Analysis of the 11B(7Li,7Be)11Be reaction at 57 MeV in a microscopic approach

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Nuclear Physics A 739 (2004) 30–56

www.elsevier.com/locate/np

Analysis of the11B(7Li, 7Be)11Be reaction at57 MeV in a microscopic approach

F. Cappuzzelloa,∗, H. Lenskeb, A. Cunsoloa,c, D. Beaumeld,S. Fortierd, A. Foti c,e, A. Lazzaroa,c, C. Nociforoa,

S.E.A. Orrigoa,c, J.S. Winfielda

a INFN–Laboratori Nazionali del Sud., Via S. Sofia 44, 95123 Catania, Italyb Institut für Theoretische Physik, Universität Giessen, Heinrich–Buff-Ring 16, D-35392 Giessen, Germ

c Dipartimento di Fisica e Astronomia, Universitàdi Catania, Via S. Sofia 64, 95123 Catania, Italyd Institut de Physique Nucléaire, IN2P3-CNRS, 91406 Orsay cedex, France

e INFN–Sezione di Catania, Via S. Sofia 64, 95123 Catania, Italy

Received 25 June 2003; received in revised form 26 March 2004; accepted 30 March 2004

Available online 23 April 2004

Abstract

The 11B(7Li, 7Be)11Be charge exchange reaction has been studied at an incident energyMeV. Spectra were measured at forward angles,θcm 35. The good energy resolution (∼50 keV)allowed the identification of transitions both to the7Be (3/2−, gs) and7Be(1/2−, 429 keV)exit channels and hence the direct measurement of the ratio of the respective cross sectangular distributions. Besides the bound ground and first excited state of11Be several low lyingexcitations just above the neutron threshold are observed. A structure seen atE∗ = 9.4 MeV withFWHM ∼7 MeV is compatible with the spin dipole resonance (SDR). The data are analysa many-body approach. For the projectile transitions shell model results are used. In orderaccount properly for the special features of the weakly bound11Be system the target transitionare described microscopically by Hartree–Fock–Bogoliubov (HFB) and quasi-particle randomapproximation (QRPA) theory. The HFB ground state densities andthe QRPA transition densities,respectively, are used in folding calculations for the optical potentials and transition form faSpectra andβ-decay transitions strengths are reasonably well described. The measured crossare well reproduced by one-step direct charge exchange distorted wave born approximation (Dcalculations. A dominance of unnatural parity transitions is found. This is explained in terms of thspin transfer behaviour of the nucleon–nucleon isovector interaction at low bombarding energ

* Corresponding author.E-mail address:[email protected] (F. Cappuzzello).

0375-9474/$ – see front matter 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.nuclphysa.2004.03.221

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F. Cappuzzello et al. / Nuclear Physics A 739 (2004) 30–56 31

2004 Elsevier B.V. All rights reserved.

PACS:21.10.-k; 21.10.Pc; 21.30.Fe; 21.60.Jz; 23.20.En; 23.40.Hc; 24.10.Eq; 24.50.+g; 25.70.kk; 27.20.

Keywords:NUCLEAR REACTIONS+11B(+7Li, +7Be),E = 57 MeV; measured particle spectra,σ(E, θ). +11Bededuced levels, configurations. Quasiparticle RPA analysis.

1. Introduction

Over the last decades the(n,p) and (p,n) charge exchange reactions (CEX)nuclear targets have been the major source of information on charge changing nisovector excitations. Because of their selectivity they also have provided much significainformation on nucleon–nucleon isovector interactions [1,2]. Heavy ion charge excreactions are useful for obtaining complementary information on states which are notexcited with the elementary probes. This includes the excitation of high spin stathe highly peripheral heavy ion collisions and the selection of Fermi and Gamow–transitions by an appropriate choice of projectile-target combinations or selecting suprojectile transitions, respectively.

One aim of such reaction studies is to derive information onβ-decay strengthdistributions which otherwise are not accessible. However, non-elastic nuclear reactiotypically proceed at a finite, albeit small, momentum transferq while β-decay is a procesat vanishingq , at least in the Fermi-approximation withq MW → ∞, whereMW is themass of theW boson. Investigations for nucleon induced CEX at intermediate ene(Elab > 100 MeV) are supporting, on an empirical level, a close relationship betwβ-decay probabilities and nuclear charge exchange cross sections at low momtransfer [3,4]. Such relationship has been observed, although more approximatefor heavy ion charge exchange reactions in certain cases [1].

An unwanted but unavoidable complications for heavy ion induced charge excreactions are admixtures from more complicated reaction mechanisms, typically givensequential transfer or inelastic processes of second or higher order[5–9]. The competitionof one-step direct and two-step sequential transfer charge exchange processes are discuin detail, e.g., in Ref. [9].

A typical heavy-ion CEX reaction investigated in recent years is (7Li, 7Be). In genera7Li beams are available in many laboratories at a broad range of energies. Also it is ua simple task to identify the7Be ejectiles using standard detection equipment. On the ohand the non-zero values of the transferred angular momentum between the7Li 3/2−ground state and the7Be 3/2− ground or 1/2− first excited state makes the spectroscosomewhat complicated.

The competition between direct one- and two-step sequential mechanisms f(7Li, 7Be) reaction has been discussed for long time [10–15]. Different argumentsbeen used in order to determine the conditions under which direct process dominappears that these conditions depend on the particular nucleus studied, and sometimesthe particular excited state. However, at least the intermediate transfer channel inv
8Be is particle unstable, thus having a reduced overlap with the bound initial and finalprojectile states. In general, calculations based on the assumption of direct CEX give

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DWBAsis.rste

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32 F. Cappuzzello et al. / Nuclear Physics A 739 (2004) 30–56

angular distributions with a steeper decrease at backward angles than the experdata indicating the increased importance of higher-order processes at large momtransfer [9,11,12]. For (7Li,7Be) reactions it was shown by Clarke and Cook [11] thatunresolved theoretical ambiguities inhibit clear statements on the significance of twcontributions. Typically, one finds that the agreement with data is hardly improved.

Quantitative studies of the two-step reaction channels in (7Li, 7Be) reactions aremissing, mainly because reliable methods for treating three-body continuum transfchannels are lacking. However, the totaltwo-step transfer contribution in the (7Li, 7Be)reactions can be expected to be small at low energies (below 10 MeV/u) and forwardangles, as indicated by the good results of data analyses based on one-stepcalculations, see, e.g., Refs. [11,12]. This is the view also taken in the present analy

A distinct feature of the (7Li, 7Be) reaction is the significant population of the fiexcited 1/2− (and only) bound state of7Be at 0.43 MeV. If properly separated in thexperiments, the relative yields measured in the7Be (gs−0.43 MeV) doublets can bdirectly connected to the ratio of spin–flip to non-spin–flip interaction [16–19]. Theimportant information on the spin structure of the isovector nucleon–nucleon interais obtained. This connection to the spin-transfer probability is most clearly evidendirect mechanism transferring a small amount of linear momentum driven by a cforce is the dominant contribution to the cross section [2]. Such conditions are mostaccomplished at forward angles and small reactionQ-values. Although the best realizatiois obtained at incident energies well above the Coulomb barrier, comprehensive studbe done already at energies not far from the Coulomb barrier.

The separation of the7Be doublet has been achieved in experiments at eneabove 20 MeV/u through the coincident detection of 0.43 MeVγ -rays [18,19]. Thistechnique is particularly useful when the energy resolution or the natural width opeaks does not allow the separation of the7Be doublet in the inclusive spectra. Howevthe coincidence requirement reduces the yield and introduces supplemental uncerin the measurement of the cross sections. At low incident energy, with the7Be ejectilesdetected by silicon telescopes or magnetic spectrometers [12,20], the separationdoublet is obtained in the inclusive spectra. In particular, the use of a magnetic spectrallows the exploration of the important region at very forward angles, including 0. Thistechnique is therefore very interesting for the spectroscopy of light exotic nuclei, wnarrow resonances are often present well beyond the particle emission thresholdhigh resolution measurement of the energy of the excited states can provide a fundatest of modern theories of nuclear structure.

In Ref. [20] the above technique has been used to explore the spectrum of11Be up to15 MeV excitation energy. The resolution obtained (∼ 50 keV) allowed the identificatioof the single particle excitations below 2 MeV and a group of narrow resonancescontinuum, some of which were observed for the first time. A QRPA approach, bastwo quasi-particle excitations, gives an appropriate description of the resonances obbelow 2 MeV, while it fails in describing the fragmentation of the strength distribuobserved at higher excitation energy. This fragmentation cannot be associated to
particle excitations and needs more sophisticated interpretation, such as the ones withinthe dynamical core polarisation model of Refs. [21,22].

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In this paper a deeper insight into both the spectroscopy of11Be and of the (7Li, 7Be)reaction mechanism beyond that discussed in [20] will be given. The paper is organfollows. In Section 2 some of the experimental data on the11B(7Li, 7Be)11Be reactionat 57 MeV, not previously presented in Ref. [20], are shown. In Section 3 amicroscopic approach, based on the QRPA theory, to the charge exchange tranin exotic nuclei is described. Thereafter the calculated response functions are comwith the experimental spectra. In Sections 4 and 5 the CEX cross sections are calculawithin the framework of the optical model and the DWBA theory and compared withmeasured angular distributions. Finally the main results obtained in this work and thedevelopments of the theory and the experiments are discussed in Section 6.

2. Experimental results

The experiment was performed at the IPN-Orsay Tandem laboratory. A7Li+++ beamat 57 MeV was focused on a 95% enriched11B self supporting target about 130 µg/cm2

thick. Additional measurements were done with a12C target of about 100 µg/cm2. The7Be ejectiles were detected by the split-pole magnetic spectrometer. The focaldetector consisted of a position and angle sensitive multi-wire drift counter, a proportionacounter and stopping plastic scintillator. Further details of the experiment may be foundRefs. [20,23].

In Ref. [20] the shapes of the excitation energy spectra were discussed and comto the results of QRPA calculations of transition level densities. In the spectra thebody phase space, taken from Ref. [24], was normalized to the high excitation eneeach spectrum. This procedure, although usefulto analyze centroids and widths of narroresonances, is less adequate for a precise estimation of the cross sections andanalysis of broad resonances at high excitation energy. Instead in this work, thethree-body phase space of Ref. [24] for the7Be+ 10Be+ n exit channel is transformeinto 11Be excitation energy and normalized in order to fit the high excitation energy reof the spectrum at 18 (the most backward angle measured). The result of this procefor the spectrum at 9 is shown on the left panel of Fig. 1. In order to reduce statistfluctuations, this spectrum is compressed by a factor 20 compared to the high resone, at the same angle, shown in Ref. [20].

All the measured spectra are consistent with the existence of a broad bump waverage centroid and width of 9.4± 0.5 MeV and 7.0± 0.5 MeV, respectively. The errobars on the measured parameters depend on the uncertainty in the fit procedurethe excitation of the doublet in7Be and the estimated three-body phase space. Tvalues agree with those (centroid energy between 9 and 10 MeV and width around 7previously measured in the11B(d,2He)11Be reaction at 70 and 270 MeV [25,26] for thspin dipole resonance (SDR). A broad structure centred at 9 MeV has also been obin the11B(t, 3He)11Be reaction at 127 MeV/u [27].

After peak integration and kinematic transformation the absolute cross sectionscenter of mass reference frame are extracted. It is interesting to compare these qu
for the11B(7Li, 7Be)11Be and the12C(7Li, 7Be)12B reactions at 0, similarly to what Sakaiet al. did in Ref. [25] (see Table 1). In the (7Li, 7Be) reaction the distribution of the counts

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Fig. 1. Excitation energy spectra of11B(7Li,7Be )11Be reaction at 57 MeV. (a) Spectrum at 9. The histogramis obtained by compressing of a factor 20 that presented in Ref. [20]. The solid line is the result of a peaanalysis. The dashed and the dotted lines represent the 3-body phase space and the Gaussian model of the bresonance at 9.4 MeV, respectively. The structurearound 15 MeV is due to a fault in one of the wires of tfocal plane detector and has no physical meaning. (b) Detail of the fitted structurearound 4 MeV in the spectrumat 2.5 . The peaks marked with an asterisk refer to the excitation of7Be at 0.429 MeV. The shaded histograrepresents the normalized background of12C(7Li, 7Be)12B reaction, the continuous line is the sum of 3-bophase space and the broad resonance as in panel (a).

over the7Be doublet must be considered. Consequently, the total transition strengeach of the11Be and12B states is derived by integrating over both the peaks.

In Ref. [25] the GT strength for11Be is extracted from the integrated counts of the 02.67 and 3.89 MeV states, while for the12B only the ground state is considered. The eneresolution of the present work allows, for the first time in a CEX experiment, to sepin the11Be spectra the doublet at 3.89 and 3.96 MeV, as shown, for example, on thepanel of Fig. 1. In Ref. [28] the state at 3.89 MeV is assigned to be as 3/2+, while thatone at 3.96 MeV as 3/2−. As a consequence we consider this latter state as relevaGT excitation, rather than integrating the doublet, as previously done [25–27]. Thezero value of the linear momentum transfer isconsidered to have very similar effect o11Be and12B cases, since the bombarding energy, the scattering angle and the reQ-value are very similar. Theβ-decay strengths B(GT) and B(SD), listed in Table 1Gamow–Teller (GT) and spin dipole (SD), are calculated in the framework of the Qtheory discussed in Section 3. A useful quantity that can be evaluated is the reducesectionσ , defined as the ratio between the CEX reaction cross section measured at andtheβ-decay strengths for the same transition [3].

A linear relation between measured cross sections and the associatedβ-decay strengthholds for the GT transitions. This is shown in Table 1, where the reduced cross sefor GT transitions to the11Be and12B differ less than the experimental uncertainty. Tcalculated B(GT) strength distribution for the12Cgs → 11B transition is almost entirelyconcentrated in one sharp peak, corresponding to the transition to the12B ground state
For 11B → 11Be case, even if the different GT transitions are separated in the experiment,it was not possible to disentangle them in the calculations and the obtained B(GT) was

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integrated over 4 MeV. This introduces only a minor source of uncertainties in the recross section if the integrated B(GT) is compared with the sum of the cross secti0.32, 2.67 and 3.96 MeV, as discussed before. The values of the reduced cross secabout a factor 40 smaller than those obtained via the(n,p) reaction at 96 MeV [29] andthe (t,3He) at 127 MeV/u on 11B target [27]. This may be explained by the larger linmomentum transferred in the (7Li, 7Be) reaction at 8 MeV/u (q = 0.55 fm−1 at 2 MeVexcitation energy and 0) compared to the previously mentioned cases, e.g.,q = 0.14 fm−1

at 2 MeV excitation energy and 0 for (n,p). For an approximate estimation of this effeon the cross sections one could consider the relationσ(q) ∝ exp(−1

3q2〈r2〉), which isvalid at intermediate energy and low momentum transfer [2]. The (7Li, 7Be) cross sectionis reduced by about one order of magnitude compared to the(n,p) one, assuming typicavalues of〈r2〉 = 16.7 fm2 for (n,p) and 20 fm2 for (7Li, 7Be) [2]. The comparison odistorted and plane wave Born approximation calculations in the approach of Sechas shown a further reduction of about two for the (7Li, 7Be) cross section due to thstronger distortions [2] produced by the optical potential for the heavy ions compathe (n,p) case. Of course, these analyses, being valid in the intermediate energy domaiare here only intended as guideline to explain the order of magnitude of the measuresections. The fact that this is possible within a factor two or less represents an inteempirical observation which certainly will help to estimate experimental yields in fuexperiments.

Proportionality relations, similar to those discussed above for GT transitions, holfor the SD ones. They involve the excitation of the11Be ground state and 1.77 MeV staand partly the broad resonance, while the states at 1.67, 2.62, 4.5 MeV and theexcitation energy up to 15 MeV are relevant for the12B case. These comparisons mayaffected by noticeable uncertainties, mainly connected to the subtraction of the three bodphase space and to the contribution from the strengths different than SD ones. Noneas argued in Ref. [25] an accuracy within 20% may be obtained, providing that thefor the 12B and11Be cases are analyzed in a consistent manner, as is done here. Fbroader resonances a supplemental source of uncertainty comes from the estimthree-body phase space to be subtracted. In the calculations for the SD excitations i11Be,two sharp peaks appear at 0 and 2 MeV together with a broad continuous distributio15 MeV, so it was possible to make a comparison with the single resonances obsethe experiment. For the case of12B, on the other hand, only a global comparison is don

As shown in Table 1 the reduced cross sections for different SD transitions buthe 12C and 11B ground states agree within the uncertainty, supporting the ideaa proportionality law holds also for the SD transitions. The broad bump in the11Bespectrum deviates from this behavior well beyond the uncertainty of this analysismay be because of an increased strength for transitions with higher multipolarities atexcitation energy. Such an effect may be more pronounced for11Be, being excited from11B with a 3/2− ground state, as compared to the production of12B from 12C(0+, gs). Inthe latter case, in fact, the strength of a fixed transition operator is limited, especiahigh multipolarity, by the reduced density of levels with the unique allowed values of
and parity, as compared to the broader phase space accessed by the same operator in thecase of the non-zero spin of11B ground state.

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Table 1Cross sections for (7Li, 7Be) reaction on11B and12C targets at 57 MeV and 0 andβ-decay strengths for GTand SD transitions. The uncertainty of the cross sections for narrow resonances is estimated to be arouThe last two columns refer to the (d,2He) data taken from Ref. [25]

GT (7Li, 7Begs) + (7Li, 7Be0.43) (d,2He)

Target Ji Jf Ex (MeV) dσ/dω

(µb/sr)B(GT)(QRPA)

σGT(µb/sr)

B(GT)(shell model)

σGT(µb/sr)

12C 0+ 1+ 0.0 235 1.01 233 0.98a 17.911B 3/2− 1/2−, 3/2−,

5/2−0.32, 2.67,3.96

149 0.75 199b 0.73c 20.9

SD (7Li, 7Begs) + (7Li, 7Be0.43) (d,2He)

Target Ji Jf Ex (MeV) dσ/dω

(µb/sr)B(SD) (fm2)(QRPA)

σSD(µb/sr) c

B(SD) (fm2)(shell model)

σSD(µb/sr)c

12C 0+ 0−, 1−, 2− 1.67, 2.62,4.5–15

570 44.3 74.9 31.0 9.0d

1/2+ 0.0 36 3.22 68.611B 3/2− 3/2+, 5/2+ 1.77 75 7.55 60.9 24.4 8.24e

9.4 1067 40.2 162.9

a See Ref. [30].b Adding the state at 3.89 MeV one obtains a value of 268 for the reduced cross section.c The reduced cross sections are scaled with the quantities〈rgs〉2, where rgs is the QRPA ground stat

transition density radius. This is equal to 2.41 fm for the12C case and 2.47 fm for the11B one.d In Ref. [25] the B(SD) is taken from Ref. [31] and refers to the interval between 2 and 30 MeV.e In Ref. [25] the B(SD) is taken from Ref. [32] and refers to the interval between 2 and 23 MeV.

As mentioned in the introduction, the separation of the7Be doublet of states allowa direct estimation of the spin transfer component of the isovector nucleon–nuinteraction. In reference [18,19] it was shown that the ratioG = σ (7Beex)/(σ (7Beex) +σ (7Begs)), whereσ (7Begs) andσ (7Beex) are the cross sections for the transition to7Be ground and first excited state respectively, should be 0.46 for pure GT transitions azero for Fermi ones. This is valid in the case of a small transfer of linear momentunegligible contribution of the tensor force and for a direct exchange reaction mechaAll of these conditions are best satisfied at very forward angles. In the case o11Bgs → 11Be transitions, the 3/2− spin-parity nature of11B ground state does not impostrong selection rules on spin transfer for every transition. In contrast for the12Cgs → 12Bunnatural transitions, the 0+ nature of12C ground state selects more efficiently spin transdynamics.

As shown in Table 2, the extracted values ofG around 0 for the transitions to al11Be states are definitely different from 0, and close to the 0.46 limit of GT transiin the projectile. DWBA calculations for the reaction cross sections show (Sectiondominant direct mechanism at forward angles, however, with a non-negligible roletensor force. This in principle makes the situation discussed here somewhat differenthat of Refs. [18,19]. Nevertheless it will be shown in Section 4, that the tensor forceminor effect on theG factor at 0. Therefore our results do indicate the dominance of stransfer even though the projectile transitions are not pure GT.

In the same table theG factors for the transitions from the12C ground state to theexcited states of12B are also reported. It is interesting to note that some of these deviate

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Table 2Values of G measured at around 0 for (7Li, 7Be) reaction on11B and 12C targets.σG represent the totauncertainty in theG parameter

11Be 12B

Jπ a 1/2+ 1/2− 3/2+5/2+

3/2− 3/2+ 3/2− – 1+ 2+ 2− 1− 1−, 2−, 4−

Ex [MeV] 0 0.32 1.77 2.67 3.89 3.96 6.05 0 0.95 1.67 2.62 4.5G (θ ∼= 0) 0.38 0.35 0.46 0.35 0.47 0.4 0.56 0.15 0.17 0.40 0.33 0.34σG 0.04 0.03 0.04 0.04 0.04 0.07 0.03 0.03 0.03 0.06 0.07 0.03

a From Refs. [28,30].

from the behaviour observed at higher bombarding energies [13,28,29]. In particulresults at higher energy indicate a prevalent GT nature of the transition to the 1+ state anda minor contribution of spin transfer for the transition to the 2+. On the contrary, at thenergy of the present data, theG parameters for the two transitions to12B do not show adistinguishable difference and exhibit an intermediate value between 0 and 0.46. A pinterpretation of this fact is the increased role of the two-step mechanisms for the tranto the well bound12B (see, e.g., Ref. [9]) that hide, in this case, the simple relation betwetheG factor and the spin–flip probability.

In Figs. 2–5 the experimental angular distributions for the11B(7Li, 7Be)11Be and12C(7Li, 7Be)12B are shown for both the population of the ground and first excitedof 7Be. These distributions are in general weakly oscillatory and peaked towards foangles. However, in detail they differ considerably from each other in regard to sespecially at very forward angles. This behaviour is not unusual for CEX reactioinvolving light nuclei and is generally related to a direct process which dominates aforward angles, superimposed on a flat two-stepcross section which contributes especiaat larger angles. In addition the many possible values of angular momentum transferredespecially in the11B(7Li, 7Be)11Be reaction, combine to smear out oscillations inobserved angular distributions.

3. Nuclear structure aspects of charge exchange transitions in exotic nuclei

Microscopic methods have been an important tool in understanding the reactiodynamics and the nuclear structure aspects of the(n,p) and (p,n) reactions, as well aof the complementary heavy ion counterparts. The detailed description of the ntransition matrix elements in terms of interactions and consistently derived wave funtogether with the explicit reference to nucleon–nucleon interactions, well testedin free space or in the nuclear medium via structure calculations, have provthe past to remove most of the uncertainties [5,9,13]. Here we follow these foexperiences by applying microscopic many-body methods in analysing the strand reaction aspects of the present data. Similar to previous applications shell modresults are used for theA = 7 systems. For the only slightly heavierA = 11 target-like
nuclei HFB and QRPA many-body methods are already applicable. Here, HFB theoryis used to obtain an appropriate single particle basis for the11B ground state and the

hangems by(QP)

applied

l two-e specialf asectionmics of

roachel


Fig. 2. Angular distributions for the transitions to the11Be ground and the excited states atE∗ = 0.32, 1.77, 2.67,3.41, 3.89, 3.96, 6.05 and 9.4 MeV, connected to the7Be ground state.

particle and hole spectra entering into the QRPA description of the charge excexcitations. QRPA theory represents the appropriate class of next-order diagracomplete resummation of particle–hole type loop diagrams for two quasi-particleconfigurations.

For stable nuclei both the structure and the reaction approaches have beenbefore, e.g., for the low energy (7Li, 7Be) reaction [33] and the (12C,12B) [12],(13C,13N) [34] and (42Ca,42Sc) reactions [35] in the range ofE/A between 10 and100 MeV/u. In these references and in [5] the theory of nuclear excitations ofβ±-typein stable nuclei applications to heavy ion CEX reactions, including also sequentiastep transfer processes, have been discussed. Here, the main emphasis is on thproperties of neutron-rich exotic nuclei such as11Be, where in this case the existence osingle neutron halo and the level inversion are the most outstanding features. In thisthe theoretical description of the nuclear structure aspects and the reaction dynaCEX processes will be discussed subsequently.

3.1. Description of CEX transitions in unstable nuclei

It is generally agreed that the multi-configuration shell model is an appropriate appto the structure of light mass nuclei,A 16. In fact, also here we will use shell mod
transition amplitudes for the (7Li, 7Be) transitions, taken from Ref. [12]. However, theshell model typically relies on empirical matrix elements assumed to behave regularly

old fareid

stablewhichles dueroach

keeptionsneralchangeby

by a


Fig. 3. Angular distributions for the transitions to the11Be ground and the excited states atE* = 0.32, 1.77, 2.67,3.41, 3.89, 3.96, 6.05 and 9.4 MeV, connected to the7Be first excited state.

and smoothly as functions of mass and charge. That assumption must no longer hfrom stability. Irregular behavior must be expected from theweak binding of the valencparticles and because of admixtures of continuum states. Both effects will cause rapvariations of matrix elements hardly being accounted for by extrapolations from theregion. For the weakly bound nuclei we therefore choose a mean-field approachproperly takes care of modification of matrix elements and related spectral observabto the irregular behavior of wave functions close to the drip-line. Also, such an appis rather independent of pre-selections on the active shells. However, in order tothe calculations on a feasible level we have to restrict the complexity of configurato the lowest level, namely to 1 particle–1 hole (1p–1h) states, or, in a more genotation, to 2 quasi-particle (2QP) states. Since we are going to describe charge extransitions converting protons into neutrons and vice versa the 2QP states are givenangular momentum coupled proton–neutron configurations

Q†JM(jpjn) = [

α†jp

⊗ α†jn

]JM

, (1)

whereα†j = uja

†j + vj aj are the one quasi-particle (1QP) state operators obtained

2 2
Bogoliubov transformation with coefficients obeyinguj + vj = 1 from the normal state
operatorsa†j . In practice, the 1QP states are obtained self-consistently by means of HFB

f

to

RPA,ewly

articleoverticallyis the[39],


Fig. 4. Angular distributions for the transitions to the12B ground and excited states, connected to the7Be groundstate. The angular distributions for the transitions to the12B state at 3.76 and 3.39 MeV with7Be excited aresummed (see text).

calculations. We have to take into account the possibility ofa coherent superposition osuch states which leads us to use a QRPA description

Ω†JM(c) =

∑jpjn

(xcJ (jpjn)Q

†JM(jpjn) − yc∗

J (jpjn)QJM(jpjn))

(2)

including the contributions from ground state correlations by the terms proportionaly∗.The ansatz of Eq. (2) is general in the sense that the state operatorsΩ† describe bothβ+ andβ− transitions. The QRPA approach—and refinements thereof—is widely used incurrentβ-decay studies, e.g., [36]. Recent applications of an extended version of Qincluding the coupling of up to 3 phonon configurations (i.e., 6QP sates), to the ndiscovered 2− twist mode in58Ni and the dipole pygmy resonances in208Pb are foundin [37,38], respectively.

A special feature of weakly bound systems is their closeness to the single pcontinuum which, for our purposes, implies that Eq. (2) in fact includes an integralthe unbound proton and neutron states continuously distributed in energy. A theoreappropriate approach which is able to account for continuum effects in observablesGreen function formulation of QRPA. This has been previously applied, e.g., in Ref.
to the description of the response of, medium and heavy nuclei in inelastic scattering atintermediate energies and, in Ref. [40], to exotic nuclei as light as8B. The propagation

ded in

onnuum

set of

atrixtedidetain


Fig. 5. Angular distributions for the transitions to the12B ground and excited states connected to the7Be firstexcited state.

of the statesΩ† is described by the QRPA Green functionG(ω) [39,41]. From the Dysonequation forG(ω) one finds the response functions for an external one-body fieldF [39,41]

R(ω,F ) = 1

πIm〈0|F †G(ω)F |0〉 (3)

given as expectation values over the QRPA ground state|0〉. A convenient way todetermineR(ω,F ) is to solve the coupled-Dyson equations forG(ω) by matrix inversionin coordinate space as in Ref. [39] where the original approach of Ref. [42] was extenvarious aspects. Similar to Ref. [39] we use a 2QP interaction obtained from aG-matrix butincluding additional density dependent pieces inorder to reproduce properly the saturatiproperties of infinite symmetric and asymmetric nuclear matter [43–45]. The contipart of the QRPA Green functions are constructed by representations in terms of adiscretized continuum single particle HFB wave functions.

Of practical interest are solutions of Eq. (3) for multipole operators of Fermi-type,F ∼F±

LM = τ±rLYLM , and the Gamow–Teller type operatorsF ∼ F±JM = τ±rL[Y L ⊗ σ ]JM ,

where J = L, L ± 1 and τ± are the isospin ladder operators. The (reduced) melements of these operators determine theβ-decay properties of nuclear states being relato the longstanding question to what extend hadronic charge exchange reaction provcomplementary information toβ-decay measurements [1,2]. From Eq. (3) we also obimmediately the sum rules∫

Sn

(F±) = R

(ω,F±)

ωn dω, (4)

ies

00014

rules,oflateces ofesekness

Vfarticle

telyonalee-d. Thed fromalheesd

nergiesfrom

usingtes

n intolly, thus

d stateraction


Table 3State dependent pairing for single proton and neutron orbitals, occupation probabilities and mean field energare listed

Protons Neutrons

Single particle(NLJ ) 1s1/2 1p3/2 1p1/2 2s1/2 1d5/2 1s1/2 1p3/2 2s1/2 1p1/2〈Nj 〉 [%] 0.99748 0.74746 0.00665 0.00012 0.00027 0.99999 0.99991 0.00005 0.Emf [MeV] −35.673 −19.636 −11.755 −1.894 −0.169 −31.017 −6.812 −0.458 −0.163

wheren = 0,1 denote the non-energy (NEWSR) and energy weighted (EWSR) sumrespectively, which, in practice, are of specialimportance. In particular, the differencethe NEWSR forβ± transitions lead to the well known Ikeda sum rules [46]. They rethe dynamical response of a nucleus to ground state properties, namely differenradial moments of the proton and neutron ground state densities. In fact, in Ref. [47] threlations were used for the dipole case to obtain information on the neutron skin thicin Sn isotopes.

3.2. QRPA response functions

Single neutron and proton orbitals with the10Be core are calculated up to 100 Meexcitation energy andL = 6 by enclosing the10Be system in a box with a radius o60 fm. This large radius was chosen such that the proton and neutron single pcontinua are represented with sufficiently highdensities of states, also resolving accurasingle particle potential resonances, e.g., in thed5/2 neutron channel. Proton and neutrstates are obtained from HFB calculations for10Be with an interaction obtained in locdensity approximation from aG-matrix as in Ref. [43], complemented by additional thrbody terms such that nuclear and neutron matter properties are correctly describepairing problem was solved with a density dependent zero-range interaction obtainethe Bonn-B [48] singlet-even NN potential. For10Be the HFB calculation gives a totbinding energy per particle ofB/A = 6.36 MeV which agrees reasonably well with texperimental valueB/A = 6.50 MeV [49]. In fact, the HFB results show that pairing donot play a significant role in theA = 10 nuclides. The10Be HFB potentials are then useas reference for the mean-fields of the valence proton and neutron states in11B and11Beby adjusting the depths of the potentials such that the experimental separation ewere reproduced. In this way we account for energy shifts in single particle statescore polarization which is known to play an important role in 1p-shell nuclei, e.g., cathe parity inversion in11Be(1/2+, gs) [50]. The spectrum of bound single particle staentering into the QRPA calculations is given in Table 3.

The dissipation of the QRPA states into higher order configurations was takeaccount by dispersive self-energies. The imaginary parts are obtained phenomenologicafrom optical potentials [36,39]. The real parts were calculated by dispersion theoryassuring analyticity. A subtracted dispersion relation was used leaving the groununaffected by the polarization self-energies by choosing the Fermi level as the subt
point. The results of the QRPA calculations are plotted in Fig. 6, for multipolaritiesJπ = 0±, . . . ,4±, both for natural and unnatural parities. Displayed are the QRPA level

onse

isve thether

s.ulated

(GT)s.99

disour

h theose to

ajorityd.orm aesorused

uct thee NN

h for

tshells intoip and


density distributions (per energy), obtained by normalizing the QRPA multipole respfunctions to the NEWSR of the same operator:

d(ω,F ) = R(ω,F )/S0(F ),

∫dωd(ω,F ) = 1. (5)

The folding with the experimental energy resolution of about 50 keV at FWHMincluded. In both cases narrow states are found at energies below or slightly aboneutron separation energy in11Be. At higher energies the response functions are rafeatureless. The broad structures are mainly due to the continuum decay widths of the11Bestates with minor contributions from the imaginary part of the dispersive self-energie

The response functions to the Fermi and Gamow–Teller like operators were calcaccording to the Eq. (3), both for the11B → 11Be and the12C → 12B transitions. Theseallow a comparison with known experimental B(GT) [26,27,29,51] and calculated Band B(SD) within shell model [30–32]. A B(GT)= 1.01 is obtained in our calculationfor the 12C → 12B transition, in agreement with the experimental value (B(GT) = 0±0.01) from Logf t values of Ref. [51]. For the11B → 11Be case we obtain B(GT)= 0.75to be compared with contradictory experimental results of 0.47± 0.08 from Ref. [27],0.58± 0.06 from Ref. [29] and 0.85± 0.05 from Ref. [26]. Our calculated B(GT) anB(SD) are typically slightly higher than theequivalent from shell model [30–32]. Thatprobably a consequence of the broad excitation energy range (80 MeV) spanned byQRPA approach allowing the excitation of two quasi-particle configurations from botvalence shells and the core, while in the shell model calculations only nucleons clthe valence shells are involved.

The microscopic transition form factors are described in a folding model. The11B →11Be target transitions were described by the QRPA theory discussed above. The mof the 11B →11Be transitions lead to states above the11Be neutron emission thresholHence, we are confronted with a new situation where the final states no longer fdiscrete spectrum but are given by spectral distributions, as in Fig. 6. The special featurintroduced by continuum effects and spectral functions in general are well accounted fby the QRPA Green function formulation discussed above. Eq. (3), in fact, could beto calculate the direct charge exchange cross sections provided we would constrappropriate hadronic transition operator from the in- and outgoing waves and thinteraction.

As discussed in Ref. [39] for inelastic excitations a more practical approaccontinuum transitions is to use one-body transition densities (OBTD)

ρ(ω, r) = 1

πIm〈0|F †G(ω)ρ|0〉/√R(ω,F ) (6)

defined in terms of the one-body density operatorρ and weighted by the matrix elemenof beta-decay multipole operatorsF . Obviously, the presently available schemes for smodel calculations are not able to provide reliable descriptions of such excitationthe continuum. Fermi and Gamow–Teller results were checked by the non-spin–fl
spin–flip Ikeda sum rules [46] which were fulfilled by better than 95% when integratingthe strength up to 30 MeV excitation energy.

ownwith

hargeseith thenergye

e


Fig. 6. QRPA level densities for charge exchange transitions from11B(3/2−, gs) to the excited states of11Be.

4. Optical model and DWBA approach to heavy-ion CEX reactions

4.1. Choice of interaction

In view of the fact that we are considering a reaction involving nuclei of rarely knstructure and unknown interaction potentials we have to take special care alsorespect to the choice of interactions in the initial and final channels and for the cexchange reaction process. In order to keep uncertainties on a controllable level we uNN interactions whose momentum structure and other properties are compatible winteractions used in the structure calculations. Taking advantage of the low incident ewe can extrapolate the density dependentG-matrix interaction into the region above thparticle threshold, as, e.g., by Khoa and von Oertzen in Ref. [44]. Since theG-matrix, infact, is aK-matrix we can retrieve the full complex NNT -matrix tNN in momentum spacby means of the relation [48] ∫

tNN(k,q) = K(k,q) − iπ d3q ′ K(k,q′)δ(E(q) − E(q′)

)tNN(q′,q)/(2π)3 (8)

diihe

s

),

Ytralwere

ulationse andospin

r force

calt forerefore


Table 4Total reaction cross sections, volume integralsJU,W (in MeV fm3) per nucleon and root-mean-square ra〈RU,W 〉 (in fm) for the 7Li + 11B (in) and 7Be+ 11Be (out) optical potentials used in the calculations. TsymbolU andW refer to the real and imaginary part of the potential, respectively

σR

(in)σR

(out)JU

(in)JW

(in)JU

(out)JW

(out)〈RU 〉i 〈Rw〉i 〈RU 〉O 〈Rw〉O

1285.3 1379.7 −278 −166 −260 −170 3.91 3.67 3.82 3.80

Table 5Volume integrals for the nucleus–nucleus interaction at vanishing density

Central (S,T ) interactions in (A,B) rest frame:

TNN atp = 0 [MeV fm3] (00) (01) (10) (11)DIRECT 130.1 239.3 154.1 175.8EXCHANGE −834.4 60.9 −15.1 −6.2

Tensor (S,T ) interactions in (A,B) rest frame:

TNN atp = 0 [MeV fm5] (10) (11)DIRECT −17.8 −150.1EXCHANGE 89.8 −30.7

given here for the “half off-shell”T -matrix whereE(q) is the NN on-shell energy adefined by the incident energy. Obviously, the density dependence of theG- or K-matrix,respectively, implies a density dependentT -matrix. Numerical calculations are simplifiedby expressing the density dependence in a separable local density approximation (LDAas, e.g., in Refs. [43,44].

The charge exchange interaction was taken from the isovector parts of the D3G-matrix [43] including spin-dependent and spin-independent direct and exchange ceninteractions together with second rank tensor. The NN spin–orbit interactionsneglected. Hence, we are using the same interaction as in the HFB ground state calcand the QRPA description of excitations thus retaining the consistency of structurreaction calculations. The global properties of the interaction in the various spin–ischannels (S,T ) are summarized in Table 5.

Inspecting of the interactions in momentum space one notices that the tensocannot be neglected in the region around a momentum transfer of about 1 fm−1, wherethe reaction mostly takes place. The exchangecomponents are treated here in the lomomentum approximation (LMA), e.g., in Ref. [43,44]. They are especially importanthe isoscalar central force, indicating a pronounced momentum dependence and thnon-locality in that channel, but play only a minor role for the isovector interactions.

4.2. Optical model description of elastic scattering for incident and exit channel

Low-energy optical potentials are known for the stable7Li nuclide for elastic scattering
on stable target nuclei, e.g., Refs. [52,53]. But such information can obviously not beobtained empirically for the scattering of unstable nuclei as in the7Be + 11Be exit

e

elastic

lnnelsleon–line

srms

ed tof haloof thisak-upngng

lue foreerp inartcross

ergiesetitionnt for

eptedr waves a


channel. Because reliable empirical optical potentials for7,11Be are not known we usa double folding approach. For consistency this was done also for the7Li and 11Bincident channel. Since density dependent effects were found to be small forscattering at forward angles, we used the free nucleon–nucleonT -matrix, derived byFraney and Love [54] for energies at and aboveElab = 50 MeV, for calculating the reaand the imaginary part of the optical potentials in the incident and the exit chain the double folding approach in lowest order impulse approximation. The nucnucleonT -matrix tNN(p,E) at the current incident energy was obtained by a spextrapolation in momentum space. Including isoscalar (τ = 0) and isovector (τ = 1)

central interactionstτNN(p,E), respectively, and accounting for exchange contributionby the already mentioned LMA we find by double folding over the Fourier transfoof the isoscalar and isovector projectile and target ground state densitiesρτ

a,A(p),respectively,

Uopt(r,E) =∑

τ=0,1

∫d3pρτ

a (p)ρτA(p)tτNN(p,E)eipr/(2π)3. (9)

In addition, the semi-classical model of Bonaccorso and Carstoiu [55] was usestimate the long range contributions to the optical potential due to the coupling owave functions to the breakup channels. According to this model the imaginary partpotential is simply related to the properties of the halo wave function through the breprobability. The real part, accounting for the extra polarization induced by the halo at lorange, is obtained through a dispersion relation. The resulting breakup potentials, includithose for7Be+ 11Be atE = 57 MeV, are discussed in Ref. [55].

In Table 4 the global properties of these potentials are displayed. The larger vathe reaction cross section in the outgoing channelσR (out) compared with the ingoing onσR (in) reflects the halo structure of the11Begs density distribution, extending much furthinto space than the11B one. In addition, the long range potential accounting for break-uthe outgoing channel slightly increasesσR (out). Indeed, both the imaginary and real pof this break-up potential were found to give a contribution to the calculated CEXsections, improving the agreement with the data.

4.3. Reaction mechanism of the (7Li, 7Be) CEX reactions

The reaction dynamics of heavy-ion CEX processes at low and intermediate enhave been investigated extensively before in Refs. [9,33,52,56]. While the compof one-step direct and two-step transfer CEX routes was found to be importareactions with an incident12C projectile [9] this seems not be the case in (7Li, 7Be)reactions [11–13]. Although a study of comparable detail as in Ref. [9] for the12Cprojectile is missing—and is not the purpose of this paper—the suppression of two-stprocesses in (7Li, 7Be) reactions, also found in previous work [13,56], is very likely relato the fact that unbound intermediate states are involved. Since the overlap of theifunctions with bound, localized configurations is reduced the two-step amplitudes awhole seem to be of very small magnitude.

In order to give stronger support to these qualitative arguments—and neglecting thecompletely unsolved normalization problem due to the largely unknown spectroscopy

ng

e


Fig. 7. Estimate for the two-step transfer contributions to the11Begs + 7Begs exit channel via intermediate6Li

and 12B channels. The angular distribution is normalized arbitrarily to the data point at the largest scatteriangle.

Fig. 8. Estimate for the two-step transfer contributions to the11Be(0.32)+ 7Begs exit channel via intermediat
6Li and 12B channels. The angular distribution is normalized arbitrarily to the data point at the largest scatteringangle.

geutron-

but

at theed two-r than

sonable

redom-c-RPAthers and

)e

ludedalso

y

ential

the


of the intermediate channels—we have performed two-step transfer charge exchancalculations for some selective cases. In Fig. 7 we show results for the sequential “nepickup–proton-stripping” two-step route via intermediate states in6Li and12B and leadingto the ground states of11Be and7Be, respectively. Two-step results for the same routeending instead in the first excited state in11Be at 0.32 MeV are displayed in Fig. 8.

In both cases the angular distribution was normalized arbitrarily to the data pointlargest scattering angle, which should be considered as the upper limit for the expectstep strength. It is seen that the shape of the angular distributions is obviously flattethe measured ones. Hence, we might conclude that at least at forward angles (θcm < 15)the direct one-step processes will exhaust the measured cross sections to a readegree.

Under these assumptions the reaction mechanism is considered here as being pinantly given by a one-step direct process,for which DWBA theory as discussed in Setion 4.2 is providing the appropriate descriptions. In Ref. [20] we argued how the Qtheory successfully describes the11Be single particle strength including the ground andexcited states at 0.32 and 1.77 MeV. Results, obtained with microscopic form factoresponse functions and using the HIDEX code [57], are presented below.

4.4. The CEX reaction form factors and cross sections

The transition form factors are obtained by folding the nucleon–nucleon (NNinteraction in momentum space with the Fourier transforms of theβ± charge exchangtransition densities, either the shell model OBTD for the projectile (ρτ

λa) [12] or the QRPAOBTD of Eq. (6) for the target nucleus (ρτ

λA), respectively

F(r,E) =∫

d3pρτλa(p)ρτ

λA(p)tτNN(p,E)eipr/(2π)3. (10)

As in the optical potentials exchange effects due to anti-symmetrization are incin the local momentum approximation (LMA), see, e.g., Refs. [32,43,44]. Here, weincluded transitions into the7Be(1/2−) excited state atEx = 429 keV as they are clearlvisible in the data.

These form factors are then used in DWBA calculations to obtain the double differdirect charge exchange cross section

d2σ

dΩ dE=

∑λ(aA)

SλA(E)dσλ(aA)

dΩ(11)

summed over all multipolaritiesλ(aA), the discrete projectile states a and weighted by
target response functions per energySA(E). The reduced DWBA cross sections calculatedwith the form factors, Eq. (10), are denoted by dσλ(aA)/dΩ .

that

ctionfor thisn 5.3.

ayegion

ions fordata,

ithoutent is

inuumationsr the

atione often


5. Discussion of results

5.1. Angular distributions to bound and unbound states in11Be

In Figs. 9–11 the angular distributions for the transitions to the analysed11Be statesare shown for7Begs and 7Be0.43 states, respectively. The results are in agreement withe measured cross sections for the11Be ground state (Fig. 9) and the excited state1.77 MeV (Fig. 11), while for the state at 0.32 MeV (Fig. 10) the calculated cross seunderestimates the data by about a factor between two and three. The likely reasonirregular behaviour lies in the optical model potential as will be discussed in SectioA sudden change of the reaction mechanism for the different states of11Be would bepuzzling. At backward angles (θcm > 20) a systematic underestimation of strength mindicate a non-negligible contribution from the two-step processes in this angular r(see Figs. 7 and 8). It is interesting to note that in the past the calculated cross sectthe (7Li, 7Be) reaction typically have required large scaling factors to agree with thesee, for example, Refs. [10,11,13,19].

Instead the present calculations describe on average the data surprisingly well wthe need for introducing large scaling factors. This rather satisfying overall agreemmainly due to two reasons; namely, the realistic description of the single particle conteffects and the large 2QP configurations space in solving the QRPA Dyson equand the use of a density dependent in-medium NN interaction [43]. In particulaD3Y G-matrix interaction of Ref. [43] seems to account in a better way for the situencountered at low energy in a halo systems, as considered in this article, than thused Love and Franey interaction [54].

Fig. 9. Angular distributions for the11B(7Li,7Begs)11Begs and11B(7Li,7Be* )11Begs transitions. The calculatedcross sections are not scaled.

terac-r the

y ionutione

2.s of


Fig. 10. Angular distributions for the11B(7Li, 7Begs)11Be1/2− and 11B(7Li, 7Be* ) 11Be1/2− transitions. Thecalculated cross sections are not scaled.

Fig. 11. Angular distributions for the11B(7Li, 7Begs)11Be5/2+ and 11B(7Li, 7Be* )11Be5/2+ transitions. Thecalculated cross sections are not scaled.

5.2. Central and tensor interactions

The cross sections are not simply determined by only the central direct intion but the tensor becomes crucial, as illustrated, for example, in Fig. 12 fo11B(7Li, 7Begs)11Begs and the11B(7Li, 7Be0.43)11Begs transitions.

The results of Fig. 12 confirm the importance of the tensor interaction in heavcharge exchange reactions, in line with the findings in Ref. [11]. Note how the contribof the tensor force to each of the two angular distributions in Fig. 12 is almost the samat zero degree, thus having a minor influence on theG ratio as mentioned in SectionIt is worthwhile to investigate in more detail the contribution to the cross section
the different combinations of angular momentum. These are shown in Figs. 9–11 for the11Bgs → 11Be couplings. It is evident that the unnatural parity transitions account for the

not

r

. Thisn the

ed, whichof the

tes and2,

pin

of thecross

ctionor theed. Ther


Fig. 12. Contribution of the central and tensor forces on the angular distribution for the11B(7Li, 7Begs) 11Begs

transition on the left, and the11B(7Li, 7Be0.43)11Begs one on the right. The calculated cross sections are

scaled.

major part of the observed cross sections. In particular the 2− coupling dominates fothe 11Be ground state, the 1+ for the 0.32 MeV and the 2− and 4− for the 1.77 MeV,indicating more favourable conditions for the mechanism of nucleonic spin transferis confirmed by the comparison of the different angular momentum couplings i7Li → 7Be transitions. These findings reveal a predominance of 1+ and 3+ couplings,as shown in Fig. 13 for the unnatural parity couplings between11Bgs and11Be states.

5.3. Angular distributions for theG-ratios

In Fig. 14 the angular distributions for theG-ratio factor are presented. As opposto the cross sections, these quantities are almost insensitive to the optical modelis approximately cancelled out in the ratio. In this manner a supplemental testmicroscopic theory of the form factors presented in this article is made.

The results show a good agreement with the experimental data for the11Be singleparticle states, giving strong support to the assumptions on the structure of the stathe reaction mechanism. They also confirm the phenomenological argument of Sectionbased on the measuredG factors at 0, about the large contribution of the nucleonic stransfer dynamics, for the isovector interaction at low bombarding energy.

The agreement of the theoretical and the measuredG-ratios for 11Be single particlestates deserves some further discussion. On first sight it is unexpected becausepronounced differences in the cross sections. In our microscopic approach thesections are calculated in DWBA approximation where initial and final state interaamong the incoming and outgoing ions are described by an optical potential. Fcharge exchange transitions form factors obtained in double folding approach are usstructure parts include shell model OBTD for the7Li → 7Be transitions, QRPA OBTD fo
the11B → 11Be ones, and a D3YG-matrix interaction. Potentially each of these ingredientinfluences the final result of the calculations for the cross sections. The shell model OBTD

lculated

ei ase

theng,angingtrong


Fig. 13. Decomposition of the calculated cross sections in terms of the angular momentum JB transferred in theprojectile7Li →7Be transitions. The shown examples referto the unnatural parity transitions to the11Be ground(panel (a)), the first (b) and the second excited state ((c) and (d)). Nth is an overall scaling factor for the cacross sections.

for projectile transitions are well known and very accurate for p-shell mirror nucl7Li and 7Be. In addition, together with the D3YG-matrix interaction they are the samfor all the calculations. So it would be very surprising if they would only fail forangular distribution of the 1/2− state in11Be. Also the structure calculations regardiQRPA succeed to explain both the11Be low lying spectrum and theβ-decay strengthsand also verify the sum rules. Hence, the only ingredient that can be reasonably chfor different transitions is the optical potential where rapid variations indicate a s
influence of coupled channels effects. The problem in fact is due to the proximity of the0.32 MeV state to the continuum (only 180 keV) which makes the coupling to break-up

the

blemlyl state

romgnetict ofed0–d crosstics ofd inrder ofctor ofare

only ofurtween

the

ecty


Fig. 14. Angular distributions for theG factors relative to the11Be ground, first and second excited states. Inplots the analysis in terms of angular momentum transfer JB in the projectile transitions are also shown.

channels very important in this case. A full quantum mechanical treatment of this prois beyond the scope of this article. The observation that theG ratios behave more regularleads us to conclude that these quantities are only weakly affected by initial and finainteractions which would explain the nice agreement with the data.

6. Conclusions

The11B(7Li, 7Be)11Be reaction has been studied at 57 MeV bombarding energy. Fan experimental point of view the use of a Tandem beam and of a high resolution maspectrometer allows a clear identification of several states and the direct measurementhe spin flip probabilities. An empirical relation of proportionality between the measurcross sections at zero degrees and theβ-decay strengths is found to be valid within 230%, both for GT and SD transitions. In addition the measured values of the reducesections for GT transitions are found to be deducible from the well known systemathe same transitions driven by(n, p) reactions. Of course, these analyses, being valithe intermediate energy domain, are here only intended as guideline to explain the omagnitude of the measured cross sections. The fact that this is possible within a fatwo or less indicates that the microscopic processes that govern the reaction mechanismfollowing essentially a pure one-step CEX while higher order transfer processes areminor importance. Nevertheless, dealing with7Be doublets one cannot exclude, from oresults, that two-step mechanisms could still contribute to redistribute the strength bethese two states, having less influence in the sum.

The analyses of the doublet shapes trough theG factors at zero degrees, enforceevidence of direct one-step mechanism for the case of11B → 11Be transitions whilehampering it for the12C → 12B ones. Most likely, the dominance of the one step dircharge reaction mechanism for11Be, as compared to the12B case, at low incident energ
is a consequence of the much narrower distribution of11Be wave functions peaked atlow values in the momentum space. This gives a larger overlap with the one-step CEX

Theseratorss.

beenlation

n ofy. Therectinghalo

eleyondne for

thin–flipatabef spingy, oft lowainly

logicaloscopic

erest(lesss andouble. Theorptiveakups wellthens ofent. Arectlyed byd out

.


operators, especially at low momentum transfer, i.e., at forward scattering angles.characteristics will reduce the overlap with the more broadly distributed two-step opethus reducing the overall transfer charge exchange contribution to the cross section

A many-body approach, appropriate for the weakly bound residual nucleus, hasformulated in order to analyze the data. The use of the QRPA theory for the calcuof the microscopic transition densities for the11Bgs → 11Be coupling and of the D3YG-matrix interaction of Ref. [43] allows an upgraded and fully consistent descriptiothe data compared to traditional analyses of CEX reactions at low incident energcross sections for the single particle states are then calculated assuming a one-step dimechanism for the CEX, in the framework of the DWBA theory with a double foldoptical model and a specific term accounting for the break-up of the loosely boundstates. The analysis of the other states would involve a more complicated structure modapproach including a detailed treatment of the 2p–2h (4QP) excitations. This is bthe scope of the present article, although calculations of this kind have been doexcitations in stable nuclei [37,38].

The results show the success of our method, which reasonably well reproduces bothe angular distributions for the cross sections at forward angle and also the spprobabilities.In general, the calculations for thecross sections underestimate the dat backward angles, where the contribution of two-step mechanisms are expected tonon-negligible. On the other hand, at forward angles, they show a general trend otransfer dynamics. This agrees with the known prevalence, at low excitation enerGT and SD transitions, observed in CEX reactions at higher bombarding energy. Aincident energy and for weakly bound systems the role of the tensor interaction, mdriven by long range pion exchange, becomes crucial and the traditional phenomenopotentials need to be substituted by the more accurate treatment of the modern micrinteractions.

Finally, the calculations for the 0.32 MeV state point to a problem of general intfor analyses of reactions with exotic nuclei. The close vicinity to the continuumthan 0.2 MeV) makes this channel very peculiar with respect to reaction dynamicstructure. Both the diagonal density distribution, entering into the calculation of the dfolding potential, and the breakup contribution to the optical potential are not knownmagnitude of the charge exchange cross section is in particular affected by the abspart of the optical potentials in the incident and the exit channels to which the brepolarization self-energies mostly contribute. While we seem to have these quantitieunder control for the transitions involving the11Be ground state this must not be true onsame level of confidence for the 0.32 MeV state. In fact, fully microscopic calculatiothe break-up contributions to the exit channel optical potential are not feasible at presconclusive results, however, is that the microscopic calculations for this transition correproduce theG-ratio angular distribution while the cross sections are underestimata factor of two. Most likely, this indicates that reaction dynamical effects are cancelefrom theG-ratio but fully contribute to the cross sections.

The challenge for the future exploration of the (7Li, 7Be) reaction is twofold
Experimentally one has to provide high resolution spectra also at intermediate energy, forexample, by dispersing matching technique. Theoretically the goal is to build a microscopic

etailedory.

sier,work

k,


approach, where the model wave functions for the structure of exotic nuclei and the ddescription of the reaction mechanism are consistently derived from many-body the

Acknowledgements

The authors wish to thank Dr. M. Khaled, Dr. H. Laurent, Dr. J.M. Maison, Dr. L. RoDr. C. Stephan and Dr. L. Tassan-Got for their help during data taking. Part of thewas supported by DFG under contract Le439/5.

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Analysis of the 11B(7Li,7Be)11Be reaction at 57 MeV in a microscopic approach

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Transcript of Analysis of the 11B(7Li,7Be)11Be reaction at 57 MeV in a microscopic approach