Exact and analytic solutions of the generalized bohr-mottelson model for spherical nuclei

Nuclear Physics A491 (1989) 91-108 North-Holland, Amsterdam

EXACT AND ANALYTIC SOLUTIONS OF THE GENERALIZED BOHR-MO-ITELSON MODEL FOR SPHERICAL NUCLEI* F.G. SCHOLTZ’,

Institut fiir

Theoretische

G. KYRCHEV’ Pl~~‘sik, Universittit

and

Amand

Tiihingen,

FAESSLER

D-7400

Tiihingen,

FRG

Received 21 July 1988 (Revised 23 September 1988) Abstract: The generalized

Bohr-Mottelson model (GBMM), which encompasses isoscalar and isovector collective vibrations, is solved exactly and analytically in the harmonic approximation for proton and neutron vibrations, coupled via the symmetry energy. To this end we utilize the underlying group structure (which is Sp(4,R) in this case) to construct a unitary transformation that diagonalizes the hamiltonian. Closed expressions for the eigenvalues and eigenstates are given. It is shown that the symmetry energy plays a role analogous to the Majorana interaction in the proton-neutron interacting boson model (IBM-2), namely, it shifts the isovector modes up in energy. The Ml operator in the framework of the GBMM is introduced and the Ml strengths are computed.

1. Introduction Although experimental well-known

the 1’ and 2+ isovector modes only started to attract attention from an point of view in recent years ‘-6), isovector vibrations have been a theoretical concept for many years ‘-‘). As a matter of fact, the idea of an isovector 2+ mode was already advocated some 20 years ago by one of the authors (A.F.) in the framework of a two-fluid geometrical model ‘). This model is an extension of the one-fluid Bohr-Mottelson model ‘0,“) to incorporate both proton and neutron degrees of freedom. In particular the 2+ isoscalar and isovector modes are respectively interpreted in this model as “in-phase” and “out-of-phase” quadrupole vibrations of the neutron and proton fluids, coupled by the symmetry energy. Since it has become customary to refer to this model as the generalized BohrMottelson model (GBMM), we shall use this terminology throughout the present paper. More recently, the l+ and 2+ isovector modes were also predicted as “mixed symmetry states” in the framework of the proton-neutron interacting boson model ‘2-‘4), usually referred to as IBM-2. The subsequent discovery of these modes I-‘) and their identification as IBM-2 mixed symmetry states IS.“) once again l Work supported by the Alexander von Humboldt foundation and the BMFT under contract No. 06Tii778. ’ Fellow of the AvHumboldt. Permanent address: Institute of Theoretical Nuclear Physics, University of Stellenbosch, Stellenbosch 7600, South Africa. * Permanent address: Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia 1784, Bulgaria.

03759474/89/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

92

F.G. Schltz

ei al. / Exact and andytie

emphasized the importance of low-lying algebraic, in nuclear structure theory.

collective

solurims

mad&,

be it geometrical

or

Despite the fact that the idea of isovector 2’ modes was we11 established in the framework of the GBMM, the recent trend has been to use the IBM-2 to calculate the properties of these modes, while the GBMM has been largely neglected in this respect, except for some recent applications in the sphericai regions “-‘I). The major reason for this is probably due to the relatively simple, unitary group structure “,‘“f associated with the IBM-2, enabling one to identify dynamica symmetries “-“), which in turn makes analytic solutions in a variety of cases possible 72W7*).For the GBMM on the other hand the associated group structure is much more complicated and involves a non-compact, symplectic group 17) which is much more difftcult to treat from a dynamical symmetry point of view. The difference between the group structures associated with the IBM-2 and the GBMM stems from the fact that the IBM-2 is a boson number conserving model, while the GBMM is not. This in turn is due to the different physical interpretations underlying the two models. In IBM-2 the bosons represent fermion pairs and, hence, real particles which have to be conserved. In the GBMM on the other hand the bosons create and annihilate phonons associated with the surface vibrations and are not necessarily conserved. In particular the GBMM hamiltonian does not conserve the number of bosons. In contrast to the one-fluid Bohr-Mottelso~ model this is even true in the harmonic approxil~ation for the proton and neutron collective patentials since the symmetry energy term violates boson number conservation’) (see sect. 2). Furthermore, the usual procedure of diagon~lizing in a truncated boson Fock space is invalid here precisely because of the strong mixing caused by the symmetry energy term between states with different numbers of bosons. In this situation an exact diagonalization of the symmetry energy term is called for, and that is one of the major aims of the present paper. Only after this has been done the possibility of diagonaiizing in a truncated boson Fock space can be considered. However, such a procedure is also not without problems. One would expect it to work reasonably well for spherical nuclei with weak anharmonicities, however, for strong anharmonicities and deformed nuclei a fairly large number of bosons should be included to render such an approach trustworthy [in the calculations of refs. 1x-3’) for the one-fluid BohrMotteison model up to 33 bosons were included]. Unfo~unately the dimension of boson Fock space grows very fast with increasing boson number in the case of GBMM (even taking into account the SO(J} symmetry), and hence this approach seems to be unpractical in this case. Therefore, before attempting applications of the GBMM to strongly anharmonic and deformed nuclei, alternative ways of solving for the eigenvalues and eigenstates in these regions have to be developed. All these factors conspired to hamper applications of the GBMM. Despite the difficulties faced by the GBMM, two factors have recently revived the interest in this model very much. The first is the fact that several groups have shown that a close relationship exists between the algebraic IBM models and the

EC. Scldtz

geometrical

models

et a/./ Ewct ad

[see, for instance,

analytic

soltctiom

refs. 22.23,3’,3’)and references

therefore to be expected that since IBM-2 the isovector modes, the GBMM should

93

therein].

It is

accounts quite well for the properties of meet with an equal amount of success.

Furthermore, the geometrical models in general, a much more transparent physical interpretation

and the GBMM in particular, have than the algebraic IBM models.

Hence, because of their close relationship, the geometrical models provide a simple physical framework in which the qualitative features of the algebraic models can be understood. In the present paper an example of this is the close analogy between the Majorana interaction in IBM-2 and the very simple, physically founded symmetry energy in GBMM. The second factor is the recent, intensified interest in symplectic into the explicit construction nuclear models ‘““f. This has stimulated investigations of matrix representations and basis states for the irreducible representations in the 7hmJ’). These investigations culmipositive discrete series of the symplectic groups nated in the study of dynamical symmetries associated with symplectic models, and in particular with Sp(4,R) modelsJ’). These studies are of interest to the GBMM since the most general GBMM hamiltonian is linear and quadratic in the generators of the semi-direct sum algebra “) sp(20,R)@h( IO), with h( 10) the Heisenherg-Weyl algebra of a lo-dimensional oscillator’“). However, a large class of hamiltonians which are such that they contain only even powers in the boson operators, are linear and quadratic in the generators of the symplectic group _“) Sp(20,R). In the spherical limit the situation simplifies even more drastically and the GBMM hamiltonian becomes linear and quadratic in the Sp(4,R) generators (see sect. 2). The work done in ref. “;) opens up the possibility of analysing the GBMM along similar lines than the IBM-2. It should, however, be remarked that in ref. “) only boson number conserving hamiltonians have actually been considered from the point of view of dynamical symmetries. The reason for this (as has been pointed out in ref. “)) is that realistic boson number non-conserving hamiltonians single chain of nested groups, and consequently analytic possible in these cases. It is in the light of this renewed GBMM

interest

cannot be described by a solutions seem not to be

that a more detailed

analysis

of the

seems worthwhile. As a first step we consider in this paper nuclei close to a filfed shell where the harmonic approximation in the proton and neutron collective potentials is fairly good. As already remarked, the model is not trivial in this approximation since the symmetry energy term is not boson number conserving. However, as we show in this paper, it is still simple enough to be solved exactly and analytically (see sect. 3). We show that the solutions provide the correct qualitative physical picture. In particular we find a vibrational spectrum, albeit harmonic, having predominantly isoscalar states as low-lying states, while isovector states appear higher up in energy. We show that it is the symmetry energy that shifts the isovector states to higher energies. In this respect the symmetry energy plays a role analogous to the Majorana interaction in IBM-2. We also show that for the Ml-transition strengths a reasonable qualitative physical picture results in the sense

FIG. Scholtz

94

that only one state obtains Of course, potential

to fit experimental

et al. / E.wact and anal_vtic solutions

all the Ml strength

in this approximation

data one needs to include

which lead to anharmonicities

in the spectrum

(see sect. 4).

anharmonic

terms in the

and cause fragmentation

of the Ml strength. Finally,

we would

like to comment

on sect. 3. In this section

we present

and

discuss the method enabling us to solve the GBMM exactly and analytically in the harmonic approximation. The method of solution is well known in the sense that it basically involves the determination of a Bogoliubov transformation that diagonalizes the hamiltonian by solving a random phase approximation type of eigenvalue equation. However, we set up the relevant equation in two different ways, both of which differ somewhat from the usual approaches 34,35,42,43*42,47). In particular, we make use of group theoretical concepts in this process. The reason for doing this is not to translate old ideas in group theoretical language, but we believe that these complementary formulations favour a deeper understanding which may hopefully lead one to extensions of the above approach which go beyond the harmonic approximation. A particular interesting possibility is to extend the present approach to investigate anharmonic potentials for which analytic solutions are already known. Such potentials for the one-fluid Bohr-Mottelson model, as well as their solutions, were discussed in ref. ‘I). The present paper is organized in the following way. In sect. 2 we briefly review the generalized Bohr-Mottelson model. Next we specialize to the harmonic approximation of the GBMM and we discuss the Sp(4,R) group structure associated with it. In sect. 3 this Sp(4,R) based GBMM is solved in closed form and the properties of the solutions are discussed. In sect. 4 the Ml-operator is introduced and the Ml-strength of the lowest l+ state is calculated. Finally a discussion and conclusions are presented.

2. The model The

generalized Bohr-Mottelson model has been discussed in several papers y,‘7-“), therefore we content ourselves with a very brief review here. The starting point is to define the radii for the proton and neutron mass distributions separately via y,‘7-2’)

(2.1) where, as usual, only quadrupole deformations are included. Since R, is real the collective coordinates CZ,~must satisfy cy$ = (- l)“cyt,_, . Furthermore, the rotational invariance of R, implies that LY,,must transform according to the I = 2 representation of SO(3). The collective variables cy,, provide the generalized coordinates describing the surface motion of the proton and neutron mass distributions. In terms of them the most general Hamilton function which is invariant under time reversal and

EC. Schdtz et al. / Evact and

rotations

95

solutions

is of the form H =;

. ci,+

1 B,,+, ,=p,n + G(a,

Because

analytic

of time

- a,) . (a,-

reversal

(2.2)

a,) .

invariance

derivatives, (Y,, are allowed. rotational invariant functions form K(a,)

1 B,.I((Y,(Y,)~.~,+t...+Vp((Yp)+V((Y,) f mp,n

only

terms

with

an even

number

V,( a,,) and

Furthermore, the potentials of LY,,and cy,,, respectively.

In general

of time V,(LY,,) are

they have the

Lu,$-C,,,(a,a,)‘.a,+C,,,(a,.a,)(cu,.cu,)+...l.

=3C,,,a,.

(2.3)

The last term in (2.2) is obtained by expanding the symmetry energy in terms of the collective quadrupole variables. In this way the coupling constant, G, is also determined ‘),“-“). For spherical nuclei we make a harmonic approximation for the proton and neutron vibrations and specialize to the Hamilton function H =;

To quantize canonically

=z B,&; [ I P.”

ci,+

C

,Lp.n

C,a;a,

I

+G(a,-cu,,).(a,-a,,).

this Hamilton function one introduces conjugate to the collective quadrupole

the generalized variables LY,~

rr+, = B,&:; =(-1)+&b, The usual quantization

condition

is then imposed,

(2.4) momenta,

~.

(2.5)

namely,

[a Ilr) 77,,,,I = ih6,,,6,,, and all other commutators

vanish.

Next we introduce

bj, = (A,/2h)“‘(N,,

- iA,&l)%_J

b,, = (h,/2h)“‘((-l)%_,

7r+,,

(2.6) the quadrupole

bosons

)

+ ihJ’(-1)%,&J

,

(2.7)

where A, = ( B,C,)‘/2, C,=C,+2G, t = p, n and p = -2, -1,. H =

. . , 2. This leads to the second x

I=,,.”

E,(b;

. 6,+&.4(b,+&,).

(2.8) quantized (b;,+&,)

hamiltonian (2.9)

with E, = h( c,/ B,)“2,

A = hG(A,A,,)-I”, 6”,,,=(-1)&b,_,.

(2.10)

F.G. Scldiz

96

et al. / E.xact and anal_vtic .solution.s

It turns out to be more appropriate isovector, t:, bosons defined by

to introduce

the

isoscalar,

de,

and

d;=&b;,+b;,), z;=&(b;,-b;,). The annihilation of these bosons

operators are given by the hermitian conjugate the hamiltonian of (2.10) acquires the form

(2.11) of (2.11). In terms

(2.12) with

Ed=~(Ep+E,)-A, E,=t(E,+E,)+il=E,+2n,

D=:(E,-E”).

(2.13)

One of the reasons for introducing the isoscalar and isovector bosons is that the parameters of (2.12) (related to the original parameters via (2.13)) are often more easily determined from experimental considerations than the original parameters themselves. Finally, upon quantization the classical form of the angular momentum 4R) acquires the form L, =-./lo

1 (bi;&); - L,, + L,, =ji7i[(d’&;+(z’z’);]. 1=p,n

(2.14)

It is very simple to identify the algebraic structure associated with the GBMM hamiltonian (2.12). First we note that it is linear in the set of 10 operators {d”. d’; Zf. z;; d’ . $; 2. 2; ;. g; 2. e J+ . ;; z”-. 2; d; . 2; zi-. ;f .

(2.15)

Utilizing the boson commutation relations for the d- and z-boson operators, we easily infer that the operators (2.15) close on the symplectic 4h) sp(4,R) algebra. An inherent feature of the GBMM hamiltonian (2.12) is that it is linear in the generators of Sp(4,R). This observation provides the key to solve for the eigenvalues and eigenstates of the hamiltonian (2.12) in closed form. Let us emphasize again that sp(4,R) is not the algebraic structure associated with the most general CBMM hamiltonian. Generally one requires combinations of the type given in (2.15), but coupled to all possible allowed angular momenta. Furthermore one also requires single boson creation and annihilation operators. These operators close on the semi-direct sum algebra sp~20,R)~h~lO). Finally, in general

F.G. Scholtz

the hamiltonian enveloping

is not linear

algebra

er al. / E.-tact and analytic solutions

in the generators,

but include

97

terms belonging

to the

of sp( 20,R) 0 h( 10).

3. Solutions

of the CBMM in the harmonic approximation

To simplify the discussion of this. section notation for the d- and z-bosons:

we introduce

the following

unifying

for T=d, for 5-=z,

(3.1)

and the hermitian conjugate for the annihilation operators. Since the hamiltonian of (2.12) is linear in the Sp(4,R) generators, we can take the following approach to solve for the eigenvalues and eigenstates. We seek a unitary transformation, U, which is also a Sp(4,R) group element, i.e., U=erp{i,f,

(3.2)

0,X,},

where the 0, are real parameters and the X, are hermitian linear combinations of the generators (2.15). This transformation should be such that UHU-’ is a linear combination of elements belonging to the Cartan subalgebra only. Since the Cartan subalgebra is spanned by d’ . d’ and z’ . 2 this implies I/H”-’

We must now determine

=

1

z -r,d

the unknown

C,bi

. & + const .

coefficients

(3.3)

on the right of (3.3) as well as

the transformation U. At this point it should be remarked that since the Sp(4,R) generators of (2.15) are all scalars with respect to the SO(3) of angular momentum defined in (2.14), U is also an SO(3) scalar. This implies that the operators T: and UT: Urn’, where TL is any SO(3) tensor operator, have precisely the same SO(3) tensorial properties. To determine the unknowns in (3.3) we note that for any T and p the relation (3.3) is equivalent to

Relations

[ “HU-‘,

b;,]

= C,b;,

[ “HU-‘,

b,,]

= -C,b,.

(3.4a) can be rewritten

,

(3.4a)

as follows:

[H, Um’b&“]=C,” [H, Um’b,“]

‘b;,U,

= -C,“-lb,,“.

(3.4b)

F.G. Scholtz et al. / Exact and analytic solutions

98

Using the well known BCH-formulas ‘0, one derives that for any fixed p, the four boson operators bl, , gT, for T = d, z transform under Sp(4,R) according to the fundamental 4-dimensional representation, i.e., V’b:$J

=

1

x,,.(O,)bi,,,+

T’=d,z

U-%&J

=

1

1

YT&)&,=

B;,,

r’=d,z

_&(

O,)b&, +

r’-:d,z

C

x:,(0&,,

= &.

(3.5)

T’2d.z

As indicated the coefficients appearing in (3.5) are functions of the real parameters 0, occurring in (3.3). These functions can be calculated explicitly, however, their explicit form is unimportant for our present discussion and we therefore refrain from listing them here. Eq. (3.5) has two important properties that should be noted. Firstly, because of the SO(3) scalar property of U the linear combinations of (3.5), which we denote by BL, and l?_, have exactly the same SO(3) transformation Secondly, particular relations exist between the properties as the b, bosons. coefficients of (3.5). These relations can be obtained by calculating the coefficients of (3.5) explicitly as functions of 0,. It is, however, easier to obtain these relations by realizing that since U is a unitary transformation, the linear combinations B& must also satisfy boson commutation relations. This leads to the following and i, relations between the coefficients x and y of (3.5): c (XT+%, 71

-Y:T,Y.,.,)

c (X,,,Y,+, -x,,,lY,,,)

= a,,,,

= 0.

(3.6)

7,

Here, and in what follows, we drop the explicit 0; dependence of these coefficients. equation. For the Using (3.5) and (3.4b), eq. (3.3) is reduced to an eigenvalue hamiltonian (2.12) this eigenvalue equation explicitly reads

[:

;

Ii

$j=c[;].

(3.7)

In the above eigenvalue equation the coefficients C, appear as the eigenvalues, U, while the coefficients x,,, and yTil, which in turn determine the transformation are determined by the corresponding eigenvectors. Formulated differently: if one multiplies (3.3) from the left with Urn’ and the right with U, one has H =

1 T=z,d

C,U

‘b:U.

W’&U+const=

1

C,B%, &+const.

(3.8)

T=l.d

In the light of eq. (3.8) the above procedure amounts to determining a set of transformed bosons, defined by (3.5), such that when the hamiltonian is written in terms of them it becomes diagonal. These bosons, as well as their excitation energies, are determined by the eigenvalue equation (3.7).

FG.

Schok et

ul. / E.-tact and anal)Tic

solutions

99

The above procedure is very well known in another context. The reader familiar with Bogoliubov transformations “) will recognize (3.5) as such. Furthermore, he will recognize

the role played

by the eigenvalue

the RPA eigenvalue equation which essentially formation that diagonalizes the RPA hamiltonian

equation

(3.7) as similar

to that of

determines the Bogoliubov trans47). Finally the relations (3.6) will

be recognized as the orthogonality conditions satisfied by the solutions of the RPA equations “). Of course, the above procedure is not an approximation, but exact. The main difference between the approach presented here and the conventional approach is that we explicitly exploit and emphasize the group theoretical structure underlying the Bogoliubov transformations and the RPA equations. Before we solve the eigenvalue equation (3.7) explicitly, some remarks on its general properties and its relation to the unitary transformation of (3.3) should be made. Once again it should be said that the properties of (3.7) are well known in the context of RPA, however, for the sake of completeness and consistency we prefer to review them briefly within the present framework. The first remark that should be made is that only solutions of (3.7) with real eigenvalues are compatible with (3.3). This stems from the hermiticity of (3.3). If the hamiltonian is such that the eigenvalues of (3.7) become complex, the transformation U does not exist. In the context of RPA complex eigenvalues signal that the Hartree-Fock (Bose) solutions do not correspond to a minimum in the energy surface, but to a maximum or saddle point “), i.e., the RPA ground-state becomes unstable. Within the present framework it is clear that if this happens the hamiltonian cannot be diagonalized by means of a Bogoliubov transformation. We therefore only consider solutions with real eigenvalues as valid. In the particular case of (3.7) where the matrix is real, the reality of the eigenvalues implies that the eigenvectors can also be chosen real. To see this one simply takes the complex conjugate of (3.7). This shows that for an eigenvector with real eigenvalue, the eigenvector is also an eigenvector of (3.7) with the linear combinations which are real can be formed. This transformation U. One can choose the transformation

the complex conjugate of same eigenvalue. Hence also reflects back on the such that the coefficients

appearing in (3.5) are real. Only four of the ten real parameters 8; feature in this case. The second remark is that one expects, as in RPA “‘), two real positive eigenvalues and their negatives as the solutions of (3.7). The reason for this, as can be seen from (3.4b), is that if a particular linear combination of the type (3.5) satisfies (3.7), so will its hermitian conjugate, but with negative eigenvalue. This can also be seen directly from (3.7). If a particular column vector satisfies (3.7) the column vector obtained from it by exchanging the x and _v will also satisfy (3.7), but with the negative eigenvalue. The next point that should be raised, is the orthogonality and completeness relations satisfied by the solutions of (3.7). If the eigenvalue equation (3.7) is indeed equivalent to determining the unitary transformation U of (3.3), it should lead to the same relations as given in (3.6). If the relations of (3.6) are incompatible with

F.G. Scholtz et al. / Exact and anal_ytic solutions

100

the eigenvalue

equation

U. The orthogonality correspond

(3.7) it would

and completeness

to real eigenvalues

E,

the completeness

relations

satisfied

of (3.7) are well known

freedom to choose the eigenvectors .in the form

Similarly,

again have implications

real, the orthogonality

(Xi&T

relations

-w?J

=

for the existence

by the eigenvectors from RPA47). relations

of

which

Using

the

can be written

0.

(3.9)

read

(3.10)

j; 2 (xi~Yi$- X,,.J’ir) = 0 .

In (3.9) and (3.10), i = I,2 tabet the two eigenvectors corresponding to the two real, positive eigenvalues of (3.7). We note that (3.9) is simply the real form of (3.6). The eigenvalue equation (3.7) is easily solved. We give the solutions in terms of the original parameters E,, E, and 11 of (2.9) since the expressions simplify considerably in terms of these parameters. On the other hand, the parameters Ed, E,, D and A appearing in (2.12) are probably more easy to determine from experimental considerations. The relations between these parameters are, however, given by (2.13) so that the relevant equations are easily derived if another set of parameters is preferable. For the eigenvalues one finds

C=*E,, E,=J1[-cu~((n2-4P)"']"', a=-(E;+E;), p = (E,E,-4n’&!?,.

(3.11)

It is also easy to solve for the eigenvectors. We give the results only for the eigenvectors corresponding to positive eigenvalues, since those with negative eigenvalues

can easily be obtained

from these, as was already

discussed.

We find

x,,=A~'[(E,5E,)(E,+E+)(E,-A-Ei)-2A'E,Jx,,, Y id=

-A~'[A(E,+E,)(E,,-E,)t2A'E,,]x,,,

y,,=A-'[A(E,+E,)(E,-E,)-2n'E,lx,,, A=(E,+E,)(E,+E,)(E,+n-E,)-2,1'E,,

(3.12)

F.G. Scholl

where the coefficients

x,~ are to be determined

(3.9). As was discussed before, it is important eigenvalues are real since the transformation stances. The conditions

ctnalyricsolutiom

et al. / EYUCCand

from the normalization

P>O,

condition

to check under what conditions the U exists only under these circum-

under which the eigenvalues a
101

(3.11) are real, are the following:

a”-4BZO.

(3.13)

The first condition in (3.13) is obviously satisfied. To check the other two conditions one calculates B and ~y’--4B explicitly in terms of the original mass parameters B,, , B,, the stiffness parameters C,, , C, and the coupling strength, G, of the symmetry energy. Since these parameters are all positive quantities in physical situations, the other two conditions in (3.13) can be checked. One finds

P= ~[C,,C,+2G(C.,,+C.,)l>O, n P (3.14) Hence, in all physical situations the transformation U exists. We are now in the position to write down the diagonal form of the hamiltonian. We denote the bosons corresponding to the positive solutions of (3.11) and (3.12) (see also (3.5)) by Bi,,, respectively. The solutions corresponding to the negative eigenvalues represent the annihilation operators B,,,. From (3.8) we then have for the hamiltonian, expressed in terms of the transformed bosons, H=E,Bi,.B++E_Bi.B_+const.

(3.15)

When the expressions for the eigenvalues E, and for the B,,, bosons in terms of the d- and z-bosons are inserted in the hamiltonian (3.15), one indeed recovers the original hamiltonian (2.12). Note from (3.15) that the transformation U enabled us to eliminate the non-compact generators of Sp(4,R) appearing in the hamiltonian (2.12). Usually this step is not undertaken in the established algebraic approach to nuclear symplectic models “,“). Instead the matrix elements of the non-compact generators are computed in the relevant basis. On the other hand it is a standard procedure in RPA, but not formulated in algebraic terms there “) [see also ref. “), p. 541. The eigenstates In+,-,;.

of (3.23) are given by

. f n t,2; n-,ml;. . . HL)=~;,,,__~

(n,..!n~,,.!)~““(B;,,)“+.~(Bi,,,)”~.~~lO) (3.16)

where for all p

B,,,lO)=o.

(3.17)

102

F.G. Scholtz ef al. / Exact

The vacuum

and anulvtic

IO) can easily be related to the vacuum,

.solutions

IO},of the d- and z-bosons

since

b, IO)= 0; Vr = d, z; tfp d.-‘b,UU~‘~O)=O +43,,,u--‘/o)=o; This implies

IO) = V’IO).

,vp.

The corresponding

(3.18)

eigenvalues

are given by

E = E, C n,., + E- C n-,,. EL &

(3.19)

Here we have chosen the energy of the ground state, which is simply the B, boson vacuum IO), at E = 0. The eigenstates (3.16) do not have good angular momenta. In order to give a complete classification of the eigenstates with good angular momenta, it is convenient to introduce the following group chain: SU,(5) The generators

X SU_(S) 2 SU(5) = SO(5) 2 SO(3) 3 SO(2).

of the different SU,(5):

An important (3.21) is exactly angular property

momentum

groups

appearing

(3.20)

in this chain are the following:

(Bi,E*);;1=1,2,3,4,

SU(5):

(B:&);+(B”k);;

l-1,2,3,4,

SO(5):

(Bi~.,):+(B”~-):;1=1,3,

SO(3):

Jlo[(B.$+);

+(

sik$1 .

(3.21)

point that should be realized here is that the SO(3) defined by the same as that of (2.14) and hence it represents the physical operators.

The reason

for this is again due to the SO(3j scalar

of U. We have from (2.14)

L, = U-‘L&f=

C hk-‘[(b;t;,);]U T-&Z

=,iiii[(B:i+);+(Siii_);].

(3.22)

Here, UU-’ has been inserted between the boson operators and (3.5) was used. Since the hamiltonian (3.15) is written in terms of the first order Casimir operators of SU,(S), it is diagonal in the basis ciassified by the chain (,3.20). A complete classification scheme for the eigenstates is therefore 22m2h):

l[N+l,[N~l,[N++N~-~;f;fl,

Y,(v,, 44

L, ML

f=o, lJ...min{N+,

N-1. (3.23)

Here, N, and N_ are the total number of B, and B_ bosons, respectively. They are also the eigenvalues of the first-order Casimir operators and label the completely symmetric representations of SU+(S) and SU_(S), respectively. The allowed representations of SU(5) are labelled by EN+ + N_ --if], where the range off is given

F.G. Scholtz et al. / Euoct and analytic

solutions

103

The SO(5) by (3.23). The quantum numbers (v,, v2) label the SO( 5) representations. representations contained in the SU(5) representations of (3.23) are obtained by a Schur function division ‘6.4”). The angular label the SO(3) and SO(2) representations,

momentum,

L, and its z-projection, The reduction

from SO(5)

to SO(3) is also well known ‘(‘). Finally, the two quantum numbers multiplicity indices labelling repeated SO(5) and SO(3) representations. values of (3.15) in the basis (3.23) are

y and 6 are The eigen-

E(N+,

respectively.

M,

(3.24)

N )=E+N++KN_.

Once again the ground-state energy has been chosen at E = 0. The ground-state itself is represented in the classification scheme of (3.23) by the state having all the vacuum IO). The classification of some of quantum numbers equal to zero, i.e., the low-lying states is listed in table 1. Next we study the influence of the symmetry energy on the solutions. For this purpose we imagine for the moment that the parameter G is made large and then we consider the behaviour of the eigenvalues (3.11) and eigenfunctions (3.12) to

TAI3LF

Classification

[N+l

IN-1

LOI

LOI

[k 01

[II LOI

LOI Cl1

[21

LOI

111

[II

[N++N_-t;fl

[31 [?I

[21

LOI [II

E

co,01

0

0

[t,ol [l, 01

11.0) (1,O)

7 2

E+

[I, 01

(0, 0)

[I> 11 [L 01

[3,01

[LIl [3,01

[II

[21

f’. 11 f3.01

101

[31

(v,,+)

L

[2,01

LOI

1

of the lowest eigenstates

[3,01

0

(2,O)

2,4

Cl,11 co,01

I, 3

(2.0)

2.4

2E, E++E

0

(0.0)

0

(2,O)

2.4

(1,O) (3,O) (I,(), (3, I) (1.0) (3,(J) (1,O) (2, I) (I,O) (3.0)

E-

2E

2

3E+

0, 3,4,6 2

2E++

I, 2,3,4,5 2 0, 3,4,6 2

E++2E-

I, 2,3,4,

s

i

(1.0)

0.3,4,6 2

(3,O)

0,3,4,

3E 6

Em

F.G. Scholtz et al. / Exact and analytic solutions

104

leading order in G. Using (2.8) and (2.10) differently in G. In particular, one obtains

one finds that the eigenvalues

scale

B= B,+B,, E-.-l.

(3.25)

Now we study the properties of the eigenstates corresponding leading order in G. For the state corresponding to E_ one finds

to E, and

E- to

(3.26)

For the state corresponding

P+~=

-$A-’

to E, we obtain

[(~)“4(l-(~)“2)(I+(~)‘/2)+I]x+=,

A++(~)“2)(,+(~)“2)(l_(~)“2-~(y4)-~. For &,= B, equations (3.26) and (3.27) lead for the eigenstate E- to vanishing isovector amplitudes, i.e.,

corresponding

?&=y,=o

and for the eigenstate

corresponding

to E, to vanishing x,=4fd=o.

(3.27) to

(3.28) isoscalar

amplitudes,

i.e., (3.29)

We have thus shown that the states E, whose energy scale like G”’ (i.e., the states that are being shifted up in energy by the symmetry energy), have predominantly an isovector nature. On the other hand the states which are not influenced by the symmetry energy to leading order in G (i.e., the low-lying states, E_), have predominantly an isoscalar nature. In this sense the role played by the symmetry energy is analogous to that of the Majorana interaction in IBM-2. It is worthwhile to note that the matrix featuring on the left-hand side of the eigenvalue equation (3.7) is nothing but a particular matrix representation of the GBMM hamiltonian (2.12). We proceed to show that this matrix representation is directiy related to the 4-dimensional fundamental representation of sp(4,R) which is the algebra underlying the GBMM hamiltonian (2.12). Without going into too

EG.

Scholtz ef al. / Exact and analytic

105

.solu/ions

much detail, we briefly indicate this result. Following ref. “‘) one can introduce following 4-dimensional matrix representation for the sp(4,R) algebra:

the

(3.30) wherej = 1,2,3; I is the 2 x 2 unit matrix, U, and a, are the 2 x 2 Pauli spin matrices and their complex conjugates, respectively. The isomorphism between the matrix representation (3.30) and the corresponding boson realization of the sp(4,R) algebra is explicitly given by j4) N++;(d’.

Bjt,-;(d’ B,--;(&

For reasons

of economy

Li+z’

. z”),

d’-_z’ d-5.

. z’), 5).

we listed here only those 5 generators

actually enter in the hamiltonian can be rewritten as

(2.12). Indeed,

H=(E,+E,)N+(E,,-E,)J,+2DJ,+2A(Bj+B,).

(3.31) of (3.30) which

using (3.31) the hamiltonian

(2.12)

(3.32)

Inserting the matrix representation (3.30) in (3.32) one finds the matrix representation of H, which turns out to be exactly the matrix appearing in the eigenvalue equation (3.7).

As before we diagonalize the Hamilton matrix obtained from (3.30) and (3.32) (see eq. (3.7)). The resulting diagonal matrix can be written as a unique linear combination of the two diagonal matrices N and J1 of (3.30). Because of the isomorphism (3.31) we recover the diagonal form (3.3) of the transformed boson hamiltonian. Furthermore, the expansion coefficients of this diagonal matrix in terms of N and JJ are nothing but the one-boson excitation energies (as seen from (3.3)) and indeed coincide with our previous result (3.11). Finally, as discussed in (3.8), the original hamiltonian can be expressed in terms of the transformed bosons B,, of (3.5) (see eq. (3.15)).

EC.

106

Scholtz et al. / E.wct

and annl.vtic solutions

4. Ml operator and Ml strengths In the framework

of GBMM

the Ml operatordX)

assumes

a form similar

to that

in IBM-2 Tt(Ml)=g,L,,,+g,L,.,,

(4.1)

where L,,, and L,,, are defined in (2.14). The operator (4.1) is in fact the most general one-body boson operator transforming according to the I = 1, SO(3) representation. In terms of the isoscalar, d, and isovector, z, bosons (4.1) becomes T:,(Ml)=f(~~+g,,)L,+fjfO(g,-g,,)[(d’~):+(z~~d):].

(4.2)

Here L, is the total angular momentum operator defined in (2.14). With the aid of the orthogonaiity and completeness relations (3.9) and (3.10), one can express the d- and z-bosom in terms of the B, and B- bosons (see eq. (3.5)). This leads to

The coefficients x and y are given in (3.12). As was already discussed in sect. 3 the ground-state of the hamiltonian (2.12) is simply the vacuum of the B, and B- bosons. In terms of the classification scheme of (3.23) this state has all quantum numbers equal to zero. From (4.3) it is clear that the Ml operator can connect only one state with the ground state. In terms of the classification out to be the following state: ][N.,=l],[N_=l],[l, Calculating

l],(l,

the Ml strength

scheme

l), L=l,

of (3.23) (see table

M)=(B:,B’)AIO).

1) this turns

(4.4)

of this state one finds

1~[11,[1lJ, 11,(1, 11,L= w-‘//w = 15~g,-g,~“r~~-,Y+,+~-,Y+d~-~~+~.Y-~+~+,Y~~~lZ*

(4.5)

5. Summary and conclusions We have solved the GBMM exactly and analytically in the harmonic approximation for the proton and neutron vibrations, coupled via the symmetry energy. For spherical nuclei this is a realistic approximation which leads to a reasonable qualitative physical picture regarding the energy spectrum as well as Ml strengths. To make detailed comparisons with experiment further refinements of the model, such as the inclusion of anharmonicities, are necessary. However, one can expect

F.G.

that this will only involve quite capable

of describing

Scholtz

et al. / Eract

“fine tuning” the properties

and

analytic

of the model

solutions

107

and that the model

of the isoscalar

and isovector

will be

modes

in

spherical nuclei. Investigations in this direction is currently being undertaken. Of course we realize that the IBM-2 accounts for most of the properties of the isoscalar and isovector modes in spherical and deformed nuclei. However, as we have pointed out in the introduction, the present geometrical model has a much more transparent physical interpretation than the algebraic IBM-2. The close relation between the geometrical and algebraic models makes it possible to understand the qualitative physical features of the algebraic models in the very simple physical framework of the geometrical models. Here an example of this is the close analogy between the very physical symmetry energy of the GBMM and the more abstract Majorana interaction of IBM-2. To solve the present model the properties of the underlying sp(4,R) algebra have been used explicitly. The similarities and differences with other algebraic and RPA-type of approaches have been indicated. Currently the possibility of extending the present procedure to include anharmonicities is being investigated. To this end we hope to exploit the point of view presented at the end of sect. 3 of the present paper. A research grant from the Alexander von Humboldt foundation and the support by the BMFT is gratefully acknowledged by two of us. The second author gratefully acknowledges partial support by the Bulgarian Committee of Science under contract N330.

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