M1 transitions in the (sdg) boson model

Volume 202, number 2

PHYSICS LETTERS B

3 March 1988

MI TRANSITIONS IN THE (sdg) BOSON M O D E L S. K U Y U C A K School of Physics, University of Melbourne, Parkville, Victoria 3052, Australia t and Institut ~ r Theoretische Physik, Universitdt Tiibingen, D-7400 Tiibingen, Fed. Rep. Germany

and I. M O R R I S O N School qfPhysics, University of Melbourne, Parkville, Victoria 3052, Australia

Received 4 November 1987; revised manuscript received 17 December 1987

Using the 1/N expansion technique we derive expressions for fl-~g, ?-~g and y ~ ? M 1 transitions in a general boson model. The M1 matrix elements in the sdg-boson model are similar in form to those in the neutron-proton IBM. Comparisons are made to some selected M 1 data exhibiting collective character.

The discovery of a strongly excited K" = 1 + band around Ex = 3 MeV [ 1 ] has led to a renewed interest in the description of M1 properties. In the neut r o n - p r o t o n interacting boson model (IBM-2), this band is assigned to mixed-symmetry states with Fspin F = F m a × - - 1. Breaking the n e u t r o n - p r o t o n symmetry o f the hamiltonian gives rise to F-spin mixing o f states. In particular it leads to small admixtures o f states with ~ ' m a x - - 1 in the low-lying collective states, which generates M1 transitions a m o n g the bandmembers via a one-body M1 operator. In the original IBM- 1, a two-body operator with three parameters is needed to describe the M1 transitions, and this is not easily systematized [2]. Thus, the one-body M1 transition operator in IBM-2 with a single parameter is quite welcome and such studies have been made by several groups [ 3-10 ]. A c o m m o n result of these calculations is that in order to reproduce the experimental data in the deformed region the n - p asymmetry in the Z parameter o f the quadrupole operator is required to be about AX= [Xn-Xp[ -~ 1. (There seems to be a sign ambiguity in AX, cf. refs. [7,9] and refs. [ 8,10].) This result is hard to reconcile with the mi-

Permanent address. 0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

croscopic calculations, and alternative explanations for M 1 transitions would be useful. There is growing evidence that the simple sd-boson model is not adequate to describe deformed nuclei and has to be extended to include g-bosons [ 11-13 ]. However, calculational complexity o f the sdg-boson model (32 hamiltonian parameters and much larger basis space) has prohibited its widespread use so far. In fact, most o f the applications are limited to the S U ( 3 ) limit, although phenomenological analyses indicate that this limit is far from being realized [ 14 ]. To overcome the above problems, we recently introduced an approximate algebraic method for solving general IBM problems [ 15]. The method is based on angular-momentum projection before variation in the intrinsic state formalism and leds to a 1/Nexpansion for quantities of physical interest. N corresponds to the boson number and the expansion coefficients are simple algebraic functions o f the hamiltonian parameters. Thus the method allows correlation o f experimental data directly in terms o f hamiltonian parameters which is very useful for systematic analysis. Another attractive feature o f the method is that higher spin bosons (e.g. g, i .... ) are as easily handled, and effects o f the g-bosons can be studied nonperturbatively. 169

Volume 202, number 2

PHYSICS LETTERSB

The details of the method and its application to excitation energies, quadrupole properties and g-factors were given in ref. [ 15 ]. In this work, we will give leading order expressions for fl--, g, ?--. g and y ~ y M 1 transitions and compare the results (in the case of the sdg-model) with those of the IBM-2 and also with experiment. Introducing the boson creation (b~,,) and annihilation (bz,,,) operators, where I= 0, 2, 4,... correspond to s, d, g,...-bosons, the intrinsic boson operators b,~,, are given by

bt,,,= Z x, mb~,n, E (Xlm)2=X'n'X" = 1 I

l

(1)

The ground-, 13- and ~,-band intrinsic states can then be written as

10~) = ((b;) u

y= ~ Z fx~, , b= Z ix,x~, I

l

b,-- E ~/lx, x,l , b z = E ~/{(F-Z)x,x,2. l

(4)

/

Choosing g1=[~/3, T(M1) becomes the angularmomentum operator, and the matrix element in eq. (3) vanishes as it should. In the sdg-model, the M 1 operator can be rewritten as T(M1)=gL+g'[gt~] (l) where g=gz/~F(-6 and g' = -,,/~gz +g4. Microscopically, the g-boson is less spin saturated than the d-boson and is expected to have a different g-factor. Thus g' is in general nonzero and the second term in T(M 1 ) can generate M 1 transitions. From eq. (3), we obtain

_ g, £,/-£ , / ~ 2 yx4 [,,/$6x'4- ( b/ bl ) X,i ]

[ - >,

A(E2/M1)=C~N ( 2 L - 1 ) ( 2 L + 3 )

+ (b~) N-2 ~ ~,'~b*,btm +...11 - >,

'

(6)

/

m#O

I~.~) = ( (b*) ~-' b~ X

(5)

for theC/~g M1 transition. Since most of the M1 data are in the form of E2/M 1 mixing ratios, we also give A(E2/M1)

[q)~> = ( ( b~)N-l b~t

2 bm t b~_,n+... ~m

m~0,2

+ (K-~-K))I->.

(2)

For a given hamiltonian H, the structure coefficient X~m are determined by varying ( H ) after angularmomentum projection which leads to an eigenvalue equation. Matrix elements of a general one-body M 1 operator T(M1)=ZlgI[b~] (I) between the states in eq. (2) can be evaluated using projection techniques [ 15]. For t h e / ~ g M1 transition, we find

i,,/Z

< ~ , L]I T(M1 ) [10g,L > = ~

× [ ]-xtx~- (b/bt)~l-xlXll 170

where £ = L ( L + 1 ), a n d y and b are defined as

(~a, L[I T(M1)II0g, L>

+(b~)~-~ ~ ~mb,nb-,, t t +...) ,n~o

+(b~) N-2

3 March 1988

] ,

g,

which utilizes the E2 matrix elements given in ref. [ 15 ]. Here the single parameter Cp contains all the structure dependence through Xlm as well as the effective charge and g-factor. The mass and L-dependence, which are the main interest of this study, are clearly separated in eq. (6). A similar calculation for the y ~ g M1 transition gives <,~, Z' IIT(M1 )[10g, L>

C ~ g, = 2y,f~ 2 L,/-~ x[~/~]--2)xzxlz-(b2/b,)~llxtxzll

(7)

In the sdg-model, the above expression reduces to < ~ , L' ILT(M1 )II0g, L> =-g'/~E-

(3)

.

2

X4 < L 0 1 IIL' 1> 2--y

X [~T8x4zv- (bJbl)x41l ,

(8)

Volume 202, number 2

PHYSICS LETTERS B

Table 1 Reduced mixing ratios A(E2/M1 ) for tSnGd. Transition

2:.--*2g 3:.--,2g 3:.~ 4g 4: -,4g 5..~4g, 5~.--,6~ 6;,-~6~ 7:.--*6~ 7:.--,8~ 2~--,2~ 4/~ 4~ 6/~--6~ 8/~8~ lOis-*lOs

A(e

In the sdg-model, eq. (10) gives for the 7 - ' 7 M1 transitions ( ~ , , L + 1[/T(M1 )][0~,, L )

b/,tt N )

experiment "~

theory

- 13.3 _+0.7 -- 8.9 _+0.5 -- 8.9 _+0.3 -- 5.5 _+0.5 -- 4.9 +14 2.9 - 11.6_+~~ - 4.2+~5.8 - 3.0_+°157 -5.721:8 14.4+~:6 5.3 _+2.3 2.9_+0.3 +0.8 2.3_o6 2.4 +_o.7L°

- 13.3 b) -- 12.4 -- 9.0 -- 6.9 --7.1 -5.9 -4.7 - 5.0 -4.4 14.4 °) 7.5 5.1 3.9 3.1

, / ' ( L + 1 ) 2 - 4 ) 1/2

× ( - x F y + ~x~2).

g(K, L) =g +g'x/~3[x]/y+(K2/[,)(--x]/y+~oX]K)].

which gives the following E2/M 1 mixing ratios: I/2

1/2

where C~, is a parameter similar to C~. In table 1, eqs. (6) and (9) are compared with the experimental values in 154Gd which has the most extensive A-data. The data agree qualitatively with the predicted I/L fall off which indicates the collective nature of these transitions. We note that in the sdIBM with a two-body M1 operator, A(E2/M1 ) has a similar L-dependence to eqs. (6) and (9) but differs in mass dependence (i.e., independent o f N) [ 16]. The available A-data seem to indicate an increase in A with increasing N. Next, we consider the intraband M1 transitions. Here it is possible to give a single expression for an arbitrary (single phonon excited) K-band (0K, L + 11[T(M1 )[[OK, L5

× [ - ( i/2y)x] +X~K].

(11)

Also o f interest here are the g-factors for K-bands in the sdg-model

.I Ref. [2]. ~1Fitted.

=K( (L+l)2-K2)'/2Z;[ J

3 March 1988

~g' (1 O)

(12)

For the ground and fl-bands K = 0, and the last term in eq. (12) vanishes. From fits to other M I properties (e.g. g-factor variations in the ground band, E2/M1 mixing ratios) g' is expected to be negative. Since the coefficient of the K 2 term is always positive, eq. (12) predicts a decrease in the g-factors with increasing K. Existing data for g(2y) confirms such a reduction [ 17 ]. It would be interesting to see whether this prediction also holds for the higher K-bands. Another general result that follows from eqs. (8) and (11) is that interband M I matrix elements are suppressed by a factor of 1/x/N compared to inband transitions. (In fact, the above conclusion is quite general and applies to any multipole operator [ 15 ]. ) Again empirical y ~ ? and y--,g M1 transitions in deformed nuclei are in support o f the above result [ 9]. Since the majority of calculations on M1 transitions were done in the IBM-2 framework, it will be instructive to compare the two approaches. In a recent IBM-2 calculation, analytic expressions were obtained for 7--, 7 and y - , g M 1 transitions using perturbation theory [ 9]. The matrix elements given in ref. [ 9 ] have similar mass dependence and identical L-dependence as in eqs. (8), (11 ) and (12). Thus it is very difficult to differentiate the two models on the basis o f nuclear systematics as long as parameters (g' in the sdg-model, AZ in the IBM-2) are fitted to the M 1 data. In order to assess the relative contributions, it would be useful to have an independent property to fix the M1 parameters. In the case of the sdg-model, measurement of the B(M1; 0 + ~ 1 +) transition strength to the symmetric K = 1 + band (F=Fmax) would provide such a property [ 15 ]

B(M1;O+-,l+)=(3/4zOg'2(N/y)(x2x4) 2 .

(13) 171

Volume 202, number 2

PHYSICS LETTERS B

Since the structure coefficients xz,, are d e t e r m i n e d from a fit to the energies a n d q u a d r u p o l e properties, eq. (13) directly determines g' once the B ( M 1 ) value is measured. We stress the i m p o r t a n c e o f this B ( M 1 ) m e a s u r e m e n t because it will turn all the M 1 expressions given in this work (plus the g-factor v a r i a t i o n s in the g.s.b. [ 12]) into p r e d i c t i o n s without any free parameters, thus allowing an u n a m b i g u o u s determination o f the c o n t r i b u t i o n o f the g-boson in M1 properties. Study of a q u a d r u p o l e h a m i l t o n i a n reveals that the splitting between the K = 2 + - 0 + and K = 3 + - I + bands are similar [ 15 ] which puts the excitation energy o f the s y m m e t r i c 1 + b a n d a r o u n d 2 M e V in the d e f o r m e d rare-earth region. Collective K = 1 + b a n d s have already been observed at the right energy in some o f these nuclei (e.g. El = 2 . 1 3 3 M e V in 168Er). N o t e that the m i x e d - s y m m e t r y 1 + states lie a r o u n d 3 M e V a n d hence are well separated from the s y m m e t r i c ones. Blaming the y ~ y M 1 transitions on g-bosons (i.e. ignoring the c o n t r i b u t i o n due to F - s p i n mixing) it is possible to give an u p p e r b o u n d for the M1 strength. F r o m eqs. (10) a n d (13) we o b t a i n B ( M 1 ;

3 March 1988

T(MIJ x 10

o,s ~g

o,c J

-05

o,t~

0,6

O,B

q

f~g

Fig. 1. Plot of the reduced (i.e. g', n and L dependence factored out) fl~g, 7~g and ~ ? MI matrix elements as a function of the quadrupole parameters, q24 is fixed at the SU(3) value as required by energy fits, and q22 and q44 are varied simultaneously from 0 to the SU(3) value (normalized to 1). From energy systematics in the rare-earth region, qe2 and q44 have values around (20-40)% of the SU(3) value.

0~--,1+) ~ 1 z~. D u e to the u n c e r t a i n t y in the value o f g ' , we refrain from doing detailed fits to M 1 p r o p e r t i e s in individual nuclei. However, it should be fairly clear from the a b o v e discussion that using the sdg-model one can o b t a i n fits o f similar quality to IBM-2 calculations [3-10]. Finally, we discuss the d e p e n d e n c e o f the structure factors on the h a m i l t o n i a n p a r a m e t e r s in the case o f a q u a d r u p o l e interaction in the sdg-model. The fl ~ g and ~ ~ g M 1 m a t r i x elements show a s m o o t h behavior a n d are relatively stable across the p a r a m e t e r range, whereas the ~ - ~ M1 m a t r i x e l e m e n t is quite sensitive to details in H, having " d i p s " for certain values o f p a r a m e t e r s (fig. 1 ). This m a y be relevant in the analysis o f some nuclei (e.g., 168Er) a n d also explain the discrepancy in a recent calculation o f 7 ~ 7 M1 transitions in the sdg-model [ 18]. In conclusion, we have presented analytic expressions for M 1 transitions in the sdg-boson model which will be useful in the systematic analysis o f M 1 data. The leading o r d e r results, in general, should involve errors o f order 1/N. However, c o m p a r i s o n with exact d i a g o n a l i z a t i o n results show that the errors i n v o l v e d 172

are consistently smaller than 1/N [ 19 ]. Thus the 1/N expansion m e t h o d offers a simple yet reliable tool for the analysis o f d e f o r m e d nuclei.

References [ 1] D. Bohle et al., Phys. Lett. B 137 (1984) 27; B 148 (1984) 260; U.E.P. Berg et al., Phys. Lett. B 149 (1984) 59. [2] P.E. Lipas, P. Toivonen and E. Hammar6n, Nucl. Phys. A 469 (1987) 348. [3] P.O. Lipas and K. Helim~iki,Phys. Lett. B 165 (1985) 244. [4] D.D. Warner, Phys. Rev. C 34 (1986) 1131. [ 5 ] H. Halter, P, von Brentano and A. Gelberg, Phys. Rev. C 34 (1986) 1472. [6] H.C, Wu, A.E.L. Dieperink and O. Scholten, Phys. Lett. B 187 (1987) 205. [ 7 ] H. Hatter, P. von Brentano, A. Gelberg and T. Otsuka, Phys. Lett, B 188 (1987) 295. [8] W. Gelletly et al., Phys. Lett. B 191 (1987) 240. [ 9 ] A.E.L. Dieperink, O. Scholten and D.D. Warner, Nuel. Phys. A 469 (1987) 173. [ 10] P. Van Isacker et al., preprint Jyv~skyl~i(1987), [ 11 ] N. Yoshinaga, Y. Akiyama and A. Arima, Phys. Rev. Lett. 56 (1986) 1116.

Volume 202, number 2

PHYSICS LETTERS B

[ 12] S. Kuyucak and I. Morrison, Phys. Rev. Lett. 58 (1987) 315. [ 13 ] Y. Akiyama, P. von Brentano and A. Gelberg, Z. Phys. A 326 (1987) 517. [14] R.F. Casten and D.D. Warner, Prog. Part. Nucl. Phys. 9 (1983) 311. [15] S. Kuyucak and I. Morrison, Phys. Rev. C 36 (1987) 774; Ann. Phys. (NY) 181 (1988), to be published. [16] D.D. Warner, Phys. Rev. Lett. 47 (1981) 1819.

3 March 1988

[ 17 ] A.E. Stuchbery et al., Z. Phys. A 320 (1985) 669; C.E. Doran et al., Z. Phys. A 325 (1986) 285. [ 18 ] Y. Akiyama, to be published. [ 19] S. Kuyucak and I. Morrison, in: Proc. Intern. Conf. on Nuclear structure, ed. H. Bolotin (Melbourne, 1987).

173