1N expansion in the interacting boson model II. The neutron-proton degree of freedom

ANNALS

OF PHYSICS

195, 126-166 (1989)

l//V Expansion in the Interacting Boson Model The Neutron-Proton Degree of Freedom S. School

KUYUCAK

AND

I.

of Physics, University Parkville, Victoria 3052,

II.

MORRWN of Melbourne, Australia

Received March 8, 1989

The l/N expansion method is used to study the neutron-proton degree of freedom in a general boson model. Employing a standard IBM-2 hamiltonian, analytic expressions for energies and electromagnetic transition rates are derived both for the symmetric and mixedsymmetry states. A formalism for F-spin analysis is developed. Effects of the g-boson and F-spin breaking in various quantities are discussed. The method is not restricted to dynamic symmetries and allows for explicit breaking of the F-spin symmetry. Thus, the formulae derived here should be useful for systematic analysis of deformed nuclei in realistic situations. 0

1989 Academic Press, Inc.

1.

INTRODUCTION

The neutron-proton interacting boson model (IBM-2) is an extension of th interacting boson model (IBM-l) in which a distinction is made between neutro and proton bosons. In addition to the totally symmetric (under interchange ( neutron and proton bosons) states which are in one-to-one correspondence wit those in the IBM-l, the IBM-2 also contains states of mixed symmetry characte Originally, there was no knowledge of the mixed symmetry states, and the IBMhad been proposed as a link to the underlying shell model [ 11. However, wit the recent discovery of the mixed symmetry states [2], it has gained tirr phenomenological roots of its own and there has been a surge of interest in th IBM-2 calculations. As in the IBM-l, there are two avenues for solving IBM-2 problems. In genera a hamiltonian must be diagonalized numerically using the available computer code [3], or, in special cases corresponding to dynamic symmetries of the hamiltbniar analytical solutions can be obtained using group theoretical techniques [471 Although the IBM-2 is richer in dynamic symmetries, allowing two different classes one F-spin conserving (i.e., symmetric under interchange of neutron and proto, bosons) and another F-spin broken, in practise their application is limited becaus of physical considerations. The first class of dynamic symmetries requires F-spi symmetry, in contrast with the microscopic basis of the model which suggests strong neutron-proton interaction. The breaking of the F-spin symmetry is als 126 OOO3-4916/89

$7.50

Copyright 0 1989 by Academic Press, Inc. All rights ol reproduction m any form reserved

1 /N EXPANSION

IN

IBM-II

needed to explain Ml transitions among low-lying bands. The second class, though it allows for F-spin breaking, does not admit a Majorana force, resulting in a very low-lying 1 + state in conflict with the experimental data. Thus dynamic symmetries are less likely to be useful in the IBM-2 and are not much emphasized, instead, numerical calculations have been the main thrust. On the other hand, there are both microscopic [8] and phenomenological [S-15] indications that the sd-boson basis is not adequate to describe deformed nuclei and should be extended to include g-bosons. The sdg-IBM-l hamiltonian contains many more parameters (32 compared to 9 of the &-model), and selection and determination of a simple set of parameters through numerical analysis are difficult. Another concern with the numerical calculations in the sdg-basis is that the basis dimensions are too large to allow systematic analysis of deformed nuclei which typically have N = 12-16 bosons. Due to above difficulties, numerical calculations in a full sdg-basis are rarely performed, and one rather resorted to the perturbation theory [9] or the SU(3) limit [l&12] which can only provide a qualitative picture. Clearly, such calculations would be prohibitively difficult in the case of the IBM-2 and are unlikely to be attempted. It appears that there is a genuine need for approximate analytic methods in the IBM which are capable of handling the above extensions without any restrictions and yet are reliable enough for systematic analysis of nuclear data. Towards this end, we have recently proposed a l/N expansion technique which is based on an angular momentum projected mean field theory and leads to (non-perturbative) analytic expressions for energies and transition rates away from the limiting symmetries [14153. In a previous publication [ 151, hereafter called I, we have demonstrated the technique in detail by solving an IBM-l quadrupole hamiltonian in a generalized boson basis. Mean field techniques for boson systems were initially introduced to study the classical limit of the IBM and its relationship to the Bohr-Mottelson model [ 161. Extensions of the intrinsic state formalism to the neutronproton and g-boson degrees of freedom were carried out in the SU(3) limit [17], or using the Tamm-Dancoff and random phase approximations [18]. Exact angular momentum projection in mean field theory is quite involved, and the first attempts used the self-consistent cranking model [19-201 and the generator coordinate method [21]. More recently, an exact albeit numerical treatment was given in connection with the fermion-boson mapping for deformed nuclei [22]. The purpose of this paper is to extend the l/N expansion technique to neutron proton boson systems governed by more general hamiltonians. In a preliminary version of this work [23], F-spin symmetry was assumed, and its breaking was treated in perturbation theory. Here, we treat the neutron-proton degree of freedom on an equal footing with the boson ones and determine all states from variational equations in a self-consistent manner. Since the goodness of the F-spin symmetry has become an important issue in study of the Ml transitions [24-271 and F-spin multiplets [28], we also develop a projection technique which enables rapid F-spin analysis of various states. In Section 2, we introduce the general for595,195/l-9

128

KUYUCAKANDMORRISON

malism. Algebraic expressions for energies of various bands and electromagnet transitions among these bands are derived in Sections 3 and 4, respectivek Section 5 gives a summary and concluding remarks. Throughout the text, the following shorthand notation is used for angule momentum eigenvalues: t = [2L + l] 1’2, E=L(L+l),andL-=L(L-l).Thek (or K-) quantum number is suppressed as an argument or index when M = 0, unlez it is needed to distinguish the P-band from the ground-band. Since our experiment; knowledge of mixed symmetry states is limited to those with F= F,,, - 1, the ten “mixed symmetry” specifically refers to such states. 2. FORMALISM

We introduce the spherical boson creation (b:,) and annihilation (b,,, operators, where p = (rc, v), I = 0,2,4, .... p, correspond to s, d, g, ...-bosons and m : the projection quantum number. The standard hamiltonian used in the IBMcalculations can be written as H=CE~,~~[+ICQ,.Q,+K~Q~.Q~+~,Q,.Q,+K’L.L+~M.

(2.1

PI Here, npl, Qp, L and M are the number, quadrupole, angular momentum, an Majorana operators, respectively, generalized to arbitrary kinds of bosons. In tern of the boson operators, they are given by

(2.2

*PI= c b;,,nbp,, m

Qp,,= c qpjJ3;j~prlj?,

(2.:

ii L,,=c

[l(l+

1)(21+ 1)/3]“2

[b;,6,,];‘,

(2.L

PI

M= N,N,-

c b~,~b~,.,~b,,,,,b,,~. Iml’m’

(2.:

The quadrupole operator contains the further parameters qpj,; that is, qpz2 = xp i the sd-model and qp22, qp24, and qp+, in the sdg-model. The L . L term has n dynamical content other than fixing the moment of inertia and therefore will not t discussed further. The Majorana operator, M, is chosen so that the mixed syn metry states are pushed up in energy to > 3 MeV and almost decouple, leaving tl low-energy bands as predominantly symmetric IBM-l states. For simplicity and t avoid introducing further parameters, we follow the convention and choose th Majorana operator proportional to the F-spin Casimir operator (see Appendix , for a derivation of Eq. (2.5)). The effect of M is then to split the symmetric an mixed symmetry states approximately by an amount aN, which becomes exact i the limit of good F-spin symmetry.

l/N

For deformed boson operators

EXPANSION

IN

systems, a more appropriate

b;,,, = c x,,b;,,,,, I

IBM-II basis is provided

129 by the intrinsic

x0,,,. xpm = 1.

In this basis, intrinsic states I$K) for low-lying collective bands correspond to a condensate of N bosons and its simple excitations. The structure coefficients xplm are dynamically determined from (H) (2.7)

by variation after projection (VAP) as discussed in I, Section 3.2. Here Pi, is the angular momentum projection operator. When the quadrupole interaction is dominant, approximate solutions to x0,,,, can be obtained using an eigenmode condition (I, 2.11). In general, however, variation leads to a set of coupled non-linear equations which have to be solved using a Hartree-Bose condition as is shown in the next section. Once the intrinsic states are determined, matrix elements of other operators can be calculated using Eq. (I, 2.12). For an axially symmetric system, intrinsic states for the ground- and single phonon excited-bands, to leading order, are given by

IdK) = ~0sybi, IN, - 1,N,) + sin ybt, IN,, N, - 1 ), I~Kw)=si~~b~,IN,-l,N,)-~~~yb~,~N,,~,-l),

(2.9)

(2.10)

where the subscript M denotes mixed symmetry states with F= F,,,,, - 1 and “7” is the F-spin mixing angle. As shown in I, the leading order expressions (2.8)-(2.10) are sufficient to calculate energies and intrabrand transitions to order 0( l/N*). Higher order terms represent orthogonality and mixing effects from other K-bands and are only needed for interband transitions. The above choice of trial states allows an independent variation of the F-spin configuration mixing and the mean field coefficients. In the limit of good F-spin, cos y = J-N; sin y = aN, and the excited bands correspond to simple symmetrised (antisymmetrised) single phonon excitation of the ground-state configuration for the symmetric (mixed symmetry) states. The variation of the phase angle “y” from the limit value is related to the degree of F-spin breaking and is known to be small from analysis of Ml transitions between low-lying states [2627], rendering F-spin as a good quantum number. Further, this variation effects mostly Ml properties, and only slightly energies and E2 transitions. We will therefore use the limit value in the discussion of excitation energies and B(E2) rates with the advantage that simple algebraic results with N,, N, scaling can be obtained, permitting a direct extraction of equivalent IBM-l parameters.

130

KUYUCAK

AND

3.

MORRISON

ENERGIES

3.1. Ground-Band

Since the angular momentum projection techniques are discussed in detail in here we give only a short derivation of the energy formula, emphasizing th differences. Substituting Eq. (2.8) in (2.7), gives for (H),,,

x (-I (by.

(b,p

He-‘pLJ (by

(by

I-),

(3.1

where JV($~, L) is the normalization obtained from (3.1) by putting H= 1 an ,V = 1. Matrix elements between intrinsic states are evaluated by defining rotate intrinsic operators as

and by using differential techniques for boson commutators. one obtains

where Z,(B) = C, xz, d&(b). Introducing [Z,(/?)-p

r.z(x,, . xp

e-p”Fp,

For the normalizatio

a gaussian approximation

r, = 21~p N, 2

Y,

for [Z,]“p,

= f 1 ix;,hp . x,, I

(3.4

where i= f(1+ l), and defining the average quantities y = C, N, y,/N, r= 2/yN, th b-integral reduces to a standard form (since the exponential is negligible for /? B n/z the integral limit can be extended to co) F(I’, L) = loK db sin /? dho(/l) eeBzir,

(3.5

which is evaluated in I, Appendix A, as k(T,L)=;{l-$(L+;)+~(P+2L+$-...}. Thus the normalization

(3.f

becomes

“/lr($dR’L) = (2L + 1)(x,. X,)N. (x, . X,)NUF(r, L).

(3.;

l/N

EXPANSION

IN

IBM-II

131

Substituting Eq. (2.1) in (3.1) for H and calculating the matrix elements along the lines of (3.2)-(3.7) yields (from now on xP . xP is suppressed for convenience)

+ C upNp(Np- 1) qpj~gpj’l’xpjxplxpj’xpl’ 1

P

x (jOj’O~JO)(lOl’O~JO)~

(JOLO~10)2F(T2,Z) I

(3.8) where r,, = 2/Y(N - n). In applying the gaussian approximation (3.4)-(3.6), we have used l/r,,,, + l/r,, x l/r,,, + ,,. In principle, this should introduce errors of the order 0(1/N). However, because F-spin breaking in the mean fields is small, one has roughly r, x r,, and the error involved is much smaller. The angular momentum sums over Z and J in Eq. (38) can be evaluated using the formulas given in I, Appendix B. Dividing by the normalization (3.7) and rearranging terms, we obtain to the first layer (i.e., to order l/N and t/N’)

x(,..x,-y).....},

(3.9)

132

KUYUCAK

where the quadratic suppressed if zero)

AND

MORRISON

forms Apmn, C,, E,,,, and y,,

A pmn= Cjmp

CiO 10 120)

9pjixpjxpl,

C,

=

9

E,, = 1 fn&I, I

are defined as (m, n arc

C

5

j, 21+ 1

(qpj/Xp,)',

(3.10

Y,” = ; c kdx,,. I

As shown in I, the gaussian approximation affects only the terms that are of highe order than those considered in Eq. (3.9), and hence it is free from such errors. The hamiltonian (2.1) is F-spin symmetric when E,/ = E,, = E[, rc, = rc, = 42, am qnj, = qvjr. Then the mean fields carry the symmetry with x, = x, = x and y,, = J Introducing, further, A, = A, = A, C, = C, = C, and E, = E, = E, Eq. (3.9) reduce, to

which is the IBM-l expression. (In I, only a quadrupole force was considered, sc the one-body term in (3.11) was not given before.) Notice that the Majorana energ! identically vanishes consistent with the Casimir operator evaluation from Eq. (A.3) 3.2. K-Bands Next, we consider the energy formulae for the single-phonon excited K-band: (including K=O, the /I-band). Due to the presence of the boson operator b,, calculations in this case are considerably more complicated. Again, we give here only the main steps necessary to point out the differences and refer to I for technica details. Substituting Eq. (2.9) in (2.7),*gives

x <-I C,/% ~0sYb,,(bzJNn-’ (b,lNv + fi x [A

sin y(b,)Nn b,,(b,)Nvp ‘1 He-‘@b cos y b~,(b~)Nv + &

sin y(bL)Nx b~,(b~)Nvp’]

I-).

(3.12

Evaluating the matrix elements in (3.12) with H= 1, we obtain for the normaliza. tion JIT(ffSK,L)

l/N

Jlr(d,,L)=(2L+l)

EXPANSION

IN

IBM-II

C(cos2y~~,K+sin2yxt,,)C i /

133 (ZKL-KJZO)2F(f,,Z)

I

+ c C(N, - 1) cos2 Y x,P,~~x,~~x,~~~ + (NV - 1) sin2 Y ~~~~~~~~~~~~~~~~ II’

xc

(10i’K~JK)(1KI’OJJK)(JKL-K(10)2

F(T,,Z)

(3.13)

JI

Carrying out the angular momentum sums and terminating first layer, the normalization becomes

the expansion at the

(3.14)

where we define a,,=1

ix:lK,

/

b,,=x

fi I

[i-fi-]1’2Xp/Xp/K,

n=l

bpo = C &,xp,m

(3.15)

/

and the average quantities aK = cos 2 y aaK + sin’ y avK,

b,=mNcosyb,,+fiNsinyb,,.

(3.16)

The extra terms (cf. Eq. (3.7)) proportional to K, aK, and b, in Eq. (3.14) come from the rotation of the intrinsic states and are of purely kinematic origin. These terms are exactly cancelled with similar terms coming from the expectation value of H and hence do not contribute to energies at the level of the first layer considered here. One of the K = 1 intrinsic states corresponds to a rotation of the groundband and is spurious. As shown in I, this state can be eliminated most easily by noting that its normalization should vanish. From (3.14), this amounts to l-~=O+~[~-1~2[~cosyxnlx~II+\sin~xV,xV,,]=1. An approximate

(3.17)

solution of (3.17) is (exact in the good F-spin limit)

f L-1 112

X plls

=

2y,

(3.18)

xp’,

where the subscript s denotes spurious state. Test studies show that even for large F-spin breaking, the solution (3.18) satisfies Eq. (3.17) to a high accuracy (better than 0.1 Oh). Thus its effect is negligible compared to the presumed l/N error

134

KUYUCAK

AND

MORRISON

associated with the leading order results. For the other (physical) K= 1 solutions X pl, one has b,,(x,,) = 0 from orthogonality (i.e., xpl . xplS = 0), which simplifies the K = 1 expression considerably. Vanishing of b,, ( xpl ) also has a simplifying effect or the interband electromagnetic transitions, as is discussed in the next section. Performing similar steps, we obtain from Eq. (3.12) (we refrain from giving an! intermediate steps here since the extra terms mentioned above lead to extreme11 long expressions which, however, are cancelled out at the end)

+$(A.(K)-A,)A,f$$A,(K)-A,)A. n

Y

+ 2 cos* y N ((A,(K) - A,) A, + B:(K)) II +aN,N,, 1-(x;x,)~-$

1

+ (n -+ v, cos y + sin y) + K’L

1 .X”J2- (%I .xJ2 ~,I

l/N

Here A,(K),

EXPANSION

IBM-II

IN

135

B,(K), EP(K), and Eb are given by

B,(K)

= c

W

10

12K)

(3.20)

qpjlXpJKXpl~

Jl E,(K)

=

c

&pIX;,K,

Eb

=

/

c

Ep~Xpi*pto~ I

In the limit of F-spin symmetry, following Eq. (3.11) and introducing

in addition

A,(K) = A,(K) = A(K), B,(K) = B,(K) = B(K), etc., Eq. (3.19) reduces to the IBM-l

expression (H)K,L=ZV[E+~(l-&)(E-+‘~6K0)+~(E(K)-E)] +;N2 ~~‘-~(~A,-~A)A-AB(O)~C&,,

+;(AA(K)--A’+B”(K))

I

+k.‘L.

1 (3.21)

Equation (3.21) differs from the corresponding expression in I (Eq. (3.56)) by the presence of the one-body terms. The extra 6, terms in (3.21) are also generated by the single boson energies. They, however, vanish after variation due to the orthogonality of the ground- and beta-bands and do not appear in the final energy formulas [29]. 3.3. Mixed Symmetry K-Bands

Calculations for the KM-bands are very similar to those for the K-bands. Comparing Eqs. (2.9) and (2.10), it follows that one needs to substitute cos y + sin y and sin y + -cos y in the calculations of the last subsection. For example, the normalization .N(f$,, L) is still given by Eq. (3.14) with aK and b,, Eq. (3.16), replaced by the corresponding mixed symmetry averages a& = sin2 y anK + cos2 y avK,

b,,=@sinyb,,--mcosyb,,.

(3.22)

Similarly, the energy expression (H) KM,L can be easily obtained from Eq. (3.19) with the above substitutions and is not repeated here. The F-spin symmetric limit of (H)Knr,~ is of some interest and is given by

136

KUYUCAK

AND

MORRISON

‘.

Notice that the terms proportional to B(K) and b,, (3.21)) because they are antisymmetric under F-spin of F-spin symmetry. Another difference from the Majorana energy does not vanish but gives UN, operator evaluation from Eq. (A.3).

(3.2:

do not appear in (3.23) (c and hence vanish in the lim symmetric case is that th consistent with the Casiml

3.4. Variation after Projection

For a simple quadrupole interaction, variation to determine the optimal mea fields is relatively easy to apply, and it does not affect the leading order results i energies and E2 transitions. Therefore, it was rather briefly treated in I. Variatio of the hamiltonian (2.1) is considerably more involved due to the presence of one body and Majorana terms, which destroy the simple eigenmode condition anI couple the proton and neutron mean fields. A discussion of variational effects i, some depth, therefore, is in order. Applying the variational principle to the ground-band energy leads to the set c equations

&

Pi

p=n,v,

CW)g,~-~,x,~x,l=O~

1=0,2 )...) p,

(3.24

subject to the condition g, = xP . xP = 1. An advantage of the l/N expansion is tha the resultant equations can be solved order by order using the following ansatz fo the structure coefficients 1 L 0 xpl= Xpl + i y,/+ p =pl,

p = 71,v, I = 0, 2, ...) p.

(3.25

Denoting the leading order terms in the energy formula (3.9) by Ho, the next orde by H’, etc., Eq. (3.24) can be cast into the form &

[Ho-l,x,.xJx;=O, P’

& &

[H’-i,x,.x,li,,,~+[~] PI

Pi x;

=O,

(3.26b

IH”-l,.x,.x~l~~+=~~,~z+[~]

=O, PI x”,

(3.26~

Pi

l/N

EXPANSION

137

IN IBM-II

where we have used the fact that Eq. (3.24) should be satisfied independently for each order (1, l/N, L/N’) leading to Eqs. (3.26a), (3.26b), (3.26c), respectively. Restoring the normalization factors gP = x, . xP in Eq. (3.9), we have for H”

(3.27b)

-E NH’-;,,N,N( 2yNZ

HZ=

-($~+~)}+dL,

(3.27~)

and D, = where we have introduced D,, = A,A,, -t A,A,, - 3A,A, (2A,, - 3A,) A, for simplicity. The derivatives in Eqs. (3.26) are evaluated most conveniently by treating gP and y as independent variables and using the chain rule. Thus from Eqs. (3.26a), (3.27a), we obtain for x”,/ (the superscript 0 is suppressed for convenience),

. (KN,A, +2~3+4,)

c qnj,x, -AA, + (&,I-En)x,/ i

where (tPjl= (j0 10 (20) qpjl. A second equation for x,, is obtained from Eq. (3.28) through (n c) v). For hamiltonians which are dominated by the quadrupole force and have a small F-spin breaking, an approximate solution to Eq. (3.28) is given by the eigenmode condition, cj qPjlxPj= ApxP,. In general, however, the above set of coupled non-linear equations has to be solved using a Hartree-Bose condition. Notice that multiplying Eq. (3.28) by x,! and summing over 1 gives 1, = 0. Similarly, one obtains A, = 0. Thus Eq. (3.28) and its counterpart for p = v can be rewritten in the form

1 ff,,Xrj-hnXr,=aNvxn ~x,x,/, i

where H, H,=

C Hvj/Xvj-hvxv,=aNnxn ‘x,x,I,

(3.29)

i

and h, are defined from Eq. (3.28) as (~NvAv+2~nNnAn)

qnjl+En/hj,,

h,=(KN,A,+2lc,N,A,)A,+E,-aN,(x;x,)*,

Hvj, = Hnj,(n * “1, h, = h,(n c1 v).

(3.30)

Equations (3.29)-(3.30) can be solved by iteration. As an initial guess one can use the eigenmode solutions for xP[ in H,, h,, and on the right-hand side of (3.29).

138

KUYUCAK

AND

MORRISON

The dimension of the matrices in (3.30) is 2 for the &model and 3 for the sdj model. Hence the set of inhomogenous linear equations posed in (3.29) can 1 easily solved. The resulting values of xP, are then fed back into Eqs. (3.29)(3.3( and this process is iterated until a self-consistent set of solutions is found. I practice, we find that the convergence is quite rapid, and due to the simplicity ( the equations a minimal amount of numerical effort is required. Once the leading order terms xPl are determined, the higher order coefBcients y and z,,[ are relatively easier to obtain because Eqs. (3.26b-3.26c) lead to a set ( linear equations which are straightforward to solve (no iteration is required). Fror Eqs. (3.26b), (3.27), we obtain for y,,,

x W, 6,,+ &J, &J c 4pji~xprc - UN, I’

-

KN,D,,+~K,N,

[

’ &Tqij4-cn] [ J

(

x,/ - ~‘%NY

D,+;2yA:

R

xx/ + Chd(2Y + 0 - (2Y-% + J%,)I xx/

-aN,C(2y,,+b,.x,)x,,-4~,,x,.x,x,,l

- KN,N,D,, + 1 (K&D, P

+ N,&) (3.31

In deriving Eq. (3.31), we made use of the Hartree-Bose normalization condition gp=xp.xp+

0 l 0 xP+~xP.YP+g-Pp

1

equations (3.29) and the

O.z,+...=l,

(3.32

which implies that (since x! . xz = 1) xz . yP = x”, . z, = 0. Another set of equation! is obtained for p = v by interchanging (x H v) in Eq. (3.31), forming a set o

l/N

EXPANSION

IN

IBM-II

139

p + 2 inhomogenous

linear equations which uniquely determines the coefficients { ypi, p = zn, v, I= 0, 2, . . . . p>. From Eqs. (3.26)-(3.27), it is clear that one has a similar set of equations for zP,. On the left-hand side of Eq. (3.31), the only change is y, + zPj, the coefficient in the bracket remaining the same. On the right-hand side, one needs to delete the terms indicated in Eq. (3.27~) and multiply the coefficient by - 1/2y before differentiating. The next step in the variation is substituting x,! (3.25) in (3.27) to obtain the final energy formula. This need only be done for the leading term Ho, since the others (Hi, H2) contribute to higher order than the first layer. Note that the L/iv2 term can be obtained from the l/A’ term through the substitution yP -+ zp. Substituting (3.25) in Ho (3.27a), we obtain to leading order in yp and zpr HO(x;, Yp, zp) =H”(xE)+$

i

KN,N,[(A.(xO,,y,)-A,xS1.y,)A,+(~c*v)]

+ 1 C2~,~347bO,, P

Y,) - A,$

.Y,) A, + &(E,bO,,

Y,) - E,xO, .Y,)I

-aN,N,~x~~x~(x~~y,+x~~y,) (3.33) +${Yp+z,i, I where we have used the obvious notation ,4,(x. y) =Cj,4Pj~xjy, and similarly for E,(x, y). Equation (3.33) can be rewritten in terms of the matrices H,, and h, (3.30) as

-6:

H’(x;,

~x:)‘(xon~Y,+x~~Y”)1

yp, zp)

=H’(x;)+$ I

~(H,,-h,6j,)xrjj-aN,x0,-x~x~, i

1

The terms in the brackets in (3.34) vanish by virtue of the Hartree-Bose equations (3.29), irrespective of the values of yp and zP, proving that “variation does not affect the energy expressions to the first layer.” Thus, after variation, the groundband energy is still given by Eq. (3.9). Since all bands have the same leading term Ho, this result also applies to the excited band energies. It should be fairly clear from (3.34) that this property is independent of the hamiltonian considered and can be shown to hold in general [29]. Calculation of the coefficients yP and zp, however, is not entirely futile, because such a cancellation does not occur in electromagnetic transitions (except in the case of E2 transitions in a pure quadrupole hamiltonian), and variational effects should be included there for exact first layer results.

140

KUYUCAK

AND

MORRISON

Next, we discuss variation of the excited band energies. Since the K= 0 bands al special, they will be considered separately. For the K# 0 bands, the energy formul Eq. (3.19) can be written as (f0

,.=H’+H’+H:,+H*,

(3.3!

where H” are the same as in (3.27) and HL, factors gP, gP, = xpK. xpK, is given by

after restoring the normalizatio

1

+(7c-+v, cosy+siny) +aN,N,

Equation (3.35) is to be varied with respect to xP, xBK, and y. Since (3.35) differ from the ground-band expression only in the first order term, varying with respec to x, one still obtains Eqs. (3.26a), (3.26~). Thus the ground-band solutions for x and zP remain intact for the K #O bands. The coeffkients ypK are different an require xpK and y for the solution to which we turn. The x,, and y dependence is confined to the Hi term which is of first order Therefore it is sufficient to obtain leading order solutions for these coefficients Varying Eq. (3.36) with respect to x,,~ subject to the condition gnK= xXK. xZK= gives ~os*y(~N,4+2%JJ,)

I(-)” (

W~-Kl20)

~nj,~niK-4AK)x,lK >

.i

+(rc,/msiny

cosyB,(K)+2rc,N,cos2yB,(K))

X 1
qnj/Xnj-

B,(K)

-aJmsiny

+ COS*

X,/K

Y(En,-

En(K))

X,/K

>

i

cosyx,

. X,(X,IK

-

x,K.

X,KX,IK)

-

IznKx,,K

=

O.

(3.37

l/N

IN IBM-II

EXPANSION

141

A second equation for xUIK is obtained from Eq. (3.37) through (no v, cos y c* sin y). As before, for hamiltonians with a dominant quadrupole interaction, the eigenmode condition cj (-)” (jK I- KI 20) qpj,xpjK = lPKx,IK provides an approximate solution to Eq. (3.37). In general, one has to resort to iterative methods for solution. Multiplying Eq. (3.37) by xzIK and summing over 1 gives A,, = 0 (similarly, A,, = 0). This allows further simplification of Eq. (3.37) by factoring out cos* y and uniformly substituting u = tan y. Thus Eq. (3.37) and its counterpart for p = v can be put in a form similar to Eq. (3.29) 1

Hnj/KXnjK

-

hnKXn,K

=

2.4

~~C(-KX"jKXnlK+axn.x,6j,)X,i~, i

j

where

Hpj/K,

hpKI

and

are defined from Eq. (3.37) as

X,/K

Hj,K=(KN,A,+2K,N,A,)(-)K

(jK1-K120)q,jl+2~,N,X,jKXnlK+~IIl~jl

hn,y= (fcN,A, + 2rc,N,A,) + HvjIK

=

E,(K)

H/K

(7~ t-f

A, + (K JN,N,

a JN,N,

UX,.

(K&W) +

[[(~Nvk

B,(K)

’ x,K,

(3.39)

qpjlxpj.

Varying Hi with respect to y and substituting equation in U, $?t

+ 2tc,N,B,(K))

h,, = h,, (n t* v, u -+ l/u),

Vh

X~IK = 1 (j0 lKl2K)

x,x,K

u&(K)

B,(K) +

+ ~K,N,B~(K)

-

.X,&K.

x,K)(u2

kz~,A&k(K)

+

For an F-spin symmetric becomes

ax,

-

-

1)

A,)

E,(K) - E, + aN,(x,

hamiltonian,

u = tan y gives a quadratic

.x,)~] - (7~++ v)] u = 0.

Eq. (3.40) decouples from

JN,N,(u2-l)+(N,-N&GO,

(3.40)

(3.38) and (3.41)

which has the positive definite solution, uK = ds, in agreement with the group theoretical result. In general, Eq. (3.40) has one positive solution uK and another solution given by Us = - l/u,. This second solution is associated with the mixed symmetry states and is already implemented in Eq. (2.10) through the orthogonality condition. Equations (3.38)(3.40) have to be solved iteratively. One can start with the eigenmode solutions for xplK and determine uK from (3.40). These values are then used in h,, and on the right-hand side of (3.38). For K= 1,2, the dimension

142

KUYUCAK

AND

MORRISON

of the matrices in (3.39) is 1 for the &-model and 2 for the sdg-model. Hence sol1 tion of the inhomogenous linear equations in (3.38) is trivial in the sd case (xPz2= and no iteration is needed) and rather simple in the sdg case. The resulting value of XpIK are fed back into Eqs. (3.38)(3.40), and this process is iterated until a se1 consistent set of solutions is found. Although the process for the K-bands is moi involved due to the double iteration in xPK and uK, the convergence is still quil rapid. This happens because in practical cases, F-spin breaking is small and u varies little from its limit value. With xpK and uK determined, we can now solve for the coefficient ypK. Variatio of Eq. (3.35) with respect to xP leads to a first order equation similar to (3.26t with changes y,+y,,and H’+H’+Hi. Thus the right-hand side of Eq. (3.31 is modified by adding ( -N/2N,) times the derivative of Hk, t3H:,

-=~{[KN”U:(A.(K)-A.)t2K,N,(A,(K)-A,)1 ax,,

(~~njiXnjeA~X.i)

K

+

i

C~J;Y,N,UKB,(K)+~K,N,B,(K)I

C

(j~loI2K)

~n,,xnjK-Bn(K)xz/ >

i

-ak/%%

uKxrK.

‘“K

-N,~~X,.X”l(X”I--X..X”X,,)

(3.42

Again, we used the Hartree-Bose equations (3.29) to simplify the expression iI (3.42). The corresponding derivative for p = v is obtained from Eq. (3.42) througl the substitution (R ++ v, uK --Pl/uK). The resulting system of inhomogenous linea equations uniquely determines the coefficients { yplK, p = X, I= 0,2, .... p}. For all other K-bands (e.g., fl, /I’, y’, etc.), orthogonality of the solutions to thosl of the lowest K-band is not guaranteed, and a condition must be included in thi variation through a Lagrange multiplier 1,, (qdK(q5K.)=~~~ y,cos ~~.x,~.x,~+sin

yK sin yK’xYK.xY,.=O.

(3.43

Equation (3.43) also applies to the K = 0 bands if we set cos y = m sin y = m for the ground-band. Here, a little digression to the situation ii IBM-l will be useful. In order for the l/N expansion to make sense it is essentia that the N dependence of the hamiltonian overlap between the K- and K-bands i; smaller than N. Otherwise, the band mixing contribution to the excitation energie: is equal to or larger than those of the diagonal elements. In IBM-1 the orthogonality condition is simply given by xK. xK, = 0. The leading order overlaI for the IBM-1 equivalent of the hamiltonian (2.1) is (angular momentum projectior here is not necessary since the leading order results are the same)
H

14~‘)

=hKXK.XK’~

!I,=

z~N(hi(K)

+

B’(K)) + E(K),

(3.44

where the Hartree-Bose energy h, corresponds to the F-spin limit of h,, (3.39). FOI

l/N

EXPANSION

IBM-II

IN

143

K = 0, a similar result is obtained with the N dependence modified to N312. Thus

in IBM-l, the orthogonality condition automatically ensures the vanishing of the leading order overlap. In IBM-2, the overlap becomes (q5J H IcjKc) =cos y,cos yKr hnKx,Ke~,Ks +sin yK sin yK’ hvKxvK .x,~.

(3.45)

Since in general h,, # hvK, Eq. (3.45) does not vanish with (3.43) and has to be included in the variation as an additional constraint. From Eqs. (3.43), (3.45), an equivalent set of conditions (similar to IBM-l) follows, xpK .xpK’ = 0, p = 71,v, which are much simpler to implement. As an example, we consider the b-band. The energy expression Eq. (3.19) can be written for K = 0 as (3.46)

(H),,,=H”+H’+H:,+H2+Hp,

where all the terms are as in (3.35) except H, which, using the Hartree-Bose equations (3.29)-(3.30) can be written as H,=(

l-- 2;N

1

: CJ%@COS

Y h, x x -x,,+~sinyh,x,~xVo]

- a[N, cos* y(x, .xno)* + N, sin2 y(x, .x,,)~].

(3.47)

The first term in Eq. (3.47) is the same as (3.45), hence it vanishes and can be absorbed into the Lagrange multiplier 1,. The variational equations in (3.38) become 7

Hkj,ox,jo

- hiox,/o

~~nX~~5UJ;IV,NV~(~lCXyiOXn~O+ax~~x~6,~)x~jO~ i

(3.48) where H,,

and hnO are modified by the second term in (3.47) to Kj/o = Hnj/o - aN,xvjx,,7

h;, = h, - aN,(x, . xzo)*.

(3.49)

The p = v equation is obtained from (3.48)-(3.49) with the usual substitutions (rc++ v, u + l/u). The y equation (3.40) is similarly modified by the addition of the term a[ - N,(x, . xzo)* + N,(x, . x~)~]u. The solution of the mean fields xPlo is trivial in the sd model (one simply takes the orthogonal combinations to xp,), therefore, we discuss the sdg case. As before, one can start with the eigenmode solutions and iterate until self-consistency is achieved. The initial values of xp,,, are fed into the y equation to solve for uo, and also into the right-hand side of (3.48). The orthogonality and the normalization conditions can be used to eliminate, for example, xPM)and xPzo. The three equations in (3.48) are reduced to a single equation by eliminating hho and ,?.,, which then determines xPdo. In fact, the xPa equation is quadratic, and the two solutions correspond to the jI- and /Y-band mean fields. 595/195/1-IO

144

KUYUCAK

AND

MORRISON

The solution of the remaining coefficients ypO and zPOare similar to those for th bands (Eq. (3.42) is modified with the additional terms coming from th derivative of (3.47)), and therefore is not repeated. Finally, we discuss the variation of the KM-bands. The energy formula is given b Eqs. (3.35)-(3.36) with Hi replaced by Hk, = Hi(cos y + sin y, sin y + -cos y Varying H j,, with respect to xpK and y, one obtains similar equations to K-band with the substitution u + -l/u in (3.38)(3.40). The y equation (3.40) remain invariant under u -+ - l/u; thus it has the same solutions uK, uM. Rememberin that the mixed symmetry solution is given by u ,,, = - l/u,, the double inversion c u also leaves the Hartree-Bose equations (3.38k(3.39) invariant, and hence on obtains the same mean field solutions for the mixed symmetry bands as for th symmetric bands. K#O

3.5. Discussion of Energy Systematics

Even for the simplified hamiltonian (2.1), the number of parameters are toI many to consider them all, and a selection must be made. The F-spin decomposi tion of the parameters will be helpful in this respect. Thus, we introduce the F-spil scalar (e.g., q= (qz + q,)/2) and vector (q” =q,-qv) quantities. The behavior c the scalar parameters N, IC, and E are well known from the IBM-l studies (al equivalent IBM-l hamiltonian exists for an F-spin scalar IBM-2 hamiltonian) ant need not be considered again. Of the vector parameters, we concentrate here OI (i) quj,, because of its significance in generating Ml transitions, (ii) N,, to study th’ relevance of the concept of F-spin multiplets in nuclear spectra, and (iii) “a,” tc investigate its effect on F-spin mixing of states. The vector single boson energies E, have little effect on the spectrum of deformed nuclei, and, in the absence of an: phenomenological indications, are not considered further here. So far, two versions of the hamiltonian (2.1) has been used in the literature. II the original (microscopic) version one has rc, = rc, = 0 which breaks the F-spin ii the Q . Q interaction. More recently, an F-spin symmetric form with K, = rcc,= K/: has been introduced especially for the phenomenological analysis of Ml data. Sine the choice should ultimately be based on experiment, we have kept both forms am pointed out the differences which may facilitate a decision. The parameters used in the calculations, typical of deformed nuclei, are N = 14 N, = N, = 7, edn = cdv= 100 keV, sgn= .sgv= 1000 keV, a = 170 keV, xi = 0.5, x2 = 1 x, =OS, xl” =0.6, x2” = -0.6, x3” = -0.6, and two sets of K: (a) K = -80 keV IC, = K, = 0; (b) K = -40 keV, IC, = K, = -20 keV, corresponding to the microscopic and symmetric versions of the hamiltonian. Here the x’s represent the quadru pole parameters normalized to the SU(3) values, i.e., x, = q2Jq2JSU(3)) and x3 = qJqJSU(3)). The scalar parameters are held lixec x2 = q24/924(sw))~ at the above values in all calculations. The vector parameters have the above value: unless varied in a plot or otherwise specified. The Majorana parameter “a” i! chosen so that the mixed symmetry 1 +-band occurs at around 3 MeV for the sym’ metric set (b). This leads to a slightly higher I+ -band for the microscopic set. Since

l/N

EXPANSION IN IBM-II

145

the effect of the Majorana parameter on the other (symmetric) observables is minimal, we have used the same value for both forms of the hamiltonian. The same set of parameters is used in both the sd- and sdg-model calculations except E,, x2, and x3, which are specific to g-boson. The calculations are carried out using the computer code IBM-2/N [30] which solves the Hartree-Bose equations either for the sd- or sdg-b&on system and gives the excitation energies and transition rates for the low-lying bands. As a prelude, we discuss the F-spin purity of the “symmetric” states which is essential for the current formalism to be meaningful. For this purpose, we employ Eqs. (A.12)-(A.13) which give the F-spin probability in the ground- and various K-bands. Figure 1 shows the probability of F,,,,, in the ground-, /I-, and y-bands as a function of xy in the sd-model. Even for the maximum breaking x0 = 1, which puts the protons in the W(3) limit and the neutrons in the O(6) limit, the bands are pure in F,,, better than 95%. Notice that the symmetric form (b) is more effective in breaking the F-spin than the microscopic one (a). Relevance of this in the Ml-transitions is discussed in the next section. In Fig. 2, we show the variation probability in an F-spin multiplet with xv = 0.6 which roughly reproduces of Fmax the Ml-strength. Again, the bands remain pure in F,,, better than 95 %. Also there is a marked difference between the two forms; (a) shows a rapid change with N, and has a minimum amount of F-spin breaking at N, = N,, whereas (b) is smooth and has a maximum F-spin breaking at N, = N,. This may be important in distinguishing the two forms of the hamiltonian through an analysis of the Ml-transitions in an F-spin multiplet. The third quantity to be studied is the

0.99.

0.98-

K=2

:

-k a z

2 0.97.

0.96.

FIG. 1. The F,,,,, probability in the ground-, b-, and y-bands in the sd-model as a function of x, (normalized to 1 in the SU(3) limit). The two figures correspond to the microscopic (a) and the symmetric (b) choices of the hamiltonian. The j-band is indicated with a dashed line (overlaps with the groundband in (b)). The parameter set is given in the text.

146

KUYUCAK

AND

MORRISON 1.00

-

0.96

,

K=2

0.98.

.-I .

o.9s2

a

b

4

6

8

*

10

12

0.95f

4

YT

j 8

6

j 10

12

Nn

FIG. 2. Same as Fig. 1 but for the F-spin multiplet N= 14. The quadrupole parameter is fixed a x, = 0.6.

Majorana parameter “a.” Figure (3) indicates that the F-spin would still be a usefu quantum number for “a” around 50 keV which corresponds to having the mixec symmetry states as low as 1.5 MeV (see Fig. 8). In the &g-model, there are three 1” parameters to be considered. Individua variations (e.g., x1”: 0 + 1, x2” = x3” = 0) are similar to Fig. 1, with xiv providing thl same amount of breaking, x2” slightly larger, and ‘x3” much smaller. An interestin question here is the relative phasing between the three xv’s in order to achieve thl

0.5i a(keV)

50

’ 100 a(keV)

’ 150

FIG. 3. Same as Fig. 1 but for the Majorana parameter “a.” The boson numbers are N, = N, = 7 ant x, = 0.6.

l/N

EXPANSION

IN

IBM-II

147

X2” FIG. 4. parameters line).

X2”

The F,,,,, probability in the y-band in the sdg-model are x ,” = 0.6 and three choices for x3,,; -0.6 (full line),

as a function 0 (dashed line),

of x2.. The other 0.6 (dashed-dotted

maximum F-spin breaking. Figure 4 shows the F,,,,, probability in the y-band for three choices of x3”, - 0.6, 0, 0.6, as a function of x2”. The microscopic form (a) clearly prefers an opposite phase for both x2” and x3” (hence the choice in the above parameter set). The symmetric form (b) does not distinguish between the relative phases as long as x2” and x3v have the same phase. Variation of the F,,, probability in an F-spin multiplet is shown in Fig. 5. In the sdg-model, one has naturally more

0.98 -

0.96

K=2

-k I! Lr, z

0.94.

0.94 \\ S__' ,

0.92

0.92

a 0.9Oj

b 4

6

8

10

12

OH2

4

6'

8'

10

12

Nn FIG. 5. Same and x3” = -0.6.

as Fig. 2 but for the sdg-model.

The

quadrupole

parameters

are x1. = 0.6, ,Q~, = -0.6,

148

KUYUCAK

AND

MORRISON

scope to break the F-spin which is reflected in the doubling of the scale. Otherwis the essential features noted for the sd-model (Fig. 2) remain the same. Having established the goodness of the F-spin symmetry, we proceed with th study of energies. In the last subsection, we have shown that variation does nc effect the energies to first layer. Thus the excitation energies for various single phonon excited bands are directly given by Eq. (3.36) (plus the second term i Eq. (3.47) for K = 0). Since the variation of the phase angle “y” from the limit valu (cos y = ,/m; sin y = m) is small, in the following we use the limit valu which leads to more transparent formulas. (Even for relatively large F-spi breaking, energies are not affected by this approximation because the effects of th variation from the limit value are washed out by the averaging process for th scalar quantities.) We thus obtain for the excitation energy of the K-bands tm leading order

E,(K)=CN,CE,(K)--E,+2h-,N,C(A,(K)-A,)A,+B~(K)II PN +JCN+

[(A,(K) - 44 A, + W,(K) - A,) A, + 2&(K) B,(K)1

-a?

[2(x,K.x,K-x,~x,)x,

‘X” + C(X”%d2 + (77%o)21 ~,I. (3.50

The corresponding expression for the mixed symmetry K-bands can be obtainer from Eq. (3.36) through the substitutions sin y = a; cos y = -m, NN E.xW,) = y

i

4 $ Ir,(K)--E,I+2n,l(A,(K)-A,.)A,+B:(K)l] [

+JC $” (ArW) - 4

A, + (71~ 4]-2&W)

n

k(K)]

N2,+Nt N N x,.x, II Y .x,J2+(71+-+v)

1 11 s,

.

(3.51

Test calculations indicate that the difference between the exact (3.36) and the approximate expressions (3.50)-(3.51) is at most a few percent which could bc hardly discerned on figures. The calculations are done with the computer code [30: which utilizes the exact expression. In Figs. (69), the dependence of the bane excitation energies on the vector parameters are studied in a similar fashion tc F-spin. Figure 6 shows the band excitation energies in the &-model as a function

1/N

1.4.

EXPANSION

K=O

IBM-II

IN

149

_

___.__-----------

K=l

3.4

3.2 a

b

0.8 0

0.2

0.4

0.6

0.X

K=l

t

1 I 0

i

I. 0

0.2

I

I

,.I

0.4

0.6

0.8

1.0

X”

X”

FIG. 6. Excitation energies of the symmetric (a) and mixed symmetry (b) bands in the &model as a function of xv. Results for both the microscopic (full line) and the symmetric (dashed line) forms are presented. The ordering of the bands are K = (1). 0, and 2 in all cases.

of xv. Since energy is a scalar quantity, they are hardly affected by the F-spin breaking in x in both the microscopic (full line) and the symmetric (dashed line) cases. The large difference between the energy of especially the K = 1 +-bands is due to vanishing of the average of B(K) in the symmetric case (see Eq. (3.23)). Measurement of the amount of splitting between the mixed symmetry K = 1 +- and 2+-bands would thus provide useful information in choosing the form of the quadrupole interaction. The difference between the two forms becomes most prominent in the behavior of the band excitation energies in an F-spin multiplet (Fig. 7). The quadratic form of

a

2

b

4

6

8 Nn

10

12

_______-_.-----’ 3.02 4

K=l __._...__..---6

8

10

N,

FIG. 7. Same as Fig. 6 but for an F-spin multiplet. The quadrupole parameteris x, = 0.6.

150

KUYUCAK

AND

MORRISON

the energies in the microscopic case is a simple reflection of the N,N, dependent in Eqs. (3.50)-(3.51), with the F-spin breaking in x causing a slight skewing. In con trast, the linear dependence in the symmetric case is entirely due to having xv # 1 (for xv = 0, all the curves would be flat). Experimentally, there are F-spin multiplet in the deformed region where the P-band goes through a maximum at N, = N, However, in the same nuclei, the y-band goes through a minimum (though les pronounced), and it would be difficult to reconcile the opposing behaviors of th two bands using the standard hamiltonian (2.1). In Fig. 8, dependence of the bane energies on the Majorana parameter is shown. In line with the F-spin purity o states for “a” above 50 keV, energy variations remain close to the Casimir values i.e., flat for the symmetric bands, and vary as UN for the mixed symmetry bands. As in the &model (Fig. 6), the band energies in the sdg-model vary little wit1 x1, and are not pursued further. Behavior of energies in an F-spin multiplet is o some interest and is shown in Fig. 9 (p’- and y’-bands are not included in the ligurc to avoid cluttering). The quadratic (microscopic) versus linear (symmetric dependence of the energies in the sd-model (Fig. 7) is preserved in the sdg-model Some qualitative differences in the skewing and slope of the curves are due to usin! opposite phases for x2Uand x~~,.Dependence of the band energies on the Majoram parameter is similar to the sd-model (Fig. 8) and is not repeated. The other physical quantity of interest is the moment of inertia of various band! which is proportional to the inverse of the t term in the energy formulas. Fran Eqs. (3.27), (3.35), (3.47), it is seen that, to leading order, all bands have the same moment of inertia given by 1

Ep-ZyE,,+2~,N,

+ KN,N,

2A,A,-1

2Y

(A,Avl + A,A,,

- 2yN*~’ - 2aN, N,x, . x, (xn.x”-y)]-‘.

1

2A:,-2J:(2A,1-3A,)A,

-3&A,)

)I

> (3.52:

The next order term in the l/N expansion is different for each band and could explain the small variation observed in the moment of inertia of low-lying bands which is roughly on the order of l/N. Compared to the band energies, the moment of inertia is even less affected by the F-spin breaking due to x or “a” and therefore is not pursued. Figure 10 shows the variation of the moment of inertia in an F-spin multiplet in the sd- (a) and sdg-models (b). The (sdg) parameters for x, are reduced by half in order to have a similar amount of F-spin breaking as in the sd-model. The curves in Fig. 10 roughly correspond to the inverted band energies (Fig. 7), though much flatter, and have the same underlying explanation. Experimentally, the situation is opposite to that of the microscopic curve (full line), i.e., the moment

l/i’/

EXPANSION IN IBM-II

K=O

1.4.

0

151

50

100 a(keV) FIG.

150

200 a(keV)

8. Same as Fig. 6 but for the Majorana parameter “a.”

of inertia has a minimum at N, = N,. Thus, summing up the band energy and moment of inertia systematics, it seems that the concept of the F-spin multiplet is untenable especially for the microscopic form of the standard hamiltonian (2.1). Including extra (n,)’ terms in the hamiltonian appears to remedy this problem [28]. Alternatively, one may give up a constant parameter description of F-spin multiplets and allow N dependence in the single boson energies [31].

----------3.02

K=l 4

-----_.__ 6

8

10

12

Nn FIG. 9. Same as Fig. 7 but for the sdg-model. The quadrupole parameters are x,. = 0.6, ,yzO = -0.6, and x3. = -0.6.

152

KUYUCAKANDMORRISON

0.1 b

FIG. 10. Variation of moment of inertia in an F-spin multiplet in the sd- (a) and the sdg-models (b; Both the microscopic (full line) and the symmetric (dashed line) forms are shown. The quadrupoli parameters are 1, = 0.6 (sd) and ,yI. = 0.3, x2” = -0.3, and xxu= -0.3 (sdg).

4. ELECTROMAGNETICTRANSITIONS In this section, we calculate electromagnetic transitions among various band; considered in the last section. Transitions between the symmetric states is shown tc obey a simple scaling law and thus can be easily obtained from those of the IBM-’ expressions given in I. Transitions involving the mixed symmetry states can bc obtained from the corresponding symmetric IBM-2 expressions through a simple substitution and hence obey a similar scaling law. A novel feature in the IBM-2 i: the F-spin mixing effects in Ml transitions which is discussed in some detail. The electromagnetic multipole operator for neutron-proton systems has the general form

Tjl;’= C r$[b;jZ,r] g’,

(4.1

PiI

which is electric for even k and magnetic for odd k. In order to limit the number of parameters, we employ the consistent Q formalism for the E2 transition operatol [32], i.e., tij,= e,qpjl and T(E2) = C, e,Q,, where err (e,) is the effective E2 charge for the proton (neutron) bosons. The Ml operator has only diagonal entries in the parameter matrix which is denoted by tbji = ajrgp,. It becomes equal to the angular momentum operator when g,, = [ r(21+ 1)/3] 1’2. 4.1. E2 Transitions We demonstrate

the scaling property in a few examples. First, we calculate tht

1/N

EXPANSION

IN

IBM-II

153

reduced matrix elements of the E2 operator between the ground-band Eq. (I, 2.9), we have (La

*w2)

(IL,) = 2[,j/“(ql,,

states. From

i’(2L + 1) L’) J/-(& L)]‘lZ 1M (LM2-MJL’O)

(4.2) with the normalization as given in Eq. (3.7). The intrinsic Eq. (4.2) can be calculated as

matrix

element in

Substituting (4.3) in (4.2), and after the usual steps of (i) combining the d-matrices, (ii) doing the M-sum, (iii) evaluating the /?-integral using the gaussian approximation, we get

<&II %=I

IIL,)

JJ(2L’+

1)(2L+

1)

= C”(4gV“1 N(dg* L)l”z xc

(j0 L’O)JO)(10 J

LO)JO)

C

pj,

Npep~pjiXpiXpl

1;

y

;}FtL

J).

(4.4)

In deriving Eq. (4.4), as in the case of the energies, we have made the simplifying approximation I/T,, + l/T, z l/T, + l/T,, x l/T,. Doing the angular momentum sum over J and dividing by the normalization, we obtain to the first layer
IlLg)

where Apmn are defined in Eq. (3.10). In case one wants to use a different parametrization for the E2 operator, Apmn must be evaluated with the replacement qpj, + tzj,. Substituting L’ = L in Eq. (4.5), we obtain for the quadrupole moments

154

KUYUCAK

Q&l=

AND

-,/=?$$x

N,e,

1 -2ylvz

MORRISON

( A,+N

’

1

[ A,-~(A,,-3A,)

P

11

A,+l(A,Z-A,I,-10Ap,+12Ap)

8~ and using L’ = L + 2 gives for the ground-band

,

(4.6

B(E2) values

B(E2; L + 2, -+ L&J 3(L+ l)(L+2) = 2(2L + 3)(2L + 5) ~~N~e~{A~+~[A~-~ia,,-3A,)] -

L(L + 3)

2yN2

2

1 A,-24y(Ap*-Apll+6Ap,-12Ap)

Ii1 .

(4.7

Comparing Eqs. (4.6), (4.7) with the corresponding ones in I, (4.6), (4.9), it is seer that they can be obtained from the IBM-l expressions through the scaling eAmn+C,W,IN) epApmn. Notice that y is already averaged in evaluating the /?-integral and therefore is not scaled. The above scaling holds for all quadratic forms in xP, but must be modified fol transitions involving the excited bands (i.e., xpK) to include F-spin breaking effect! in configuration mixing. As an example, we calculate the K-r g E2 transition. A: noted in I, for interband transitions, it is essential to include mixing of other K-bands in order to avoid superfluous contributions due to non-orthogonality o states. Thus, Eq. (2.9) for the K-band intrinsic states is modified to I~K)=~~~yb~KIN,-l,N,)+sinyb~,IN,,N,-l)

+A

c 5:CJN,-Icosyb~,bH,,IN,-2,N,) mm’

m+m’=K

+,,/~sinyb~,,,b~,, + (A

IN,, NV-2)

sin y bL,,,bi,. + fi

cosyb;,,b:JIN,-l,N,-l)],

(4.8:

where
M(i,,d,,L)=~~n

(4Ki

e-iBLy

Idg>=o?

(4.91

0

which gives for K = 0 and 2, t’i =

bo/(b,,b;,),

t: = -M%J;J,

6, = 1 N,b,,slN P

(4.10)

l/N

EXPANSION

IN

IBM-II

155

where 6, are as in Eqs. (3.15)(3.18) but averaged with y = yK and not yM. It may appear that there are other (physical) K= 1 intrinsic operators to mix in; in fact, b, = 0 for these solutions and hence they have zero amplitudes. For the same reason, calculation of ri is more complicated (one has to go to the next order). Since its explicit form is not required, we do not pursue it further. The K -+ g E2 matrix element is given by [2 - 6Ko]“* i’(2L

+ 1)

In evaluating the intrinsic matrix element in Eq. (4.1 l), two of the terms coming from (d/iYbi a/abpRT(2,,,) vanish due to the orthogonality condition (4.9). Of the remaining terms, the ones proportional to (2 are suppressed through the to sums over the C-G coefficients, and only the terms proportional (a/abL, alab,, T?‘,) contribute to the leading order giving

XC

(4.12)

(jKI-M-K12-M)d’,+~o(P)~e,qpj,xpj,x,,. P

PII

Substituting in (4.11) and performing order K + g E2 matrix element

the usual steps, we obtain for the leading

(LX11 T(E2) IIL,) = [2-6,]“*i(LO2KIL’K)

x CA

cosyKe,UK) + &

sin y,eJ,(K)l,

(4.13)

where B,(K) is defined in Eq. (3.20). Comparing with I, we see that quadratic forms in x, xK scale as eB(K) + (m cos yK e, B,(K) + m sin yK e, B,(K)). Intraband transitions in K-bands involve quadratic forms A(K) (3.20), which clearly scale as (cf., Eq. (3.22)) eA(K) + (cos2y,e,A,(K) + sin* y,e,A,(K)). Finally, interband transitions among the K-bands involve quadratic forms in xK, xK which would scale as eB(K, K’) -+ (cos y,cos yK’ e,B,(K, K’) + sin y,sin yKe,B,(K, K’)). (In Z, the y + /I E2 transition was calculated approximately. In a full calculation, the second term in Eq. (I, 4.25) is cancelled and the matrix element becomes proportional to B(2, O).) Scaling becomes particularly simple if one uses the F-spin limit value for the mixing angle y. Then differences between various bands disappear, and all transitions have the same scaling property as those for the ground-band. The above discussion for the symmetric states can be easily extended to the mixed symmetry states. To replace a K-band with the corresponding mixed sym-

156

KUYUCAK

AND

MORRISON

metry band, one simply needs to substitute (cos yK -+ sin yK, sin yK -+ --OS yK) ii the symmetric matrix element. For example, the matrix element for the K, -+ g E: transition is found from Eq. (4.13) to be (L&II

T(E2) llL,> = [2 - 6,]“’ x [JNn

L(LO 2KI L’K) sin y,e,B,(K)

- &co,

y,e,B,(K)].

(4.14

Matrix elements for other E2 transitions from the mixed symmetry states can bc obtained using the same prescription. In the limit of F-spin symmetry, the cross band transitions K,,,, -+ K’ scale as eB(K, K’) -+ (dK/N)(e, - e,) B(K, K’) whicl vanishes for a symmetric operator (i.e., e, = e,), in agreement with the groul theoretical result that a symmetric operator cannot connect states with differen symmetry characters. 4.2. Electromagnetic Excitation of K-Bands The scaling property, int~roduced above in the discussion of E2 transitions applies equally well to other electromagnetic transitions. Here, we give the excita tion strengths for various K-bands which are experimentally important. Fron Eq. (I, 4.30), we find for the excitation of a K-band via the operator TCK’ (4.1),

(Lkll P) IIL >

’

[Aces

YKt~lXnjKXnl+

&

sin

(4.15

YK t$xujKxv,l.

The expression for the g -+ K, transition follows from Eq. (4.15) with the usua substitution (cos yK + sin yK, sin yK -+ - cos yK). Of particular interest here is the Ml excitations of the K= 1 +-bands. In the sdg-IBM-2, there are three such bands (one symmetric and two mixed symmetry) which have the following transitior matrix elements

= -

T[

3 4lt 30=

yN I

II2 c N&Jp2Xp4, P

&WL Ckyn- guy,)+ lW&

= -

30 91’2

Jm

- g:x$)],

(g;xn2xn4 - g:x,2x,q),

(4.161

where we have introduced the boson g-factors g,V) = g,,cw

+ 1 Y31 -“2,

gp = g,(d),

g; = g,(g) - g,(d).

(4.171

l/It’

rN IBM-II

EXPANSION

157

Here gb measures the defect between the g-factors of d- and g-bosons. In the sdmodel, there is only one mixed symmetry 1 +-band whose matrix element can be obtained from (4.16) by setting gb =0 in the middle expression. In deriving Eq. (4.16), we have used the F-spin limit value for y, and also Eq. (3.18) to eliminate xpl in favor of xp. These approximations lead to simpler expressions, and for small F-spin breaking, do not introduce any appreciable errors. The transitions given in (4.16) were previously calculated in the SU(3) limit of IBM-2 [33, 341. 4.3. Ml

Transitions between Symmetric States

Contrary to energies and quadrupole properties which are almost independent of F-spin breaking, Ml transitions between symmetric states are partly generated by F-spin breaking and therefore it is necessary to keep the general form of the expressions. The intraband Ml transitions in a K-band can be obtained from the IBM-l expression [35] using the scaling property as

CL+ l,ll T(Ml) IILK> =

(L+ 1)*-K* L+l

I’* 1

’ [ -1 2yN- (N,g n,& + N, g,,x:,) + (cos2 yK g,&,

+ sin* yK wt,,)

1 .

Note that Eq. (4.18) vanishes if the Ml operator is proportional to the angular momentum operator. In the sdg model, with the boson g-factors as given in (4.17), the Ml matrix element becomes

CL + 1,ll T(Ml) IILK> (L+l)‘-K* L+l

y,+1%$x:,) I[ yecPN,(g, ‘I*

-1

+(cos’ Y&, + &&,) +sin*yK(g, +g:x2 mw))]. The relative g-boson and F-spin breaking contributions (4.19) can be resolved by substituting

to the Ml

(4.19)

matrix element

(4.20) where q is a measure of F-spin breaking and is determined from (4.20) as q = (ug - uK)/( 1 + ugK) with ug = ,/m and uK = tan yK. Neglecting second and higher order terms (i.e., q*, qgb, etc.), we obtain from (4.19)-(4.20),

158

KUYUCAK

AND

MORRISON

Here the first two terms are due to F-spin breaking in the phase angle “y” and tb mean fields xP, respectively, and the last term is due to g-bosons. Setting gb = 0 ii Eqs. (4.19), (4.21), one obtains the sd-model results. As in the case of the energie and E2 transitions, the above inband Ml matrix elements are complete to leadinl order. Calculation of the interband Ml transitions are complicated by the fact that con tributions coming from the AK= 1 band mixing [36] are of the same order (in N as the direct term. (This is to be contrasted with the interband E2 transitions when the band mixing effects come into play only in the next to leading order [37]. Since our aim here is to derive simple expressions for transitions among the band generated by the mean fields, we postpone a complete calculation including the band mixing effects to a future publication. Calculation of the direct part is stil involved because the mixing terms in the intrinsic state (4.8) also contribute to the leading order matrix element. Further, these terms involve two phonon operator, and hence the simple scaling property described above for the single-phonon band; has to be modified. Rather than going through lengthy calculations and describing a new scaling rule, we simply give the IBM-1 and the corresponding IBM-2 expres sions which should be enough to illustrate the point. The p + g and y + g M transitions in IBM-l [35] can be combined in to a single expression given by

(LXII zvf1) II&> = -JqGc

[l +&J/2

x c gl[i(21+

L’;;-;1’2

(LO 11 IL’ )

1)/3] P1’2 [[cl’- K)]“2 x!x,~ -F

f1’2~I~,ls

/

(4.22

IS

In IBM-2, the above matrix element becomes


x [r(i-K)1”2 C,/~COs

YK&lXnlXalK

+

msin

YK&lXvlXvlKl

{

--

6,

2b,s& +b;&/N,INcos

il’*

bl, [

(

N” NE Yg gn/xn/xn/1s + -$- g”lx”/x”~ls

YK gniX,&nlls

+

msin

YK

> gv/X,/Xvlls

)I> 3

(4.23

l/N

EXPANSION

IBM-II

IN

159

where b;, is defined in (4.10). Note that Eqs. (4.22b(4.23) vanish if the Ml operator is proportional the angular momentum operator. The relative g-boson and F-spin breaking contributions to (4.23) can be resolved by following a similar procedure to the inband Ml transitions. Thus, we obtain in the sdg-model

x rlJmmL)++CNp i J18

Xp4K

i

P -

2

xp41s

g, b,,,-F bp,s IS ( 2

IS

where the various terms are identified as in Eq. (4.21). As before, the &model result follows from Eq. (4.24) by setting gb = 0. It is clear from Eqs. (4.18k(4.24) that in the sd-model, F-spin breaking in the hamiltonian is the only way to generate Ml transitions with one-body operators, In contrast, in the &g-model, it is possible to generate Ml transitions even in the limit of F-spin symmetry if one assumes a defect between the g-factors of the d- and g-bosons; i.e., g,(d) # g,(g). Effects of such a defect on the Ml transitions are discussed in the next subsection. A final Ml observable of topical interest is the g-factors for various states. Since the diagonal Ml matrix elements can simply be obtained from the corresponding IBM-l expressions [15, 351 through the scaling described above, they are not repeated here. 4.4. Discussion of Electromagnetic Transitions The E2 transitions among the symmetric states are affected little by F-spin breaking in the hamiltonian and such a study is not pursued. Instead, we consider the effect of F-spin breaking in the E2 operator due to having unequal proton and neutron effective charges as indicated by systematic studies [38-391. Figure 11 shows the B(E2, 2, + 0,) values in an F-spin multiplet in the sd-model for the cases of equal effective charges (a), and e, = 0.15 eb, e, = 0.11 eb (b) which are extracted from the systematics of rare-earth nuclei [39]. The effect of having unequal effective charges is seen to increase the upslope of the lines. Experimentally, the B(E2) values in an F-spin multiplet go through a maximum at N, = N,, and hence, a description of the B(E2) values with constant parameters does not seem to be tenable. The microscopic and symmetric forms in Fig. 11 almost overlap, which indicates the insensitivity of the E2 properties on the form of the hamiltonian. The Ml transitions considered above were not all complete, therefore, we first discuss the accuracy of the results. Comparison with the numerical calculations in the sd-model [40] shows that the y + y Ml transitions are reproduced within the 595/195/l-11

160

KUYUCAK

AND

80. N

MORRISON

80. __--,

al

FIG. 11. B(E2, 2, -+ 0,) values in an F-spin multiplet in the sd-model. The effective charges used i. the E2 operator are e, = e, = 0.13 eb (a), and e, = 0.15 eb, e, = 0.11 eb (b). The other E2 parameters i: the consistent Q formalism are x = 0.5 and x0 = 0.6. The microscopic (full line) and symmetric (dashel line) forms almost overlap.

expected l/N error level. Equation (4.24), on the other hand, accounts for only ha1 of the y -+ g Ml matrix element, the rest presumably coming from the AK= mixing of the K= l,& band. Calculations are done using the standard bosom g-factors g, = 1 pLNand g, = 0. Figure 12 shows the behavior of the 3, -+ 2, (a) ant 3, + 2, (b) Ml matrix elements in an F-spin multiplet in the sd-model. As expected the Ml amplitudes are roughly proportional to the amount of F-spin breaking it

2

4

6

8

10

12

Nn FIG. 12. the &model. microscopic

Behavior of the 3, + 2, (a) and 3, - 2, (b) Ml matrix elements in an F-spin multiplet it The boson g-factors are g, = 1 pN and g, = 0. The full and dashed lines correspond to thl and symmetric forms, respectively.

l/N

EXPANSION

IN

IBM-II

161

the y-band (cf. Fig. 2). In contrast to the E2 transitions which do not distinguish between the microscopic and the symmetric forms, the Ml transitions are very sensitive to the choice of the hamiltonian. Notably, the microscopic form varies rapidly with N,, vanishing at N, = 8, whereas the symmetric form shows a smooth variation and has a maximum at N, = 8. Experimental information on Ml transitions is rather meager to reach a definite conclusion in this respect; nevertheless, the available data seem to favor the symmetric form. As a final example, we discuss the effect of g-bosons on the Ml transitions. In comparison to the s&model, band mixing effects in the &g-model are smaller for Ml transitions. Thus, the AK= 1 mixing contribution to the y -+ g Ml matrix element is suppressed, and the results given in Eqs. (4.23)-(4.24) are more reliable. Here, one further needs to specify gb. Measurement of the excitation strengths to the 1 + states indicated in Eq. (4.16) would uniquely determine gb. In the absence of such information, we use the values obtained from a lit to the Ml transitions g; = g: = -0.2 [23]. Figure 13 shows the 3, -+ 2, (a) and 3, + 2, (b) Ml matrix elements in the sdg-model as a function of xzv for fixed x10=0.6 and xjv=O. A defect in the g-factors of magnitude gb = -0.2 pLNhas a marginal effect on the y + y transitions, but it could play an important role in the case of the y --f g transitions (see Ref. [35] for a plot of the matrix elements in the sd-IBM-l). Notice also that, in the symmetric case (dashed line), the two transitions have opposite behaviors; the y -+ y transition increases in magnitude with x2” whereas the y + g transition decreases. This extra degree of freedom, which is not available in the s&model, would be useful in tuning the relative amplitudes of the two Ml transitions.

0,.

I,.

”

”

”

I

”

,

FIG. 13. Same as Fig. 12 but for xzU in the s&-model. and xsv = 0. The lines with g’ = -0.2 show the contribution of magnitude gb = g,(g)g,(d) = -0.2 I”~.

The other quadrupole parameters are x,” = 0.6 of g-bosom due to a defect in the g-factors

162

KUYUCAKANDMORRISON

5. CONCLUSIONS We have extended the l/N expansion technique to study the neutron-protol degree of freedom in the interacting boson model, Using a standard IBM-: hamiltonian, we have derived general algebraic expressions for energies and elec tromagnetic transition rates. The formalism was then used in a systematic study a F-spin vector parameters. Our main results are (i) F-spin is established as a gooc quantum number, (ii) a description of F-spin multiplets with constant parameter is found to be in conflict with the energy and B(E2) systematics, (iii) an experimen tal study of Ml transitions in an F-spin multiplet would be very useful in regarc to the form of the hamiltonian and also in determining the relative g-boson am F-spin breaking contributions. With its analytic formulation, the l/N expansion method has all the appealinl features of the dynamical symmetries, yet it is not restricted to a special paramete set. Another advantage of the method is that it can easily handle arbitrary kinds o bosons and interactions. This is important especially for the IBM-2 calculation where a numerical diagonalization of the hamiltonian in the &g-model space i, prohibitively difficult. Thus, the l/N calculations would be helpful in the search fo new collective bands, for example the mixed symmetry K= 3 +-band [41]. Comparisons with numerical calculations show that the relative error involved i; roughly l/N. In general, the inband quantities are more accurate than the interbanc ones which are more complicated due to the orthogonality and band mixing effects The error level slightly increases with F-spin breaking reflecting the AK # 0 mixinl effects which are not accounted by the mean field approach. In comparison to the &model, the ljlv results in the &g-model are in better agreement with the numeri cal calculations, indicating a better and earlier established band structure. Error; are least therefore in the area of physical interest. Overall, the l/N expansior appears to be an ideal tool for large scale analysis of deformed nuclei.

APPENDIX

A: F-SPIN FORMALISM

The neutron-proton degree of freedom is most conveniently dealt with by intro ducing a formalism called F-spin [l]. It is similar to isospin (has the same SU(2 structure), but is applied to a system of bosons with F0 = 4 for proton bosons am F, = - $ for neutron bosons. For a system of N= N, + N, bosons, F-spin takes the values F,,, = W, + N,P, F,,,,, - 1, .... F,, = IN, - N,l/2. The states with F= F,, are totally symmetric under interchange of neutron and proton bosons, ant correspond to the low-lying IBM-l states. States with F-c F,,,,, are of mixed sym, metry character. So far, only the members with F= F,,, - 1 have been identifiec experimentally [2]. Denoting the quantum numbers (Im) by i, the generators of the SU,(2) algebra are given by

l/N

EXPANSION

IN

F. = ; .& (biibni - blibJ. I

Fp = (F+)+,

F+ = 1 b:ib,, 3 I

163

IBM-II

For a general boson system, a special choice for the Majorana M=

1 (bhibzj- biibLj)(b,ib,-

(A.11

operator is

b,ib,),

(A.21

i>j which is related to the F-spin Casimir operator through l)-F.F.

M=F,,,,,(F,,,+

(A.3)

Using Eq. (A. 1 ), F . F can be explicitly evaluated F.F=F+Fp+F,(F,,-l), =I

ij

bLibzjbnjbyi+C

I

bLib,i+

Fo(Fo-

1).

(A.4)

Substituting (A.4) in (A.3), we obtain another expression for M which is more suitable for a direct evaluation of matrix elements, M = N, N, - 1 b;ib;jb,b,i. q

(A.5)

In general, IBM-2 hamiltonians are not F-spin symmetric, and intrinsic states generated by such hamiltonians contain admixtures of states with different F-spins. An important question, therefore, is the F-spin purity of various bands in Eqs. (2.8)-(2.10). Since F-spin has the same W(2) structure as angular momentum algebra, using similar techniques, one can derive simple expressions for F-spin projection. Denoting the F-spin projection operator by PLM, we have for the ground-band, Eq. (2.8 ),

x (-I tb,P @,P e-i&(bt,)“~ (b;)NuI-), where M= F. and /I denotes the rotation F-spin rotated boson operators as

(A.61

angle in the F-spin space. Defining the

bLR = e-r~Fvb~ei~Fy= 1 xp, dA$ b:,, al and using differential techniques for boson commutators, (A.6) can be evaluated as

(A.7) the matrix element in

(-I (b,P (b,P U&J”” tbh?‘v I-> =~(-)fl(~)(~)(cos~)N~*“(xz.xVsin~)2’.

(A.8)

164

KUYUCAK

AND

MORRISON

Substituting (A.8) and the explicit form of the d-matrix integral is standard and given by the /?-function [42]. f = F,,,,, -F, we obtain (&I

CA4

14,)

=

;Iy;l1

1 n

C-1”

(x,

in (A.6), the resultin Simplifying and usin,

. X,)2n

xc C-1”(“;;‘)( m

Nv

N7x (

n

Nv;f)/(

)(

n

)

I;;).

(A.9

Using the combinatorial identities (we were unable to find them in the literature they may be of interest to mathematicians),

(A.10

; (-Jm-(;)(;)l(f)= -N,N,/(;), and others that can be derived from them, the m-summation an expansion in f and n,

can be evaluated ac

; (-)m(Nn;f)(N,f)/(;;;) = (-)” (;)I(

ir)

,+(-W-W-2) 2(N,-

Substituting

l)(N,-

“’

1)

1 .

(A.111

(A.1 1) in (A.9) and rearranging terms, we obtain, after some algebra,

( Nn;f)( “;‘)/( xc” (x*.Xd2”

;j.

64.12)

In the limit of good F-spin, x, .x, = 1, and Eq. (A.12) vanishes unless f = 0 (F = F,,,), in which case it gives 1. A calculation along the same lines yields a similar expression for the F-spin pro-

l/N

EXPANSION

IN

165

IBM-II

jection of the K-bands, Eq. (2.9). One only needs to make the following replacement in Eq. (A.9), (x, . XJ2n -+ (x,. x,)ZX

l

>I

x;x,+2n

sin y cos y Jm

x,K’xvK

Since performing the m-sum leads to a more complicated final expression in this case, it is not pursued. For the KM-bands, Eq. (2.10), it is easily seen from the symmetry of the states that one needs to substitute cos y + sin y and sin y + -cos y in Eq. (A.13). In the F-spin limit, cos’ y = NJN, sin* y = NV/N, and (A.13) collapses to 1, giving the same results as the ground-band. For the KM-bands, Eq. (A.13) instead reduces to (1 - nN/N,N,), which summed over m and n in Eqs. (A.9)-(A.11) gives 1 for f = 1 (F= F,,, - 1) and vanishes otherwise. In the above formulation, we have not considered angular momentum projection because effects of rotation appear only at the O( l/N*) level. From Eq. (A.13) it is seen that F-spin projection of different bands vary at the 0(1/N) level. Thus the results given here indicate-a small cross-band and a much smaller inband variation in F-spin projection of states, consistent with the earlier numerical calculations ~71. REFERENCES 1. A. ARIMA, T. OTSUKA, F. IACHELLO, AND I. TALMI, Phys. Let?. B 66 (1977), 205; T. OTSUKA, A. ARIMA, F. IACHELLO, AND 1. TALMI, Phys. Lett. B 76 (1978), 139. 2. D. BOHLE et al., Phys. Letf. B 137 (1984), 27; 148 (1984), 260; U. E. P. BERG et al., Phys. Left. B 149 (1984), 59. 3. T. OTSUKA, Computer code NPBOS, University of Tokyo, 1978; I. MORRISON, Computer code BOZO, University of Melbourne, 1983. 4. I. MORRISON, Phys. Rev. C 23 (1981), 1831. 5. A. E. L. DIEPERINK AND R. BUKER, Phys. Left. B 116 (1982). 77; A. E. L. DIEPERINK AND I. TALMI, Phys. La. B 131 (1983), 1. 6. 0. SCHOLTEN, K. HEYDE, P. VAN ISACKER, J. JOLIE, J. MOREAU, M. WAROQUIER, AND J. SAU, Nucl. Phys. A 438 (1985), 41. 7. P. VAN ISACKER, K. HEYDE, J. JOLIE, AND A. SEVRIN, Ann. Phys. (N.Y) 171 (1986), 253. 8. N. YOSHINAGA, A. ARIMA, AND T. OTSUKA, Phys. Lett. B 143 (1984), 5. 9. P. VAN ISACKER, K. HEYDE, M. WAROQUIER, AND G. WENES, Nucl. Phys. A 380 (1982) 383. 10. H. C. WV, Phys. Left. B 110 (1982), 1. 11. Y. AKIYAMA, Nucl. Phys. A 433 (1985), 369. 12. Y. AKIYAMA, P. VON BRENTANO, AND A. GELBERG, Z. Phys. A 326 (1987), 517. 13. N. YOSHINAGA, Y. AKIYAMA, AND A. ARIMA, Phys. Rev. Let?. 56 (1986), 1116. 14. S. KUYUCAK AND I. MORRISON, Phys. Rev. Let?. 58 (1987), 315; Phys. Rev. C 36 (1987), 774. 15. S. KUYUCAK AND I. MORRISON, Ann. Phys. (N.Y.) 181 (1988), 79. 16. A. E. L. DEEPERINK AND 0. SCHOLTEN, Nucl. Phys. A 346 (1980), 125; J. N. GIN~CCHIO AND M. W. KIRSON, Nucl. Phys. A 350 (1980), 31. 17. H. C. WV, A. E. L. DIEPERINK, AND S. PITTEL, Phys. Rev. C 34 (1986). 703; N. YOSHINAGA, Nucl. Phys. A 456 (1986) 21.

166

KUYUCAK

AND

MORRISON

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