fulltext

Lagrange-mesh calculations of halo nuclei

( 

)

D. BAYE

Physique Nucl´eaire Th´eorique et Physique Math´ematique C.P. 229, Universit´e Libre de Bruxelles - B 1050 Brussels, Belgium (ricevuto il 24 Giugno 1997; approvato il 15 Ottobre 1997)

Summary. — The Lagrange-mesh technique is applied to a calculation of cluster + n + n three-body systems with effective cluster-neutron and neutron-neutron forces. A variational calculation takes the form of a mesh calculation with the help of the Gauss-Laguerre quadrature, without losing its accuracy. The6He,11Li and14Be halo nuclei are discussed. Matter radii agree with experiment when the binding energies are cor-rectly reproduced.

PACS 21.45 – Few-body systems. PACS 21.60.Gx – Cluster models.

PACS 21.10.Dr – Binding energies and masses. PACS 27.20 –6A19.

PACS 01.30.Cc – Conference proceedings.

1. – Introduction

Reaction data reveal large nuclear radii for different unstable nuclei with a large neu-tron excess [1]. For example, 6He, 11Li and 14Be present much larger radii than the corresponding stable isobars (the ratios are about 1.1, 1.3 and 1.2, respectively). Their bound-state properties are consistent with a large probability of finding two neutrons far from the other nucleons: this is the famous halo effect. Two-nucleon halo nuclei, such as

6_He,11_{Li and}14_{Be, are one of the most interesting topics of present-day nuclear physics.}

A unique definition does not exist. Besides a large radius, their typical features are a rather small binding energy, a dissociation involving two free neutrons, and a lack of bound excited states. To a good approximation, these halo nuclei can be treated as three-body systems composed of a point cluster and two neutrons interacting through effective forces [2, 3].

( 

)Paper presented at the 174. WE-Heraeus-Seminar “New Ideas on Clustering in Nuclear and

Atomic Physics”, Rauischholzhausen (Germany), 9-13 June 1997.

Lagrange-mesh calculations combine the accuracy of the variational method with the simplicity of numerical techniques on a mesh [4-6]. They involve a variational calcula-tion with a basis, the Lagrange basis, where matrix elements are obtained with a Gauss approximate quadrature. This method does not require analytical calculations of matrix elements. Its simplicity is therefore similar to the simplicity of standard mesh methods, such as finite differences. However, contrary to those methods, its accuracy remains com-parable to the accuracy of the original variational calculation. In other words, the Gauss quadrature does not cause any significant loss of accuracy [6].

Here the Lagrange-mesh method is applied to three-body systems. It is a simplified version of the method presented in ref. [7]. A spherical-harmonic basis describes the angular part of the wave functions, and Lagrange functions associated with a Laguerre mesh, i.e. with zeros of a Laguerre polynomial, describe the radial part. This leads to an eigenvalue problem involving real symmetric matrices which are both sparse and fast to generate. The computation time is determined by the diagonalization time.

In sect. 2, the Lagrange-mesh method is described. A simple example is given in sect. 3. The three-body model of sect. 4 provides physical results in sect. 5. Concluding remarks are presented in sect. 6.

2. – Lagrange-mesh method

The Lagrange mesh is formed of

N

mesh points

x

iassociated with an orthonormal set

of

N

indefinitely derivable functions

f

j[4-6]. At mesh points, these functions satisfy the

Lagrange condition

f

j (

x

i )=



,1=2 i



ij (1)

(where the



iare defined below), i.e. the Lagrange functions

f

jvanish at all mesh points

but one.

For a radial equation, a variational calculation is performed with the trial function

(

r

)= N X j=1

C

j

f

j (

r=h

)

;

(2)

where the

C

j are linear variational parameters and the scale factor

h

is a non-linear

pa-rameter aimed at adjusting the mesh to the domain of physical interest. With (2) and (1), the coefficients

C

i =



1=2 i (

hx

i ) (3)

provide a direct picture of the wave function at the mesh points. However, contrary to other mesh methods, the wave function is also defined between mesh points by (2).

The

x

iand



iare connected with a Gauss quadrature formula Z 1 0

g

(

r

)d

r

 N X k=1



k

g

(

x

k )

:

(4)

Here, we consider the case of the Gauss-Laguerre quadrature. The Gauss formula (4) is exact when

g

(

r

)is a polynomial of degree2

N

,1at most, multiplied byexp[,

r

]. The

orthonormality relations between the corresponding Lagrange functions

f

i

(

r

)are exactly

obtained with the Gauss approximation (4) as a consequence of (1). With (4), radial kinetic matrix elements are given by

T

ij = Z 1 0

f

i (

r

)  , d2 d

r

2 

f

j (

r

)d

r

,



1=2 i

f

00 j (

x

i )

:

(5)

This compact expression is exact for some Lagrange meshes. Potential matrix elements are given at the Gauss approximation by

V

ij = Z 1 0

f

i (

r

)

V

(

r

)

f

j (

r

)d

r



V

(

x

i )



ij

:

(6)

The potential matrix is simple and diagonal.

With (2), (5) and (6), the variational method provides a system of

N

mesh equations

N X j=1 [

h

,2

T

ij +

V

(

hx

i )



ij ,

E

ij ]

C

j =0

:

(7)

In this system, the first term has a simple standard form (see eq. (10) below) and the second one is diagonal and only involves values of the potential at scaled mesh points.

3. – A simple example

The Laguerre mesh is based on zeros of a Laguerre polynomial of degree

N

[4],

L

N (

x

i )=0

:

(8)

The explicit form of the corresponding Lagrange functions is given by

f

i (

r

)=(,1) i

x

,1=2 i

r

(

r

,

x

i ) ,1

L

N (

r

)exp [,

r=

2]

;

(9)

which is simply a polynomial of degree

N

, multiplied by an exponential function. In fact, most calculations can be performed without explicit expressions of



iand

f

i

(

r

).

At the Gauss approximation, the kinetic matrix elements (5) read [6]

T

ij = ( (,) i,j (

x

i

x

j ) ,1=2 (

x

i +

x

j )(

x

i ,

x

j ) ,2 (

i

6=

j

)

;

(12

x

2 i ) ,1 [4+(4

N

+2)

x

i ,

x

2 i ] (

i

=

j

)

:

(10)

Equation (7) is solved for the Morse potential

V

(

r

) =

D

[exp[,2

a

(

r

,

r

0)], ,2exp[,

a

(

r

,

r

0)]] for

D

= 0

:

10262,

r

0 = 2,

a

= 0

:

72, and 2

m

= 1836. Its

eigen-values are known analytically. The errors on a mesh calculation with

h

0

:

05are given

in table I. The obtained accuracy is excellent with rather small numbers of mesh points. It is essentially equivalent to the accuracy of a similar variational calculation without the Gauss approximation.

TABLEI. – Errors on mesh eigenvalues for different states with quantum number. N =0 =5 =10 20 410 ,7 10 ,3 – 40 <10 ,15 ,210 ,10 ,310 ,7 60 <10 ,15 <10 ,15 510 ,13 4. – Three-body model

According to ref. [2], the Hamiltonian describing the relative motion of the nucleons with respect to the cluster reads

H

= p21 2



+ p22 2



+ p1
p2 (

A

c +1)



+

U

cn(r1)+

U

cn(r2)+ (11) +

V

nn(jr1,r2j)+ X FS j FSih FSj

;

where

m

,

A

c

m

and



=

A

c

m=

(

A

c

+1) are respectively the nucleon, cluster and

re-duced masses. The cluster-neutron (cn) interaction is described by potential

U

_cnwhile the neutron-neutron (nn) interaction is represented by

V

_nn. The eigenvalues of

H

provide binding energies with respect to the decay into the cluster and two free nucleons. The potential

U

_cn often displays non-physical bound states whose role is to simulate Pauli-forbidden states. Forbidden states (FS) are eliminated with a pseudopotential [8].

The wave function of the neutron pair involves Lagrange functions for the radial part and spherical harmonics for the angular part. For

J

=0, they are defined as

= X LSli1i2

C

LS li1i2  LS li1i2 = X Sli1i2

C

LS li1i2 [[

Y

l (1)

Y

l (2)] L



S ] 00

_F

i1i2 (

r

1

;r

2)

;

(12)

where

L

and

S

are respectively the total orbital momentum and total spin of the two nucleons and



S

is a spin state. The two-dimensional Lagrange functions are defined as

F

i1i2 (

r

1

;r

2)/

f

i1 (

r

1

=h

)

f

i2 (

r

2

=h

)+

f

i2 (

r

1

=h

)

f

i1 (

r

2

=h

) (13)

with

i

₁

i

2because of the identity of the neutrons. The basis functions are not orthogonal

but they are orthonormal at the Gauss approximation (4),

h LS li 1 i 2 j L 0 S 0 l 0 i 0 1 i 0 2 i



LL 0



SS 0



ll 0



i1i 0 1



i2i 0 2

:

(14) The coefficients

C

LS li 1 i 2

determine the wave function anywhere. They also allow

calculat-ing observables such as radii or densities.

The different terms of the Hamiltonian lead to a simple block structure for the Hamil-tonian matrix. The kinetic matrix elements only appear in diagonal blocks with

l

=

l

0

. The central part of the cn interaction leads to fully diagonal matrix elements like in eq. (14). They only depend on the

N

values of this interaction at scaled mesh points. The spin-orbit part also appears on the diagonal of the blocks coupling the

S

=0and

S

=1sectors in the

LS

coupling scheme. The nn interaction is diagonal with respect to the mesh indices

i

1and

i

2but not with respect to

l

. The numerical calculation of these matrix elements requires a multipole expansion of the interaction [7]. It involves an evaluation of the multipole terms at mesh points and some simple angular-momentum coupling coefficients. The matrix elements of thep1
p2 term in eq. (12) do not vanish for

l

0

=

l

1. They give rise to

off-diagonal blocks which yield the largest contribution of non-zero matrix elements.

5. – Halo nuclei

The angular momenta and parities of the6He, 11Li and 14Be ground states and of their core clusters4He,9Li and12Be are respectively0+,3

=

2

,

and0+. One can therefore

assume

J

= 0for the neutron pair, which leads to

L

=

S

= 0or 1. Parity imposes a

common orbital momentum

l

for both neutrons.

As nn interaction, the central part of the Minnesota force [9] is employed in all cases with an exchange parameter

u

= 1. This interaction reproduces the deuteron binding

energy and presents good nucleon-nucleon scattering properties at low energies. It is parametrized as a sum of Gaussians. The cn potentials are selected as follows. For6He, the



+ n potential is taken from ref. [10], as in refs. [2] and [7]. It fairly reproduces the



+ n phase shifts with an

s

forbidden state. For14Be, the12Be + n interaction cannot presently be deduced from experiment. An interaction fitted [11] to a resonating-group calculation [12] involving

s

and

p

forbidden states is selected. For11Li, two9Li + n interactions are considered: from ref. [13] (interaction K) and from ref. [14] (interaction P2). They both reproduce some properties of10Li and present

s

1

=

2and

p

3

=

2forbidden

states. However, the force P2 has been fitted so as to introduce a low-lying virtual state in the

s

wave. Finally, a large value= 5000 MeV is employed for the pseudopotential

constant [7, 11]. The conditions of the calculation are

N

= 20 and a maximum orbital

momentum

l

_max = 18throughout. The corresponding matrices have dimension 7770.

About 3 % of the matrix elements do not vanish. Choosing a good scale parameter

h

is important but

h

needs not be optimal. Here the value is

h

=0

:

3.

The6He binding energy is about 0.73 MeV with the above interactions. The discrep-ancy with respect to the experimental binding energy 0.975 MeV should be attributed to an inadequacy of the three-body model to describe perfectly the6He nucleus. In order to reproduce the binding energy, we multiply the



+ n interaction by 1.01. The binding energy then becomes 0.972 MeV (see table II). The accuracy is better than 0.003 MeV with respect to extrapolated values. With a correct binding energy, the matter radius is well reproduced and the average distance between neutrons is rather large. The domi-nant components in the wave function are1

p

(79.5 %),3

p

(15.7 %),1

d

(2.9 %),1

s

(1.1 %) in the

LS

coupling scheme or(

p

3

=2 )2(91.2 %),(

p

1 =2 )2(3.9 %),(

d

5 =2 )2(2.4 %),(

s

1 =2 )2(1.1 %)

in the

jj

coupling scheme.

The14Be binding energy is not well determined from experiment. By multiplying the

12_{Be + n interaction by 1.06, the obtained binding energy lies between the experimental}

values (see table II). The matter radius is well reproduced. The average distance between neutrons is far beyond the range of the nn interaction. The dominant components in the wave function are1

s

(75.2 %),1

d

(18.4 %),3

d

(3.2 %),1

p

(1.9 %) or(

s

1

=2 )2(75.2 %),(

d

5 =2 )2 (19.7 %),(

d

3 =2 )2(1.9 %),(

p

3 =2

)2(1.3 %). Notice that the halo neutrons mostly occupy the 1

s

1

=2subshell rather than the expected 0

d

5

=2one. Consequently, spin-1 components are

rather small.

The11Li binding energy is very small. Here we compare two plausible cn interac-tions in order to show that the results are very sensitive to the interaction choice. The K

TABLEII. – EnergiesE(in MeV), mass radiusrm(in fm), and average distancernnbetween the

neutrons (in fm).

nucleus E Eexp rm rmexp[1] rnn

6_He ,0:972 ,0:975 2.43 2:480:03 4.9

14_Be ,1:18 ,1:12=,1:34 3.15 3:160:38 7.3

11_{Li (K)} ,0:287 ,0:32 2.68 3:120:16 5.3

11_{Li (P2)} ,0:256 ,0:32 3.17 3:120:16 6.7

interaction has to be multiplied by 1.02 to reach qualitative agreement with experiment (see table II). The matter radius is however too small. With the P2 interaction multiplied by 0.98, the matter radius is well reproduced. The average distance between neutrons is much larger with the P2 interaction. The role of a virtual state in the

s

wave of the cn interaction [14] is confirmed here with a different nn interaction. The dominant compo-nents with K are(

p

1

=2

)2 (96.2 %),(

s

1 =2

)2 (1.8 %),(

d

3 =2

)2(0.9 %) in agreement with the

usual sequence of single particle levels. However, the measured radius is not reproduced. The matter radius is larger for the P2 interaction. The probabilities are then(

p

1

=2 )2(66.5 %),(

s

1 =2 )2 (27.7 %),(

d

5 =2 )2 (2.4 %),(

d

3 =2 )2 (1.4 %), (

p

3 =2

)2 (1.0 %) with a much larger

occupation of the virtual

s

level.

6. – Conclusion

The Lagrange-mesh method offers a simple and accurate way of calculating properties of a three-body system with Gaussian-like forces. The method is very simple since no analytical calculation of potential matrix elements is required. It is accurate in spite of small numbers of points in the Gauss quadrature. The approximate Hamiltonian matrix is sparse and self-similar (the numerical calculation of matrix elements remains the same for any size). Analytical wave functions are available but different physical quantities can easily be obtained with the Gauss quadrature.

The study of halo nuclei requires a renormalization of the bare cluster-neutron interac-tion. This effect is present in all cases but is well established only for6He where the cn and nn interactions reproduce two-body data. This seems to point towards antisymmetriza-tion or virtual cluster-excitaantisymmetriza-tion effects which are not described by the simple three-body model. When the binding energy is reproduced, the matter radius is in good agreement with available data. The binding in halo nuclei results from a delicate balance between different forces and the reliability of model descriptions is not clearly established yet.

  

This text presents research results of the Belgian program on interuniversity attrac-tion poles initiated by the Belgian-state Federal Services for Scientific, Technical and Cul-tural Affairs.

REFERENCES

[1] TANIHATAI., KOBAYASHIT., YAMAKAWAO., SHIMOURAS., EKUNIK., SUGIMOTOK., TAKAHASHIN., SHIMODAT. and SATOH., Phys. Lett. B, 206 (1988) 592.

[2] SUZUKIY., Nucl. Phys. A, 528 (1991) 395.

[3] ZHUKOVM. V., DANILINB. V., FEDOROVD., BANGJ. M., THOMPSONI. J. and VAAGEN J. S., Phys. Rep., 231 (1993) 151.

[4] BAYED. and HEENENP.-H., J. Phys. A, 19 (1986) 2041.

[5] VINCKEM., MALEGATL. and BAYED., J. Phys. B, 26 (1993) 811. [6] BAYED., J. Phys. B, 28 (1995) 4399.

[7] BAYED., KRUGLANSKIM. and VINCKEM., Nucl. Phys. A, 573 (1994) 431. [8] KRASNOPOL’SKIIV. M. and KUKULINV. I., Sov. J. Nucl. Phys., 20 (1975) 470. [9] TANGY. C., LEMEREM. and THOMPSOND. R., Phys. Rep., 47 (1978) 167.

[10] KANADAH., KANEKOT., NAGATAS. and NOMOTOM., Prog. Theor. Phys., 61 (1979) 1327. [11] ADAHCHOURA., BAYED. and DESCOUVEMONTP., Phys. Lett. B, 356 (1995) 445.

[12] DESCOUVEMONTP., Phys. Lett. B, 331 (1994) 271.

[13] KAMEYAMA H., KAMIMURA M. and KAWAI M., in Proceedings of the International Symposium on Structure and Reactions of Unstable Nuclei, Niigata, Japan, 1991 (World Scientific, Singapore) 1991, p. 203.