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Mass-number dependence of the moment of inertia and IBM
Volume 241, number 4
MASS-NUMBER
PHYSICS LETTERS B
DEPENDENCE
OF THE MOMENT
24 May 1990
OF INERTIA AND IBM
N. Y O S H I D A Department of Information Science, Faculty of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113, Japan H. SAGAWA, T. O T S U K A and A. A R I M A Department of Physics, Faculty of Science, University of Tokyo, 7-3-I Hongo, Bunkyo-ku, Tokyo 113, Japan Received 19 December 1989; revised manuscript received 2 March 1990
The m a s s - n u m b e r dependence of the m o m e n t of inertia obtained from the energy difference between the ground state and the first 2 + state is studied in relation with the boson number in the S U ( 3 ) limit of the interacting boson model 1 (1BM-1). The analytic formula in the limit indicates that the pairing correlation between nucleons is directly related to the m o m e n t of inertia in the IBM. It is shown that the kinks of the m o m e n t of inertia of the Yb and Hfisotopes coincide exactly with the m a x i m u m boson numbers of each kind of isotopes.
The m o m e n t of inertia is a well-known fundamental observable to characterize deformed nuclei [ 1,2 ]. The pairing correlation plays a crucial role to decrease the m o m e n t of inertia from the rigid-rotor value by a factor 2-3 [ 1 ]. The decrease due to the pairing correlation, however, has never been considered systematically in terms o f the interacting boson model [ 3 ]. In this paper we discuss how the pairing correlation is connected with the m o m e n t o f inertia in the S U ( 3 ) limit of the IBM, when the hamiltonian is solved in an analytic way, and how one can see its effects on the mass-number dependence of the moment of inertia determined from the energy difference between the ground state and the first 2 + state. We take the IBM-1 for simplicity. The model hamiltonian is written as
H=&ns +eona +xQ'Q+tc'L'L,
(1)
where ns (rid) is the s-boson (d-boson) number operator; Q is the quadrupole operator in the S U ( 3 ) limit:
Q= d's+ s , a - },,/5[d*a ] (~, ,
(2)
and L = x / / ~ [d*d] (1) is the angular m o m e n t u m operator. The expectation values o f H for states in the ground
state band, i.e., states of the SU (3) representation (2, #) = (2N, 0), are given by N(2N+I)
E=& 3 ( 2 N - 1 ) + (x'-~x)L(L+
, 4N(N-1) -~cN(2N+3) ±~d 3~-~----1) 1 ) + (e d --•s)
1 L(L+I) 6 ( 2 N - 1)
=Eo +AL(L+ 1),
(3)
where Eo = es A=I¢'
N{2N+ 1) 3(2N- 1) 3 ~
-~:~-
(& - ~ )
4N(N-1) 1 6(2N- 1) '
-~¢N(2N+3)
(4)
N is the boson number, and L is the angular momentum. In the following, we assume that the neutron number dependence of the interaction strengths is negligible. The excitation energy of 2 + is o f the form 6 1 E ( 2 + ) = ~ +8~ 2 N - 1 '
{5)
where 6
2T=6(K'-]K),
0370-2693/90/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland )
(6)
459
Volume 241, number 4
PHYSICS LETTERS B
and e~ = e d - e , .
(7)
Eq. (5) can be expressed in terms of the moment of inertia of 2+"
24 May 1990
Table 1 The parameters 6/2c~ and e~ determined from the experimental excitation energies. The parameters of Nd and Os isotopes cannot be determined because only one data point is available for each element in fig. 1. Element
N-½ 2Icrf = 2c~ N - ½+ a e a / 6 "
(8)
6/2a (MeV)
e~(dir) a) (MeV)
0.024_+0.007 -0.005_+0.015 -0.003_+0.018 0.000+_0.018 0.000_+0.013 0.052+_0.007 0.061 _+0.021
1.22_+0.15 2.14_+0.38 2.47_+0.52 2.49_+0.50 2.46_+0.38 1.16_+0.19 1.20_+0.52 -
NO
We will now use the above equations to analyse nuclei in the rare-earth region. Since the present model is valid only for well-deformed nuclei, we choose those nuclei with ratios of the excitation energy of the 4 + state to that of the 2 + state, [i.e., E(4 + ) / E ( 2 i ~) ] larger than 3.2. The boson number is counted as the number of particle (or hole) pairs with respect to the nearest magic number 50, 82 or 126 [4]. Keeping oe and e} as constants for all the isotopes of each element, we determine that value by eq. (5), using the plot in fig. 1. The obtained values are listed as 6/2c~ and ea (dir) in table 1. The statistical errors for the values ea (dir) are reasonably small; (10-20)% except W isotopes. The values of these two parameters for the Nd and Os isotopes cannot be determined, because only one data point is available for each of these isotopes. The calculated moments of inertia are shown in fig. 2 together with the experimental data [ 5,6 ].
e~(cor) b) (MeV)
-
Sm Gd Dy Er Yb Hf W Os
-
0.71 1.05 1.15 1.16 1.15 0.69 0.70 -
") Values obtained directly from fits [eq. (5) ]. b) The correction due to the coupling to the fl-band is taken into account (eq. (10) with AE= 1 MeV).
/ / / /
/
J
J
j
.-~rigid 150
I
IO0 °
\
,,.-
o
o
>O%.oo
° JS/.-'-'>,-,
¢N
--
;
/
- o., o I
• Nd o D y • Sm - E r ~ Gd o y b
,;o )
///
0[~_. /
=Sm-Er
.W
~ Gd o Y b v Os
0.05
°
,"*"*'--L. ,..,
5C "Hf * W "Os
MASS
NUMBER
Fig. 2. Mass-number dependence of the moment of inertia. The solid lines are calculated by eq. (8) using the values 2c~ and e~ determined in fig. 1 together with the effective boson number. The rigid rotor values [ 1 ] are shown by a dashed line.
1/(2N-1)
Fig. 1. The excitation energies of 2t+ states as a function of 1/ ( 2 N - 1 ). The line for each element is drawn to determine the values 2o~ and e~ in eq. (5). The lines for the Yb and Er isotopes cannot be distinguished in the present scale of the figure. The experimental data are taken from refs. [ 5,6 ].
460
The calculation reproduces the general trends of the mass-number dependence of the moment of inertia. As a function of the neutron number N,, the calculated moment of inertia increases and becomes maximal at N , = 104, which is the middle of the major
Volume 241, number 4
PHYSICS LETTERS B
shell between 82 and 126. After AT,= 104, the moment of inertia decreases since the boson number should be counted from the nearest magic number [4]. This boson number rule works well for the Yb and H f isotopes. The kink deviates slightly from the theoretical prediction in the isotopes Dy and Er. These deviations may reflect sub-shell effects. Besides these maximum points, the slopes of massnumber dependence after the maximum points are well described by the analytic formula (6), except for the W isotopes. Again sub-shell effects may play a role for i s2W. The d-boson excitation energy e~, which is the energy required to break an s boson, should be the pairing energy 2A according to a naive consideration. In fact, the values ofe~ obtained by the present analysis roughly agree with the empirical values [ 1 ] 2A= 24/ x/A MeV, except for Sm. The values, however, appear larger than typical values ( ~ 0 . 5 MeV) obtained from phenomenological fits in IBM-1 calculations (ref. [ 3 ] ). The large d-boson energy e~ breaks the SU (3) scheme and gives vibrational spectra when the strength ~c of the Q.Q force is moderate. When one adds the pairing interaction tc"P~P in 0 ( 6 ) [3] to the hamiltonian ( 1 ), eqs. (5) and (6) are modified to 6 1 E ( 2 + ) = ~aa + (eh-¼~c'') 2 N - 1 '
(5')
6 2 a - 6(~c' -3~c) - ~ c "
(6')
"
When one takes a typical value of ~c" which is close to K or K' in magnitude, the interaction ~c"P*Pdoes not change the above result. In the pure SU (3) limit, the ground state band has the (2N, 0) representation, while the fl band belongs to ( 2 N - 4 , 2), The non-vanishing d-boson energy e~ couples these two bands. The second-order perturbation due to e~ nd gives the following correction term toeq. (3): - I ( (2N, 0); Ll~'ana I ( 2 N - 4 , 2); L ) 12/zXE I --
×
(e~)
2
[2N(2N+l)-L(L+ l) ] [ 8 ( N - 1 ) 2 - L ( L + I)] 18(2N- 1)2(2N-3) (9)
24 May 1990
where AE is the energy difference between the (2N, 0) and the ( 2 N - 4 , 2) bands. Taking the difference between the energies of the 2+ state ( L = 2 ) and the ground state ( L = 0 ) , it can be shown that this effect comes out in the excitation energy of 2~- as an additional term which is approximately proportional to 1 / ( 2 N - 1 ). The excitation energy of the 2~- state is now written as 6
E(2~)= ~
+e~
1+
(10)
in the leading order of ( 1/N). The value of e~ obtained from the fit using eq. (10) is referred to as e~ (cot), as listed in table I. Note that eq. (10) becomes identical to eq. (5) by replacing e~(cor) × [ 1 +e~ (cor)/2tEl by e~ (dir). In carrying out the fitting, a typical energy difference A E = 1 MeV is adopted between the ground state band and the fl band. The values ofeh (cor) are about half of the corresponding values of eh (dir) obtained from eq. ( 5 ), and are closer to the values of e5 adopted commonly in empirical fits to data. This means that, because the second order perturbation in eq. (9) becomes more accurate for smaller eS, the present calculation becomes more reliable by including the coupling in eq. (9). Although the perturbative treatment, eq. (9), may not be adequate for e~ or Nbeing large, the basic trend in the analytic expression in eq. (10) is expected to persist. For a more quantitative discussion, however, numerical calculations are needed. We used the IBM-1 in the above discussions. The distinction between the proton and the neutron bosons (i.e., IBM-2 [4] ) might play an important role for more quantitative discussions including the fl- and y-bands. Recently, IBM-2 calculations with large e~ values varying from 0.5 to 1.66 MeV have been performed in the Sm, H f and Er isotopes, and they reproduce successfully the energy spectra, including fl and 7 bands [ 7 ]. The phenomenological value for ed which yields the energy spectra and the transition strengths, is determined to be 0.7 MeV for 154Sm. This value is very close to our value obtained by formula (10). In summary, we have investigated the relation between the moment of inertia and the pairing correlation in terms of the interacting boson model. We found that the pairing correlation can be related explicitly to the moment of inertia in the IBM. The mass 461
Volume 241, number 4
PHYSICS LETTERS B
n u m b e r d e p e n d e n c e o f the empirical m o m e n t s o f inertia is related with the boson number, assuming the SU ( 3 ) limit o f the IBM. In particular, we f o u n d that the kink o f the m o m e n t o f inertia o f Yb a n d H f isotopes coincides with the m a x i m u m b o s o n n u m b e r at the center o f the shell. The o b t a i n e d d-boson energies in our analysis are found to be ~ 2A, which is larger than the phenomenological values o f IBM- 1 calculations. The fitted value o f the d-boson energy is, however, decreased by a factor ~ by the coupling between the (2N, 0) and ( 2 N - 4 , 2) bands, moving closer to the phenomenological values. The distinction between p r o t o n and neutron bosons might also play an i m p o r t a n t role for u n d e r s t a n d i n g this difference. Actually, the recent study o f well-deformed nuclei using the IBM-2 yields large ed values, the same as ours in magnitude, and describes successfully rotational spectra, including fl- and y-bands. The p r o b l e m of the d-boson energy is an interesting one for future study in order to establish a microscopic relation between the IBM and pairing theory.
462
24 May 1990
We are grateful to Professor B.R. Barrett for careful reading o f the m a n u s c r i p t and valuable comments. The support by the G r a n t - i n - A i d for General Scientific Research (No. 63 540206) by the Ministry o f Education, Science and Culture is acknowledged.
References [ 1] A. Bohr and B.R. Monelson, Nuclear structure, Vols. 1, 2 (Benjamin, New York, 1969, 1975). [2] J.M. Espino and J.D. Garren, Nucl. Phys. A 492 (1989) 205. [3] F. laehello and A. Arima, The interacting boson model (Cambridge U.P., Cambridge, 1987 ); A. Arima and F. lachello, Ann. Phys. (NY) 111 (1978) 201. [4] T. Otsuka, A. Arima and F. lachello, Nucl. Phys. A 309 (1978) 1; T. Otsuka, A. Arima, F. Iachello and I. Talmi, Phys. Lett. B 76 (1978) 139. [5] M. Sakai, At. Data Nucl. Data Tables 31 (1984) 399. [ 6 ] S. Raman, C.H. Malarkey, W.T. Milner, C.W. Nestor Jr. and P.H. Stelson, At. Data Nucl. Data Tables 36 (1987 ) 1. [7] A. Novoselsky, Nucl. Phys. A 483 (1988) 282.
MASS-NUMBER
PHYSICS LETTERS B
DEPENDENCE
OF THE MOMENT
24 May 1990
OF INERTIA AND IBM
N. Y O S H I D A Department of Information Science, Faculty of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113, Japan H. SAGAWA, T. O T S U K A and A. A R I M A Department of Physics, Faculty of Science, University of Tokyo, 7-3-I Hongo, Bunkyo-ku, Tokyo 113, Japan Received 19 December 1989; revised manuscript received 2 March 1990
The m a s s - n u m b e r dependence of the m o m e n t of inertia obtained from the energy difference between the ground state and the first 2 + state is studied in relation with the boson number in the S U ( 3 ) limit of the interacting boson model 1 (1BM-1). The analytic formula in the limit indicates that the pairing correlation between nucleons is directly related to the m o m e n t of inertia in the IBM. It is shown that the kinks of the m o m e n t of inertia of the Yb and Hfisotopes coincide exactly with the m a x i m u m boson numbers of each kind of isotopes.
The m o m e n t of inertia is a well-known fundamental observable to characterize deformed nuclei [ 1,2 ]. The pairing correlation plays a crucial role to decrease the m o m e n t of inertia from the rigid-rotor value by a factor 2-3 [ 1 ]. The decrease due to the pairing correlation, however, has never been considered systematically in terms o f the interacting boson model [ 3 ]. In this paper we discuss how the pairing correlation is connected with the m o m e n t o f inertia in the S U ( 3 ) limit of the IBM, when the hamiltonian is solved in an analytic way, and how one can see its effects on the mass-number dependence of the moment of inertia determined from the energy difference between the ground state and the first 2 + state. We take the IBM-1 for simplicity. The model hamiltonian is written as
H=&ns +eona +xQ'Q+tc'L'L,
(1)
where ns (rid) is the s-boson (d-boson) number operator; Q is the quadrupole operator in the S U ( 3 ) limit:
Q= d's+ s , a - },,/5[d*a ] (~, ,
(2)
and L = x / / ~ [d*d] (1) is the angular m o m e n t u m operator. The expectation values o f H for states in the ground
state band, i.e., states of the SU (3) representation (2, #) = (2N, 0), are given by N(2N+I)
E=& 3 ( 2 N - 1 ) + (x'-~x)L(L+
, 4N(N-1) -~cN(2N+3) ±~d 3~-~----1) 1 ) + (e d --•s)
1 L(L+I) 6 ( 2 N - 1)
=Eo +AL(L+ 1),
(3)
where Eo = es A=I¢'
N{2N+ 1) 3(2N- 1) 3 ~
-~:~-
(& - ~ )
4N(N-1) 1 6(2N- 1) '
-~¢N(2N+3)
(4)
N is the boson number, and L is the angular momentum. In the following, we assume that the neutron number dependence of the interaction strengths is negligible. The excitation energy of 2 + is o f the form 6 1 E ( 2 + ) = ~ +8~ 2 N - 1 '
{5)
where 6
2T=6(K'-]K),
0370-2693/90/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland )
(6)
459
Volume 241, number 4
PHYSICS LETTERS B
and e~ = e d - e , .
(7)
Eq. (5) can be expressed in terms of the moment of inertia of 2+"
24 May 1990
Table 1 The parameters 6/2c~ and e~ determined from the experimental excitation energies. The parameters of Nd and Os isotopes cannot be determined because only one data point is available for each element in fig. 1. Element
N-½ 2Icrf = 2c~ N - ½+ a e a / 6 "
(8)
6/2a (MeV)
e~(dir) a) (MeV)
0.024_+0.007 -0.005_+0.015 -0.003_+0.018 0.000+_0.018 0.000_+0.013 0.052+_0.007 0.061 _+0.021
1.22_+0.15 2.14_+0.38 2.47_+0.52 2.49_+0.50 2.46_+0.38 1.16_+0.19 1.20_+0.52 -
NO
We will now use the above equations to analyse nuclei in the rare-earth region. Since the present model is valid only for well-deformed nuclei, we choose those nuclei with ratios of the excitation energy of the 4 + state to that of the 2 + state, [i.e., E(4 + ) / E ( 2 i ~) ] larger than 3.2. The boson number is counted as the number of particle (or hole) pairs with respect to the nearest magic number 50, 82 or 126 [4]. Keeping oe and e} as constants for all the isotopes of each element, we determine that value by eq. (5), using the plot in fig. 1. The obtained values are listed as 6/2c~ and ea (dir) in table 1. The statistical errors for the values ea (dir) are reasonably small; (10-20)% except W isotopes. The values of these two parameters for the Nd and Os isotopes cannot be determined, because only one data point is available for each of these isotopes. The calculated moments of inertia are shown in fig. 2 together with the experimental data [ 5,6 ].
e~(cor) b) (MeV)
-
Sm Gd Dy Er Yb Hf W Os
-
0.71 1.05 1.15 1.16 1.15 0.69 0.70 -
") Values obtained directly from fits [eq. (5) ]. b) The correction due to the coupling to the fl-band is taken into account (eq. (10) with AE= 1 MeV).
/ / / /
/
J
J
j
.-~rigid 150
I
IO0 °
\
,,.-
o
o
>O%.oo
° JS/.-'-'>,-,
¢N
--
;
/
- o., o I
• Nd o D y • Sm - E r ~ Gd o y b
,;o )
///
0[~_. /
=Sm-Er
.W
~ Gd o Y b v Os
0.05
°
,"*"*'--L. ,..,
5C "Hf * W "Os
MASS
NUMBER
Fig. 2. Mass-number dependence of the moment of inertia. The solid lines are calculated by eq. (8) using the values 2c~ and e~ determined in fig. 1 together with the effective boson number. The rigid rotor values [ 1 ] are shown by a dashed line.
1/(2N-1)
Fig. 1. The excitation energies of 2t+ states as a function of 1/ ( 2 N - 1 ). The line for each element is drawn to determine the values 2o~ and e~ in eq. (5). The lines for the Yb and Er isotopes cannot be distinguished in the present scale of the figure. The experimental data are taken from refs. [ 5,6 ].
460
The calculation reproduces the general trends of the mass-number dependence of the moment of inertia. As a function of the neutron number N,, the calculated moment of inertia increases and becomes maximal at N , = 104, which is the middle of the major
Volume 241, number 4
PHYSICS LETTERS B
shell between 82 and 126. After AT,= 104, the moment of inertia decreases since the boson number should be counted from the nearest magic number [4]. This boson number rule works well for the Yb and H f isotopes. The kink deviates slightly from the theoretical prediction in the isotopes Dy and Er. These deviations may reflect sub-shell effects. Besides these maximum points, the slopes of massnumber dependence after the maximum points are well described by the analytic formula (6), except for the W isotopes. Again sub-shell effects may play a role for i s2W. The d-boson excitation energy e~, which is the energy required to break an s boson, should be the pairing energy 2A according to a naive consideration. In fact, the values ofe~ obtained by the present analysis roughly agree with the empirical values [ 1 ] 2A= 24/ x/A MeV, except for Sm. The values, however, appear larger than typical values ( ~ 0 . 5 MeV) obtained from phenomenological fits in IBM-1 calculations (ref. [ 3 ] ). The large d-boson energy e~ breaks the SU (3) scheme and gives vibrational spectra when the strength ~c of the Q.Q force is moderate. When one adds the pairing interaction tc"P~P in 0 ( 6 ) [3] to the hamiltonian ( 1 ), eqs. (5) and (6) are modified to 6 1 E ( 2 + ) = ~aa + (eh-¼~c'') 2 N - 1 '
(5')
6 2 a - 6(~c' -3~c) - ~ c "
(6')
"
When one takes a typical value of ~c" which is close to K or K' in magnitude, the interaction ~c"P*Pdoes not change the above result. In the pure SU (3) limit, the ground state band has the (2N, 0) representation, while the fl band belongs to ( 2 N - 4 , 2), The non-vanishing d-boson energy e~ couples these two bands. The second-order perturbation due to e~ nd gives the following correction term toeq. (3): - I ( (2N, 0); Ll~'ana I ( 2 N - 4 , 2); L ) 12/zXE I --
×
(e~)
2
[2N(2N+l)-L(L+ l) ] [ 8 ( N - 1 ) 2 - L ( L + I)] 18(2N- 1)2(2N-3) (9)
24 May 1990
where AE is the energy difference between the (2N, 0) and the ( 2 N - 4 , 2) bands. Taking the difference between the energies of the 2+ state ( L = 2 ) and the ground state ( L = 0 ) , it can be shown that this effect comes out in the excitation energy of 2~- as an additional term which is approximately proportional to 1 / ( 2 N - 1 ). The excitation energy of the 2~- state is now written as 6
E(2~)= ~
+e~
1+
(10)
in the leading order of ( 1/N). The value of e~ obtained from the fit using eq. (10) is referred to as e~ (cot), as listed in table I. Note that eq. (10) becomes identical to eq. (5) by replacing e~(cor) × [ 1 +e~ (cor)/2tEl by e~ (dir). In carrying out the fitting, a typical energy difference A E = 1 MeV is adopted between the ground state band and the fl band. The values ofeh (cor) are about half of the corresponding values of eh (dir) obtained from eq. ( 5 ), and are closer to the values of e5 adopted commonly in empirical fits to data. This means that, because the second order perturbation in eq. (9) becomes more accurate for smaller eS, the present calculation becomes more reliable by including the coupling in eq. (9). Although the perturbative treatment, eq. (9), may not be adequate for e~ or Nbeing large, the basic trend in the analytic expression in eq. (10) is expected to persist. For a more quantitative discussion, however, numerical calculations are needed. We used the IBM-1 in the above discussions. The distinction between the proton and the neutron bosons (i.e., IBM-2 [4] ) might play an important role for more quantitative discussions including the fl- and y-bands. Recently, IBM-2 calculations with large e~ values varying from 0.5 to 1.66 MeV have been performed in the Sm, H f and Er isotopes, and they reproduce successfully the energy spectra, including fl and 7 bands [ 7 ]. The phenomenological value for ed which yields the energy spectra and the transition strengths, is determined to be 0.7 MeV for 154Sm. This value is very close to our value obtained by formula (10). In summary, we have investigated the relation between the moment of inertia and the pairing correlation in terms of the interacting boson model. We found that the pairing correlation can be related explicitly to the moment of inertia in the IBM. The mass 461
Volume 241, number 4
PHYSICS LETTERS B
n u m b e r d e p e n d e n c e o f the empirical m o m e n t s o f inertia is related with the boson number, assuming the SU ( 3 ) limit o f the IBM. In particular, we f o u n d that the kink o f the m o m e n t o f inertia o f Yb a n d H f isotopes coincides with the m a x i m u m b o s o n n u m b e r at the center o f the shell. The o b t a i n e d d-boson energies in our analysis are found to be ~ 2A, which is larger than the phenomenological values o f IBM- 1 calculations. The fitted value o f the d-boson energy is, however, decreased by a factor ~ by the coupling between the (2N, 0) and ( 2 N - 4 , 2) bands, moving closer to the phenomenological values. The distinction between p r o t o n and neutron bosons might also play an i m p o r t a n t role for u n d e r s t a n d i n g this difference. Actually, the recent study o f well-deformed nuclei using the IBM-2 yields large ed values, the same as ours in magnitude, and describes successfully rotational spectra, including fl- and y-bands. The p r o b l e m of the d-boson energy is an interesting one for future study in order to establish a microscopic relation between the IBM and pairing theory.
462
24 May 1990
We are grateful to Professor B.R. Barrett for careful reading o f the m a n u s c r i p t and valuable comments. The support by the G r a n t - i n - A i d for General Scientific Research (No. 63 540206) by the Ministry o f Education, Science and Culture is acknowledged.
References [ 1] A. Bohr and B.R. Monelson, Nuclear structure, Vols. 1, 2 (Benjamin, New York, 1969, 1975). [2] J.M. Espino and J.D. Garren, Nucl. Phys. A 492 (1989) 205. [3] F. laehello and A. Arima, The interacting boson model (Cambridge U.P., Cambridge, 1987 ); A. Arima and F. lachello, Ann. Phys. (NY) 111 (1978) 201. [4] T. Otsuka, A. Arima and F. lachello, Nucl. Phys. A 309 (1978) 1; T. Otsuka, A. Arima, F. Iachello and I. Talmi, Phys. Lett. B 76 (1978) 139. [5] M. Sakai, At. Data Nucl. Data Tables 31 (1984) 399. [ 6 ] S. Raman, C.H. Malarkey, W.T. Milner, C.W. Nestor Jr. and P.H. Stelson, At. Data Nucl. Data Tables 36 (1987 ) 1. [7] A. Novoselsky, Nucl. Phys. A 483 (1988) 282.