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The mass dependence of moments of inertia of rapidly-rotating nuclei
Physics Letters B 289 (1992) 267-270 North-Holland
PHYSICS LETTERS B
The mass dependence of moments of inertia of rapidly-rotating nuclei D.F. Winchell t, D.O. Ludwigsen 2 and J.D. G a r r e t t Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6371, USA Received 19 May 1992
An analysis of high-spin moments of inertia for the yrast sequences of deformed nuclei from A = 72-184 indicates that within each deformed mass region, the moment of inertia is surprisingly constant. In contrast, between the different deformed mass regions the A 5/3dependence of the moment of inertia, expected for a macroscopic rotor, is approximately restored.
The interplay between nuclear forces and the Coriolis and centrifugal effects arising in rapidly-rotating nuclei leads to a variety o f structural changes within the nucleus. Often the study o f the systematic variation o f experimental quantities over a range o f nuclei can provide insight into the microscopic structure of the nucleus. One such quantity accessible to experimental study [ 1 ] is the nuclear m o m e n t o f inertia, 0, which is a measure o f the force that must be applied to a deformed nucleus to change its rotational angular momentum. The rotational energy of a nucleus is a quadratic function o f the angular m o m e n t u m , / , E ( I ) = ( h 2 / 2 0 ) I ( I + 1 ). Therefore, it is possible to extract moments o f inertia for deformed nuclei from experimental data from both the first and second derivatives o f rotational energy with respect to angular m o m e n t u m [2], since d E / d / i s an experimentally measurable quantity, the gamma-ray transition energy. The term m o m e n t o f inertia in the present paper (unless otherwise stated) is taken to be the kinematic m o m e n t o f inertia, O ~~~, determined from the first derivative, L~(I)/h2= (2•-- 1 ) / E v ( I ) . ( E v ( I ) is the gamma-ray transition energy for the transition I - , I - 2. ) A dynamic m o m e n t of inertia, O t2 ~, also can be defined from the second derivative, 0 <2~/h2= 4 / Partial support from the Joint Institute for Heavy-Ion Research, Oak Ridge, TN 37831, USA. Participant in the University of Tennessee Science Alliance Undergraduate Summer Program from Beloit College, Beloit, WI 53511, USA.
[ E v ( I ) - E v ( I - 2 ) ]. An analysis similar to that described herein also can be made for 0(2); however, due to the increased fluctuations in 0 (2) [3] (it is a higher order derivative), the result o f such an analysis does not lend itself to such a straightforward interpretation. The m o m e n t o f inertia of a macroscopic ellipsoid is proportional to mr 2. For a nucleus the mass, m, is proportional to A and the radius, r, is proportional to A 1/3; SO the m o m e n t o f inertia might be expected to be proportional to A 5/3. O f course, the nucleus is a q u a n t u m system; hence deviations from this classical limit are expected and indeed are found. One o f the most important "microscopic" effects on the moment o f inertia is the short-range correlations between like nucleons in time-reversed orbits [ 4 ]. The superfluidity [ 5 ] that arises from such pair correlations results in a decreased m o m e n t o f inertia [ 6 ]. As the nucleus rotates, the Coriolis interaction leads to the "breaking" of pairs [ 7,8 ], thus reducing the nuclear superfluidity and increasing the m o m e n t o f inertia. At low spin the moments of inertia are greatly reduced relative to that expected for a rigid ellipsoid, see fig. 1. A dramatic increase is observed in the moment of inertia at the angular m o m e n t u m (or rotational frequency, hog) associated with breaking a single pair o f high-j particles ~l [ 1,8 ], often termed the ~l More recent works, see e.g. ref. [9], describe backbends in terms of band crossingsassociated with the excitation of a pair of highly-alignedquasiparticles.
0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
267
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h~o (MeV) Fig. 1. Plot of kinematic moments of inertia, O(l ) as a function of the rotational frequency, h09, for the yrast even-spin, positiveparity decay sequence in a typical deformed light nucleus, 76Kr, and a typical deformed heavy nucleus, '62Er. The transitions used in constructing fig. 2 are indicated, as is the rigid-body moment of inertia for an ellipsoid with deformation parameter fl= 0.4 for krypton and fl= 0.25 for erbium.
" b a c k b e n d " . M u c h o f this increase in the m o m e n t o f inertia usually is a t t r i b u t e d to reduced pair correlations; however, the intrinsic angular m o m e n t u m o f the u n p a i r e d particles also contributes. Indeed, the recent observation o f n u m e r o u s rotational sequences in neighboring odd- a n d even-mass nuclei with identical m o m e n t s o f inertia indicates that the detailed microscopic description o f nuclear m o m e n t s o f inertia is incomplete [ 10 ]. At high spin the m o m e n t s o f inertia corresponding to the yrast sequences o f e v e n - e v e n nuclei are rem a r k a b l y constant for the entire rare-earth mass region A = 1 5 2 - 1 8 4 [3], see also fig. 2. The average m o m e n t o f inertia associated with the 22+--.20 + transition in this mass region is 0 (1) = 66.6 M e V - 1 h 2 with a s t a n d a r d d e v i a t i o n o f 3.0 MeV -1 h 2 (i.e. roughly 5%), even though A 5/3 varies m o r e than 40% between A = 152 and 184. The absence o f a mass dependence for high-spin m o m e n t s o f inertia in the stably-deformed rare-earth region is a clear demonstration o f microscopic structural effects on this quantity. At large angular m o m e n t u m , where pair correlations 268
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A Fig. 2. Values of high-spin kinematic moments of inertia, 0 ¢~), plotted as a function of atomic number, A. Circles show the values of 0 °) corresponding to the 22+-,20 + transition in the A = 152-184 nuclei, and to the 18 + --, 16+ transition in the lighter nuclei (see fig. 1). The solid line is proportional to A 5/3 and normalized to pass through the centroid of the points in the rareearth region. Triangles show the values of 0 ~) for the 38 +~ 36 + transition (above the second, quasiproton, "backbend") in A= 158-162 nuclei, and the dashed line is normalized to pass through the centroid of these points.
are reduced, it is energetically favorable to occupy states associated with a large c o m p o n e n t o f angular m o m e n t u m aligned with the rotational axis. Thus, within a given mass region the various nuclei apparently all have the same valence orbitals occupied at high spin, a n d therefore, stabilize at about the same m o m e n t o f inertia [3]. Perhaps this overall constancy is not so surprising; however, the very small observed variation in the m o m e n t o f inertia over such a large mass region is striking! Similar nearly-constant m o m e n t s o f inertia have been o b t a i n e d recently [ 11 ] for rapidly-rotating e v e n - e v e n nuclei with A = 72-84; however, the m o r e restricted d a t a in this mass region is less definitive, see fig. 2. G i v e n the strikingly-constant m o m e n t s o f inertia in two specific regions o f d e f o r m e d nuclei, a question o f considerable interest is the global behavior o f highspin m o m e n t s o f inertia. Besides the A = 7 2 - 8 4 and the A = 152-184 mass regions, sufficient high-spin
Volume 289, number 3,4
PHYSICS LETTERSB
data also exist ~2 for deformed nuclei with A = 118138 for such an investigation of the moments of inertia. High-spin moments of inertia are shown in fig. 2 as a function of A for deformed even-even nuclei in the A = 7 2 - 8 4 , 118-138, and 152-184 mass regions. The open circles shown for the A = 152-184 region correspond to the 2 2 + ~ 2 0 + transition. These are for essentially the same set of points analyzed in ref. [ 3 ]. This particular transition was chosen from the region of nearly constant moments of inertia above the first and below the second backbend (see fig. 1 ) as a compromise between being consistently above the lowest-frequency backbend for the complete mass region and yet at a sufficiently low angular m o m e n t u m to allow enough data for a meaningful analysis. For the lighter nuclei (A = 72-84 and 118138) the yrast 18 +--, 16 + transition was used for the analysis. Even though this transition is just above the backbend it was chosen to increase the number of points in the analysis. Each data set was checked to insure that the 18 +--, 16 ÷ transition is outside of the backbend region and characteristic of the high-spin moment of inertia; however, some of the fluctuations observed in the data points may be due to fluctuations associated with different interaction strengths between the bands producing the backbend. Just as the moments of inertia at high spin for the A = 152-184 mass region are nearly independent of A, for the limited data shown they also are not strongly mass dependent for the two lighter regions of deformed nuclei, see fig. 2. Indeed for the A = 118-138 nuclei the lowest values are for the heaviest nuclei. The average value of O ~~) for A = 118-138 deformed nuclei is 39.7 MeV -~ h 2, and it is 23.7 MeV -~ h 2 in the A = 72-84 nuclei. The standard deviations are 3.5 and 2.1 MeV-1 h z respectively, i.e. about 9% of the average value in both cases. Although within a specific mass region, there is no systematic dependence of 0 (~) on A, on a "global" scale the A 5/3 dependence is approximately restored between the centroids of the various mass regions, see fig. 2. However, a sizeable discrepancy is observed for the lightest (A = 72-84) nuclei. On average, the moments of inertia for these nuclei lie about seven units above the A 3/3 curve (O ( 1) = 0.0120 A 5/3 M e V ~2 See the Oak Ridge - University of Tennessee High Spin Data Base, as described by Garrett et al. [ 12 ].
10 September 1992
h 2) extrapolated from the centroid of the A = 152-
184 nuclei (Ac~nt= 166.8 and O ~ t =60.9 MeV - I h2). This discrepancy is attributed to a structural difference of these light nuclei which have N ~ Z. Hence the proton and neutron Fermi levels are at nearly the same position with respect to the proton and neutron shell model states. Therefore, the lg9/2 protons and neutrons align at nearly identical rotational frequencies. The lowest backbend in A ~ 80 nuclei usually is associated with the alignment of both a pair of lg9/2 quasiprotons and a pair of lg9/2 quasineutrons [ 13 ]. (It has been verified that the experimental alignments associated with the A ~ 80 data are consistent with the alignment of both quasineutron and quasiproton pairs. ) As more pairs of nucleons are broken, the pair field of the nucleus decreases and the moment of inertia increases toward that of a rigid-body (i.e., it becomes less superfluid). Indeed, the highspin moments of inertia of the even-even A = 72-84 nuclei are in better agreement with an A 5/3 extrapolation of rare-earth data from above the second backbend (triangles in fig. 2). For these points, the 38 + --,36 + transition was used (see fig. 1 ). The yrast sequence of the A = 158-162 nuclei above the second backbend also is associated with aligned pairs of both quasiprotons and quasineutrons [ 14]. Coincidentally, the dashed curve showing the extrapolation from the triangles (0 = 0.0151 A 5/3 M e V - ~ h 2 calculated from the centroid values Ac~nt= 160.4 and ,a(~) t/cent = 71.5 M e V - ' h 2) is nearly identical to the rigid-body moment of inertia of an ellipsoid with an average radius of 1.2 A w3 fm and a quadrupole deformation of fl2=0.3. The striking deviations of the "global" moment-ofinertia data from that expected for a macroscopic object is yet another illustration of the quantal (independent-particle) nature of the atomic nucleus. However, when extended over a sufficiently-large range of nuclei, quantal fluctuations should average out and the experimental observables should approach the limiting classical values. Indeed, this seems to be the case for the moments of inertia of rapidlyrotating nuclei. For data within a specific shell highspin moments of inertia of deformed nuclei deviate from the A 5/3 dependence expected for macroscopic objects, indicating the importance of single-particle (quantal) effects. However, when such data are extended beyond a single shell, the average mass depen269
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dence of the macroscopic properties apparently are restored for a similar spectrum of excited quasiparticles, even though the rigid-body value is not achieved. The extent of the global agreement is perhaps surprising, since sizeable variations in deformations are predicted for each mass region. Discussions with C. Baktash a n d S.L. Tabor are acknowledged. Oak Ridge National Laboratory is m a n aged by M a r t i n Marietta Energy Systems, Inc. for the US D e p a r t m e n t of Energy, u n d e r contract No. DEAC05-84OR21400.
References [ 1]A. Johnson, H. Ryde and J. Sztarkier, Phys. Lett. B 34 (1971) 605. [2 ] Aa. Bohr and B. Mottelson, Phys. Scr. 24 ( 1981 ) 71. [3] J.M. Espino and J.D. Garrett, Nucl. Phys. A 492 (1989) 205.
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[4] Aa. Bohr and B.R. Mottetson, Mat. Fys. Medd. Dan. Vid. Selsk. 30 (1953) No. 1. [5] Aa. Bohr, B.R. Mottelson and D. Pines, Phys. Rev. 110 (1958) 936. [6] S.T. Belyaev,Mat. Fys. Medd. Dan. Vid. Selsk. 31 (1957) No. 11. [7] B.R. Mottelson and J.G. Valatin, Phys. Rev. Lett. 5 (1960). [8] F.S. Stephens and R. Simons, Nucl. Phys. A 183 (1972) 257. [9] R. Bengtssonand S. Frauendorf, Nucl. Phys. A 327 (1979) 139. [ 10] C. Baktash, J.D. Garrett, D.F. Winchelland A. Smith, Oak Ridge National Laboratory preprint (1992). [ 11 ] S.L. Tabor, Phys. Rev. C 45 (1992) 242. [12]J.D. Garrett et al., in: ORNL Physics Division report (1988), p. 70. [ 13] L. Luhmann, M. Debray, K.P. Lieb, W. Nazarewicz, B. Wormann, J. Eberth and T. Heck, Phys. Rev. C 31 (1985 ) 828. [14]J. Simpson, P.D. Forsyth, D. Howe, B.M. Nyako, M.A. Riley, J.F. Sharpey-Schafer, J. Bacelar, J.D. Garrett, G.B. Hagemann,B. Herskind,A. Holm and P.O. Tjom, Phys. Rev. Lett. 54 (1985) 1132.
PHYSICS LETTERS B
The mass dependence of moments of inertia of rapidly-rotating nuclei D.F. Winchell t, D.O. Ludwigsen 2 and J.D. G a r r e t t Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6371, USA Received 19 May 1992
An analysis of high-spin moments of inertia for the yrast sequences of deformed nuclei from A = 72-184 indicates that within each deformed mass region, the moment of inertia is surprisingly constant. In contrast, between the different deformed mass regions the A 5/3dependence of the moment of inertia, expected for a macroscopic rotor, is approximately restored.
The interplay between nuclear forces and the Coriolis and centrifugal effects arising in rapidly-rotating nuclei leads to a variety o f structural changes within the nucleus. Often the study o f the systematic variation o f experimental quantities over a range o f nuclei can provide insight into the microscopic structure of the nucleus. One such quantity accessible to experimental study [ 1 ] is the nuclear m o m e n t o f inertia, 0, which is a measure o f the force that must be applied to a deformed nucleus to change its rotational angular momentum. The rotational energy of a nucleus is a quadratic function o f the angular m o m e n t u m , / , E ( I ) = ( h 2 / 2 0 ) I ( I + 1 ). Therefore, it is possible to extract moments o f inertia for deformed nuclei from experimental data from both the first and second derivatives o f rotational energy with respect to angular m o m e n t u m [2], since d E / d / i s an experimentally measurable quantity, the gamma-ray transition energy. The term m o m e n t o f inertia in the present paper (unless otherwise stated) is taken to be the kinematic m o m e n t o f inertia, O ~~~, determined from the first derivative, L~(I)/h2= (2•-- 1 ) / E v ( I ) . ( E v ( I ) is the gamma-ray transition energy for the transition I - , I - 2. ) A dynamic m o m e n t of inertia, O t2 ~, also can be defined from the second derivative, 0 <2~/h2= 4 / Partial support from the Joint Institute for Heavy-Ion Research, Oak Ridge, TN 37831, USA. Participant in the University of Tennessee Science Alliance Undergraduate Summer Program from Beloit College, Beloit, WI 53511, USA.
[ E v ( I ) - E v ( I - 2 ) ]. An analysis similar to that described herein also can be made for 0(2); however, due to the increased fluctuations in 0 (2) [3] (it is a higher order derivative), the result o f such an analysis does not lend itself to such a straightforward interpretation. The m o m e n t o f inertia of a macroscopic ellipsoid is proportional to mr 2. For a nucleus the mass, m, is proportional to A and the radius, r, is proportional to A 1/3; SO the m o m e n t o f inertia might be expected to be proportional to A 5/3. O f course, the nucleus is a q u a n t u m system; hence deviations from this classical limit are expected and indeed are found. One o f the most important "microscopic" effects on the moment o f inertia is the short-range correlations between like nucleons in time-reversed orbits [ 4 ]. The superfluidity [ 5 ] that arises from such pair correlations results in a decreased m o m e n t o f inertia [ 6 ]. As the nucleus rotates, the Coriolis interaction leads to the "breaking" of pairs [ 7,8 ], thus reducing the nuclear superfluidity and increasing the m o m e n t o f inertia. At low spin the moments of inertia are greatly reduced relative to that expected for a rigid ellipsoid, see fig. 1. A dramatic increase is observed in the moment of inertia at the angular m o m e n t u m (or rotational frequency, hog) associated with breaking a single pair o f high-j particles ~l [ 1,8 ], often termed the ~l More recent works, see e.g. ref. [9], describe backbends in terms of band crossingsassociated with the excitation of a pair of highly-alignedquasiparticles.
0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
267
Volume 289, number 3,4
PHYSICS LETTERS B
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.............. ,oo~._ccc_ °
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h~o (MeV) Fig. 1. Plot of kinematic moments of inertia, O(l ) as a function of the rotational frequency, h09, for the yrast even-spin, positiveparity decay sequence in a typical deformed light nucleus, 76Kr, and a typical deformed heavy nucleus, '62Er. The transitions used in constructing fig. 2 are indicated, as is the rigid-body moment of inertia for an ellipsoid with deformation parameter fl= 0.4 for krypton and fl= 0.25 for erbium.
" b a c k b e n d " . M u c h o f this increase in the m o m e n t o f inertia usually is a t t r i b u t e d to reduced pair correlations; however, the intrinsic angular m o m e n t u m o f the u n p a i r e d particles also contributes. Indeed, the recent observation o f n u m e r o u s rotational sequences in neighboring odd- a n d even-mass nuclei with identical m o m e n t s o f inertia indicates that the detailed microscopic description o f nuclear m o m e n t s o f inertia is incomplete [ 10 ]. At high spin the m o m e n t s o f inertia corresponding to the yrast sequences o f e v e n - e v e n nuclei are rem a r k a b l y constant for the entire rare-earth mass region A = 1 5 2 - 1 8 4 [3], see also fig. 2. The average m o m e n t o f inertia associated with the 22+--.20 + transition in this mass region is 0 (1) = 66.6 M e V - 1 h 2 with a s t a n d a r d d e v i a t i o n o f 3.0 MeV -1 h 2 (i.e. roughly 5%), even though A 5/3 varies m o r e than 40% between A = 152 and 184. The absence o f a mass dependence for high-spin m o m e n t s o f inertia in the stably-deformed rare-earth region is a clear demonstration o f microscopic structural effects on this quantity. At large angular m o m e n t u m , where pair correlations 268
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A Fig. 2. Values of high-spin kinematic moments of inertia, 0 ¢~), plotted as a function of atomic number, A. Circles show the values of 0 °) corresponding to the 22+-,20 + transition in the A = 152-184 nuclei, and to the 18 + --, 16+ transition in the lighter nuclei (see fig. 1). The solid line is proportional to A 5/3 and normalized to pass through the centroid of the points in the rareearth region. Triangles show the values of 0 ~) for the 38 +~ 36 + transition (above the second, quasiproton, "backbend") in A= 158-162 nuclei, and the dashed line is normalized to pass through the centroid of these points.
are reduced, it is energetically favorable to occupy states associated with a large c o m p o n e n t o f angular m o m e n t u m aligned with the rotational axis. Thus, within a given mass region the various nuclei apparently all have the same valence orbitals occupied at high spin, a n d therefore, stabilize at about the same m o m e n t o f inertia [3]. Perhaps this overall constancy is not so surprising; however, the very small observed variation in the m o m e n t o f inertia over such a large mass region is striking! Similar nearly-constant m o m e n t s o f inertia have been o b t a i n e d recently [ 11 ] for rapidly-rotating e v e n - e v e n nuclei with A = 72-84; however, the m o r e restricted d a t a in this mass region is less definitive, see fig. 2. G i v e n the strikingly-constant m o m e n t s o f inertia in two specific regions o f d e f o r m e d nuclei, a question o f considerable interest is the global behavior o f highspin m o m e n t s o f inertia. Besides the A = 7 2 - 8 4 and the A = 152-184 mass regions, sufficient high-spin
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data also exist ~2 for deformed nuclei with A = 118138 for such an investigation of the moments of inertia. High-spin moments of inertia are shown in fig. 2 as a function of A for deformed even-even nuclei in the A = 7 2 - 8 4 , 118-138, and 152-184 mass regions. The open circles shown for the A = 152-184 region correspond to the 2 2 + ~ 2 0 + transition. These are for essentially the same set of points analyzed in ref. [ 3 ]. This particular transition was chosen from the region of nearly constant moments of inertia above the first and below the second backbend (see fig. 1 ) as a compromise between being consistently above the lowest-frequency backbend for the complete mass region and yet at a sufficiently low angular m o m e n t u m to allow enough data for a meaningful analysis. For the lighter nuclei (A = 72-84 and 118138) the yrast 18 +--, 16 + transition was used for the analysis. Even though this transition is just above the backbend it was chosen to increase the number of points in the analysis. Each data set was checked to insure that the 18 +--, 16 ÷ transition is outside of the backbend region and characteristic of the high-spin moment of inertia; however, some of the fluctuations observed in the data points may be due to fluctuations associated with different interaction strengths between the bands producing the backbend. Just as the moments of inertia at high spin for the A = 152-184 mass region are nearly independent of A, for the limited data shown they also are not strongly mass dependent for the two lighter regions of deformed nuclei, see fig. 2. Indeed for the A = 118-138 nuclei the lowest values are for the heaviest nuclei. The average value of O ~~) for A = 118-138 deformed nuclei is 39.7 MeV -~ h 2, and it is 23.7 MeV -~ h 2 in the A = 72-84 nuclei. The standard deviations are 3.5 and 2.1 MeV-1 h z respectively, i.e. about 9% of the average value in both cases. Although within a specific mass region, there is no systematic dependence of 0 (~) on A, on a "global" scale the A 5/3 dependence is approximately restored between the centroids of the various mass regions, see fig. 2. However, a sizeable discrepancy is observed for the lightest (A = 72-84) nuclei. On average, the moments of inertia for these nuclei lie about seven units above the A 3/3 curve (O ( 1) = 0.0120 A 5/3 M e V ~2 See the Oak Ridge - University of Tennessee High Spin Data Base, as described by Garrett et al. [ 12 ].
10 September 1992
h 2) extrapolated from the centroid of the A = 152-
184 nuclei (Ac~nt= 166.8 and O ~ t =60.9 MeV - I h2). This discrepancy is attributed to a structural difference of these light nuclei which have N ~ Z. Hence the proton and neutron Fermi levels are at nearly the same position with respect to the proton and neutron shell model states. Therefore, the lg9/2 protons and neutrons align at nearly identical rotational frequencies. The lowest backbend in A ~ 80 nuclei usually is associated with the alignment of both a pair of lg9/2 quasiprotons and a pair of lg9/2 quasineutrons [ 13 ]. (It has been verified that the experimental alignments associated with the A ~ 80 data are consistent with the alignment of both quasineutron and quasiproton pairs. ) As more pairs of nucleons are broken, the pair field of the nucleus decreases and the moment of inertia increases toward that of a rigid-body (i.e., it becomes less superfluid). Indeed, the highspin moments of inertia of the even-even A = 72-84 nuclei are in better agreement with an A 5/3 extrapolation of rare-earth data from above the second backbend (triangles in fig. 2). For these points, the 38 + --,36 + transition was used (see fig. 1 ). The yrast sequence of the A = 158-162 nuclei above the second backbend also is associated with aligned pairs of both quasiprotons and quasineutrons [ 14]. Coincidentally, the dashed curve showing the extrapolation from the triangles (0 = 0.0151 A 5/3 M e V - ~ h 2 calculated from the centroid values Ac~nt= 160.4 and ,a(~) t/cent = 71.5 M e V - ' h 2) is nearly identical to the rigid-body moment of inertia of an ellipsoid with an average radius of 1.2 A w3 fm and a quadrupole deformation of fl2=0.3. The striking deviations of the "global" moment-ofinertia data from that expected for a macroscopic object is yet another illustration of the quantal (independent-particle) nature of the atomic nucleus. However, when extended over a sufficiently-large range of nuclei, quantal fluctuations should average out and the experimental observables should approach the limiting classical values. Indeed, this seems to be the case for the moments of inertia of rapidlyrotating nuclei. For data within a specific shell highspin moments of inertia of deformed nuclei deviate from the A 5/3 dependence expected for macroscopic objects, indicating the importance of single-particle (quantal) effects. However, when such data are extended beyond a single shell, the average mass depen269
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dence of the macroscopic properties apparently are restored for a similar spectrum of excited quasiparticles, even though the rigid-body value is not achieved. The extent of the global agreement is perhaps surprising, since sizeable variations in deformations are predicted for each mass region. Discussions with C. Baktash a n d S.L. Tabor are acknowledged. Oak Ridge National Laboratory is m a n aged by M a r t i n Marietta Energy Systems, Inc. for the US D e p a r t m e n t of Energy, u n d e r contract No. DEAC05-84OR21400.
References [ 1]A. Johnson, H. Ryde and J. Sztarkier, Phys. Lett. B 34 (1971) 605. [2 ] Aa. Bohr and B. Mottelson, Phys. Scr. 24 ( 1981 ) 71. [3] J.M. Espino and J.D. Garrett, Nucl. Phys. A 492 (1989) 205.
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