BOUND STATE PROPERTIES OF BORROMEAN HALO NUCLEI: 6He AND 11Li
M.V. ZHUKOV, B.V. DANILIN, DV. FEDOROV The Kurchatov Institute of Atomic Energy, 123182 Moscow, Russian Federation
J.M. BANG The Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen 0, Denmark
I.J. THOMPSON Department of Physics, University of Surrey, GuildfOrd GU2 5XH, UK
and
J.S. VAAGEN SENTEF, Institute of Physics, University of Bergen, N-5007 Bergen, Norway NORDITA, DK-2100 Copenhagen 0, Denmark
NORTH-HOLLAND PHYSICS REPORTS (Review Section of Physics Letters) 231, No. 4 (1993) 151199. PHYSICS REPORTS North-Holland
Bound state properties of Borromean halo nuclei: 6He and ‘1Li M.V. Zhukova, B.V. Danilina, D.V. Fedorova, J.M. Bangb, I.J. Thompsonc and J.S. Vaagen’~ The Kurchatov Institute of Atomic Energy, 123182 Moscow, Russian Federation The Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen 0, Denmark
C Department of Physics, University of Surrey, Guildford GU2 5XH, UK SENTEF, Institute of Physics, University of Bergen, N-5007 Bergen, Norway ‘NORDITA, DK-2100 Copenhagen 0, Denmark
Received November 1992; editor G. Brown
Contents:
1. Introduction 153 4.2. Matter distribution; di-neutron and cigar struc- ii. Excitement on the neutron drip line 153 tures in the wave function 171 1.2. Borromean nuclei ‘‘Li, 6He 154 4.3. High energy 6He fragmentation and transverse 1.3. Scope of the article 155 momentum of constituents 174 2. Characteristics of currently used three-body 4.4. 4He—n and n—n momentum correlations 176 procedures 156 5. COSMA method for rapid assessment of “true” 2.1. Expansion on hyperspherical harmonics (HH) 156 wave functions for Borromean nuclei 177 2.2. Treatment of the Pauli principle and effective 5.1. Motivation for COSMA 177 interactions 158 5.2. The model and test for 6He 177 2.3. Coordinate space Faddeev approach (CSF) 160 5.3. Results of calculations for “Li 182 2.4. The cluster-orbital shell model (COSM) 161 5.4. ‘‘Li wave function structure in the framework of 2.5. Two-particle Green’s function method (GFM) 161 COSMA: spatial densities and 9Li—n and n—n 2.6. Variational approaches (VA) and method of correlated momentum distributions 184 integral equations (IE) 162 6. Strict three-body calculations for ‘‘Li 186 3. Tests against the “well known” non-Borromean 6.1. Effective interactions 186 nucleus 6Li 163 6.2. Faddeev (CSF) and HH calculations 187 3.1. Ground and excited states of 5Li 163 6.3. Matter and momentum distributions in the ‘‘Li 3.2. Electromagnetic and weak observables 165 g.s. 191 3.3. Quasielastic scattering of nucleons on 6Li 167 6.4. Predictions for momentum correlations 193 3.4. Conclusion 168 6.5. Summary 194 4. Strict three-body calculations of 6He 168 7. Conclusions and outlook 195 4.1. Energy and geometry of the ground state; wave References 196 function structure 169
Abstract: The nuclei 6He and “Li which exhibit pronounced halo-structures with two loosely bound valence neutrons, are currently being explored as secondary-beam projectiles. These nuclei are Borromean, i.e. while they are bound (only one bound state) they have, considered as three-body systems, no bound states in the binary subsystems. We argue that a three-body description is the natural one for central properties of such exotic loosely bound nuclei, and give the state of the art by comparing fully blown three-body calculations for 6He (and neighboring A = 6 nuclei) with a range of measured observables. We restrict this review to bound state properties, with emphasis on genuine three-bodyfeatures. The bound state is the initial stage ofthe various reaction scenarios that now are being studied experimentally and a main objective of these studies. Currently used procedures for solving the three-body bound state problem are outlined, with emphasis on expansions on hyperspherical harmonics and also the coordinate space Faddeev approach. Although strict calculations can also be carried out for ‘‘Li, they are inconclusive concerning the details of the structure since the available information on the binary neutron-9Li(core) channel is insufficient. Calculations for a number of plausible model interactions, including treatments ofthe Pauli principle, are presented. They all reproduce the binding energy and halo characteristics such as valence one-particle density and give about the same internal r.m.s. geometry for ‘‘Li. In spite ofthis, the wave functions have pronounced differences in their spatial correlations. The same ambiguity is also present in other inclusive observables, such as momentum distributions. We also demonstrate that candidates for the nuclear structure can be explored within an approximate scheme COSMA. Predictions of exclusive observables are discussed, and quantities such as momentum correlations in complete measurements are found to be more sensitive to the detailed features of the nuclear structure of the bound state.
0370-1573/93/$24.00 © 1993 Elsevier Science Publishers By. All rights reserved. 1. Introduction
1.1. Excitement on the neutron drip line
The recent development of sufficiently intense secondary radioactive nuclear beams (RNB), produced through high-energy fragmentation of nuclei, has opened up new and exciting possibili- ties in the nuclear physics of light radioactive nuclei, in particular of nuclei near the neutron drip line. It bears promises for many applications in different fields of nuclear science, but especially for nuclear structure and reaction studies. The first experiments measured interaction cross-sections or reaction cross-sections at high and low energies for light radioactive neutron drip line nuclei [1—4] revealing abnormally large cross-sections for neutron rich nuclei such as “Li, “Be, ‘4Be, 17B These data provided important information on the nuclear sizes, and put on the agenda the question about existence of large neutron halos or dilute neutron skins (with long neutron tails extending well outside the nucleus) in loosely bound nuclei near the neutron drip line, in particular for the “Li nucleus. It should be noted that extended neutron halos had not been seen before in nuclei. Subsequent experiments with break-up of loosely bound neutron-rich projectiles on light targets [5—7]supported the existence of neutron halos in these systems. In particular, a very narrow °Li transverse momentum distribution (as compared to stable nuclei) was found from “Li fragmenta- tion at high energy [5]. Extremely narrow neutron angular distributions were also found from ~1~i break-up at low energy [6]. A very recent measurement of the 9Li longitudinal momentum distribution [8] shows an even narrower 9Li momentum distribution than that found in [5]. These experiments give substance to the idea of a neutron halo, and questions about the detailed structure of the neutron halos should now be asked. Based on the neutron halo hypothesis, existence of a new low-lying giant dipole resonance mode, the so-called soft dipole mode, has been suggested [9] for such systems. Its appearance is connected with suggested low-frequency oscillations of the halo neutrons against the core, giving rise to low-lying dipole excitations. Assuming that “Li is composed of a 9Li core and a dineutron, large electromagnetic dissociation (EMD) cross-sections for ~ incident on heavy targets were predicted [10, 11]. The large cross-sections expected for Coulomb dissociation of ~ have been confirmed experimentally(at least qualitatively) at high [12], intermediate [13] and low [6] energies. It is very important that the EMD cross sections of “Li on a lead target (at high energy [12]) are approximately 80 times larger than those of the nearby normally bound projectile nucleus ‘2C after appropriate scaling of the cross section by Z2 of the projectile. Similarly large EMD cross sections were also obtained at high energy on heavy targets for two other neutron-rich nuclei, 6He and 8He [14]. The current rapid development of radioactive nuclear beam techniques at many nuclear facilities in the world [15] promises good progress in the near future. Some of the recent experiments with RNB and some of the experiments in progress or being planned are: (a) experiments with high intensity polarized beams and their applications to studying nuclei far from the stability line (the first results of measurements of g-factors of exotic nuclei were reported in [16]),
153 6He and ‘‘Li 154 MV. Zhukor eta!., The Borromean halo nuclei (b) reactions of astrophysical interest are studied at many laboratories, and many new results were reported at the Second International Conference on Radioactive Nuclear Beams (Louvain- la-Neuve, Belgium, 1991), (c) elastic scattering of exotic nuclei (low energy 6He scattering on light and heavy targets was measured in [17, 18] as well as low energy 13N + ‘2’13C elastic scattering [19], and the first results for intermediate energy “Li + p elastic scattering were reported in [16]), (d) inelastic scattering and transfer reactions with RNB, (e) measurements of EMD cross sections at different energies, (f) measurements of pion charge-exchange and double charge-exchange reactions (the first results on the “B (~—, ~ ~ ~ reaction have been published [20]), (g) beta-decay studies may also provide very important information about neutron halos in drip-line nuclei. First found was the beta-delayed deuteron emission from the decay of 6He [21], and subsequently the beta-decay of the neutron-rich nuclei 6’8He and 9”Lj was measured [22]. In all cases strong transitions were observed to states that lie close in energy to the initial state. More experiments concerning beta-decay studies ~ofneutron halo nuclei are planned for the near future. (h) Correlation studies of RNB fragmentation. The first results on neutron-neutron coincidence for (‘‘Li, 9Li) fragmentation on different targets at low energy were reported in [23]. Measure- ments of correlations between neutron and core-fragment have recently been carried out for 1’Li at low and high energies [24—26].They are in progress for the 6He case.
1.2. Borromean nuclei “Li, 6He
All existing experimental data exhibit the presence of pronounced neutron-halo effects in nuclei near the neutron drip line. This means that, compared with nuclei near the stability line, neutron-rich nuclei in many respects represent qualitatively different many-nucleon systems, having other important degrees of freedom. First of all the correlations between the neutrons in the halo become very important and on equal base with those between these neutrons and the core. The assumptions of conventional shell-model and Hartree—Fock (mean-field) approaches have to be questioned close to the neutron (and also proton) drip lines. In fact self-consistent Hartree—Fock calculations (including the relativistic mean-field approach) fail to reproduce the separation energies, r.m.s. matter radii, as well as observed interaction cross sections for “Lj and 6He [27—29].Neither are conventional shell-model calculations [30] able to correctly reproduce the anomalous properties of “Li. These results are not very surprising as it is known in the framework of mean-field approximations to be difficult to account for the properties of the nuclear surface and correspondingly the neutron- halo structure. A modification of conventional Hartree—Fock was introduced in [29] by adjusting the separation energy of the last neutron to measured values. Doing this, it was possible to reproduce the large radii for “Li and 14Be, but then some discrepancies appeared with lighter isotopes Li and Be. Without going into more details of such approaches, we only like to mention that the same authors [31,28] stressed that it is necessary to go beyond the mean-field approxima- tion and include additional correlations, or to use three-body approaches for such peculiar systems as ‘‘Li. Within the framework of a two-cluster approach (dineutron + core) [10], it has been possible to account for essential features of “Li. These successes definitely point to the existence of a neutron halo — a concerted motion of the two valence nucleons over a wide spatial region. The dineutron structure is, however, an ad hoc assumption and its structure can obviously not be derived within such a formulation. An investigation of “Li within the three-body approach, 9Li + n + n, is more 6He and ‘‘Li 155 M.V. Zhukov et a!., The Borromean halo nuclei promising, as has already been argued for in the literature. Additional evidence for a three-body structure of ‘1Li was recently found in quadrupole moment measurements of “Li and 9Li [32], giving nearly equal quadrupole moments for these nuclei. This is in accordance with models describing the “Li nucleus in the framework of three-body approaches. We have, as reflected in the title, devoted this report to so called Borromean*) nuclei; cluster-like nuclei which are bound (but normally weakly) from hadronic point of view, but where there are no bound states in any binary subsystems. So far most attention has been paid to “Li. We emphasize however.that there exists (at least) one more neutron-rich light nucleus (which for a long time was not thought of as a halo nucleus) namely 6He which clearly exhibits a Borromean three-body halo structure. Numerous three-body calculations for A = 6 nuclei [33—36]show that the ground and low-lying excited states of these nuclei have well-developed three-body 4He + N + N structures. The “Li and 6He nuclei have in fact much in common from the point of view of three-body models: (i) they are as already pointed out, both Borromean; (ii) the ratios of the separation energy of the two valence neutrons to the one-neutron separation energy from the core, are close, respectively 1 MeV/20 MeY (6He) and 0.3 MeV/4.1 MeV (“Li);
(iii) the prevailing total J of the two valence neutrons is J = 0 for both cases; (iv) the transverse core-fragment momentum distributions resulting from ~ and 6He break- up in collisions with light targets and at high energies, have a peculiar two-component structure in both cases [5, 20]. For 6He this structure was found very recently [20]; (v) the normalized EMD cross sections are nearly equal [14]; (vi) in both cases there are unusually large changes in the r.m.s.matter radii for the core plus two neutron systems compared with the core nuclei. Thus Rrms(Q() 1.46 fm ~ Rrms(6He) 2.5—2.6 fm [2,37], and Rrms(9L~) 2.32fm [2], ~‘~i~Rrms(”L~) 3.2fm [2]. In the following we will investigate the neutron halo structure for these two Borromean systems
— “Li and 6He — in different three-body approaches. In nuclear physics there are other examples of
Borromean systems: the 9Be and 9B nuclei, and other neutron drip line nuclei — 8He, ‘4Be and ~ [38]. Therefore, such typical few-body-like behavior seems to be very natural for nuclei on the neutron drip line (and perhaps to some extent on the proton drip line also). Interest in Borromean loosely bound states of three particles is currently also strong in molecular physics and chemistry [39] with experimental searches for such systems. In nuclear physics such systems exist.
1.3. Scope of the article
In the next section, we outline the methods that have been used to solve the three-body problems that arise in the analysis of Borromean nuclei. These will first be tested against the 6Li nucleus, which shows many similar characteristics, and then for 6He and “Li. With the rapid ongoing developments it is impossible to present a fully understood picture, not even a consistent one. The correlation data which at this very moment are taken off the data tapes at various laboratories will certainly throw new light on the fragmentation scenarios at various energies and for different targets, and probably also raise new questions.
~°The Borromean rings, the heraldic symbol of the Princes of Borromeo, are carved in the stone of their castle in Lake Maggiore in northern Italy. The three rings are interlocked in such a way that if any of them were removed, the other two would also fall apart. In nuclear physics “Li and 6He have been found to have this property (although for quite different physical reasons) when described in a three-body model. 156 MV. Zhukov et a!., The Borromean halo nuclei °Heand ‘‘Li
Hence, our review has in some sense to be a preliminary one and we have chosen a part of the complex where we feel that the theoretical methods are reliable although they may be far from explaining the full physics. Thus we only address the ground state properties of the halo nuclei, their internal momentum distributions and correlations, both in coordinate and momentum space. To what extent these properties are directly seen by the detectors is somewhat of an open question, but it ought be a reasonable starting point for high beam energies (0.5—1 GeV/A) where the Serber mechanism (sudden approximation) or direct breakup from the ground state appears to be a natural initial assumption. Final state interactions may still play a role, in particular for the lightest outgoing particles, the neutrons. We will not address how the Coulomb field affects the projectile: some still preliminary but quite comprehensive discussions of the electromagnetic dissociation of the halo-like nuclei have already been given by the other authors [40—43]. Our ambition is to point out some solid ground from which new and unknown territories can be explored. The A = 6 nuclei seem to constitute such a base. Recently we have also performed three-body continuum calculations for such systems. These results will not be reviewed here although they have increased our understanding of these nuclei. Although fully fledged three-body calculations now are technical possible and most of the results we show do correspond to fully converged real calculations, we will also discuss some approximate schemes and their applicability for less ambitious practitioners. Efforts to make the three-body problem more accessible, within a qualitative approach, have been made recently in [44]. In spite that there seems to be a common belief among physicists (as was pointed out in [44]) that the three-body problem is generally so difficult that significant results can only be obtained by means of exhaustive numerical calculations, we hope to demonstrate that our theoretical approaches are surprisingly transparent and that the essential limitations are the lack of physical binary neutron- core information needed for conclusive three-body calculations. This review will also aim at the insight in the physics gained so far. For extensive descriptions of the various formalisms we refer the reader to the literature.
2. Characteristics of currently used three-body procedures
2.1. Expansion on hyperspherical harmonics (HH)
In a three-body core + N + N model, the total wave function (WF) ~P= &(~)~‘jM(l, 2) is assumed to have a product form, where 4~,is the (usually) inactive core intrinsic wave function (depending on intrinsic core coordinates ~), while ~I’(1,2) is the active part. It depends on the relative coordinates and spins (often suppressed) of the two valence nucleons, and hence is translation invariant and the object of the calculation. ~P(1,2) is solution of the three-body Schrödinger equation
(T+ V—E)~P(1,2)=0, where V= V~, + V~ 2+V,2, (1) with binary interaction potentials being assumed and the energy measured from the three-body threshold.
It is convenient to introduce translation invariant sets of Jacobi coordinates Xk = ~ r~,
(r,3 = r — r3), Yk = ~./~A~kr(,J)k, where A1~= A~A~/(A+1 A3) is the reduced mass for pair (i,j) 6He and “Li 157 M.V. Zhukov et a!., The Borromean halo nuclei (measured in units of the nucleon mass m) and similarlyfor the CM of(i,j) with respect to particle k. With 1 and 2 referring to the two valence neutrons and 3 to the core, we get
x 3 = r,2/~/~, y~= ~2A3/(A3 + 2) r~,213
x, = ~/A7(A3 + 1)r23, y, = ~/(A3 + 1)/(A3 + 2)r23(,), (2) and similarly for (x2, y2). Notice that the Jacobi coordinatey, is co-linear with the coordinate r in
CM system: r• = —.J(A~+ Ak)/[A,(A, + A3 + Ak)]yj.*) In the HH method, the wave functions in LS representation are
~JM = i,_5/2~ x~1~[~!4~’(Q5)®Xs]JM(~) XTMT (3) where the hyperradius p given by
2 = (x2 + y2) = ~ = ~ (4) p
is a collective translationally, rotationally and permutation invariant variable (A = A 3 + 2). The hyperharmonic basis functions &~‘~(Q5) are functions of the five hyperspherical polar angles
= (~, ~, 5’) containing the hyperangle u = arctan(x/y) which extracts part of the radial correla- tions. The corresponding angle for conjugated momenta distributes the total energy between the x and y degrees of freedom. X5(1, 2) is the coupled two-nucleon spin function and likewise for
isospin. The extra quantum number K = l~+ l~,+ 2n (n = 0, 1,2, ...) related to ~ is called the hypermoment. For non-interacting particles K is conserved, for harmonic oscillator interactions (6D H.O.) K is connected with the principal quantum number. It should be noted that every hyperharmonic term in (3) corresponds in shell-model language to an infinite series of radial excitations. Hyperspherical harmonics have the explicit form
~KLML(~5) ~ (cc) [Ye(S) ® Yjy (Y)]LML: (5)
~ (cc) = N K (sin cc)” (coscc) P,~” ‘~ (cos 2cc)
where P~ is the Jacobi polynomial, N~”’~a normalizing coefficient and n = (K — 1,, — When expansion (3) is inserted in eq. (1) a set of one-dimensional coupled differential equations in the variable p is obtained. These are solved numerically as explained in more detail in refs. [36,45]. For bound states, the solutions go sufficiently fast to zero for large interparticle
*iCorrespondingly, Jacobian type momenta are introduced. The relative momenta corresponding to (2) are simply
P12 = (P, — P2)/2 and P(,2)3 = [A3(P, + P2) — 2P3]/(A3 + 2) where the notation refers to lab. frame momenta P. which add up to the total momentum P P, + P2 + P3. If alternatively CM momenta k, are used, with the property2.k,The+ k2kinetic+ k3 =energy0 we getis nowthe
simplerseparableexpressionsin relative coordinatesP12 = (k1 —ink2standard)/2 and ~way with= —correspondingk3 = k, + k2inertial, and vicemassesversaA, k, = P12 + P1,2)3/ 2 and A,,213 and converted into an operator with gradients V,2 and V,,213. Ifalternatively the scaledJacobian coordinates (2) are used, we may associate the corresponding gradients with
scaled Jacobian momenta p,,, = p3 = ..J~p,2and p~= = ~/(A3 + 2)/(2A3)p,,213. 6He and ‘‘Li 158 M. V. Zhukov ci a!., The Borromean halo nuclei separations. In the HH method this simplymeans that x(~)—~ 0 for p —÷ cxc. This set of equations, in which every diagonal term contains an effective three-body centrifugal repulsive potential (propor- tional to (K + 3/2)(K + 5/2)/p2), is similar to asingle-particle problem where the particle moves in a deformed mean field. We note here that the three-body effective centrifugal barrier does not vanish ifthe angular momenta of subsystems l. and l~,are equal to zero. The three-body mean field
(matrix elements of V in the HH basis) behaves asymptotically as VKK’(P —~ cc) ‘~ p_fl with n 3
(n = 3 for the diagonal terms) for the short range nuclear pairwise interactions. This power behavior reflects the peculiarity of three-body problem, namely the possibility of two particles to interact far away from the third particle [44]. The asymptotic form of the three-body WF, given by (3) and (1) is, for finite range two-body potentials,
X(p)—*exp(_Kp), where h2ic2/2m = —E. (6)
This has been proved in [46] (see also [34]) to give the correct asymptotic form in all regions of six-dimensional space when there are no bound states in any two-body subchannels. The proof depends crucially on the Borromean property. For other nuclei where two fragments are bound together in a cluster when the third is removed, the asymptotic form of the three-body WF will in certain regions of configuration space, be dominated by the two-body asymptotics. Since the Borromean nuclei are very weakly bound, the region where the WF is well approxim- ated by (6) is very extended. It will therefore also contribute significantly to most measurable properties of these nuclei (where core degrees of freedom not are involved). This contrasts with other nuclei where the asymptotic properties are generally only important in calculations of transfer and break-up amplitudes. This underlines the need for proper three-body calculations in the quest to understand the properties of such nuclei. It also underlines the central importance of reproducing the energy correctly in such calculations. The common feature of all HH calculations is a very rapid convergence of the WF, and a rather slow one for the binding energy. So, if we restrict ourselves to evaluation of the WF to within say 1 % accuracy (for A = 6 nuclei), we need only calculate up to K 6, but if wewant to investigate the convergence of the binding energy and therefore the fine details of nuclear dynamics, we must extend HH basis up to K = 16—26 (depending mainly on the complexity of the NN interaction type). The transparent nature of the HH method as well as a priori correct asymptotic behavior (6) of the WF makes it more convenient for qualitative estimates for Borromean nuclei; often essential physical information is contained in only a few or even one K-component in expansion (3).
2.2. Treatment of the Pauli principle and effective interactions
When an A-nucleon problem is reduced to a core + N + N system, a most important aspect compared to usual three-body calculations is the treatment of the Pauli principle between the valence nucleons and the nucleons of the core. It is well known that taking the Pauli principle into account in a complete way with complex clusters leads to non-local interactions between them. Different approximations to the complete antisymmetric approach have been suggested. In the limit where the interaction between the valence nucleons of our model is negligible, only the centre of mass motion gives rise to some small deviations from the ordinary shell model. In this model antisymmetry is automatically fulfilled, by constructing Slater determinants from the relevant orbits. Looking for example at the harmonic oscillator shell model, this centre of mass problem may be solved completely or approximately. When residual NN interactions are taken into 6He and “Li 159 M. V. Zhukov et a!., The Borromean halo nuclei account, the situation becomes more complicated, since they give rise to mixing of configurations, including those which in the pure shell model were occupied. Residual interactions actually partially lift the forbiddenness of such states, as known for example in the random phase approximation and pairing theory. This partial allowance is, however, in general a small effect, appreciable only in the neighbourhood of the Fermi surface. Here the configuration admixtures caused by interactions between valence nucleons (like the two nucleons in our model) are also small far from the Fermi surface, and it is therefore a possible approximation to neglect the Pauli principle for extremely loosely bound systems. The opposite approximation is to replace it with a complete exclusion of some occupied states. It seems actually that the exclusion approximation must be good for the Os state which corresponds to the very tightly bound alpha-cluster.
If, therefore, we want to treat the A = 6 nuclei as a three-body problem with an effective local czN interaction and take into account of the Pauli principle effects there are some alternatives (see also [47]): (a) to project out occupied or partially occupied orbits (Pauli forbidden or almost Pauli forbidden) if the three-body WF is generated with deep effective intercluster interactions which produce such orbits; (b) alternatively we may add inhomogeneous terms to the Schrodinger equation, a procedure used below in section 2.3; (c) for deep potentials to add a homogeneous term in the intercluster interaction, a projector on forbidden states multiplied with a constant which in principle should go infinity [48]; (d) by introducing an additional repulsive interaction (a ‘Pauli core’) in the partial components of effective intercluster interactions, and fitting these interactions to the experimental phase shifts. This approximation is one of the ways to take the Pauli principle into account in a three-body model of the A = 6 nuclei, which was used in HH calculations. It should be noted that there are recent results [49, 50] connecting the last two approaches via a supersymmetric transform. Effective NN interactions are generally derived from nuclear matter, thus they contain Pauli principle effects to some extent. To implement nuclear matter interactions in finite nuclei is connected with uncertainties. All these methods nevertheless only lead to approximately antisymmetric solutions of the full A-body problem.
Choice of core-N and NN interactions The peculiarity of the three-body problem is its sensitivity to the off-shell behaviour of the two-body interaction. So to check this, different choices of NN and ccN were used in calculations of
A = 6 systems. Two types of cN interactions were tried. Both fit cN scattering phase shifts [51] satisfactorily. The first, V,,,,(SBB) of the same Gaussian shape (b = 2.3 fm) for all potential components, has p-wave and is parameters as originally introduced [52], but a slightly attenuated d-wave to reproduce the phase shift (V,,,, = —47.32, — 23 and — 5.855 MeV for p, d and is respectively). The second choice V,,~,(WS)employs a Woods—Saxon of range 2.0 fm and WS-derivative form for the corresponding potential components as used in the coordinate space Faddeev calculation [53] for 6Li. For both choices of Vna, the HH calculations included a purely repulsive s-wave component of
J/,,,~with the same Gaussian shape as given above, but with V~= + 50 MeV. This component of the J’~interaction reproduces the experimental s 1~2phase shifts well. This “Pauli core” is the way the method takes the Pauli exclusion principle into account, excluding from the ccN potential a Os state occupied by the cc-core neutrons. 6He and ‘‘Li 160 M. V. Zhukov et a!., The Borromean halo nuclei For the NN interaction four variants were used: a simple central Gaussian (G) [54]; and three “realistic” ones, the GPT [55], the SSC [56] and the RSC [57]. The realistic potentials include both I~s and tensor components. The SSC has a core which reaches out to larger distances, it is also steeper at the core surface. All four choices reproduce NN scattering phase shifts successfully within a reasonable energy range.
2.3. Coordinate space Faddeev approach (CSF)
In the framework of the coordinate space Faddeev approach (CSF) the three-body WF ~I’from (1) is represented as a sum of three components
(7) and the three-body eigenvalue problem is rewritten as a set of coordinate-space coupled Faddeev equations [53]:
(E — T— V, 2)çlt,2 = V,2(1//~,+ ‘I’c2),
(E — T— V~,)i/i~,= V~,(iIi~+2 ~//12)+ P~,, (8)
(E — T— V~2)lJI~2= V~2(1/i~,+ 1//12) + P~2.
Here the terms P,.,, (i = 1,2) take care of the Pauli exclusion principle between valence neutrons and neutrons from the core. It should be noted that the CSF calculations employ a different treatment of the Pauli principle as compared to the HH method (see detailed discussion in previous section). Here the occupied core-states are projected out, i.e. the Faddeev components are required to lie entirely in the allowed space. Technically this is done [53] by finding functions F,(y,) in the source term P,, = ~~~(i)F1(y,) so that for each corresponding core state (nif), <~1~(i)IW> = 0. Note that P,,, has a finite range in the core-particle coordinates hence it does not influence the asymptotic behavior. Using different treatments of the Pauli principle in the CSF and HH methods enables us to compare these approaches for calculations6He,withthe precisethe samewaybinaryin whichpotentialsthe exclusion(see below).is takenBecauseinto accountthe admixturesis presumablyof Os arenotanyhowso crucial.small in In the coordinate-space Faddeev formulation, the tj components in eq. (7) are expressed in their natural Jacobi coordinates and given bipolar angular momentum decompositions
~JM(’f) = ~ ~ (x,y)(Y,~(5’)®{Xk ® [Y 1~(~)® Xs~J]J}E)JMXTMT. (9)
When these expansions are inserted into eq. (8), a set of two-dimensional coupled differential equations are obtained. They are solved numerically as explained in more detail*) in [34, 53]. Note that the different components in eqs. (7), (8) are appropriately given in different coordinates
(x,,y,) (2), since V1 = V,(x,). In the actual CSF calculations hyperspherical coordinates p, cc (eq. (4))
*) Notice that the Jacobi coordinates employed in [53] are obtained from (2) by multiplication with an overall scale factor + 1)/A3. 6He and “Li 161 MV. Zhukov eta!., The Borromean halo nuclei are again introduced. Each of the Faddeev components again have the asymptotic dependence in p given by eq. (6). Apart from this, the use of the hyperspherical coordinates highly facilitates the transformation between the coordinate systems eq. (2) appropriate for different components, since it becomes simply arotation in six-dimensional space (x,,y,) with fixed p. For bound states the CSF and HH methods give in principle identical solutions, including the same asymptotic behavior. This is guaranteed by requiring that the solutions go sufficiently fast to zero for large interparticle separations. Although the HH method appears simpler, the CSF approach also has certain advantages. It enables us to choose a variable grid spacing and hence gives a better numerical treatment of the variable y/x for which the HH method uses an expansion which may be slowly convergent for binary interactions such as the RSC and SSC with strongly repulsive cores.
2.4. The cluster-orbital shell model (COSM)
The cluster-orbital shell model was suggested in [58] to describe a system of several valence nucleons coupled to a core. The COSM introduces translationally invariant coordinates between the core and valence nucleons. One of the advantages of the COSM over conventional shell model is that excitation of the valence nucleons does not lead to spurious center of mass excitations. In this model single-particle orbits can be determined consistently with underlying potentials between the core and the valence nucleon. It means that the potential between the core and the valence nucleon may be different from the potential for a nucleon in the core (the main problem of the conventional shell-model). The COSM is therefore an extended version of the shell model, suitable for a nucleus which can be described by an inert core + valence nucleons and can thus in principle be extended somewhat beyond the three-body problem. However, this model has some problems with slow convergence of the binding energy of the three-body system. These problems are mainly connected with the choice of the translationally invariant coordinates between the core and valence nucleons. These coordinates are not orthogonal like the conventional Jacobi coordinates for the three-body problem (eq. (2)), and there appears a mixing term V,,~V 21, in the three-body kinetic energy operator which is responsible for a low rate of convergence of the binding energy in COSM [59]. The antisymmetry of the valence nucleons is taken explicitly into account in this model and the Pauli principle between the valence nucleon and the core is treated in an approximate way by imposing that the valence nucleon6HeWFandshould“Libe[59,60].orthogonal to the WF of a nucleon from the core. This model has been applied to 2.5. Two-particle Green’s function method (GFM)
To study pairing effects in weakly bound nuclei, and in particular the “Li nucleus, it was recently proposed [61] to use a three-body model Hamiltonian with an interaction between valence neutrons which contains a density dependent term. This density dependence was chosen to fit the empirical pairing in the nuclei ‘4C and ‘2Be, as these contain the same number of the neutrons as 1tLi. The interaction between valence neutrons and the core was taken as a Woods—Saxon potential with parameters as close as possible to standard shell model values. The numerical method used to solve the suggested three-body Hamiltonian is the two-particle Greens function method in coordinate space. This method is however practical if the pairing interaction between the valence neutrons is a contact (zero-range) interaction: only this type of pairing interaction was considered in [61]. The Pauli principle between valence neutrons and neutrons from the core enters in two ways. It is contained in the density-dependence in the NN 162 MV. Zhukov ci a!., The Borromean halo nuclei ‘He and ~ interaction, and furthermore, the bound states occupied by the core neutrons were explicitly excluded from the wave function. By fitting the potential parameters to the tentative resonance energy of ‘°Li(assuming that the valence neutron is nearly a P,/2 state with energy —.~0.8MeV) the model is able to reproduce the small binding energy of “Li, as well as the large electric dipole strength in “Li found experi- mentally. It should be noted however that the model contains some approximations which may to some extent affect the final results. The model ignores finite-range (and hence p-wave) effects in pairing interactions, and although using core-particle coordinates as in [59] the mixing term (see section 2.4) in the three-body kinetic energy operator is omitted, disregarding some effects of core recoil.
2.6. Variational approaches (VA) and method of integral equations (IE)
To some extent all above mentioned three-body methods can be called variational methods. In all of them the strict three-body equations are solved using different types of wave function (or Greens function) expansions. Since these expansions in practice are limited to a finite number of terms, from the point of view of binding energy calculations all results obtained are variational ones. Below we shall however classify as variational approaches (VA) mainly only two methods used to solve three-body equations for composite particles systems [35,62, 63]. In the first [63] the neutron drip-line nucleus “Li was carefully studied, while in the second [35,62] only the A = 6 system has been examined. In [63] the three-body WF is found by expanding in terms of coupled harmonic oscillator wave functions depending on the Jacobi variables of eq. (2). The oscillator scale parameters are also treated as variational parameters in addition to the expansion coefficient to obtain the binding energy of the system. The neutron—neutron potential was chosen as a pure central potential which reproduces the low energy scattering data [54]. The core-neutron potential was also chosen as purely central without any spin-spin or spin-orbital dependence. The Pauli principle between the valence neutron and a neutron of the core was imitated by using a shallow core—neutron potential which has no bound or narrow resonance states. It should be noted that in spite of the simplicity of this approach for taking into account the Pauli principle, it may be quite reasonable. The point is that the binding energy for this system is extremely small and the average distance between the core and the valence neutron is large compared to the size of the core. This suggests that there may only be a small influence on the results of the calculations for ‘‘Li from the precise method of antisymmetrization between valence and internal neutrons. In [35,62] the three-body WFs were found by solving a set of two-dimensional Hill—Wheeler integral equations on the quadrature Chebyshev grid. A nonorthogonal many-dimensional Gaus- sian basis was taken to be the expansion basis. As the authors use core-nucleon interaction potentials which contain Pauli forbidden states they take the Pauli principle into account by the method mentioned above (section 2.2, (c)). Note that the three-body WF will be strictly orthogonal to the Pauli forbidden states of the two body interaction potentials only when the orthogonalizing coupling constant goes to infinity. In the integral equation (IE) method [33, 64,65] the three-body WFs for both 6He and 6Li were constructed by solving the Schrodinger equation in momentum space with Jacobi momentum coordinates. This leads to the set of coupled integral equations for unknown amplitudes which depend on Jacobi momentum coordinates. The two-body interactions used in the three-body Schrödinger equation were taken as separable potentials which fit two-body data at low energies. MV. Zhukov et a!., The Borromean halo nuclei ‘He and “Li 163
The Pauli principle between a valence nucleon and a nucleon from the core was treated in two ways (as a repulsive core in the ctN interaction or as a deep cN potential with forbidden state). The IE method was applied to study many properties of A = 6 systems, mainly for ground state WFs. It has been shown that different treatments of the Pauli principle have only small influence on the structure of the ground state WFs. However, in the IE method only low-rank separable interac- tions were used, which in particular do not contain the short-range repulsion of NN interactions, nor d-waves in the cLN interaction. The tensor forces in the NN channel are only included in an approximate way.
3. Tests against the “well known” non-Borromean nucleus 6Li To obtain reliable knowledge on fundamental properties of the lightest nuclei we have to eliminate computational and model uncertainties, and test the wave functions against numerous experimental data. The presence of the two-body deuteron tail in the asymptotics of the 6Li ground state WF (T = 0) does not permit us to treat this nucleus in the rigorous way as a pure Borromean system. For this case with a closed cLD channel, due to the completeness of the HH basis we can nevertheless reach an arbitrary accuracy in the interior region by taking into account a sufficient number of HH components. Note that the CSF method has no difficulties in incorporating two-body subchannels, and this makes the comparison with results of the CSF method very useful. Thus the variety of available experimental data enable us to examine the details of cluster dynamics in all kinds of processes — nuclear reactions, weak and electromagnetic observables. Most relevant for the following 6He studies are:
(i) The 6Li 0~(T = 1) excited state belongs to the same isobaric triplet as the 6He(g.s.) and 6Be(g.s.), so, except for the Coulomb violation of isospin symmetry, we can deduce some of the 6He properties by studying the 6Li(O~)state in electromagnetic processes and inelastic (p,p’) scattering where the reaction mechanism and its dependence on the nuclear structure is rather clear; (ii) calculations of ground (6Li) and other excited-states, electromagnetic static characteristics and elastic electron scattering, which give information about the validity of the “frozen” cluster approximation. A number of advanced microscopic investigations [53, 65, 66,62] have been performed (elastic and inelastic form-factors, magnetic and quadrupole moments etc.). Here we will discuss in detail recent results of solving the bound-state three-body problem with the HH method and CSF approaches.
3.1. Ground and excited states of 6Li
The realistic NN and ccN interactions that enter as input in the calculations were described in the previous chapter. All methods give underbinding of 6Li for all plausible interactions: HH: ~—~O.4—O.8MeV; CSF: —~O.3—0.4MeV; VA: ,-..CO.8 MeV; IE: ‘—~0.5MeV. In the HH method we can only estimate the binding energy from the asymptotic approach to convergence. The physical reasons for underbinding may be three-body forces, and the presence of other cluster components (3He + 3H etc.) corresponding to the closed channels, and not explicitly included in our model space. Within the three-body model, these can be treated as a weak polarization of the cc-particle in the field of the valence nucleons, and taken into account by increasing the core size and consequently the ccN interaction range by 1.5% in the CSF approach, and 3% in the HH method. The sensitivity of the binding energy to the Ncc interaction radius is large, about 12 MeV/fm. 164 MV. Zhukov ci a!., The Borromean halo nuclei ‘He and ‘‘Li
Table 1 Relative weights of the ‘Li (ground state) WF components for various NN and Na potentials. For HH’ the asymptotic region for convergence of the binding energy was not reached. HH and CSF* include a-core scaling.
NN: GPT GPT SSC SSC SSC SSC RSC RSC - -
‘Li Na: SBB WS SBB WS WS WS SBB SBB - - Comp. Method HH HH HH’ HH’ CSF CSF* CSF VA lE rep. IE proj.
L = 0, S = 1 1, = 0, 1,, = 0 92.09 91.00 87.87 86.04 91.05 89.29 90.19 95.54 91.78 91.48
I, = 2, 1,. = 2 0.99 1.09 1.16 1.19 0.30 0.39 0.22 — — — L=1,S=0 !,=1,l~=1 2.85 3.64 4.58 6.07 3.53 5.36 3.89 1.08 4.01 4.65
I, = 3, 1, = 3 0.09 0.15 0.13 0.22 -- - — — —
L=1,S=l !~=2,!,=2 0.20 0.45 0.21 0.48 — — — — 0.50 0.48 L=2,S=i !~=2,!,=0 3.68 3.51 5.94 5.74 4.98 4.70 5.65 ~=3.38 E=3.7l ~=3.40
= 0, !~= 2 0.11 0.16 0.11 0.26 0.12 0.25 0.09 — — —
We find that the main dynamics is governed by the p-component of the Nc interaction and the s-component of the NN interaction. We have studied the sensitivity of the binding energy to changes in the partial potential depths for the cLN interaction SSB and a central Gaussian NN interaction and found leading gradients for these partial components. For (s, p, d, Is) the gradients are (0.082,0.384,0.0007,0.143) and (0.143,0.006,0.004,0.006) for Ncc and NN respectively. 6Li(ground state). In the HH, IE and The WF (see table 1) is dominated by L = 0, S = 1 for the CSF methods the norms of admixed components depend on the NN and cLN interactions only very weakly. Note that the L = 1, S = 0 admixture in 6Li has a large weight (3—6%) in contrast with the result of ref. [62]. The hypermoment (K)-decomposition of the WF in the HH and CSF methods gives an almost Pauli forbidden K = 0 state with weight 4—5% only, the main K = 2 component with norm of about 95% being an approximately good quantum number. The main components have l = 1 in the one-particle-core motion as is expected from conventional shell model. For NN forces from GPT and scaled (3%) SBB and WS interactions for N~,the asymptotic binding energy of 6Li in HH calculations was determined to be 3.5 ±0.1 MeV. The deviation from the experimental value (3.7 MeV) may here be explained by a formation of an asymptotic deuteron cluster with its internal attraction insufficiently accounted for in our WF structure in the external region. This is suggested by a predominance of 1,, = 0, 2 (deuteron internal l) for K> 10, actually with ~ = 0 corresponding to a dominant s-state in the relative ccD motion. It should be noted that for the SSC nucleon—nucleon potential, which has a large repulsive core, an extremely slow convergence of the binding energy is found (as in three-nucleon calculations) in spite of a rather fast stabilization of the wave function. Nevertheless, a comparison with CSF calculations of 6Li with the same type of ccN and NN forces shows close agreement in the component structure of the WF for the two cases (see table 1). Another example is the almost Pauli forbidden component K = 0; = l~= 0 (not shown separately in table 1). In the HH calculations this component has a weight of 5% which is in agreement with the result of CSF calculations.
The first excited state of 6Li with 0~T = 1 has no hadronic decay channels, being an analog to the 6He ground state. It is shifted relative the 6He(g.s.) by a Coulomb energy of 0.837 MeV. The partial weights of the WF of 6Li (O’, T = 1) coincide with those of 6He (see below) to within an accuracy of 0.1—0.4%, and the main components of the WF are contained in K = 0,2 (96%). The
L = 1, S = 1 admixture in the WF of this state is rather large, 13—14%. Comparing the 1 + and 0’ states, the relative signs of the main and admixture amplitudes are opposite. The weights of these amplitudes are determined only by the c~Nspin-orbit interaction as is easily seen from comparison MV. Zhukov eta!., The Borromean halo nuclei ‘He and ‘‘Li 165
Table 2 (Quasi) binding energies of A = 6 nuclei, calculated for ‘Li (T = 0) and ‘He (T = I) ground states correspondingly.
NN: GPT GPT SSC Sq. well Na: SBB WS WS SBB Rep F Method HH HH CSF* VA IE Exp. [67]
3T= 0 2.25 2.22 2.13 2.32 1.96 2.186 2~T=0 3.86 4.03 4.84 5.00 4.31 2~T=1 ‘He 1.84 1.78 1.737 1.60 1.49 1.8
2~T=1 ‘Li 2.62 2.56 1.81 — 2.64 2~T=1 ‘Be 3.92 3.86 3.72 — 4.02
of purely central and realistic NN forces [45]. The two types of ccN interactions give very similar results. With the same realistic interactions, excited states have been calculated as quasibound states. This is possible for the 3~ and 2~(T= 0) and 2~(T= 1) states in 6Li, as they have small widths compared with the level spacing, and significant effective 3-body centrifugal barriers in the HH representation, the lowest hypermoment being K = 2 in all cases. The binding energies in these calculations have an intrinsic error which is less than the level width. Relative energies (calculated from the 6He ground state) for the T = 1 case are reproduced quite well
(see table 2). Calculations of the 3~and 2~states (T = 0) in the HH method give an overbinding of these states. By decreasing the cc-particle polarization from 3% to 0.5% the correct value for the 3 + energy is reproduced. The d-wave component of the ccN interaction is very important in this case. This component for the WS (taken from literature) should be reduced to at least half its value as it gives an incorrect description of the d partial wave phase shifts (and consequently additional overbinding). Spin-orbit 2~—3~splitting indicates the advantage of a surface-type (1. s)-force for ccN, and the 0.3 MeV undersplitting may have its origin in the d-wave spin-orbit component, that should be enhanced to reproduce experimental d-phase splitting (see [62]). Nevertheless oversplitting in VA and IE calculations leaves the question of high partial ccN dynamics open. As for the ground state of 6Li, the HH calculations for other states have also demonstrated that the hypermoment K = 2 is a very good quantum number: in all considered cases the total norm of this component is about 95%.
3.2. Electromagnetic and weak observables
Static quantities such as magnetic and quadrupole moments are strongly integrated nuclear characteristics and depend mainly on the weights and signs of the partial amplitudes which, in turn, are defined by the three-body dynamics. They are nevertheless, basic measures for the applicability of the cc + N + N model and the role of various components of ccN and NN forces. Form factors contain more information on the currents and charge distributions and in principle give us the possibility to examine in a consistent and independent way the contributions of valence nucleons and cc-particle to these characteristics. In this sense the Ml inelastic formfactor for the excitation of the 0~(T = 1) state is of great importance, the WFs describing these states in 6Li and 6He being very similar because of spin-isospin symmetry. This gives an additional independent test on the 6He structure. 166 MV. Zhukot’ ci a!., The Borromean halo nuclei ‘He and ‘‘Li
1 0 2~
- Ml NEL. FORMFACTOR OF 6Li
Fig. I. The inelastic transverse Ml form factor of ‘Li. Experimental data are taken from ref. [68].
Table 3 Calculated magnetic moments of the ‘Li system. The HH include K 16. The HH and CSF* include a-core scaling. The experimental magnetic moment is 0.822.
NN GPT GPT SSC SSC SSC SSC RSC Na SBB WS SBB ws ws WS SBB Method HH HH HH HH CSF CSF* VA 0.834 0.8402 0.8267 0.8195 0.828 0.828 0.8563
3.2.1. Inelastic transverse MI .formfOctor of 6Li By the isospin selection rule, only the magnetic (spin) current contributes to the inelastic Ml formfactor. The Ml formfactor is a superposition of transition densities which, in turn, depend on the correctness of geometrical characteristics of the valence nucleons. These transition densities have been calculated with the HH WFs mentioned above. All necessary formulae are given in [66]. Figure 1 compares the calculated Ml formfactor with experimental data [68]. The theoretical curve reproduces the location of the minimum and the second maximum up to q 2 fm~,while well-known contributions from relativistic corrections and mesonic exchange currents, essential in this region, have been left out. Almost the same conclusions are arrived at for the elastic Ml form factor. The contribution from the orbital current is very small 5% and its magnitude is similar to the orbital contribution to the magnetic moment. Regarding the sensitivity of the Ml formfactor to the NN tensor force we mention that, in spite of the binding energy and geometrical characteristics being fairly well reproduced in calculations with purely central NN forces, the location of the minimum of the Ml inelastic formfactor is shifted to q = 1.6 fm -‘ and the second maximum is significantly less than with a tensor force. Knowing the inelastic transverse Ml formfactor it is easy to estimate the radiative width of the 6Li Ml transition
0~(3.56 MeV) —* 1 + (g.s.) which is defined by the transverse formfactor value at the photonic point. The calculated value F,, = 8.13 eV coincides within experimental errors with the measured value
= 8.13 ±0.2 eV [67].
3.2.2. Magnetic and quadrupole moments of 6Li Calculations of the magnetic moment in standard non-relativistic reduction are given in table 3. It is seen that: (1) the calculated magnetic moment agrees with the experimental value to within 2—5% (the deviation for HH calculations is probably due to spin-orbit corrections and relativistic M. V. Zhuko,’ ci a!., The Borroinean halo nuclei ‘He and ‘‘Li 167
Table 4 Calculated ft-value and BGT for 3-decay of ‘He. The HH include K 16 and a-core scaling.
NN GPT GPT SSC SSC Na SBB WS SBB WS Method HH HH HH HH
ft 789 791 823 819
BGT 4.85 4.84 4.68 4.70 effects); (ii) the magnetic moment is very sensitive to the tensor NN-force and ccN (1.s)-interaction; (iii) the magnetic moment is, in general, dominated by the spin part of the valence nucleon wave function (95%); (iv) the cc-particle orbital motion contributes very weakly to the magnetic moment (0.55%). Making use of the quadrupole component of the 6Li density the quadrupole moment Q has been calculated to be 4 mb in the HH method with GPT + SBB interactions, and 0.6 mb in the CSF method. Q = 2.4 mb is obtained from the variational method [62], with the same sign and close to our results. The very small experimental value (Q = —0.8 mb) [67] and the negative sign may reflect a quadrupole self-polarization (well known to be quite big in heavy nuclei [69], < —20 mb) of the cc-particle in the field of the valence proton.
3.2.3. j3-decay of 6He A detailed investigation of all aspects of 6He 3-decay was given in [70]. Theft-values with the parameters ft(0~— O~)= 3072.4 and g~/g~ = —1.26 that were obtained (see table 4), coincide with experiment (812.8 ±3.8 s) within 3—4% accuracy. Comparison of the calculated results listed in table 4 with the experimental data and previous results [70], leads to the following conclusions: (1) the realistic nucleon-nucleon potential, GPT, provides better agreement with experiment than pure central NN forces, supporting that the tensor NN force plays an important role in the 6Li WF structure; (2) polarization of the cc-particle hardly affects the n-decay rate ( <1 % decrease offt); (3) a variation of the (1. s)-force by 10% does not change the ft value very much; (4) the cases SSC + SBB and SSC + WS (last two columns of table 4) are rather instructive. Due to incomplete convergence of the HH expansion for binding energies for these potentials the ft-values have somewhat unphysical values larger than the experimental one, which lead to a ratio being larger than the vacuum value. (5) in spite of the good agreement of the calculated ft-values with the slightly experimental data, the theoretical results in the two first columns of table 4 are lower than experiment. This indicates a possible renormalization of the axial-vector weak constant A. (taking into account the good description of the Ml inelastic form-factor) to a smaller value which is consistent with the trend of the gaxjai(A.) value in nuclear matter; on the whole the results of the 6He f3-decay calculations show the good quality of the WF.
3.3. Quasielastic scattering of nucleons on 6Li
Unlike the weak and electromagnetic processes which are well understood from a theoretical point of view, the description of quasielastic reactions of nucleons on nuclei contains a large 168 MV. Zhukov ci a!., The Borroinean halo nuclei ‘He and ‘‘Li
10 2 d 0,dc(nlh/sr)
q(fm- ) -
000.2 -~ -, ~ 0
Fig. 2. Comparison between self-consistent DWIA calculations and experimental data (ref. [71]) at E,,,, = 280 MeV. Curve 1, ‘Li(p, n)’Be; 2, ‘Li(n, p)’He; 3, ‘Li(p, p’)’Li’ reactions.
number of assumptions, where at least some require a better justification from first principles. In the intermediate energy region and for small transferred momenta, the description is somewhat simplified, implying fewer approximations ~and a corresponding increase in the reliability of the theoretical analysis. 6Li(p, n)6Be, 6Li(n,p)Therefore,6He, andwe restrict6Li(p,p’)ourselves6Li(O~,3.56to MeV)the investigationat energies larger[66] ofthanthe100reactionsMeV. In this energy region at small momentum transfer, the dynamics of the process can be described as a single-step transition, and the theoretical analysis can be carried out in the framework of the distorted-wave impulse approximation (DWIA). The results shown in fig. 2 correspond to “self-consistent” microscopic DWIA calculations of quasielastic reactions, with ingredients tested against available weak and electromagnetic data as discussed above. In this sense there are no free parameters. The self-consistent evaluations of cross sections reproduce experimental [71] behavior reason-
ably well up to transferred momentum q 1 fm 1 for (p, p’), (p, n) and (n, p) reactions on 6Li, populating the isospin triplet.
3.4. Conclusion
The three-body cc + N + N approaches discussed above seem to be well applicable for the
investigation of A = 6 nuclei. In particularly the HH and CSF methods enable us to calculate the WFs of such systems with high accuracy, as well as the amplitudes of many processes with strong, electromagnetic and weak interactions. All the results taken together suggest a broader applicabil- ity of HH and CSF approaches to 6He and other Borromean systems.
4. Strict three-body calculations of 6He
The lightest Borromean nucleus is 6He. From the point of view of a three-body calculation this is the most clear case as the experimental information about the ccn and nn subsystems is sufficiently complete to construct ccn and nn potentials and there are no long range Coulomb interactions in any of the binary subsystems. All ccn and nn potentials used for calculations of 6He were described in chapter 2 (section 2.2). Below we present the results of strict three-body calculations of 6He with these potentials, mainly those obtained with the HH and CSF methods. Having chosen the ccn and nn potentials the wave functions obtained allow one to perform parameter-free calculations of all interesting observables of 6He. Thus 6He may serve as a reference point for other Borromean systems.
. MV. Zhukov et a!., The Borromean halo nuclei ‘He and “Li 169
4.1. Energy and geometry of the ground state,’ wave function structure
The calculations, with both the HH and CSF methods and corresponding variants of ncc and nn potentials, lead to some underbinding for 6He (as was also the case for 6Li). They give bindings of about —0.4 MeV instead of the experimental value —0.973 ±0.04 MeV. This observation is common to all realistic calculations so far [34, 35, 64, 59]. In the HH method this is not related to lack of convergence in the calculations. Although the HH energy was still not fully convergedfor all
potential variants at K = 20, we can assess the convergence since we know that for Gaussian and Woods—Saxon form factors the energy deviation from the asymptotic value is exponential exp(—ccK) in K. Energy convergence was essentially obtained except for the interaction SSC. In the case of the CSF method, convergence is more readily obtained. To understand the physics of the underbinding one probably has to go beyond the simplest three-body model. The controlled accuracy of our calculations allow for such extensions. Presence of three-body forces or core degrees of freedom have to be explored: a polarization of the cc-particle is again to be expected. Since the cc-particle is tightly bound, a small polarization may be sufficient. A simple increase of the ncc interaction radius by 1.2% gives the correct binding in the CSF approach, in the HH a slightly larger increase of about 3% is needed. This means that the two valence neutrons ‘see’ a slightly larger polarized core than just a free 4He nucleus. The sensitivity of the binding energy to the ncc interaction radius is large, again about 12 MeV/fm. Correspondingly we should not expect the phase shifts to be exactly given by the free case, the calculated deviation is however rather small [45].
Taking into account that the ground state of 6He has Jfl = 0~and that the total spin of the two valence neutrons has only two possible values S = 0, 1, it is easy to see that in the framework of the
LS-coupling the total orbital momentum L also has only two values L = 0,1. So in LS-coupling,
the WF of the 6He(g.s.) should contain only two terms: L = S = 0, L = S = 1. If in addition the strict antisymmetry between the two valence neutrons is taken into account, it is easy to show that for the first choice of Jacobi coordinates, eq. (2) (where x is proportional to the distance between the two valence neutrons) the orbital momenta (lx. 1,,) in the corresponding subsystems have to be equal
(I~= l,, = 1) and to have a parity equal to the parity of the total L.
The WF (see table 5) in LS-coupling is dominated by L = S = 0 for 6He. In the HH and CSF methods the norms of different components were found to be very close to each other (in spite of quite different methods for inclusion of the Pauli principle between core and valence neutrons), and depend only very weakly on the choice of nn and ccn interactions. The component structure of the WF is just the same as in the case of a “vacuum” ccn interaction, so only weights summed over K are
presented in table 5. Note that the admixture of the L = S = 1 component in the 6He WF has
Table 5
Relative weights of the WF components (l = I = I for ‘He). HH and CSF* include a-core scaling. For HH’, the asymptotic region for convergence of the binding energy was not reached.
on: GPT GPT SSC SSC SSC SSC RSC RSC ‘He na: SBB WS SBB wS WS WS SBB SBB component Method: HH HH HH’ HH’ CSF CSF* CSF VA
S = 0, 1 = 0 84.02 83.56 84.75 83.51 87.42 86.00 87.59 94.60 S = I, 1 = 1 13.54 13.81 12.91 13.73 10.77 12.09 10.82 4.29
S = 0, 1 = 2 1.93 1.97 1.78 1.71 2.35 2.17 2.35 1.11
S = 1, 1 = 3 0.51 0.63 0.56 0.69 — — — — 170 M. V. Zhuko,’ ci a!., The Borromean halo nuclei ‘He and ‘‘Li
Table 6
K-decomposition of the IS components (I = I, = 1,) of the ~ wave function for the two interaction pairs (G, SBB) and (GPT, SBB).
S=0 S=1 S=0 S=l S=0 S=1 S=0 S=l K on !=0 !=l 1=2 1=3 1=4 1=5 1=6 1=7
0 G 4.41 GPT 4.68 2 G 78.93 139! GPT 78.10 13.41 4 G 0.02 0.12 0.51 GPT 0.03 0.15 0.61 6 G 0.01 0.01 1.15 0.52 GPT 0.003 0.003 1.43 0.67 ~8—16 G 0.24 0.002 0.13 0.02 0.02 GPT 0.58 0 0.25 0.003 0.08 0.06 0.002 0.004
a large weight (13—15%) in contrast to the result of [35]. Notice also that the contribution of the 6Li WF component with S = 1 extracted from experimental data on ~t radiative capture by (12—16%) [72] corresponds very well to our values. Such admixtures are also important for the f3-decay of 6He. Since 1,, = 0 dominates the relative motion between the neutrons, the motion of their CM(1,2) with respect to the core is dominated by l~= 0. The K-composition of the various (I, S) components of the 6He WF is given in table 6 for the interaction pairs (SBB, G) and (SBB, GPT). The hypermoment (K)-decomposition of the HH wave function gives an almost Pauli forbidden K = 0 component with weight 4—5%. The main component K = 2 has a weight of 91—92% and
K = 2 is again nearly a good quantum number. K = 2 is the lowest value not attenuated by the Pauli principle. The calculated contributions of the different components of the WF agree well with the results of the experimental investigation of the 6Be ground state decay in cc + 2p [73] (assuming that both nuclei have a similar structure, as they are members of the same isospin triplet). Having WFs of 6He in both HH and CSF approaches for different two-body potentials, observables for this three-body system have been calculated without any additional fitting para- meters, first of all the geometrical structure of 6He i.e. the different r.m.s. distances between the particles including the r.m.s. matter radius. In the framework of the HH method it is easy to prove a simple formula which connects the r.m.s. matter radius RA of three-body system with the r.m.s. matter radius A~of the core and the r.m.s. matter radius of the three-body system containing only point particles (the latter corresponds to the variable p)
(A~+ 2)R~= A~’R~+
cc 2> = ~ IdpP2[X~,~(P)]2. (11)
Table 7 Calculated r.m.s. separations between various parts of the ‘He system. The HH include K 20. The HH and CSF* include a-core scaling.
nn GPT GPT SSC SSC SSC SSC SBB WS SBB wS ws wS Coordinate Method. HH HH HH HH CSF CSF*
4.83 4.72 5.03 4.99 4.77 4.58 4.43 4.36 4.50 4.50 4.52 4.19 ~CM 3.46 3.40 3.54 3.53 3.67 3.43
1vCM 1.24 1.22 1.25 1.25 1.29 1.18 2.59 2.53 2.65 2.64 2.57 2.45
6He, formula (10) gives R(6He) = {[4R(4He)2 +
transform the HH basis functions ~ (Q5). They transform via Raynal—Revai coefficients [74]
KLML(5) = ~ ~ . (12)
Conservation of the quantum numbers K, L, ML and parity implies the selection rules: I,, + l~even and I,, + l,, even. These formulae provide 6theHe possibility(the last rowto calculatein table all7) maynecessarybe comparedvalues inwithtablethe7. The calculated r.m.s. matter radius of value (Rexp = 2.57 ±0.1 fm) obtained in a model independent way [37] from the measurement of the total interaction cross-section [1]. All results for the matter radius of 6He are in rather good agreement with this experimental value. Note that for the G and GPT interactions between the valence neutrons the calculations gave a decrease from 2.66 fm to 2.59 fm with cc-core scaling, both numbers are however within the experimental limits. The CSF method gives 2.57 fm, and 2.45 fm with cc-core scaling. The value obtained in [59] is somewhat smaller, 2.40 fm. As is seen from table 7 the average distances between the two valence neutrons and between the cc-particle and a valence neutron are quite large. From this one might conclude that in this three-body system correlations are absent. But that this is not true will be shown in the next section. The large average distance between the cc-particle and a valence neutron gives an explanation for why the different inclusions of the Pauli principle between the cc-particle and a valence neutron have only little influence on the binding energy and the WF structure.
4.2. Matter distribution,’ dineutron and cigar structures in the wave function
As we saw in the previous section CSF and HH approaches give very similar results for the 6He WF structure as well as for the geometrical structure. Hence, for calculations of different 172 MV. Zhukov eta!., The Borroniean halo nuclei ‘He and ‘‘Li
LU’ 6 He
(fm)
Fig. 3. Neutron and a components of the ‘He matter density. observables for 6He, we use the simplest procedure. The results presented in this section are obtained by the HH method mainly [75, 76, 66]. First of all we present calculations of the matter density of the 6He ground state, referred to its CM. The density operator for such calculation was chosen also taking into account the nucleon- density inside the core (cc-particle)
n(r) = i=~,2 ~(r — r~)+ j’dzpc(r — z)~(z— rj, (13) where Pc and r~are the core (cc-particle) density and the distance of the CM of the core from the CM of three-body system. So in calculations of the matter density of 6He, we correct for the size of the cc by folding the squared wave function of the cc center-of-mass motion with the internal cc density, which was taken as a Gaussian with width parameter reproducing the r.m.s. matter radius of 4He. The most convenient way to calculate the matrix element of the density operator (13) with a three-body WF is to choose for every part of this operator the “eigen” coordinate system. In such coordinate systems r, andr~are proportional to the corresponding Jacobi coordinatesy, andy 3 (2) and most of the integrations (in six dimensional coordinate space) can be done analytically leading to one-dimensional numerical integration [66, 96]. 6He matter density are shown in fig. 3. The calculationThe valenceusedneutron-the potentialand cc-setcomponents(GPT, SBB)offorthethe (nn, ncc) interactions. Replacing GPT with a simple Gaussian gives negligible change. The other potential pairs discussed above lead to essentially the same results. From fig. 3 it is seen that for r> 2.6 fm the total density is connected mainly with the valence nucleon densities. Thus three-body calculations reproduce in anatural way the main feature of neutron drip line nuclei — the existence of a neutron halo in these systems (here M. V. Zhukov et a!., The Borromean halo nuclei ‘He and ‘‘Li 173
6He spatial correlations
R(fln~core)f 0 0
Fig. 4. Correlation density plot for the ground state of ‘He in the (nn) and (nn)a variables. in particular for 6He). The presence of three-body correlations are reflected in these densities, but less strikingly than in the momentum-space correlation densities to be discussed below. Thus the slope of the neutron density is substantially steeper than that defined by an exponential tail of the WF with half the two-neutron separation energy as often used in a shell model treatment. This is a natural consequence of the three-body model where the WF asymptotic behavior is given by exp(— 1~p),,~= ~ where S is the two-neutron separation energy (i.e. Ibinding energyI). Note that the numerical calculations show a power-law behavior of the valence nucleon density in the small r region (—~r” with y —‘ 2) reflecting the Pauli principle influence. This is easy to understand in a shell model picture where the valence neutron are situated mainly in the p-shell. A fascinating aspect of the neutron drip line nuclei is the possibility to explore correlations between the particles in the neutron halo. We have already in table 6 given decompositions of the wave function in terms of the hypermoment K, which is an approximately good quantum number with K = 2. In principle these decompositions contain all necessary information about particle correlations in 6He. Additional insight is gained by plotting the correlation density (fig. 4) (which is the probability to have definite distances between the particles in the three-body system) defined by
P(r~~,r(flfl)C) = ~ 2J + l~Jd~ 3d5’3 I ~jM(X3, Y3)I, (14) where ~ and r(flfl)C are the neutron-neutron separation and the distance from the cc-core to the (nn) CM respectively. The correlation plot (fig. 4) exhibits two prominent peaks, a di-neutron-like peak with the two valence neutrons located together outside the cc-particle (r~~< r(flfl)a) and a cigar-like peak with the valence neutrons positioned on opposite sides of the cc-particle (r~~> rtnnia). While the former is smaller in extension than a free deuteron, the latter is larger. Qualitatively, the same result was obtained in CSF and VA [35] approaches. The origin of these configurations is the so 174 MV. Zhukov ci a!., The Borromean halo nuclei ‘He and ‘‘Li
called Pauli focusing quantum effect [77], caused by approximate dominance of a single K = 2
(L = 1 = 0) component in the framework of the HH method (the lowest K = 0 component is
suppressed by the ncc s-wave “Pauli core”, see table 6). The K = 2, S = 1 component gives rather small and uncorrelated contribution between the two peaks in fig. 4. It is necessary to stress that these correlations are connected mainly with taking the Pauli principle between the nucleons of the cc-particle and the valence nucleons into account. The result is almost independent of the exact way the Pauli principle was included. The difference in heights of the peaks (fig. 4) is mainly due to the nn interaction making the di-neutron configuration more probable. Such correlation patterns are washed out in plots like fig. 3 of the usual radial density as well as from the average geometry characteristics (table 7) which are highly integrated characteristics. Only correlation experiments with registration of two fragments can explore details of these correlations. The correlations are exhibited with best possible clarity in the chosen Jacobi coordinates, other choices of coordinates wash them out to various degrees. The shapes (e.g. the asymmetry of the peaks) carry information about the underlying correlations in the system (mainly connected with details of the nn and ccn interactions).
4.3. High energy 6He fragmentation and transverse momentum of constituents Fragmentation experiments with the exotic nuclei of interest as beams, appear to be the best (and perhaps the only) way to really explore the correlations between their constituents. The necessary information about the momentum distributions (and momentum correlations, see next section) of the constituents of 6He is obtained by transforming WF (3) to momentum representation. For the HH method this gives formally an essentially identical WF of 6He in the conjugate momenta Pnn,Pa [76]
— V ~jrLS ( ~na, ~ ly j(-)P~ y 1 v ~‘JM~Pnn,Pa) — ~ KL - ~5)5~~’SiJM~TMT’
= iKp2 ~~1/2 x~1~(P)JK+2(PP), (15)
where pn 0 andPc, are the Jacobi momenta of the nn and (nn)cz subsystems (conjugated to the Jacobi 2 = pc,~+ p~,.The HH basis functions depend on the angular variables of thecoordinatessix-dimensionalpx3,y3) andvector,p JK+ 6He are calculated without any additional2 is afreeBesselparameters.function. Thus the momentum distributions in Detailed experimental data have previously not been made available on the cc-particle transverse momentum distribution from 6He fragmentation on light targets at high energy, although they have existed. Only a Gaussian fit to the data was published in [5]. Recent experiments [20] have however shown that the cc-particle transverse momentum distributions from 6He fragmentation measured on a C target at energy 400 MeV/A with good statistics, exhibit a two-Gaussian component structure (as for ~ case). Since the momentum kc, of the cc-particle in the 6He CM frame is associated with the Jacobi momentum i’ 2 (kc, = — 2pc,/.,/i), a comparison with the experimental distribution of cc-particles in kc,x can now be performed if one integrates the squared WF (15) over variables which are unobservable in the experiment. As discussed in the introduction, this comparison assumes a direct MV. Zhukov ci a!., The Borromean halo nuclei ‘He and “Li 175
1000 6He breakup 800 400 MeV/A
600 0) C C.)0 400
200 4 a.
0 . -250 -150 -50 50 150 250 4He transverse momefltum (MeV/c)
Fig. 5. The a-particle transverse momentum distributions. Experimental data are taken from [20].
Serber type break-up. The inclusive cc-particle transverse momentum distributions is now cal- culated as
dN/dp~ Jdp~dp~dpnnI~(pnn,pa)I2. (16)
In [76] only the WF components with K = 2 and K = 0 were used, which allowed the majority of integrations in eq. (16) to be performed analytically. The result of the calculation is given by the solid line in fig. 5. The results obtained for the GPT nn-potential and for the one with Gaussian shape, are very close to each other. The WF components with K = 2 are decisive, the component with K = 0 has, practically, no influence on the distribution. This means that the rest of the 6He WF components (with K > 2) which have a summed weight of only ‘~ 3—4% cannot noticeably change the results. The theoretical results (see fig. 5) are close to the recent experimental data [20] especially in the region Ik~I 100 MeV/c. Thus, the strict three-body calculation (without any free parameters except total normalization to experimental data at the point k~= 0) gives a good description of the experiment in the region which is statistically most significant. The good agreement argues for the applicability of the Serber treatment for this case, a heavy fragment and high energy. As can be seen from fig. 5 the strict three-body result cannot be fitted with just one Gaussian. The curve has a characteristic two-Gaussian component structure (narrow in upper part and broad in lower part) which reveals the peculiar structure of 6He, strongly supported by recent experimental data [20]. From the three-body WF (15) the momentum distribution of neutrons mayalso be calculated. In the 6He fragmentation, valence neutrons may be emitted, (but also neutrons from the cc-core). To calculate the valence neutron momentum distribution from WF (15) a transformation from p,,,,,pc, to another set of Jacobi momenta pc,~,p,, of the ncc and n(ncc) subsystems is performed. It can be done using the transformation of the HH basic functions (12) via the Raynal—Revai coefficients 176 MV. Zhukov eta!., The Borromean halo nuclei ‘He and ‘‘Li
[74]. To obtain the neutron momentum distribution from the cc-cluster in 6He, one should: (i) calculate the neutron momentum distribution in 4He; (ii) take into account the cc-particle motion in 6He. The details of these calculations are described in [76], where also the results for neutron transverse momentum distributions in Serber approximation are shown. The theoretical studies show that the transverse momentum distributions do reflect to some extent (mainly through the widths and shapes) the structural peculiarities of this nucleus. However, this structure appears more clearly in exclusive distributions, for example, in the ccn and nn momentum correlation functions.
4.4. 4He—n and n—n momentum correlations
The transverse momentum distributions of the cc-particle and neutrons have been obtained using the WF (15) in momentum representation. The corresponding general formula for the cc-particle
and neutron momentum correlation function [76] in the CM system (with J = M = 0, L = S and
1,, = ~ = l for the ground state of 6He) is:
d2N ______— ~ ~rL ~ ~ ~iL’ ( \ t~j sJ r ii ~ (i+I’)/2 ,~ , — ~-1~ Kl’~P)’-~K’!”~P)~’K11”K’l’LZ~1 — ~‘~2’~fl KK’LL’I!’
(1 + 1/2), (1 + 1/2)~’, 1 ~(!‘ + 1/2), (1’ + 1/2)~., v11’ I x P(K
21)/2 k — I tK’—21’)/2 y_Z — I LL’~Y
NK1 — {(2K + 4)[~(KF(~(K+3))— 2l)]![~(K + 21) + 1]!}u/2 ‘ 17
K~.(y)= öLL (21 +1)(21’ + ~~_1)i[
where kc, and k~are the cc-particle and neutron CM momenta. They are connected with pc, and Pnn of the previous section by (see footnote in section 2.1)
kc, = — (2/~/~)pc,, k~= (,~/~)_1p~~— (1/2)kc,. (18)
(1+ t/2).tI+ 1/2) . 2 2
Again P(K21)/2 (2z — 1) are Jacobi polynomials, z = pc,/p ,y is the angle between~ andpc,. F is the usual gamma-function, W(jll’L;1’!) are Racah coefficients and P~(cosy) Legendre polynomials.
In the calculations only terms with K = 0 and K = 2 were taken into account. Other terms of the WF4a/dEc,dQadEndQnexpansion do notin theaffectlab. thesystemresultsfor asignificantly.6He beam atFigure300 MeV/A6 showsandtheforcalculationdetecting theof
cc-particled and neutron in coincidence at Oc, = = 00. No data exist so far. This distribution has a characteristic complex shape which reflects the exotic structure of the neutron halo in 6He. Two prominent peaks are clearly seen, which correspond to the “cigar” configuration, and also two small peaks corresponding to the “dineutron” configuration. The suppression of the “di-neutron”
configuration is caused by interference of two hyperspherical harmonics with K = 2 and K = 0
(both of them have L = S = 0). In spite of the small contribution of the term with K = 0 to the total WF it appreciably affects the form of the momentum correlation. The reason is the kinematical enhancement of the CM low momenta part of the cross section, due to the transition from CM to
lab-frame system. There is no contribution from the term with K = 2, S = 1 because neutrons and MV. Zhukov et al., The Borromean halo nuclei ‘He and ‘‘Li 177
-‘1
5~
0
\
‘9 ~. ~ __ , ç,G:c
‘9 ~7.o i:~- ~ ~
Fig. 6. The na correlation momentum distributions calculated for fragmentation experiment at the ‘He beam energy of 300 MeV/A.
cc-particles are assumed to be detected at the same angle (in this case the angle ~‘ = 0). If the realistic wave function is replaced by a simple Gaussian, a rather dull featureless correlation plot results. This striking difference may be possible to explore experimentally.
5. COSMA — method for rapid assessment of “true” wave functions for Borromean nuclei
5.1. Motivation for COSMA
In the previous sections we have described results of strict 3-body cc + n + n calculations for 6He, the lightest Borromean nucleus. We now like to continue with a discussion of “Li, the principal case, in a (9Li + n + n) model. Contrary to 6He however, where the cc—n interaction is known quite well, the 9Li—n binary channel and its interaction is insufficiently known and the subject of current investigations. This prevents for the moment reliable strict fully blown calcu- lations of “Li; they can technically be carried out, but the result depends very sensitively on the potentials used (we will return to this point later). Thus, although strict calculations are possible it would be more economic to have an approximate procedure where test WFs of transparent structure could be tried out. COSMA, described below is such a procedure. While 4He is a 0 + nucleus, the ground state of 9Li has quantum numbers 3 / 2, and so has ‘1Li.
Our discussion of 1’Li will (like for 6He) assume J~= 0 + for the active part of the wave function included in our 3-body formulation.
5.2. The model and test for 6He
The COSMA model draws its inspiration from the strict few-body formulation COSM [58] discussed in section 2.4 for systems consisting of valence nucleons coupled to a core. This scheme is 178 M. V. Zhukor’ ci a!., The Borromean halo nuclei ‘He and ‘‘Li translationally invariant and combines advantages of the shell and cluster models. Unlike COSM, where the Schrödinger equation with pairwise interactions was solved for both wave functions and binding energies, we construct a similar phenomenological wave function with a few free parameters, determined from essential experimental data. The resulting wave functions are sub- sequently used to calculate other observables. For the radial motion of n + core, phenomenologi- cal oscillator wave functions are used. To simplify the following formulae we consider the most transparent case with total spin S = 0 for the two valence neutrons and a total orbital momentum L = 0 for the three-body system. 9Li) is assumed inert and its degrees of Again, in a c + N + N model, the core nucleus (here c = freedom left out of the formulation. In such a case the relative motion wave function (WF) for the system has the form
= >C~jInl(r,~),nl(r 2~);0>. (19)
Here n and 1 are nodal and orbital angular momentum quantum numbers for a neutron relative the core. The translation invariant coordinates ~ i = 1,2 give the positions of the two neutrons with respect to the core. For simplicity we leave out the spin-part of the wave function. (It should be pointed out that all particle coordinates are measured in r0 units, where r0 is the oscillator parameter (length)). The coupled two-nucleon oscillator wave function is denoted as I ni, ni;0)’, in agreement with the definition of [78],
Inl(r,~),nl(r2j;0>=~
where Inlm> is the usual one-particle oscillator function [78]. As an example, the Op (n = 0, 1 = 1) oscillator wave function looks like 14(8/3)’12mexp(—r2/2) Y lOim> = i(’ 1m(1) (21)
COSMA deviates from the usual oscillator shell model at a few points. Firstly,9Li) core.theSecondly,neutron thecoordinatesoscillatordoparameternot refer to(r the CM of the (“Li) system, but to the CM of the ( 0) for the valence neutrons is not connected with the structure para- meters (i.e. the size of the average field) for the nucleons belonging to the core. In addition to r0, the fitting parameters of the model are the mixing coefficients C,,, of the wave functions. The first condition which we shall use to fix the free parameters is the requirement that the model wave function (19) describes correctly the r.m.s. matter radius of the nucleus; for “Li,
R(’ ‘Li) = 3.2 ±0.1 fm [2].*) Taking into account that the total wave function has the product form & ® W(r,~,r2~)where & describes the nucleonic motion9Li,within the core, it is easy to obtain the relation connecting the matter r.m.s. radii of ‘Li and
11R2(’1Li) = 9R2(9Li) + ~[10 *)Although r.m.s. matter radius is not directly measured it seems to be in a one to one correspondence with the cross sections [1,2]. MV. Zhukov eta!., The Borromean halo nuclei ‘He and ‘‘Li 179 Here ~ means the corresponding quantity averaged over the wave function ~ r (19). 9Li) = 2.32 ±0.02 fm is taken from experiment [2]. 2~) whileAssumingR( that the c-fragment momentum distributions from reactions at high energy corres- pond to the core momentum distributions in the projectile nucleus, we will also use the transverse momentum distribution [5] dN/dp~of 9Li from “Li fragmentation at E = 790 MeV/nucleon on carbon target to fix the trial WF (19). The transformation properties of harmonic oscillator WFs are very convenient for this purpose. In the CM frame the coordinates of the core r~and valence neutrons r• = r~+ r,~are expressed in terms of r,,, and r 2~in the following way 1 (A~-~-1’\ 1 1 ‘\ r~= —(r,~+ r2c)A + 2’ r, = r,e~A + 2) r2c~A + 2)~ (23) —The corresponding relative variables are given by: r,2 = (r,~— r2~), r(,2)C = (r,~+ r2~)/2. (24) With9Li + nthe+ helpn systemof theseusingformulaethe WF ofit eq.is easy(19) afterto estimatehaving fixedall geometricalall fitting parameterscharacteristicsin it. of the To obtain the 9Li momentum distribution we need the WF(19) in momentum representation. The momentum representation of (19) is essentially straightforward. The one particle oscillator wave functions have in momentum representation just the same form as the initial spatial ones, to within a phase factor (—i)’. When a COSMA two-particle configuration is transformed it is again form invariant to within a phase (— 1)’, but note that the proper conjugate momentum arguments are the CM momenta k, and k 2, and not those relative to the core. Shell model thinking is here easily misleading. Thus instead of the coordinates r1~we just9Liputtransversethe variablesmomentumk, (i = 1,2).distribu(All- momentation) is proportionalare measuredto thein PodN/dp~= h/movalue,units).forThewhichdN/dk~we have(the the formula ~ JdP~dp~dPnnI~(k,,k 2)I2, (25) where cli(k, , k2) is the Fourier transform of ~1’(r,~, r2,~).The transition in ~t(k, , k2) from momentum coordinates k, , k2 to -the Jacobi momentum coordinates can easily be done using Talmi— Moshinsky transformation coefficients [78]. Before proceeding*) to calculations of momentum distributions for “Li we try to limit the number of free parameters in our problem invoking simple physical ideas. Three L = S = 0 candidates I, II and III for the physical nature of “Li have been selected: *) It should be mentioned, however, that there is no straightforward connection in COSMA between the neutrons occupying the (Os) shell in ‘Li and the valence neutrons in the (Os) state, because of quite different r0 parameters for internal and valence neutrons. The small probabilities to find one valence neutron (or two valence neutrons) inside the ‘Li core, as given in table 9, show that the Pauli principle is obeyed with good precision.2 wave functionCalculationsfor thebothouterofneutrons,the ‘Li transversemade orthogonalmomentumto thedistributioncorrespondingand of(Os)‘Li—n2 statemomentumfor the innercorrelationneutronswithby aaddingCOSMAa correspondingpure (Os) correction term, show no significant changes in the calculated distributions. The weight of the orthogonalizing term is about 3%. 180 M. V. Zhukov ci al., The Borromean halo nuclei ‘He and ‘‘Li (I) The outer neutrons are mainly in (Op)2 as standard shell model prescribes with a small admixture of (Os)2. The COSMA (Os)2 state is not forbidden by the Pauli principle because the COSMA Os state for outer neutrons and the core Os state have rather different oscillator radii (—~4.88 fm and —~ 1.8 fm) hence they belong to different full sets. (II) The outer neutrons are in an admixture of (Op)2 and (Od)2 configurations. The Od orbital is the next one in a standard oscillator basis and might in accordance with some speculations, lie even lower than the Op orbital due to spin-orbit forces and residual interactions. (III) The outer neutrons are mainly in (Op)2 with an admixture of (ls)2. Explicitly, = cxIOl,01O> +flIOO,00;O> , (26) = ccIOl,O1; 0> +ThO2,02;O> , (27) ¶P(r,~,r 2~)=ccIOl,01;O>+fl~1O,lO;0>. (28) Taking into account that a common phase multiplier does not contribute to any observable value and that the WF has to be normalized, we choose the following parametrization cc = cos 0, f3 = sin 0 (29) This leaves only one free parameter: 0 (because for fixed 0, r~is determined simply by (22)), which will be fitted to the experimental data. Carrying out all necessary transformations and integrations with WF9Li(26—28)transverseinsertedmomentumin formulaedistribution(22) and (25)canonebe writtenfinds, foras all choices under consideration that the = W~(s)= cos2O W 2O Wb(s) + 2cosOsinO M’~b(s), (30) 5(s) + sin where s = k~ro/197.33/,.,/~,k~is measured in (MeV/c) units. The additional expressions needed to pin down r 0 and 0 are now 2(’’Li) — 9R2(9Li) (I) r ( llR 20) + ~(\/~/4)sin0cos0) 0 = ~ + cos Wa(S) vr’~[Woo(s)+ W 1o(s)] (31) Wb(s) = Woo(s) ~‘~b(5) = 1-~V~~W00,o(s) 2(’’Li) — 9R2(9Li) (II) r ( 11R2 0) + ~~sinOcos0) 0 = ~ + sin W~(s)= ~( Woo(s) + W,o(s) , Wb(s) = ~ W 00(s) + ~ W2o(s) + ~ W02(s) + ~ W,o(s) , (32) MV. Zhukov et a!., The Borromean halo nuclei ‘He and “Li i8i Wab(S) = ~ W0010(s) + ~ W,~(s), 2”’Li’—9R2’9Li’ \1J2 (III) r / 11R 0 = I 2 0)~ + -?r(2/.,/~)sin0cos0!‘ Wa(S) = ~(W00 + W,0) , \~(~ + sin ~ = ~ W,~ 2~(s)— (1/6~) W0o10(s). -- The functions Wm(S) in eqs. (31)—(33) are explicitly given by: 2, W00(s) = ~e22; W,0(s) = ~ — ~2 + 5/4)e~ Wo,(s) = ~ + l)e~2 W 2 + 2)e’2, 02(s) = ~ + 2s W 6 + ~s4 — ~s2 + q-~)e~2, (34) 20(s) = 12(~8 — 6s W — 1 L.J~ _s2 OOiO(S)~~( 22s — 1 )e w1020 S ~— ~16 — 144~ t728 S — 16 e Having fixed all parameters in WF(26)—(28) we also have the possibility to calculate the valence neutron momentum distribution without any free parameters. For the neutron transverse mo- mentum distribution dN/dk~in CM it is also easy to obtain ageneral formula for all WFs (26)—(28): 20 Wa(s) + sin2 0 Wb(s), (35) = Wa(s) = cos where s = k~ro/197.33,k~is measured in (MeV/c) units. For the three cases (26—28) W~(s), Wb(s) are given by (I) W 5(s) = W01(s) , Wb(s) = W00(s) , (36) (II) W~(s) = W01(s), Wb(s) = W02(s), (37) (III) Wa(s) = W01 (s) , Wb(s) = W10 (s) , (38) where W00(s), W01(s), W,0(s), W02(s) are defined by expressions (34). It is worthwhile to mention that the dN/dk~distributions do not depend on the sign of 0, only on its value and on m0. 182 A’LV. Zhukov ci a!., The Borromean halo nuclei ‘He and ‘‘Li dN/di~,a.’~, ~ g) dNJdA~.~.,~t.’tt.- - ISO —Th 0 75 150 -150 -75 0 75 150 k,~,HeV/c A~r,t1~’/C Fig. 7. Comparison between COSMA calculations (solid line) and strict calculations (dashed line) for ‘He. (a) a transverse momentum distribution, (b) neutron transverse momentum distribution. which experimental data on 4He momentum The COSMA has been tested in [79] for 6He for distributions are known [5] and for which precise three-body calculations of both 4He and neutron momentum distributions have been carried out. In the case of 6He the situation is particularly simple, for there is a pure (Op)2 configuration that fits the r.m.s. radius and also gives a quite satisfactory fit to the momentum distribution of exact three-body calculations. This is displayed in fig. 7. The schematic model accounts rather well for both theexperimental data and for the features of the more advanced theory, indicating this to be a viable approach also for 11Li and heavier Borromean systems such as ‘4Be and ‘7B. 5.3. Results of calculationsfor “Li For the distributions corresponding to the simplest model where the valence neutrons occupy pure (Os)2, (Op)2, (Od)2 or (is)2 configurations relative to the core there are no free parameters in the WF, the r 0 parameter is fixed9Liimmediatelymomentumbydistributions(22): r0(Os) can= 4.88befm;calculatedro(Op) = straight3.78 fm;- forwardlyr0(Od) = r0from(ls) =(31)—(33)3.2 fm. withoutHence theany additional parameters. The results have low sensitivity to small variations of these radii. None of the pure configurations describe the experimental data for all k~well (only the (Op)2 configuration comes close to the experimental data) [80].In the two-body subsystem (9Li + n) the Os component is in a shell model suppressed by the Pauli principle. It could be expected that this component is suppressed also in the three-body system. The results of strict three-body calculations of 6He explicitly show a small contribution (4—5%) of such a component [36]. The results for configuration admixtures — case II is shown in fig. 8, are rather sensitive to the choice of the fitting parameter (0) especially to its sign. Not only the width of the distribution but also the curve itself changes significantly if the sign of the mixing amplitude is reversed. Cases I and III give fits of comparable quality to that of II, see [80]. MV. Zhukov eta!., The Borromean halo nuclei ‘He and ‘‘Li 183 -- —200 200 9Li transverse momentum distributions of COSMA for case ii. The experimental data are taken from [5]. Curve 1 is Fig. 8. The calculated for the parameters r 0 = 3.44 fm, 0 = arccos( —0.6). Curve 2 is calculated for the parameters rc, = 3.31 fm, 0 = arccos( + 0.6), notice the reversed sign. For the COSMA I fit the parameters are (3.84 fm, —0.97) while for COSMA III they are (3.68 fm, — 0.9). The corresponding curves are not shown. Table 8 Table 9 Calculated r.m.s. distances between2, (Op)particles.2. (Od)2 The probabilitiesinside and ofoutsidefindingofvalencethe core.neutrons Diagonal caseand is(Is)for2 configurations.the pure (Os) Case ~r P°~°’ Case r,, (fm) r~ 0(fm) ~ (fm) I 0.025 0.001 4 0.950.975 Diag.I 5.986.01 8.468.31 4.234.33 IIIII 0.0130.05 —.3x1O0.005 0.91 II 6.1 6.71 5.1 III 6.03 7.78 4.61 The r.m.s. distances between valence neutron and core, between the two valence neutrons and between the core and CM of the two valence neutrons are given in table 8 for pure configurations and configuration mixtures I—Ill. For cases I and III there is good correspondence with the Values of [60], obtained in extended cluster shell model calculations and for case II they are close to the values of [63].In table 9 the probabilities to find only one valence neutron inside the core (9Li)—Pr and the probabilities to find the two valence neutrons outside (inside) the core (9Li)_P0 t(P~’)are given for I—Ill. 2~1 The probabilities P~are close to values easily estimated from a core + n relative motion wave function [60] and smaller than P’,” —~ 0.16 obtained in [63].It should be noted that the probability to find two valence neutrons outside the core (9Li) in ref. [63]is about 0.80. The small probabilities to find one valence neutron or both valence neutrons inside the 9Li core show that the ~Pauli principle between the core and the neutrons is obeyed with good precision. The internal transverse momentum distributions of the neutrons are also contained in [80]. It should be mentioned that the pure (Os)2, (Op)2. (Od)2 configurations, which describe rather well the upper part of the 9Li momentum distribution, give quite different widths (but approximately the 184 M. V. Zhukor ci a!., The Borroinean halo nuclei ‘He and ~ same forms) of the transverse neutron momentum distributions. For the pure (Is)2 state it has a very peculiar structure. 5.4. “Li wave function structure in the framework of COSMA: spatial densities and 9Li—n and n—n correlated momentum distributions COSMA seems to be rather useful for simple and quick estimations of the WF of neutron-rich nuclei. In the framework of this model it is possible to calculate not only inclusive momentum distributions, but also the correlated momentum distributions. We showed in the last section that for “Li ambiguities are present: different configuration compositions which give different neutron halo structures for the ‘1Li nucleus, describe equally well the 9Li inclusive transverse momentum distributions and simultaneously the r.m.s. matter radius of t1Li Thus correlation experiments rather than inclusive experiments, seem required if detailed features of the neutron halo structure are to be disclosed. We will briefly discuss how 9Li—n and n—n correlation functions have been calculated within COSMA for all versions of the “Li trial wave function discussed previously. A measure for spatial densities of”Li in the framework of the three-body approach is defined by inserting VP(ri~,r in2 (14). These distributions are, for all choices (I—Ill), given in [81]. The spatial densities for2~different)I W differ significantly from each other in spite of almost equally good descriptions of inclusive momentum data and radius. Each case has a specific neutron halo structure. We will return to and illustrate this point in the context of strict calculations in section 6. For W (I) the density looks very similar to the 6He case (see fig. 4) with obvious scaling. Pronounced “dineutron” and “cigar” type configurations are again seen. In case III, these configurations have changed in relative magnitudes and forms, while a three-peak structure is present for the configuration mixture II. These specific correlations between the particles are almost washed out in the inclusive particle transverse momentum distributions, which are highly integrated characteristics. The CM momenta are the following combinations of the lab momenta of the core P~and the valence neutrons P 1, P2: = P, — (1/l1)P , P, k, + (l/ll)P , = P2 — (1/ll)P, P2 = k~+ (1/1i)P, (39) P—P,+P2+P~, P~—(9/11)P—k,—k2. 9Li) momentum correlations are now simply given by the following The n—n and core—n (c = expressions: P~~(P,,P 2, (40) 2=) VP(P, — (i/11)P, P2 — (1/l1)P)1 PCfl(Pl,PC) = ~(P, — (1/11)P, (1O/11)P— P~— P,fl~. (41) Figure 9 shows the n—9Li momentum correlations for the cases I—Ill from [81]. All distributions have characteristic features making measurements of such distributions interesting and important. For convenience the figures are plotted against the lab energies of the corresponding fragments. MV. Zhukov ci al., The Borromean halo nuclei ‘He and ‘‘Li 185 -~ ~OP’.-I0S~ k ~t. Op -ItS) ~ “5.,, ~ .‘~ ~~<. ~ C;’- .d~ 4~?0 ~ ,.‘~ Fig. 9. The ‘Li—n correlated momentum distribution at zero angles in the laboratory frame at E(’ ‘Li) = 300 Mev/nucleon for the COSMA cases I—Ill. The total momentum P corresponds to a ~ projectile energy of 300 MeV/nucleon,9Li and neutron)that ofwerethe bothpresentchosenDarmstadtto be zerofacility.degrees.The registration angles of the emitted particles ( The neutron halo structures of the WFs of cases I and III, which are hard to distinguish in inclusive experiments (see [80]), appear to be separable in correlation experiments, their correlated momentum distributions having quite different features. The differences are the following: The central peaks of the case I are clearly separated because of a small weight (—~0.06) of the (Os)2 component, the only one which contributes between the two central peaks. A pure (Op)2 config- guration gives a valley between the central peaks down to zero. For case III the (is)2 configuration has a weight about 4 times that of the (Os)2 configuration in case I. This fact together with a reduced weight of the (Op)2 configuration results in a complete filling of the gap between the two central peaks, hence only a composite central peak is seen in fig. 9 for case III. The momentum correlations for both particles detected at zero degrees, display the most pronounced features. Additional test calculations for the Darmstadt energy show that the specific structures of the distributions are washed out if the angle of neutron registration is more than 2°in the laboratory frame (keeping the core registration angle at zero). The n—n momentum correlations are nearly replicates of those for 9Li—n, but rotated by approximately 90°and slightly distorted compared with 9Li—n in accordance with the coordinate replacement in formulae (40) and (41). In a full three-body approach where an equation of motion is solved, it is obvious that the weights of the different components in the trial wave function (19) depend essentially on the 9Li—n potentials put into the calculation. Lacking sufficient experimental information on 186 M. V. Zhukot ci a!., The Borromean halo nuclei ‘He and ‘‘Li these potentials from 9Li—n scattering COSMA has proven useful to explore the approximate structure of ‘Li. Finally we mention that in refs. [60,61] (connected with the 9Li—n potentials used) the main contribution to the “Li WF is the (0p 112)Lo configuration containing both S = 0 and 1. From the point of view of the conventional shell model this seems most reasonable. COSMA also allows us to consider this choice of trial WF without introducing more fitting 1parameters.Li radius, butThusthe resultingwe have investigated9Li transversethemomentumpure ~/2 distribution)~= ~ configuration.is essentiallyWe canbroaderreproducethenthethe ‘experimental one. Inclusion of configuration admixture could give better agreement. Thus the 9Li—neutron and neutron—neutron momentum correlations, as calculated in COSMA, seem able to distinguish between different configuration mixtures. “True” wave functions, cal- culated by solving the dynamical three-body problem should also be tested against these observ- ables which are now being measured. 6. Strict three-body calculations for “Li A number of calculations of the ‘‘Li ground state have already been performed [29,30,63,82,83,61,60,85], treating ‘‘Li as an inert 9Li core plus two valence neutrons. The importance of the correlation energy of these neutrons means that mean-field approximations [29] have restricted validity, and their weak binding means that shell models (e.g. [82,30, 60]) require many orders of hw excitations to be accurate. The best calculations will therefore treat the three-body dynamics explicitly. For example, we wish to avoid assuming any dineutron moving as a unit. Rather, we want to see to what extent the dineutron configuration arises naturally from the dynamics of the model. The weak binding also implies that observables such as matter radii will be strongly dependent on changes in the binding energy, and hence on approximations in solving the three-body models. All the three-body modelswill also lead to re-examination of the various model-dependent forms for the neutron density used in the extraction of “experimental” r.m.s. radii from nuclear [2]and pion-exchange [86]scattering experiments. 6.1. Effective interactions The approximations made when “Li is treated as a three-body problem are mainly the neglect of the degrees of freedom of the 9Li core. This may seem a serious approximation in the light of Hartree—Fock calculations and of the fact that the first excited state of °Liis at 2.6 MeV. Compared to the similar 6He—4He problem, the core of ‘‘Li seems soft. This can be compensated, however, because the polarisation of the core is already includedif we use an effective core-neutron potential which reproduces scattering data. We mainly assume that this potential has a local equivalent of a Woods—Saxon shape. We also ignore the small off-diagonal effects on one neutron of the core polarisation due to the other one. In principle this would be (at least) a three body interaction. For the potentials in the three-body model we shall therefore use those local potentials which reproduce scattering phase shifts, i.e. we use for Va,, between the neutrons the super soft core potential SSC [56], or the Reid soft-core potential RSC [57]. Some approaches, e.g. [61],need to modify V~,,inside the core. Such modifications are usual and pretend mainly to simulate effects of the exclusion principle and of polarisation of the core matter. These modifications are generally derived from nuclear matter calculations and therefore very uncertain when applied to light nuclei MV. Zhukov eta!., The Borromean halo nuclei ‘He and ‘‘Li 187 where a surface polarisation may dominate. We shall here stay with the free nn interaction, sincewe shall treat the Pauli principle in a more fundamental way. The main additional effect of the Pauli principle is now not the exchange interactions in Va,,, but the exclusion of filled orbits, as in [60,61]. The main diagonal effects of core polarisation are included in our calculations by using effective core-neutron potentials characterised by their scattering effects. For the determination of I/a’, the scattering data are sparse, and not unambiguous. Some experiments [87, 88] suggest a resonance in ‘°Liat —~ + 0.8 ±0.25 MeV, whereas others [89] suggest a resonance at ‘-~ + 0.2 ±0.2 MeV. We will see below which energy leads to the best description of “Li. 6.2. Faddeev (CSF) and HH calculations Thethree Faddeev components l/J~~~(if = nn, ci, c2) of(7) are represented in Jacobian coordinates. The l~and l~are then coupled to the intrinsic neutron spins, using eitherjj or LS coupling. The total angular momentum is set to J = 0 and T = 1, assuming that the total angular momentum of “Li 9Li core only as a spectator. (~) is due to the The CSF equations are solved by the method of ref. [84], with boundary conditions giving exponential decay in the hyperradius defined previously, and with the constraint of orthogonality of W to a set of occupied states of core neutrons (using the method of [53] rather than of [34]). The first class (I) of calculations will follow [61, 85], and assume that the ‘°Liresonance is of a OP1/2 type, with the OP3/2 existing as a bound state of 9Li (S~= 4.1 MeV). The 9L1 core is now assumed to have a full set of Op 312 neutrons, and the three-body wave function11Li waveis orthogonalizedfunction will beto predominantlythe occupied Os(Op,/2)and OP3/22, whichstatescorrespondsof neutron-core(for harmonicmotion. Theoscillators)resultingto a linear combination of + \/~7i3p, states of neutron—neutron motion. This implies that the “Lj ground state looks like a di-neutron with a probability starting at about 33%. The class I calculations adopt the prescription for V~of choosing the central and spin-orbit strengths to give °P3/2 and OP,/2 eigenstates at specified energies. The potentials consist of a central Woods—Saxon potential and a spin-orbit term of derivative Woods—Saxon shape. We have kept the °P3/2 at — 4.1 MeV, the separation energy of neutrons in 9Li, and tried various OP1/2 resonance energies between 0.2 and 0.7 MeV (c.m.). The different potential geometries are given in table 10, along with the strengths required to fit the given OP3/2 and OP 1/2 eigenStates. The freedom to choose the geometries means that we can see which interactions give the best description of 1’Li. The second class (II) of calculations follow [82,86] in taking the pairing rather than the spin-orbit forces as dominant in the 9Li core. A Cohen—Kurath type of calculation for 9Li gives [82]a ground state that is 93% of the symmetry [f] = [432],which can couple only with the spatially symmetric ([f] = [2])neutron pair to form “Li with a closed shell structure. This means that, in LS coupling, the two valence neutrons are almost entirely in a ~ state, with the S = 1 configurations being blocked by the core neutrons according to the Pauli principle. The neutron- core spin-orbit force can therefore be omitted, as it is largely damped in the interior, and does not destroy the coherence of the \/~7i(Op3/2)2 + ~Ji7~(Op,/2)2 state. It causes merely a weak splitting of ‘1Li states by introducing small amounts of S = 1 configurations exterior to the core. Because the Vt,,, potential is not now constrained by a resonance position, we adjust its geometry and strength (table 11) to fit plausible properties for the 1’Li ground state. 188 MV. Zhukov ci a!., The Borromean halo nuclei ‘He and ‘‘Li Table 10 Potential geometries for ~ required central and spin-orbit strengths, and “Li binding energies from class I calculations. A is 2E~, 12— E. All calculations used the SSCC nn potential, except9Li,0E4withwhichR followed ref. [61] (see text). The matter r.m.s. radius Rm uses a radius of R2 = 2.32 fm for 2_ defined by llR~,= 9R~+ 2R~,. V2_ geometry - E(p,2) V E(”Li) A R20 Rm r a (fm) (MeV) (Mev) (MeV) (MeV) (MeV) (fm) (fm) 09 1.27 0.67 0.20 39.40 35.20 —0.262 0.66 5.22 3.05 M8 1.1 0.75 0.20 48.34 36.92 —0.252 0.65 5.36 3.10 Q9 1.45 0.75 0.20 31.70 43.05 —0.296 0.70 5.47 3.14 010 1.27 0.67 0.25 39.33 35.77 —0.164 0.66 5.40 3.08 M9 1.1 0.75 0.25 48.23 37.64 —0.141 0.64 5.61 3.18 Q10 1.45 0.75 0.25 31.63 43.76 —0.199- 0.70 5.68 3.20 0E4 1.27 0.67 0.70 38.79 40.67 —0.127 1.57 5.27 3.07 Table II Potential geometries for V~,,,required central strengths, and ‘‘Li b,nding energies from class II calculations. All calculations used the SSC nn potential. Potential geometry E(p,12) V E(’’Li) A R2 R,, r a (fin) (MeV) (MeV) (MeV) (MeV) (fm) (fm) L5D 1.27 0.75 1.7 25.2 —0.198 3.6 5.16 3.04 L6B 1.45 0.75 1.5 19.88 —0.202 3.2 5.34 3.09 L5C 1.27 0.75 1.45 25.6 —0.372 3.3 4.85 2.94 L6A 1.45 0.75 1.33 20.18 —0.348 3.0 5.09 3.02 A third class (H) of calculations takes the same potentials as used in the HH calculations of[83]. They use shallow gaussian potentials from [63](see table 12), so that most of the strength is in the Os channel. This means that the potential supports no occupied states, and would thus be some kind of effective interaction already including a repulsive “Pauli core”. The HH and CSF methods can be directly compared in this case. Results The two-particle binding energies E calculated by the class I models are given in table 10, with the potentials listed there (these results are different from those in ref. [85],because of an error there in the sign of the p-tensor force). Radii p (from eq. (4)) up to 35 fm were included. The number of components {1~,l~} was varied, so as to ensure that inclusion of further components did not change the results significantly. The main components have, in agreement with shell model expectations, l~= l~ 2, with 1x(cfl) = i as the dominant term [60].In table 13 we give the squares of amplitudes of the different angular momentum components in the two bases neutron—(core—neutron) and (neutron—neutron)—core for several class I total wave functions. The dominant role of the P’/2 in the first basis is evident, as is the large p-wave component in the cluster basis. The probability densities 2are shown in fig. 10, for the three different cases (I), (II) and (H). of thewave functions I ~ r~~)I MV. Zhukov et a!., The Borromean halo nuclei ‘He and ‘‘Li 189 Table 12 Potential geometries and strengths for V from [63. 83], and ‘‘Li binding energies from class Z calculations. Potential geometry E(’’Li) R 2 R V (MeV) r (fm) V (MeV) (fm) (fm) Z2B 7.80 2.55 G ‘S, —0.302 6.26 3.39 Z4 7.80 2.55 SSC ‘5, 3P, —0.301—0.308 6.266.25 3.393.38 Z4B 7.80 2.55 SSC ‘S,, Table 13 Percentage probabilities of different partial waves from Faddeev calculations. 0R9 is 09 repeated using the RSC potential, and 0T9 uses the GPT nn potential. = 2 shell model basis: n—(cn) i = 1 cluster basis: (nn)—c E - ______Calculation (MeV) 05,12 00,/2 0P312 S P D 09 —0.262 3.0 92.3 0.82 37.6 55.5 1.4 0R9 —0.271 3.0 92.3 0.79 37.7 55.5 1.4 0T9 —0.296 3.0 92.1 0.82 37.8 55.5 1.3 Q9 —0.296 3.7 89.6 0.9 39.9 54.8 1.2 010 —0.164 3.6 91.7 0.9 38.2 54.5 1.5 Q1O —0.199 4.4 89.0 1.0 40.6 53.8 1.2 0E4 —0.127 6.7 88.5 1.2 42.8 53.4 0.9 In case (II) we see distinct “dineutron” and “cigar” peaks, whereas in case (I) these peaks are joined by the addition of the density for the 3P, component. The case (H) gives a single structureless peak. The currently accepted experimental values for E(1tLi) are — 0.25 ±0.10 MeV [91] or —0.34 ±0.05 MeV [92]. The table 10 also gives the r.m.s. internal and matter radii from the calculations. The experimental Rm is 3.10 ±0.17 fm from [2], or 3.02 ±0.20 fm from ref. [86] using their R 2~= 5.1t~~fm. Table 10 shows that all the class I calculations all easily fit the weaker binding energy and the matter r.m.s. radii together. If the stronger “Li binding energy is required, then a potential with a larger r.m.s. radius might be indicated. It is thus not difficult in these Faddeev calculations to fit the matter radii for the deeper binding energies around 350 KeY, provided an energy of around +0.2 MeV is accepted for the g.s. of ‘°Lirelative to the breakup threshold. This appears to be inconsistent with the results of [61],who find a realistic “Li binding energy is possible with a~ resonance energy of + 0.8 MeV. However, ref. [61]assumes that the nn strength3isP increased somewhat to give an infinite scattering length, and that the repulsive potential in the 1 channel can be neglected. The Faddeev calculation OE4 in table 10 is an attempt to simulate these assumptions. Several class II calculations are listed in table 11, first for the weak and then the stronger binding of “Li. The energy and radii of these calculations are very similar to their class I counterparts, but now that the neutrons are almost entirely in an 1,, = 0 relative state, the binding increment A is larger by at least the 100%/38% factor to be expected from table 13. The class II 190 M. V. Zhukov ci al., The Borromean halo nuclei ‘He and ‘‘Li JiLl spatial correlations: Q5 ilLi spatial correlations: L6A d 0 - .-. rI 4- C’s C’s .~ o 0 L 0 0 10 10 8 8 2 0 0 2 ~~‘(flfl.core)~ ‘° 2 0 0 ~ 8 10 liLi spatial correlations: Z2 0 5- a5~ 4} ~0 5- CS .0a 0 10 8 2 0 0 2 ~~ 15~.cOTe) Fig. 10. Spatial densities of the ‘‘Li ground states calculated by the Faddeev method. The cases are (I) “spin-orbit” case Q10, (II) “pairing” case L6A, and (H) “shallow potential” case Z2. models therefore9giveLi, butthelosesamethebehavioursimple resonancefor “Li, andinterpretationare perhapsofmore‘°Liasconsistenta single-particlewith previous°P,/2 resonance.models [82]for Use of the Reid soft core potential RSC [57] gives at most 10 keV change in the binding energy. The MT13B [93] and the G Vnn potentials give 300 keY more binding in the case I calculations, largely due to the missing P-wave force. It is perhaps surprising that these 3P, potentials are significant in this case, but they do account for around 55% of the class I wave functions. MV. Zhukor et a!., The Borromean halo nuclei ‘He and ‘‘Li 191 6.3. Matter and momentum distributions in the “Li g.s. We plot in fig. lithe radial matter densities for the CSF wave functions compared with those of HH (the Z2 case) and COSMA calculations. The various matter r.m.s. radii from these curves are shown in table 14, along with those of other calculations from the literature. The data with error bars describe the band of densities permitted by a Glauber model fit to reaction cross section data [95]. The HH and Faddeev calculations have the correct asymptotic decay rates at large radii, though at slightly different rates because the angular momentum content are slightly different. The Faddeev density for the valence neutrons has a node near the origin as these particles are in Op rather than s states. The COSMA curve, by contrast, is based on harmonic oscillator basis states, and even though large oscillator radii are used, the densities eventually decay at too large a rate. Overall, we see that these quite different models produce largely the same densities over much of the radial range. This means that other observables will have to be used to distinguish between the models. The other experimental data available at present, are the 9Li momentum distributions in breakup reactions of ‘‘Li on different targets [5]. To calculate this we shall first use the Serber model: an extreme variant of the sudden approximation with plane waves, valid for high projectile energies: the momentum distributions in CM system of ~ is taken to be the same as in the initial state. An improved model would consider the possibility of final-state interactions in the n + 9Li channel influencing the distributions. In the Serber model, the inclusive transverse momentum distribution of 9Li reflects the momentum distribution of the core in “Li. The latter is isotropic; the transverse is obtained by projection on some region of a plane perpendicular to the beam direction. If the beam direction is the z-axis we can use eq. (25). Here we assume that the integral is over all p” (rather than merely pY = 0) because [94] in the experiments [5] the opening in the detectors is rather large compared with the typical widths of the momentum distributions. The results are shown in -fig. 12 for several Faddeev calculations, and the dot-dashed line gives the distributions for the shallow s-wave potentials used in [63, 83]. The upper set of data points are from [5, 97] and [8]. We find that the wings of the distribution represent mainly the part of the wave function which is not in an s state in the coordinate r(fln). between the core and the neutron pair, and this is typically (table 13) larger than 55% for class I calculations. Classes II and Z calculations have almost entirely s states here, and hence give narrower distributions. Since all the class I calculations produce distributions which are too wide, this would appear to support class II models for the angular momentum structure. More recently Orr et al. [8] have measured the longitudinal momentum distributions at 65 MeV/A. In the Serber model these should be the same as the transverse distributions, but here they are significantly narrower. The data of [5] may still have to be corrected for multiple scattering within the thick target [97], so it may be that neither set of data can be fitted by any of our three classes of models. If positive parity levels from the sd shell were important in the low part of the ‘°Lispectrum, then a rather different structure would be indicated for the ~ ground state. There may, for example, be low-lying resonances among the excited states of ‘1Li. If there were to be a three-body resonance in ~lLi* within 1 MeV of the ground state, this could lead [98] to narrower distributions for both the neutrons and the 9Li fragments. There could also be final-state interactions between the neutrons and 9Li, in particular concerning resonances at + 0.2 and/or + 0.8 MeY, may enable class I models to produce narrower distributions, in particular the very narrow neutron distributions seen by [6]. As was shown recently [100], this effect is very important for reproducing the narrow neutron momentum distributions from 6He fragmentation. 192 M. V. Zhukov ci a!., The Borromean halo nuclei ‘He and ‘‘Li 10’ 10~1 - ---. - core HHvalence ~ 1O~ -~..., --.. —-.—.- COSMAvalence 0 2 46 8 12 Radius (fm) Fig. 11. One-particle radial densities of ‘‘Li from [96]: (a) densities, provided by the core nucleons and the valence neutrons, (b) total one particle densities. The HH calculation is calculation Z2 from section 6, the COSMA model is case I from section 5, and the CSF calculation is case L5D from section 6. Also shown are statistical estimates of the ‘‘Li one-particle density, obtained in [95] from the interaction cross sections at different targets by Glauber types of calculations. Table 14 Predictions for the r.m.s. average internal geometry of ‘‘Li. The experimental matter radius is 3.16 ±0.11 fm for ‘‘Li and 2.32 fm for ‘Li. Coordinate JJH [63] TS [60] BES [61] CSF (H) CSF (Q9) CSF (L6A) COSMA, COSMA 2 7.1 7.73 6.24 7.8 6.7 5.7 8.31 6.71 4.9 4.6 5.0 4.6 4.7 4.33 5.1 5.97 5.84 6.4 5.8 5.5 6.01 6.1 j~CM 5.8 5.2 4.8 j~CM 0.89 0.95 0.83 0.85 0.79 0.93 R_,,,~, 3.19 3.32 3.14 3.02 3.2 3.2 10Li resonances,The resolutionlookingof intheseparticularquestionsat thewillbehaviourdepend onof clarificationthe three-bodyof themodelsnaturein theof thebreakup continuum. The resolution will also depend on reaction models [99] using the “Li wave functions calculated here. MV. Zhukov ci a!., The Borromean halo nuclei ‘He and ‘‘Li 193 0 20 40 60 80 100 120 140 9Li), MeV/c p( Fig. 12. Breakup momentum distributions of ‘Li core (all normalised to 4.0 in arbitrary units) from the M9 (dotted line), L5D (solid line) and QlO(dashed line) calculations, and from Z2 (dot-dashed line). The curves for the deeper ‘‘Li binding are approximately 10% wider than those shown here. The squares are the Pi <0 data from [5, 97], and the circles p 1 > 0. The triangles show the longitudinal momentum distributions p, from [8]. 6.4. Predictionsfor momentum correlations 9Li pairs in theAllSerberof ourmodel.wave functionsThis modelleadusesto correlatedthe Fouriermomentumtransformdistributions~(k, , k for n-n and n- 2) in eqs. (40) and (41). The distributions depend on the angular separation of the measured9Li detectors.particles,Thesesoareweforpresentthe samein observablesfig. l3a—c theas resultspredictedforinforward-anglefig. 9 from the(0°)simplerneutronCOSMAand models. Here we give the correlated distributions for the neutron and 9Li fragments following the breakup of “Li at 300 MeV/A as predicted by the three kinds of models (I, II, and H). We see that the details of the correlated momentum distributions, in particular the smaller peaks around the central maximum, are rather sensitive to the model used. In order to demonstrate that these distributions are also sensitive to the 9Li—n potential, we have also carried out a COSMA calculation where the (Os)2 state (as permitted for example by the shallow (H) potential) is the dominant term in expansion (19). Figure 13d shows the 9Li—n correlated momentum distribution obtained with such a pure (Os)2 COSMA configuration. In this case only one parameter — the oscillator radius r 0 (‘—j 4.88 fm) is needed to fit the experimental matter radius of “Li. Comparison of figs. 13c and d shows that both the Z2 and the COSMA calculations give single peaks, rather than the several peaks in figs. 9 and 13a, b. The width of the COSMA distribution is however larger than that of the strict Z2 calculation. This is mainly due to the extremely long tails of the strict WF which cannot be simulated by the single term COSMA WF (19). 2 COSMA state (roWe= 4.88alsofm)testedwith athesmallcase(—~where3%) admixturethe outer ofneutronsthe coreare(Os)2mainly(r in a (Os) 2 state as the Pauli principle demands.0 =This1.8casefm) socorrespondsthat total WFto theis simpleorthogonal9Li—ntos-wavethe coreattractive(Os) potential [63] and does not exhibit any specific structure in the correlated momentum distributions. 194 MV. Zhukoi’ ci al., The Borroinean halo nuclei ‘He and ‘‘Li (a) Spin-orbit case I: Q5 (b) Pairing case II: L6A 240 ______240 ______Ne 300 2550 UIP 01~, 300 -~ 2550 ~‘) ~ 9L~ (MeV) ~ (iI~~, ~ (MeV) (c) Shallow potentialcase H: Z2 (d) COSMA: Os states 240 A 240 3~53 ..~ 2550 300 — 2550 C ~ !~eut~” ~te’V) ~ ~ V~ 36d~ 9U~C(MeV) 9L1 fragments at ft. following the breakup of ‘‘Li at 300 MeV/A. (a) Spin-orbit case Q9,Fig. (b)13.theCorrelatedpairing caseenergyL6A,distributions(c) the shallowfor npotentialand case Z2, (d) ‘Li—n correlated momentum distribution obtained with a pure (Os)2 COSMA configuration. 6.5. Summary In this section we have presented solutions in coordinate space to the hyperharmonic equations (HH) and to the Faddeev three-body equations (CSF). They used standard realistic neutron- neutron potentials with repulsive cores (contra [61,63,83]), and use neutron-core potentials that reproduce possible ‘°Li resonance structures. These solutions correctly describe core recoil, and the CSF method includes the Pauli principle by explicit orthogonalization. The existence of these Faddeev treatments enables comparison with experiment to highlight not MV. Zhukov ci a!., The Borromean halo nuclei ‘He and ‘‘Li 195 restrictive approximations, but the physical inputs such as the model potentials, and the nature of shell closure in 9Li. We then find that three quite different classes of potentials all reproduce the r.m.s. matter radius and the binding energy of “Li. The cases, however, assume quite different characters for the nature of ‘°Liand any resonances in the 9Li + n channel. They will therefore have to be distinguished, not by inclusive measurements of matter densities, but by measurements of breakup fragmentation. The different classes do give somewhat different distributions for the production of 9Li in the Serber model of breakup, but none of them reproduces the longitudinal 9Li distributions of [8] or the neutron distributions of [6]. This calls into question the validity of the simple Serber model, and means that detailed reaction studies will now be required. 7. Conclusions and outlook Our growing understanding of exotic halo-like neutron-rich nuclei is mainly a result of frag- mentation studies, where such loosely bound systems are broken up in encounters with target nuclei. While only inclusive (strongly integrated) observables were available up to very recently, at the moment of completing this article also exclusive quantities such as momentum correlations in complete measurements are starting to be measured. These data will definitely shed new light on the whole break-up scenario. We choose to limit this review to the initial stage only, and to Borromean three-body-like nuclei 6He and “Li where the ground state is the only bound state. This state, and the determination of its structure, was from the very beginning a main objective of the experimental investigations. To which extent and how the properties of the bound state are reflected in the fragmentation data will vary with relative energy and target nucleus, but this question is not addressed in this review. Neither did we discuss the structure of the three-body continuum, although recent calculations will be published in nearfuture. We have reviewed a number of three-body procedures currently used, but most of the results shown refer to calculations performed within our own collaboration with the hyperharmonic expansion method or the coordinate space Faddeev pro- cedure. We have also, with reference to recent measurements, presented evidence for 9Li being a reasonably good core-nucleus for “Li. We have tried to argue that Borromean nuclei, where the three-body system is bound (although loosely), while none of the binary subsystems have bound states, are genuinely non-mean-field systems. A three-body treatment is the natural approach, using ‘realistic’ binary interaction potentials taken from two-body scattering studies, and we have demonstrated that such calcu- lations are feasible these days. As an example of the state of the art we have reviewed the A = 6 nuclei, containing the Borromean nucleus 6He, and showed that a three-body treatment is quite satisfactory. The success obtained for 6He provides a benchmark for studies of other Borromean nuclei. Although 6He is a Borromean nucleus with fully developed halo characteristics, most of the excitement has been sparked off by “Li, the cardinal example. For a three-body (or any) treatment the starting point for ‘‘Li is more uncertain than for 6He, as information on the (valence)-neutron—core(9Li) interaction or the resonance structure in this channel is still incom- plete. In a three-body treatment, the binary interactions also reflect the Pauli principle to some extent, or it has to be taken into account by some other recipe. Alternatives were discussed and evaluated. In general we feel that the (approximate) ways we treat the exclusion principle are satisfactory. 196 MV. Zhukov eta!., The Borromean halo nuclei ‘He and ‘‘Li We have demonstrated that a number of plausible interaction models all reproduce the important halo features as they are reflected in inclusive observables, such as binding energy, geometrical characteristics, and also one-particle densities. We have also described an approximate method COSMA, which employs simple trial wave functions (configuration mixtures in core- neutron coordinates) and which makes it possible to explore possible features of the nuclear structure without having to carry out full blown three-body calcu- lations. Three different simple trial wave functions for 1’Li were found which all reproduce the r.m.s. radius and the observed transverse 9Li-momentum distribution at high energies. They give similar geometrical characteristics to those found for the various strict three-body calculations. We have also shown that the information content in spatial- and momentum correlations carries more clear fingerprints of the underlying nuclear structure. Again results of both COSMA and strict calculations were discussed. Such quantities are now being measured, and this may help solving the ambiguity left by the inclusive measurements. Without going into details, we like to remark that also our preliminary calculations for the continuum structure of ‘1Li carry substantial sensitivity to the n-core interaction model used to generate the bound state, in spite of the fact that all these models give comparable reproduction of the energy and geometry of the ground state. The future development will have to encompass both the adventure of exploring new Borromean (or Borromean-like) nuclei such as ‘4Be and ‘7B as well as a clarification of the break-up mechanisms as a function of energy and target charge. Complete measurements, although hard, seem to become a necessity, and the field will have to develop in a close interplay between experiment and theory. Such experiments will also reveal whether and when the assumption of a structureless core becomes insufficient. A number of interesting reaction scenarios with (sufficiently intense) radioactive beams can be envisioned. These may clarify reaction mechanisms which ordinary nuclear collisions have been unable to discriminate. Such detailed studies will, however, rely on the detailed structure informa- tion for the state of the projectile halo-nucleus discussed in this review. Acknowledgements The authors would like to express their gratitude for discussions with many colleagues both about experiments and theoretical ideas. The collaborations with F.A. Gareev, A. Korsheninnikov and L. Chulkov have been particularly helpful. We have obtained financial support from a number of sources including our own institutions. 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