- Email: [email protected]
Scissors mode and nuclear deformation in the generalized coherent state model
Physics Letters B 300 (1993) 195-198 North-Holland
PHYSICS LETTERS B
Scissors mode and nuclear deformation in the generalized coherent state model N. Lo Iudice, A.A. Raduta Dipartimento di Scienze Fisiche, Universith "'Federico H'" di Napoli, and Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, Mostra d'Oltremare, Pad 19, 1-80125 Naples, Italy
and D.S. Delion Institute of Atomic Physics, P.O. Box MG 6, Bucharest, Romania
Received 25 June 1992; revised manuscript received 3 December 1992
The magnetic dipole strength of the scissors mode is computed within the generalized coherent state model and found to be quadratic in the deformation parameter in good agreement with experiments and with the results obtained earlier within collective and shell model schemes.
After the discovery that the s u m m e d strength o f the low-lying M 1 excitations, observed in d e f o r m e d nuclei [1,2] and known as scissors m o d e [3], is quadratic in the d e f o r m a t i o n p a r a m e t e r ~ [4] and saturates with it [5], several m o d e l descriptions o f the m o d e have been re-examined a n d found to be consistent to more or less extent with such a quadratic law [6-12]. In this p a p e r we will investigate if such a consistency is met by the generalized coherent state m o d e l ( G C S M ) [ 13 ], which p r o v e d to be successful in describing the properties o f the m o d e throughout the periodic table in conjunction with the fl and 7 bands. The G C S M is an interacting boson m o d e l closely related to the IBM-2 [ 14 ] or to the n e u t r o n - p r o t o n def o r m a t i o n ( N P D ) m o d e l [6] in the classical limit [ 15 ] but differing from them at the q u a n t u m level. Unlike the IBM-2 which deals with a conserved number o f m o n o p o l e and q u a d r u p o l e bosons and is essentially a truncated shell m o d e l scheme, the G C S M is based on a n u m b e r nonconserving interacting quadPermanent address: Institute of Atomic Physics, P.O. Box MG 6, Bucharest, Romania.
rupole bosons h a m i l t o n i a n and corresponds to a higher quasi-particle r a n d o m - p h a s e a p p r o x i m a t i o n ( Q R P A ) . In this respect, it is closer to the N P D model but is simpler to deal with and has more predictive power. The energy and the M 1 strength o f the m o d e are c o m p u t e d once all model parameters are fixed by an i n d e p e n d e n t fit. As the above models it has the two-rotor model ( T R M ) [3 ] as its geometrical limit [ 15 ]. F o r all these reasons the present analysis m a y usefully i m p l e m e n t the other studies. We will see that the G C S M yields results in satisfactory agreement with the observed d e f o r m a t i o n laws and with the resuits obtained previously within microscopic [ 7 - 1 0 ] as well as collective models [ 11 ]. The basic ingredients o f the G C S M are six mutually orthogonal intrinsic states qb~x which diagonalize to a large extent the model hamiltonian. The physical states are obtained by projecting out the good angular m o m e n t u m
~"to~JMK=Nc~JP JMKI~ocK ,
( 1)
where N ~ j is a normalization factor and P S K a projection o p e r a t o r o f standard form. The scissors M 1 state ~ul = ~ u = j = ~ is projected out of
0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
195
Volume 300, number 3
PHYSICS LETTERSB
of the GCSM operator, we obtain the standard expression
qh = ( b+ ®b + )~%', q)o,
Cbo:exp(~2~Pdb.~-bm))lO),
(2)
where ~ = ~ , = uK= ~ and ~o is the intrinsic ground state, a coherent state in the deformation parameters p~ and r=p, n. We will assume equal deformation for protons and neutrons, so that pp---pn=p. The quadrupole boson creation and annihilation operators, b~+ and b~ respectively, are related to the shape variables a~u through a~u = [b+u+ ( - )"b~_u]/k~x/2, where k~ in the harmonic limit is just k~ =x/-B,C,, with B, and C, the usual mass and restoring force constants. The M 1 transition from the ground (To) to the M 1 state ~L is promoted by a liquid drop M 1 operator, to lowest order in the shape variables, with a strength B ( M 1 ) I ' = B ( M 1 ; 0 +~1 +)
9
(
- 40u (gp-gn)2p4
1+
1 ~ 2 (N"~2p~, 2 Io] k.Noo]
(3)
where L,=Ij(p 2)=fldxPJ(x) exp[pZP2(x)] are overlap integrals with Pj(x) Legendre polynomials and
(Nod)-2= ( 2 J +
1 )/j exp( _ p 2 ) ,
(4)
(N,~)_2 = (l+~p2)N5o: + ( ~ + ggp3Z)No2-2 ~2~-2 .t • 04
-[- ~ f f ~ p
(5)
•
For small deformations (p<< 1 ) I2/lo=~p 2 and (NL,/Noo) a ... 1 - ~oP 2. In this vibrational limit the strength becomes Bvib(M1 )$ -~ ~
9
(gp -g,)2p41t~.
(6)
For large deformations (rotational limit) we have instead [ 16] I2/ Io~ 1 - 1/p 2 and ( Nll/ Noo)2 = 20/9p 2 which yields B~ot(M 1 )1" 9
- 8~z (gp_g,)2
[p
z _ -~+O
(pl)]
~t~v.
(7)
One can relate p to the deformation parameter fl= ( ~ ) ~/z5 through the E2 strength. Taking the liquid drop E2 transition operator, which is the leading term 196
11 February 1993
B(E2; 0 + ~ 2 +) = [ (0 +I[Q2112+) I
=(-~)2Z2R41~(p)2e2,
(8)
with the deformation parameter replaced by fl(P) = (0+llaplt2 + ) _ F(p)
k.'
fNo2 IN ooh V(p) = ~ - p ~,~oo + 5 Noz]"
(9)
We used eq. (C4) of ref. [ 13 ] to compute the matrix element of ap between the J ~ = 0 + and the J~=2 + projected ground band states (eqs. ( 1 ), (2)). In the rotational limit fl(p) can be identified with the static deformation fl extracted from the experimental E2 strength. Using eqs. (4), (5) in this limit, we have F(p) -~p and therefore p ~-kpfl so that eq. (7) becomes Brot(M 1 )~ ~ ~ k2pfl2(gp- - g n
) n ~ 2N .
( 10)
This assumes the same form of the NPD [ 6 ] strength if we put k~ = ~ ~ ~x/~Bc or the TRM expression [ 15 ] if we further use the relations ~ = 3Bfl 2 and o) = x/-~B. It is also closely related to the M 1 strength derived within the IBM2 context [ 11 ]: B(M1 )~ -~ ( 9/2zORa(NpN,/N 2) (gp -g,,) 2/t2N,where Np and N, denote the number of valence protons and neutrons respectively, N = N; + N, and/Va is the average number ofquadrupole bosons in the ground state. Putting indeed Na=Zub-~bu and taking the unprojected ground state q~o (eq. (2)), we get f a = p 2 = (kpfl) 2. For Np = N . = i N as in lS4Sm we recover the GCSM strength (10). The f12 law yields a linear relation between the M1 and the E2 strengths consistently with their observed saturation properties [ 5 ]. Inverting eq. (8) with respect to f12 and inserting into eq. (10) we get Brot(M 1 ) T-~ 2z~~ B(E2; 0 + ~ 2 + ) z/,( × (gp -gn)2/t2
e2fm4
(11)
Volume 300, number 3
PHYSICS LETTERS B
11 February 1993
,
,
•
,
,
r
,
.
I
'
'
'
,
'
'
'
I
'
Th Exp
3
:{
"R
1
J ,
,
,
,
i
,
,
,
,
1
i
,
,
,
,
2
i
,
,
,
0
,
3
0.00
4
i
I
i
i
0.0E
0.04
0.06
0.08
P
Fig. 1. Plot o f F ( p ) (eq. ( 9 ) ) v e r s u s p .
Fig. 2. Plot of the GCSM and experimental M1 strengths versus
8. Table 1 Theoretical versus experimental energies (co) and M 1 strengths. The fitted p's and the values of fl corresponding to the O's used in ref. [ 4 ] are also shown.
J4SSm 15OSm 15ZSm 1548m
p
fl
o9
ogexp
B(M1 )t
YBexp(M 1 )t
1.71 2.03 2.69 3.23
0.128 0.173 0.263 0.289
3.064 3.148 3.024 3.121
3.07 3.18 2.98 3.09
0.53 0.90 1.98 3.18
0.51 0.97 2.35 2.65
This relation holds for large deformation only. In the vibrational limit the model predicts correctly a finite E2 strength but a vanishing M I transition probability in agreement with recent shell model results [ 10 ]. Using eqs. (4), (5) in this limit we get from eq. (9) fl(P) = 13o+ fl= x/5/2kp kp. The first piece flo represents the contribution to the transition coming from the zero point fluctuation of the shape variable otu, since f12 = 5 / (2kp) 2_~ 5 / (4BC) l/2 as in standard vibrational models. This is independent of p, which, by definition of coherent state, is solely related to the static deformation fl~_p2/2x/5 kp. This relation yields (eq. (6) )
+P2/2~/~
Bv,b(M1 )1-~
9kEv,g2(gp-g,)Elz~.
(12)
The strength is quadratic in fl as in the rotational limit but with a different slope. The numerical analysis was carried out for all Sm isotopes studied experimentally except for 144Sm,
which lacks the data necessary for determining the model parameters. We fixed p by an overall fit of the fl band and determined the four parameters entering into the hamiltonian by a fit of the two lowest ground band levels and the fl and 7 band-head levels. We finally put gp= 1 and g , = 0 for the gyromagnetic factors as in other interacting boson models [ 6 ]. To check the relation between fl and p we computed exactly F (p) (eq. (9) ). As shown in fig. 1 F(p) grows quadratically from its zero point motion value ½x/5 but becomes linear very soon and coincides with p already at p-~ 1.5. The fitted values o f p (table 1 ) fall in the range where F(p) ~-p, so that p~-kpfl. The table shows however, that the fitted p are not exactly linear in the experimental fl's. Apparently the fitting procedure has selected different values of kp for different nuclei. Consequently the computed M1 strength, though exactly quadratic in p in the range of the fitted values, is only roughly quadratic in fl as shown in fig. 2. The deviations from such a law and from experiments are modest however. This is saris197
Volume 300, number 3
PHYSICS LETTERS B
factory especially since the model parameters were fitted independently of the mode. The computed M1 strengths are reasonably close to the values obtained microscopically [ 7 - 1 0 ] . This implies that pairing correlations, which proved to be essential to the quadratic deformation law in these calculations, are accounted for effectively in the present model. Being indeed the model especially suitable for describing the properties of low-lying states, the quadrupole bosons are to be thought of as highly correlated two-quasi particle rather than particle-hole states. Numerically one of the requests of the fitting procedure was to reproduce the first 2 + level of the ground (rotational) b a n d which defines the nuclear m o m e n t of inertia, a quantity very sensitive to pairing correlations. That these are accounted for is finally supported by the close connection of the GCSM M1 strength with that obtained in IBM-2 [11], a model explicitly based on pairing correlations. The similarity with IBM-2 is encouraging also in view of a quantitative study of the saturation properties of the M 1 strength. The term ATaappearing in the IBM2 strength and found to correspond to p2 is essential to the saturation of the IBM-2 M 1 strength [ 1 1 ]. We may conclude that the success of the model in describing the deformation dependence of the mode support the scissors character and the collective nature of the observed low-lying M1 excitations. The model is in fact of collective nature by construction and, as its close relation to the T R M shows, describes these excitations as rotational oscillations of protons versus neutrons. These behave like superfluids as suggested by the important role played by pairing correlations here taken effectively into account. Such
198
11 February 1993
a picture is meaningful only in well deformed nuclei. The success of the model description outside this region indicates that even in weakly deformed nuclei a scissors-like correlation is responsible for the lowlying M 1 excitations.
References [ 1] D. Bohle, A. Richter, W. Steffen, A.E.L. Dieperink, N. Lo Iudice, F. Palumbo and O. Scholten, Phys. Lett. B 137 (1984) 27. [2] For a summary see A. Richter, Nucl. Phys. A 507 (1990) 99. [3] N. Lo ludice and F. Palumbo, Phys. Rev. Lett. 41 (1978) 1532; G. De Franceschi,F. Palumbo and N. Lo Iudice, Phys. Rev. C29 (1984) 1496. [4] W. Ziegler, C. Rangacharyulu, A. Richter and C. Spieler, Phys. Rev. Lett. 65 (1990) 2515. [ 5 ] C. Rangacharyulu,A. Richter, H.J. W~Srtche,W. Zieglerand R.F. Casten, Phys. Rev. C 43 ( 1991 ) R949. [6] S.G. Rohozinski and W. Greiner, Z. Phys. A 322 (1985) 271. [7 ] I. Hamamoto and C. Magnusson,Phys. Lett. B 260 ( 1991 ) 6. [8] E. Garrido, E. Moya de Guerra, P. Sarriguren and J.M. Udias, Phys. Rev. C 44 ( 1991 ) R1250. [9 ] K. Heyde and C. De Coster, Phys. Rev. C 44 ( 1991) R2262. [ 10] L. Zamick and D.C. Zheng, Phys. Rev. C 44 ( 1991 ) 2522. [ 11 ] J.N. Ginocchio, Phys. Lett. B 265 ( 1991 ) 6. [ 12 ] N. Lo Iudice and A. Richter, to be submittedfor publication. [ 13 ] A.A. Raduta, A. Faessler and V. Ceausescu, Phys. Rev. C 36 (1987) 2111. [14] F. Iachello and A. Arima, The interacting boson model (Cambridge U.P., Cambridge, 1987). [ 15 ] N. Lo ludice, in: New trends in theoretical and experimental nuclear physics, eds. A. Raduta et al. (World Scientific, Singapore, 1992) p. 41. [ 16] A.A. Raduta and C. Sabac, Ann. Phys. 148 ( 1983) 1.
PHYSICS LETTERS B
Scissors mode and nuclear deformation in the generalized coherent state model N. Lo Iudice, A.A. Raduta Dipartimento di Scienze Fisiche, Universith "'Federico H'" di Napoli, and Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, Mostra d'Oltremare, Pad 19, 1-80125 Naples, Italy
and D.S. Delion Institute of Atomic Physics, P.O. Box MG 6, Bucharest, Romania
Received 25 June 1992; revised manuscript received 3 December 1992
The magnetic dipole strength of the scissors mode is computed within the generalized coherent state model and found to be quadratic in the deformation parameter in good agreement with experiments and with the results obtained earlier within collective and shell model schemes.
After the discovery that the s u m m e d strength o f the low-lying M 1 excitations, observed in d e f o r m e d nuclei [1,2] and known as scissors m o d e [3], is quadratic in the d e f o r m a t i o n p a r a m e t e r ~ [4] and saturates with it [5], several m o d e l descriptions o f the m o d e have been re-examined a n d found to be consistent to more or less extent with such a quadratic law [6-12]. In this p a p e r we will investigate if such a consistency is met by the generalized coherent state m o d e l ( G C S M ) [ 13 ], which p r o v e d to be successful in describing the properties o f the m o d e throughout the periodic table in conjunction with the fl and 7 bands. The G C S M is an interacting boson m o d e l closely related to the IBM-2 [ 14 ] or to the n e u t r o n - p r o t o n def o r m a t i o n ( N P D ) m o d e l [6] in the classical limit [ 15 ] but differing from them at the q u a n t u m level. Unlike the IBM-2 which deals with a conserved number o f m o n o p o l e and q u a d r u p o l e bosons and is essentially a truncated shell m o d e l scheme, the G C S M is based on a n u m b e r nonconserving interacting quadPermanent address: Institute of Atomic Physics, P.O. Box MG 6, Bucharest, Romania.
rupole bosons h a m i l t o n i a n and corresponds to a higher quasi-particle r a n d o m - p h a s e a p p r o x i m a t i o n ( Q R P A ) . In this respect, it is closer to the N P D model but is simpler to deal with and has more predictive power. The energy and the M 1 strength o f the m o d e are c o m p u t e d once all model parameters are fixed by an i n d e p e n d e n t fit. As the above models it has the two-rotor model ( T R M ) [3 ] as its geometrical limit [ 15 ]. F o r all these reasons the present analysis m a y usefully i m p l e m e n t the other studies. We will see that the G C S M yields results in satisfactory agreement with the observed d e f o r m a t i o n laws and with the resuits obtained previously within microscopic [ 7 - 1 0 ] as well as collective models [ 11 ]. The basic ingredients o f the G C S M are six mutually orthogonal intrinsic states qb~x which diagonalize to a large extent the model hamiltonian. The physical states are obtained by projecting out the good angular m o m e n t u m
~"to~JMK=Nc~JP JMKI~ocK ,
( 1)
where N ~ j is a normalization factor and P S K a projection o p e r a t o r o f standard form. The scissors M 1 state ~ul = ~ u = j = ~ is projected out of
0370-2693/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
195
Volume 300, number 3
PHYSICS LETTERSB
of the GCSM operator, we obtain the standard expression
qh = ( b+ ®b + )~%', q)o,
Cbo:exp(~2~Pdb.~-bm))lO),
(2)
where ~ = ~ , = uK= ~ and ~o is the intrinsic ground state, a coherent state in the deformation parameters p~ and r=p, n. We will assume equal deformation for protons and neutrons, so that pp---pn=p. The quadrupole boson creation and annihilation operators, b~+ and b~ respectively, are related to the shape variables a~u through a~u = [b+u+ ( - )"b~_u]/k~x/2, where k~ in the harmonic limit is just k~ =x/-B,C,, with B, and C, the usual mass and restoring force constants. The M 1 transition from the ground (To) to the M 1 state ~L is promoted by a liquid drop M 1 operator, to lowest order in the shape variables, with a strength B ( M 1 ) I ' = B ( M 1 ; 0 +~1 +)
9
(
- 40u (gp-gn)2p4
1+
1 ~ 2 (N"~2p~, 2 Io] k.Noo]
(3)
where L,=Ij(p 2)=fldxPJ(x) exp[pZP2(x)] are overlap integrals with Pj(x) Legendre polynomials and
(Nod)-2= ( 2 J +
1 )/j exp( _ p 2 ) ,
(4)
(N,~)_2 = (l+~p2)N5o: + ( ~ + ggp3Z)No2-2 ~2~-2 .t • 04
-[- ~ f f ~ p
(5)
•
For small deformations (p<< 1 ) I2/lo=~p 2 and (NL,/Noo) a ... 1 - ~oP 2. In this vibrational limit the strength becomes Bvib(M1 )$ -~ ~
9
(gp -g,)2p41t~.
(6)
For large deformations (rotational limit) we have instead [ 16] I2/ Io~ 1 - 1/p 2 and ( Nll/ Noo)2 = 20/9p 2 which yields B~ot(M 1 )1" 9
- 8~z (gp_g,)2
[p
z _ -~+O
(pl)]
~t~v.
(7)
One can relate p to the deformation parameter fl= ( ~ ) ~/z5 through the E2 strength. Taking the liquid drop E2 transition operator, which is the leading term 196
11 February 1993
B(E2; 0 + ~ 2 +) = [ (0 +I[Q2112+) I
=(-~)2Z2R41~(p)2e2,
(8)
with the deformation parameter replaced by fl(P) = (0+llaplt2 + ) _ F(p)
k.'
fNo2 IN ooh V(p) = ~ - p ~,~oo + 5 Noz]"
(9)
We used eq. (C4) of ref. [ 13 ] to compute the matrix element of ap between the J ~ = 0 + and the J~=2 + projected ground band states (eqs. ( 1 ), (2)). In the rotational limit fl(p) can be identified with the static deformation fl extracted from the experimental E2 strength. Using eqs. (4), (5) in this limit, we have F(p) -~p and therefore p ~-kpfl so that eq. (7) becomes Brot(M 1 )~ ~ ~ k2pfl2(gp- - g n
) n ~ 2N .
( 10)
This assumes the same form of the NPD [ 6 ] strength if we put k~ = ~ ~ ~x/~Bc or the TRM expression [ 15 ] if we further use the relations ~ = 3Bfl 2 and o) = x/-~B. It is also closely related to the M 1 strength derived within the IBM2 context [ 11 ]: B(M1 )~ -~ ( 9/2zORa(NpN,/N 2) (gp -g,,) 2/t2N,where Np and N, denote the number of valence protons and neutrons respectively, N = N; + N, and/Va is the average number ofquadrupole bosons in the ground state. Putting indeed Na=Zub-~bu and taking the unprojected ground state q~o (eq. (2)), we get f a = p 2 = (kpfl) 2. For Np = N . = i N as in lS4Sm we recover the GCSM strength (10). The f12 law yields a linear relation between the M1 and the E2 strengths consistently with their observed saturation properties [ 5 ]. Inverting eq. (8) with respect to f12 and inserting into eq. (10) we get Brot(M 1 ) T-~ 2z~~ B(E2; 0 + ~ 2 + ) z/,( × (gp -gn)2/t2
e2fm4
(11)
Volume 300, number 3
PHYSICS LETTERS B
11 February 1993
,
,
•
,
,
r
,
.
I
'
'
'
,
'
'
'
I
'
Th Exp
3
:{
"R
1
J ,
,
,
,
i
,
,
,
,
1
i
,
,
,
,
2
i
,
,
,
0
,
3
0.00
4
i
I
i
i
0.0E
0.04
0.06
0.08
P
Fig. 1. Plot o f F ( p ) (eq. ( 9 ) ) v e r s u s p .
Fig. 2. Plot of the GCSM and experimental M1 strengths versus
8. Table 1 Theoretical versus experimental energies (co) and M 1 strengths. The fitted p's and the values of fl corresponding to the O's used in ref. [ 4 ] are also shown.
J4SSm 15OSm 15ZSm 1548m
p
fl
o9
ogexp
B(M1 )t
YBexp(M 1 )t
1.71 2.03 2.69 3.23
0.128 0.173 0.263 0.289
3.064 3.148 3.024 3.121
3.07 3.18 2.98 3.09
0.53 0.90 1.98 3.18
0.51 0.97 2.35 2.65
This relation holds for large deformation only. In the vibrational limit the model predicts correctly a finite E2 strength but a vanishing M I transition probability in agreement with recent shell model results [ 10 ]. Using eqs. (4), (5) in this limit we get from eq. (9) fl(P) = 13o+ fl= x/5/2kp kp. The first piece flo represents the contribution to the transition coming from the zero point fluctuation of the shape variable otu, since f12 = 5 / (2kp) 2_~ 5 / (4BC) l/2 as in standard vibrational models. This is independent of p, which, by definition of coherent state, is solely related to the static deformation fl~_p2/2x/5 kp. This relation yields (eq. (6) )
+P2/2~/~
Bv,b(M1 )1-~
9kEv,g2(gp-g,)Elz~.
(12)
The strength is quadratic in fl as in the rotational limit but with a different slope. The numerical analysis was carried out for all Sm isotopes studied experimentally except for 144Sm,
which lacks the data necessary for determining the model parameters. We fixed p by an overall fit of the fl band and determined the four parameters entering into the hamiltonian by a fit of the two lowest ground band levels and the fl and 7 band-head levels. We finally put gp= 1 and g , = 0 for the gyromagnetic factors as in other interacting boson models [ 6 ]. To check the relation between fl and p we computed exactly F (p) (eq. (9) ). As shown in fig. 1 F(p) grows quadratically from its zero point motion value ½x/5 but becomes linear very soon and coincides with p already at p-~ 1.5. The fitted values o f p (table 1 ) fall in the range where F(p) ~-p, so that p~-kpfl. The table shows however, that the fitted p are not exactly linear in the experimental fl's. Apparently the fitting procedure has selected different values of kp for different nuclei. Consequently the computed M1 strength, though exactly quadratic in p in the range of the fitted values, is only roughly quadratic in fl as shown in fig. 2. The deviations from such a law and from experiments are modest however. This is saris197
Volume 300, number 3
PHYSICS LETTERS B
factory especially since the model parameters were fitted independently of the mode. The computed M1 strengths are reasonably close to the values obtained microscopically [ 7 - 1 0 ] . This implies that pairing correlations, which proved to be essential to the quadratic deformation law in these calculations, are accounted for effectively in the present model. Being indeed the model especially suitable for describing the properties of low-lying states, the quadrupole bosons are to be thought of as highly correlated two-quasi particle rather than particle-hole states. Numerically one of the requests of the fitting procedure was to reproduce the first 2 + level of the ground (rotational) b a n d which defines the nuclear m o m e n t of inertia, a quantity very sensitive to pairing correlations. That these are accounted for is finally supported by the close connection of the GCSM M1 strength with that obtained in IBM-2 [11], a model explicitly based on pairing correlations. The similarity with IBM-2 is encouraging also in view of a quantitative study of the saturation properties of the M 1 strength. The term ATaappearing in the IBM2 strength and found to correspond to p2 is essential to the saturation of the IBM-2 M 1 strength [ 1 1 ]. We may conclude that the success of the model in describing the deformation dependence of the mode support the scissors character and the collective nature of the observed low-lying M1 excitations. The model is in fact of collective nature by construction and, as its close relation to the T R M shows, describes these excitations as rotational oscillations of protons versus neutrons. These behave like superfluids as suggested by the important role played by pairing correlations here taken effectively into account. Such
198
11 February 1993
a picture is meaningful only in well deformed nuclei. The success of the model description outside this region indicates that even in weakly deformed nuclei a scissors-like correlation is responsible for the lowlying M 1 excitations.
References [ 1] D. Bohle, A. Richter, W. Steffen, A.E.L. Dieperink, N. Lo Iudice, F. Palumbo and O. Scholten, Phys. Lett. B 137 (1984) 27. [2] For a summary see A. Richter, Nucl. Phys. A 507 (1990) 99. [3] N. Lo ludice and F. Palumbo, Phys. Rev. Lett. 41 (1978) 1532; G. De Franceschi,F. Palumbo and N. Lo Iudice, Phys. Rev. C29 (1984) 1496. [4] W. Ziegler, C. Rangacharyulu, A. Richter and C. Spieler, Phys. Rev. Lett. 65 (1990) 2515. [ 5 ] C. Rangacharyulu,A. Richter, H.J. W~Srtche,W. Zieglerand R.F. Casten, Phys. Rev. C 43 ( 1991 ) R949. [6] S.G. Rohozinski and W. Greiner, Z. Phys. A 322 (1985) 271. [7 ] I. Hamamoto and C. Magnusson,Phys. Lett. B 260 ( 1991 ) 6. [8] E. Garrido, E. Moya de Guerra, P. Sarriguren and J.M. Udias, Phys. Rev. C 44 ( 1991 ) R1250. [9 ] K. Heyde and C. De Coster, Phys. Rev. C 44 ( 1991) R2262. [ 10] L. Zamick and D.C. Zheng, Phys. Rev. C 44 ( 1991 ) 2522. [ 11 ] J.N. Ginocchio, Phys. Lett. B 265 ( 1991 ) 6. [ 12 ] N. Lo Iudice and A. Richter, to be submittedfor publication. [ 13 ] A.A. Raduta, A. Faessler and V. Ceausescu, Phys. Rev. C 36 (1987) 2111. [14] F. Iachello and A. Arima, The interacting boson model (Cambridge U.P., Cambridge, 1987). [ 15 ] N. Lo ludice, in: New trends in theoretical and experimental nuclear physics, eds. A. Raduta et al. (World Scientific, Singapore, 1992) p. 41. [ 16] A.A. Raduta and C. Sabac, Ann. Phys. 148 ( 1983) 1.