fulltext

Numerically exact description of multicluster bound states

(  ) R. G. LOVAS( 1 ), K. ARAI( 2 ), Y. SUZUKI( 2 )and K. VARGA( 1 )( 3 ) ( 1

) Institute of Nuclear Research of the Hungarian Academy of Sciences

P. O. Box 51, Debrecen, H-4001, Hungary

( 2

) Department of Physics, Niigata University - Ikarashi, Niigata 950-21, Japan (

3

) RIKEN - Hirosawa, Wako, Saitama 351-01, Japan (ricevuto il 10 Luglio 1997; approvato il 15 Ottobre 1997)

Summary. — We present the variational few-body approach to multiparticle bound states called the correlated-Gaussian stochastic variational method (CGSVM) and show how it can be applied to systems of nuclear clusters. It will be demonstrated that the treatment of the intercluster motion is numerically exact and the method provides a virtually perfect realization of the conventional cluster model.

PACS 21.45 – Few-body systems. PACS 21.60.Gx – Cluster models. PACS 27.20 –6A19.

PACS 01.30.Cc – Conference proceedings.

1. – Introduction

Some light nuclei do not seem to obey mean-field theories; they rather behave as com-posites of smaller clusters. For example, the light multi-



nuclei and halo nuclei show clustering even in their ground states. The clusters are subsystems whose internal bind-ings are much stronger than the binding between them, and thus the behaviour of a multi-cluster system is mostly determined by the intermulti-cluster relative motion. Therefore, while the cluster internal states may be described schematically, the relative motion should be treated as accurately as possible.

It is a challenge to describe multicluster systems accurately. To cope with the technical difficulties, one may divide the system into as few clusters as possible, running the risk of including very soft clusters, or one may admit just well-behaved clusters at the cost of having many. E.g., the 9

Li+n+n model of 11

Li involves a distortable 9

Li, while an



+t+n+n+n+n model involves six clusters. We opt for the latter strategy, and present a framework in which the intercluster motion is treated virtually exactly.

In sect. 2 we outline the theoretical framework, in sect. 3 we show how the method works when all clusters are structureless, in sect. 4 we test the cluster ansatz for the example of6

He, and in sect. 5 we draw some general conclusions.

( 

)Paper presented at the 174. WE-Heraeus-Seminar “New Ideas on Clustering in Nuclear and Atomic Physics”, Rauischholzhausen (Germany), 9-13 June 1997.

2. – Theory

The approach to be presented [1] is based on a variational method with a trial function built up from generalized Gaussians called correlated Gaussians. We introduce this trial function with elementary considerations and start with

A

structureless particles.

Choosing a suitable trial function amounts to choosing a functional form that is likely to approximate a wave function of any conceivable shape with as few parameters as possi-ble. A radial wave function

F

l

(

r

)is most conveniently expressed as

F

l

(

r

)

P

k

c

k

g

a

k

(

r

),

where

g

a

(

r

)is a function containing a continuous parameter

a

. A nodeless

harmonic-oscillator (h.o.) function

g

a

(

r

) =

r

l

exp[, 1 2

ar

2

] is qualified for this role because any

(square-integrable) wave function with angular momentum

lm

can be approximated, to any desired accuracy, by a linear combination of such functions:

F

lm

(r) 

P

M

k

=1

c

k

exp[, 1 2

a

k

r

2 ]Y

lm

(r), withY

lm

(r) =

r

l

Y

lm

(^

r ). For an

A

-particle problem, one should

in-troduce a set of intrinsic Jacobi coordinates x 1

;:::;

x

A

,1, to be regarded as a column

vectorx. We exclude the vector pointing to the total centre of mass (c.m.) from the outset

to emphasize that the dynamics of the problem can be solved without reference to the c.m. The expansion may then take the form

F

LM

L (x) X ff

k

gf

l

gg

c

f

k

gf

l

g exp " , 1 2

A

,1 X

i

=1

a

f

k

g

i

x 2 #



f

l

g

LM

L (x) (1) = X ff

k

gf

l

gg

c

f

k

gf

l

g exp  , 1 2 ~ x

A

f

k

g x 



f

l

g

LM

L (x)

:

Heref

k

gf

k

1

;:::;k

A

,1

glabel the different sets of product Gaussians, which are, in the

second line, written as a quadratic form with

A

f

k

g a diagonal matrix:

A

f

k

g

ij

=

a

f

k

g

i



ij

.

The tilde marks transposition, and the quadratic formx~

A

f

k

g

xinvolves scalar products

of the Cartesian vectors: x~

A

f

k

g x = P

i

x

i

( P

j

A

f

k

g

ij

x

j

). The factor



f

l

g

LM

L (x)may

be imagined as a vector-coupled product of the solid spherical harmonics for the individ-ual Jacobi vectors; then the labelf

l

gmay denote a set of orbital momentum quantum

numbers:



f

l

g

LM

L (x)=  Y

l

1 (x 1 )

:::

Y

l

A,1 (x

A

,1 ) 

LM

L

:

The sum may contain far fewer terms than if there were

A

,1uncorrelated particles,

owing just to the correlations that blur the mean-field picture.

To conform to some correlations or to some part of the asymptotic region, one may alternatively choose other sets of relative coordinates. E.g., for four identical particles there are two topologically different sets of independent relative vectors: those which form a K pattern (the Jacobi vectors) and those which form an H, with one particle sitting at the end of each arm. E.g., an asymptotic partitioning into two pairs of particles is described most economically with the H-shaped set of vectors. To have good asymptotics everywhere, it is economical to include terms with all possible relative vectors. Such terms will overlap with each other substantially, but since even the terms with the same relative vectors are non-orthogonal, this does not pose extra difficulties.

This scheme becomes simpler by two straightforward generalizations.

First, all possible relative coordinate systems, and even their mixtures, may be con-tained in a single sum as in eq. (2) if we allow

A

f

k

gto be arbitrary positive-definite

sym-metric matrices, or, equivalently, if they can be constructed via an orthogonal transforma-tion via a matrix

G

from a diagonal matrix with positive elements:

A

f

k

g = ~

G

f

k

g

A

(0) f

k

g

G

f

k

g

;

where

A

(0) f

k

g

ij

=

a

f

k

g

i



ij

; a

f

k

g

i

>

0

:

(2)

That is so because the relative coordinates transform into each other and, a fortiori, into the set of Jacobi vectors,x, by orthogonal transformations. The terms of eq. (2), exp[, 1 2 ~ x

A

f

k

g x]



f

l

g

LM

L (x), with

A

f

k

gof eq. (2), are called correlated Gaussians.

The second generalization is that we allow



f

l

g

LM

L

(x)not to carry definite values of the

intermediate angular momenta. Then, for states of parity(,1)

L

, the function



f

l

g

LM

L (x) may be chosen as [2]



f

l

g

LM

L (x)=

v

2

K

f

l

g Y

LM

L (v f

l

g )

;

withv f

l

g =

A

,1 X

i

=1

u

f

l

g

i

x

i

:

The values of the coefficients

u

f

l

g1

;:::;u

f

l

g(

A

,1), which may be arranged in a one-column

matrix

u

f

l

g, are parameters of the basis. The case of (,1)

L

+1

is also simple [2].

For this scheme to be applicable to nuclear problems, one has to include the intrinsic degrees of freedom of the particles and to impose the proper antisymmetrization. The trial function will thus look like =

P K

i

=1

c

i

i

, with

i

 f

k

igf

l

ig(

L

i

S

i)

JMTM

T (x)=A n

e

, 1 2 ~ x

A

fk i g x 



f

l

ig

L

(x)



S

i 

JM

X

TM

T o

;

where



andX are the spin and isospin vectors andAis the antisymmetrizer.

The matrix elements are calculated via a generating function technique. The cor-related Gaussians are generated by an integral transform of shifted Gaussian intrinsic states, which are, in turn, expressed in terms of s.p. states. Any matrix elements between shifted Gaussian s.p. states can be calculated by standard techniques [3].

A shifted Gaussian state is

'

s i (r

i

) = (

=

) 3

=

4 exp[, 1 2



(r

i

,s

i

) 2 ], with



=

m!=



h

ands

i

called a ‘generator coordinate’. Their product can be expressed as Q

A

i

=1

'

si (r

i

)= Q

A

i

=1

'

S i

(x

i

), whereS

i

are Jacobi generator coordinates. The c.m. factor

'

S

A

(x

A

)can be

shifted in front ofAand omitted. It is straightforward to verify that a correlated Gaussian

is generated by

g

(

;

w ;

u;

x

;A

)=exp[, 1 2 ~ x

A

x+



v
w ]as follows:

e

, 1 2 ~ x

A

x



KLM

(

u;

x)= 1

B

KL

Z dw^

Y

LM

(w)^  d 2

K

+

L

d



2

K

+

L

g

(

;

w ;

u;

x

;A

)     =0 w jw j=1

;

where

B

KL

=4



(2

K

+

L

)!

=

[2

K

K

!(2

K

+2

L

+1)!], and

g

is related to Q

A

,1

i

=1

'

Si (x

i

)as

g

(

;

w ;

u;

x

;A

)= h (4



) 1 2 (

A

,1) det

C

i ,3

=

2

e

, 1 2 ~ W

C

W Z dS

e

, 1 2 ~ S

Q

S+ ~ W S "

A

,1 Y

i

=1

'

Si (x

i

) #

;

where

C

=



,1 (

I

,



,1

A

),

Q

=

C

,1 ,

I

,W =



,1 w

C

,1

u

, with

C

and

Q

being

The shifted Gaussians will generate a wave function of

A

s.p. ‘clusters’ if allS

i

are kept

different, but they will generate states with composite clusters whose internal states are

0

s

h.o. shell-model states, if some ofS

i

are constrained to coincide. That is how they can

be used in cluster models.

To perform well, the trial function must have many terms, and each term contains a number of parameters: the coefficient

c

i

, the with parameters

a

f

k

i g

j

(

j

= 1

;:::;A

,1), the 1 2

A

(

A

,1)parameters of

G

f

k

ig defined in eq. (2) and

u

f

l

ig

j

(

j

=1

;:::;A

,1). Since,

however, the basis elements are singled out from a continuous set of largely overlapping functions, there are infinitely many almost equally good parameter sets, and it would be useless to find the very best among them even if it were feasible. Good parameters can be obtained by properly varying thstatese linear parameters only, while the non-linear parameters are sampled stochastically term by term and a new term is admitted only if it lowers the energy substantially enough [4]. The admittance criterium may then be loosened gradually, whereby rapid convergence to the same energy value can be achieved. This trial and error procedure can be regarded as an approximate optimization, and hence the term ‘stochastic variation’ [5]. As will be seen in sect. 3, the numerical values obtained agree with the exact model values whenever those are known.

3. – Performance of the approach

We have made calculations for more than a dozen of various systems. The examples range from three-quark systems to nuclear body systems, neutron-halo nuclei, few-electron–few-positron systems, muonic molecules etc. We now show two examples, which are most representative of the performance of the method. Both involve systems of struc-tureless few particles.

Table I shows results for few-nucleon systems. We see that our results agree with the variational Monte Carlo method (VMC) acknowledged to be numerically exact. Our claim that the method is numerically exact is based on these and on many similar comparisons. For

A

=7there are no rival results.

The second example is meant to illustrate what accuracy can be attained in repro-ducing the few-body asymptotics. It is well-known that, if the two-body subsystems of a three-body system have zero-energy

s

-states, then the three-body system has an infinite

TABLEI. – Energies and radii ofA-nucleon systems interacting via the Malfliet-Tjon potential [6]. VMC: variational Monte Carlo method [7]

. A (L;S)J  Method E(MeV) hr 2 i 1=2 (fm) K 2 (0;1)1 + numerical ,0.4107 3.743 CGSVM ,0.4107 3.743 5 3 (0; 1 2 ) 1 2 + VMC ,8.26890:03 1.68 CGSVM ,8.2527 1.682 80 4 (0;0)0 + VMC ,31.30:05 1.39 CGSVM ,31.360 1.4087 150 5 (1; 1 2 ) 3 2 , VMC ,42.980.16 1.51 CGSVM ,43.48 1.51 500 6 (6 He) (0;0)0 + VMC ,66.340.29 1.50 CGSVM ,66.30 1.52 800 7 (7 Li) (1; 1 ) 3 , CGSVM ,83.4 1.68 1300

number of bound states accumulated at zero energy, and these are extremely extensive spatially. Such three-body states are called Efimov states [8]. Any minute increase of the attraction will make the Efimov states immediately disappear. The Efimov effect is a genuine three-body effects, and to produce it, one has to observe three-body asymptotics as accurately as in Faddeev calculations. In a Poeschl-Teller potential that produces a zero-energy

s

-state we have, however, found two excited

L

= 0states in the three-body

problem at,0

:

008and at,0

:

000015MeV, with radius 25 and 6000 fm, respectively. They

disappeared upon deepening the potential, which proves they are indeed Efimov states. Thus the method is accurate enough to yield Efimov states.

4. – The description of6

He

Knowing that the relative motion is described correctly, one can apply the approach to systems of physical interest. The nucleus6

He, which is one of the simplest non-trivial applications at hand, lends itself to a thorough-going test of the multicluster model itself.

The interaction should in principle be appropriate for the restricted relative motion of the nucleons within the clusters as well as for the unrestricted relative motion of nucleons between different clusters. As a compromise between these contradictory requirements, we used the Minnesota force [9] with a spin-orbit force constructed independently [10]. The Coulomb force was also included.

We performed the calculations in five models: i) the basic model implied by our ap-proach, i.e., a pure



+n+n model with the



-cluster described by a single0

s

configura-tion; ii) an



+n+n model with breathing distortions of



; iii) anf



+n+n;t+tgmodel with

breathing distortions of



and single 0s configuration of t; iv) af



(t+p)+n+n;



(h+n)

+n+ngmodel (



composed of two clusters) with breathing distortions of t and h; and v)

the previous model augmented with a t+t component of the type of model iii). In all mod-els all independent relative coordinates have been included, with all angular momenta that are expected to contribute appreciably. Models ii) and iii) are the same as those in-troduced by Cs´ot´o [11], and the two versions of the Minnesota force adopted are also the same. In addition, we show the results of model i) with force I readjusted so as to produce the correct



+n+n separation energy

"

(model i0

)).

The results are summarized in table II. One can observe that the



-breakup deepens the energy much more than the breathing. At the same time, there is an overall under-binding. This is mostly due to the treatment of the t and h clusters; indeed, the breathing t and h are underbound with respect to the exact solutions by2

:

4MeV, while the same

number for model v) of6

He is larger just slightly,3

:

4MeV [1]. On the other hand, since

the force parameters were chosen so as to reproduce

"

in less complete state spaces, there is an overbinding with respect to the



+n+n threshold.

A closer look at the results shows that the binding in model iv) is some 1.12 MeV stronger than in model iii), but the respective separation energies are much the same, which suggests that the improvement comes mainly from the treatment of the



-cluster itself. Indeed, the binding energy of the free



-particle is pushed down from,25.595 to ,26.549 MeV, while the first excited state is shifted from 34.043 MeV to the realistic value

of 20.21 MeV. The little change in the separation energy suggests that the two ways of breaking up the



-cluster opens up very much the same segment of the state space. This is corroborated by the finding that, when the t+t term is added to the



+n+n model (be-tween models ii) and iii)), there is an 0.680 MeV gain in the intercluster binding, whereas, when the same term is included in the modelf



(t+p)+n+n;



(h+n)+n+ng, its

TABLEII. –6

He binding energiesE,+n+nseparation energies"(in MeV) and point nucleon (N), proton (p) and neutron (n) r.m.s. radii (in fm) (experiment: [12]).

Model E " Radii

Force I Force II Force I Force II Force N p n

i) ,25.349 ,24.909 ,0.660 ,0.220 I 2.52 1.83 2.81 ii) ,26.333 ,25.914 ,0.738 ,0.319 I 2.49 1.82 2.77 iii) ,27.013 ,26.557 ,1.418 ,0.961 II 2.42 1.81 2.68 iv) ,28.127 ,27.699 ,1.578 ,1.151 II 2.35 1.76 2.60 v) ,28.222 ,27.794 ,1.673 ,1.245 II 2.34 1.76 2.58 i0 ) ,25.684 ,0.995 I 2.42 1.77 2.69 Exp. ,29:271 ,0:975 2.33 1.72 2.59

The radii change monotonously with the



+n+n binding. Since the separation energy happens to be closest to the experimental value for model iii), that value is most realistic. It is interesting that, when the separation energies are set equal by adjusting the force (model i0

)), the radii and, indeed, all other important properties (except the total binding energies) are more or less the same in all models, and differ very little even from the exact six-nucleon calculations that produce the same separation energies.

5. – Conclusion

The correlated-Gaussian approach has been shown to be feasible. It owes its feasibility and accuracy to a fully analytical scheme. The Minnesota force used has proved to be realistic both in an exact few-body problem and in a hierarchy of cluster models.

For the specific system of6

He it has been shown that the binding energy is appreciably deepened by the breakup of the



-cluster, but otherwise the behaviour of the nucleus is not much affected. A comparison of the results with the exact six-body calculation shows that all cluster models considered are sound, including the simplest



+n+n model.

REFERENCES

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[3] BRINKD. M., in Proceedings of the International School of Physics “Enrico Fermi”, Course XXXVI, edited by C. Bloch (Academic, N. Y.) 1966, p. 247.

[4] VARGAK., SUZUKIY. and LOVASR. G., Nucl. Phys. A, 571 (1994) 447. [5] KUKULINV. I. and KRASNOPOL’SKYV. M., J. Phys. G, 3 (1977) 795. [6] MALFLIETR. A. and TJONJ. A., Nucl. Phys. A, 127 (1969) 161. [7] WIRINGAR. B., private communication.

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[9] THOMPSOND. R., LEMEREM. and TANGY. C., Nucl. Phys. A, 286 (1977) 53. [10] REICHSTEINI. and TANGY. C., Nucl. Phys. A, 158 (1970) 529.

[11] CSOT´ O´A., Phys. Rev. C, 48 (1993) 165.

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