Nuclear Physics A 705 (2002) 335–351

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Asymptotics of three-body bound state radial wavefunctions of halo nuclei

R. Yarmukhamedova,b, D. Bayeb,∗, C. Leclercq-Willainb

a Institute of Nuclear Physics, Uzbekistan Academy of Sciences, Tashkent 702132, Uzbekistanb Physique Nucléaire Théorique et Physique Mathématique, C.P. 229, Université Libre de Bruxelles,

B 1050 Bruxelles, Belgium

Received 17 January 2002; received in revised form 30 January 2002; accepted 4 February 2002

Abstract

Asymptotic expressions for the radial partial waves of a bound-state wave function of a three-bodysystem in relative coordinates are obtained in explicit form, when the relative distance between twoparticles tends to infinity. This formula can be applied, for instance, to wave functions of halo nucleifor large distances of one of the valence neutrons and the core. Besides a well-known exponentialdecrease as a function of a hyperradius, the derived asymptotic expressions involve factors thatcan influence noticeably the asymptotic values of the three-body radial wave functions for somedirections in the configuration space. The obtained asymptotic forms are applied to the analysis ofthe asymptotic behaviour of accurate6He three-bodyαnn wave functions derived with the Lagrange-mesh method. The agreement between the calculated wave function and the asymptotic formula isexcellent up to distances close to 20 fm. Information is extracted about the values of the three-bodyasymptotic normalization factors. 2002 Elsevier Science B.V. All rights reserved.

1. Introduction

Halo nuclei are one of the most interesting topics of nuclear physics [1]. Their wavefunctions extend to much larger distances than for normal nuclei. Two-neutron halonuclei are particularly striking since the lowest breakup channel is a three-body channel.Considered as a three-body system, such a nucleus is bound in spite of the fact that its two-body subsystems are unbound. The asymptotic form of its wave function must thereforehave properties which are not often encountered in quantum mechanics.

* Corresponding author.E-mail address: dbaye@ulb.ac.be (D. Baye).

0375-9474/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0375-9474(02)00672-3

336 R. Yarmukhamedov et al. / Nuclear Physics A 705 (2002) 335–351

Early models of halo nuclei concentrated on explaining the large radius of these nucleiand on deriving the breakup probability distributions [1]. Little attention has been devoteduntil now to the asymptotic properties of the wave function. In two-body systems however,the asymptotic normalization constant is an important characteristics of a bound-state wavefunction [2,3]. The study of the asymptotic properties of the wave function of a halonucleus is however not simple because it requires accurate wave functions up to largedistances.

Since 1990, various analytic and semianalytic three-body approaches were developedfor a description of the structure of light nuclei, and especially of halo nuclei. Letus mention, e.g., the multicluster stochastic variational method [4–7] and the methodof hyperspherical harmonics [1]. The microscopic cluster model offers a many-bodydescription of the same nuclei [8–11]. All these models lead to approximate bound-statewave functions whose validity does not necessarily extend to large distances. Analyticapproximations in the asymptotic regions are therefore useful either to test the accuracy ofapproximate wave functions or to correct them in the asymptotic region. Recently, a newapproach, the Lagrange-mesh technique [12,13], allowed determining accurately the wavefunction up to large distances. The aim of the present paper is to derive an approximateexpression for the asymptotic form of the wave function of a halo nucleus and to compareit with Lagrange-mesh results.

The asymptotics of bound states of a three-body system was first studied by Merkur’ev[14]. The derived expression describes the asymptotic behaviour of the full wave function.In Ref. [15], the asymptotic expression has been derived for a three-bodyradial wavefunction in the case of short-range (nuclear) interactions between three particles. Thisresult was established in the context of hyperspherical coordinates for large values of thehyperradiusR. It concerns partial waves in hyperspherical harmonics. The limitR → ∞means that one of the Jacobi coordinates, or both, tend to infinity. The obtained asymptoticexpression contains an exponential function depending on the hyperradius [14] but alsoinvolves a factor that can influence noticeably the asymptotic values of the three-bodywave function for some directions in the configuration space. In Ref. [15], this asymptoticexpression has been compared with the asymptotic behaviour of a three-bodyαnn wavefunction of the6He nucleus derived in Ref. [5] within the framework of the multiclusterdynamical model. As a result, in Ref. [15], information has been obtained about thevalue of the three-body asymptotic normalization factor as a function of the hyperangleϕ = arctan(y/x), where x and y are a pair of modified Jacobi coordinates [14]. Inparticular, it was shown that the value of the three-body asymptotic normalization factoris rather sensitive to the form of theαn potential. However, it should be noted that inRef. [5] the binding energyε calculated for the ground state of6He differs significantlyfrom the experimental value (εexp = 0.975 MeV) for the employed potentials. It shouldbe emphasized that the three-body asymptotic normalization factor is a fundamentalcharacteristic of three-body bound systems which plays the same role as the asymptoticnormalization coefficient of the radial wave function for a two-body system [2,15].

In Refs. [12,13], the regularized Lagrange-mesh technique [16] has been applied tothe calculation of the wave function of the6He halo nucleus within the framework of thethree-bodyαnn model. In this method, the relative coordinatesr31 andr23 providing thelocations of the halo neutrons with respect to the4He core have been chosen as independent

R. Yarmukhamedov et al. / Nuclear Physics A 705 (2002) 335–351 337

variables of the three-body wave function. The wave function is expanded in partial wavesin this system of relative coordinates. With a slight increase of a realisticαn interaction[17], this approach provides an accurate binding energy for the ground state of6He(ε = 0.972 MeV). Testing the accuracy of the asymptotic behaviour of such a model wavefunction is important. It is also a source of information about the three-body asymptoticnormalization factors.

In the present work, the asymptotic behaviour of the radial wave functions of a three-body bound(123)-system is studied for large values of the relative coordinatesr31 andr23.The results are compared with an improved version of the Lagrange-mesh wave function ofRef. [13], involving more mesh points. The contents of this paper is as follows. In Section 2,the asymptotic expression for the radial component of the three-body wave function isderived. In Section 3, an improved version of the three-bodyαnn wave function obtainedin Ref. [13] for the ground state of the6He nucleus on the basis of the Lagrange-meshtechnique is described. In Section 4, the asymptotic behaviours of the partial waves of thiswave function are tested with respect to the obtained formula. The information about thethree-body asymptotic normalization factors is analyzed and discussed. Conclusions aregiven in Section 5.

2. Asymptotic behaviour of three-body radial wave functions

2.1. Fourier transform of partial waves

Let us consider the bound three-body systema = (123) consisting of two “valence”neutrons (say, particles 1 and 2) and of a core 3. Let us writer ij for the relative coordinatebetween the centres of mass of the particlesi andj andqij for the corresponding relativemomentum. Ifmj is the mass of particlej , we denote asµ(ij) = mimj/mij the reducedmass of the(ij) subsystem andµ(ij)k = mijmk/m the reduced mass of the(ij)k system,wheremij =mi +mj andm=m1 +m2 +m3.

The Fourier transform of the total wave functionΨ (r23, r31) of the three-body boundsystem is defined by

Ψ (q23,q31)=∫dr23dr31e

i(r23·q23+r31·q31)Ψ (r23, r31), (1)

whereΨ (q23,q31) is the three-body wave function in the momentum representation.The wave functionsΨ (r23, r31) and Ψ (q23,q31) can be expanded in series on

the complete orthonormalized sets of coupled spherical harmonicsYl23l31LML of theirrespective spaces. For example, in momentum space, the eigenstates of the square of thetotal angular momentumL of the three-body (123) system and of its projectionLz alongthez axis [15] read

Yl23l31LML(q23, q31)=∑ν23ν31

CLML

l23ν23l31ν31Yl23ν23(q23)Yl31ν31(q31), (2)

338 R. Yarmukhamedov et al. / Nuclear Physics A 705 (2002) 335–351

whereCcγ

aαbβ is a Clebsch–Gordan coefficient andqij = q ij /qij . The partial waves inmomentum space read

Ψν(q23, q31)= (4π)−1∫dΩq23

dΩq31Y ∗l23l31LML

(q23, q31)Ψ (q23,q31), (3)

with ν = l23l31Ls12S. Hereinlij is the relative orbital angular momentum of particlesi

andj , L = l23 + l31, s12 = s1 + s2 andS = s12 + s3, wheresj is the spin of particlej .Then, in momentum space, the three-body radial wave functions become

Ψν(q23, q31) = 4π2il23+l31

∞∫0

dr23r223jl23(q23r23)

×∞∫

0

dr31r231jl31(q31r31)Ψν(r23, r31), (4)

wherejl(x) is a spherical Bessel function [18].From relation (4), one obtains the following symmetry properties of the partial wave

functions in momentum space

Ψν(q23, q31)= (−1)l23Ψν(−q23, q31)= (−1)l31Ψν(q23,−q31), (5)

sincejl(−x)= (−1)ljl(x). Then one can invert Eq. (4) as

Ψν(r23, r31) = i−l23−l31

(4π2)2

∞∫−∞

dq23q223jl23(q23r23)

×∞∫

−∞dq31q

231jl31(q31r31)Ψν(q23, q31). (6)

In this expression, the integration domains have been extended by using Eq. (5). This isconvenient for the derivation of approximate expressions but one has to keep in mind thatsome parts of the integrands are redundant.

2.2. Vertex function

The momentum-space wave functionΨ(q23,q31) is related to the noncovariant vertexfunctionW(q23,q31) for the virtual decay

a ≡ (123)→ 1+ 2+ 3 (7)

by the relation [19]

Ψ (q23,q31)= −W(q23,q31)

ε + εa. (8)

Herein the energyε reads [12]

R. Yarmukhamedov et al. / Nuclear Physics A 705 (2002) 335–351 339

ε = q231

2µ(23)1+ (q23 + λ2q31)

2

2µ23(9)

= q223

2µ(31)2+ (q31 + λ1q23)

2

2µ31, (10)

whereλ1 =m1/m31 andλ2 =m2/m23, andεa is the binding energy of the systema withrespect to the virtual decay (7)(εa > 0). We useh= c = 1 throughout.

The vertex functionW(q23,q31) is determined by the residue at the pole singularity(ε = −εa) of the amplitude of the three-body(123) scattering [19]. If, in the bound(123)system, two-particle subsystems (αβ) (αβ = 12, 23 or 31) can be bound with bindingenergyεαβ with respect to theα + β channel, the vertex functionW(q23,q31) has so-called two-particle poles as a function of the relative kinetic energyEαβ of the particlesα andβ at Eαβ = −εαβ . According to Eq. (8), the wave functionΨ (q23,q31) has thesame poles as the vertex functionW(q23,q31) and possesses the so-called three-body polesingularity atε = −εa .

Taking into account expression (8), one can perform the integrations over the variablesΩq23

andΩq31in Eq. (3). To this end, one makes use of the partial wave expansion

W(q23,q31)= 4π∑

l23l31LML

Yl23l31LML(q23, q31)Wl23l31Ls12S(q23, q31), (11)

whereWl23l31Ls12S(q23, q31) is a partial vertex function, and of the relation [20]

1

ε + εa= 4πm3

q23q31

∑lm

(−1)lQl

[ζ(q23, q31)

]Ylm(q31)Y

∗lm(q23). (12)

Herein,Ql(ζ ) is a Legendre function of the second kind [18] and

ζ(q23, q31)= q231 + λ1λ

−12 (q2

23 + σκ2)

2λ1q23q31,

whereκ = √2µ(23)1εa andσ = µ23/µ(23)1 = (1− λ1λ2)λ2/λ1. Note thatζ(q23, q31) > 1

for realq23 andq31. Furthermore, after performing an integration over the angle variables,one can reduce the radial wave function (6) to the form

Ψν(r23, r31) = − m3

(4π2)2

√l31l23

r23r31(−1)l31+l23+L

×∑

l′23l′31L

′L′

√l′31l

′23Il23l31l

′23l

′31L

′L(r23, r31)

×(l23 l′23 L′0 0 0

)(l31 l′31 L′0 0 0

)l31 l′31 L′l′23 l23 L

, (13)

with j = 2j + 1 and

Il23l31l′23l

′31L

′L(r23, r31) =∞∫

−∞dq23e

iq23r23fl23(q23r23)

∞∫−∞

dq31eiq31r31fl31(q31r31)

×QL′[ζ(q23, q31)

]Wl′23l

′31Ls12S

(q23, q31). (14)

340 R. Yarmukhamedov et al. / Nuclear Physics A 705 (2002) 335–351

In deriving (13) and (14), the representation

jl(x)= 1

2x

[i−l−1eixfl(x)+ (−i)−l−1e−ixfl(−x)

](15)

has been used for the spherical Bessel function [18,20], where

fl(x)=l∑

n=0

(l + n)!n!(l − n)!

1

(−2ix)n. (16)

Because of properties (5) of the partial wave functions, both terms of the r.h.s. of Eq. (15)lead to the same integrals overq23 (or q31) in (6) and (14).

2.3. Asymptotic behaviour

The asymptotic form of the radial wave functionΨν(r23, r31) atr23 → ∞ (or r31 → ∞)

is determined by the two-body cluster poles of theWl′23l′31Ls12S

(q23, q31) partial vertexfunctions that are associated with a formation of possible bound states in the two-bodysubsystems [19] and by the three-body singularity (branch point) arising due to a presenceof the functionQL′ [ζ(q23, q31)] in the integrand of Eq. (14). The latter singularity isdefined from the equation

ζ(q23, q31)= ±1. (17)

The solution of Eq. (17) leads to the branch type of singularities on the variableq23, locatedat

q(1,2)23 = ±λ2q31 + i

√σ

√q2

31 + κ2 (18)

and

q(3,4)23 = ∓λ2q31 − i

√σ

√q2

31 + κ2, (19)

in theq23 plane (see Fig. 1).An explicit form of the cluster asymptotics can easily be derived by using the results of

Ref. [14]. Here, we are interested in an asymptotic expression forΨν(r23, r31) at r23 → ∞determined by means of the extraction of contributions from the three-body singularities,given by relations (18) and (19), into the radial wave functionΨν(r23, r31) determinedby Eqs. (13) and (14). To this end, we can employ the following standard techniques inEq. (14). First, a deformation of the contour of integration into the upper half of theq23plane is carried out as shown in Fig. 1. Then, in the integrand of the obtained integral, thefunctionQL′(ζ ) is written as (Eq. 8.834.2 in Ref. [20])

QL′(ζ )= 1

2PL′(ζ ) ln

(ζ + 1

ζ − 1

)−

L′∑n=1

1

nPn−1(ζ )PL′−n(ζ ),

wherePn(ζ ) is a Legendre polynomial. As a result, one can separate the part of the integralrunning along the cutC1 in the q23 plane starting fromq(1)23 till q(2)23 that corresponds tothe sought asymptotics. When, forr23 → ∞, one extracts in the integral contourC1 the

R. Yarmukhamedov et al. / Nuclear Physics A 705 (2002) 335–351 341

Fig. 1. Contour in theq23 complex plane.

contribution from the singular pointsq23 = q(1,2)23 , expression (14) can be reduced to the

form

Il23l31l′23l

′31L

′L(r23, r31)≈2∑

j=1

(−1)j+1I(j)

l23l31l′23l

′31L

′L(r23, r31), (20)

with, for r23 → ∞,

I(j)

l23l31l′23l

′31L

′L(r23, r31) = π

r23ξ(j)

L′

∞∫−∞

dq31eiq31r31+iq(j)23 r23

× fl31(q31r31)fl23

(q(j)

23 r23)Wl′23l

′31Ls12S

(q(j)

23 , q31), (21)

whereξ(1)L′ =1 andξ(2)

L′ = (−1)L′.

At the limit r23 → ∞, the integration overq31 in Eq. (21) can now be performed byusing the saddle-point method [21]. The saddle point is a solution of the equation

dS(j)(q31)

dq31= 0, (22)

where

S(j)(q31)= i

(q(j)

23 + r31

r23q31

)(23)

andq(j)23 is defined in expression (18). The saddle point is given by

q(j)31 = i2µ(23)1

√εa

| r31 − (−1)jλ2r23 |R(j)(r23, r31)

, (24)

342 R. Yarmukhamedov et al. / Nuclear Physics A 705 (2002) 335–351

where

R(j)(r23, r31) =√

2µ23r223 + 2µ(23)1(r31 − (−1)jλ2r23)2 (25)

≡√

2µ31r231 + 2µ(31)2(r23 − (−1)jλ1r31)2 (26)

is a modified hyperradius. The expressionsR(1) andR(2) are respectively the maximumand minimum values of the usual hyperradiusR [15] (or

√2ρ [1]) for fixed values of

r23 and r31, when the angle between the relative coordinates varies. They coincide withR when the three particles are aligned:R(1) = R when 1 and 2 are on opposite sides of3 andR(2) = R when they are on the same side. Whenm1 = m2, one easily verifies thatthe equivalent expressions (25) and (26) are symmetric inr23 andr31. Inserting expression(24) into (18), one obtains

q(j)

23 = i2µ(31)2√εa

| r23 − (−1)jλ1r31 |R(j)(r23, r31)

. (27)

As a result, the proper three-body asymptotic form of the radial wave functionΨν(r23, r31) is derived forr23 → ∞ as

Ψ (as)ν (r23, r31) = 1

r23r31

C(1)ν (r)fl23

(q(1)23 r23

)fl31

(q(1)31 r31

)exp[−√

εaR(1)(r23, r31)

][R(1)(r23, r31)

]3/2

−C(2)ν (r)fl23

(q(2)23 r23

)fl31

(q(2)31 r31

)exp[−√

εaR(2)(r23, r31)

][R(2)(r23, r31)

]3/2

.

(28)

The same expression is valid forr31 → ∞ since the saddle points and modified hyperradiido not change. This means that the asymptotic formula (28) is valid when bothr31 andr23

tend to infinity. However, in the derivation of the saddle pointsq(j)31 andq(j)23 , one implicitly

assumes that the ratior is larger thanλ2 and smaller than 1/λ1. Hence, as also shown bynumerical examples, Eq. (28) is not valid when one of the relative coordinates is too small.

In Eq. (28), the asymptotic normalization functionsC(j)ν (r) are related to the partial

vertex functions by

C(j)ν (r) = N(−1)l31+l23+L ∑

L′l′23l′31

ξ(j)

L′ L′√l31l23l

′23l

′31Wl′23l

′31Ls12S

(q(j)

23 , q(j)

31

)

×(l23 l′23 L′0 0 0

)(l31 l′31 L′0 0 0

)l31 l′31 L′l′23 l23 L

, (29)

where the factorN is given by

N = (2π)−5/2m3(m1m2m3/m)1/2ε

1/4a . (30)

The coefficientsC(j)ν depend on the ratio of relative coordinates

r = r31/r23 (31)

R. Yarmukhamedov et al. / Nuclear Physics A 705 (2002) 335–351 343

sinceq(j)23 andq(j)31 only depend on this ratio. Notice that Eq. (28) is symmetrical inr23 andr31 whenm1 =m2.

The present asymptotic expression is not directly comparable to the relation derivedin Ref. [15] since the partial wave expansions are different. However, a comparison ispossible when particle 3 is very heavy because the Jacobi and relative coordinates coincidein that case. Whenm3 → ∞ (λ1, λ2 → 0), one hasx → √

2m1r31 andy → √2m2r23. The

expressionsR(1) andR(2) tend toR and the ratior is simply related to the hyperangle.Moreover, one obtainsC(1)

ν = C(2)ν sinceL′ = 0 in Eq. (29). Indeed, the latter is easily

derived from the right-hand side of Eq. (12) at the limitm3 → ∞ by making use of formula8.771(2) of Ref. [20]. Hence, by applying the L’Hospital rule, Eq. (28) becomes equivalentto the asymptotic formula (15) of Ref. [15]. Whenm3 is not large, the two expressions maybehave quite differently.

3. Approximate 6He wave functions in the three-body model

In this section, we test the asymptotic behaviour of a6He wave function, derived withinthe framework of theαnn three-body approach based on the Lagrange-mesh technique[12,13]. As mentioned in the Introduction, this wave function accurately corresponds tothe experimental binding energy.

Within the framework of the Lagrange-mesh technique [16,22,23], a partial wave of theαnn wave function for the ground state of6He is represented as [13]

ΨllLSS(r23, r31)= (r23r31)−1

N∑i1i2=1

cLli1i2Fi1i2(r23, r31). (32)

In this expression, theFi1i2 are Lagrange basis functions and thecLli1i2 are variational

coefficients. Since the total angular momentumJ of 6He is zero, its total orbital momentumL and its total spinS (which is equal to the total spins12 of the neutrons) are equal(L= S = s12 = 0 or 1). Since the parity is positive, the relative orbital momental23 andl31

take the common valuel. The symmetrized two-dimensional Lagrange functions read

Fi1i2(r23, r31) = [2(1+ δi1i2)

]−1/2h−1

× [fi1(r23/h)fi2(r31/h)+ fi1(r31/h)fi2(r23/h)

], (33)

with the one-dimensional Lagrange–Laguerre functions defined as [13]

fi(x)= (−1)ix−1/2i x(x − xi)

−1LN(x)e−x/2, (34)

where LN(x) is a Laguerre polynomial and the Laguerre zerosxi are solution ofLN(xi)= 0 [13,16]. The basis functionsFi1i2(r23, r31) are associated withN2 mesh points(hxi1, hxi2) where they satisfy the Lagrange property

Fi1i2(hxi′1, hxi′2)∝ δi1i′1δi2i′2, (35)

344 R. Yarmukhamedov et al. / Nuclear Physics A 705 (2002) 335–351

i.e., they vanish at all mesh points but one. The wave function is normalized according to

∑Ll

∞∫0

dr23r223

∞∫0

dr31r231

[ΨllLSS(r23, r31)

]2 = 1. (36)

With the Gauss–Laguerre quadrature associated with the mesh, this normalizationcondition reads∑

Ll

∑i1i2

(cLli1i2

)2 = 1. (37)

Thanks to the Lagrange property (35) and the use of the Gauss quadrature associated withthe mesh, the variational calculation is replaced by a much simpler mesh calculation witha diagonal potential matrix, without apparent loss of accuracy [23]. The factorh scalesthe Laguerre zeros to the dimensions of the physical problem. It can be considered as anapproximate non-linear variational parameter. Note that all the basis functions in (33) havethe same exponential decrease exp(−r/2h). Therefore, one cannot expect them to have acorrect asymptotic behaviour beyond abouthxN−1.

The values of the coefficientscLli1i2 were first obtained in Ref. [12] with theαn interactionof Ref. [17]. However the binding energy is then underestimated. In Ref. [13], theexperimental binding energy was accurately reproduced by multiplying theαn interactionby 1.01. The corresponding wave function contains partial wavesl = 0 to 18 and isobtained withh = 0.3 fm andN = 20 (210 basis states per partial wave for a total of7770 basis states). In the following, we shall make use of an improved wave function withthe same partial waves, same scale factor andN = 30 (465 basis states per partial wavefor a total of 17 205 basis states). This case corresponds tohxN−1 ≈ 27 fm and should bevalid up to much larger distances than the previous wave function for whichhxN−1 ≈ 16fm.

The dominant partial waves for6He are, in decreasing order of importance,(L, l) =(0,1) (79.5 %), (1,1) (15.7 %),(0,2) (2.9 %) and(0,0) (1.1 %). The(1,2) componentwhich is also discussed in the following amounts to 0.1 %.

4. Results and discussion

First we have to find a procedure to determine the asymptotic normalization functionsC(j)Ll (r) (from now on, the quantum numbers summarized by indexν are fully determined

by the valuesL and l). Because of symmetry, only ratiosr 1 will be considered. Wedo not discussr larger than about 2.5 because these values correspond either to smallr23values for which the asymptotic expression (28) is not valid, or to larger31 values for whichthe approximate wave function (32) becomes inaccurate.

After a number of tests, we have found that the most convenient procedure is to fix thetwo coefficientsC(1)

Ll andC(2)Ll by fitting them to two values of the partial wave function

separated by a distance9r23 for a fixed ratior = r31/r23. The optimal value of9r23 mustnot be too small, nor too large. The values 3 to 4 fm seem to be adequate. When fitting theasymptotic normalization functions, the obtained values depend on the chosen points. In

R. Yarmukhamedov et al. / Nuclear Physics A 705 (2002) 335–351 345

each case, we have searched for a region where the obtained values are stable within 10 %and we have then tested the validity of the obtained fit inside that region.

Examples of the search are displayed in Table 1. The obtained values are shown as afunction of different choices for the fitting points(r23, r31) and(r23 +9r23, r31 + r9r23)

for the dominant partial waves(L, l)= (0,0), (0,1) and (0,2) for spin 0, and (1,1) and (1,2)for spin 1. As shown in Table 1, different choices can be found which lead to similar values.Strikingly, the ratioC(2)

Ll /C(1)Ll is very stable. This indicates that the shape of the asymptotic

form is well fixed by the numerical wave function. Only the overall normalization is moresensitive to the fitting points. We shall take that into account later in our presentation ofthe asymptotic normalization functions which will be expressed as an interval of possiblevalues. One observes in Table 1 that the region where the fit is performed moves to largerdistances whenl increases. This reflects the fact that the centrifugal barrier pushes theasymptotic region to larger distances with increasingl.

The validity of the fits performed in Table 1 is illustrated in Fig. 2. In this figure aredisplayed the ratios of the spin-zero numerical partial waves to the asymptotic expression(28) as a function ofr23 for fixed values of the coordinate ratior. In each case the wave-

Table 1Three-body asymptotic normalization factorsC(1)

Ll(r), C

(2)Ll

(r) of the 6He ground state and ratios

C(2)Ll

(r)/C(1)Ll

(r) for different choices of the matching points(r23, r31) and(r23 +9r23, r31 + r9r23) at fixedvalues of the ratior = r31/r23 for the quantum numbersL= 0, 1 andl = 0, 1 and 2

(L, l) r r23 r31 9r23 C(1)Ll C

(2)Ll C

(2)Ll /C

(1)Ll

fm fm fm fm−1/4 fm−1/4

(0,0) 1.0 3.0 3.00 3.0 −87.6 −55.4 0.6335.0 5.00 3.0 −86.6 −54.7 0.6313.0 3.00 4.0 −87.3 −55.2 0.6335.0 5.00 4.0 −83.5 −53.0 0.634

1.5 3.0 4.50 3.0 −66.7 −43.3 0.6494.0 6.00 3.0 −63.5 −41.5 0.6543.0 4.50 4.0 −64.8 −42.1 0.6504.0 6.00 4.0 −61.5 −40.3 0.656

2.1 3.0 6.30 3.0 −52.1 −34.9 0.6704.0 8.40 3.0 −49.3 −33.3 0.6763.0 6.30 4.0 −50.5 −33.9 0.6724.0 8.40 4.0 −47.8 −32.4 0.679

(0,1) 1.1 4.0 4.40 3.0 80.7 39.3 0.4876.0 6.60 3.0 76.9 37.8 0.4914.0 4.40 4.0 79.7 38.8 0.4876.0 6.60 4.0 76.2 37.5 0.492

1.7 4.0 6.80 3.0 61.0 26.7 0.4395.0 8.50 3.0 57.1 25.4 0.4444.0 6.80 4.0 59.4 26.1 0.4405.0 8.50 4.0 56.2 25.0 0.445

2.3 4.0 9.20 3.0 54.8 19.4 0.3545.0 11.50 3.0 49.9 18.0 0.3613.5 8.05 4.0 56.5 19.8 0.3514.5 10.35 4.0 50.6 18.2 0.359

346 R. Yarmukhamedov et al. / Nuclear Physics A 705 (2002) 335–351

Table 1 (continued)

(L, l) r r23 r31 9r23 C(1)Ll

C(2)Ll

C(2)Ll

/C(1)Ll

fm fm fm fm−1/4 fm−1/4

(0,2) 1.3 8.5 11.05 3.0 −17.9 −4.93 0.2759.5 12.35 3.0 −19.7 −5.36 0.2718.0 10.40 4.0 −17.7 −4.89 0.2709.0 11.70 4.0 −19.5 −5.31 0.273

1.7 8.0 13.60 3.0 −14.7 −3.16 0.2159.0 15.30 3.0 −16.4 −3.47 0.2127.5 12.75 4.0 −14.3 −3.09 0.2168.5 14.75 4.0 −16.5 −3.50 0.212

2.3 6.5 14.95 3.0 −10.4 −1.37 0.1327.5 17.25 3.0 −11.9 −1.55 0.1306.5 14.95 4.0 −11.2 −1.47 0.1317.5 17.25 4.0 −11.8 −1.54 0.130

(1,1) 1.1 5.5 6.05 3.0 −16.7 −8.74 0.5236.5 7.15 3.0 −15.3 −8.20 0.5355.0 5.50 4.0 −16.8 −8.76 0.5216.0 6.60 4.0 −15.4 −8.21 0.533

1.9 6.5 12.35 3.0 −14.4 −6.73 0.4687.5 14.25 3.0 −12.7 −6.25 0.4916.0 11.40 4.0 −14.5 −6.75 0.4657.0 13.30 4.0 −12.8 −6.24 0.488

(1,2) 1.0 9.5 9.50 3.0 2.79 0.83 0.29610.5 10.50 3.0 3.12 0.91 0.2929.0 9.00 4.0 2.80 0.83 0.297

10.0 10.00 4.0 3.12 0.91 0.2921.5 9.0 13.50 3.0 3.40 0.83 0.243

10.0 15.00 3.0 3.78 0.90 0.2398.5 12.75 4.0 3.40 0.83 0.2449.5 14.25 4.0 3.77 0.90 0.240

2.3 8.5 19.55 3.0 3.84 0.50 0.1299.5 21.85 3.0 4.37 0.56 0.1278.0 18.40 4.0 3.90 0.50 0.1299.0 20.70 4.0 4.32 0.55 0.129

function ratios are presented for the pair of fitting points of Table 1 which provides themost central values for theC(j)

Ll (r). For l = 0 andr31 = r23, the two expressions agreewithin 10 % from about 4 to 12 fm and within 25 % up to 20 fm. The same behaviour isobserved forr31 = 1.5r23. For r31 = 2.1r23, the fit is excellent up to about 14 fm whereoscillations start to appear in the numerical wave function. Notice that this corresponds toabout 30 fm forr31, i.e., to values beyond the next-to-last pointhxN−1 of ther31 mesh.

For the dominantl = 1 partial wave, the agreement is even better. An accuracy ofabout 10% is obtained up tor23 equal to about 18 fm forr31 = 1.1r23 and 1.7r23. Forr31 = 2.3r23, the region where the wave function and the asymptotic approximation closelyagree is reduced to values smaller than about 13 fm. For the smalll = 2 component, thequality of the agreement is less good: from 8 to 16 fm forr31 = 1.3r23 and smaller rangesfor higher ratiosr. Notice the more apparent oscillation of the numerical wave functionstarting aroundr23 = 10 fm for r31 = 2.3r23.

R. Yarmukhamedov et al. / Nuclear Physics A 705 (2002) 335–351 347

Fig. 2. Ratios of the spin-zero numerical partial waves to the asymptotic expression (28) as a function ofr23 forfixed values of the coordinate ratior . Each case is calculated for the pair of fitting points which provides the most

central values for theC(j)Ll (r) in Table 1.

The curves displayed in Fig. 2 confirm both the validity of the asymptotic approximation(28) and of the approximate wave function (32). Let us now emphasize that this validitydoes not depend only on the exponential factors of Eq. (28). Forl > 0, the multiplicativefactorsfl23(q

(j)23 r23)fl31(q

(j)31 r31) also play an important role. This point is illustrated by

Table 2, where are displayed ratios of these multiplicative factors for the minimum distancermin23 and maximum distancermax

23 delimiting the domain in which the approximate wavefunction and the asymptotic form agree within 10%. One observes that the multiplicativefactors contribute for a factor 3 to 5.5 to the good agreement obtained in Fig. 2. The 10%agreement observed in that figure for broad ranges ofr23 values would not be possiblewithout those factors.

348 R. Yarmukhamedov et al. / Nuclear Physics A 705 (2002) 335–351

Table 2Influence of the factorsfl(q

(j)23 r23)fl(q

(j)31 r31): ratiosB(j)

l(r) =

fl(q(j)23 r

min23 )fl(q

(j)31 rmin

31 )/fl (q(j)23 r

max23 )fl (q

(j)31 r

max31 ) in intervals

rmin23 r23 rmax

23 for various values ofl andr (L= 0)

l r rmin23 rmax

23 B(1)l

B(2)l

1 1.1 3.4 19.4 4.60 5.451 1.5 3.3 16.9 4.04 4.921 2.1 3.5 14.5 3.25 4.142 1.0 8.7 17.1 3.66 4.232 1.3 8.2 15.2 3.20 3.712 1.9 7.2 12.8 2.98 3.56

Fig. 3. Ranges of obtained functionsC(j)Ll

(r) as a function of the coordinate ratior (delimited by dashed lines)and recommended value (full lines) as a function of the coordinate ratior for L= S = 0 andl = 0, 1 and 2.

R. Yarmukhamedov et al. / Nuclear Physics A 705 (2002) 335–351 349

We have applied the fitting technique described above to the determination of theasymptotic normalization functions of the6He halo nucleus under the model conditionsdescribed in Section 3. The ranges of obtained functionsC

(j)Ll (r) as a function of the

coordinate ratior are delimited by dashed lines in Fig. 3 for spin 0 and in Fig. 4 for spin 1.The recommended value is indicated as a full line. In all cases, one has

∣∣C(1)Ll

∣∣ > ∣∣C(2)Ll

∣∣.One observes that the asymptotic normalization functions are large and negative for thespin-zeros wave and large and positive for the spin-zerop wave. Thes wave values canbe considered as especially large when one takes into account that the probability of thatpartial wave is rather small (about 1 %). Spin 1 components are reduced as expected fromthe smaller probabilities of the spin-one components.

Fig. 4. Same as Fig. 3 forL= S = 1 andl = 1 and 2.

350 R. Yarmukhamedov et al. / Nuclear Physics A 705 (2002) 335–351

5. Conclusion

In this work, we have determined the asymptotic form of the wave function of halonuclei in core-nucleon coordinates. The obtained asymptotic expression (Eq. (28)) is oneof the central results of this paper. Although such an expression was already known forpartial waves in hyperspherical coordinates, we think that the present formula is interestingbecause it allows a simpler physical visualization of the asymptotic behaviour as a functionof the two distancesr23 andr13.

This asymptotic form allowed us to test the validity of a Lagrange-mesh approximatewave function for the6He bound state. As expected from the distribution of mesh points,this analytical approximation is valid over large ranges ofr23 and r31 values, up to orbeyond 20 fm in some cases. For larger values, the order of magnitude of the partial wavefunctions becomes less good and unphysical oscillations may appear.

We have used the regions where the agreement between a partial wave function andits asymptotic expression is excellent to deduce values for the asymptotic normalizationfunctionsC(2)

Ll (r) andC(1)Ll (r) depending on the coordinate ratior31/r23. The shape of

the asymptotic behaviour is determined by the ratioC(2)Ll (r)/C

(1)Ll (r), which is very well

defined by the approximate wave function. The overall normalization is obtained withinabout 10 %.

The asymptotic normalization functions are in principle observable quantities andit would be interesting to compare the present results with experiment. One shouldhowever keep in mind that the present model results depend on assumptions on theαnand nn potentials, in addition to the more general three-body approximation. It wouldbe interesting to test the sensitivity of the asymptotic normalization functions to theseassumptions.

Acknowledgements

This text presents research results of the Belgian program P4/18 on interuniversityattraction poles initiated by the Belgian-state Federal Services for Scientific, Technicaland Cultural Affairs. R.Y. acknowledges financial support from the Fonds National de laRecherche Scientifique. He also thanks the Theoretical Nuclear Physics group, PNTPM-ULB, for the hospitality during the autumns 1998, 2000 and 2001 where this work wasperformed.

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