Vibrations and rotations for the proton-neutron interacting boson model

Nuclear Physics A483 (1988) 23-49 Noah-HoIiand, Amsterdam

VIBRATIONS

AND ROTATIONS FOR THE PROTON-NEUTRON INTERACTING BOSON MODEL F.J.W.

Institut fiir

Theoretische Physik, Uniuersitiit Frankfurt, 06000 Frankfurt am Main 11, West Germany F.G. SCHOLTZ

~nstjtute of ~eorefical ~ue~ear Physics, University of Steile~bosch, Ste~Zenbosch7600, South Africa Received

23 November

I987

Abstract: The interacting proton-neutron boson model (IBM-2) is analyzed in terms of the concepts of a geometrical picture. The centre of mass and relative-motion deformations are determined, and for the deformed systems, rotational and vibrational modes are identified. The parameters in an intrinsic hamiltonian which govern these modes are calculated. For the SU(3) dynamical symmetry a one-to-one mapping is given between low-lying basis states in the geometrical and the algebraic model. The magnetic dipole operator in the geometrical model is derived from its counterpart in IBM-2. This serves as an example for the calculation of static and transition operators, generally.

1. Introduction

The study of nuclear collective motion with special emphasis on the proton and neutron degrees of freedom was initiated in a framework of geometrical concepts many years ago rW3)an d 1‘t is continuing “)_ More recently, with the avalanche of the interacting boson model 5, (IBM) unto nuclear structure physics, the proton and neutron collective motion has received widespread attention “). This version of the interacting boson model ‘,‘), called IBM-2, has led to the successful search of new collective states referred to as mixed-symmetry states “). All along there have been efforts to argue in terms of concepts of both the geometrical as well as the algebraic models. The initial stance of two opposing models has given way “) to a “workable marriage” which holds promise for a deeper understanding of the physics involved. In this spirit we have recently developed a method *‘-‘*) which enables us to translate from the algebraic to the geometrical model. For the proton-neutron system, the basic ideas of the method were explained 12) by studying a simplified model, namely the U(3) x U(3) model. It was pointed out that these methods could be generalized to other groups and in this paper we do this explicitly for the U(6) x U(6) group, which is the group of IBM-2. t Permanent Africa.

address:

Department

of Physics,

0375-9474/88/$03.50 @ Ekevier Science Publishers (Nosh-Holland Physics ~blishing Division)

University

B.V.

of Stellenbosch,

Stellenbosch

7600, South

24

F.J. W. Hahne, F.G. Scholtz / Vibrations and rotations

Many researchers have attempted to find the correspondence between the algebraic and the geometrical model. Some of this work has been discussed in our previous papers l”-12) and elsewhere 9Y’3-‘5).We do not repeat this discussion here, However, we do reiterate that in our method we remain fully in the domain of quantum mechanics and also that we take special care to protect exact symmetries, when approximations are made, in order not to break them. At this point we also repeat that our method is based on a l/N expansion, where N is the number of bosom. The approximation which is made is that this expansion is terminated at some point. In the present paper only the first order terms are retained. The required background on IBM-2 is presented in sect. 2 and in sect. 3 the positional representation for IBM-2 is discussed. It is also shown how a general operator is written in terms of rotationaf modes and vibrational bosons. This method is developed further in sect. 4 where it is also applied to different forms of the IBM-2 hamiltonian. In this discussion the importance of F-spin invariance is demonstrated by means of examples. In sect. 5 attention is paid to the construction of the wave functions for the geometrical picture. It is then shown that there exists a one-to-one relation of low-lying states in the geometrical and the algebraic model as one also sees from table 1. In sect. 6 we present the calculation of Ml strengths in the geometrical model. It serves as an example for caiculations of transitions generally. Sect. 7 contains the conclusion. Some of the technical detail can be found in appendices A and B.

2. The IBM-2 model

In the IBM-2 proton s- and d-bosons created by s’, and dLfi,(p = -2, -1,. . .2) as well as neutron s- and d-bosons created by s’, and dt,(p = -2, -1,. . .2) are introduced ‘x8).The traditional interpretation is that the proton and neutron bosons are associated with proton and neutron pairs, respectively. With this interpretation in mind the number of proton and neutron bosons have to be conserved separateiy 7,8).This gives rise to the dynamical group structure U,(6) x U,(6) where U,(6) and U,(6) are generated by the creation-annihilation pairs b’,‘b&, and b’y’,bLwT (I = 0 denotes a s-boson and I = 2 a d-boson), respectively. All physical operators can be written in terms of the generators of this group. Of particular importance are the total angular momentum operators given by L,=~[(d%~~):,+(dtci,)~]~L,,+L,

(2.1)

and which generate an SO(3) subgroup. For later reference we also list the CartanWeyl basis of this algebra, namely, L* = FL*, ,

&=La,

(2.2)

F.J. W. Hahne, F.G. Scholtz / Vibrations and rotations

and the Cartesian

2s

basis Lx =&L++ Ly = -i&L+

L-), - L_) ,

L, = Lo.

(2.3)

The IBM-2 hamiltonian is then chosen as the most general one-plus-two-body operator which conserves the number of proton and neutron bosons separately, and which is an SO(3) scalar. We observe that for a fixed number of proton and neutron bosons N,, and NV, the IBM-2 basis states span the direct product space [N,] x [NV] where [NT] and [NV] are the symmetric representations of U,(6) and U,(6), respectively 7*8,16). Just which basis choice terms

as in the case of IBM-l there are several group chains of U,(6) x U,(6) terminate in SO(3) 1 SO(2) and which can be used to classify the IBM-2 a particular states 13-18). To each of these group chains there corresponds of interaction strengths for which the IBM-2 hamiltonian can be written in of the Casimir operators occurring in that particular group chain only. In

these cases, referred to as dynamical symmetries, the ‘3-18). One such dynamical symmetry down analytically case for the formalism we develop here, is the SU(3) which we discuss in more detail below. A concept of central importance in IBM-2 is that of terms of an SU(2) group generated by lo) F+=st,s,+dt,.

eigenvalues can be written which provides an ideal test dynamical F-spin

symmetry

13-18)

7,‘o). It is defined

in

L?y,

F_=s;s,+d:& Fz=~[(s~s,+d~~L?,)-(s~s,+d~~~v)].

(2.4)

One can also introduce the SU(2) Casimir operator *-” with eigenvalues F(F + 1). For fixed number of proton and neutron bosons the eigenvalues of F, are fixed on 5 (N,, - NY) and F is clearly restricted to 31N,, - N,] G F s i (N,, + NV). The states with maximum F-spin, i.e., F,,, =$N (N being the total number of bosons), are completely symmetric in both proton and neutron degrees of freedom. The formalism developed here is applicable to any system which is well deformed in the c.m. coordinates, However, to ensure maximum clarity and simplicity, we illustrate it here only for two hamiltonians most commonly used in applications to deformed nuclei with axial symmetry, and for the Ml transition operator. As an example of an F-spin invariant hamiltonian we consider H,=K(Q,+Q,).(Q,+Q,)+K’L’L+M, where the quadrupole

operators

(2.5)

are given by

o,,=(s~L?p)2+(d~sP):+xP(d~L?p)~,

(P=~,v),

(2.6)

26

EJ. W. Hahne, EC.

Scholtz / Vibrations and rotations

and the Majorana term by 20) (2.7) k=1.3

Calculating the commutators with (2.4), one finds that one must take x,, = xy to ensure that the quadrupole-quadrupole part of H, is an F-spin scalar. From (2.4) one also notes that L and i%fare already F-spin scalars. Note also that H, commutes with the total angular momentum operators, but not with the proton and neutron angular momentum operators separately. This F-spin invariant hamiltonian has been used in applications to axially deformed nuclei ‘I). In ref. “) it was also shown that the breaking of F-spin symmetry by taking xn Z xy does not really lead to any improvements in the fits to energy spectra. As an example of an F-spin breaking hamiltonian we consider &=K&’

t&+K’L.

L+i%f.

053)

Here we take in general xrr # xy but, of course, even when ,Y_= x,, the hamiltonian (2.8) still breaks F-spin symmetry. The hamiltonian (2.8) has also been used in applications to axially deformed nuclei ‘I). We remark that when one chooses in (2.7) h = 5, = 5; = 6, the Majorana term can be written in terms of the F-spin Casimir operator s2, one has 18*2o) M=h[F,,,(F,,,-tl)-~*I.

(2.9)

In the case of an F-spin invariant hamiltonian f; is a good quantum number and the eigenvalues of (2.9) which are given by 18) M(F) = h[F,,X(Fln,,+

I)- F(F+

1)l

(2.10)

appear as an additive term which does not depend on the other details of the dynamics. As already remarked, a particularly useful test case for the formalism is the SU(3) dynamical symmetry. This is obtained by setting xv =xy = -4fi in (2.5). In this case one can write for the hamiltonian (2.5) (.$ = 0, Vi) (see ref. ‘“>) &&=$Kc,(su(3))+(K’-;K)b

L.

(2.11)

Here C,(SU(3)) is the second order Casimir operator of the SU(3) group generated by

0, = Q-w + Q,

(,Yw=xy=-$7)

(2.12)

and the total angular momentum operators of (2.1). The hamiltonian (2.11) is clearly diagonal in the basis classified by the group chain U,(6) X U,(6) = U(6) = SU(3) = SO(3))

(2.13)

F.J. W. Hahne, F.G. Scholtz / Vibrations and rotations

where U(6) is generated values 16)

by b&b&,+

27

b~~~~~‘. One readiiy finds for the eigen-

EfA,& L)=~K[h2+11;*+h/&+‘3(ti-i-~)]+(K’-~K)~(~+1).

(2.14)

Here (A, p) labels the SU(3) representations contained in the direct product representations [N,] x [N,]. These are easily found to be the SU(3) representations obtained from the reduction of the Kronecker products (A,, pW) x (A,, py), where (A,, p,) and (A,, pV) are the SU(3) representation contained in the symmetric U(6) representations [NT] and [NV], respectively. It is well known that the symmetric U(6) representation [NJ contains the following SU(3) representations 22): (h,,~)=(2N-2k,k-35’)

(2.15a)

with s=0,1,2,...

*(N-2)

or

$(N-1)

or

$N

(2.15b)

and k=3$3S+2,3S+4

,...,

N-l

N

for N + S even

for N+Sodd.

(2.15~)

The reduction of the SU(3) representations with respect to SO(3) is also well known. The SO(3) content of the SU(3) representation (A, p) is given by 22,23) L=K,

K+l,.._,

K+max{A,y}

forKf0,

(2.16a)

where K=min{A,~},min{h,~}-2

,...,

1 or

0.

(2.16b)

L=max{A,~},max{A,~}-2

,...,

1 or

0.

(2.16~)

For K = 0 one has

3. Positional representation Since the formalism is well established by now ‘o-*2), we give just a very brief summary of the main results. The positional representation is based on going over to the Holstein-Primako~ realization for U,(6) and U,(6) by eliminating the proton and neutron s-bosons, respectively. The remaining bosons are then related to shape operators in the following way: cylpF=&d&,+&J, The simultaneous

p== T, v_

(3.1)

eigenstate of the operators (3.1) is given by

+J-i(n, - d:+ a, + dt,))lO)

(3.2)

F.J. W. H&me, F.G. Scholtz / Vibrations

28

and rotations

and satisfies GpPIo,, CX,)=N~~~cY,,q,) with aPP * =(-l)IL~P_P (for more detail see ref. ‘I)). The state (3.2) involves a particular choice for the scale of the shape variables. This choice leads to unfamiliar expressions for the transformation to centre-of-mass and relative coordinates, but it simplifies the expressions and makes comparisons with already existing results on the IBM easier. The familiar expressions for the transformation to centre-of-mass and relative coordinates can be obtained by a simple resealing of the shape variables occurring in (3.2). One can introduce centre of mass, mMp, and relative, crRIL,shape coordinates in the following way:

(3.3) Simultaneously

one introduces the new bosons d,,=$$:,+$$CP, 13.4)

In terms of these the state (3.2) becomes ’ ffY,+~R.~R))exp(-~(d~.dtM+d~.d:,)

I%, *R)=--$exp(-f(ru, rr +Jz(ar,

* &+a,

* d;))[O) 1

(3.5)

It is now possible to define an intrinsic frame as follows: aMfi

=c

h

~~(nbMA,

aM2

=

aM-2,

aMI=uM-,=O.

(3.6)

The physical content of (3.6) is simply that the intrinsic axes are chosen to coincide with the principal axes of the mass distribution in the centre-of-mass frame. Presently the a& refer to the laboratory frame. They can also be transformed to the intrinsic frame via tY&= c f)ZW)%h. A Introducing

(3.7)

(3.6) and (3.7) in (3.5) one has ‘I)

xexp(JZ(aM,dtMo+uM2(d‘iM2+dR_-2)+tYk.

&))I@

FY. W. ~uh~~

F.G. Schottz ,I

~hratjons arrdro!utjo~

29

with R(f2) =exp (-if?, L2) exp (-i&L,)

exp (-i&t,).

(3.9)

Here LX, LY and L, are the Cartesian components of the total angular momentum operators. For convenience we drop the prime on the ty,+ and simply keep in mind that they now refer to the intrinsic frame. The completeness relation reads “)

with (3.11) and (3.12) Furthermore dT(a

MOr

h4.2)

=JZaM2&G40-N.d

dsMO

daM2

d.C?= sin f&d& d& de,

(3.13)

and the region of integration in (3.10) is (a MO1 aMZ)E(E0,a3)Xt0,~a,,/~f)eSSimilar to the discussion in ref. I*) one can now introduce deformations. To discuss the hamiItonians of sect. 2 it is adequate to introduce deformations in the aMOand +@ directions only as will become clear in sect. 4. One therefore shifts the reference points to PM and &, i.e.,

(3.14) and introduce the new bosons b;=-&&6,~+d~, +=--

With these transfo~at~ons

&4&-t

d&z

the state I&,,,, aMZtcyR)becomes

(3.15)

EJ. W. Hahne, EG. Scholtz / Vibrations and rotations

30

+

C V#O

xexp

exp (-t(;

1)

(-f)“~~v~~-~

(-I).(b:bl.+etcl,))

>

~(5Mb~+5Rc~+aM2(b:+bt_2)+ c oRvcL)16) “f0

(3.16)

with I~>=~~P(-~(P~+P~))~~~(J~(PM~~O+PR~~O))~O),

b,~i5)=cpp>=o,

v/L,

(qo)

The completeness

relation

=

1.

(3.17)

still holds

(3.18) with

dT(&, aM2>= 3&'/3$

aM2[(l+fj’-$jf]

In order to treat the axial symmetry introduce rotations and to define

around

the intrinsic

of the system z-axis

d[,da,,

properly

it is convenient

to

into the state (aM2, &, &, (YRIL#;O)

where L, is the z-component of the total angular momentum operator. All the results of ref. “) still hold. Of particular importance are the completeness relations

(3.22)

F.J. W. Hahne, F.G. Scholtz / Vibrations and rotations

1’=2~_~dCM~o2=d0Jo~o,,da,,J~~da,lo,,,b,C,,A,Un,rd

31

(3.23) where

1’ denotes

the identity

on the boson

Fock space spanned

by the b,,, b2, b_,

and cfi bosons. Using the fact that (a M2, 4, &,, (a, a@#,,) is the vacuum of the b, and b-, bosons and that L, = LMp + LRlr (see eqs. (2.1) and (3.4)), one finds

LbM2, 4, tM,

‘tR, aR,ZO>

={2(b:bz-b~zb-z)+2(c:c2-c~2c-~)+(c:c,

-ckd1%12,d’,

&.I,

‘$R,

aRwf0)

(3.24) Following the analysis of ref. “) one can write in the case of large deformations in the c.m. coordinates (compared to the vibrational amplitudes in the c.m. coordinates) for an operator SL

(3.25) with

xf1’2&,

a*2,2)1%42,

6

CM,

‘iR,

aR~#O

>

(3.26) where f is introduced

in eq. (3.19) and

8~,=~(b:b~,+b,b_,+b:b,+b~,b_,+l).

(3.27)

The remainder of the analysis centres around the commutation of the operators T and 6’ K. A detailed discussion can be found in ref. ‘I). Here we only note that to lowest order it is sufficient to make the replacements TT = TT, where r is any of the single boson c,(/.L=-2,... 2). The replacement

(3.28)

creation of annihilation operators bo, b,, b-, or (3.28) is exact for the c’s. The treatment of the

F.J. W. Hahne,

32 b,,

bosons

is somewhat

different

EC. Scholtz / Vibrations and rotations

than in the application

finds to lowest order (see appendix

of ref. ‘I). In this case one

A)

(3.29)

Here we have used the well known LiR_‘(0)

relations

= CR_‘(R)

24) )

i = x, y, z ,

(3.30)

where Li are the boson total angular momentum operators and il are the intrinsic components of the angular momentum operators expressed as differential operators acting on the Euler angles 24). Eqs. (3.24), (3.28) and (3.30) give rise to a result which is of profound importance in the construction of wave functions, namely, J,TR-‘(0)

= T&R-‘(0)

= TL,R-‘(0)

= i; T-R-‘(O).

(3.31)

Eq. (3.31) states that the action of the boson operator J, and the differential operator L”: on the wave functions must be the same. This imposes strict conditions on the type of wave functions that may be constructed as is discussed in sect. 5. Using the approach outlined above, one can write for an operator 6;.

X[(61,),+(61,),,,+(61,),,,+(gl,),,,_,i,+.

. .]TR-‘(~).

(3.32)

Here (6:), is a constant. The vibrational part (6k),i, depends only on the b,, b&2 and c, bosons, while the rotational part (&),,, depends only on the angular momentum operators ii. The rotational-vibrational part depends on both the boson and angular momentum operators. Taking the matrix element between IBM-2 states ((Y,JM) which transform properly under SO(3), leads to the familiar expression for a matrix element when calculated within a geometrical framework as an integral of Wigner D-functions over the Euler angles. The corresponding intrinsic operator is identified as the expression occurring in the square brackets in (3.32) and the intrinsic wave functions as Tlcz, JM). The procedure for calculating the intrinsic operator should now be clear. One expands the operator 6: in terms of the b- and c-bosons and makes the replacements (3.28) and (3.29). Usually one truncates the expansion at the lowest order necessary for the calculation of just the vibrational and rotational parts. This is in any case the order to which we calculate the intrinsic operators here.

33

F.J. W. Hahne, F.G. Scholtz / Vibrations and rotations

The precise

method

the b- and c-bosons

one uses for the expansion is immaterial.

where one uses the Glauber outlined

of the operator

One can use the method

coherent

state. Alternatively

6:

outlined

in terms of in refs. l”*l’)

one can use the method

in ref. “) where one makes an operator expansion of the square root factors of the S, and s, di . & (p = r, v) arising from the elimination

JN, - I;, with &, =

bosons in the Holstein-Primakoff realization. PM and PR by Finally, as was done in refs. ‘O-“) one calculates the deformations demanding that no terms linear in the b, and c,, bosons occur in the intrinsic hamiltonian. As is shown in the next section these are the only terms occurring linearly in a quadrupole-quadrupole interaction, all other linear terms vanish automatically. 4. Intrinsic hamiltonians At first we do not consider the Majorana term and set & = 0, Vi in the hamiltonians H, and H2 of eqs. (2.5) and (2.8). Furthermore we note from eq. (3.30) that the rotational term K’L * L can immediately be replaced by the differential form ~‘2 . 2. We can therefore, without any loss of generality, put K’ = 0 for the present investigation. The easiest

way of calculating

the intrinsic

hamiltonian

corresponding

to the

hamiltonians H, and H,, is to expand the quadrupole operators themselves in terms of the b- and c-bosons up to the lowest order necessary to calculate the vibrational part of the hamiltonian to harmonic order as well as the rotational part. Before doing this we introduce the following simplifying notation: (4.1) which lead to a compact

form of eqs. (3.3) and (3.4). We further

JT(a,MP~+apnPn)=~~,, o,=JC& Inverting

introduce

P=T,v,

(4.2)

p=TT, v.

(4.3)

(4.2) one has

(4.4) Expanding the quadrupole operators to the required the structure of the hamiltonian operator) one finds c?,,*~ = .&[apM(bL+

&)

+ a,J&+

C&r = *ELI[apM(bfl

- b,,) + a,,(&

order

(which

cdl, -

41,

is dictated

by

(4Sa) (4Sb)

34

F.J. K Hahne, F.G. Scholtz / Vibrations and rotations

(4.6) Now we can calculate the intrinsic hamiltonian corresponding to the most general quadrupole-quadrupole interaction by inserting the expansion of the quadrupole operators in the hamiltonian, multiplying, rearranging and making the substitution (3.28) and (3.29). One then solves for the yb’s from the condition that terms linear in the bosons vanish. Inserting these solutions, one can calculate excitation energies and moments of inertia. Before doing this for H, and Hz we remark that from eq. (4.5) terms of the types c CL! and fit,, occur in the intrinsic hamiltonian. They introduce a rotationvibration coupling and therefore one neglects them in the present approximation. It can also be verified directly from (3.29) that they introduce higher order corrections in N to the vibrational part.

F.J. W. Hahne, F.G. Scholiz / Vibrations and rotations

35

First we consider the F-spin invariant hamiltonian H, with & = 0, Vi, K’ = 0 and xfl = xy =x. In this case the conditions that terms linear in the bosons vanish in the intrinsic

hamiltonian

are given by Y;o+Y:o=O

(4.7a)

ym1= -Yv1= 0.

(4.7b)

or

In order for stable solutions to exist for (4.7) one must take K < 0. The corresponding solutions are then found from eq. (4.7b) +,=&34=sgn(-x)z Note that as a consequence simplifications occur:

l [ l+ &J2.

of (4.8)

and

the

(4.8)

fact that

x,, =xy,

the following

PM=J=% PlX=O, &,i = &,i = E;, %i

One then finds for the intrinsic quadrupole part of HI

=

YYI

EE

Yi 9

hamiltonian

i=1,2, i=1,...5.

(4.9)

corresponding

to the quadrupole-

(b;b2+b~2b_2)+~3(~;~2+~~2~-2) + y4(c;c1+cI-1c_l)+

ys(b:bo+c;co)

II (4.10)

We immediately note that the coefficients occurring in (4.10) do not depend on N, and N, separately, but only on N = N,, + N,,. Furthermore, we note a complete decoupling between the centre-of-mass and relative motion. The vibrational part can now be diagonalized by means of a Bogoliubov transformation of the following form: 6: = b: cash & + b_, sinh 6,,, , E/j= CAcash a,+ co sinh So.

/_&=0,*2, (4.11)

36

FJ. W. lfahne, F.G. Scholtz / Vibrations and rotations

The annihilation operators are given by the hermitian conjugates of (4.11). Note that the b. and c, parts are diagonahzed by the same Bogoliubov transformation. Note also that (4.11) preserves the SO(Z) transformation properties. The transformation (4.11) is particuiariy simple for the b,- and c,-parts, one has cash 6, = -2-42 28 sinh 6

’

92

0=2$.

(4.12)

A pa~icuIar~y usefui test case for (4.10) is the SU(3) dynamical symmetry for which 7. In this case one has # = Jg and &=&=-$zr

((Qm+Qv> * (Qm+Qu))int =~[2N*-4N(b:,b~+b,b~+c;:c~~c~c~)-6N(b:b,+b~,b_,+c:c~+~~~~_~) -3N(c:c,+cL1c_,)-10r\r(b~b,+c~c,)]-~~(~:2+~~).

(4.13)

In this case a Bogoliubgv transformation in the b,, bosons is not required. The transformation for the b. and co parts is given by cash So= 245, sinh 6, = Ji . With this transfo~ation

(4.14)

(4.13) becomes

((Qr+ Qv) - (Q,+ Qvlhnt

-3N(c:c,+c~,c-,)]-~K(t:2+~~z).

(4.15)

Comparing with the exact solution found in eq. (2.14) one finds agreement to leading order in N. In sect. 5 we show that in the present approximation the degeneracy patterns generated by (2.14) and (4.15) are exactly the same, showing that there exists a one-to-one correspondence in the large-N limit. One can now follow the same procedure for the hamiltonian Hz. Again we take [i = 0, Vi and K’ = 0, but we allow in general for ,ylr + xy. Recall also that Hz is not an F-spin scalar. Here the conditions that the terms linear in the bosons vanish in the intrinsic hamiltonian are given by ‘ynoY”l=

0

(4.16a)

or (4.16b)

F.J. W. Hahne, F.G. Scholtz / Vibrations and roiations

As was the case for (4.7) stable corresponding

solutions

only

31

exist if ri ~0.

One finds for the

solutions (4.17)

Note that when x,, = xy, one has 41r = &, and that the same simplifications as in (4.9) occur. Recall, however, that with this choice H2 is still not an F-spin scalar. One finds for the intrinsic hamiltonian corresponding to the quadrupole-quadrupole part of H2 (0,

+ Qv)int = KN,NLJ%o?‘,o+ dHm+

Here the hamiltonian H E.nl.S-+

for the centre-of-mass

Kotl .

HA+ Hcoupf

motion

(4.18)

is given by

rz~y~(b:b12+b2b_2)+(~~o~vz+tYvo?l~2)(b~b~+bobo)

+((~*o~Y3+~Yo~?I3+2~,2~,2)(b:b2+bbl2b-b)+((~rro~vs+~uo~xs)b~bol.

(4.19) The hamiltonian

for the relative

motion

is given by

2N

N

(4.20) In this case one also finds a coupling

between

the centre-of-mass

and relative motion

given by m =N[Em2 H COUp + Z(Nvy,o~ry,z

E,~( NV - Nr)( b; c?, + b’!, c; + b-2 c, + b, c-2) - N,ry,,y,,)(b:

x (c~b,+b~c,)+(N,~,oy,,x

c:+

&co)

+ (Nu~uo~m5

Nm~~o~v3+(Nu

(b: c2 + b?, c_~ + c: b, + 2, b-z)] ,

- Mr~rno~v5)

- N71)~,2~v2)

(4.21)

F.J. W! Hahn+ F.G. Scholtz / Vi~ratjons and rotations

38

The rotational part is given by (4.22) We observe that in contrast to the F-spin conserving hamiltonian H1, the coefficients in (4.19)-(4.22) depend on N,, and N, separately. Consequently the excitation energies also depend on N,, and NV separately. In particular we note that the centre-of-mass excitation energies are proportional to N, NV/ N. Furthermore a coupling between the centre-of-mass and relative motion is present. Analogous results were found for the U(3) x U(3) model discussed in ref. I*). The analogy between the results for HZ found here and those of the U(3) x U(3) model of ref. ‘*) becomes even clearer if one chooses X~ =xy. In this case F-spin is still broken by HZ, but (4.19)-(4.22) simplifies somewhat. In particular one notes from eqs. (4.9) and (4.21) that the coupling between the relative and centre-of-mass motion is in this case propo~ional to (NV - N,) and hence it vanishes for N, = NV. We conclude therefore that as iong as ,&, = xy, which appears to be the case in many applications *I), and N,, = NY, the coupling is small. Finally we discuss the Majorana term. In order to determine the effect of the Majorana term on the energy spectrum we consider the F-spin invariant hamiltonian H, and take & # 0 and K’ = 0. One finds that the conditions that the terms linear in the b0 and c, bosons vanish are stili given by eq. (4.7) and the corresponding solutions are given by eq. (4.8). One then finds (M)i,t=

NY,&+

N[82~~+(~~1+~~3)~2](c:c,+c~,

+ N[~2~2+~3~2](~:c2+cf2~_2).

c_,) (4.23)

Here we have introduced the Bogoliubov transformation of eqs. (4.11) and (4.12) for the co part. This transformation diagonalizes the co parts of both the quadrupolequadrupole interaction of (4.10) and the Majorana term. We note that the Majorana term influences only the energies of the relative excitations. In particular they can be shifted up in energy with the correct choice of the 5’s. This was indeed the purpose for introducing the Majorana term. We also note that the coefficients occurring in (4.23) do not depend on N,, and N, separately, but only on N. The physical content of F-spin is illustrated very clearly when the choice A = [I = t2 = 5; is made in the Majorana term. One then has for the eigenvalues of M M(n,)=

NAn,,

(4.24)

where n, is the total number of relative excitations, i.e., the total number of E. and c,(p f 0) bosons. As was discussed in sect. 2 this choice enables one to express the Majorana term in terms of the second order Casimir operator, 6’, of the SU(2) algebra (2.4). Setting F = F,,,,, - m = i N - m one has from eq. (2.10) to leading order in N M(m) = Nhm.

(4.25)

F.J. W. Hahne,

Comparing relative

F.G. Scholtz / Vibrations

(4.25) and (4.24) one notes

bosons

and F-spin.

a simple

In particular

and rotations

relation

between

39

the number

we note that the states having

of

no relative

excitations correspond to maximum F-spin states. We also note that with this choice all the c-excitations are shifted by the same amount. However, in contrast to the U(3) x U(3)

model

discussed

in ref. ‘*), the Majorana

term can also shift the c-

excitations relative to each other. In particular we note that by setting e2 = 0 the E0 excitations are not influenced by the Majorana term. This choice gives rise to the presence of a K = 0 band (see sect. 5 for more detail on the wave functions) with mixed F-spin in the vicinity of the p- and y-bands. As a matter of fact they are degenerate in the SU(3) limit (see eq. (4.15)). This observation has already attracted some attention in the literature ‘“). Finally we remark that when the Majorana term is added to an F-spin breaking hamiltonian such as H2, the conditions that the terms linear in the co and b, bosons vanish are no longer given by (4.16). Furthermore one has to diagonalize since the normal modes appearing in M are no longer normal modes of H2. 5. Construction of wavefunctions Now that we have calculated the intrinsic hamiltonians we are in a position to construct eigenfunctions with the correct SO(3) transformation properties. To simplify the discussion we consider the intrinsic hamiltonian (4.10) corresponding to the F-spin invariant case with a possible Majorana term, i.e., (Hi)i,t=((Q,+QV)

(5.1)

* (Qw+Qu))int+(M)int-

In this case the normal modes can easily be identified from a Bogoliubov transformation as was discussed in eq. (4.11) and accordingly the eigenfunctions are particularly simple. For a more general (4.18), a diagonalization is outlined here to construct a mation properties and then From

the discussion

which transform

properly

hamiltonian, such as the F-spin breaking case of eq. in general required. In this case one uses the method set of basis functions having the correct SO(3) transforthe hamiltonian is diagonalized in this basis.

in sect. 2 we can write down under

the eigenfunctions

of (5.1)

SO(3). They are

11, M, K, n)= #&In~Z,

no,

~LY~,, Q, n,,, n,,

n,_,, nc-J

“D&(n).

(5.2)

Here the bar denotes the Bogoliubov transformed bosons of eq. (4.11). Recall that this transformation preserved the SO(2) transformation properties. The relations (3.24) and (3.31) relate the quantum numbers K and ni in the following way: K =2(n~~--~-_,)+2(n,-n,,)+(n,,-n,,). Note that because of the relation (5.3) and the fact that (5.1) commutes the quantum number K is conserved.

(5.3) with J,,

40

F.J. W. Hahne, F.C. S&oh

/ Vibrations and rotarims

As usual, one still has to symmetrize the wave functions (5.2) under interchanges of the intrinsic axes ‘1,24).In particular one requires invariance under a rotation of $7~ around the intrinsic z-axis and a rotation of +r around the intrinsic x-axis 24). We denote them by T, and T2, respectively. In this case where one expands around an equilibrium in the potential energy function invariance under a permutation of the three intrinsic axes is not required. A detailed discussion of this point can be found in ref. 24). Making use of the SO(2) transformation properties of the bosons occurring in (5.2), one finds that invariance under T, requires that K - (nC,-n,_,) is even. This condition is already satisfied if the relation (5.3) is taken into account. To construct wave functions which are invariant under T,, one uses Ti = 1 and constructs Ir,M,K,n)=Z)~KIn)+T,D~,Jn).

(5.4)

Then one uses

and

T26; T;' = i?,,

f_L=O,*2,

Tz CL7-T’= cl,,

_SX

T,E;T;‘=?

=*1,12,

(5.5b)

0’

Eq. (5Sa) is standard24) and eq. (5Sb) can easily be verified from (4.11) as well as the SO(3) transformation properties of the d-bosons in terms of which the band c-bosons are defined (see (3.15)). Using (5.5) in (5.4) one finds for the symmetrized states 11,M, K, n) = D& In>+(-l)‘D$_.K

1-n).

(5.6)

Here I-n) denotes the ket obtained from In) by making the interchanges ng2*n6_,, and nc, ++ nc__,, and it carries angular momentum projection -K. n, ++ k, Note that except for a possible global phase factor the states obtained by replacing K + -I( coincide with the states (5.6). For this reason we must impose the restriction K 3 0. Note also that because of (5.3) this restricts the allowed combinations of the ni’s. The case K = 0 needs some further consideration. From eq. (5.3) we note that K = 0 does not necessarily imply In) = I-n} as was the case in the treatment of IBM-l in ref. ‘I). For instance, one has K = 0 for (nt;z= 1, n,, = 2), but 1-n) = Ini;_,= 1, nC,= 2) f )n). From (5.6) we can write 11,M, K =O, n) = D&(ln)+(-l)‘I-n))

.

(5.7)

The following possibilities can now occur in (5.7). When In)= I--n> the wave functions are non-vanishing only when I is even. However, when In) f 1-n) the wave functions are non-vanishing for all values of f. In the latter case, however,

F.J. W. Hahne,

F.G. Scholtz / Vibrations and rotations

the even and odd values of I belong independent

intrinsic

II, M, K =O, n) = and for odd values

to two different

wave functions.

rotational

For even values

D;,,(ln)+J-n)) ,

41

bands built on linear

of I one has from (5.7). I = 0,2,4,

...

I = 1,3,5,.

..

of I

II, M, K =O, n)=D~&z)+n)),

(5.8b)

Note that (5.8a) always occurs, i.e., when In) = j-n) and In) f 1-n). On the other hand, the wave functions of (5.8b) only occur when In) # ( -n). In IBM-2 rotational bands with K = 0 and odd values of angular momentum can therefore occur. This is in contrast to IBM-1 where K = 0 implies In) = ( -n) and hence (5.8b) cannot occur. In conclusion, we have for the completely symmetrized and normalized wave functions II, M, K>O,

I=K,K+l,...

n)=

(5.9a) I = 0,2,4,

I = 1,3,5,. where S,_,

has the value of one if I-n) = In) and zero otherwise

...

(5.9b)

..

(5.9c)

(see the discussion

below eq. (5.6)). To illustrate the correspondence between the wave functions of (5.9) and the IBM-2 wave functions, we compare the degeneracy patterns of the vibrational parts generated by the IBM-2 hamiltonian on the one hand and the corresponding intrinsic hamiltonian of (5.1) on the other. For this purpose we consider the SU(3) dynamical symmetry

where an analytic

expression

is given by eq. (2.14). The eigenvalues are obtained from eq. (4.15) E(rz)-2N’~

= -6N~(n~+rq_~

for the eigenvalues

of the IBM-2 hamiltonian

of the corresponding

intrinsic

+n~~+nn,+n,_,+n,,)-3N~(n,,+n,,).

hamiltonian

(5.10)

Our modus operandi for comparing the degeneracy patterns is as follows. We choose an excitation energy as given by (5.10), e.g., 0, -INK etc. Then we determine from (5.9) and (5.10) the different bands together with their angular momentum content having this particular excitation energy. Accordingly we determine the IBM-2 bands together with their angular momentum content which have this particular excitation energy by calculating (2.14) to leading order in N and using eqs. (2.15) and (2.16). The result is summarized up to an excitation energy of -12N~ in table 1. We note a one-to-one correspondence in the large-N limit.

42

EJ. W. Hahne, F.G. Scholtz / Vi&rat~ons and rotations TABLE

Comparison

of degeneracy

1

patterns

in the SU(3) limit

IBM-2

Geometrical Ground

n,=O,Vi K=O,f=0,2,4

(2N,+2N,,O) Comes from the reduction of: (2N,, 0) x (2Nw 0) Contains the bands: K=O,I=O,2,4 ,.I. 2(N,+N,,) E = -3(N,+ (2N,,+2N,,-2,l) Comes from the reduction

state

,_..

N,)K n,+n,,=l K = 1, I = 1,2,3,.

of:

.

W’L, 0) x UN,, 0) Contains the bands: K=1,1=1,2,3 ,... Z(N,+N,,-1 E = -6(N,,+ (2N,+2Nv-4,2) (3 times) Comes from the reduction of: (2Nw, 0) x (2N,, 0) (2N,-4,2)x(iN,,O) (2N,, 0) x (2N, -4,2)

N,)rc n,, + n,, = 2 K=0,1=0,2,4 ,... K = 2, I = 2,3,4,. ..

nh2+nb-,

Contains the bands: K=O,f=0,2,4 ,... 2(N,+N,)-4 K=2,1=2,3,4 ,... 2(N,+N,,-2

=

1

K=2,1=2,3,4

,...

nC2-6 nC_ = 1 X=2,1=2,3,4

,...

ncQ+ nFO= 1 K=O,I=0,2,4 ,... (2times) Total number of bands: K = 0, I = 0,2,4,. (3 times) K = 2, I = 2,3,4, . . (3 times) E = -9(N,,+ (2N,+2Nv-6,3) (3 times) Comes from the reduction of: (2N,, 0) x (2Nw 0) (2N, -432) x (2N,, 0) (2N,,O)x(2N,--4,2) Contains the bands: K=l,I=l,2,3 ,I.. 2(N,+N,)-5 K=3,1=3,4,5 ,.,. 2(N,+N,)-3 ON_+2N..-5.1) (2 times) \-,. ”

1..

Comes from the reduction of: (2N, -432) x (ZN,, 0) (2N=,O)x(2~~-4,2) Contains the bands: K=l,i=l,2,3 ,... Z(N,+N,)-4 Total number of bands: K = 1, I = 1,2,3,. . . (5 times) K=3,1=3,4,5 ,... (3times)

N,)K nc, + n,, = 3 K=l,i=f,2,3 K=3,1=3,4,5 nhz+

nb_,=

,... ,...

1, %,+

K=l,I=t,2,3 K=3,1=3,4,5

%r = 1

,... ,..I

nh + nr_>= 1, q,+ n,, = 1 K=l,I=1.2,3 ,... K=3,1=3,4,5 ,... ncr,+n,~=l,n,+n K = 1, I = 1,2,3,.

=l

fY’(2times)

Total number of bands: K = 1, I = 1,2,3,. . . (5 times) K = 3, I = 3,4,5,. . (3 times)

F.3. W. Hahne,

F.G. Schoifz / Vibrations and rotations TABLE

l-continued

E=-12(N,+N,)K (2N,,+2Nv-8,4)(6times) Comes the reduction of: (2% 0) x (2% 0) (2N, -4,2) x UN,, 0) (2N,, 0) x (2Nv -492) (2N,,-4,2)x(2N,-4,2) (2N, - 8,4) x (2N,, 0) (2N,, 0) x PN, - 8,4) Contains the bands: K=O,I=0,2,4 ,._. 2(N,+N,)-8 K=2,1=2,3,4 ,... 2(N,+N,)-6 K=4,I=4,5,6 ,... 2(N,+N,)-4 (ZN,, +2N, -6,0) (5 times) Comes from the reduction of: (2% -4,2) x (2N,, 0) (2N,, 0)x (2N,-472) (2N,-4,2)x(2N,-4,2) (2N, -6,0) x (2Nw 0) (2N,, 0) x (2Nv -690) Contains the bands: K=O,I=0,2,4 ,... 2(N,+N,)-6 (2N,+2N, -?,2) (3 times)

Comes from the reduction of: (2N, -492) x (2N,, 0) (2N,vO)x(2N,-4,2) (ZN,-4,2)~(2N,-4,2) Contains the bands: K=O,I=1,3,5 K=2,1=2,3,4

,... ,..,

2(N,+N,)-7 2(N,,+N,)-5

nq + %_, = 4 K=O,I=0,2,4 ,._. K = 2, I = 2,3,4, . . , K = 4, I = 4,5,6, , . . nh,fnb_,=i,n,+n,,=2

K=O,I=0,2,4 ,... K=O,i=1,3,5 )... K = 2, I = 2,3,4, . . . K=4,1=4,5,6 ,,.. n,,+ n,_, =1,n,,+n,_,=2 K=O,I=0,2,4 ,... K=O,I=l,3, 5 ,... K = 2, I = 2,3,4, . . K = 4, I = 4,5,6, . . . ngOi-nT,=l,n,,+n,_,=2 K = 0, I = 0,2,4, . . , (2 times) K=2,1=2,3,4 ,... (2times) K = 0, I= 0,2,4, . . . K=4,1=4,5,6 ,...

K = 0, i = 0,2,4, K=4,1=4,5,6,...

..a

nh2fnh_2=l,nC2+n,_,=1 K = 0, I = 0,2,4, . K = 0, I = 1,3,5, . . . K = 4, I = 4,5,6, . . .

n,,+n,_,=l,nh-,+n,=l K=2,1=2,3,4

,...

(2times)

nrz+ nr_*=l,ngOi-n,,=I

Total number of bands: K=O,I=0,2,4 ,... (Iltimes) K = 0, I = 1, 3, 5, , . . (3 times) K = 2, I = 2,3,4, . . . (9 times) K=4,1=4,5,6 ,,.. (6times)

K=2,1=2,3,4

,...

(2times)

nC0+n,,=2 K=O,f=O,2,4

,...

(3timesf

Total number of bands: K=O,I=O,2,4 ,... (lltimes) K = 0, I = 1,3,5, , . (3 times) K=2,1=2,3,4 ,... (9times) K=4,1=4,5,6 ,... (6times)

43

44

EJ. W. Hahne, F.G. Scholtz / ~b~atjons 6.

and rotations

Ml strengths

Finally, we illustrate how transition rates can be calculated in the present framework. For this purpose we consider the most important transitions in IBM-2, namely, the Ml transitions. The magnetic dipole operator is taken as 14*‘8) T:=

(6.1)

where g, and g, are the g-factors for the proton and neutron bosons, respectively. The operators L, and L, are defined in eq. (2.1). In the formalism outlined in sect. 3, the operator (6.1) is replaced by (see eq. (3.32)) T:+C

DL*K(TL)int

(6.2)

K

and matrix elements are calculated as in the usual geometrical framework using geometrical wave functions such as (5.9). The main task is to calculate the intrinsic components of T: . This is done in appendix B and we just list the result here. (T:)~“,=A,ZI+B~(~~+C-~L)~C~L~,~++~(L~,~,~+LL~,~,~).

(6.3)

Here sI are the spherical components of the angular momentum operators (in differential operator form) which are related to the Cartan-Weyl basis as in (2.2). The coefficients occurring in (4.3) are given in eq. (B.8) and the operators L+, Lb,c,rr and Lc,b,p in eq. (B.9). We can now calculate the Ml strengths. We consider the F-spin symmetric case for which the wave functions are known from eq. (5.9). The ground state is given by

From eqs. (6.3) and (B.8) we note that the only state which obtains Ml strength is the state with ncl+ n,_, = 1.)i.e. If=l,M,I(=l,n,+n,,=l)=

& J---

I~Ll~,,

= I>- NLl~c,

= 01 *

(6.5)

Calculating the Ml strength one has from eqs. (6.3), (4.8), (4.9) and (B.8) /(f=l,Ic;:=l,n,,+n,,=l/lT1(lG)l’

For the SU(3) case with x = -ifi

=~i,,_,.,2~[1+~]. this gives

19(Ml, 0+ + l+) = ~(gn-gJ2 which to leading order in l/N

(6.6)

coincides with the exact result 14).

(6.8)

F..i. W. Hahne, F.G. Scholtz / vibrations

45

and rotations

7. Conclusion When different

groups use different

formalisms

to analyze

the same experimental

results, one is interested in the underlying connection between the models. Success which is achieved in the one model may then indicate extensions and refinements which

are necessary

here should

in the other

be extremely

useful

model. to obtain

The methods parameters

which

we have developed

for the geometrical

model

when one has a successful IBM-2 fit. Furthermore it has been noted “) that when employing IBM-2 one loses much of the simplicity which contributed to the popularity of IBM-l. Although IBM-2 appears much clearer and more acceptable when viewed from the standpoint of a microscopic foundation, its actual application to spectra is less transparent as it often requires a large computational effort. The analysis presented in this paper should serve to understand and predict the results of such computer calculations. It is important also to have these guidelines extended to matrix elements. We have indicated with the calculation of the Ml strength how one can do this. We have treated the deformed system, and in particular, we have tacitly assumed that both the proton part and the neutron part are deformed in much the same way. This made our basis of centre-of-mass and relative bosons a convenient choice, as is clearly seen by observing that the different vibrational modes essentially decouple in the models we investigated. In other models, for instance when xv and xv have opposite sign, the coupling is expected to be very strong and one would then fare better with a reformulated geometrical picture. At this stage, where experimental evidence is still scarce, it does not seem justified to develop detailed formulations of other models based on various geometricai ~onst~~tions. However, when the need should arise one could do this by following the general procedures outlined in this paper. F.J.W. Hahne would like to thank Walter Greiner for the kind hospitality extended to him in Frankfurt and acknowledges support by the Deutsche Forschungsgemeins&aft. Support by the Foundation for Research Development of the South African Council for Scientific and Indust~al Research is also acknowledged. Appendix A In order to determine

the action

of the b,$ boson

operators

on the operator

T of

(3.26), one calculates the action of the total angular momentum operators of (2.1) on T. To do this it is necessary,to calculate the action of the totai angular momentum operators on the bra state (uM2, #, &, , .&, aRp~o~exp (b, b_,). For this purpose the following operator identity turns out useful. Suppose f( dP,+) is an analytic function of the d-boson annihilation operator only, then

(A.11

46

FX W. Hahne, EC.

S&&z

/ Vi~rafio~s and rotations

with p =7r,vandp=-2,..., 2. The boson operators are treated as ordinary complex variables in the derivative. Next one realizes that the creation operators db,, form the components of a I = 2 tensor operator under SO(3) for each value of p. Using this one finds for the commutators in (A.I):

,

CcIp,fi,L+j=~~~6-~(~-I)d,,,-,

[dp,,.s, L-1=dJti-cLb

f lMp,,+,, (A.21

[4,,mLl= &,p 1

Eq. (A.2) was obtained by taking the hermitian conjugate of the commutation relations of the angular momentum operators with the boson creation operators. These commutation relations are known from the tensor character of the creation operators. The uncommon factor J$ in (A.2) is due to the definition of the CartanWeyl basis in (2.2). Using (A.l), (A.2), Li 10)= 0; Vi (where 10) is the d-boson vacuum) as well as the fact that (3.16) is an eigenstate of the position operators (b: + b-,)/a (,u = 0, *2) and (c’, + (-l)“c_,)/fi (E.L= *2, il, 0), one finds (aM2, 4b,lM9 &+t%+fOI exp (bt b-1)L*=(%2, =+fi

6M,

tR,

aRp&fie*ichLlt

e*“(%%

CM,

&Rt

e”L”

%&Ol

x[~(~~~~~(~~+~~))~~,

-hm:,+

Wb*,

+&p,+J~(c~cc*))c,, t-42(&, -c*,)cTZ]eirbLz.

Using the SO(2) transformation

rearrangement

properties

(A3)

of the bosons and after some simple

one finds

@if

(b~2ib**)bT,_(C:2+C*,)C71

hdb,,

fiPM

-

ah.4

&PM (A.4)

Eq. (A.4) lends itself to an iterative procedure since the b,,-bosom on the right of (A.4) can be replaced repeatedly by the expression given in (A.4). It is easily

F.J. W. Hahne,

verified

that the leading

F.G. Scholtz / Vibrations

order term to the rotational

47

and rotations

part comes from the first term

on the right of eq. (A.4). Similarly one finds that except for the term PR/2&(cz, + c&r) all other terms introduce corrections of the order l/N and higher to the vibrational PR - pM -m terms

part.

The

above

in general

mentioned

term

must

be treated

diifferently

since

and hence it is of a lower order than the other vibrational

in

(A.4). In the present approximation only the terms L,/&& and + c,~) are therefore kept in (A.4). Making the replacement (3.28) and PFJ2P&L (3.30) leads to the second equation given in (3.29). The first expression in (3.29) for the creation operators is derived from the observation that

Appendix B The Ml operator

is given by

(B-1)

T:= To calculate and relative

its intrinsic components bosons dL,, and dk+

we first express it in terms of the centre-of-mass of (3.4), finding

IN,gv - - 1 +(Ng, Kg,+ N

T; = +a% 7

N,g, N

L

M’CL

(%r -&)[Lhl,R,,

+

N

-L N

LR,M,fi

)

R.w

(B.2)

with L M+ =~(d?,d,):, L M,R,,i

Lr+ =fi(d:,&);,

=d=%d;l,&):,

&,M,,,

=m(d:,&,):.

03.3)

We recall from eq. (2.1) that L, = L,, Using

+ L,,

= Lt+

+ LrQ .

(B-4)

(B.4) yields T:, =

3 J-K47T m-37 + --jy

(g7r -&)(LM,R,p

+ LFwp

03.5)

F.J. W. Hahne, F.G. Scholtz / Vibrations and rotations

48

To obtain the final form for the intrinsic components one expresses (BS) in terms of the b- and c-bosons of eq. (3.15) and makes the replacements L, + fi: )

/A-.0,*1

Here fi:, are the spherical components of the angular momentum operators in differential operator form. They are related to the Cartan-Weyl basis as in (2.2). Substitnti~g the necessary Clebsch-Gordan coefficients one finds for the final expression 03.7) with

Furthermore L,,, =dia

z; (2Y,2Y*1lp)ct,(-l)“c_, VI v2

References 1) 2) 3) 4) 5)

W. Greiner, Phys. Rev. Lett. 14 (1965) 599 W. Greiner, Nucl. Phys. 80 (1966) 417 A. Faessier, Nucl. Phys. 85 (1966) 653 S.G. Rohozinski and W. Greiner, 2. Phys. 322 (1985) 271 A. Arima and F. Iachello, Ann. Rev. Nucl. Sci. 31 (1981) 75

,

F.J. W. Hahne, F.G. Scholtz / Vibrations and rotations

49

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