The interacting boson-fermion model

The interacting boson-fermion model

Nuclear Physics A347(1980).51-65.@North-Holland Publishing Co., dmsterdam Not to be reproduced by photoprint or microfilm without written permission f...

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Nuclear Physics A347(1980).51-65.@North-Holland Publishing Co., dmsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher.

THE INTERACTING

BOSON-FERMION

MODEL*

F. Iaehello Physics Department, Yale University, New Haven, Ct. 06520 and Kernfysisch Versneller Instituut, Groningen, The Netherlands

I introduce a new theoretical framework for the description of collective quedrupole states in odd-A nuclei and discuss the main features of the three limiting cases which emerge from this approach.

1.

I~RODUCTION

fn addition to experimental information on even-even nuclei, a considerable amount of new experimental information has been accumulated .inthe last 20 years on collective spectra in odd-A nuclei, This information has been analyzed in terms of somewhat unrelated models, each valid in an appro riate mass region, such as the particle-vibrationmodell), the Nilsson model2P and the particletriaxial rotor model3). In this lecture, I will describe an alternative approach to collective states in odd-A nuclei, which promises to provide a unified description of collective states in these nuclei, irrespectively of their nature (vibrational,axially symmetric rotationa triaxial,...). This approach, called the interacting boson-fermi n modeltl (IBFA) is a natural extension of the interacting boson model5P (IBA) introduced a few years ago in order to describe collective states in even-even nuclei. In the interacting boson-fermion model, an odd-A nucleus is treated as a collection of bosons (the correlated pairs) plus the odd fermion. The corresponding Hamiltonian is H = HB + HF + VBF, where HR is the boson part of R, HF

PJ:

Ej "jrn ajm*

and VBF is the boson-fermion interaction, Although, for a detailed comparison with experiment, one needs to perform complete calculations (Sect.6) even before attempting any of such calculations, it is interesting to discuss those limiting situations which yield spectra with simple properties. These sftuations are the corresponding situations, in odd-A nuclei, of the three limiting cases of the interacting boson model. The main purpose of my lecture is to describe the major features of the spectra which arise in these limiting situations. A more complete exposition, together with a comparison with experiments, will be presented in a series of forthcoming papers, which will parallel the existing series for even-even nuclei. is existing series wil 5 ; 11) the Sgj3) lim&+; be denoted in the fo owing by I) the W(5) limit6 III) the SO(6) limit8 ; IV) the transition from W(S) to SU(3) ; and V) the transition from SIJ(3)to SO(6)lO).

2. TlW INTERACTING BOSON-FRRMION MODEL, As indicated above, the starting point of this model is the Hamiltonian

F. IAC?iELLO

52

H=H

B + $

+ 'BF

*

In calculating the spectra of odd-A nuclei, hB is fixed by requiring that its eigenvalues describe accurately the adjacent even-even nucleus, which, in this approach, is calculated simultaneously with its odd-A partner. The spectra of odd-A nuclei thus depend only on the boson-fermion interaction VBF. In princi le, several terms may contribute to VBVll). However,we have suggested!) that three terms may dominate the spectra of odd-A nuclei, pairing and quadrupole coupling terms and an exchange term. In the case in which the odd particle may occupy several single-particle levels, j, j',..., these three terms can be written as follows

'BF

(0) = 5 Aj [(stxs)(o)x(a~~~j)(o)] +

(0) T j,j'

: [ (alx,)(j”)x (dtxij) (j”) ]

lfFi

-

T j,j'

:

(2.2)

where

(2) QB

I

(s+x;+d+xs) (2) + X(d+xd) (2) ,

(2) (2) QF = (aixaj,)

.

(2.3)

The operators st, dt, ai are defined in Refs. 4 and 5,

- (-)ud4

* ajm

=(-)-l-maj ,-m'

the superscripts denote tensor products and normal ordering (:) has been introduced in the last term &n (2.2 in order to remove the contribution arising from the commutator of d and d1 . In the case in which only one single particle level is available to the odd particle (as it is for the unique parity levels g 12 in the shell 28-50, -126) the matrices Aj, hllf2 in thz shell 50-82 and i13/2 in r A the shell 85: rjj* and Ajj:become numbers A, * * Thus, for these levels, only three numbers are needed in order to calculate properties of the spectra. When several single particle Xevels are available to the odd particle, the full matrices A rjj~ and Ajj, would be, in principle, needed. However, an analysis o3'the eqerimental spectra in terms of all the numbers appearing in Aj, rjj' and hjj * is very difficult return to the microscopic structure of derive the j-dependence of the matrices lations13) within the framework of the genera suggest that the j, j’, j” dependence of Aj, approximation, by

53

THE INTERACTING BOSON-FEBMION MODEL

A, = a (2j+l)

9

rjj' - Y < jlIY211j8>

,

A$,

= x



.

(2.4)

With (2.4), the experimental data can be discussed in terms of only three numbers a, y and A, even in the case in which the odd fermion can occupy several single particle levels. Moreover, it can be easily seen that the monopole term 5 is relatively unimportant in determining the structure of the . . __ spectrum, since its effect is that of compressing or expanding the spectrum without affecting the ordering of levels. Therefore, in the present description, spectra in odd-A nuclei, are, to a large extent, determined by the interplay of two terms, the quadrupole coupling, y, and the exchange term, X. The interplay of these two terms, corresponds to the interplay between deformation 8, and Fermi energy, XE, in the usual description in terms of shape variables.

3.

THE W(5) LIMIT.

I begin by considering the W(5) limit6) . In this limit, the boson core is characterized by a set of quantum numbers [[N] nd v nAUl>. Its energy levels are given by the formula

E([N]n v n LM) = E nd + a k nd(ndfl)+ B(nd-v)(nd+v+3)+ y[L(L+l)-6nd]. d A

(3.1)

In order to discuss the main features of the coupling of an odd fermion to the SU(5) core, I will now consider the case in which the odd-particle can occupy only one single particle level, j. When E Is large, only those terms in VBF which do not change the number of s (or d osons separately will contribute. tlP, may be rewritten as These are three such terms. The first, VBy A =-

nnz-

(N-nd) nj

,

(3.2)

G where ns, nd, nj are number operators. Since - 1, this term gives rise, apart from an overall shift AN/m, to a renormalizationof the boson energy E, which becomes c-(A/m). For example, the multiplet with nd=l is shifted by an amount



where J * j+2, j+l, j, of the multiplet. by the interaction i@

(3.3)

j-l, j-2, is the angular momentum of the various members e (3.3) is independent of J, the multiplet is not split (Fig. 1). The second term, VA:), is given by

54

F. IACHELLO

(2) "BF

(0) = (XT) 6

.

[ (d+xd)(2)x(a+x;l )(2) ]

1

(3.4)

j

This term gives rise to a splitting, but no shifting, of the multiplets. The splitting of the multiplet with nd=l can be easily calculated

0

Fig. 1.

--o---4

-o--

I

9,

--o---4

,

Splitting and shifting of the boson multiplet with nd=l due to the monopole (a), quadrupole (b) and exchange (c) interactions, for j=9/2.

and it is given by

J+j AE(2) (J) = (xr) 5 nd=l

2

_I

j

2

(-) (3.5)

Finally, the third term, VL:), is given by

(3) =-A/Q? "BF

:

[(alxd)(j).(dtxaj)(j)](O) :

.

This term gives rise to a splitting and a shifting of the multiplets. The splitting of the multiplet with nd=l can be calculated as

(3.6)

55

TEE INTERACTING BOSON-FEBMIONMODEL

AEC3)(J) Pl

n

= - A (2j+l)

(3.7)

d

I note that, in most cases, AE (3) gives the largest contribution to the splitThe splitting and shifting of the multiplets due to 8 o~@ea~l#e~~*shown in Fig. 1 for j=9/2. The nature of the spectrum uEFtA the nd=2 multiplet is shown in Fig. 2 (for quadrupole coupling) and Fig. 3 (forexchange coupling). The limiting case discussed in this section, corresponds to the particle vibration coupling scheme of the geometrical description. However, in most phenomenological studies done using this scheme, the collective-singleparticle coupling (when translated into a boson language) is assumed to be

VBF =

z

rjj,

6

[(~+&++3)(~)~ (ai&j.)(2)]

(0) .

(3.8)

_iJ As one can see from (2.2), this is only part of the full interaction, VBF, and in fact, according to the reasoning presented above, that part which is the least important in the limit of large E. The presence and importance of the other terms in VBF (especiallyof the exchange term which plays a very dominant role in the interacting bos -fermion model) has been emphasized several times by Bohr and Mottelson18 in the context of the particlevibration coupling. These terms fol ow automatically from the nuclear field theory (NFT) o collective spectral5$ and have been used previously by severa1 authors16f.

Fig. 2.

Typical spectrum of an odd-A nucleus (j=9/2) in the W(5) limit with diagonal quadrupole interaction (XT=-1 I&V). Only levels up to n 52 are shown. The lines denote allowed E2 transitions. d

F. IACHELLO

'56

Fig. 3

Typical spectrum of an odd-A nucleus (j-9/2) in the SU(5) limit with exchange interaction (A=1 MeV). Only levels up to n -2 are d shown. The lines denote allowed E2 transitions.

ecause of their diagonal nature, the effect of the terms is that of splitting and shifting, but not mixing, the various nd multiplet, thus giving rise to a coupling scheme which, although not being strictly speaking a weak-coupling scheme, retains many properties of it. In order to avoid confusion, this scheme will be denoted by diagonalcoupling scheme in the following sections. The diagonal-coupling scheme includes obviously the weak-coupling scheme which occurs when the coupling constants 5, y and A are small.

4.

THE SU(3) LIMIT.

Next, I consider the SU(3) limit7) . In this limit, the boson core is characterized by the quantum numbers \[N] (X,u) KLM>. The corresponding spectrum is given by WiNI

(bid

KM

= C$

K-K’)

L(L+l)

+

~~~2+u2+~*+3(~+~)]

.

(4.1)

Moreover, the boson quadrupole operator, QB (2), tends to its W(3) value,

(2) QB

IP

(dtxs+&& (2)- $- ($x;t) (2)

.

(4.2)

57

THE INTERACTING BOSON-FEHMIONMODEL

Thus x--012 here. We study once more the structure of the spectrum in the case in which there is only one single particle level available to the odd particle, j-912. We first consider the case in which A=O, A=0 and r
2

0

Ko=+

Fig. 4.

(0)

A typical spectrum in the SU(3) limit of the interacting bosonfermion model. The number of bosons is N-6, the odd particle has j-9/2 and the energy levels are calculated the Hamiltonian HI-K& [Qi2)x Q&2)](O) + r&IQ, bd):$n;'?$ng with ~-12.5 Ice",P-220 keV. Only a selected number of levels is shown. The levels have been arranged into bands denoted by the lowest value of the angular momentum, K, contained In the band. This quantum number is only approximately equivalent to the quantum number K in the Nilsson model. In the inset, the corresponding situation in the Nilsson model is shown.

We now study the effects of adding the exchange term, A. If, for fixed r, we Increase &the internal structure of the bands remains, to a large extent, unchanged. However, the relative position of the band heads changes. This Is shown in Fig. 5, where now the band with Ko-7/2 is the lowest, follorjedby Ko=512, 312, l/2, 912. Thus, the effect of increasing the exchange term, A, is similar to that of raising the Ferml energy, XF, in the Nilason model, as shown in the inset in Figs. 4 and 5.

58

F. IACHELLO

3

E (MN)

Fig. 5.

Tfi[Qi2)x Ql$2)](o)+ keV, l'=-220 keV owest state of each band is shown.

1:

with

~=12.5

We also note that, in treating even-even nuclei within the framework of the interacting boson model, we switch from a description in terms of particles to one in terms of holes when reaching the middle of the shell. We apply a similar prescription in odd-A nuclei. Switching from particles to holes has the effect of reversing the sign of T. On the contrary, A does not change sign when going from one description to the other. Finally, we have also studied the effects of adding a monopole term, A. This term does not appear to change appreciably the nature of the spectrum. It only expands or compresses the energy scale. In concluding this section, we note that the interacting boson-fermion model produces, in its SU(3) limit, spectra similar to those of the Nilsson model. However, contrary to the Nilsson model, it calculates at the same time also other bands (8, y, 28, 2y,...). It would be interesting to see whether or not this close relationship between ground and excited bands is experimentally observed.

5.

THE SO(6) LIMIT.

In this limit, the boson core is characterized8) by the quantum numbers I[N] u T vALM>. The corresponding spectrum is given by

E([N] a T vAL M) - $ (N-o) (N+o+4) + ; T (r+3) + c L (L+l),

(5.1)

59

THE INTERACTING BOSON-FERMIONMODEL

and the boson quadrupole operator tends to its SO(6) value

f.Jc2) (d+xs+s+xd) (2) B I

.

(5.2)

Thus x=0

here. We now study once more the case of one single particle level, j-912. We first set A=O,A -0 in VBF. The resulting spectrum is shown in Fig. 6. Its structure is very regular and simple, with several bands connected by large E2 transitions. These bands have a "triangular"-likestructure which resembles neither of the previous two coupling schemes (Sects, 3 and 4). Nonetheless, es in these two cases, the entire low-lyin spectrum can be constructed by means of some simple rules which I now listf7):

(1) there are first three bands, denoted by To, T2 and T4 in Fig. 6;

the lowest angular momentum in these bands is 912, 512, l/2 respectively; (the_correspondingrule for a single particle level, j # $, is that there gre n bqdsl T+, with lowegt angular momentum given by j = j - n, n -"0,2,4,... (j>O) );

(ii) at higher excitation energies there are two additional bands, denoted by Rl and R3 in Fig. 6; the lowest angular momenta in these bands are 712 and 3/2 respectively; (the corresponding rule for a single particle level with angular momentym j is chat there are n bands, G, with lowest angular momenta given by j = j - II,n = 1,3,5, ...(j>O) );

Fig. 6.

Typical O(6)-like spectrum in an odd-A nucleus. The number of bosons is N=6, the odd particle has j-912 and the energy levels are calculated by die ona ising the +,' " + C C3 + r& [(d+xs+stx!)(2fx (a]xiJ)?8it8nian * A' t: withH%200 eV B - 225 keV, F - 0 and I - 220 ke . The lines connecting the levels denote large E2 transitions.

60

F. IACUELLO

(iii) within each band, states can be classified by*a quantum number ; = 0,1,2, .... the bands stop at some value 'I=rmax related to the number of bosons in the core; the angular momenta J contained in each &multiplet are given by the rule

J - j+2;, j+2;-1, . . . . j+;

;

(5.3)

(iv) the energy levels are approximately given by the formula ,.. L,. ..,... . E(n,r,J) - A(n) + Br(r+3) + C J(J+l)

,

(5.4)

1 n ,. . * where A {n] depends only on n = 0,1,2, ... and B and C are appropriate constants; large deviations from this formula appear only in the band with .j= l/2. Next, we study the effects of adding an exchange term, A. This is shown in Fig. 7, where it can be seen that the effect of this term is to lower the I$ bands with respect to the T; bands, thus playing a role somewhat similar to that played in the SU(3) limit. Finally, we note that the effect of the monopole term, A, is again that of compressing or expanding the spectrum, without affecting its nature. In concluding this section, I note that, since the O(6) limit of the interacting boson model corres onds18), within certain approximations to the y-unstable model of Wilets and Jeax$) (which in turn shares some properties with the rigid triaxial rotor mode120)),some features of the coupling scheme discussed here should also a pear in both the y-unstable21) and triaxial-rotor plus particle calculations3P. A detailed comparison between the two approaches would be very illuminating.

Fig. 7.

THE INTERACTING BOSON-PENMION MODEL 6.

61

INTERMEDIATE SITUATIONS.

The major advantage of the method presented here is that it can describe in an equally simple manner, other, intermediate situations, in addition to those discussed above. In order to do this, one proceeds as follows. One calculates first the spectrum of the adjacent even-even nucleus. This calculation fixes Hg, the boson part of II. One then returns to (2.1), (2.2) and (2.4) and adjusts the parameters 5, y and X appearing in VHFby requiring that the calculation describes the observed spectrum in the odd-A nucleus. In most cases, a monopole term, a, is not needed and thus only y and 1 are adjustable parameters. Moreover, if the single particle levels are completely empty, y=O, and in that case only y remains to be determined. Calculations of spectra of odd-A nuclei usin this method have already begun. The first calculat one were performed by us 49 by Casten and Smith22) and by Gelberg and Ka~p~~f. We calculated the low-&g negative parity states in the odd 63Eu-isotopes built on the hll/2 proton single particle level. The boson part, HD, of H was taken from a previous calculation of the 62Sm-isOtOpesg) which are the cores for these odd-A nuclei. The results are shown in Fig. 8. In this particular in which there ie only one single particle level, the ma;;~om;l~;;:;~n;, I’ and A. The strengths A, r and A in under, since we were mainly interested

E (Me\

Tt

63EU

63EU

EXP

1.0

13 5

15

I! 3

7,b

0.5

13 -11

0

NEUTRON NUMBER

Fig. ,8.

NEUTRON NUMBER

Low-lying negative-parity states in the odd Eu-isotopes. The numbers next to each level denote the values Of 23. The-experimental location of the levels J” = 9/2 , 1312 , and 1512 in 155Eu is not known. The theoretical curves are calculated by use of (2.2) with A-300 KeV, y-800 keV, x- -1.32, AwlgO keV.

62

F. IACHELLO

standing the general features of the spectra. These features which are a change from a diagonal-coupling-like(Sect.3) structure for 129Eu, to a Nilsson-like (Sect.4) structure for 1553 are reproduced by the calculation. In their calculation, Casten and Smith22P'discussed the negative parity states in the odd 46Pd-isotopes built on the hlll2 neutron single particle level. The boson part, HB, of H was taken from a r vious calculation of the even Pd-isotopes performed by Hasselgren et al.$17. Here again the strengths of the interactions A, T and A were kept the same for all isotopes. Finally, in the calculation of Gelberg and Kaup23), the positive parity levels in the 37Eb isotopes, built on the g9i2 single particle level were discussed. More recently, we have begun to test the assumption (2.4) by performing calculations with several ingle particle levels. Preliminary results obtained by Blasi and LoBianco248 who have calculated both positive (8712, d5/2) and negative (hill?) parity ievels in 6?Eu, by Wood and Braga25), who have calculated 60% positive (d3/2 ~17;) and negative (hg/2, hll/2) parity levels in 79Au, and by Bijkere61, who has calculated the same positive (d3/2, 611~) and negative (hg/2, hll/2) parity levels in 77Ir, seem to indicate that (2.4) is a good approximation in the case in which several single particle levels are available to the odd particle.

7.

CONCLUSIONS.

The calculations performed so far seem to indicate that the interacting bosonfermion model may provide the framework for a unified description of collective states in odd-A nuclei. Up to now, these calculations have dealt only with energy levels. It is clear that, before drawing any definitive conclusion, the calculations must be extended to include other properties, such as electromagnetic transition rates. This is presently being done. I would like to conclude this talk by mentioning that the possibility of describing both even and odd-A nuclei within the same theoretical framework, opens an intriguing perspective into the study of symmetries in physical systems. The symmetries encountered so far are symmetries of either purely bosonic or purely fermionic systems. It has been suggested by several authors27) that there may exist symmetries in which both bosonic and fermionic degrees of freedom are linked together in a single group theoretical framework. Since the interacting boson-fermion model describes both fermionic and bosonic degrees of freedom, it may be possible that examples of this new kind of symmetries, called supersymmetries, exist in nuclear spectra. In some preliminary studies, I have been able to construct a supersymmetric solution of the Hamiltonian (2.1). This solution corresponds to bosons with SO(6) symmetry (Sect. 5) and a single particle with angular momentum j=3/2.28) For nuclei with N bosons and either 0 or 1 fermions (even-even or odd-even nuclei) the corresponding energy formula is

(E([N],(ol,02,03),(rl,r2),vA,J,M)= - g4 [Ol(Ul+4) + 02(u2+2) + a321 +

+;

[~l(rl+3) + T~(T~+~)] + c J(J+l)

.

(7.1)

THE INTERACTING BOSON-FERMIONMODEL

63

This formula is a generalization of Eq. (5.1). The single quantum number u is replaced here by (~1, 42, ~3) and z by (~1, ~2). Rules to construct the values of ol, o2$ a3 and ~1, r2 are given in Ref. 28. Here, it is sufficient to say that in even-even nuclei o2'03'G, ul=o, ~2-0, 71-z and J=L. It is eag to see then that Eq. (7.1) reduces to Eq. (5.1), except for a constant term+ N (N+4). The spectrum corresponding to Eq. (7.1) is shown in Fig. 9.

Fig. 9.

Typical even-even and even-odd spectra in the supersyzzzetric situation described by Eq. Q.l), N=3. The energy levels are given by Eq. (7.1), with A/4= 80 keV, B/6= 60 keV and C= 10 keV. The ground state is taken as zero of the energy. The numbers in parenthesis next to each level denote the quantum numbers (~1, 12). The numbers on the top of the figure denote the quantum numbers (01, 42, 03). The lines connecting the levels denote large E2 transitions.

A remarkable feature of the observed spectra in the region of the Platinum nuclei is that they display the supersymmetric structure of Fig. 9 *29p ';;I region, the even-even nuclei are well described by an O(6) symmetry odd-proton nuclei have, among other states, a well developed structure built on the d312 level. This structure appears to be intimately related to that of as shown in Fig. 10 for the pair of nuclei, ~~~*~~~~ ?!&I;" nue1eus* Not only the observed states can be classified according to the group theoretical chain of Fig. 9 but also their energies are well reproduced by Eq. (7.1). From the practical point of view, the presence of supersymmetrieswill open the way to a simple and yet very detailed study of spectra in odd-A nuclei. For example, in addition to closed formulas for energies, one can construct closed expressions for electromagnetic transition rates, transfer intensities, etc., which can be easily checked by experiment. From the conceptual point of view,

64

F. IACHELLO

the occurrence of dynamical supersymmetries is the latest, most complex and most intriguing example of the role played by symmetry considerations in the description of physical systems.

Fig. 10.

An example of supersynmretricstructure in heavy nuclei: the experimental spectra of lqgPtl14307and 1QIrl1431j The lines connecting the levels denote observed electromagnetic transitions (E2 and Ml).

THE INTERACTING BOSON-FRRMIONMODEL

65

REFERENCES ARD FOOTNOTES.

*Work supported in part under USDOE Contract No. RY-76-C-02-3074.

1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30)

31).

A. Bohr, K. Dan. Vidensk. Selsk. Mat.-Fys. Medd. 26, No. 14 (1952). S.G.Nilsson, K. Dan. Vidensk. Selsk. Mat.-Fys. Medd. 9, No. 16 (1955). J. *Meyer-ter-Vehn,Nucl. Phys. E, 111 (1975). F. Iachello and 0. Scholten, Phys. Rev. Lett. 43, 679 (1979). A. Arima and F. Iachello, Phys. Rev. Lett. 35, 1069 (1975). A. Arima and F. Iachello, Ann. Phys. (N.Y.) 99, 253 (1976). A. Arima and F. Iachello, Ann. Phys. (N.Y.) 111, 201 (1978). A. Arima and F. Iachello, Ann. Phys. (N.Y.) Scholten, F. Iachello-and A. A&a, Ann. Phys. (N.Y.) 115, 321 (1978). 0. R. F. Casten and J. A. Cizewski, Nucl. Phys. A309, 477 (1978). 761-76). A. Arima and F. Iachello, Phys. Rev. w, A. Arima, T. Otsuka, F. Iachello, and I. Talmi, Phys. Lett. m, 205 (1977); T. Otsuka, A. Arima, F. Iachello and I. Talmi, Phys. Lett. m, 139 (1978). 0. Scholten and I. Talmi, private comitunication. A. Bohr and B.R. Mottelson, in Nuclear Structure, Vol. 2 (W.A.Benjamin, Reading, Mass, 1975), p. 425. P.F. Bortignon, R.A. Broglia, D.R. Bes, and R. Liotta, Phys. Rep. E, 305 (1977). 0. Civitarese, R.A. Broglia and D.R. Bee, Phys. Lett. z, 45 (1977). F. Iachello and 0. Scholten, to appear in Phys. Lett. J. Meyer-ter-Vehn, Phys. Lett. w, 10 (1979). L. Wilets and M. Jean, Phys. Rev. 102, 788 (1956). A.S. Davydov and G.F. Filippov, Nucl. Phys. 8, 237 (1958). G. Leander, Nucl. Phys. A273, 286 (1976). R.F. Casten and G.J. SmirPhys. Rev. Lett. 43, 337 (1979). A. Gelberg and U. Kaup, in Interacting Bosons in Nuclear Physics, ed. F. Iachello, (Plenum Press, New York, 1979), p. 59. N. Blasi and G. LoBianco, private communication. J. Wood and R.A. Braga, private communication. R. Bijker, private communication. L. Corwin, Y. Ne'eman and S. Sternberg, Rev. Mod. Phys. 7, 573 (1975). F. Iachello, to appear. J. A. Cizewski, R. F. Casten, G.J.Smith, M.L.Stelts, W. R. Kane, H.G. B8rner and W. F. Davidson, Phys. Rev. Lett. 40, 167 (1978). M. Finger, R. Foucher, J.P. Husson, J. Jastrzebski, A. Johnson, G. Astzer, B.R. Erdal, A. Kjelberg, P. Patzelt, 1. Hoglund, S.G. Malmskog and R. Henck, Nucl. Phys. A188, 369 (1972). J. Lukasiak,R.Kaczarowski, J. Jastrzebski, S. Andre' and J. Treherne, Nucl. Phys. A313, 191 (1979).