Revista Mexicana de Física 40, Suplemento 1 (J 994) 63-73

80ft dipole mode in 11Li and three body continuum

YU. F. S~IIIlNOV'Instituto de Fisica, UNAM

Apartado postal 20-364, 01000 México. D.F., México

YU. A. LURIE ANO A.M. SlIIIlOKOVInstitute f01. Nuclear Physics al Moscow State University

Moscow 119899, RussiaReceived 11 February 1994; accepted 4 June 1994

ABSTRACT. The properties of the neutron rkh II Li nucleus are ealculated in a framework of acluster llIodel 9Li + n + n. The formalism of the harmonic oscillator representation is used forthe descriptioll of bound and eontiuuum states. The last .ones are found in a "true" three-bodyscattering approximation. It is shown that this approach allows to take iuto account adequatelya long asymptotic tail of the 11Li wave function (a neutron halo) and to reproduce correctlythe binding energy, radius aJl(1cross-section of clectromagnetic dissociation of 11Li. A shape andposition of the peak corresponding to the soft dipole Illode are also in agreement with experimento

RESUMEN. Se calculan las propiedades del núcleo de 11Li rico en neutrones en el marco de refer-encia de un modelo de cúmulos, !JLi+ 1l + n. Se usa el formalismo de la representaci6n del osciladorarmónico para la descripción de los cstados ligados y del continuo. Los últimos se calculan en unaaproximación de dispersión de tres cuerpos vcrdadera. Se demucstra que esta aproximación nospermite tomar cn cuenta adecuadamcnte la larga cola asintótica de la función de onda del 11Li,(el halo de los neutrones) y reproducir correctamente la energía de ligadura, el radio y la seccióneficaz de disociación electromagnética del 11Li. La forma y posición de los picos que correspondenal modo dipolar suave están también cn acuerdo con el experimento.

rAes: 24.90.; 21.60.G; 27.20

l. INTRODUCTION

Recently secoudary beallls of radioactive heavy ions beco me llvailable. As a resnlt onegets a tool for experimental studies of light neutron-excess short-lived nueleL 11Li is oneof the most interesting nuelei of the type, and its properties have being intensivoly studiedexperimentally 11-41 aud theoretically [5-121.The Illaiu pecnliarity of the 11Li nueleus is the so-called ueutron halo due to the availabil-

ity of two weakly-bouud ueutrons in it (two-neutron separation energy '2" = 247:1:80 keV;note, that there are no bound states in two-body subsystems 9Li + n and n + n [13]).The binding energy of the Ileutron pair is small, so the wave functiou decreases slowlyat large distances. As a result, the r.m.s. radius of IlLi is anolllalously large: (1"2)1/2 =

• On leavc of absence from Institutr for Nuclear Physics at 11osrow State University, Moscow119899, Russia.

63

64 Yv. F. SMIRNOV ET AL.

3.16:1:0.11 fm 13](to be compared to the one of9Li (r2)1/2 = 2.32:1:0.02 fm [3]). Evidently,the anomalously large electromagnetic dissociation (ED) cross section of 11Li beams onheavy target nuclei is the manifestation of the neutron halo (the 11Li ED cross section onthe Pb target is 20 times the one for the carbon beam [4]). To explain this effect it wassuggested [14,15] that oscillations of the halo neutrons with respect to the core 9Li giverise to the specific low energy branch of the giant dipole resonance (soft dipole mode) withexcitation energy less than 1 MeY. Soft dipole mode is supposed to exhaust about 10%of the dipole energy-weighted sum rule (EWSR).

The shell model [6,7,11], cluster [9] and combined cluster-shell model [8,10) approacheshave been used in theoretical studies of the IILi neutron halo properties. In a part of thesecalculations a simple diagonalization approach was used which does not take into accountthe continum effects. Also there are calculations with a correct description of the continumstates in binary [11] and three body break up [12) channels. Here we study neutron haloproperties of 11Li in a three-body cluster model 9Li + n + n. We make use of a three-bodywave function expansion in six-dimensional harmonic oscillator eigenfunctions. Our atten-tion is focused on the effects of the three-body continuum. The continuum spectra effectsare accounted for in the framework of the oscillator representation of the scattering the-ory [16]. This method has been generalized [17] to the true three-body scattering [18,19]and has been successfully used in the study of the 12C monopole excitations in the clustermodel Ú' + Ú' + Ú' [20]. As it has been noted, there are no bound subsystems like 9Li + nor n + n in the system. Therefore only so called "democratic" three body decay [21] isinherent to this system. It can be adequatcly described in a framework of the "true" threebody scattering approach [18]. In the study of the ground and excited state properties wego beycind the pure diagonalization of the Hamiltonian matrix on the basis of oscillatorfunctions. We search for the S-matrix pole corresponding to the ground state, and con-struct an infinite series expansion in oscillator functions for the ground and excited statewave functions. This enables us not only to calculate the IILi ground state energy withhigher accuracy, but to describe also the wide spatial distribution of two valence neutronsin 11Li (neutron halo).

2. TIIE MODEL

The 11Li ground state and the continuum spectrum wave functions are calculated in theframework of the three-body cluster model 9Li + n + n. The model assumptions are:

1) The cluster 9Li is supposed to be structureless and the excitations of its internaldegrees of freedom are not considered.

2) We don't account for non-central components of the interaction between two valen ceneutrons and between valence neutro n and the cluster 9Li. Therefore, the wave functioncan be characterized by the orbital angular momentum L the three-body total angularmomentum J, and its projection M.

3) Only sta tes with the total spin of the valence neutron pair S = O are considered, andthe ground state three-body orbital angular momentum is supposed to be equal to zero:L =0.

The wave function of the system 9Li + n + n is expanded in three-body hyperspherical

80FT DIPOLE MODE IN "LI AND TfIREE BODY CONTINUUM 65

functions 4>1.1, L ML (íj):

(1)

where X/~3/2 is an interna! wave function of the 9Li core, K is the hyperrnornenturn, Ixand Iy are the angular momenta corresponding to the Jacobi coordinates

= VIS mw (r1 +r2y llf, 2 (2)

respectively, m is the neutron mass, ro are coordinates of the valence neutrons (i = 1,2)and the cluster 9Li (i = 3), and p = (x2 + y2)1/2 is a three-hody hyperradius.

In the C.rn. frame the Hamiltonian is of the form

(3)

where T is the kinetic energy of three-body relative rnotion, and V;j are the two-body po-tentials. For the radial wave functions ¡plfL 1, (p) we have the usual set of the K -harmonicmethod coupled equations 1191.The equations are solved using the expansion of the three-hody wave function

in the six-dimensional harrnonic oscillator eigenfunctions

(4)

2nl K K+2 2 2r(n+K+3)P Ln (p)exp(-p/2), (5)

where L~(x) is the associated Laguerre polynomiaI.For large values of a total number of oscillator quanta N = 2n + K the potential

energy matrix elements are small cornpared to the matrix elements of kinetic energy whichincrease linearly with N. Therefore in 'practical calculations of scattering characteristicsthe potential energy matrix V = Vl2 + V13 + V23 on the basis of functions (5) can betruncated, Le. all Illatrix elelllents Vj.¡V?' for N > IV or N' > ií' are neglected (/ir denotesthe truncation boundary). The kinetic energy Illatrix TfJt is an infinite tridiagonal rnatrixfor any value of the multi-index r = {K, Ix,ly} which labe1s dilferent decay channels inthe hyperspherical representation. The eigenvectors of this matrix, which are illlportantfor the description of the wave function asymptotic behaviour at p - 00, are of the formin the oscillator representation 117):

(6)

(7)

66 Yv. F. SMJI{NOV ET AL.

whcre Sr r' is the lIlatrix element of the S-lIlatrix, and ar is the normalization constantof the bound state wave function for the channel r.There are analytical expressions for the functions C~';J(E) [1il:

wherc

1S"K(E) =-

~

(8)

(9)

(la)

(11 )

The functions S" K(E) and C" K(E) give rise to functions with the following asymptoticbehavior in the coordinate space:

00

L S" K(E) 'Pn K(p)n=O

00

L Cn K(E) 'Pn K(p) '"n=O

1~ 2 NK+2(Qp).,wp

( 12)

(13)

The asymptotic expressions (6) for D~Jk1,1, (E) correspond to the account of democraticdecay channels only iI91, i.e. the wave function in the aSYlllptotic region is a superpositionof an outgoing six-dimensional spherical wave in the channel r' and ingoing six-dimensionalspherical waves in all channels r. The applicability of the democratic decay approximationfm various nuclear reactions with few particles in a final state has been discussed in [19,211.In the framework of the oscillator representation of scattering theory this approximationhas been used [201 in the stndy of monopole excitations of 12C in a cluster model a+a+a.The analysis of experimental data for 2+ states in A = 6 nuclei within the a + N + Ncluster model assumptions has been performcd using this approximation in Ref. 1221.The energy spectrum ami the wave functions are calculated in the following way [171.

First of all, one should find eigenvalues Ef) ami eigenvectors h~~1(J)} of the truncatedHamiltonian matrix {H~'t (J»), 2n + /( :;; /ir, 2n' + /(' :;; /ir; ¡.e. one should solve aneigenproblem for the set of equations:

2n'+K'$.Ñ'" [¡In' r' (J) _ Ó Ó' E(J)] (» (J) _ ()~ n r n, n' r. f' A 1'n' r' - 1

ni, 1"

2n + /( :5 Ñ. (14)

The usual approximation is to treat E¿J=3/2) and the eigenvector h~,of(J=3/2)} as theground state energy and (he corresponding wave function in the oscillator representation.

50FT OIPOLE MOOE IN •• LI ANO TIIREE BOOY CONTINUUM 67

Such approximation is unable to describe the slowly dying asymptotic tail of the haloneutron space distribution. Thus, in order to describe the neutron halo properties of the11Li nucleus, one is pushed to account for the asymptotic region N ~ Ñ. For boundstates this account is equivalent to the location of S-matrix poles. The S-matrix can becalculated by the equation [17]:

where the matrix elements of matrices A (+) and A (-) are of the form

(,,) ('")() ñ',r'( r' C('") ()Arr, = .Ir, r' Cñ,K E + Pñ, r E) Tñ',ñ'+I ñ'+l,K' E ,

(15)

(16)

the kinetic energy rnatrix elements T[,n+I = - h; J(n + l)(n + [( + 3), ñ = (Ñ - [()/2,ñ' = (Ñ - [(')/2, and

(Al (A)pn',r'(E) = '\' !'n'r'!'nrn,r L EA- E '

A(17)

To calculate the bound state energy, Le, to locate the corresponding S-rnatrix pole, oneshould solve the non linear equation [17]

detA(+) = 0, (18)

which is easily obtained from (15), The coefficients D~Jk1,1, (E) of the expansion (4) in theasyrnptotic region N ~ Ñ are to be obtained using (6) or (7) for the continuum spectraor bound states, respectively,

The coefficients D~Jk1,ly(E) in the inner regio n N :s; Ñ are calculated by the expression:

(J) () _ L n r ( ) r' (J)DnK11 E - - Pñ'f' E Tñ, ñ'+lD-'+l K'IIt,(E),z 11 • n, z 11r'

(19)

where ñ' = (Ñ - [(')/2 and Di1I, K 1,1, (E) is given by (6,7). The asyrnptotic normalizationconstants for the bound states (see (7» can be found by

Qr = QSrr(E), (20)

where SrrCE) are the matrix elements of the scattering matrix for E < 0, and Q is fixedby the numerical normalization of the wave function.

68 Yu. F. SMIRNOV ET AL.

3. TIIE RESULTSOF THE CALCULATIONS

The interactions of the valence neutrons with each other and with the cluster 9Li aredescribed by the potentials V12(rI2) and VI3(r13) = V23(r23), respectively. We use thefol!owing parameterization of the potentials [9]:

V;j(r) = VS) exp [- (r/b~;)f] + V;}2) exp [- (r/b~~)f] ,

\/,(1) (2) (1)12 = -31 MeY, V12 = 0, b12 = 1.8 fm;

Vl(32) = -1 MeY, b(l) - 2 4 f13 - . m,

(2)b13 = 3.0 fm.

In the external asymptotic region N ~ Ñ we consequently al!ow for channels r charac-terized by K = Kmin, K~in + 2, ... (Kmin is the mini mal possible value of K for a givenJ) until the convergence for al! physical properties under consideration is achieved. Theconvergence is found to be very rapid, and the al!owance for the decay channels withK > Kmin + 2 does not yield any visual variation of the results. So, we consider in theexternal asymptotic region N > Ñ the channels with K ~ Kmin + 2 only. Note, thatcomponents with al! possible values of K ~ Ñ are accounted for in the calculation of thewave function in the inner region N ~ Ñ.The parameter ñw is set to be equal to 7.1 MeY in our calculations. This value corre-

sponds approximately to the minimum of ground state energy Eo.

3.1. The ground state

The results for the 11Li ground state for different values of the truncation parameterÑ are presented in the tableo The ground state energy values E~d) obtained by the purediagonalization of the truncated Hamiltonian matrix are listed in the second column, whilethe values Eo which are the solutions of the Eq. (18) are listed in the third column. It isseen that by locating the S-matrix pole using Eq. (18) which is equivalent to the al!owancefor the long asymptotic tail of the wave function, we improve essential!y the convergencefor the binding energy.The mean square radius of the neutron halo is defined by the equation:

2 2 9 2 ñ 2ó(r ) = (r )¡1 -ll(r )9 = 11m)p ), (21)

where (r2):{2 and (r2)~/2 are the mean square radii of 11Li and 9Li, respectively. The values(r2)1/2 (d) and (r2)1/2 obtained by the pure diagonalization of the truncated Hamiltonianmatrix and with the al!owance for the asymptotic tail of the wave function, respectively,are presented in the 4-th and the 5-th columns of the table.It is seen from the table that (r2)1/2 (d) and (r2)1/2 converge to the experimental value

3.16:1:0.11 fm [21 from opposite directions: while (r2)1/2 (d) increases, (r2)1/2 decreases withÑ. Such a behaviour is easily explained by the fol!owing simple reasoning. To describe the

50FT DIPOLEMODEIN "LI ANDTHREEBODYCONTINUUM 69

TABLE 1. 11Li ground state properties.

Ground state energy, Neutron haloTruncation MeV mean square radiusboundary Ñ ( 2)1/2 fT 11 1 m

E(d) E. ( 2)1/2 (d) ( 2) 1/2• T 11 T 11

12 -0.012 -0.150 2.83 3.31

16 -0.116 -0.199 2.91 3.29

20 -0.171 -0.225 2.98 3.31

24 -0.202 -0.240 3.04 3.32

Experiment -0.247:1: 0.080 3.16:1: 0.11

wave funetion at larger and larger values of the hyperradius pone is pushed to allow forthe oseillator basis funetions eharaeterized by larger and larger values of n = (N - K)/2[161. Thus, enlarging the value Ñ we aeeount for larger values of p in the diagonalizationproeedure. As a result, (r2)1/2 (d) inereases. At the same time, in the ealeulation of (r2)1/2we allow for an infinite number of basis funetions in the deeomposition (4). Thus, theenlarging of Ñ eorresponds to the more complete deseription of the attraetive interaetionin the system. This results in the squeezing of the system, and in the deerease of (r2)1/2.

Figure 1 presents the transverse momentum distribution of the cluster 9Li in 11Li. Thismomentum distribution is eurrently supposed (see, e.g., [12]) to be proportional to the 9Litransverse momentum distribution ddN in the fragmentation of high-energy 11Li beams

Plon target nucleL The experimental data for 11Li fragmentation [21 are also presented onthe figure. The results of both ealculations (pure diagonalization and the allowanee forthe asymptotie tail of the wave funetion) are seen to be in good agreement with theexperimento

3.2. The soft dipote mode

The dipole transition operator in our model is of the form

M(El¡,) = - N~Z ey YI~(!i), (22)

where e is the proton eharge, A = 11, Z = 3 and the number of valenee neutrons Nv = 2.The operator (22) eorresponds to the excitation of the three-body cluster oseillation modeonly. The excitation energy of the first exeited state of 9Li is relatively high (~ 4 MeV). So,low-energy El-transitions eorrespond to the exeitation of the cluster degrees of freedomonly and should be deseribed by the operator (22). The cluster EWSR associated withthe operator (22) is of the fonn

(23)

dN Idpx, arb. units

70 Yu. F. SMIRNOV ET AL.

2.0

1.8

1.6

1.4

1.2

1.0

0.80.6

0.40.2

0.0-200 -150 -100 -50 O 50

Px, MeV le100

Ñ= 8-Ñ=!6 -<>-

Ñ=24 -

¡ ¡¡;

150 200

FIGURE l. The transversalmornentum distribution for the cluster 9Li in the ground state of 11Li.The experimental points are taken frorn the paper [2).The curve l corresponds to the diagonaliza-tion approach, the curve 2 is calculated with taking account of the asyrnptotic tail of the groundstate wavc function.

where the reduced probability of the El-transition from the ground state to the state withthe energy El

6(EI; El - Eo) = JI :L I(Jlr.M(El)IIJo)12,2. 0+1 JI

(24)

and Jo = 3/2 and JI = 1/2, 3/2, 5/2 are the total three body angular momenta for theground and excited states, respectively.

The strength function for the cluster EI-transition is displayed in Fig. 2. The calcula-tions were carried out with Ñi = 20, and ÑI = 21. The curve 3 on Fig. 2 is obtained withallowance for asyrnptotics of the wave functions in both the ground and excited states;curve 2 corresponds to the allowance for the wave function asymptotics for excited statesonly. Curve 1 presents the results of the diagonalization calculations. It is Seen from thefigure that the allowance for the ground state wave function asyrnptotics yields consider-able changes of the strength function, e.g., it gives a decrease of 0.9 MeV in the energy ofthe maximum, corresponding to the 80ft dipole mode.

Figure 3 shows the comparison of the results of our calculations of cluster 6(El; EI-Eo) with the parametrization of experimental data of Ref. [231. The agreement is reason-able. The form of the 6(El; El - Eo) peak is well reproduced, the discrepancy in theposition of the 6(E!; El - Eo) maximum is supposed to be eliminated by the adjustmentof the potentials. The results of the 6(E!; El - Eo) calculations of Refs. [12,13] are also

50FT DIPOLEMODEIN "LI ANDTIlREE BODYCONTINUUM 71

8(El; i -> j), e2 fm2/ MeV

1.6

1.4

1.2

1.0

0.8

0.6

0.40.2

0.0O 1 2 3

E, MeV4 5

l +-2-3-

6

FIGURE2. The distribution of strength of El transitions.

depieted on Fig. 3. The three-body cluster ealculations with the allowanee for demoeratiedeeay ehannels of Ref. [12J nieely reproduce the energy of the soft dipole mode. Neverthe-less, the form of the peak in our ealculations is reprodueed better. Note, that the authorsof Re£. [121 used another set of the potential parameters. The calculations of Re£. [131 withthe allowance for two-body decay ehannels failed to reproduce both the position and thefonu of the B(E1; El - Eo) peak.

The soft dipole mode exhausts about 90% of the cluster EW5R assoeiated with theoperator (22), 5du,,(E1). Nevertheless it is easy to obtain that

(25)

where 5'o,(E1) is the total EWSR aeeounting for excitations of all nucleons. So, thecontribution from the soft dipole mode to the total EWSR is relatively small. In thevicinity of the sharp B(E1; El - Eo) maximum at the excitation energy E", 1-2 MeVonly ~ 8% fraetion of the total EWSR is exhausted. Nevertheless, the aceount for the softdipole mode results in an essential inerease of the electromagnetic dissoeiation cross seetionof 0.8 CeV jnucleon 11Li beams on Pb and Cu targets. The wide space distribution of thehalo neutrons density is also well-manifested in the eleetromagnetie dissoeiation of Il Libeam. The eontribution of the large-distance part of the wave funetion to the cross seetionis about 50%. The calculations have been performed by the equivalent photon method [241.The only parameter of the method is the minima! value of the impact para meter bmin. \Veuse for bmin (he values of 9.0 fm for Pb and 6.8 fm for Cu target nuclei, respectively. Thesequantitics are tite SUlIlS of thc 11Li and target Iluelen::,; chargc radiL \Vith thcse values ofbmin we obtain for the electromagnetic dissociation cross sections the values of 0.966 barnfor the Pb target and 0.132 barn for the Cu target; the corresponding experimental values

72 Yu. F. SMIRNOV ET AL.

B(Elj i --+ f), e2 fm2 / MeV1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.00.0 0.2 0.4 0.6 0.8 1.0 1.2

E, MeV1.4 1.6

1-2 -><-3 -<>-

4-

1.8 2.0

FIGURE 3. Comparison oí our results íor B (El; g.S. ~ continuum) in 11Li with results oí otherauthors. 1. this work (J-matrix method), 2. Reí. [12]' 3. Reí. [13]' 4. experimental data parametriza-tion oí Reí. [23).

are 0.890:f: 0.110 barn and 0.21 :f:0.04 barn, respectively [41. EO- and E2-transitions giveonly 1.2% contribution in the cross sections.

E1-transitions in JI Li have been studied in the framework of RPA + two-body-contin-uum model in Refs. [11,25]. Though our model assumptions differ significantly from theones of Refs. [11,251, the results are in good agreement. For example, the excitation energyvalues corresponding to the peaks of the function E(E1; El - Eo) displayed on the Fig. 2,are very close to the values that one can find in Refs. [11,25].

4. CONCLUSIONS

It is shown, that cluster model 9Li + n + n yields a good description of the ground stateproperties and E1-transitions in the JI Li nucleus. The oscillator functions expansion tech-nique may be used in the studies of weakly-bound systems with long-tailed wave functions,e.g., in the study of neutro n halo properties. For both bound and continuum states thecorrect account of the wave function asymptotics in the framework of the oscillator repre-sentation of scattering theory is very important in such studies. Low-energy E1-transitionsin JI Li are of the cluster nature. The widths and the position of resonant states calculatedin the democratic decay approximation are in a reasonable agreement with experimento

SOFT DIPOLE MODE IN •• LI AND TIIREE BODY CONTINUUM 73

ACKNOWLEDGMENTS

We are thankful to Profs. J. Dang, B. Danilin, !. Thompson and J. Vaagen for valuablediscussions.

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