Description of the 11Li(p,d)10Li transfer reaction using structure overlaps from a full three-body model

Physics Letters B 767 (2017) 307–313 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Description of the 11Li(p, d)10Li transfer reaction using structure overlaps from a full three-body model J. Casal a,b,∗, M. Gómez-Ramos a, A.M. Moro a aDepartamento de Física Atómica, Molecular y Nuclear, Facultad de Física, Universidad de Sevilla, Apartado 1065, E-41080 Sevilla, Spain bEuropean Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT∗) and Fondazione Bruno Kessler, Villa Tambosi, Strada delle Tabarelle 286, I-38123 Villazzano (TN), Italy a r t i c l e i n f o a b s t r a c t Article history: Received 21 November 2016 Received in revised form 23 December 2016 Accepted 7 February 2017 Available online 14 February 2017 Editor: J.-P. Blaizot Keywords: 10,11Li Transfer DWBA Overlaps Three-body Recent data on the differential angular distribution for the transfer reaction 11Li(p, d)10Li at E/A = 5.7MeVin inverse kinematics are analyzed within the DWBA reaction framework, using the overlap functions calculated within a three-body model of 11Li. The weight of the different 10Li configurations in the system’s ground state is obtained from the structure calculations unambiguously. The effect of the 9Li spin in the calculated observables is also investigated. We find that, although all the considered models succeed in reproducing the shape of the data, the magnitude is very sensitive to the content of p1/2wave in the 11Li ground-state wave function. Among the considered models, the best agreement with the data is obtained when the 11Li ground state contains a ∼31% of p1/2wave in the n–9Li subsystem. Although this model takes into account explicitly the splitting of the 1+and 2+resonances due to the coupling of the p1/2wave to the 3/2−spin of the core, a similar degree of agreement can be achieved with a model in which the 9Li spin is ignored, provided that it contains a similar p-wave content. ©2017 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 1. Introduction Halo nuclei have triggered intensive work in the nuclear physics community since their discovery back in the eighties [1,2]. The case of neutron Borromean nuclei, consisting of a compact core plus two valence neutrons [3], keeps being the subject of a considerable amount of experimental and theoretical studies. In these three-body systems, any two-body pair is unbound, which poses a real challenge from the theoretical point of view [4]. Examples of two-neutron halo nuclei are 6He, 11Li, 14Be or 22C. The understanding of their structure requires solid constraints on the neutron unbound binary subsystems, i.e. 5He, 10Li, 13Be or 21C. Our present understanding of the peculiar properties of these exotic systems largely stems from the analysis of reactions in which these nuclei are part of the colliding systems or appear within some of the reaction products. In the former case, the experiments must be performed in inverse kinematics, and have therefore only become possible since the development of rare isotope beam facilities in the late eighties. Examples of these reac- *Corresponding author at: European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT∗) and Fondazione Bruno Kessler, Villa Tambosi, Strada delle Tabarelle 286, I-38123 Villazzano (TN), Italy. E-mail address: [email protected] (J. Casal). tions are nucleon-removal (also named knockout) reactions [5,6], Coulomb dissociation [7], single- and multi-particle transfer and, most recently, quasi-free breakup reactions of the form (p, pn)or (p, 2p)[8–11]. The outcomes of these measurements are complementary to one another. Theoretical works have shown that the structure of twoneutron halo nuclei results from the delicate interplay of several factors, such as the binding effect due to the pairing interaction between the halo neutrons, the coupling to the core collective excitations [12], the effect of Pauli blocking and the role of tensor correlations [13]. In the case of 11Li, many theoretical and experimental efforts have been devoted to understanding its conspicuous structure. This system is bound by only S2n=369.15(65)keV [14]. Its large spatial extension was first evidenced in the pioneering knockout experiments performed by Tanihata and collaborators [1] using an energetic 11Li beam on a 12C target. The analysis of the momentum distributions from subsequent fragmentation experiments [15–17], including the angular correlations of the fragments [18], revealed an admixture of sand pwaves in the 11Li ground-state and permitted to extract their relative weights. Reaction cross section data of these high-energy experiments have been also used to constrain the radius and s-wave content of the 11Li ground state [19,20]. http://dx.doi.org/10.1016/j.physletb.2017.02.017 0370-2693/©2017 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 308 J. Casal et al. / Physics Letters B 767 (2017) 307–313 A proper understanding of the 11Li structure requires an accurate knowledge of the 10Li subsystem. This is an interesting system by itself, located just beyond the drip-line for N=7 isotones, and displaying an inversion of the 2s1/2and 1p1/2levels. The location and properties of these levels has been studied by means of several knockout and transfer experiments (see [21] and references therein). Most of these experiments predict a near-threshold concentration of  =0strength, consistent with the presence of a virtual state, and one or more narrow  =1 resonances with a centroid around ∼0.5 MeV. However, the detailed parameters for the virtual state (e.g. scattering length) and the actual position and width of these resonances vary widely from one experiment to another and are still a matter of debate [22]. The relative amount of s- and p-waves in the 11Li ground state varies also significantly between different works. Furthermore, none of these experiments was able to resolve the doublets arising from the coupling of the s-wave and p-wave to the 3/2−spin of the 9Li core. This introduces an additional ambiguity in the theoretical interpretation of these data, since the observed energy distributions are usually compatible with different model assumptions. Many of these experiments provide information on the continuum spectrum of the isolated 10Li, but not on the structure of 10Li within the 11Li nucleus. One of the few exceptions is provided by a recent measurement of the 11Li(p, d)10Li stripping reaction, performed at TRIUMF at E/A =5.7MeV[21]. The measured excitation spectrum of the produced 10Li exhibits a resonant-like structure located at Er=0.62(4)MeV. The comparison of the angular distribution of this peak, after subtraction of the non-resonant background, with standard DWBA calculations, was consistent with an  =1 configuration with a spectroscopic factor of 0.67 ±0.12. This corresponds to a 33% content of (p1/2)2in the 11Li ground state. In addition to the assumptions inherent to the DWBA reaction framework, the theoretical analysis was based on a simple model for the overlap function of the transferred neutron, which was described using a Woods–Saxon well with standard parameters. While this model can be roughly justified for two-body systems, the underlying picture of a neutron orbiting a 10Li core seems more questionable. Consequently, some of the conclusions of Ref. [21] could be biased by this oversimplification of the structure model. In the present work, we reexamine the same data using also the DWBA method, but replacing the simple Woods–Saxon model by a more sophisticated and, in principle, realistic description of the 11Li and 10Li systems. The former is treated within a threebody model, with effective n–nand n +core interactions, whereas the 10Li is described using n +9Li scattering states generated with the same n–core interaction as that used for 11Li. Our goal is to clarify the influence of the structure model on the extracted properties and, in particular, to see whether the conclusions of [21] are affected by the use of a more realistic structure model. 2. Reaction formalism Let us consider a reaction in which an incident composite nucleus (A)collides with a proton target, which picks one neutron N1from the projectile, giving rise to a residual subsystem (B)and a deuteron (d). We focus on the particular case of a three-body projectile comprising an inert core (C)plus two valence neutrons (N1,N2), so the reaction takes the form (C+N1+N2)   A +p→(C+N2)   B +d,(1) which is schematically depicted in Fig. 1. If the nucleus Bdoes not form bound states (e.g., the composite Ais a Borromean system) the products of its decay after one neutron removal will provide spectroscopic information on the original projectile wave function. Fig. 1. Diagram for a (p, d)reaction induced by a three-body projectile in inverse kinematics. For this process, the prior-form transition amplitude can be formally reduced to an effective few-body problem, leading to Tif =√2(−) f( x, R, r)|VpN1+UpB −UpA|A( x, y)χ(+) pA ( R), (2) where Arepresents the ground-state wave function of the initial three-body composite (11Li in our case), χ(+) pA is the distorted wave generated by the auxiliary potential UpA, and (−) fis the exact four-body wave function for the outgoing d–Bsystem. The ± superscript refers to the usual ingoing or outgoing boundary conditions. Notice the explicit factor √2arising from the two identical neutrons in the initial wave function. The origin of this factor is further discussed in Ref. [23]. In writing the transition amplitude in the form (2), we implicitly use a participant/spectator approximation, assuming that the reaction occurs due to the interaction of the incident proton with a single neutron (N1)of A(the participant), whereas the system B =N2+Cremains unperturbed. To reduce (2) to a tractable form, we approximate the exact wave function (−) fby the factorized form, (−) f( x, R, r)≈ϕ(−)  q,σ2,ζ ( x)χ(−) dB ( R)φd( r), (3) where φdis the deuteron wave function, ϕ(−)  q,σ2,ζ is a two-body continuum wave function with wave number  qand spin projections σ2,ζ of the binary subsystem B(10Li), and χ(−) dB is a distorted wave describing the d–Brelative motion in the exit channel. The function ϕ(−)  q,σ2,ζ is the time-reversed of ϕ(+)  q,σ2,ζ , which can be written as (cf. [24], p. 135) ϕ(+)  q,σ2,ζ ( x)=4π qx  LJJ TMT iLY∗ LM( q)LMs2σ2|JMJ ×JMJIζ|JTMTfJT LJ(qx)YLs2J( x)⊗κIJTMT, (4) where, for each component, the orbital angular momentum  L and the spin  s2of the neutron N2couple to  J, and  JTresults from coupling  Jwith the spin  Iof the core. Note that  xcontains also the internal coordinates of C. The radial functions fJT LJ are obtained by direct integration of the two-body Schrodinger equation for the N2+Csystem subject to the boundary condition fJT LJ(qx)−→ i 2H(−) L(qx)−SJT LJH(+) L(qx),(5) where qis related to the N2+Crelative energy as q =√2μxε/¯ h, with μxits reduced mass, and H(±)are Coulomb functions [25]. The ground state wave function of the initial three-body composite Ais described within a full three-body model [26–28]. In this work, this wave function is calculated as an expansion in hyperspherical harmonics using a pseudostate basis for the radial part [29], which has been successfully applied to describe J. Casal et al. / Physics Letters B 767 (2017) 307–313 309 the structure and reaction observables for exotic nuclei [30,31]. This three-body wave function is most naturally obtained in the Jacobi-T set, but for the purposes of computing the required overlap functions, it is then transformed into the Jacoby-Y set, where the xcoordinate relates the core and one neutron (see Fig. 1), and we choose a coupling order compatible with that of the two-body continuum wave function given by Eq. (4). After diagonalization of the three-body Hamiltonian, the wave function can be written schematically as jμ A( x, y)= β3b wj β3b(x,y) ×Ylxs2jx( x)⊗κIj1⊗Yly( y)⊗κs1j2jμ, (6) where β3b={K, lx, jx, j1, ly, j2}is a set of quantum numbers coupled to a total angular momentum j. In this set, Kis the hypermomentum, lxand lyare the orbital angular momenta associated with the Jacobi coordinates xand y, respectively, and jxresults from coupling lxwith the spin s2of a single neutron. Consistently with our spectator approximation for the composite B, we assume that the interaction does not change the state of this system, and the overlap between the two- and three-body wave functions (given by Eqs. (4) and (6), respectively) contain all the relevant structure information. If we denote by ψLJJ TMTthe part of this overlap containing the spatial dependence in Eq. (4), we have ψLJJ TMT(q, y)=fJT LJ(qx) xYLs2J( x)⊗κI∗ JTMTjμ A( x, y)d x, (7) and the transition amplitude Tif can be expanded as Tif =√24π q LJJ TMT (−i)LYLM( q)LMs2σ2|JMJ ×JMJIζ|JTMTTLJJ TMT if , (8) depending on a set of DWBA-like amplitudes, TLJJ TMT if ≡χ(−) dB φd|VpN1+UpB −UpA|ψLJJ TMTχ(+) pA .(9) These amplitudes enable the description of the process using a consistent structure input, in which the three-body projectile and the binary fragment incorporate the same core-neutron interaction. From the transition amplitude, and after integrating over the angles  qof the relative wave vector  q, the double differential cross sections as a function of the C–N2relative energy and the scattering angle of Bwith respect to the incident direction can be written as dσ2 dBdεx=32π2 q2ρ(εx)1 2(2j+1) μiμf (2π¯ h2)2 kf ki × LJJ T νTLJJ TMT if  2 ,(10) where ρ(εx) =μxq/[(2π)3¯ h2]is the density of Bstates as a function of the C–N2excitation energy εx, μi,fthe projectile-target reduced mass in the initial and final partitions, and ν≡{MTσd} represents the spin projections of the final products. 3. Structure model To describe the 11Li system, we need the n–nand n–9Li interactions. For the former, we use the well known GPT potential [32], which reproduces the scattering length for this system. For n–9Li, since the properties of 10Li are less known, we consider four different choices, which will produce four different ground state wave functions of 11Li. This will allow us to study the influence of the underlying properties of the 11Li nucleus on the calculated cross sections. The first two models, labeled P1I and P2I hereafter, are based on the following parametrization of Garrido et al. [33], with parity-dependent central, spin–orbit and spin–spin components V(L) 9Li-n(x)=V(L) c(x)+V(L) ss (x) s2· I+V(L) so-c(x) L· I+V(L) so-v L· s2.(11) These models take into account the actual 9Li spin (I=3/2−), which results in a splitting of the single particle orbital s1/2into 1−, 2−states, and p1/2into 1+, 2+resonances. The radial functions follow the Gaussian shape V(L) i(x) =v(L) iexp{−(x/R)2}, with R =2.55 fm and the strength parameters listed in Table 1. The other two models, labeled P3 and P4, are based on the parametrization of Thompson and Zhukov [34]. They ignore the spin of the 9Li core and contain central and spin–orbit terms, V(L) 9Li-n(x)=V(L) c(x)+V(L) so-v L· s2,(12) where the central term follows the usual Woods–Saxon form V(L) c(x) =v(L) c/(1 +exp{(x −R)/a}), the spin–orbit is given by a Woods–Saxon derivative, and their parameters are R =2.642 fm and a =0.67 fm. The strengths are also shown in Table 1. With these potentials, the ground state of 10Li is a s1/2virtual state. In the models including the spin of the core, the strengths of the different terms are chosen to ensure that, after the spin– spin splitting, the 2−state is the ground state of the system and the 2+resonance appears higher in energy than the 1+resonance. For each choice of the binary interactions, we compute the ground state of 11Li in our full three-body model. Potentials P1I and P2I include repulsive terms to suppress the p3/2Pauli forbidden states, which is referred to in the literature as the repulsive core approach [35]. On the contrary, P3 and P4 produce a p3/2 bound state that needs to be removed in order to avoid unphysical states in the three-body calculations. Here, we use the adiabatic projection method [36]. Following previous works, to recover the ground-state energy we include also a three-body force to account for effects not explicitly included [29,37], such as pairing correlations or core excitations. For these calculations, we adopt the ground state energy −0.37 MeV [14]. The resulting properties of the 10Li and 11Li systems obtained with these four parametrizations are shown in Table 2. For P1I, the 2−ground state of 10Li is characterized by a scattering length of as=−37.9fm, and the 1−states correspond to non-resonant continuum. For the p1/2resonances, this model provides two states at 0.37 and 0.61 MeV with spin and parity of 1+and 2+, respectively. The three-body wave function probabilities in the n–9Li subsystem are given by 31% of p1/2components and 67% of s1/2 components. In this model, the d5/2contribution is negligible. The weights of the individual 1−, 2−, 1+, 2+configurations are 27%, 40%, 12% and 19%, respectively. Taking these values into consideration, the effective scattering length of the 2−ground state is reduced to aeff =−29.3fm, and the two presonances have their centroid at 0.52 MeV. This is consistent with recent experiments being unable to resolve the doublet. The three-body calculations using this n–9Li potential provide a 11Li ground state characterized by matter and charge radii of 3.2 fm and 2.41 fm, respectively. As we will see below, this potential including the 9Li spin is found to give the best agreement with experimental data. The other potentials give different properties for 10Li and hence for 11Li. The potential P2I is chosen to produce two s1/2virtual states with smaller scattering lengths in absolute value and the 310 J. Casal et al. / Physics Letters B 767 (2017) 307–313 Table 1 Potential strengths for the 11Li models used in this work. Potentials P1I and P2I follow Eq. (11) and have a Gaussian radial shape with a =2.55 fm. Potentials P3 and P4 follow Eq. (12) and use Woods–Saxon forms with R =2.642 fm and a =0.67 fm. Model v(0) cv(1) cv(0) ss v(1) ss v(1) so-cv(1) so-v P1I −5.4 260.75 −4.50 1.00 1.00 300 P2I −5.0 260.25 −2.00 1.00 1.00 300 P3 −50.5−39.0– –– 40.0 P4 −49.6−39.4– –– 35.5 Table 2 Features of the 10Li structure for the different potentials employed in this work. The second column shows the energy of the p1/2resonance while the third one shows the scattering length of the s1/2virtual state. Note that for the model with spin, both the resonance and virtual state are split. The fourth and fifth columns show the weights of the p1/2and s1/2waves in the 11Li ground state respectively, while the last two columns show its matter and charge radii. Er(MeV) a(fm) %p1/2%s1/2rmat (fm) rch (fm) 1+2+1−2− P1I0.370.61– −37.9 31 67 3.2 2.41 P2I 0.30 0.55 −1.1−6.7 44 54 3.0 2.40 P3 0.50 −29.8 30 64 3.6 2.48 P4 0.23 −16.2 67 27 3.3 2.43 two p1/2resonances at slightly lower energies. This enhances the p1/2content in the 11Li ground state up to 44%, and reduces its radius by about 10%. Regarding the interactions neglecting the spin of the core, P3 is chosen to give a similar p1/2content as P1I, while in P4 the s1/2content is strongly reduced. This is reflected by the different values of the scattering lengths shown in Table 2. Note that P3 and P4 predict a small but not negligible d-wave component, so the s1/2and p1/2contributions add up to ∼94% only. Notice also that P3 provides the largest matter distribution for 11Li. In the following, the effect of these structure properties on the reaction dynamics will be discussed. 4. Application to 11Li(p, d)10Li We apply the present formalism to the reaction 11Li(p, d)10Li and compare our results with the recent experimental data by Sanetullaev et al. [21]. As in Ref. [21], we describe the reaction mechanism in DWBA and use the same potentials for the p-11Li, d-10Li and p-ninteractions as in Ref. [21]. Therefore, the main difference between our calculations and those of [21] relies on the choice for the ψLJJ TMT(q,  y)overlaps. Whereas in [21] they were approximated by a single-particle wave function of the transferred neutron in a Woods–Saxon potential, we obtain these functions from the three-body model of 11Li described in the preceding section. We must also remark that, while the calculation presented in [21] assumes a definite energy for the 10Li resonance, the calculations in this paper are performed for a range of n–9Li continuum energies extending from 0 to 3 MeV, and integrations over energy have been performed when necessary. In Fig. 2, the computed n–9Li relative energy spectra are presented using the preceding sets of potential parameters for the n–9Li interaction. In these calculations, it has been assumed that the two halo neutrons of 11Li have the same angular momentum  J= L+ s2. This condition is strictly fulfilled when the spin of the core is neglected and is also a good approximation when using I=3/2−. Only the configurations (p1/2)2and (s1/2)2have been considered, since other contributions, while present in the three-body calculations, give negligible components in the ground state of 11Li. The four panels in Fig. 2 correspond to the results using potentials P1I, P2I, P3 and P4. Here, the shaded peaks correspond to the p1/2resonances. The splitting of the resonances and virtual states can be seen in the upper two spectra. A folding of the total distribution with the experimental resolution given in Ref. [21] is presented for all four calculations (solid line). It is Fig. 2. The relative n–9Li energy spectrum for the different s1/2(green) and p1/2 contributions (red). The angle-integrated relative-energy cross section before (blue dashed) and after (blue solid) the convolution with the experimental resolution are also shown. Potentials P1I, P3 and P4 have been selected for the calculations in the top, middle and bottom panels respectively. For P1I, the 1−(green dashed) and 2− (green dotted) contributions are shown separately as well as 1+(red dashed) and 2+(red dotted). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) rather remarkable how the low-energy peaks are smoothed out by the experimental resolution leading to a relatively featureless distribution, which could explain why the fine details of 10Li are not observed experimentally. The peak widths and heights, however, can be noticeably different. In general, the effect of the spin–spin splitting broadens the energy distributions. In Fig. 3 the energy-integrated angular distribution of the outgoing deuteron is presented. Calculations have been performed using again the four discussed potential models. It has been assumed that the subtraction of the non-resonant background performed in the analysis of the data completely excludes the s-wave com- J. Casal et al. / Physics Letters B 767 (2017) 307–313 311 Fig. 3. Energy-integrated angular differential cross section for the 11Li(p, d)10Li reaction at 5.7 MeV/u. The lines are the results of DWBA calculations populating the p1/2orbitals in 10Li, using four different 11Li models: P1I (blue solid), P2I (red solid), P3 (magenta dashed) and P4 (green dashed). The cross section for neutrons transferred from the s1/2orbitals is also shown, for P1I, in the blue dotted line. Experimental data are from Ref. [21]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) ponent, so experimental data are compared only with the cross section corresponding to neutrons transferred from the p1/2or- bitals. Here we must remark that no fitting to the data has been performed, since the spectroscopic factor is given by the p-wave content in the three-body calculation. Our calculations provide absolute cross sections, in contrast to the theoretical curves in Ref. [21] and previous works in which the cross sections are renormalized to fit the experimental data and extract spectroscopic factors. The results for P1I and P3 give the best agreement with the experimental data, while the shape of all four calculations is rather similar in spite of the different structure properties of the 11Li ground state. This seems to portray the energy-integrated angular distribution as an observable dependent mostly on the angular momentum of the extracted neutron and its weight in the ground-state of 11Li. Note that the experimental data have been measured at angles for which the s1/2contribution is minimal, as shown in Fig. 3, where the s-wave contribution corresponding to the model P1I is presented. This may explain why in our energy distribution the contribution from swaves is quite important at low energies, while experimentally there is no direct sign of this effect. The p-wave content suggested by the present analysis (∼30%) is very close to the value extracted in the original analysis of the same data [21]. However, it is somewhat smaller than the value of 45% extracted from the analysis of the momentum distributions in the 11Li fragmentation on carbon [18]. A reanalysis of the same data, along with the data from Ref. [16], performed by Garrido et al. [38] using a 11Li three-body wave function similar to our PI models within a participant-spectator reaction model, suggested a p-wave content of about 40%. Another inert core model proposed by Vinh Mau and Pacheco [39], which uses a density-dependent pairing interaction between the halo neutrons, gave a p1/2content 27.6% for a separation energy of 0.375 MeV, in better agreement with our result. A more elaborated three-body model, including core tensor and pairing correlations [40], predicted 44% and 46.9% of 2s1/2and 1p1/2admixtures, respectively. Furthermore, the calculation performed by Barranco and co-workers [12], describing the neutrons in a mean-field potential supplemented by a pairing interaction and the coupling to the core collective excitations, resulted in 40% and 58% for 2s1/2and 1p1/2. For the present calculations to reproduce the (p, d) experimental data, our model requires a smaller p-wave content that those in Refs. [12,40], thus providing a larger s-wave contribution. Other theoretical works predict different numbers for the s-wave and p-wave admixture [41,33]. Clearly, this disparity of values calls for further experimental and theoretical work. It can be argued that the DWBA framework may be overly simplistic to accurately describe the reaction process, since the low beam energy favors higher-order processes, such as initial- and final-state interactions. These effects are not taken into account in the present formalism, which might hinder the effect of the 11Li structure on the angular distribution. However, the agreement of the calculations with the experimental data is rather satisfactory, which could be an indication that these effects are not crucial in the transfer process. Moreover, some of the configurations resulting from these higher-order processes could be included as part of the non-resonant continuum responsible for the bump at high n–9Li energies that appears in Fig. 2 of Ref. [21]. This background was already subtracted from the experimental angular distribution shown in Fig. 3, thus reducing the degree of unsuitability of the DWBA formalism. 5. Summary and conclusions To summarize, we have studied the 11Li(p, d)10Li reaction using the DWBA framework, computing the required 11Li|10Lioverlap functions from a three-body model calculation of the 11Li ground state. Our model provides absolute cross sections, in contrast to previous approaches. The comparison of our calculations with the angular distribution reported in Ref. [21] confirms the  =1dom- inance of this distribution. While different three-body models are found to explain equally well the shape of the angular distribution, we find a strong sensitivity on the p1/2-wave content, with the best agreement provided by a model with ∼31% for this configuration. The coupling with the intrinsic spin of the 9Li core does not affect the overall shape of the angular distribution, but changes the height and width of the relative energy spectra. Experimental data on other reactions involving 10,11Li are to be tested against the same structure input to find better constraints on their properties. The methodology employed in the present case could be useful to analyze similar data involving other Borromean nuclei, such as 6He, 14Be or 22C. Moreover, the calculated three-body overlaps can be also used in other reactions, such as quasifree (p, pn) breakup processes or (d, p)reactions. Calculations of this kind are in progress and will be presented elsewhere. Acknowledgements We are grateful to Dr. R. Kanungo for providing us the experimental data in tabular form and for useful discussions on their interpretation. 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