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The competition between the shears mechanism and core rotation in a classical particles-plus-rotor model
18 March 1999
Physics Letters B 450 Ž1999. 1–6
The competition between the shears mechanism and core rotation in a classical particles-plus-rotor model A.O. Macchiavelli, R.M. Clark, M.A. Deleplanque, R.M. Diamond, P. Fallon, I.Y. Lee, F.S. Stephens, K. Vetter Nuclear Science DiÕision, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA Received 14 December 1998 Editor: W. Haxton
Abstract We present a model study of the competition between the shears mechanism and the rotation of the core. This is done in the framework of a classical solution of a Particles-plus-Rotor Model. The competition is governed by the relative strength of the residual interaction between the blades in the shears and the rotational energy of the core. The results are summarized in terms of a ‘‘phase-diagram’’ of these two modes of rotation that is applicable for all mass regions. We find that in order for the shears mechanism to dominate, the deformation of the core has to be e Q 0.12. q 1999 Published by Elsevier Science B.V. All rights reserved.
The rotational behavior of the M1 bands observed in neutron-deficient Pb isotopes w1x, came both as a surprise and as an intriguing puzzle for the nuclear structure community. The small E2rM1 branching ratios observed and the large values of the ratio of the moment of inertia to the reduced E2 transition probability, J Ž2.rB ŽE2., gave hints of a different kind of rotational motion. It was S. Frauendorf who, based on the Tilted-Axis-Cranking ŽTAC. model w2x, proposed the following picture for the Pb nuclei: h 9r2 and i 13r2 protons and i 13r2 neutron-holes align their respective spin vectors, jp and jn , to generate the total angular momentum in a way that resembles the closing of a pair of shears. After several years of intense work, the shears mechanism is now supported by a wealth of experimental data. In this context, the significant fingerprint for the shears picture is the behavior of the reduced M1 transition probabilities, and precise lifetime measurements of
states in the M1 bands of 193y199 Pb isotopes w3x carried out with Gammasphere provided the first sensitive experimental test of this new mechanism. This mode of rotation is not limited to the Pb region. In fact, structures with similar properties have been observed in other mass regions Žfor example in neutron-deficient Cd, Sn, In and Sm nuclei w4x. and can be explained within the same new picture. The shears mechanism represents a basically different way to generate angular momentum than the rotation of a core, and in general we will expect an interplay between these two modes. It is then important to establish when and how each of these mechanisms will dominate. In this letter we try to answer this question by studying a classical solution of a Two-ParticleŽblade.-plus-Rotor Model. In a previous work w5x we presented evidence that the shears angle, u , is the important degree of freedom in describing the electromagnetic properties of
0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 1 1 9 - 7
2
A.O. MacchiaÕelli et al.r Physics Letters B 450 (1999) 1–6
these M1 bands. In close analogy with the work of Schiffer w6x in the analysis of effective interactions from two-nucleon spectra, we proposed that the rotational-like behavior can be interpreted as a consequence of an effective interaction of the form V2 P2 Ž u ., between the proton and the neutron blades w7x. Characteristic of the long-range part of the nuclear force, this term splits the otherwise degenerate I s < jp y jn < . . . Ž jp q jn . multiplet. The excitation energy EŽ I . along the shears band is given by the change in potential energy caused by recoupling the angular momenta of the blades, E Ž I . s V2 P2 Ž u .
Ž 1.
where the sign of V2 describes particle-particle Žnegative. or particle-hole Žpositive. interactions. We show in Fig. 1 the properties of this interaction, assuming for simplicity that jp s jn s j and measuring the angular momentum in units of the maximum spin Ž2 j . available from the two blades, i.e. Iˆs Ir2 j. The parabolic behavior around the minima at Iˆs 0 and Iˆs '2 r2 is apparent and one can show w5,8x
that the resulting rotational-like properties are characterized by an effective moment of inertia given by J s j 2r3V2 . The known M1 bands can be described by the particle-hole case, and in particular by the region to the right of the minimum Žblades perpendicular., which is likely to be near-yrast. Note that the number of g-ray transitions in the band is n ; 2 j y '2 j, thus stressing the importance of large j values to generate long cascades. It still remains an interesting experimental challenge to look for the states on the unfavored side, to the left of the minimum. To study the competition between the shears mechanism and the rotation of the core we add to Eq. Ž1. a term representing the rotational energy of the core EŽ I . s
R2 2I
q V2 P2 Ž u .
Ž 2.
This expression for the total energy looks like the familiar Hamiltonian of the Particle-plus-Rotor
Fig. 1. Effective interaction, P2 Ž u ., as a function of Iˆs Ir2 j and u ŽTop axis. for both particle-particle Ždotted line. and particle-hole Žsolid line. cases. The latter is applicable to the known M1 bands; the minimum at Iˆs '2 r2 is indicated by the vertical dashed line.
A.O. MacchiaÕelli et al.r Physics Letters B 450 (1999) 1–6
Model w9x. Using the standard substitution, R s I y jp y jn , it goes into EŽ I . s
Ž I y jp y jn . 2I
2
q V2 P2 Ž u .
Ž 3.
which we solve in the classical limit by the requirement that at each spin the shears angle minimizes the energy. It is convenient to introduce the dimensionJ that relates less parameter x s I rŽ j 2r3V2 . s I rJ the moment of inertia of the core Ž I . to the effective moment of inertia of the shears Ž J ., or in other words, x s E2q Ž shears .rE2q Ž core . the ratio of the energies of an effective 2q of the shears to the 2q of the core. We can now rewrite Eq. Ž3. in the dimensionless form Eˆ Ž Iˆ. s E Ž I . r Ž 2 j 2rI .
ˆ s Iˆ 2 y 2 Icos q 12 y
u 2
q 12 cos u q
x 4
cos 2u
x
12 ˆ The minimization condition Ž EEuE .Iˆs 0 determines the function u Ž I,ˆ x . that gives the shears angle at each Iˆ for a given x . Intuitively we expect that the competi-
3
tion between these two modes should be governed by the parameter x measuring the relative strength of the residual interaction to the rotational energy of the core. Depending on the sign of V2 , x - 0 describes the particle-particle case and x ) 0 the particle-hole case. In this paper we study only the latter case, which, is the one relevant to the known M1 bands. From Eq. Ž4. and the minimization condition we obtain Iˆs cos
u 2
Ž 1 q x cos u .
Ž 4.
In Fig. 2, the shears angle satisfying Eq. Ž5. is shown as a function of the angular momentum for different values of x . Two limiting cases can be easily understood. If x ™ 0, the energy needed to recouple the blades to a given angular momentum is small compared to the energy associated with the core rotation and the solution of Ž5. gives exactly the angle between two angular momentum vectors coupled to spin I, i.e. cos u ™ 2 Iˆ2 y 1. If x ™ `, then cos u ™ 0; in this case it is more favorable for the core to rotate and for the shears to remain static at the minimum of the potential energy that occurs at u s 908 ŽSee Fig. 1.. The gradual transition between the
Fig. 2. The shears angle, obtained from the minimization of Eq. Ž4., as a function of Iˆ for different values of x as indicated.
A.O. MacchiaÕelli et al.r Physics Letters B 450 (1999) 1–6
4
two limiting cases discussed above is clearly seen. Notice that, independent of the value of x , the shears angle stays at 908 for Iˆs '2 r2. Also, it is interesting to calculate the spin at which the shears close, by setting u s 0 in Eq. Ž5. we get Iˆs 1 q x . This simple expression gives us a way to estimate x from the experimental data. If we know the bandhead spin configuration we can derive the maximum spin for the band Imax and compare it to the maxiobs obs mum spin observed Imax , then x f Imax rImax y 1. From the shears angle it is possible to calculate the core contribution to the total angular momentum as shown in Fig. 3. This behavior follows naturally from Figs. 1 and 2; when x ™ 0 it is so expensive for the core to rotate that its contribution is almost zero until Žof course. the shears close at Iˆs 1. From then on, the angular momentum can only be generated by the core but with an alignment of 2 j as indicated by the inset and therefore Iˆs Rˆ q 1. When x ™ `, following the arguments presented above, the shears like to stay at u f 908 and contribute an alignment of '2 2 j, thus Iˆs Rˆ q 0.707. As seen in 2
the figure, for other values of x the core contribution increases smoothly with the total angular momentum, between the limits discussed above. A sim-
ple estimation of this contribution can be obtained by using a linear approximation, in this case we have Rˆ f
x x q 0.293
Ž Iˆy 0.707 .
We see in Fig. 3 that for angular momenta below the minimum of the potential energy, the core is actually counter-rotating to compensate the spin of the blades and at the same time minimizing the total energy. In an effort to present our results in a concise way we would like to introduce a ‘‘phase diagram’’ between shears and core. An important issue here is how to define the transition between the two modes. A priori, one may want to define it at the point where the shears close. It then follows from the result above that even for large values of x there is always shears, a conclusion which seems intuitively wrong and argues against such a definition. On the other hand we know that the dynamic response of a rotating system is related to changes in angular momentum, D I. In terms of inertias, it is the dynamic moment that governs those changes. If we characterize these two phases in terms of their contributions to the total change in angular momentum, D I s D RŽ core . q D jŽ shears ., we can now properly
Fig. 3. The contribution of the core Ž Rˆ. to the total angular momentum as a function of I.ˆ Values of x are the same as those in Fig. 2.
A.O. MacchiaÕelli et al.r Physics Letters B 450 (1999) 1–6
define the transition point Ža given Iˆ and x pair of values. when each of the contributions is 50%. This ‘‘phase diagram’’ is shown in Fig. 4, where the shaded region corresponds to a dominant shears mechanism Ži.e. more than 50% contribution to the total change in angular momentum is provided by the shears.. When x is small the system prefers to generate angular momentum by the shears and the transition to core rotation occurs once the spins from the blades is exhausted. As the value of x increases it is more economical to follow the rotation of the core while the shears contribute less and less to the generation of angular momentum. The transition occurs much earlier than the spin at which the blades close: they are still closing, but ‘‘ very slowly’’! The core region at low spin and x f 0.5 can be traced back to the fast change in the counter-rotation of the core observed in the unfavored region in Fig. 3. Indicated by the the dashed line is the bandhead spin Ž'2 j .; the region above this line is that accessible experimentally since the states below it, having a lower spin and higher excitation energy, are likely non-yrast Žsee Fig. 1.. Qualitatively, we propose to associate a quenching of the shears mechanism when,
5
already at the bandhead, the contribution from the core is more than 50%. As seen in Fig. 4 this happens at x ; 0.5. From the arguments presented in Ref. w5x we expect both I and J to follow an A5r3 dependence and therefore the value of x should be common to all mass-regions. Using estimates from the Pb region, where we know that the shears mechanism dominates, we have E2q Ž shears . ; 150 keV requiring E2q Ž core . R 300 keV and a deformation w10x of the core e Q 0.12, in good agreement with the experimental observations. We can now summarize the main ingredients that are required for the shears mechanism to dominate: Ži. particle-hole coupling that provides angular momentum at the bandhead and thus favors near yrast structures; Žii. large j values that can give rise to rather long sequences of transitions; and Žiii. small deformation of the core. These requirements delineate the regions around doubly-magic nuclei as the most favorables for the appearance of shears bands. Similar conditions are expected from the calculations based on the TAC model w11x. To conclude, we have addressed the important question of the competition between the shears
Fig. 4. A ‘‘phase transition’’ diagram between the shears mechanism and the core rotation in the Iˆ versus x space. The dashed line at Iˆ s '2 r2 indicates the bandhead spin, the region above it is expected to be near-yrast. ŽSee text for details..
6
A.O. MacchiaÕelli et al.r Physics Letters B 450 (1999) 1–6
mechanism and the core rotation within the framework of a classical particles-plus-rotor model. The competition between these two modes depends on the relative strengths of the residual interaction and the rotational energy of the core given by the parameter x . Our results indicate that in all mass regions the shears mechanism should dominate for x - 0.5. Using data from the Pb region, this translates into deformations e Q 0.12.
Acknowledgements This work was supported by the U.S. Department of Energy under grant DE-AC03-76SF00098.
References w1x R.M. Clark et al., Phys. Lett. B 275 Ž1992. 247; G. Baldsiefen et al., Phys. Lett. B 275 Ž1992. 252, A. Kuhnert et al., Phys. Rev. C 46 Ž1992. 133. w2x S. Frauendorf, Nucl. Phys. A 557 Ž1993. 259c. w3x R.M. Clark et al., Phys. Rev. Lett. 78 Ž1997. 1868; Phys. Lett. B, in press. w4x See BAPS 43 Ž1998. 1098. w5x A.O. Macchiavelli et al., Phys. Rev. C 57 Ž1998. R1073. w6x J.P. Schiffer, W.W. True, Rev. Mod. Physics 48 Ž1976. 191, and references therein. w7x A.O. Macchiavelli et al., Phys. Rev. C 58 Ž1998. R621. w8x A.O. Macchiavelli et al., Phys. Rev. C, in press. w9x A. Bohr, B.R. Mottelson, Nuclear Structure, vol. II, W.A. Benjamin, New York, 1974. w10x L. Grodzins, Phys. Lett. 2 Ž1962. 88. w11x S. Frauendorf, Z. Phys. A 358 Ž1997. 163.
Physics Letters B 450 Ž1999. 1–6
The competition between the shears mechanism and core rotation in a classical particles-plus-rotor model A.O. Macchiavelli, R.M. Clark, M.A. Deleplanque, R.M. Diamond, P. Fallon, I.Y. Lee, F.S. Stephens, K. Vetter Nuclear Science DiÕision, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA Received 14 December 1998 Editor: W. Haxton
Abstract We present a model study of the competition between the shears mechanism and the rotation of the core. This is done in the framework of a classical solution of a Particles-plus-Rotor Model. The competition is governed by the relative strength of the residual interaction between the blades in the shears and the rotational energy of the core. The results are summarized in terms of a ‘‘phase-diagram’’ of these two modes of rotation that is applicable for all mass regions. We find that in order for the shears mechanism to dominate, the deformation of the core has to be e Q 0.12. q 1999 Published by Elsevier Science B.V. All rights reserved.
The rotational behavior of the M1 bands observed in neutron-deficient Pb isotopes w1x, came both as a surprise and as an intriguing puzzle for the nuclear structure community. The small E2rM1 branching ratios observed and the large values of the ratio of the moment of inertia to the reduced E2 transition probability, J Ž2.rB ŽE2., gave hints of a different kind of rotational motion. It was S. Frauendorf who, based on the Tilted-Axis-Cranking ŽTAC. model w2x, proposed the following picture for the Pb nuclei: h 9r2 and i 13r2 protons and i 13r2 neutron-holes align their respective spin vectors, jp and jn , to generate the total angular momentum in a way that resembles the closing of a pair of shears. After several years of intense work, the shears mechanism is now supported by a wealth of experimental data. In this context, the significant fingerprint for the shears picture is the behavior of the reduced M1 transition probabilities, and precise lifetime measurements of
states in the M1 bands of 193y199 Pb isotopes w3x carried out with Gammasphere provided the first sensitive experimental test of this new mechanism. This mode of rotation is not limited to the Pb region. In fact, structures with similar properties have been observed in other mass regions Žfor example in neutron-deficient Cd, Sn, In and Sm nuclei w4x. and can be explained within the same new picture. The shears mechanism represents a basically different way to generate angular momentum than the rotation of a core, and in general we will expect an interplay between these two modes. It is then important to establish when and how each of these mechanisms will dominate. In this letter we try to answer this question by studying a classical solution of a Two-ParticleŽblade.-plus-Rotor Model. In a previous work w5x we presented evidence that the shears angle, u , is the important degree of freedom in describing the electromagnetic properties of
0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 1 1 9 - 7
2
A.O. MacchiaÕelli et al.r Physics Letters B 450 (1999) 1–6
these M1 bands. In close analogy with the work of Schiffer w6x in the analysis of effective interactions from two-nucleon spectra, we proposed that the rotational-like behavior can be interpreted as a consequence of an effective interaction of the form V2 P2 Ž u ., between the proton and the neutron blades w7x. Characteristic of the long-range part of the nuclear force, this term splits the otherwise degenerate I s < jp y jn < . . . Ž jp q jn . multiplet. The excitation energy EŽ I . along the shears band is given by the change in potential energy caused by recoupling the angular momenta of the blades, E Ž I . s V2 P2 Ž u .
Ž 1.
where the sign of V2 describes particle-particle Žnegative. or particle-hole Žpositive. interactions. We show in Fig. 1 the properties of this interaction, assuming for simplicity that jp s jn s j and measuring the angular momentum in units of the maximum spin Ž2 j . available from the two blades, i.e. Iˆs Ir2 j. The parabolic behavior around the minima at Iˆs 0 and Iˆs '2 r2 is apparent and one can show w5,8x
that the resulting rotational-like properties are characterized by an effective moment of inertia given by J s j 2r3V2 . The known M1 bands can be described by the particle-hole case, and in particular by the region to the right of the minimum Žblades perpendicular., which is likely to be near-yrast. Note that the number of g-ray transitions in the band is n ; 2 j y '2 j, thus stressing the importance of large j values to generate long cascades. It still remains an interesting experimental challenge to look for the states on the unfavored side, to the left of the minimum. To study the competition between the shears mechanism and the rotation of the core we add to Eq. Ž1. a term representing the rotational energy of the core EŽ I . s
R2 2I
q V2 P2 Ž u .
Ž 2.
This expression for the total energy looks like the familiar Hamiltonian of the Particle-plus-Rotor
Fig. 1. Effective interaction, P2 Ž u ., as a function of Iˆs Ir2 j and u ŽTop axis. for both particle-particle Ždotted line. and particle-hole Žsolid line. cases. The latter is applicable to the known M1 bands; the minimum at Iˆs '2 r2 is indicated by the vertical dashed line.
A.O. MacchiaÕelli et al.r Physics Letters B 450 (1999) 1–6
Model w9x. Using the standard substitution, R s I y jp y jn , it goes into EŽ I . s
Ž I y jp y jn . 2I
2
q V2 P2 Ž u .
Ž 3.
which we solve in the classical limit by the requirement that at each spin the shears angle minimizes the energy. It is convenient to introduce the dimensionJ that relates less parameter x s I rŽ j 2r3V2 . s I rJ the moment of inertia of the core Ž I . to the effective moment of inertia of the shears Ž J ., or in other words, x s E2q Ž shears .rE2q Ž core . the ratio of the energies of an effective 2q of the shears to the 2q of the core. We can now rewrite Eq. Ž3. in the dimensionless form Eˆ Ž Iˆ. s E Ž I . r Ž 2 j 2rI .
ˆ s Iˆ 2 y 2 Icos q 12 y
u 2
q 12 cos u q
x 4
cos 2u
x
12 ˆ The minimization condition Ž EEuE .Iˆs 0 determines the function u Ž I,ˆ x . that gives the shears angle at each Iˆ for a given x . Intuitively we expect that the competi-
3
tion between these two modes should be governed by the parameter x measuring the relative strength of the residual interaction to the rotational energy of the core. Depending on the sign of V2 , x - 0 describes the particle-particle case and x ) 0 the particle-hole case. In this paper we study only the latter case, which, is the one relevant to the known M1 bands. From Eq. Ž4. and the minimization condition we obtain Iˆs cos
u 2
Ž 1 q x cos u .
Ž 4.
In Fig. 2, the shears angle satisfying Eq. Ž5. is shown as a function of the angular momentum for different values of x . Two limiting cases can be easily understood. If x ™ 0, the energy needed to recouple the blades to a given angular momentum is small compared to the energy associated with the core rotation and the solution of Ž5. gives exactly the angle between two angular momentum vectors coupled to spin I, i.e. cos u ™ 2 Iˆ2 y 1. If x ™ `, then cos u ™ 0; in this case it is more favorable for the core to rotate and for the shears to remain static at the minimum of the potential energy that occurs at u s 908 ŽSee Fig. 1.. The gradual transition between the
Fig. 2. The shears angle, obtained from the minimization of Eq. Ž4., as a function of Iˆ for different values of x as indicated.
A.O. MacchiaÕelli et al.r Physics Letters B 450 (1999) 1–6
4
two limiting cases discussed above is clearly seen. Notice that, independent of the value of x , the shears angle stays at 908 for Iˆs '2 r2. Also, it is interesting to calculate the spin at which the shears close, by setting u s 0 in Eq. Ž5. we get Iˆs 1 q x . This simple expression gives us a way to estimate x from the experimental data. If we know the bandhead spin configuration we can derive the maximum spin for the band Imax and compare it to the maxiobs obs mum spin observed Imax , then x f Imax rImax y 1. From the shears angle it is possible to calculate the core contribution to the total angular momentum as shown in Fig. 3. This behavior follows naturally from Figs. 1 and 2; when x ™ 0 it is so expensive for the core to rotate that its contribution is almost zero until Žof course. the shears close at Iˆs 1. From then on, the angular momentum can only be generated by the core but with an alignment of 2 j as indicated by the inset and therefore Iˆs Rˆ q 1. When x ™ `, following the arguments presented above, the shears like to stay at u f 908 and contribute an alignment of '2 2 j, thus Iˆs Rˆ q 0.707. As seen in 2
the figure, for other values of x the core contribution increases smoothly with the total angular momentum, between the limits discussed above. A sim-
ple estimation of this contribution can be obtained by using a linear approximation, in this case we have Rˆ f
x x q 0.293
Ž Iˆy 0.707 .
We see in Fig. 3 that for angular momenta below the minimum of the potential energy, the core is actually counter-rotating to compensate the spin of the blades and at the same time minimizing the total energy. In an effort to present our results in a concise way we would like to introduce a ‘‘phase diagram’’ between shears and core. An important issue here is how to define the transition between the two modes. A priori, one may want to define it at the point where the shears close. It then follows from the result above that even for large values of x there is always shears, a conclusion which seems intuitively wrong and argues against such a definition. On the other hand we know that the dynamic response of a rotating system is related to changes in angular momentum, D I. In terms of inertias, it is the dynamic moment that governs those changes. If we characterize these two phases in terms of their contributions to the total change in angular momentum, D I s D RŽ core . q D jŽ shears ., we can now properly
Fig. 3. The contribution of the core Ž Rˆ. to the total angular momentum as a function of I.ˆ Values of x are the same as those in Fig. 2.
A.O. MacchiaÕelli et al.r Physics Letters B 450 (1999) 1–6
define the transition point Ža given Iˆ and x pair of values. when each of the contributions is 50%. This ‘‘phase diagram’’ is shown in Fig. 4, where the shaded region corresponds to a dominant shears mechanism Ži.e. more than 50% contribution to the total change in angular momentum is provided by the shears.. When x is small the system prefers to generate angular momentum by the shears and the transition to core rotation occurs once the spins from the blades is exhausted. As the value of x increases it is more economical to follow the rotation of the core while the shears contribute less and less to the generation of angular momentum. The transition occurs much earlier than the spin at which the blades close: they are still closing, but ‘‘ very slowly’’! The core region at low spin and x f 0.5 can be traced back to the fast change in the counter-rotation of the core observed in the unfavored region in Fig. 3. Indicated by the the dashed line is the bandhead spin Ž'2 j .; the region above this line is that accessible experimentally since the states below it, having a lower spin and higher excitation energy, are likely non-yrast Žsee Fig. 1.. Qualitatively, we propose to associate a quenching of the shears mechanism when,
5
already at the bandhead, the contribution from the core is more than 50%. As seen in Fig. 4 this happens at x ; 0.5. From the arguments presented in Ref. w5x we expect both I and J to follow an A5r3 dependence and therefore the value of x should be common to all mass-regions. Using estimates from the Pb region, where we know that the shears mechanism dominates, we have E2q Ž shears . ; 150 keV requiring E2q Ž core . R 300 keV and a deformation w10x of the core e Q 0.12, in good agreement with the experimental observations. We can now summarize the main ingredients that are required for the shears mechanism to dominate: Ži. particle-hole coupling that provides angular momentum at the bandhead and thus favors near yrast structures; Žii. large j values that can give rise to rather long sequences of transitions; and Žiii. small deformation of the core. These requirements delineate the regions around doubly-magic nuclei as the most favorables for the appearance of shears bands. Similar conditions are expected from the calculations based on the TAC model w11x. To conclude, we have addressed the important question of the competition between the shears
Fig. 4. A ‘‘phase transition’’ diagram between the shears mechanism and the core rotation in the Iˆ versus x space. The dashed line at Iˆ s '2 r2 indicates the bandhead spin, the region above it is expected to be near-yrast. ŽSee text for details..
6
A.O. MacchiaÕelli et al.r Physics Letters B 450 (1999) 1–6
mechanism and the core rotation within the framework of a classical particles-plus-rotor model. The competition between these two modes depends on the relative strengths of the residual interaction and the rotational energy of the core given by the parameter x . Our results indicate that in all mass regions the shears mechanism should dominate for x - 0.5. Using data from the Pb region, this translates into deformations e Q 0.12.
Acknowledgements This work was supported by the U.S. Department of Energy under grant DE-AC03-76SF00098.
References w1x R.M. Clark et al., Phys. Lett. B 275 Ž1992. 247; G. Baldsiefen et al., Phys. Lett. B 275 Ž1992. 252, A. Kuhnert et al., Phys. Rev. C 46 Ž1992. 133. w2x S. Frauendorf, Nucl. Phys. A 557 Ž1993. 259c. w3x R.M. Clark et al., Phys. Rev. Lett. 78 Ž1997. 1868; Phys. Lett. B, in press. w4x See BAPS 43 Ž1998. 1098. w5x A.O. Macchiavelli et al., Phys. Rev. C 57 Ž1998. R1073. w6x J.P. Schiffer, W.W. True, Rev. Mod. Physics 48 Ž1976. 191, and references therein. w7x A.O. Macchiavelli et al., Phys. Rev. C 58 Ž1998. R621. w8x A.O. Macchiavelli et al., Phys. Rev. C, in press. w9x A. Bohr, B.R. Mottelson, Nuclear Structure, vol. II, W.A. Benjamin, New York, 1974. w10x L. Grodzins, Phys. Lett. 2 Ž1962. 88. w11x S. Frauendorf, Z. Phys. A 358 Ž1997. 163.