen not larger than required by the condition of vanishing of the nuclear (A–6Li)-interaction [15]. According to this condition the value R0 is taken to be equal √ to 2 /4E < b [13,14], where a value R is taken as R = r (3 6 + = R + πZ Z e R√ 6 0 N i N N N Li A 3 208 ), b is an impact parameter, Z e is a charge of the particle β, E = k 2 /2µ β i A6 Li and i µβγ are the reduced mass of particles β and γ . For the calculation of amplitude (1) it is necessary to know the exact three-body wave (−) function Ψkαd kf (r, R) and the overlap integral Iαd (r). In Eq. (1) the presence of the overlap integral Iαd (r) cuts off the integration region over variable r due to its asymptotic behaviour at r → ∞, given by relation: Iαd (r) ' Cαd;lαd

W−ηαd ;lαd +1/2 (2καd r) Ylαd ναd (ˆr), r

r → ∞,

(4)

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where Wα;β (x) is the Whittaker function, ηαd = Zα Zd e2 µαd /καd is the Coulomb √ parameter for the (α + d)-bound state, καd = 2µαd εαd , lαd is the relative orbit angular momentum of the deuteron and the α-particle in the (α + d)-bound state with the binding energy εαd , and Cαd;lαd is the asymptotical normalization coefficient, related to the nuclear vertex constant (NVC) Gαd;lαd for the virtual decay 6 Li → α + d as [16]: √ π exp(iπηαd /2)Cαd;lαd . (5) Gαd;lαd = −i µαd But in Eq. (1) the integration over variable R does not protect the analogous condition for the overlap integral Iαd (r) due to an oscillation of the integrand at R → ∞. Therefore, in amplitude (1) the integration region includes the asymptotic region mainly with r/R  1 and R > R0 > RN . On other hand, we are interested in the kinematic region of the 208 Pb(6 Li, αd)208 Pb Coulomb breakup with rather low energies E of the breakup fragments. Consequently, due to the strong Coulomb repulsion between the three (αparticle, deuteron and 208 Pb) particles in the final state and well large value of cutoff (r, R) parameter R0 (R0 > RN ), in amplitude (1) the three-body wave function Ψk(−) αd kf can also be replaced by its purely Coulomb asymptotic expression in the region r/R  1 (r, R) have been found in recent papers [17,18]. and R > R0 . Such asymptotics of Ψk(+) αd kf It has the following form [17]: (−)

(+)∗

Ψkαd kf (r, R) = Ψ−kαd −kf (r, R),

 (r, R) ' Ψq(+) (r)eikf R exp iηAα ln(kAα R Ψk(+) αd kf αd (R) + iηdA ln(kdA R + kdA R) ,

r/R  1,

(6) − kAα R)

R → ∞.

(r) Herein Ψq(+) αd (R)

is the Coulomb wave function corresponding to the relative motion of the α-particle and deuteron in the presence of the Coulomb field of the 208Pb-ion, qαd is the local relative momentum of the α-particle and deuteron at a distance R from 208Pb-ion [17]: ˆ (7) qαd (R) = kαd + a(R)/R, ˆ is determined by Eqs. where kβγ is the relative momentum of the particles β and γ , a(R) ˆ (80), (82) and (83) of paper [17] and R = R/R. Since inequalities mA  mα , mA  md and kαd  kf hold in the kinematic region of the Coulomb breakup considered below, hereafter one can use the approximations: md mα kf , kAα ' − kf , kdA ' m6 Li m6 Li Zd Zα ηf , ηAα ' ηf , (8) ηdA ' Z6 Li Z6 Li ˆ can be presented in the where ηf = ZA Z6 Li e2 µA6 Li /kf . Then the explicit expression a(R) form [17]   ˆ ˆ Zα µαd Zd ˆ = Kηf R + kf , − K= . a(R) ˆ Z6 Li md mα 1 + kˆ f R

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If the mass defects of α-particle and deuteron are directly taken into account then the value K is estimated to be equal to 1.43 × 10−4 [9]. It should be emphasized that the choice of the asymptotical expression (6) is based on the following considerations. Firstly, this asymptotical expression has the correct asymptotic behavior in the asymptotic regions of the configuration space where values of relative distance r are rather less than those of relative distance R. Therefore it makes possible to take correctly into account a contribution of the integration area over the variable r at r < R0 in the amplitude (1). Secondly, this asymptotic involves the important three-body Coulomb effect [17], that is, the presence of the Coulomb field of the multicharged Pbion in the final state of the breakup considered changes the relative momentum kαd of the α–d-pair by the momentum qαd (R) given by (7). For the overlap integral Iαd (r) one can make use of the result of the multicluster dynamic model with Pauli projection which was proposed in papers [19,20] within the framework of the three-body (α–n–p) model. The overlap integral Iαd (r) has the following form [20]: Z X 1M6 Li Iαd;lαd (r)Clαd ναd r). (9) Iαd (r) = ψd (x)ψ6 Li (x, r) dx = 1Md Ylαd ναd (ˆ lαd

Herein ψd (x) (ψ6 Li (x, r)) is the wave function of a bound state of the deuteron (the 6 Li in cγ the three (α–n–p)-model), Caαbβ is the Clebsh–Gordon coefficient, Ma is the projection of a spin of particle a, and Iαd;lαd (r) is as [20]: Iαd;lαd (r) = r lαd

N X

Ci;lαd e−bi;lαd r , 2

(10)

i=1

where Ci;lαd and bi;αd are linear and nonlinear variational parameters, respectively. It should be noted that the overlap integral (9) and (10) describes energies of low-lying states of the 6 Li, the r.m.s. radiuses, the magnetic moments and the electromagnetic form factors of the 6 Li ground state without any free parameters [20]. It should also be stressed that this overlap integral also has both the correct asymptotical behavior given by Eq. (4) and an absolute normalization factor value of its tail for the S-wave [20], defined by the value of the ANC Cαd;0 or the NVC Gαd;0 (Section 3). In this calculation the contribution from D-wave (lαd = 2) has not been taken into account since it is negligible [19,20]. Taking into account the Eqs. (6) and (9) as well using the leading terms of the (R) asymptotical expression at R > R0 → ∞ for the Coulomb wave function Φk(+) i c after an expansion of W (r, R) into the multipoles [21], one can write the amplitude M c (ki ; kf , kαd ) as √ X 4π c Mlml (ki ; kf , kαd ), (11) M (ki ; kf , kαd ) = 2l + 1 lml Z dR i[(ki −kf )R+ηi ln(ki R−ki R)] e Mlml (ki ; kf , kαd ) = e−iν ZA e R l+1 R>R0 ∗ × eiηf ln(kf R+kf R) Ylm l



ˆ M αd qαd (R) , R lml

(12)

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 Z
Zd (−)∗ αd l l Zα Mlm (R) = eµ + (−1) q dr ψqαd (R) (r)r l Iαd;0 (r)Ylml (ˆr), αd αd l mlα mld

(13)

where ν = ηf [Zd ln(md /m6 Li ) + Zα ln(mα /m6 Li )]/Z6 Li . As is seen from Eqs. (11)–(13) the amplitude of the Coulomb breakup considered is not expressed through the amplitudes of the photodisintegration process γ + 6 Li → α + d. The reason for this is that the wave function of αd-scattering depends on the local momentum qαd (R) in the integrand of Eq. (13). Such dependence arises due to an influence of the Coulombic field of 208 Pb-ion on the relative momentum of the breakup fragments [17]. It should be noted that if one neglects this influence in the Eq. (13), by replacing (αd) l represents the qαd (R) to kαd , then the obtained amplitude, Mlml (kαd ) multiplied by kαd 6 amplitude of the El-multipole transition for the Li + γ → α + d process in the long-wave approximation. The kinematics of the Coulomb breakup considered is chosen as in a paper [5]. That is the relative momentum kαd is parallel to momentum kf . Then for the local momentum qαd (R) in the approximation up to terms of the order O(R −2 ), one has   qαd (R) ' qαd (R) = kαd 1 + C(kαd ) , (14) R where C(kαd ) = 1.43 × 10−4 ηf /kαd . The approximation (14) is also valid at rather small values of kαd (or Eαd ) when the condition C(kαd )/R0  1 is satisfied. (r) in partial waves and taking into account Further, expanding the wave function Ψq(−) αd (R) Eq. (14) as well the determination for the a(R) of paper [17], and after performing integration over the angle variables of the vector r, one can reduce the expression for αd (qαd (R)) to the following form Mlm l αd αd (qαd (R)) = 4πi −l e−iδl M˜ lm (kαd ; R)Mlml (kαd ; R), Mlm l l   Z∞
Zα αd l l Zd (k ; R) = eµ + (−1) dr Fl qαd (R)r r l+2 Iαd;0(r), M˜ lm αd αd l l mlα md

(15) (16)

0

 1/2   X ˆ C(kαd ) −l l! Mlml (kαd ; R) = 1 + R (2l1 )!(2σ2 )!σˆ1 ! l1 +l2 =l;σ1 +σ2 =l2 l2  C(kαd ) ˆ × Cllν1 0l2 ν Cσl21ννσ2 0 Yσ2 ml (R), 2R cos2 (θR /2)

(17)

where Flf (qαd (R)r) is the radial Coulomb wave function of the relative motion of the α-particle and deuteron taking into account the Coulomb field of 208 Pb-ion, δlf is αd the ˆ kˆ αd = kˆ f , σˆ 2 = 2σ + 1 and the direction of the vector Coulomb phase shift, cos θR = kˆ f R, kf (or kαd ) is aligned along the Oz axis. The triple-differential cross section in the c.m.s of the Coulomb breakup considered can be represented as m26 µαd kαd 2 d 3σ = Li 5 M c (ki ; kf , kαd ) , dΩ6 Li dΩαd dEαd (2π)

(18)

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Fig. 1. The triple-differential cross section of the 208 Pb(6 Li, αd)208 Pb Coulomb breakup at E6 Li = 156 MeV. Data points are from paper [5]. The solid curve is our result and the dashed curve is the result of a paper [5].

where the amplitude M c (ki ; kf , kαd ) is determined by expressions (11), (12) and (15)– (17). It can be seen from (11), (15)–(18) that the TDCS for the 208 Pb(6 Li, αd)208 Pb Coulomb breakup can not be expressed only in terms of the astrophysical S-factor for the Elmultipole: El (E) = EσγEl (E)e2πη , Sαd

(19)

where σγEl is the cross section for the El-multipole of the direct radiative capture reaction α + d → 6 Li + γ and η = Zα Zd e2 µαd /kαd is the Coulomb parameter for the αdscattering. That is the result of the presence of the interference of E1- and E2-multipoles into TDCS as well as of the influence of the three-body (d–α–208Pb) Coulomb effects in the final state. Besides, as is seen from Eqs. (15), (16) and (18) the strength of the E1contribution into TDCS is mainly defined by the fact that to what extent taking into account of the value difference 1 = Zα /mα − Zd /md , due to a presence of the defects of masses for the α-particle and the deuteron, becomes importance for the interference term of E1and E2-multipoles. The calculated triple-differential cross section (solid line) and the experimental data [5] are shown in Fig. 1. The calculation has been performed at the kinematics condition of a paper [5]. This is projectile energy 156 MeV for 6 Li and θd = θα = 3◦ , where θd (θα ) is the detected angle of the deuteron (α-particle). For these kinematics and within admissible range of rN = 1.36 fm, the calculated value of the cutoff parameter R0 is to be 12.4 fm, while a value of the radius RN of the nuclear (6 Li–208Pb)-interaction is about

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10.5 fm. The relative contribution of the E1- and E2-multipoles to the TDCS has also been estimated. The calculation showed that the contribution of the E2-multipole is dominated into TDCS in the energy range E > 250 keV, but the contribution of the E1-multipole is enhanced with decrease energy E and reaches more than 20% at astrophysically relevant energies (6 170 keV). That is due to the presence of the interference term for E1- and E2-multipoles in expression (18). For a comparison, a result of the semiclassical method (dashed line) of paper [5], which has been obtained taking into account the contribution only from the E2-multipole, is displayed in Fig. 1. It can also be seen from Fig. 1 that for the energy range E 6 250 keV there is a good agreement between our results and experimental data. The observed discrepancy between our results and experimental data at E > 250 keV can apparently be connected with an influence of the nuclear interactions. In this connection it should be emphasized that for a reliable estimation of contribution of nuclear interactions the method proposed should be improved by taking into account the nuclear part of the amplitude for the breakup considered. But this task is rather difficult and requires a specific consideration. At present time such work is in progress. Nevertheless this assumption does not contradict with the results of papers [5,21]. In paper [21] an analysis of the contributions of the Coulomb and nuclear interactions to the breakup of the light nuclei in field of 208Pb-ion was carried out within the framework of DWBA using the eikonal approximation for the initial and final states. It was shown that at small scattering angles and in a very narrow energy interval E, the Coulomb contribution to the breakup processes considered can be reasonably singled out of the nuclear contribution. But as the energy E increases, the influence of nuclear contribution increases too and this in turn can lead to suppression of the purely Coulomb contribution due to the presence of their interference [21]. Besides, it would like to note also that, according to paper [15], within the framework of the semiclassical method, where the projectile is assumed to move on a Rutherford orbit, the expression for TDCS corresponding to the El-multipole at fixed angles θd and θα of breakup fragments as function of the energy E can be factorized by the cross-section σγEl (E) for the El-multipole and by the virtual photon number per unit solid angle Ωf , dnEl /dΩf . From here it follows that within the framework of the semiclassical method used in paper [5] in the TDCS calculating the information of a contribution of the nuclear interaction between the breakup fragments is only contained to the cross section σγ (E) of the d(α, γ )6 Li reaction. Therefore, for estimation of an influence of nuclear αd- interaction on calculated TDCS in paper [5] it is sufficient to calculate the ratio Rγ = σγCN (E)/σγC (E), where σγCN (σγC ) is the total cross section of the d(α, γ )6 Li reaction taking into account the Coulomb–nuclear (Coulomb) αd-interaction. The calculation showed that with increase of the energy E an influence of nuclear αdinteraction on the cross section σγ (E) consequently, and on TDCS is noticeably increased in the energy range with E > 250 keV. But its influence is unnoticeable at E 6 250 keV (Rγ < 1.1), that is, less than 10 percent. These facts confirm our assumption that in TDCS calculation of the Coulomb breakup considered at astrophysically relevant energies (6 250 keV) it can be restricted by taking into account only Coulomb interaction in the final states. Thus, from here it follows that in the energy range (say, at energies E more than a some value E0 ) where the contribution from the E2-multipole into TDCS and the

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cross section σγ (E) is dominant the results of paper [5] are reliable both for TDCS and for the astrophysical S-factor of the d(α, γ )6 Li reaction. However, the presence of a good agreement between the experimental data for the TDCS of the Coulomb breakup considered and the results of paper [5] obtained without taking into account the contribution of the E1-multipole in the energy range with E 6 250 keV, where the contribution of the E1-multipole to TDCS must be not disregarded in reality, can be apparently associated with following circumstance. In paper [5] a calculation of TDCS has been carried out using the linear interpolation formula Sαd (E) = [(0.91 ± 0.18) + (2.92 ± 0.66)E] × 10−5 MeV mb for the astrophysical S-factor anywhere in the energy range pointed out in Fig. 1. But this interpolation formula reproduces the energy dependence of the astrophysical S-factor, Sαd (E), for the d(α, γ )6 Li reaction at energies E > E0 , where the contribution of the E1-multipole is sufficiently unnoticeable. Therefore, using this extrapolation formula can hardly justified for the energy range where the contributions from both the E1- and E2-multipoles are noticeable (E 6 E0 ), because the analytical properties of the function Sαd (E) (or the crosssection) and the used interpolation formula on the E-plan may not be the same, so that the interpolation formula could be applied for the extrapolation of these data [22]. Besides, the calculation showed that the interpolation formula Sαd (E) = [(0.91 ± 0.18) + (2.92 ± 0.66)E × 10−5 MeV mb leads to high values for the astrophysical S-factor in a part of the energy range (E 6 E0 ) where the contribution of the E1-multipole becomes the same or even greater than the contribution of the E2-multipole. This in turn leads to low values of the virtual photon number, due to the absence into the TDCS of the contribution of the exp E1-multipole. Since in paper [5] the experimental astrophysical S-factors, Sαd (E), are presented as the ratio of the experimental TDCS to the virtual photon number, dnE2 /dΩf [23], then an application of this relation for getting the information of the astrophysical S-factor for the d(α, γ )6 Li reaction in the energy range E 6 E0 may give high values for exp the astrophysical S-factor, Sαd (E), due to low values of the real virtual photon number. Consequently, in paper [5] in the used expression for the TDCS dnE2 d 3σ ∼ Sαd (E) dΩ6 Li dΩαd dEαd dΩ6 Li at energies with E 6 E0 each of the multipoliers in it r.h.s. do not correspond to a dominating mechanism of the Coulomb breakup considered, due to a importance of the contribution of E1-multipole. Therefore a good agreement between the experimental and the calculated TDCS at energies E 6 300 keV obtained in paper [5] can apparently be explained by the fact that the overestimated values of the Sαd (E) compensate the underestimated values of the real virtual photon number. Perhaps that is a true reason of the observed discrepancy between the values calculated [10,11] and the extracted experimental data [5] for the astrophysical S-factors of the d(α, γ )6 Li reaction in the energy range with E 6 300 keV. One would like to draw attention to the following. There is a contradiction between our deduction, concerning influence of nuclear α–d interaction in the final state of the 208 Pb(6 Li, αd)208 Pb Coulomb breakup, and results of papers [24,25]. To find out a reason

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of this, one should discuss the approximations used by the authors of papers [24,25]. In those works the three-body approaches [8,26] have been applied for a comparative analysis between TDCS calculated and the same experimental data of the 208 Pb(6 Li, αd)208 Pb Coulomb breakup in the energy range beginning from E ' 70 keV up to a vicinity of the resonance peak at energy E = 710 keV. The resonance peak observed in the experimental TDCS is assumed to correspond to the mechanism, going through a formation of the resonance 6 Li∗ (2,186 MeV; 3+ ) in the intermediate state. In paper [25] the resonance amplitude [24] is added to the amplitude of the direct Coulomb breakup, which has the same form as that of Eq. (1). As a result in paper [25], a good agreement between TDCS calculated and the experimental data in the energy range 70 6 E 6 710 keV has been achieved. But in paper [25] a value of the cutoff parameter R0 was taken to be equal to the radius RN (6 11 fm), that is a violation of the condition R0 > RN has been admitted. On our opinion in Eq. (1) if one put R0 = RN then taking into account the nuclear part of the breakup amplitude is also required for the process considered [21]. Besides in works [24,25] it was assumed that a contribution from the nuclear αd-interaction between the breakup fragments is negligibly small everywhere in the energy range E 6 710 keV [25]. However, our calculation and the results of a paper [5] show that in reality in the energy range E > 250 keV an influence of the nuclear αd–interaction in the continuum both for Coulomb breakup considered and for the d(α, γ )6 Li reaction is not negligibly small, especially, when a value E is going to the vicinity of the resonance energy at E ' 710 keV.

3. Astrophysical S-factor for the d(α, γ )6 Li reaction at astrophysically relevant energies In this section we discuss a problem of the extraction of the astrophysical S-factor, exp Sαd (E), for the direct radiative capture d(α, γ )6 Li reaction using the results of calculation of Section 2 for the TDCS of the 208 Pb(6 Li, αd)208 Pb Coulomb breakup in the energy

range E 6 250 keV. The expression for TDCS, given by (18) which further is denoted by d 3 σ th (E), as a function of the energy E at the fixed angles θα and θd , can be presented in the following form d 3 σ th (E) = d 3 σE1 (E) + d 3 σE2 (E) + d 3 σint (E).

(20)

Here the terms d 3 σE1 and d 3 σE2 are related to the contributions from the pure E1- and E2-multipoles, respectively, the third term d 3 σint (E) is related to their interference. As is seen from Eqs. (11), (12), (15)–(18) there are two reasons not allowing directly to get the information about the astrophysical S-factor, E1 E2 (E) + Sαd (E), Sαd (E) = Sαd

(21)

for the d(α, γ )6 Li reaction from the TDCS analysis. First of them is arisen due to a correct taking into account of the three-body Coulomb effects in the final state of the Coulomb breakup considered. As was mentioned above in this case the Coulomb breakup amplitude

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Table 1 Results of the calculation of the triple-differential cross-section (TDCS) and the astrophysical SE2 (E)) is the TDCS (astrophysical S-factor) for factors as functions of energy E; d 3 σE2 (E) (Sαd 3 the E2-multipole; d σ (E) (Sαd (E)) is the TDCS (the astrophysical S-factor) for the E1 + E2multipoles; the figures in brackets correspond to the E1-contribution (on percent) for the respective magnitudes E (keV) 70.0 90.0 110.0 130.0 170.0 210.0 230.0 250.0

d 3 σE2 (E)



mb MeV sr2

1.37 × 10−2 4.69 × 10−2 1.16 × 10−1 2.35 × 10−1 6.67 × 10−1 1.40 1.89 2.46



d 3 σ (E)



mb MeV sr2



E2 (E) (MeV nb) Sαd

Sαd (E) (MeV nb)

1.07 1.34 1.68 2.06 2.93 4.19 4.48 5.08

2.57 (140%) 2.82 (110%) 3.39 (102%) 3.89 (89%) 4.99 (70%) 6.60 (58%) 6.89 (54%) 7.59 (49%)

1.82 × 10−2 (33%) 6.05 × 10−2 (29%) 1.46 × 10−1 (26%) 2.89 × 10−1 (25%) 7.91 × 10−1 (19%) 1.62 (16%) 2.16 (14%) 2.79 (13%)

is not expressed through the amplitude of the 6 Li(γ , α)d photodisintegration (see Eq. (16)) since in the integrand of the Eq. (16) the wave function of the αd–scattering depends on the local momentum qαd (R) but not only on the momentum kαd . The second reason is th (E), in the r.h.s. of connected with a presence of the important interference term, d 3 σint Eq. (20), which doesn’t allow us to express the TDCS in terms only through the values of E1 (E) and S E2 (E). the Sαd αd Consider every case from these ones separately. For this aim at first, one should estimate an influence of the three-body Coulomb effects upon absolute values of the TDCS of the Coulomb breakup considered. The calculation showed that in the energy range 70 6 E 6 250 keV for the Coulomb breakup considered the contribution of the three-body Coulomb effects to the TDCS is unnoticable and to be less than 5%. Then in the amplitude (16) one can substitute the local momentum qαd (R) by kαd :   Z∞ elαd (kαd ; R) ' Mlαd (kαd ) = eµlαd Zα + (−1)l Zd Fl (kαd r)r l+2 Iαd;0(r) dr. (22) M mα md 0

It should be also noted that this contribution to the TDCS is enhanced with a decrease of the energy E and reaches about 10% at E ' 25 keV. In this case the first two terms of the r.h.s. E1 (E) and S E2 (E) as a factors, respectively. of Eq. (20) can be expressed through the Sαd αd However the E1- and E2-multipoles give different relative contributions for the TDCS, d 3 σ th (E), and for the astrophysical S-factor, Sαd (E). For a visual clearness Table 1 of the calculation results both for the TDCS and for the astrophysical S-factor [10,11], obtained with taking into account the E1- and E2-multipoles (E1 + E2) as well only the E2-multipole, are presented. The figures in brackets correspond to the E1-contribution values (on percents) for the respective magnitudes. As is seen from Table 1 a degree of an influence of the E1-contribution into values of Sαd (E) is more large than into values of d 3 σ th (E). As is also seen from Table 1 the contribution of the E1-multipole

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Fig. 2. The triple-differential cross section for the 208 Pb(6 Li, αd)208 Pb Coulomb breakup as a function of the cutoff radius r0 .

into the TDCS of the 208 Pb(6 Li, αd)208 Pb Coulomb breakup enhances with decrease of the energy E and it reaches more than 25% in the energy range with E 6 70 keV. Such noticeable contribution of the E1-multipole into the TDCS of 208 Pb(6 Li, αd)208 Pb Coulomb breakup points out on an importance of taking into account the interference of the E1- and E2-multipoles to extract reliable values of the astrophysical S-factor, Sαd (E), for the d(α, γ )6 Li reaction at astrophysically relevant energies. Namely, this important circumstance has been mentioned by authors of a paper [5] but it was not taken into account in their consideration. With this aim below the analysis result of Section 2 has been applied for getting the information about the Sαd (E) for the d(α, γ )6 Li reaction in energy range with E 6 250 keV. But in Eq. (20) a presence of the interference term doesn’t enable us to express the experimental TDCS d 3 σ exp (E) only through the astrophysical S-factor, Sαd (E), as a factor form. However one should grow an attention to the following circumstance. It is turn out that a contribution from the region 0 6 r 6 4 fm into the TDCS given by Eqs. (11), (12), (18) and (22) is strongly suppressed. To demonstrate this fact one calculated the dependence of the TDCS d 3 σ th (E) on cutoff radius r0 in the lower limit of integration over the variable r in the Eq. (22). As an illustration, in Fig. 2 a dependence of the TDCS, d 3 σ th (E), on the cutoff parameter r0 only for the two energies is plotted. The same dependence occurs at another energies up to 250 keV. It is seen that the value of d 3 σ th (E) is virtually independent of the radius r0 in the region r0 6 4 fm. Then, at r > 4 fm according to a paper [20], in Eq. (22) the Iαd;0 (r) can be approximated by the asymptotic expression (4). This circumstance enables us to factorize every term of the sum in the r.h.s. of Eq. (20), according to the relations (4) and (5) as well Eqs. (11), (12) and (22), by the

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modulus squared of the value NVC for the virtual decay 6 Li → α + d, |Gαd;0|2 : d 3 σEl (E) = |Gαd;0|2 d 3 σ˜ El (E),

d 3 σint (E) = |Gαd;0|2 d 3 σ˜ int (E),

(23)

where the explicit expressions for the functions d 3 σ˜ El (E) (l = 1, 2) and d 3 σ˜ int (E) can be derived from the Eqs. (11), (12), (18) and (22) with taking into account a substitution of the overlap integral Iαd;0 (r) (10) by the asymptotical relation (4) and (5). Thus inserting the relations (23) into (20) the expression for the TDCS can be presented in the following form: d 3 σ th (E) = |Gαd;0|2 d 3 σ˜ th (E),

(24)

where the d 3 σ˜ th (E) is the known function, which determines an energy dependence of the TDCS. On the other hand, according to papers [10,11] a surface character of the d(α, γ )6 Li El (E), in reaction at rather low energies (6 250 keV) allows us to present the S-factor, Sαd the following form: El El (E) = |Gαd;0|2 S˜αd (E), Sαd El (E) (l = 1, 2) can be found from Eq. where the explicit expression for the S˜αd the formulae (6) and (7) of a paper [10] 1 . From Eqs. (24) and (25) one find:

d 3 σ th (E) , d 3 σ˜ th (E) d 3 σ th (E) ˜ Sαd (E), Sαd (E) = 3 th d σ˜ (E)

|Gαd;0|2 =

(25) (19) as well

(26) (27)

E1 (E) + S˜ E2 (E). where S˜αd (E) = S˜αd αd The expressions (26) and (27) allows us to obtain values of the NVC |Gαd;0|2exp for the exp virtual decay 6 Li → α + d and of the astrophysical S-factor, Sαd (E) for the d(α, γ )6 Li reaction at astrophysically relevant energies (6 250 keV) using the experimental TDCS d 3 σ exp (E) [5] instead of the d 3 σ th (E) and the adopted value of the cutoff parameter exp R0 = 12.4 fm. The results given here for |Gαd;0|2exp and Sαd (E) can be considered as an “indirect measurement” of the NVC for the virtual decay 6 Li → α + d and of the astrophysical S-factor for the d(α, γ )6 Li reaction. The experimental data d 3 σ exp (E) have less than 30% uncertainty in the absolute cross-section in the energy range 130 6 E 6 250 keV and more than 40% at E 6 110 keV. It should be noted that at present these data are the only available those of d 3 σ exp (E) at extremely low energies. Therefore they are used by us for obtaining exp “indirect measurement” of the values of the |Gαd;0|2exp and of the Sαd (E) from the same experimental data putting d 3 σ th (E) = d 3 σ exp (E) in the r.h.s. of Eqs. (26) and (27) in the energy range where experimental uncertainty is less than 30% (130 6 E 6 250 keV). In Figs. 3 and 4 the results are presented. In Fig. 3 we compare our results for the |Gαd;0|2exp (circle points) with most available estimations of the other authors (solid line) [27–29]. 1 It should be noted that there is a misprint in formula (9) from [10]: the factor µ αd should be replaced by √ µIαd · (µαd c/h¯ π).

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Fig. 3. The value |Gαd;0 |2 for the virtual decay 6 Li → α + d. The square points are our results. The solid line is the average value of the |Gαd;0 |2 from papers [27–29].

As is seen from Fig. 3 our results of an “indirect measurement” of the |Gαd;0|2exp are in a good agreement with the independent estimations: |Gαd;0|2 = 0.42 ± 0.02 fm [27]; 0.41 ± 0.06 fm [28] and 0.41 fm [29], the latter of them within the framework of the three-body (6 Li = α + n + p) method of calculation of NVC Gαd;0 was obtained with direct use of the same three-body wave function Ψ6 Li (x, r) from paper [20]. It should be also stressed that in the relation (26) the fraction d 3 σ exp (E)/d 3 σ˜ th (E) doesn’t practically depend on the energy E in the energy range 130 6 E 6 250 keV although absolute value of the d 3 σ exp (E) noticeably depends on the energy and can be changed up to two times. This fact and a presence of a good agreement between the |Gαd;0|2exp and values of the |Gαd;0|2 obtained by other authors allow us to conclude that the experimental TDCS d 3 σ exp (E) can be used as a independent source of getting the information about the NVC for the exp virtual decay 6 Li → α + d. The results for the Sαd (E) (triangle points) are displayed in exp Fig. 4. The obtained new data for the astrophysical S-factor, Sαd (E), show a significant energy dependence in the contrast to the data (square points) of a paper [5], which were obtained from analysis of the same Coulomb breakup with taking into account only the exp E2-multipole. The obtained here data for Sαd (E) are also in a good agreement with the theoretical predictions (solid line) [10,11]. In Fig. 4 the result of our calculation for the Sαd (E) of the d(α, γ )6 Li reaction with taking into account only the Coulomb α–dscattering is also presented (dotted line). The calculation shows a noticeable contribution of the nuclear α–d-scattering for the Sαd (E) in the energy range E > 250 keV. It should exp be stressed that our data for Sαd (E) experimentally confirm also the conclusions of the authors of papers [4,11] that the production rate of 6 Li via the d(α, γ )6 Li reaction in the big bang nucleosynthesis is sufficiently small. Therefore this reaction gives a small

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Fig. 4. The astrophysical S-factor for the d(α, γ )6 Li reaction. The triangle points are our results and the square points are the data from a paper [5]. The solid line shows the results for the E1 + E2 multipoles from [10,11] taking into account Coulomb–nuclear αd-scattering and the dashed line shows our result taking into account only Coulomb αd-scattering. The dashed band is the results of the interpolation formula Sαd (E) = (9.1 ± 1.8 + (29.2 ± 6.6)E) MeV nb [5].

contribution to the abundance of universal 6 Li that may remove a long-standing uncertainty in big-bang nucleosynthesis about a quantitative creation of 6 Li. But for giving a final answer on this question a further improvement of the experimental uncertainty for the Coulomb breakup considered with more than 30–40% till less than 10% is desirable. It should be allowed to obtain the “indirect measured” data of the astrophysical S-factor, exp Sαd (E) at astrophysically relevant energies with smaller experimental uncertainty. For this aim the 6 Li Coulomb breakup experiment should be repeated.

4. Conclusion In this study the triple differential cross section (TDCS) of the 208 Pb(6 Li, αd)208 Pb Coulomb breakup at astrophysically relevant energies E (6 250 keV) of the relative motion of breakup fragments has been investigated in detail within the three-body (208 Pb–α–d) approach with correct taking into account both the three-body Coulomb effects in the final state and the contributions from the E1- and E2-multipoles, including their interference. The influence of the Coulomb field of the Pb-ion, which has been revealed by authors of paper [17], upon the state corresponding to relative motion of the breakup fragments of the Coulomb breakup considered has been evaluated. It is found out that these purely

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three-body Coulomb effects do not practically change absolute values of the TDCS even at extremely low energies. It is shown an importance of a correct taking into account of the contribution from the E1- and E2-multipoles into the TDCS of the 208 Pb(6 Li, αd)208 Pb Coulomb breakup to get a reliable information about the astrophysical S-factor, Sαd (E) for the d(α, γ )6 Li reaction in the energy range with E 6 250 keV. The obtained new exp data for the Sαd (E) show significant energy dependence in the contrast to the data of Ref. [5], which were obtained within the framework of the semiclassical method using the same Coulomb breakup reaction but with taking into account only the E2-contribution. Our exp new data for the astrophysical S-factor Sαd (E) are in a good agreement with theoretical predictions [10,11] and confirm also the results of Refs. [4,11] about the production rate of the 6 Li in the big-bang nucleosynthesis. It is also demonstrated that at extremely low energies E the experimental TDCS for the 208 Pb(6 Li, αd)208 Pb Coulomb breakup [5] can be used as an independent source of information on the values of NVC for the virtual decay 6 Li → α + d. The main advantage of the method proposed is that it allows to determine both the absolute value of NVC for exp the virtual decay 6 Li → α + d and the astrophysical S-factor Sαd (E) at astrophysically relevant energies by means of an analysis of the same experimental TDCS in a correct way. From the theoretical point view it is felt that there is a necessity in the future improvement of this method within the framework of the distorted-wave approach with taking into account nuclear interactions both in the initial and in final states. This would make possible, firstly, to avoid a possible unresolved uncertainty from the nuclear breakup amplitude, especially, in that region of the energy E (> 250 keV) where the Coulomb and the nuclear breakup coexist, secondly, to carry out the comparative analysis between the method being developed and the DWBA with an eikonal approximation [21] (or its different modifications) for estimating the contribution of the three-body Coulomb effects (so-called high-orders) in the continuum by means of including to the amplitude of breakup processes all the terms of the expansion in the potential W C (r, ρ). Besides, it would also allow to find the overall energy range where the method of developing and the semiclassical exp method can give the same results for the Sαd (E). In this case the value of the cutoff parameter R0 is expected to be chosen as R0 = RN [21] since an accurate account of nuclear interactions leads to, as a rule a strong suppression an internal regions of the colliding nuclei. In this direction additional experimental and theoretical efforts are also required to have reliable data on the elastic 6 Li–208 Pb, α–208 Pb and d–208Pb scattering. Besides it is of great interest to carry out a similar study for the other breakup processes (like, (7 Li, αt), (7 Be, α 3 He), (8 B, p7 Be) etc.) for astrophysical applications. For these processes the influence of the three-body Coulomb effects is noticeable larger than for the 208 Pb(6 Li, αd)208 Pb breakup [30]. At present such studies are under consideration.

Acknowledgments The authors are deeply grateful to Profs. E.O. Alt, G. Baur, D. Baye, L.D. Blokhintsev, J. Kiener, V.I. Kukulin and Ch. Leclercq-Willain for many fruitful discussions and general

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encouragement. The work has been supported by the International Science Foundation (grant No Ru 8000) and by the Fund of Fundamental Research of Uzbekistan Academy of Science (grant No 18-98). One of authors (R.Y.) thanks the Abdus Salam ICTP for the hospitality in Trieste during autumn 1999, where the work has been completed. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]

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