1.D.2 ] Nuclear Physics A309 (1978) 1 7 7 - 1 8 8 ; ( ~ ) N o r t h - H o l l a n d P u b l i s h i n g Co., Amsterdam I

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TIME-DEPENDENT HARTREE-FOCK POLARIZABILITY AND RANDOM PHASE APPROXIMATION SUM RULES

(I). Theory

S. S T R I N G A R I , E. LIPPARINI , G. O R L A N D I N I , M. T R A I N I and R. L E O N A R D I

Dipartimento di Matematica e Fisica, Libera Univerita' di Trento, Italy

Received 20 April 1978

Abstract: The lineal' response of a nucleus in an external oscillating field has been studied in the frame- work of T D H F theory. The dynamic polarizability evaluated in T D H F theory has been related to RPA energies and matrix elements and a method to obtain RPA sum rules from the T D H F polarizability has been explicitly developed. This method can be applied to isoscalar as well as to isovector and spin excitations. Finally, a prescription to identify mass and restorting force parameters of various collective models with RPA sum rules is given.

1. Introduction

Much theoretical effort is presently being made to find connections and links between different microscopic and macroscopic theories of nuclei. The aim is to better understand the microscopic foundations of phenomenological models (hydro- dynamics, vibrational models...) as well as to give an intuitive description of micro- scopic theories.

Particular attention has been drawn to the study of the connections between time- dependent Hartree-Fock theory (T.DHF) and the random phase approximation (RPA) [see for example ref. 1) for an exhaustive discussion]. It is now clear that a way to derive the RPA equations consists in linearizing the TDHF equation.

In this work we want to discuss further the link between TDHF theory and the RPA, by studying the linear response (dynamic polarizability) of a nucleus to an external oscillating field 2).

It is well known that if one uses first-order perturbation theory to determine the solution of the Schroedinger equation when an external field interacts with a nucleus, one finds for the dynamic polarizability the expression

( E , - Eo)l(OIFIn)l 2 2

(E, - - Eo) 2 - - ~ 2

where E. and In) are the eigenvalues and the eigenstates of the nuclear Hamiltonian, ~o is the frequency of the external field and F the operator which couples to the field to generate the interaction Hamiltonian.

In order to extend this relation to the RPA-TDHF scheme we have evaluated the 177

178 S. STRINGARI et al.

dynamic polarizability in the framework of the T D H F theory and we have related it to RPA energies and matrix elements. In this way we prove that the previous expression for c4(o) can explicitly be used when one works within the T D H F theory and the RPA, i.e. we have proved the following theorem

'~TDHF((0) ~-" 2 (~n (En=E°)I<0IFIH>I2~ (E, , - Eo) 2 - (,)2 /RPA"

It follows that a method to evaluate RPA sum rules consists in calculating the T D H F polarizability 3). In fact the expansion of c~(~o) for o --, 0 leads to odd inverse-energy- weighted sum rules and the expansion for co ~ o¢ to odd energy-weighted sum rules.

We have studied in more detail the limiting case o) --, ~ because it can be handled analytically. Furthermore, we suggest a method to evaluate explicitly the dynamic polarizability ~TDHF(CJ)) up to terms in 1/0 4. This corresponds to the evaluation ot the sum rules S~ PA and S~ PA. The method and the formalism are quite general and apply to isoscalar as well as to isovector and spin excitations. (A numerical applica- tion to isovector dipole, quadrupole and monopole excitations will be treated in a separate paper.)

Finally, we point out how a collective Hamiltonian can be constructed in a natural way in the formalism of T D H F theory and consequently how its parameters (mass and restoring force) are related to RPA sum rules.

2. Sum rules and dynamic polarizability

2.1. SUM RULES

Let F be a Hermitian one-body excitation operator, and H 0 and 10> the Hamilto- nian and the ground state of a nuclear system. The following different inverse-energy- weighted and energy-weighted operator strengths can be defined:

I<0[Fln>l 2 S_ k = ~ ( E _ E o ) k (1)

S k = ~ ( E . - Eo)kl(OIFln>l z, (2) n

where k is a positive integer laumber. The quantities S k lead to the (formally) well- known energy-weighted sum rules since the following identities can be established:

S k = (OlFnkorlO>, (3)

and in turn (O{FHRoFI O) can be expressed through commutators and anticommutators of F and H o. For example:

S 1 = ½<01IF, [H0, F]]I0>, (4)

S 2 = ½(0[{[V, Ho], [H o, FILL0>.

TDHF POLARIZABILITY (I) 179

Furthermore the S k sum rules for k odd can, independently, be estimated using a method involving the dynamic polarizability. An important part of this work is devoted to show how in practice this last possibility may be even more interesting than the direct evaluation of S k through eq. (4). On the other hand the k-odd S_ k set of relations can be reduced to a set of sum rules only through the dynamic polarizability.

In the following we will discuss odd sum rules only.

2.2. DYNAMIC POLARIZABILITY

The dynamic polarizability of a nuclear system is the linear response of the system to an external oscillating field 2 cos cot with which it interacts through the interaction Hamiltonain Hi. I = - 2 F cos cot and it is defined as

( ~( t)lFl~P( t ) ) - (0[FI0) ~(co)-~ lim , (5)

~ o )~ cos tot

where I@(t)) is the solution of

& Hick(t)) = (H 0 - 2F cos cot) lO(t)) = i ~t Iff(t))' (6)

Using first-order (in 2) perturbation theory to express I~(t)) the polarizability becomes

( E , , - E o ) ] ( o l g l n ) l z (7)

~(co) = 2 ,, (E,, - Eo) 2 - co2 '

where E, and In) are the eigenvalues and the eigenstates of H 0. As mentioned before or(co) can be connected both to Sk and S - k (k odd). This connection can be established studying ~(co) in the two limiting cases co --, oc (Sk) and co -* 0 (S-D, respectively. In fact, expanding ~(co) around co - ~ one has:

lI' 1 [ ~ ( c o ) ] c , ~ -- -- ~ S1 -~ ~ $ 3 - ~ . . . , (8)

while expanding ~(co) around co ~ 0 one has:

[~(co)]c,~0 = 2[S_~ + coZS~ 3 + . . . ] . (9)

To understand more clearly the connections between eqs. (5), (8) and (9) on the one hand and (1) and (2) on the other we now show how operators F and H 0 determine explicitly the wave function I~(t)).

For this purpose it is useful to write [~b(t)) as

tO(t)> = e Ar~'}[0>, (10)

where 10) is the ground state of H o and A r ( t ) is an appropriate time-dependent anti- hermitian operator, which can be determined by means of perturbation theory. If

180 S, S T R I N G A R I et al.

one writes

At(t) = A; e-i°"-Ave i''', (11)

then one easily obtains, by solving eq. (6) in the limit of small deformation (2-~ 0),

<,,IFIO> <n[A(: I0> = ½2 - - + higher order terms in 2, (OnO -- O)

<nlAvl0) = - ½2 <nlFI0)__ O)nO Jr- (0

+ higher order terms in 2.

(12)

These formal expressions can be studied in two interesting limiting cases" ~o ~ 3c and ~o ~ 0.

In fact, expanding <nlA~ 10> around e) = ~c we obta in ,

( ' t I 2@[ F + [Ho, F ] + . . . [0), (13) <nlA; 10> - 2 c,~ (o ,

so that we are lead to define

Ar = - 1 2 ( o 2 ( F+ col [ H ° ' F ] + ' " ) '

1 2 ( F - 1 [Ho, F ] + . . . ) .

(14)

(15)

A similar expansion can be performed around (o = 0.

3. Polarizability and sum rules within the TDHF-RPA scheme

It is clear that any approximate expression for Av(t) allows an evaluation of ~(~o) (eq. (5)) and, through the expansions (8) and (9), an estimate of the odd sum rules. The crucial question, however, is to control the accuracy of these estimates.

The aim of this section is to prove that it is possible to find an "approximate" expression for At(t), such that sum rules may be estimated with RPA accuracy from eqs. (8) and (9) for the most general one-body operator F. For this purpose one has to reconsider the definition of the dynamic polarizability within the TDHF and RPA schemes.

Let IHF> be the HF ground state of our target. The TDHF solutions ]t~(t)> of a nuclear system coupled with an oscillating field 2 cos eJt are defined by the variational principle,

.8 (~t~(t)lH-t (?t I~(t)) = 0, (16)

If the unperturbed nuclear target is in a state [m) then the relevant matrix element is <nlA+vlm> etc,

T D H F P OL AR IZ AB ILITY (I) 181

where H = H o-)~ F cos elt. The solution ]~(t)) is a Slater determinant and can be written by means of a one-body operator:

.4e(t) = A ; e - i ' ° t - A v e i''', (17)

a s

I/~(t)> = e zr(') IHF>. (18)

Once eq. (18) is introduced in eq. (16), picking up the terms linear in 2 and separating the terms with phases + ielt and - ie l t , one obtains

<HF[[gAv, [Ho, A+]]IHF> -co<HFI[fAF, A ; ] IHF> = ½2<HFI[fAv, F]IHF>, (19)

- <HFI[f,4F, [Ho, AF]]IHF> - el<HVl[6Ar,/~v]tHF> = ½2<HUI[f,4F, F]IHF>.

To find the solutions of eqs. (19) one now expands the operator A~ in terms of a complete set of RPA excitations * 0 + k~Ok

A-~ = ~ (o~ <nFl[Ok, A~]IHF>--Ok<HFI[O ~, A~-]IHF>), (20) k

and utilizes the RPA equations"

<HFI[-6AF, [-Ho, O~--]]IHF> = el~<HFI[,~AF, O+]IHF>,

<HFI[f/~F, [H0, Ok]]IHF> = _ elk(HFl[OJ~v, Ok]IHF>, (21)

where elk is the energy of the kth RPA excitation. After having chosen 6 A F = eO k and 6A v = eO + one finally gets, for the solutions

of eqs. (19):

<HFI[Ok, A+]IHF> = ½2 <nFl[Ok' F ] IHF) = ½2 <~IFI0>, ( D k - - O ) (2) k - - O )

( H F [ [ O [ , / I ; ] I H F ) = -½)~ (HFI [O[ , F ] I H F ) = +½2 (0[FIE) e l k + e l e l k + e l '

where 0~> and I/~> are RPA eigenstates. We have written

(22)

( 0 1 F f ) = <HFI[F, O[ ] IHF> = ~ (Ym,(k)F* i + Z.,,(k)F*.,), mi

(23)

where Fmi is the single-particle matrix element <mlFli> = ~qJ,,F~k i. Eqs. (22) corre- spond to eqs. (12) of the microscopic theory developed in the previous section.

We have now the possibility to express the dynamic polarizability within the

t Here the O + (Ok) a r e the operators creating (annihilating) the kth RPA excitation and are generally + + + +

written as O I = Zmi( Yrai( k )a,. a i - Zmla i a , . ) where the a m a i ( a i am) a r e particle-hole creation (annihilation) operators.

182 S. STRINGAR1 et al.

T D H F approximation; in fact

~TDHF(fD) = lim (~b(t)lFl~(t))- ( H F I F I H F ) , (24) ~ o )~ COS cot

(~(t)[Fl~(t) = ( H E I e ~"(')F e "~':(° [HF)

= (HFIFIHF) + (HFI[F, Adt ) ] IHF) + higher order terms in 2, (25)

Using then eqs. (20) and (22) we get

(KHVlE , O; IHF>I 2 KHFIEF, O; ]IHF>I 2] (HFI[F, A ; ] I H F ) = ½z ~ \ COk--CO + (J)k -~-Oo / , (26)

and from eqs. (23)~25), we finally obtain the following expression for the dynamic polarizability:

cokl(0lFf)[ 2 ~TDHF(CO) = 2(k~ COk--CO--2~ )RPA" (27)

The static limit (co = 0) of this identity has been already established by Marshalek and Da Providencia 4) who related the static polarizability, evaluated performing a constrained HF calculation, to the RPA inverse-energy-weighted sum rule. The generalization of this identity to the dynamic case implies that every method for evaluating the dynamic polarizability, in the framework of the T D H F scheme, is a way to evaluate RPA sum rules. This follows from a suitable co-expansion in the right-hand side of eq. (27). In fact, in the two limiting cases co ~ ~ and co --* 0 we get:

( 1 sRPA+" ) , [7TonF(CO)],~ ~ -- 2 ~2 SR1PA + ~ "

[0(TDHF(CO)m~ 0 ~ 'S RPA- 2sRPA-- ~- --Zl 1 +CO -3 ~- ' ' ' ) ,

respectively. We can now find a direct connection between the operator A v, determining the

temporal evolution of the wave function I~,(t) and the dynamic polarizability ~TDHF(CO)" To do this let us evaluate the double commutator (HF[[,4 F, [H o, AF]]IHF). By expanding the one-body operator "4F in terms of RPA excitations (eq. (20)), one has

(HFI[,4F, [Ho, ,4v]]IHF) = 2 ~ COk(O[AFI~)('~I,'4F[O ). (29) k

Using eqs. (22) we conclude, by comparison with eq. (27)

4 CqOnF(CO) -- 22 <HFI[Av, [Ho, ,4v]]IHF>- (30)

In this way the problem of evaluating odd energy-weighted sum rules with RPA

T D H F P O L A R I Z A B I L I T Y (I) 183

accuracy has been reduced to the problem of finding an explicit expression for A v. In the following we study the expressions (22) around o~ = ~ . A simple Taylor

expansion leads to

- 2 ~ (0 2 (HFI[O,, F]IHF) - O,(HFI[O, +, F]IHF))

+ ~(o).02(MFI[O.,F]IHF)+m.O.(HFI[O2, F]IHF)) (31) (D n

= - ~ F+o9 + ' " '

where* G = ~,(og,O~(HFI[O,,F][HF)+~o,O,(HFr[O+,, F]fHF)). We note that eq. (22) defines only particle-hole components of the operator A +. This is due to the nature of the RPA operators O~- which contain only terms like a+~ai or a~-am.

When written in the coordinate space, one-body operators contain not only particle-hole components but also terms like a+mam or a~ai. It follows that the identi- fication of the operator A + of eq. (31) with one-body operators written in the coor- dinate space is meaningful only for the particle-hole component part. In this respect we have proved that the other component parts (particle-particle and hole-hole) do not give any contribution to the physical quantities we want to evaluate, i.e. to the TDHF dynamic polarizability ~(co). We conclude that the operator A + can always be written in terms of one-body operators defined in the coordinate space. The search of an explicit expression for G in the coordinate space will be the object of the last part of this section and will give us a practical way to evaluate S RPA.

Eq. (31) corresponds to eq. (14) of the previous section. One has to keep in mind however that Ae, by definition (see eqs. (17) and (18), is a one-body operator for any one-body excitation operator F, so that in eq. (31) it must be expressed through explicit one-body operators like F, G etc. On the contrary Av (see sect. 2) is not necessarily a one-body operator since [H o, F] could contain a two-body contribution (for example when F is isovectorial and H o contains exchange terms).

To complete our program (to give an explicit expression for AF so that sum rules may be estimated with RPA accuracy) we have to write down explicitly also G. In practice this is related to the possibility of finding an expression connecting directly "4v to the Hartree-Fock Hamiltonian and can be done as follows. One starts from the equation

H n v - i & I~( t ) )=0 , (32)

t In order to preserve the analogy with the exact theory developed in sect. 1, one could write G = [HRv A, F ] with Hru, A = ~.~o.O~+O. and boson commuta t ion rules for the operators O ÷ and O; we note, however, that commuta t ion rules are guaranteed only when the commuta tor [O ÷, O] is evaluated on the H F ground state.

184 S. S T R I N G A R I et al.

where HHF is the Hartree-Fock Hamiltonian and Lt~(t)) is a Slater determinant; eq. (32) defines the T D H F solutions. By writing the wave function I~(t)) as e~F"IIHF ), one finds an equation for the operator Ae(t) which can be determined starting from the H F Hamiltonian. The H F Hamiltonian depends, by definition, on the density matrix of the Slater determinant over which it operates. For this reason H.v, in eq. (32), depends on the operator AF(t) through the changes that the transformation e ~"1 produces on the density matrix of the unperturbed Slater determinant IHF>. We call 6HHv(AF) the changes linear in A r in the Hartree-Fock Hamiltonian produced by the transformation e ~m. Then, to the first order in AF(t),

nnr en~"~lHF ) = H°vIHF) + 6HHF(~,(t))IHF) + H°F~,(t)IHF) - 2Fcos t~)tlHF), (33)

where H~v is the HF Hamiltonian relative to the unperturbed Slater determinant. Putting eq. (33) in eq. (32) and developing the term (~/?,t)l~(t)) one then obtains:

([H°F,/iF(t)] + ~)HHF(AF(t))-- i gt ~ F ( t ) - I~F c o s (Dt IHF> = 0. 34)

Looking for solutions of the type t]r(t ) = 4 + e -~°" -A e ~°~t, eq. (34) can be written as

{([HOv, - + ~ + A v ] + 6Hnv(A r ) - o~A~ -½2F}IHF> = 0. (35)

When ~o -~ oc this expression yields the simple solution A[ = -~2/og)F; one can also easily determine higher order terms by putting in eq. (35)

A ' /~- 2co2 ( F + co-lG) "

One finally gets

G = [H°E, F] + 6HnF(F ). (36)

Eqs. (36) represents one of the main goals of our work. In fact this relation contains an explicit prescription to construct, starting from the H F Hamiltonian, the operator ,4r which permits the estimate of our sum rules with RPA accuracy. Substituting (31) in eq. (30) we obtain

0~TDHF(LO) = - - - - (HF][F, [H o ,F ] ] IHF) ¢t) 2

+ ~,1 ½<HFI[G + ' [Ho, G]]JHF) + . . . ] , (37) %

/

and by comparision with eq. (28)

S RPA = ½<HFJ[F, [H o, F] ]JHF) , (38)

S~ PA = ½<HF[[G +, [Ho, G]]IHF>. (39)

We point out that eq. (38) is the well-known Thouless theorem 5) while eq. (39) is an

TDHF POLARIZABILITY (1) 185

important result of this work because it permits the evaluation of S ReA not only

for isoscalar modes involving only the excitation of spatial coordinates [for which eq. (39) becomes very simple (G = [T, F]), which has been already discussed in the literature 6)] but overall for isovector modes and in general for operators F which do not commuta te with the nuclear potential.

4. Collective Hamiltonians and sum rules

In this section we want to develop and discuss a prescription to construct collective Hamiltonians from our sum rules.

In the previous section we have shown explicitly how to evaluate S 1 and $3, etc. (limit co --, 00) once an excitation operator has been chosen; similarly one could

evaluate S_ 1 and S_ 3, etc. (limit ~0 - , 0). Of course, sum rules are an economical approach to the excitation spectrum in case that the excitation operator is a "resonant operator"; in this case the giant resonance energy co R and the strength of the operator are two relevant quantities characterizing the excitation spectrum. In terms of sum rules these quantities could be defined through ~0 a = Sx/S~3/S ~ and S~ or ~0 a =

Sx/S~-~/S-3 and S_ 1 etc. In order to study these giant resonances one can directly construct a collective Hamiltonian whose parameters (spring constant and mass) define co R and the strength of the operator:

\ dt ,/ c% = .

In the following we want to show how a collective Hamiltonian can be deduced in the framework of the T D H F theory.

Let F be the excitation operator (isoscalar or isovector) and let HF be the H F Slater determinant of our unperturbed target. In the previous sections we have studied the explicit form of the T D H F solutions I~(t)) = e 2~(') IHF) in the limiting case o~ --* ~ . From eq. (31) one gets:

,o ) I ~ ( t ) ) o ~ = exp f sin o ) t - ~ cos cot IHF). (40)

The mean value of the nuclear Hamil tonian on the state I~(t)) is (we keep the leading terms in 2)

(~(t)lHol~(t))- ( H F I H o l H F ) = 2 sin ~ot (HFI [F , [H o, F ] ] [ H F )

+ - ~ cos o~t (HFI [G÷, [Ho, G] IHF) (41)

2 ~ 2 = (~ sin o~t)sIRPA"~-(--~ COS O)[ ") $3 RPA.

186

Furthermore,

S. STRINGARI et al.

,~ - ( ~(t)Wl~(t) ) - ( H F I F I H F ) - 2

0) 2 sRPA2 COS (Dr, (42)

d

d l 0)

Substituting in eq. (41) one obtains

(@(t)lHol@(t))- ( H F I H o I H F ) =

]

. ~ = s1RPA2 s in (Dt. (43)

) sRPA 1 d 2 1 j _ ! ~ - 2 " 3 _ (44) 2 dt ' y 2S~ ~ - A - 2 , 2(sRPA) 2'

Relation (44) clearly suggests the interpretation of .W as the collective coordinate and 1/2S RPA and RP'~. RPA 2 $ 3 / 2 ( S 1 ) as, respectively, the mass and the restoring force constants of our collective Hamiltonian, so that

/TkPWJ ~ , ~ (D R = N/D3 /01 . (45)

The following comments are in order here: as it is evident from formula (40) the unitary transformation consists of two parts.

The first one, involving the operator F, is a t ime-odd transformation and generates variations in the energy connected with a collective kinetic energy contribution (see the first term of formula (41) or (44)). The second one, involving G, is a time-even transformation and induces variations in the energy connected with the collective potential energy (arising from the restoring force). As it is clear from eq. (44) the first part leads to the introduction of the mass parameter 1/2S~ PA and the other of the restoring c o n s t a n t sRPA/2(S RpA~2 3 / 1 I -

A similar development can be performed around (D--+ 0. Once more one can evaluate the mean value of the nuclear Hamiltonian obtaining

(~(t)lHol~(t)) - ( H F I H o I H F ) --- (2 c o s (Df)2sR_ei~' +( -- ).(D sin (Dtj2sR_e3 A, (.46)

one has

so that

?T = <~(t)[Fl~9(t)> - <HFIFIHF > = 22SR_P~ cos (Dt. (47)

d dt ~ = - 2J~(DSR-P~ sin rot, (48)

@ ( t ) H ° I ~ ( t ) ) - ( H F I H ° I H F ) = ½~212SR_ PA + 21( ddt '~ )2 ~-3~'RPA 2(sR~A)2. (49)

In the latter case the collective Hamil tonian governing the collective coordinate ,~- is defined by 1/2sRF1A (spring constant) and sRPA/2tsRPA~Z-3 ." ~ _~ j (mass, parameter) and

/~RPA/~RPA (DR = x /~- 1 /~ - 3. (50)

TDHF POLARIZABILITY (I) 187

The evaluation of eq. (50) for isoscalar monopole excitations has been performed, in the framework of T D H F theory in ref. 7).

As it is clear from eqs. (45) and (50) the definition of the collective energy in terms of sum rules, differs in the two limiting cases, One should keep in mind that a collective description of the excitation F rests on the assumption that a highly collective state absorbs most of the strength so that in the limiting case of a delta excitation, eqs. (45) and (50) lead to the same COR' For a finite width distribution, however, the two defini- tions can differ considerably.

5. Conclusions

In this paper the familiar formula for the dynamic polarizability, as given by first-

order perturbation theory,

( E . - Eo)l(OIFIn)l 2 ~(co) = 2 , ( E , - Eo) 2-O) 2 '

has been extended to the T D H F - R P A scheme. This relation connects the T D H F dynamic polarizability (on the left) to RPA

energies and matrix elements (on the right). As a consequence, developing co(co) around eJ = 0 or co = ~ , the T D H F dynamic polarizability can be written in terms of different RPA odd sum rules. In the limit co -~ ~ the T D H F equations have an analytical solution which allows for an explicit evaluation of the T D H F dynamic polarizability and, consequently, of the energy-weighted and cubic energy-weighted sum rules with RPA accuracy.

Finally, we have shown how the T D H F solution allows the deduction, in a natural way, of a collective Hamil tonian in the framework of the T D H F theory. The collective

RPA RPA 2 parameters in the case co-+ ~ are defined as 1/2S RPA (mass) a n d S 3 /2($1 ) S RPA 2 S RPA 2 (restoring force), whereas for co 0 they are defined by - 3 / ( - 1 ) (mass) and

I/2sRP1A (restoring force).

In a separate paper 8) we have used the techniques developed in the present work to evaluate numerically the cubic energy-weighted sum rules (S~ PA) for various isoscalar and isovector modes.

We wish to thank D. M. Brink for stimulating discussions and helpful suggestions.

References

1) D. J. Rowe, Nuclear collective motion (Methuen, London, 1970) 2) D. Pines and P. Nozier6s, The theory of quantum liquids (Benjamin, NY, 1966) 3) S. Stringari, Nucl. Phys. A279 (1977) 454 4) R. Marshalek and J. Da Providencia, Phys. Rev. C7 0973) 2281 5) D. J. Thouless, Nucl. Phys. 21 (1960) 225

188 S. STRINGAR! etal.

6) E. Lipparini, G. Orlandini and R. Leonardi, Phys. Rev. Lett. 36 (1976) 660; J. Martorell, O. Bohigas, S. Fallieros and A. M. Lane, Phys. Lett. (~)B (1976) 313; R. Leonardi, E. Lipparini and G. Orlandini. Phys. Lett. 64B (1976~ 21

7) K. Goeke, Phys. Rev. Lett. 38 (1977) 212; and, to be published: D. Vautherin, Phys. Lett. 69B (1977) 393

8) S. Stringari, E. Lipparini, G. Orlandini. M. Traini and R. Leonardi, Nucl. Phys. A309 (1978) 189