Nuclear-matter radii of 7Be and 7Li and astrophysical S-factors for radiative alpha-capture reactions

Volume 202, number 4

PHYSICSLETTERSB

17 March 1988

NUCLEAR-MATFER RADII OF 7Be AND 7Li AND ASTROPHYSICAL S-FACTORS FOR RADIATIVE ALPHA-CAPTURE REACTIONS T. K A J I N O a,b, 1, H. T O K I b, K.-I. K U B O b a n d I. T A N I H A T A c a NationalSuperconducting Cyclotron Laboratory, Michigan State University, East Lansing, M148824, USA c Department of Physics, Tokyo Metropolitan University, Setagaya-ku, Tokyo 158, Japan c RIKEN, Wako, Saitama 351-01, Japan

Received 10 August 1987;revised manuscript received 4 January 1988

The nuclear-matter radii of 7Be and 7Li measured in high-energyheavy-ionreactions have been calculated and applied to the estimate of astrophysical S-factorsfor the 3He((x,'y)7Be and 3H(ct,~)TLi reactions. The absolute S(0) values in KeV b are constrained to be 0.36
A recent experiment [ 1,2] of high-energy heavyion reactions using exotic-isotope beams provides a new tool for the study of the nuclear structure of light elements He, Li and Be including several radioactive isotopes. The hadronic matter radii were determined from the measured interaction cross sections in the experiment. The data of unstable 7Be and 7Li nuclei among them are of particular interest for an application to astrophysics: The formation reactions of these nueclei, i.e. 3He(a,7)TBe and 3n(ct,7)7Li, are presumed to take one of the keys to resolve the missing solar-neutrino problem [3,4] and the problem [5,6] of big-bang production of 7Li. Although a lot of experiments have been done to measure the reaction cross sections repeatedly, some of them differ appreciably from one another, casting a doubt on the theoretical prediction of solar neutrino counting rate in 37C1 or primordial 7Li abundance for which the above two reaction rates are important input parameters. In a recent theoretical study [ 7 ] of these reactions, a very strong correlation was found between the calculated reaction rates and nuclear sizes of 7Be and 7Li. However, there were no data available for unstable 7Be, although the charge radius of 7Li was determined in electron-scattering experiments. Hence, the measPresent address: LawrenceLivermoreNational Laboratory,L413 Universityof California, Livermore,CA 94550, USA. 0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

ured matter radii, in particular, of unstable 7Be make an independent estimate of the reaction rates by the use of the theoretical correlation between them. The purpose of this letter is to constrain the absolute strength of the reaction rates (often expressed as astrophysical S-factors) for the 3He(a,y)TBe and 3H ((t, 7 ) 7Li reactions. A microscopic cluster model wave function [ 7 ] was adopted in the present calculation of the matter radii and reaction rates. The main ambiguity of the calculated quantities in the resonating group method (RGM) comes from the choice of the nuclear force. In order to assess the significance of this ambiguity one of the present authors has examined [ 7] the force dependence of electromagnetic properties by using many different types of nuclear forces which are established very well in the cluster model calculations. As the result, it was found that almost all existing data on the binding energies, the electromagnetic structures of 7Be and 7Li, and the scattering amplitudes for the alpha + (A = 3) systems are explained systematically by the use of the modified Hasegawa-Nagata force [ 8 ]. This force is used in the following calculations. See ref. [ 7 ] for details. Let us start a discussion of the matter radii of two nuclear isobars, 7Be and 7Li. The experimental measurements have indicated almost identical matter radii within their error bars although the proton and 475

Volume 202, number 4 (fm) 2.7

PHYSICSLETTERSB

RMS RADII OF 7Li AND 7Be MATTER

2.6

PROTON MATTER

NEUTRON MATTER

t-

2.~ 2.4 2.3 2.2 2.1

CHARGE

Li

Be

Li

Be

Li

Be

1 Li

Be

Fig. 1. Comparison of the charge and matter radii of the A= 7 nuclear systems between theory and experiments. Experimental values are from ref. [2l (11)and refs. [9-11] (D). neutron matter radii between the two isobars show appreciable difference as shown in fig. 1. Taking account of the nucleon finite-size effect, the measured proton matter radii lead to the RMS charge radii of 2.39 + 0.02 fm and 2.49 + 0.02 fm for 7Li and 7Be, respectively, which are consistent with the charge radius of 7Li determined in electron-scattering experiments. The theoretical calculation agrees with all these data fairly well. A close comparison between theory and experiment, however, shows a different charge dependence from each other. As naively expected, an existence of the Coulomb force and charge symmetry-breaking effects, as discussed very often with respect to the Nolen-Schiffer anomaly [ 12 ], will lead to (r2m(7Be)) 1:2 > (r2m(7Li) > 1/2 ,

(la)

(r2m(7Be)) 1/2 > < r2pm(7Li) > 1/2

(lb)

,

where and < r,m z ) are, respectively, the meansquare proton and neutron matter radii. Consequently, the matter radii defined by ~/z = [ ( Z + N ) -1 (Z<~pm > +N 1:2 > < r2(7Li ) > 1/2 .

(3)

Although the difference in this theoretical prediction is very small at the 2% level, it would be interesting to see if the hadronic matter radii of these two isobars 476

17 March 1988

remain to have the same size after performing more precise measurements and analyses to extract the radii. There are several differences and similarities of the hadronic and electromagnetic properties predicted [ 13 ] between 7Li and 7Be. The present result was obtained in the single-channel RGM calculation which satisfies the variational stability conditions [ 13,14 ] of the constituent clusters. The calculated radii are essentially the same as those of Kaneko, Shirata, Kanada and Tang [ 15 ], although they assumed a quite different nuclear force from ours and used the multi-configuration cluster wave function [ 16 ]. This is not an accidental agreement between the two independent theoretical calculations. Kaneko's multi-configuration cluster wave function takes account of the breathing mode of the constituent clusters. From analysis of the results of a precise coupled-channel study [ 13] of 7Li, it was found that the breathing mode of the cluster has a negligible effect on the nuclear structure and even on a dynamical process as the radiative alpha capture reaction (both of which are phenomena with low transfer momentum), when the variational stability conditions are satisfied; the stability conditions are equivalent to the orthogonality relation between the ground and virtual breathing excited states of the cluster, and they play an essential role to minimize the couplings between these channels in the A = 7 nuclear systems. Now the measured matter radius of unstable 7Be is available for the first time. By using this value we would like to estimate the reaction rate for 3He(ct,7)VBe. For this purpose it is worthwhile explaining the correlation which was found theoretically [ 7 ] between the astrophysical S-factors and the nuclear sizes of the A = 7 nuclear systems. At very low energies of astrophysical interest Ec~ = 1 - 100 keV, the electric dipole transition dominates the radiative capture process 3H(a,7)TBe. More than a 90% contribution to the total capture cross section comes from the extra-nuclear region of 5 < r < 30 fm because the Coulomb \barrier makes it difficult to find the two charged nuclei in close distance from each other in the scattering states. The capture cross section is, therefore, very sensitive to the amplitude of the bound state wave function in the external region. This is the main reason why the calculated radiative capture cross section depends strongly on the size of the final

Volume 202, number 4

1.0

I

I

PHYSICS LETTERS B I

I

I

I

I l-

17 March 1988

dii was also taken into account. The present estimate (4) is consistent with the range [ 17 ]

I

sssQ E

S(0) = 0 . 5 6 + 0 . 0 3 k e V b ,

> 0.5

which was determined by extrapolating the observed S-factors for the 3He(a, T )7Be reaction at the higher energies to zero energy, assuming the energy dependence of the S-factor calculated in the microscopic cluster model [14]. (See ref. [17] and references therein on the data.) A good agreement of (4) with (5) justifies the theoretical estimate [ 17 ] of the solar neutrino counting rate 6.5 +0.9 SNU (1 Solar Neutrino U n i t = 1 capture s - 1/1036 detector atoms) in the 37C1 detector; the present result of the S-factor (4) constrains the counting rate to be 4.6-7.1 SNU. Since the detected number of neutrino counting rate is 2.1 + 0.3 SNU [4], the missing solar neutrino problem persists. It is, however, confirmed here that the ambiguity of the nuclear reaction rate for 3He(~t,7 ) 7Be is not responsible for the problem. One should find a true solution elsewhere. As for the 3H(~,T ) 7Li reaction, a similar analysis gives a constraint

v

0 r,.o 0

Zl

2.0

I

i

i

i

2.2 2.4 ,v/(r 2m)(fm)

i

=

2.6

i

2.8

Fig. 2. Correlation between the astrophysical S-factor for the 3He(ct,7)TBe reaction and the matter radius of 7Be. Observed matter radius is from ref. [ 2 ]. Solid line represents an approximately linear correlation between these observables which was obtained by Z2-fitting to the calculated values of S(0) and (V/~m>. Dashed lines take account of + 20% uncertainty for the linearity of this correlation. Dots indicate the calculated results by using different types of effectiveinteractions. See text and ref. [ 7] for details. nucleus. Performing the calculation of both quantities, S-factors and nuclear sizes, by using many different types of effective interactions, one finds the correlation between the two observables, as shown by a solid line in fig. 2. Let us go to the discussion of the S-factor with this correlation. Using the observed RMS matter radius of 7Be, we obtain the following constraint on the absolute strength of the astrophysical S-factor ~: 0 . 3 6 < S ( 0 ) <0.63 keV b .

(4)

This is the first estimate of the S-factor for 3He(a,7)7Be by using the data of unstable 7Be nucleus. In this estimate we have considered the experimental error of the observed matter radius and a + 20% uncertainty for the linearity of the theoretical correlation as shown in fig. 2 with the dashed lines. The ambiguity arising from the difference at the 2% level between the observed and theoretical matter ra~ The constraint obtained is stronger than that [7] determined by using the charge radius of the mirror 7Li nucleus in electron-scattering experiments; the S-factor turned out to be 0.4
0.083 < S ( 0 ) <0.15 k e V b .

(5)

(6)

This range is consistent with previous values determined by direct capture gamma-ray experiments; Griffiths' data [ 18 ] extrapolated to zero energy by assuming the energy dependence of the calculated Sfactor [ 14] lead to S(0) =0.100+0.025 keV b [ 17] and Schroder's data [ 19 ] with the same theoretical S-factor lead to S ( 0 ) = 0 . 1 3 4 + 0 . 0 2 0 keV b. Although Schroder et al. have obtained another value S(0) = 0.163 + 0.024 keV b based on another theoretical energy dependence of the S-factor calculated by Williams and Koonin [20], the energy dependence referred to proved to be incorrect [ 21 ]. The inferred value 0.162 + 0.024 keV b is out of range of the present study (6). We comment that the primordial 7Li abundance calculated in the big-bang model in ref. [ 17 ] is reliable within + 40% and - 18% error, if combined with the present constraint (6) on S(0) for the 3H(a,T)7Li reaction. Schroder's measurement [ 19] of the S-factor has shown a branching ratio o f R ~ 0.32, which is smaller than the Griffiths' data, R ~ 0.42, and almost all existing theoretical calculations consistent with Grif477

Volume 202, number 4

PHYSICS LETTERS B

fiths' data and of course with the present analysis. The analysis here is based on the theoretical correlation between the total S-factor and the radius of the ground state, and it is a weak point to use the theoretical branching ratio to obtain S ( 0 ) . However, the small branching ratio R ~ 0.32 is not justified theoretically [22]. In addition, a recent new observation [23] has shown that both of the S-factors a n d branching ratios are in reasonable agreement with Griffiths' data, better than with Schroder's data. We, therefore, have not estimated the possibility of a small branching ratio here. To summarize, we have calculated the charge, proton and n e u t r o n matter radii of 7Be a n d 7Li in the microscopic cluster model. The theoretical matter radii show an interesting difference between 7Be and 7Li though the observed radii are almost equal to each other. We have also estimated the astrophysical Sfactors for the 3He(ct,y)TBe a n d 3H(ct,7)7Li reactions with the help of the measured matter radii of 7Be and 7Li. The theoretical relation between the Sfactor and the matter radius has been used. The present result strongly constrains the absolute strength of the S-factors which affect the solar n e u t r i n o counting rate in 37C1 a n d the big-bang nucleosynthesis of 7Li. One of the authors (T.K.) acknowledges Sam Austin, George Bertsch a n d Alex Brown for useful discussions a n d the support of the US National Science Foundation.

478

17 March 1988

References [ 1] I. Tanihata et al., Phys. Rev. Lett. 55 (1985) 2676. [2] I. Tanihata, in: Proc. XI Intern. Conf. on Particles and nuclei (Kyoto, 1987), to be published; RIKEN Accelerator Facility Report, RIKEN-RF-NP-59. [3] J.N. Bahcall et al., Rev. Mod. Phys. 54 (1982) 767. [4] R. Davis Jr., B.T. Cleveland and J.K. Rowley, Conf. on Interactions between particles and nuclear physics, ed. R.E. Mischke (AIP, New York, 1984). [5] S.M. Austin and C.H. King, Nature 269 (1977) 782. [6] J. Yang et al., Astrophys. J. 281 (1984) 493. [7] T. Kajino, Nucl. Phys. A 460 (1986) 559. [8 ] F. Tanabe, A. Tohsaki and R. Tamagaki,Prog. Theor. Phys. 53 (1975) 677. [9] L.R. Suelzle,M.R. Yearian and H. Crannell, Phys. Rev. 162 (1967) 992. [ 10] G.J.C. van Niftrik et al., Nucl. Phys. A 174 (1971) 173. [ I 1] F.A. Bumiller et al., Phys. Rev. C 5 (1972) 391. [ 12] J.A. Nolen and J.P. Schiffer, Ann. Rev. Nucl. Sci. 19 (1969) 471. [13] T.Kajino, T. Matsuse and A. Arima, Nucl. Phys. A 413 (1984) 323;A414 (1984) 185. [ 14] T. Kajino and A. Arima, Phys. Rev. Lett. 52 (1984) 739. [ 15] T. Kaneko, M. Shirata, H. Kanada and Y.C. Tang, Phys. Lett. B 192 (1987) 259. [ 16] T. Kaneko et al., Phys. Rev. C 34 (1986) 771. [17] T. Kajino, H. Toki and S.M. Austin, Astrophys. J. 319 (1987) 531. [ 18] G.M. Griffiths et al., Can. J. Phys. 39 (1961) 1397. [ 19] U. Schroder et al., Phys. Lett B 192 (1987) 55. [20] R.D. Williams and S.E. Koonin, Phys. Rev. C 23 (1981) 2773. [21 ] T. Kajino and G.F. Bertsch, in preparation. [22] T. Kajino, in: Origins and distribution of elements, ed. G.J. Mathews (World Scientific, Singapore, 1987), to be published; T. Kajino and G.J. Mathews, in preparation. [23] S. Burzynskiet al., Nucl. Phys. A 473 (1987) 179.