Nuclear Physics A539 (1992) 177-188North-Holland

M.V. Zhukov, D.V. Fedoiov and B.V. Danilin

The Kurchatov Institute ofAtomic Energy, 123182 Moscow, USSR

J.S . Vaagen'

NORDITA, DK-2100 Copenhagen 0, Denmark

J.M. Bang

Received 28 August 1991(Revised 4 November 1991)

1. Introduction

P

Neutron halo structure, 9 i-neutron and neutron-neutronmomentum correlatio s in I1 i

The Niels Bohr Institute, University ofCopenhagen, DK-2100 Copenhagen 0, Denmark

s Permanent address: SENTEF Institute of Physics, University of Bergen, Norway.

0375-9474/92 ;X05.00 O 1992 - Elsevier Science Publishers B.V. All. rights reserved

UCLYSICS

Abstract: The approximate three-body approach COSMA is used

) explore the structure of 'r Li .This "Borromean" system is discussed with reference to the similar system 6i.iv Lr whk-hinformation on the binary subsystems is more complete . Spatial dc rsities anJ a variety ofcorrelated momentum distributions for 91i-n and nn are calculated. The latter seem toprovide the essential information to pin down the r r Li (neutron halo) struc"ur, --. An estimateof the energy of the soft dipole mode is also given.

The Borromean rings, heraldic symbol of the Princes of Borromeo, are carved in thestone of their castle on an island in Lago Maggiore in northern Italy. The three ringsare interlocked in such a way that if any of them were removed, the other two wouldalso fall apart. In nuclear physics 6I-Ie and "Li have been found to have this property(although for quite different physical reasons) when described in a three-body model.We may refer to them as "Borromean" systems.These neutron-nich radioactive nuclei have been actively explored in recent experi-

mental investigationn, 1 .2 ) . They have several common features . T®.vir binding energiesare small, their radii are large, and there are no bound states or narrow resonances inthe two-body subsystems 9 Li + n, a + n and n + n. Conventional shell-model 3 ) andI-Iartree-Fork calculations 4 ) seem unable to reproduce such anomalous properties .

Some sort of three-body approach, core + n + n. here with the cores 9Li and 41-îe seemsmore appropriate. The main reason is that the nn interaction in such a loosely boundsystern cannot be treated in the mean field approximation, aahough it i :esponsiblefor the binding of the system . Exact three-body calculations are complicated and suffer

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from some uncertainties, particularly in the 11 Li case . Here the n-9Li interaction isinsufficiently known, and cakuh6ons with different interactions lead to quite differentstructures of the 11 Li nucleus (see, for example refs.")).With the help of available information on momentum distributions of the constituent

particles one can nevertheless try to explore and estimate approximately the structureof the wave function of "Li . Assuming that we understand the reaction scenario, thisfunction should describe the experimental momentum distributions and other availableexperimental data and should be chosen properly in terms of a three-body approach .In an effort to assess the "true" wave function (WF) we have previously proposeda simple schematic model 9-11 ) COSMA [the cluster (oscillator) orbital shell modelapproximation, sometimes also referred to as OCSM - oscillator cluster shell model]with analytic wave functions. Although being an approximate scheme, it neverthelessincludes essential features of three-body wave functions; appropriate translation in-variant intrinsic coordinates, antisymmetrization of valence neutrons and neutron-corePauli blocking although in an approximate manner. COSMA has previously been testedagainst the best known light "Borromean" nucleus with approximate "core + n + n"structure - the 6He nucleus" ) . The COSMA approximation is inspired by the strictthree-body procedure ~:OSM (cluster orbital shell model), developed in ref.'' ) .

We showed in ref 9 ) Chat for the "Li nucleus ambiguities are present: Wave functionsof ditTerent configuration compositions which give different neutron halo structuresfbi- the "Li nucleus, describe equally well the 9Li inclusive transverse momentumdistributions'-°') and simultaneously the r.m.s . matter radius of "Li [ref. 13) 1 . Thecorresponding spatial densities are discussed in sect . 2.

ence, correlation experiments rather than inclusive experiments, seem required ifdetailed features of the neutron halo structure are to be disclosed. Such 'Li-n- and nncorrelation functions are calculated in this paper (sect. 3) for all versions of the "Litrial wave function discussed previously 9-10 ) . Some of our new results are given inref. 14

) . A comment on the Pauli principle is given in sect . 4, while '-sect . 5 contains anestimate of the soft dipole mode.

w ve-Nuction s re in the framework of COSMA and spatial densities

As in refs . ' - ' 0 ) we treat the 11 Li nucleus as a three-body system 9Li + n + n. Intro-ducing translation invariant coordinates n, and q, pointing from the core Q=TO tothe outer neutrons we exp.-ess the three-body wave function of "Li as follows:

ere n and I are nodal and orbital angular momentum quantum numbers for a neutronrelative to the 9Li core and jn1(rj c ),n/Q c );O` configurations composed of ordinaryoscillator wave functions with an oscillator parameter rc, T'his expression assumes thatthe total orbital momentum and total son of the outer , neutrons both e4ual zero,hence, that the total momentum of % is that of the `Li core (I - ) corresponding2

M. V. Zhukov et al. / Neutron halo structure

179

to experimental data (see also refs. 5.7)) . The free parameters in WF (1) are the

mixing coefficients C�, and ro. The fitting parameters are required to reproduce boththe experimental radius of "Li and the transverse momentum distribution of the 9 Linucleus from a "Li fragmentation reaction .Assuming that only a few (in fact only two at the time, hence two fitting parameters ro

and the configuration mixing angle) lowest terms in (1) give the main contribution tothe WF, we found previously 9 ) that three combination of (0s)', (0p)2 , (Od)2 and (IS)2

configurations describe equally well both the experimental rms matter radius of ° ° Li(ti 3.2-3.3 fm) [ref. 13) ] and the transverse momentum distribution of the 9Li nucleus*from a 1 1 Li fragmentation reaction on a carbon target at 790 MeV/nucleon ref.' ):

Having fixed the parameters for the different COSMA w^ve functions we can calculate

other observables and distributions without free parameters . In ref. 9 ) we showed thatneutron inclusive momentum distributions are not very informative and have simi-lar structures for all cases considered, hence we have to proceed to investigations ofcorrelated observables.Before doing so, we display spatial features of cases

A measure for spatialdensities of the "Li in the framework of the three-body approach is defined as

2p(rnn, rc(nn) ) ^'

drnn drc(nn) rnnrc(nn)Iw(rc z , r'c!~

where rnn = (r)c - r2c )) rc(nn) = i (r:, + ~c ) . This distribution was calculated for allthree choices (0-011) . The results are plotted in figs . 1-3.Note that the spatial densities given by different P differ drastically from each other

in spite of almost equally good descriptions of inclusive momentum data and radius.Each case has a specific neutron halo structure. For YJ, case (I), the density (fig. 1)

looks very similar to the 6He case' 5 ) with obvious scaling. Pronounced "dineutron"and "cigar" type configurations are seen (see again ref. )5 ) ) . In fig. 3, case (III),

these configurations have changed in relative magnitudes and forms. Fig. 2 gives thethree-peak structure of configuration mixture (II) .The specific correlations between the particles seen in figs. 1-3, are almost washed out

in the inclusive particle transverse mornentunl distributions, which are highly integratedcharacteristics. Only correlation experiments with registration of two fragments can

explore these correlations .

It should be added that all calculated 9 1,1 momentum distributions have the peculiar structurefound in recent experiments (with deuteron and carbon targets) 2 ) : They can not be described bya single gaussian form but reed for their reproduction at least two gaussian components (a broadand a narrow one) 1 - 2 ) .

case (I) IF = (0.97)101,01 ;0) + (-0.243)100,00;0), ro = 3.84 fm, (2)

case (11) IP = (0.6)J01,01 ;0) + (-0.8)102,02;0) . ro = 3.44 fm, (3)

case (III) IF = (0.9)J01,01 ;0) + (-0.436)J10,10,0), ro = 3.68 fm . (4)

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ALV

Fig. 1 . S

hukov et at / Neutron halo structure

atial density P

) for WF (2) (case 1, (Op )2 and (Os) 2 adn, ixture 1 .

3. -n a

Fig . 2 . Same legend as for fig. 1, but for WF (3) [case (ID, (0p)2 and (0d) 2 admixture].

correlated momentum distributions in I'Li

Assuming validity of a Serber model or I'

13A, high-energy fragmentation of lightnuclei [see refs . 16-18

) and references therein I lonely reveals the ground-state momentumdistribution for the constituents.

Fig . 3 . Same legend as for fig . 1, but for WF (4) [C.ISC (111) . (()p)2 and !,!s )1 2 adunixturej .

M. V. Zhukov et al. / Neutron halo structure

181

In principle the momentum distribution of fragments i also influenced by the reac-tion mechanism, as contained in the so-called phase-shift function . For losely boundsysiems and high projectile energies (say, larger than 100 MeV/nucleon), however,Lh;E influence seems negligible as discussed in our earlier article 9 ) and by many otherauthors Isee ref 9 )) . This is at least so, when Coulomb effects are small, i.e. for lightertargets. For small energies and/or heavy targets more detailed break-up calculationsmay be of interest.Thus, most experiments in which two [the valence neutrons or the core (91,i) and

a valence neutron] of the constituents from a break-up of "Li are detected (energiesand emission angles), are assumed to give cross sectio.s proportional to the densitiesPnn (PI,P2) or P,.� (p,,p 1 ) of "Li in momentum representation defined below.The momentum representation of the WF (1) is essentially straightforward. The

single-particle oscillator wave functions have in momentum representation just thesame form as the initial spatial ones, to within a p'.~ 3sF factor (-r)°. +When a CUSIVÎAtwo-particle configuration is transformed it is again form invariant to within a phase(-1) l, but note that the proper conjugate momentum arguments are the c.m. mornentak, and k2, and not those relative to the core . Shell-model thinking is here easilymisleading . The c.m. momenta are the following combinations of the iab momenta ofthe core p, and the valence neutrons p 1 , P2 :

kI = p1 - lL1P,k2 = P2 - I_L1 P,

P1 = k 1 + il-, P,

P2 = k2 + IIIP,

=Pl +P2+pc ,

Pc = 9P - kl-k2-

(6)

The nn and core-n (c = 9Li) momentum correlations are now simply given by thefollowing expressions:

Pnn(P19P2) = IP(PI - I-L,P, P2-1'°1p)12,

P"' (P1,PC) = I VJ (PI - -LP'

1®P -

2II

I1

PC -PI)I

-

(7)

(g)

With the explicit forms (2)-(4) we have calculated the corresponding distributionsfor a total momentum P, chosen to correspond to a 11 Li projectile energy of 300MeV/nucleon, that of the present Darmstadt facility energy. The registration angles ofthe emitted particles (9Li and neutron) were both chosen to be 0° . It should be notedthat our formalism also enables us to calculate the correlated momentum distributionsat arbitrary angles and energies .For convenience the figures presented below are plotted against the lab energies of

the corresponding fragments. Figs . 4-6 show the n9Li momentum correlations for thecases (I)-(III) . All distributions shown in figs . 4--6 have characteristic features snakingmeasurements of such distributions interesting and important.The neutron halo structure given by the WFs of cases (I) and (III), which are hard

to distinguish in_ inclusive experiments (see ref. 9 ) ), can be separated in correlationexperiments, the correlated momentum distributions having quite different features .

hukor et al. / Neutron halo structure

Fig. 4. The 'Li-n correlated momentum distribution at zero angles in the laboratory frame atE(° ° Li) = 300 MeV/nucleon predicted by WF (2) [case (1), (0p)2 and (o,,,) 2 admixture ].

Fig, j. Same iegenà as 1ûi 11g. 2, but for WF (3) [case (11), (0p)2 and (0d)2 admixture] .

The differences exhibited by figs . 4 and 6 are the following : The central peaks ofthe case (I) (fig. 4) are clearly separated because of a small weight (r., 0.06) of the(0s) -' component which only contributes between the two central peaks. A pure (0p)2

configuration gives a valley between the central peaks down to zero. For case (III)

Fig. 6. Same legend w; for fig. 2, but for WF (4) [case (III), (0p)2 and (IS)2 admixture] .

M. V. Zhukov et al. / Neutron halo structure

183

the (IS)2 configuration has a weight about 4 times that of the (OS )2 configuration incase (I). This fact together with a reduced weight of the (0p)2 configuration resultsin a complete filling of the gap between the two ccr tral peaks hence only a compositecentral peak as seen in fig. 6.

In this paper we only present the momentum correlations at 0° for both particles.This choice of angles gives the most pronounced pictu:es. Additional test calculationsfor the Darmstadt energy shows that the specific structures of the distributions arewashed out if the angle of neutron registration is more than 2° in the laboratory frame(keeping the core registration angle at zero) .The nn momentum correlations for all three cases are shown in figs . 7-9. Also here

the different configurations are easily distinguished. The nn momentum correlationsare to a large extent replicates of those for 9Li-n, they are rotated by approximately 90°and slightly distorted compared with 9 Li-n in accordance with the coordinate exchangein formulae (7) and (8) .To elucidate these results we include analytic formula for pure (0p) 2 and (Os)2

configurations . A pure COSMA configuration of equivalent n = 0 orbitals is essentiallygiven by

exp[- 2 (r

+ r2~) lri,ri,Pl (F, - r2,) .

The radii are measured in the oscillator length (ro ) and we have used the fact thatthe n = 0 harmonic-oscillator orbitals ar;,~ gaussians times harmonic polynomials. (Theexpression given above is the L = S = 0 component if we start from a (Olj) .2^0configuration.)To obtain momentum correlations, we transform (9) to momentum space and obtain

G(k l , k2) = exp[-1/2(k ; + k2)ß, for (Us)L=o,

(10)

G(ki ,k2)(k, - k2)

for

(OP) 2

(11)

We may now use eq. (6) to obtain the lab frame nn momentum correlations . Takinginto account that we consider here only the case when the two neutrons are registered

Fig . 7 . The nn-correlated momentum distribution at 0° angles in the laboratory frame at

E(I'Li) = 300 MeV/nucleon predicted by WF (2) [case (1 ), (0p)2 and (Os)2 admixture].

184

Mi'

hukov et al. / Neutron halo structure

Fig. g. Same legend as for fig. 7, but for WF (3) [case (11), (0p)2 and (0d)2 admixture] .

at 0° in the lab frame the corresponding nn correlated momentum distribution for thepure (0p)2 configuration is proportional to the value

IG(k,z,k2z)(k,zk2_)I -' .

(12)

From this formula it is easy to see (k j _ and k;_ are zero where G is the maximum)that the nn-correlated momentum distribution for the pure (0p)2 configuration shouldhave a four-peak structure in the lab frame for lab momenta of the neutrons, p, andp2. being close to !he transmission momentum ,'-I P. Fig. 7 essentially gives the resultbecause of the smaîl weight of the (0s)2 component in the WF (2).

. Remarks about the (0s) 2 configuration

One expects the dominant term in expansion (1) to be (0p)-. This follows from asimple shell model consideration in which the last two neutrons in "Li close the Opneutron shell. The results of ref. 7 ) support this, giving the (®p)2 configuration a weightof about 90%. The results of ref. 8 ) are quite similar. Hence, we have chosen the (0p) 2configuration as the main part and mixed it with one of nearest configurations . As

Fig. 9. ;âme legend as for fig. 7, but for Wr (4) [case (111) . (0p) 2 and (IS)2 admixture] .

M. V. Zhukov et al. / Neutron halo structure

185

we have discussed, the weights of the admixtures were found by simultaneously fittingthe rms matter radius of "Li and transverse momentum distribution of 9Li in the fullrange of momenta.

It should be noted that a pure (Os)2 configuration also succeeds in describing theupper narrow part but fails to describe the tails of the 9 Li transverse momentumdistribution 9 ) . If one believes that only the narrow part of the distribution should bedescribed, then one can use this configuration as a leading term in expansion (1).The question arises if this is in contradiction with the Pauli principle, which requiresorthogonality of the WF of the outer neutrons to the (Os)2 configuration occupied bythe internal neutrons . It should here be emphasized that there is no straightforwardconnection in COSMA between the occupied (Os)2 configuration in 9Li and a candidatefor a (OS )2 configuration of the valence neutrons in " Li because of quite differentro parameters for internal (ro - 1 .8 fm) and valence (ro ti 4.9 fm) neutrons. Thisleads to a small overlap between Os configurations of outer and inner neutrons . Thesmall probabilities to find one valence neutron or two valence neutrons inside the 9 Licore, reported in table 2 of ref. 9 ) substantiates that the Pauli principle is obeyed withnecessary precision. Nevertheless, we did special calculations both for the 9 Li transversemomentum distribution and for 9Li-n momentum correlation with a COSMA pure(Os)2 wave function for the outer neutrons, made orthogonal to the corresponding (Os)2

state for the inner neutrons by adding a corresponding correction term . No significantchanges ül the calculated distributions were found. T:c weight of the orthogonalizingterm is about 3%.

In a full ,hree-body approach where an equation of motion is solved, it is obviousthat the weights of the different components in our trial wave function (1) dependmainly on the 9Li-n potentials put into the calculation. Lacking sufficient experimentalinformation on these potentials from 9 Li-n scattering we have tried to explore andpredict apprcximately the structure of the 1 1 Li nucleus in the framework of COSMA.To show that the correlated momentum distributions are rather sensitive to the 9 Li-npotential, a calculation of "Li bound state with a simple attractive 9Li-n potential 5 )for all partial waves was made in ref. 6 ) . As one expects the (OS) 2 configuration takesabout 95% of the WF weight. The resulting binding energy and matter radius areboth in good agreement with experimental data . Using now the WF obtained in thiscalculation to estimate the 9Li-n momentum correlation we get the result shown infig. 10a. It is clear that this distribution differs essentially from the previous figures,there is noN only a single narrow peak:.

For comparist)n witl': the strict three-body calculation mentioned above with a simpleattractive s-wave potential in the 9 Li-n channel we have also carried out a COSMAcalculation where the (Os)2 state permitted by this potential is the dominant term inexpansion (1). Fig. l0b shows the 9Li-n correlated momentum distribution obtainedwith a pure (Os)2 COSMA configuration . In this case only one parameter - theoscillator radius ro (,., 4.88 fm) - is needed to fit the experimental matter radius of"Li. Comparison of figs . 10a and 10b shows that, COSMA also gives a single peak

hukor a al. / Neutron halo structure

Fig. 10. The 9Li-n correlated momentum distribution at 01 angles in laboratory frame atEOILO = 300 MeV/nucleon for: (a) strict three-body calculation 6) with simplest type of

the '-4 Li-n potential 5 ), (b) COSMA one component (OS)2 WE

rather than the several peaks in figs. 4-6; the width of the distribution is, however,larger than that of the strict calculation. This is mainly due to the extremely long tailsof the strict WF which cannot be simulated by the single term COSMA WF (i).

5. Estimate

We would like to add that COSMA is not only useful for quick and simple estimates ofconstituent particle momentum distributions (and correlated momentum distributions)of neutron-rich nuclei, but also gives a straightforward possibility to estimate the energyH of the soft dipole mode resonance excitation . In this ulodel JE equals the energyquantum fact of the effective oscillator potei-itial for the valence neutrons. Using theconnection between the oscillator parameter ro and hw

f energy of soft dipole excitation

hW = h2l (/Àr()2) .

where ji is the core-n reduced mass. We obtain 6E , 3 MeV for 11 Li (ro

4 fm).This value is in agreement with the one obtained in the strict calculation 20

For thee case with the oscillator parameter ro , 2.7 fm [ ref. 11 ) I the excitation energy of

soft dipole mode is 6E , 7.3 MeN also in agreement with ref. 20) .

These dipole excitation energies may be obtained directly from the - more or lessapproximate - three-body calculations or estimated by dividing the energy-weighteddipole sum rule with the non energy weighted one. In the latter case only g.s. wavefunctions are needed; note that the effective neutron charges, although neccessary forthe sum rules, do not affect the excitation ev-,ngies.

M. V. Zhukov et al. / Neutron halo structure

187

6. Summary

Having insufficient information on the potentials in the 9Li-n system one cannotbe sure that solving the Schroedinger equation of the three-body problem strictly willgive the "true" "Li wavefunction. Instead we have tried to explore approximatelythe structure of the "Li nucleus using existing experimental data, especially for the9Li transverse momentum distribution . COSMA trial functions in the relative valenceneutron - core coordinates have been constructed for this purpose. The COSMA wavefunctions reproduce the two component structure of the 9 Li transverse momentumdistribution found in recent experiments 2 ) . Three L = S = 0 candidates (1), (11) and(III) have been selected for the physical nature of "Li:

(i) The outer neutrons are mainly in (0p) 2 as standard shell model prescribes witha small admixture of (Os)2. The COSMA (Os)2 state is not forbidden by the Pauli prin-ciple because the COSMA Os state for outer neutrons and the core Os state have ratherdifferent oscillator radii (_ 4.88 and ti 1 .8 fm) hence they belong to different full sets .

(ii) The outer neutrons are in an admixture of (0p)2 and (Od)2 configurations. TheOd orbital is the next one in a standard oscillator basis and might in accordance withsome speculations, lay even lower than the Op orbital due to spin-orbit forces andresidual interactions .

(iii) The outer neutrons are mainly in (0p) 2 with an admixture Of (I S )2 .

We also tested the case where the outer neutrons are mainly in a (Os)2 COS Astate (ro = 4.88 fm) with a small (,.., 3%) admixture of the core (OS )2 (r0 = 1 .8 fm)so that total WF is orthogonal to the core (Os)2 state as the Pauli principle demands.This case corresponds to the simple 9Li-n s-wave attractive potential s ) and does notexhibit any specific structure in the correlated momentum distributions.As emphasized previously in ref. 9 ), available experimental data on the matter radius

and inclusive transverse momentum distributions do not allow us to distinguish betweenthe three possible configuration mixtures.

Finally we mention that in refs.'-'), and connected with the 9Li-n potentials used,the main contribution to the "Li WF is the (OPIi2 ) .2, -0 configuration which containsboth S = 0 and 1 . From the point of view of the conventional shell model thisseems most reasonable. COSMA also allows us to consider this choice of trial WFwithout introducing more fitting parameters . Thus we have investigated the pure

(OPIi2 )J=o configuration. We can reproduce the "Li radius, but the resulting 9Litransverse momentum distribution is broader then the experimental one. Inclusionof configuration admixture could give better agreement. This is subject for a currentinvest gation .

In conclusion we have shown that the 91i-neutron and neutron-neutron momentumcorrelations, as calculated in COSMA, seem able to distinguish between differentconfiguration mixtures . "True" wave functions, calculated by solving the dynamicaithree-body problem should also be tested against this observable which hopefully canbe measured in near future.

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.11V. Zhukoe et al. /Neu, :-cn halo structure

The calculated momentum distributions do not include influences from the reactionscenario. For higher energies our estimate appears to be resonable 21,22 ) .

We are grateful for discussions with Prof. I. Tanihata, Prof. P.G. Hansen, Dr. A.Jensen and Dr. K. Riisager concerning experimental as well as theoretical aspects ofthe subject of this work. Three of the authors (B.D., J.V . and M.Z.) acknowledgeinvitations from the 4iels Bohr Institute and Nordita, where this work was partiallydone .

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