On the microscopic theory of interacting bosons

On the Microscopic Theory of Interacting Bosons

P. RING*, Y. K. GAMBHIR**, S. IWASAKI? and P. SCHUCK$

*Physik-Department der Technischen Universit#t Mfinchen, D-8046 Garehing, West Germany **Physics Department, Indian Institute of Technology, Bombay, India ~fInstitute for Nuclear Study, University of Tokyo, Japan $1nstitut Laue Langevin, Grenoble, France

ABSTRACT

General Theories are discussed, which allow the description of collec tive excitations in a fermion system in terms of bosonic degrees of freedom. Applications of the Dyson-method in the vicinity of double magic nuclei are presented and the convergence in the number of different bosons is investigated. Using the approximation of one or several broken pairs which is based on a generalized seniority scheme, a method is presented to give a microscopic interpretation of the phenomenological IBA-model. Starting from a microscopic fermion interaction we derive IBA-interaction parameters for spherical nuclei far from closed shells.

INTRODUCTION

The introduction of the interacting boson model and its extensions by Arima and Iachello (Arima and Iachello, 1975,1976) and coworkers has greatly revived the interest in, and has provided considerable impetus to the studies of boson models for nuclear collective pheno- mena (for a recent review see Barret, 1981). The inherent simple and attractive nature of the IBM (e.g. (i) the description of nuclear properties in terms of monopole (s) and quadrupole (d) bosons only and (ii) the number of bosons being fixed and being equal to the number of valence pairs of nucleons under consideration) and its tremendous success has directed the recent attempts in this subject to two pronged investigations, namely (i) to correlate the IBM with the earlier boson models and (ii) to provide a microscopic founda- tion of the IBM. Only recently, it turned out that the IBM is essentially equivalent to the older phenomenological interacting boson models (Klein et al, 1981) and this indeed is gratifying.

We shall dwell in this contribution only to the latter aspect, i.e. to the microscopic basis of IBM. This problem essentially involves two steps. In the first one is required to generate the fermion coun- terpart of the bosons, which must be guided by the known physical ob-

465

466 P. Ring et al.

servations. For instance a coherent superposition of ph-pairs of fermions for closed shell nuclei or of pp-pairs for open shell nuclei could be generated in the framework of the Tamm-Dancoff approximation (TDA). Far from closed shell nuclei the collective pp-pairs may be constructed using number conserving quasJparticle theories. Starting from these collective fermion pairs, the second step involves the construction of ideal bosons from these correlated fermion pairs through a suitable mapping procedure. One then ends up with an Hamil- tonian in the boson space whose force parameters are determined from the microscopic fermion-fermion interaction and which depend on the number of nucleons under investigation.

There have been several procedures proposed in the literature for car rying out this mapping from fermions to bosons. In section 2 we will discuss some of the basic ideas. In section 3 we present an application of the Dyson-method for nuclei close to a double magic core. In this case the number of bosons is restricted by the number of valence nucleons. It turns out, however, that a restriction to s- and d-bosons only is a very bad approximation for these nuclei. An investigation of the convergence shows that one nee~many other degrees of freedom and the coupling to non collective twoquasiparticle configurations is rather strong. In section 4 we move towards the open shell by adding more and more particles. It then turns out that the determination of the optimal bosons is a much more complicated task. The approximation of one or several broken pairs provides, however, at least for spherical nuclei far from closed shells a managable tool to study these problems. It turns out that going away from the closed shell the lowlying monopole and quadrupole bosons get more and more collective which gives us hope that the restriction to the s-d subspace is at least a meaningful concept in the frame work of which the coupling to other configurations can be taken into account by a renormalization of the matrixelement in the boson space.

The Representation of Fermion Systems in Termes of Bosons.

The idea to represent fermionic degrees of freedom in terms of bosons is very old in physics(for a review see Garbaczewski 1978). The basic idea is that the treatment of collective excitations in a fermion space has to be carried out by collective fermion pair operators which we shall denote by capital roman letters in the following (we use summation convention)

+ 1 1 ~ a l + + 1 p Bp = ~ X 2 a 2 , B = ~ X12 a2a I (1)

= p~ v +

A~w X13 X23 a 2 a I

Here ~÷ denote single fermion creation operators with quantum numbers collectively denoted by the index i. At this point we do not yet specify the nature of the w-operators. They might by particle operators which annihilate the bare vacuum or quasiparticle operators (p- or h-operators) which annihilate an Hartree-Fock or an Hartree-Fock-Bogoliubov ground state. We also leave the question open how to generate the parameters X~ . They determine the collective degrees of freedom and it is one of the crucial points in the boson theory to determine these coefficients in an optimal way. In the following sections we will present different methods for spherical nuclei close to magic core and for those within an open shell.

On the Microscopic Theory of Interacting Bosons 467

The disadvantage of the representation in terms of the operators B e B % , A~ is the fact that these operators which form a closed Lie algebra obey very complicated commutation relations (Ring and Schuck, 198o) :

with

[B, B + ~'] = ~U'- A ~

9 . ~'y

i FP'~{ AU~' AU'~'] ~ ( ~'7 v'vy ) AyO

(2)

r~, u , ~ ~' ~ ~' ~ = X12 X23 X34 X41 (3)

~'~'y' = X12 X23 X34 X45 X56 X61

The idea is now to represent these operators by boson operators b~, b~ which obey a much simpler algebra

+ ] = 6 (4) [b, b, ~,

There are several prescriptions available in the literature for carry- ing out such a mapping. Some of them are equivalent and can be derived by various concepts:

i) Mapping of operators (Belyaev and Zelevinskii, 1962): The images of the fermion pairs in the Boson space are represented by polynomi- als in the boson operators whose coefficients are determined in such a way that the commutation relations (2) are preserved by the mapping. With the additional prescription that the fermion vacuum is the image of the boson vacuum we thus have a mapping of the full Hilbert space. One disadvantage of this procedure is certainly the fact that the boson space is larger than the fermion space. It contains a large unphy- sical part and in the calculation of physically observable quantities one has to make sure that one does not collect spurious contributions.

ii) Mapping of basis states (Marumori et al., 1964) In this case one maps an orthonormal set of basis states in the fermion space onto an orthonormal set of basis states in the boson space. This is a one-to-one mapping and on this way one does not leave the physical subspace. The operators are mapped by the prescription that matrix-elements between corresponding states should be equal.

In the boson space a natural and orthogonal set of basis vectors is given by the multiboson states

+ . b + IO ) (5) I~I"'" ~N) = NB b~4 "" ~N

where N) is a normalization constant. In the fermion space a natural basis would be the set

B + + I O) (6) I~I"'" UN> = NF ~I"'" B~N

It is, however, in general not orthogonal if Nw is only a number. In such cases one has to carry out an orthogonalization and N v becomes an operator, which is usually not determined in a unique way and it

468 P. Ring et a~.

requires additional physical intuition to choose the optimal basis set in the fermion space.

iii) Method of Generator Coordinates. The method of generator coordinates pr~ides an additional tool to introduce bosonic degrees of free dom. A set of functions depending on generator coordinates is used to construct a mapping: in a first step the fermion space is mapped onto this function space and in a second step a mapping between this function space and a boson space is introduced (Jancovici and Schiff, 1964,Dobaczewski 1981).

As long as one insists on a complete exact mapping of the full fermion space all these methods give more or less the same results (Jans- sen et al., 1971).

÷ F u Y b + b Ap~ ~ J 7

(7)

B + b + I b + b + I Fvyo b + b + blb + ÷ - FvY b + b -...

The representation of the fermion pair operator BE is not finite. The expansion coefficients can be calculated by perturbation theory (Mar- shalek, 1974). In special cases the series can be formally summed up to a square root expression, which shows the close relation of this representation with the Holstein-Primakoff representation of the SU(2) (Holstein and Primakoff, 194o). For collective bosons the P -parameters are small and a rapid convergence should be achieved (Holzwarth et ai.,1976).

On the other hand the whole boson picture is only meaningful, if the collective subspace spanned by the collective bosons is separated from the rest of the Hilbert space at least approximately, i.e. the algebra should be closed within this subspace (kinematic separation) and the Hamiltonian should not mix the collective subspace with non collective states (dynamical separation). It is not clear, to what extend this separation is really possible in realistic nuclei, because it is very hard to take into account at the same time many or- ders in the expansion (7) (horizontal convergence)and many different types of bosons b (vertical convergence).

If one does not insist on an Hermitian mapping, i.e. if one allows the creation operators B~ + to be mapped onto operators which are no longer the Hermitian conjugates of the images of the annihilation operators B~ , or in the language of Marumori, if one allows to map the bravectors (6) in a different way as the corresponding ketvec- tors, we can achieve a finite representation (Ring and Schuck, 1977) which corresponds to the Dyson boson expansion (DBE) of the SU(2) (Dyson, 1956)

B+ ÷ ~+ _= b + I ~Y b+ b+ b

(8) ~+)+ B ÷ b u @ (

In this case we have no problems with horizontal convergence. We therefore use this method in the next section to investigate vertical convergence for nuclei in the lead region. The prize we have to pay for this advantage is the Hermitizity of the boson Hamiltonian. One has to diagonalize non-Hermitian matrices and


to be careful in the evaluation of matrixelements.

If we restrict the ~f-operators to be particles, we end up with pp- bosons in the sense of Arima. In this case the Hamiltonian can be written formally as

+ + + (9) H = E ~ ~ + V ~ ~ ~

In terms of the fermion pair operators B, B +, A of eq.1 it has the form

H = ~ A + ~ B+B (10)

Using the Holstein-Primakoff expansion (7) we find formally

(11)

where we have formally summed up the infinite series. In the Dyson representation (8) we have

I ~ r b+b + H = (5 + ~) b+b - ~ bb (12)

In this case the expansion stops precisely after fourth order. The operator is not Hermitian, because the matrix elements ~ are no longer symmetric. We also see that H contains a single-boson part and a boson-boson interaction which has its full origin in the term V ~5 of eq.8. It accounts for the Pauli principle and we see from eq.12 very clearly the boson-boson interaction in this case is partly induced by the Pauli principle.

Application of the Dyson Method in the Vicinity of Closed Shells

We first want to study the case of only a few valence nucleons and the first nontrivial case is the one of four particles outside a

• ~I~ 1,12, closed shell, as for instance the nuclei Pb, Rn, etc. If one stays in a certain subspace the exact shell model diagonalization might still be possible and the subsequent boson theory (two bosons in each case) can be compared with the exact results. Let us derive the Dyson boson representation for this specific example; we first go the route entirely in the fermion representation and then show how it can be viewed also from a boson description. The two-body Hamilto- nian in second quantization is as usual given by

I + + HF = ~I sT ~I + 4 V1234 ~I ~2 ~4~3 (13)

The fermion operators 0<+ represent true particles, but the theory can be equally well developed for quasiparticles, i.e. particles and holes.

We first have to determine the correlated two-particle states. We end up with the well known pp-TDA equation

I~> ~ X~2 + + ~i~2 ]0>

I (a - = v1234

(14)

470 P. Ring e~ ~.

with the completeness relation

= ~ 3 ~ - ~ ~ = X~2 X34 I 24 14 23 I2p

where I~ is the unity in the twoparticle subspace.

In a complete analogous but more tedious way we arrive at the four particle equation

(15)

Is> = ~1234 (16)

His> = E ~ Is>

s (E - ~I-<2-£3-E4) ~1234 = v1256 I2p Y5634 + exchange

I n t r o d u c i n g as i n d i c a t e d t h e 2 p - u n i t m a t r i x , u s i n g (14) and m u l t i p l y - i n g (16) from the left with X~ ~ X;~ we arrive at the following equation

(Es_~p_~v) ~ 1

with ~s = X~ ~ ~i ~ ~ 2 X34 234

and the non-Hermitian interaction

U,~, (17)

W#v~, v p* v* P' v' (18) , = X12 X34 (~p,-EI-E 4) X14 X23

It is remarkable that in (18) no bare matrix elements appear any more; it is, of course, clear that bare matrix elements can always be eliminated with eq. (14) but in any other four-particle equation many more summations over the single particle states than the four ones in eq. (18) would appear in a corresponding matrix ~/ . The dramatic simpli- fication of our matrix ~ of eq. (18) comes from the fact that (17) is equivalent to the Faddeev-series (Iwasaki et al., 1979) of the four particle system where the correlated four particle states are entirely expressed by the solutions of the two-body subsystems.

Eq. (18) can also be derived as an eigenvalue problem in the boson space with the DBE-Hamiltonian of eq. (12)

H B = £ b + b + I N # ~ W ,v, b+b+bp v u'bv' (19)

It has the advantage to be immediately applicable also to more than four particle systems.

To see why the matrix ~ is non Hermitian we go back to eq. (16). We can express the eigenstate I~> in the fermion space as

i~ > ~ ~s B ÷ B + Io> (20) ~v p V

and in the boson space after a Dyson mapping explained in eq. (8)

~+ ~+ Io> Is> ~ ~#v p V

From

<ol b ~ b ( % - E s l l ~ > = o

(21)

(22)


we obtain

I , , F v' = (Ea-~-~) FY~ (23) W v yo v ~v ~X~

Using the orthogonality relation for the twoparticle amplitudes and the antisymmetry of the four particle amplitudes ~4~ in eq. (16) we find #~=% and have again exactly eq.17. Eq.23 is of the typical form if an overcomplete non orthogonal basis is used and therefore (23) as well as (17) has spurious solutions. They can as usually (Iwasaki et al., 198o) be eliminated in diagonalizing P and eliminating its zero eigenvalues. A DBE-equation completely analogous to (23) for three valence nucleons has been used (Silvestre-Brac and Boisson, 1981). For four particles we have applied eq. (17) for the z0~ Pb+4 nucleon systems (P.Schuck et ai.,1976). A more extensive study was undertaken for weak coupling states in ~o5 Bi (Iwasaki et ai.,198o). The situation here is somewhat different from the one considered above since we have to consider 2p-lh configurations. The twobody

{MeV)'

0.2

0.1

-0.1

-0.2

0.3

--1%2 209 Bi

÷

• 2o

V)

-----1312 ~ --7/2 , 7~

--'~2 __ ,L --{k~ --~'2

--7 e .~ 10

--F~ 2 ~n Im

¢,9 LU m

¢exp boson model ,-~,~,

DBE

/ - ' NFT

I I I I I I I I exp. 0.) b) c) DBE E~Ewithout

Fig. 1

Fig.2

Spectrum of the septuplet in 2°9Bi compared to the experiment and calculations of a) Bortignon et al., 1977; b) Arita and Horie, 1971; c) Zawischa, 1974.

BE3-values of the septuplet in 2°9Bi. DBE is compared with Nuclear Field Theory (NFT, Bortignon et al., 1977) and the experiment

472 P. Ring et al.

solutions are therefore either of the pp- or ph-type. We took as in- put the pp-solutions from a G-matrix calculation (Kuo and Herling, 1971) and the ph-solutions from a Migdal-force (Ring and Speth,1974). We carefully checked "vertical" convergence, i.e. convergence with the number of different bosons. Due to the closeness of the magic nucleus 2o8 Pb there is, of course, no s-d dominance at all; we nevertheless achieve perfect convergence and a nice agreement with experiment as well for the energy levels as for the transition pro- babilities as shown in fig. I and 2.

We can say that our calculation is the most systematic among all the existing investigations on this problem and at the same time repro- duces best experiment without any adjustable parameter. In Fig.3 we show convergence of the septuplet with the number of different boson~ In principle our calculation corresponds to a full diagonalization of the 2p-lh shell model space.

-0.9 E N

(MeV) -I.0

-1.1

-1.2

-1.3

-1.4

diag. -30 -/-,0 -70 -150

~ . _ - - 15/2 +

7/2 ÷

11/2+ 5/2÷ 9/2÷

_ _ ~ 3 1 2 ÷

I I I I I 0.002 0.001 0.0002 0

Vmin

Fig. 3 Convergence of the septuplet as a function of V ;, , where all configurations which couple with a matrix element smaller than V~;. are neglected. On top the approximate number of configurations is indicated. Full lines are calculated with a diagonalization of the norm matrix, dashed lines are calculated without such diagonalization. We see that practically only the 3/2 state is affected.


Interacting Boson Theory for Spherical Nuclei far from Closed Shells

Going further and further aw@y from closed shells, i.e. filling in more and more valence nucleons the Hamiltonian (19) could still be used in principle, however, the underlying structure of the bosons will not be optimal any more since they are calculated as if there were no surrounding nucleons in the open shell; the more valence nucleons there are, the less the structure of the bosons will be determined by the TDA-equation (14) as we supposed in Sec.3.

We therefore propose the following procedure to determine the bosons in the many-valence nucleon case (Gambhir et al., 1982). We start with numberconserving quasiparticle theory, which is equivalent to the Broken Pair Approximation Gambhir et ai.,1969) or genera-

(BPA'I~o> has the same structure as the lized seniority. The ground state number projected BCS state

> = p2p H _ + v. a + a+ ) I ~o j ,m>O (u3 3 jm jm Io> (24)

where PZ~ projects or, 2p particles and ~ = ()e~-~ x +

- ~-~ We in-

troduce a collective fermion pair with angular momentum 3=0

S + = [ Xj " + + JJ' j, [aj aj,]O (25)

and find that the groundstate can be expressed as

I~0> = (s+) p Io> (26)

The coefficients

Xjj, = ~jj, ((2j+ ~ ) A / ~ ) vj/~j

can be obtained by exact minimization of the Hamiltonian H F for the wavefunction I~0>

The excited states in BPA are generated by diagonalizing H F in the one broken pair basis

lJJ JM> + ' = AjM(JJ' (s+)P-I I O> (J4J') (27)

where

+ . . ) - 1 / 2 + a + AjM(33') = (I + ~ j, [aj j']JM (28)

As a result the excited states have the form

> = [ X~. A + (j j') (S+)P -I IO> I~ j.~j, 33' JpM (29)

B + ( s + ) P -1 IO> = N

with B+ = [ X~'" + (JJ') (30) j!j' 33' Aj~M~

It is to be pointed out that this procedure is completely general and generates all types~bosons. Now we are in a positioD to construct the

474 P. Ring et al. +

boson images of B~ by a suitable mapping procedure + .

In the Marumori expansion method one retains up to fourth order terms in the boson expansion of the Hamiltonian H

= b + b + I b + b + b b (31) H B £~, ~ , 4 W~I~, ~ ~ ~| I

The coefficients 8~, and W~v~'~' are calculated by requiring that the matrix elements of H between the two- and four-fermion states be equal to the matrix elements of H between the corresponding two- and four-boson states respectively. Neglecting the normalization this means formally:

<O1 S p-w B ~ H F (B+)v'(s+)P-W'IO> (32)

(O 1 s p-w b w H B (b+) w' (s+lP-W'IO)

AS long as one takes into account only the lowest s and d bosons this procedure corresponds to the boson mapping method proposed by Otsuka et al., 1978. Our more general prescription shows that one has neglected i) sixth and higher order terms ("vertical"convergence) and ii) g, s', d~ ... and other bosons. Even if they do not show up explicitely in the lowlying spectra, they might cause a renormalization of the force parameters in the model Hamiltonian.

The terms of sixth and higher order in the Hamiltonian are hard to control. It therefore seems to be reasonable to use the Dyson method (DBE) with an Hamiltonian which breaks down exactly after fourth order. (Explicit expressions are given by Gambhir et al., 1982.)

The general formulation presented above provides a legitimate technique to study the following questions, in particular:

- How does the structure of the correlated fermion pairs and the corresponding bosons change with the nucleon number?

- How do the microscopically determined parameters a and ~ compare with the corresponding phenomenological ones? How well the experi- mental data is reproduced by the calculation?

- How good is the s-d truncation, i.e. how important are other bosons?

- Is a renormalization necessary and possible to incorporate?

- How serious is the non-Hermitizity in the Dyson method?

We have tried to answer some of this questions by studying explicitely the system of a fixed number of protons and 2-24 neutrons occu- pying the 5o-82 shell (For details see Gambhir et al., 1982).

In Fig.4 and 5 the amplitudes Xjj, are plotted for the lowest 3=O and J =2 states as a function of the number of valence pairs p. Also shown is the variation of the normalization constant No (~=O) and N (9=2) in eq.29. It is clear from these figures that the structure

+In general the operators Bm defined in eq. (3o) are neither norma- lized nor orthogonal. In such cases an orthonormalization procedure has to be used (Gambhir et ai.,1982)


10

U% w

~os X

10

W. I1~

X j: o (S -6 , (3=0 25} Neutrons

dsi2

~g7/2

S i i 2

. ~ h~t 2

I I I l I L I I ; I I l

05 No

N~

p p

Fig.4 The microscopic structure of the s-boson (eq.25) as a function of the number of valence pairs in the open shell and normalization constants N~ explained in the text.

of the correlated fermion pairs which correspond to s and d bosons in a microscopic picture, varies very smoothly with the number of valence particles except for J=2 at p=3. This large variation in X z at p=3 is due to the subshell (2d~z) closure.

In the proton-neutron IBM (also known as IBM2) the simplest Hamilto- nian which incorporates the essential features of the underlying fermionic interaction is of the form

H = ~' nd + T~" T~ (33)

where

TO K O(( S~p 2 P)21 I .

is the quadrupole operator and ~=~or~. • zs related to the dlfference in the s- and d-boson energies, n& is the number operator of d-bosons. In order to determine the constants K and ~ on a microscopic basis we only have to carry out a Marumori-mapping for the microscopic quadrupole operator ~ . The constant ~ is calculated from the equa-

476 P. Ring et a t .

8'

.4'

.2" I

- 2o x

-.4-

".6"

",8~

a - - ~ " " 1 3 1 ) . / , , LU,,mm:~::ar~o--~--j;--, ,x,-;T,x,-;-;~,, i ;- j- ; . x - - x _ _ x _ _ , , ~ ,..-.- (33! J 0 o 0 0 • ° ° 0 1 0 o o ° ° i l l ° ° l ° I I F J ~

, ~ / , , ~ . . . . . . . . . . . . . . . . . . . . 1531

IL

"" ~* (11 ~1) ' ~ ' - - o ~ ° . . . . . .

\ L" . : - : ' : - - - - --:- - --:::- - - ---:-- : - - - - - - - - - - : - - : - ,s,, °~o ~ o - - " ° o o • o ° ° ° ° ° * ( 7 3 )

\ /o/0 "~./0

z~ IS- 6) A

tion

( ( p - l ) ' )

Fig.5 The microscopic structure of the d-boson (eq.3o) as a function of the number of valence pairs. The numbers on the r.h.s, of each curve label the single particle pairs, as for instance (73)=(g7/z ,d ~/z )

" N12 <sP-IDll~plKs+)p-1D+> -I (sp-1 d I ITpl I (s+) p-I d) =

(35) and the constant ~is determined by

(P! (P-l)!) -I/2 (s p I ITpl I (s+) p-1 d) = NoN I <S p ] ]qpl I (S+) p-I D+> (36)

The results are given in Fig.6. Again subshell effects show up es- pecially in the beginning and towards the end of the 50-80 shell. It turns out that the calculated values ~v and ~, for 3 4 p < Io show the same qualitive trends as those exhibited by the corresponding phenomenological values. This in fact is quite satisfying and it is in accordance with the belief that IBM may not be expected to work well in the beginning and at the end of the major shell.


05,

-05

-10

K ~

i A L L I I I L I L

- /

, t

t

;

I 2 4 5 6 7 8 9 I0 11 12 p ,

x.

Fig.6 The microscopically determined IBM2-parameter ~ and ~, as a function of the number of valence pairs.

CONCLUSIONS

In this paper we discussed the lines along which a microscopic deri- vation of phenomenological interacting boson theories might be possible. We showed that the Dyson boson expansion can be successfully applied to situations with only a few nucleons in an open shell. We stress the point that here, of course, no s-d dominance prevails and we investigated the convergence with the number of different bosons (vertical convergence): Many types of bosons are needed and one cannot draw any advantage from grOup theory. The technique of a boson description is nevertheless still a very useful concept, because the prediagonalization in a twoparticle subspace allows to take into account many correlations in a simple way and reduces the numerical effort considerably as compared to full shell model calculation.

For situations with many valence nucleons the Dyson or Marumori technique can still be used in principle, however, the structure of the bosons cannot be of the TDA-type any longer since the influence of

478 P. Ring et al.

the surrounding valence nucleons has to be taken into account. This we achieve in adopting the broken pair approximation and we find that certain pp-bosons (mainly s- and d-) change from non collective to collective ones in going from few to many valence nucleons. This is, of course, exactly the feature one is expecting in order to understand the success of the phenomenological boson models. First microscopic calculations on this basis of the model parameters show at least qualitative agreement. There is, however, still a long way to go to understand all the details and to obtain on this way a microscopic understanding on the structure of transitional nuclei.

REFERENCES

Arima,A. and Iachello,F.;(1975)P_hhys. Lett.57B,39 Arima,A. and Iachello,F.;(1976)Ann. Phys.99,253;111(1978)2ol Barret,B.R. (1981)Rev.Mexicana de Fis.27,533 Belyaev,S.T. and Zelevinskii,V.G.; (1962)Nucl. Phys.39,582 Dobaczewski,J. (1981)N__uucl. Ph[s.A369,213 Dyson,J.F. (1956)Phys. Rev.1o2,1217 Gambhir,Y.K., Rimini,A. and Weber,T(1969)Phys. Rev. 188,1573 Gambhir,Y.K.,Ring,P. and Schuck,P.;(1982)Nucl. Phys.A384,37 Gambhir,Y.K.,Ring,P. and Schuck,P ;(1982)i5~ys Rev. C25,2858 Garbaczewski,P.; (1978)P__hhys._RRe£~36__~C,65 Holstein T. and Primakoff, H. (194o)Ph[s. Rev. 58,1o98 Holzwarth,G. ,Janssen,D. and Jolos,R.V. (1976)Nucl. Phys.A261,1 Iwasaki,S.,Ring,P. and Schuck,P. (1979)Nucl. Phys. A331,81 Iwasaki,S., Ring,P. and Schuck,P. (198o)Nucl. Phys.A339,365 Jancovici,B. and Schiff,D.H. (1964)Nucl. Phys.58,678__ Janssen,D,D6nau,F,Frauendorf,S. and Jolos,R.V. (1971)Nucl. Phys.A172, 145 Klein,A.,Li,C.T. and Vallieres,M. (1982)P__hys. Rev. C25,2733 Kuo,T.T.S. and Herling,G.H. (1971)NRL Memorandum Report 2258 Marshalek,E.R. (1974)Nucl. Phys. A22~,221 Marumori,T.,Yamamura,M. and Tokunaga,A. (1964)P_ro~r. Theor. Phys.31,1oo9 Otsuka,T,Arima,A. Iachello,F. and Talmi,I. (1978)Phys. Lett.76B,139 Otsuka,T.Arima,A., and Iachelio,F. (1978)Nucl. Phys.A3o9,1 Ring,P. and Speth,J. (1974)N__ucl. Phys.A235,315 Ring,P. and Schuck,P. (1977)Ph[s. Rev. C16,8o1 Ring,P. and Schuck,P. (198o). The nuclear manybody problem. Springer Verla~, New York Schuck,P.,Wittmann,R. and Ring,P. (1976)Lett. Nuovo Cim. 17,1o7 Silvestre-Brac,B. and Boisson,J.P. (1981)Phys. Rev. C24,717

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On the microscopic theory of interacting bosons