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High spin shapes in 182Os with density dependent forces
Physics Letters B 322 (1994) 22-26 North-Holland
PHYSICS LETTERS B
High spin shapes in 182Oswith density dependent forces J.L. Egido, L.M. R o b l e d o Departamento de Fisica Tedrica C-XI, Universidad Aut6noma de Madrid, E-28049 Madrid, Spain
and R.R. Chasman Argonne National Laboratory, Argonne, 1L 60439, USA
Received 4 November 1993; revised manuscript received 22 December 1993 Editor: C. Mahaux The first shape-constrained high-spin Hartree-Fock-Bogoliubov calculations, using the density dependent Gogny interaction, have been carded out to investigate the energy minima in 182Os. Deep, well separated minima have been found for shapes with axis ratios of 1.5:1 and 2.2:1. Corrections for rotational zero point energies and for panicle number fluctuations are taken into account. Comparisons are made with cranked Strutinsky calculations for this nucleus. Superdeformed and hyperdeformed nuclear shapes have been an area of active experimental and theoretical investigation in the past decade. The first superdeformed nuclei were discovered some thirty years ago in the heavy elements [ 1 ] and explained in terms of the shell correction method [2]. With the availability of heavy ion beams and large detector arrays, it has become possible to extend the search for such phenomena, by exciting a nucleus to high spin. Superdeformation at high spin was predicted [3-5] for the A = 150 mass region and found in high spin studies [6]. Similar predictions [7] were made for the A = 190 mass region and superdeformed shapes have been found in this region at high spin [8]. An excited superdeformed m i n i m u m has been found for nuclei in this region in calculations [9-11] carried out at I = 0. Early calculations of hyperdeformed shapes [ 12 ] suggest that such shapes will become yrast at I ,-~ 80h for nuclei in the 17°yb mass region. Very extended nuclear shapes were also found [7] in calculations at high excitation energies for nuclei near A = 180 at I = 40. Extensive enery surfaces have been presented recently for many nuclei at high spin and high deformation [ 13]. These calculations include quadrupole 22
and hexadecapole deformations. A first observation ofhyperdeformation has been reported [ 14] in 153Dy. It has been suggested that a hyperdeformed minimum may become yrast in this region [ 15 ] when octupole deformation effects are taken into account. Recently, it has been suggested [16] that there is a region of very extended nuclear shapes centered at N = 106 that may be experimentally accessible in heavy ion reactions at spins of I ~ 60h. These shapes are characterized by axis ratios of ~ 2.2 : 1 and static moments of inertia of 140 ( M e V ) - I . It was found that a deformation space consisting of only ellipsoidal and hexadecapole degrees of freedom was not adequate to characterize these shapes, and that a necking degree of freedom gives a substantial lowering of the energy. The shapes that were found to minimize the energy had negative values for the necking parameter, implying a necking out relative to shapes characterized by only ellipsoidal and hexadecapole deformations. Two methods have been used for the study of nuclear energy surfaces; the shell correction method introduced by Strutinsky and the Hartree-Fock method. Each of the methods has advantages and disadvantages. In the Strutinsky approach, the number of grid points goes up substantially (at least an order of mag-
0370-2693/94/$ 07.00 ~) 1994-Elsevier Science B.V. All fights reserved SSDI 0370-2693 ( 9 4 ) 0 0 0 0 5 - R
Volume 322, number 1,2
PHYSICSLETTERSB
nitude) with each additional degree of freedom in deformation space. This makes it barely feasible to consider more than three degrees of freedom in deformation space. A great advantage of the Strutinsky approach is that one can study the energy surfaces of many nuclei (,-~ 300) with a single set of calculations. Although constrained Hartree-Fock calculations are quite time-consuming relative to Strutinsky calculations, they have the advantage of determining the shape at a minimum, without being limited to a few specified deformation modes. Indeed, all modes that are not explicitly constrained in the HartreeFock calculation are varied automatically in such a way as to minimize the energy. By constraining one degree of freedom, such as the quadrupole moment or the elongation, one can obtain a one-dimensional projection of the energy surface and determine the positions of the relative minima, the barriers between them and the shapes at the minima. One can exploit the strengths of both methods by first carrying out Strutinsky calculations to survey a region and then choose the most interesting nuclides to study with Hartree-Fock calculations. Our motivations in studying ~SEOs are: (1) to compare the results of Gogny force and cranked Strutinsky predictions at very large deformations and (2) to see whether the three deformation modes included in the Strutinsky study, are suffÉciently flexible to adequately describe the nuclear shape in the very extended minimum. High-spin studies with the Skyrme interaction have been carried out in the A = 190 region [ 17 ]. Recently [ 18 ] the density dependent Gogny force [ 19 ] has been applied to the calculation of high spin states of heavy nuclei. In that work the selfconsistent cranked Hartree-Fock-Bogoliubov (CHFB) equations were solved with the gradient method [20]. The high spin calculations with the Gogny force were parameter free and gave results in good agreement with the known properties of the nucleus 164Erat high angular momentum. In this work, we carry out for the first time constrained quadrupole moment calculations to study the high spin energy surface of lS2Os. Since the DS1 parameterization of the Gogny force used in this calculations gives fission barriers in reasonably good agreement with experiment [21 ], we expect a good description of extended shapes. We have carried out Hartree-Fock-Bogoliubov calculations to minimize the expectation value of the
10 February 1994
hamiltonian n'
= n - 2 N - ¢Ofx - q2~)20.
(1)
The Lagrange multipliers 2, to, and q2 are adjusted in such a way that the average values for theparticle number (N) = N, angular momentum (.Ix) = V ~ I + 1) and quadrupole moment (Q20) = Q20 have the desired values. The Lagrange multiplier q2 is varied to cover the range of (Q20) from 0 to 50 barn. The wave functions obtained by the variational principle conserve the symmetries only on the average. It is well know that such symmetry violating functions contain spurious components describing a motion where the orientation of the entire deformed density is rotated against the deformed well. The contributions of these spurious components must be subtracted from the energy obtained with the cranking prescription. The rotational correction, in particular, might be quite important here, as we are analyzing very different shapes. These corrections are usually written in the form ER(q2) ---- ( ~ ° ( q E ) l ( A J ) Z l ~ ° ( q 2 ) ) 2if(q2)
(2)
with I~0(q2)/ the q2 constrained wave function and if(q2) the moment of inertia. This correction can be derived in different approximations [22] that differ in the expression for the moment of inertia. In an angular momentum projected theory one obtains the Yoccoz moment of inertia; in the Random-PhaseApproximation (RPA), the Thouless-Valatin (or dynamical) moment of inertia (for (AJx)2) and the selfconsistent one for ,/2 + jff. The RPA framework provides also a generalization of eq. (2) to the case of several broken symmetries such as rotational invariance and particle number. The expression we use to evaluate these corrections, within the RPA, is given by eq. ( 9 3 ) o f r e f . [23]. In our calculations we use a triaxial harmonic oscillator basis determined through the energy cutoff condition a x n x + ayny + a z n z <~ No.
(3)
The coefficients ax, ay and az are related to the axis ratios of the matter distribution q = R z / R x and p = R y / R x through the usual relations ax = ( q p ) 1/3, ay = q l / 3 p - 2 / 3 and az = p1/3q-2/3. We have used the values q = 1.6 andp = 1.0 which are well suited for the 23
Volume 322, number 1,2
-1401 -1402 -1403 ""
m ~"
PHYSICS LETTERS B
1820S ",, " l
-1404
-1405 -1406 -1407 -1408 -1409
//
.~
I=60 h' / -.", / "
~
Table 1 Rotational and particle number correction energies (in MeV) for several values of the quadrupole moment.
/' // :
"",-""
"
", / "/ -"~ '~, ,:,t , '.'i
t" ' x ....
t
,
~ ./
/
-1410
__ I~.HFB
-1411
,\ t/ -- --
-1412
EHFB.+ECOl
.......
d
fo'
2'0
3'0
4'0
10 February 1994
Strutlnsky
5'0
6'0'70
Q20 (barn)
Q0 (barn)
ERot(TV)
Epair(Z + N)
6.371 9.557 15.928 22.299 28.670 33.448 38.226 44.597 47.783
0.890 1.354 2.069 2.779 2.984 3.371 3.402 3.846 3.940
0.357 0.000 0.066 0.307 1.384 1.425 1.704 1.782 1.903
Fig. 1. The potential energy versus the quadrupole moment. See text for further details. 9 superdeformed region and No has been varied from 11.1 to 15.1 in steps o f one unit to check the convergence o f the calculation with the basis size. The resuits show that already at No = 13.1 the deformation energy and the other mean field quantities (except for the total energy) are almost independent of the basis size. We have also carried out some calculations with q = 2.2 and No = 11.1, 12.1, 13.1 around the region o f very extended shapes. The q = 2.2, No = 13.1 resuits were very close to the q = 1.6, No = 13.1 ones giving us confidence in the adequacy o f our basis. The results reported in this work are for q = 1.6,p = 1.0 and No = 13.1 In fig. 1, we show our m a i n result, the energy surface o f lS2Os as a function of Q20. The dotted line represents the binding energy o f the nucleus without any correction. The dash-dotted curve is the energy surface obtained with the cranked Strutinsky calculation. In the C H F B calculations, the shapes are prolate in both m i n i m a ((Q22) is very small). The striking feature here is the fact that both calculations (CHFB and Strutinsky) give m i n i m a in the energy surface at values of Q20 that are quite similar, about 15 and 39 barn. The axis ratios are also very similar at these minima. However, the relative excitation energies o f the two m i n i m a are different in the two calculations. In the cranked Strutinsky calculations, the very extended m i n i m u m becomes yrast at I = 60h, whereas in the H a r t r e e - F o c k calculations it is excited by 5 MeV relative to the m i n i m u m at Q20 = 15 barn (v2 ,~ 0.4). The H a r t r e e - F o c k calculations have not 24
,-,
78
00
182 O s
1 = 6 0 ~h / ~
Proton
---
10
20
/
/
30
v
40
50
Q2o (barn) Fig. 2. The pairing energy along the constrained path, for protons (solid line) and neutrons (dashed line). yet been extended to sufficiently large values o f Q20 to examine the energy surface in the vicinity of the third m i n i m u m found in the cranked Strutinsky calculation and to study the fission barriers. In table 1 we display the angular m o m e n t u m and particle number corrections due to the broken symmetries in the HFB approximation. The rotational correction lowers the position o f the second m i n i m u m relative to the first by about 1.3 MeV. As the deformation increases along the constrained path, one expects an increase in both the numerator and the denominator o f the rotational correction given in eq. (2). Looking at fig. 2 - where we display the pairing energy as a function o f the quadrupole m o m e n t - and table 1, we find that the pairing correlations are negligible in the first m i n i m u m while in the second one
V o l u m e 322, n u m b e r 1,2
N
12 10 $ 6 4 2 0 -2 -4 45 -8 -10 -10
0
-6
-2 2 X (fro)
PHYSICS LETTERS B
6
10
-10
-6
-2 2 X (fm)
6
10
Fig. 3. Comparison of the HFB density and the sharp surface liquid drop shape (thick dashed line) at the minima. they are not. This reduces the m o m e n t o f inertia in the second m i n i m u m and gives a larger rotational energy correction than one would have obtained in the second m i n i m u m in the absence o f pairing correlations. The particle n u m b e r correction provides an additional 1.7 MeV, caused again by the vanishing of the pairing correlations in the first m i n i m u m and the sizeable pairing energy found in the second m i n i m u m ~1. When we a d d these corrections to the HFB energies along the constrained path we obtain the full line o f fig. 1. N o w the second m i n i m u m appears at an excitation energy o f 2 MeV with respect to the first minimum, in better agreement with the Strutinsky result. The cranked Strutinsky calculations differ slightly from the the results given previously [ 16 ], because the calculations have been extended to more grid points in the three dimensional deformation space. The values of u2 have been extended from 3.75 to 4.75 in steps of 0.05; u4 was varied from 0.50 to 1.36 in steps o f 0.02; and Unk was varied from - 1.0 to 0.20 in steps of 0.2. In fig. 3, we show a two dimensional projection o f the density for the nuclear shapes found at the two minima. We compare these shapes with the liquid drop shapes found at these two m i n i m a in the cranked Strutinsky calculations. The shapes found in the Gogny calculations differ slightly from the sharp surface liquid drop shapes found in the cranked Strutinsky calculations. The elongations are very similar, but the extension in the p direction is somewhat larger in the Strutinsky calculations. As can be seen the den*~l The effect of the zero point corrections is qualitative, a more quantitative estimate could be obtained by projection before the variation.
10 F e b r u a r y 1994
sity is not uniform in the nuclear interior in the Gogny calculations, whereas a uniform density is assumed in the liquid drop calculations. F o r the very extended shape, the curvature at the ends (at z ,,~ 10 fm) is reduced substantially relative to pure quadrupole shapes, and at the same time there is no necking in. The similarity o f the H a r t r e e - F o c k and the liquid drop shapes indicate that the necking degree o f freed o m is very useful for the parameterization o f extended nuclear shapes. In conclusion, we have performed the first constrained HFB calculations at high spin with the density dependent Gogny force to study shape isomers. Our results are in semi-quantitative agreement with earlier calculation [16] based on W o o d - S a x o n plus Strutinsky shell corrections. The calculations reported here were carried out in part at NERSC in Livermore. The research o f R.C. is supported by the US Dept. of Energy, Nuclear Physics Division under contract W31-109-Eng-38. The research of J.L.E. and L.M.R. is supported in part by D G I C y T , Spain under project PB91-0006. The collaboration has been greatly facilitated by a N A T O collaborative research grant 921182.
References [1] S.M. Polikanov et al., Zh. Exsp. Teor. Fiz. 42 (1962) 1016. [2] V.M. Strutinsky, Arkiv for Fysik 36 (1966) 629. [3] K. Neergaard and V. Paskevich, Phys. Lett. B 59 (1975) 218. [4] J. Dudek et al., Phys. Lett. B 112 (1982) 1. [5] J. Dudek and W. Nazarewicz, Phys. Rev. C 31 (1985) 298. [6] P.J. Twin et al., Phys. Rev. Lett. 57 (1986) 811. [7] R.R. Chasman Phys. Lett. B 219 (1989) 227. [8] E.F. Moore et al., Phys. Rev. Lett. 63 (1989) 360. [ 9 ] C.F. Tsang and S.G. Nilsson Nucl. Phys. A 140 (1970) 275. [10] M. Girod, J.P. Delaroche and J.F. Berger, Phys. Rev. C 38 (1988) 1, 519. [11] P. Bonche et al., Nucl. Phys. A 500 (1989) 308. [12] J. Dudek, T. Werner and L.L. Reidinger, Phys. Lett. B 211 (1988) 2, 52. [ 13] T.R. Werner and J. Dudek, Atomic Data Nuclear Data Tables 50 (1992) 179. [ 14 ] A. Galindo-Uribarri et al., Phys. Rev. Lett. 71 ( 1993 ) 231. 25
Volume 322, number 1,2
PHYSICS LETTERSB
[ 15 ] S. Aberg, Lund Inst. of Technology Report Lund-Mph-
92/09. [16] R.R. Chasman, Phys. Lett. B 302 (1993) 134. [17] H. Flocard et al., Nucl. Phys. A 557 (1993) 559c. [18] J.L. Egido and L.M. Robledo, Phys. Rev. Lett. 70 (1993) 2876. [19] D. Gogny, in Nuclear Selfconsistent Fields, eds. G. Ripka and M. Pomeuf (North-Holland, Amsterdam, 1975).
26
10 February 1994
[20] J.L. Egido, P. Ring and H.J. Mang, A 334 (1980) 1. [21 ] J.F. Berger, M. Girod and D. Gogny Nucl. Phys. A 428 (1984) 23c. [22] J.L. Egido and P. Ring, Phys. Lett. B 95 (1980) 331. [23] J.L. Egido, H.J. Mang and P. Ring, Nucl. Phys. A 341 (1980) 229.
PHYSICS LETTERS B
High spin shapes in 182Oswith density dependent forces J.L. Egido, L.M. R o b l e d o Departamento de Fisica Tedrica C-XI, Universidad Aut6noma de Madrid, E-28049 Madrid, Spain
and R.R. Chasman Argonne National Laboratory, Argonne, 1L 60439, USA
Received 4 November 1993; revised manuscript received 22 December 1993 Editor: C. Mahaux The first shape-constrained high-spin Hartree-Fock-Bogoliubov calculations, using the density dependent Gogny interaction, have been carded out to investigate the energy minima in 182Os. Deep, well separated minima have been found for shapes with axis ratios of 1.5:1 and 2.2:1. Corrections for rotational zero point energies and for panicle number fluctuations are taken into account. Comparisons are made with cranked Strutinsky calculations for this nucleus. Superdeformed and hyperdeformed nuclear shapes have been an area of active experimental and theoretical investigation in the past decade. The first superdeformed nuclei were discovered some thirty years ago in the heavy elements [ 1 ] and explained in terms of the shell correction method [2]. With the availability of heavy ion beams and large detector arrays, it has become possible to extend the search for such phenomena, by exciting a nucleus to high spin. Superdeformation at high spin was predicted [3-5] for the A = 150 mass region and found in high spin studies [6]. Similar predictions [7] were made for the A = 190 mass region and superdeformed shapes have been found in this region at high spin [8]. An excited superdeformed m i n i m u m has been found for nuclei in this region in calculations [9-11] carried out at I = 0. Early calculations of hyperdeformed shapes [ 12 ] suggest that such shapes will become yrast at I ,-~ 80h for nuclei in the 17°yb mass region. Very extended nuclear shapes were also found [7] in calculations at high excitation energies for nuclei near A = 180 at I = 40. Extensive enery surfaces have been presented recently for many nuclei at high spin and high deformation [ 13]. These calculations include quadrupole 22
and hexadecapole deformations. A first observation ofhyperdeformation has been reported [ 14] in 153Dy. It has been suggested that a hyperdeformed minimum may become yrast in this region [ 15 ] when octupole deformation effects are taken into account. Recently, it has been suggested [16] that there is a region of very extended nuclear shapes centered at N = 106 that may be experimentally accessible in heavy ion reactions at spins of I ~ 60h. These shapes are characterized by axis ratios of ~ 2.2 : 1 and static moments of inertia of 140 ( M e V ) - I . It was found that a deformation space consisting of only ellipsoidal and hexadecapole degrees of freedom was not adequate to characterize these shapes, and that a necking degree of freedom gives a substantial lowering of the energy. The shapes that were found to minimize the energy had negative values for the necking parameter, implying a necking out relative to shapes characterized by only ellipsoidal and hexadecapole deformations. Two methods have been used for the study of nuclear energy surfaces; the shell correction method introduced by Strutinsky and the Hartree-Fock method. Each of the methods has advantages and disadvantages. In the Strutinsky approach, the number of grid points goes up substantially (at least an order of mag-
0370-2693/94/$ 07.00 ~) 1994-Elsevier Science B.V. All fights reserved SSDI 0370-2693 ( 9 4 ) 0 0 0 0 5 - R
Volume 322, number 1,2
PHYSICSLETTERSB
nitude) with each additional degree of freedom in deformation space. This makes it barely feasible to consider more than three degrees of freedom in deformation space. A great advantage of the Strutinsky approach is that one can study the energy surfaces of many nuclei (,-~ 300) with a single set of calculations. Although constrained Hartree-Fock calculations are quite time-consuming relative to Strutinsky calculations, they have the advantage of determining the shape at a minimum, without being limited to a few specified deformation modes. Indeed, all modes that are not explicitly constrained in the HartreeFock calculation are varied automatically in such a way as to minimize the energy. By constraining one degree of freedom, such as the quadrupole moment or the elongation, one can obtain a one-dimensional projection of the energy surface and determine the positions of the relative minima, the barriers between them and the shapes at the minima. One can exploit the strengths of both methods by first carrying out Strutinsky calculations to survey a region and then choose the most interesting nuclides to study with Hartree-Fock calculations. Our motivations in studying ~SEOs are: (1) to compare the results of Gogny force and cranked Strutinsky predictions at very large deformations and (2) to see whether the three deformation modes included in the Strutinsky study, are suffÉciently flexible to adequately describe the nuclear shape in the very extended minimum. High-spin studies with the Skyrme interaction have been carried out in the A = 190 region [ 17 ]. Recently [ 18 ] the density dependent Gogny force [ 19 ] has been applied to the calculation of high spin states of heavy nuclei. In that work the selfconsistent cranked Hartree-Fock-Bogoliubov (CHFB) equations were solved with the gradient method [20]. The high spin calculations with the Gogny force were parameter free and gave results in good agreement with the known properties of the nucleus 164Erat high angular momentum. In this work, we carry out for the first time constrained quadrupole moment calculations to study the high spin energy surface of lS2Os. Since the DS1 parameterization of the Gogny force used in this calculations gives fission barriers in reasonably good agreement with experiment [21 ], we expect a good description of extended shapes. We have carried out Hartree-Fock-Bogoliubov calculations to minimize the expectation value of the
10 February 1994
hamiltonian n'
= n - 2 N - ¢Ofx - q2~)20.
(1)
The Lagrange multipliers 2, to, and q2 are adjusted in such a way that the average values for theparticle number (N) = N, angular momentum (.Ix) = V ~ I + 1) and quadrupole moment (Q20) = Q20 have the desired values. The Lagrange multiplier q2 is varied to cover the range of (Q20) from 0 to 50 barn. The wave functions obtained by the variational principle conserve the symmetries only on the average. It is well know that such symmetry violating functions contain spurious components describing a motion where the orientation of the entire deformed density is rotated against the deformed well. The contributions of these spurious components must be subtracted from the energy obtained with the cranking prescription. The rotational correction, in particular, might be quite important here, as we are analyzing very different shapes. These corrections are usually written in the form ER(q2) ---- ( ~ ° ( q E ) l ( A J ) Z l ~ ° ( q 2 ) ) 2if(q2)
(2)
with I~0(q2)/ the q2 constrained wave function and if(q2) the moment of inertia. This correction can be derived in different approximations [22] that differ in the expression for the moment of inertia. In an angular momentum projected theory one obtains the Yoccoz moment of inertia; in the Random-PhaseApproximation (RPA), the Thouless-Valatin (or dynamical) moment of inertia (for (AJx)2) and the selfconsistent one for ,/2 + jff. The RPA framework provides also a generalization of eq. (2) to the case of several broken symmetries such as rotational invariance and particle number. The expression we use to evaluate these corrections, within the RPA, is given by eq. ( 9 3 ) o f r e f . [23]. In our calculations we use a triaxial harmonic oscillator basis determined through the energy cutoff condition a x n x + ayny + a z n z <~ No.
(3)
The coefficients ax, ay and az are related to the axis ratios of the matter distribution q = R z / R x and p = R y / R x through the usual relations ax = ( q p ) 1/3, ay = q l / 3 p - 2 / 3 and az = p1/3q-2/3. We have used the values q = 1.6 andp = 1.0 which are well suited for the 23
Volume 322, number 1,2
-1401 -1402 -1403 ""
m ~"
PHYSICS LETTERS B
1820S ",, " l
-1404
-1405 -1406 -1407 -1408 -1409
//
.~
I=60 h' / -.", / "
~
Table 1 Rotational and particle number correction energies (in MeV) for several values of the quadrupole moment.
/' // :
"",-""
"
", / "/ -"~ '~, ,:,t , '.'i
t" ' x ....
t
,
~ ./
/
-1410
__ I~.HFB
-1411
,\ t/ -- --
-1412
EHFB.+ECOl
.......
d
fo'
2'0
3'0
4'0
10 February 1994
Strutlnsky
5'0
6'0'70
Q20 (barn)
Q0 (barn)
ERot(TV)
Epair(Z + N)
6.371 9.557 15.928 22.299 28.670 33.448 38.226 44.597 47.783
0.890 1.354 2.069 2.779 2.984 3.371 3.402 3.846 3.940
0.357 0.000 0.066 0.307 1.384 1.425 1.704 1.782 1.903
Fig. 1. The potential energy versus the quadrupole moment. See text for further details. 9 superdeformed region and No has been varied from 11.1 to 15.1 in steps o f one unit to check the convergence o f the calculation with the basis size. The resuits show that already at No = 13.1 the deformation energy and the other mean field quantities (except for the total energy) are almost independent of the basis size. We have also carried out some calculations with q = 2.2 and No = 11.1, 12.1, 13.1 around the region o f very extended shapes. The q = 2.2, No = 13.1 resuits were very close to the q = 1.6, No = 13.1 ones giving us confidence in the adequacy o f our basis. The results reported in this work are for q = 1.6,p = 1.0 and No = 13.1 In fig. 1, we show our m a i n result, the energy surface o f lS2Os as a function of Q20. The dotted line represents the binding energy o f the nucleus without any correction. The dash-dotted curve is the energy surface obtained with the cranked Strutinsky calculation. In the C H F B calculations, the shapes are prolate in both m i n i m a ((Q22) is very small). The striking feature here is the fact that both calculations (CHFB and Strutinsky) give m i n i m a in the energy surface at values of Q20 that are quite similar, about 15 and 39 barn. The axis ratios are also very similar at these minima. However, the relative excitation energies o f the two m i n i m a are different in the two calculations. In the cranked Strutinsky calculations, the very extended m i n i m u m becomes yrast at I = 60h, whereas in the H a r t r e e - F o c k calculations it is excited by 5 MeV relative to the m i n i m u m at Q20 = 15 barn (v2 ,~ 0.4). The H a r t r e e - F o c k calculations have not 24
,-,
78
00
182 O s
1 = 6 0 ~h / ~
Proton
---
10
20
/
/
30
v
40
50
Q2o (barn) Fig. 2. The pairing energy along the constrained path, for protons (solid line) and neutrons (dashed line). yet been extended to sufficiently large values o f Q20 to examine the energy surface in the vicinity of the third m i n i m u m found in the cranked Strutinsky calculation and to study the fission barriers. In table 1 we display the angular m o m e n t u m and particle number corrections due to the broken symmetries in the HFB approximation. The rotational correction lowers the position o f the second m i n i m u m relative to the first by about 1.3 MeV. As the deformation increases along the constrained path, one expects an increase in both the numerator and the denominator o f the rotational correction given in eq. (2). Looking at fig. 2 - where we display the pairing energy as a function o f the quadrupole m o m e n t - and table 1, we find that the pairing correlations are negligible in the first m i n i m u m while in the second one
V o l u m e 322, n u m b e r 1,2
N
12 10 $ 6 4 2 0 -2 -4 45 -8 -10 -10
0
-6
-2 2 X (fro)
PHYSICS LETTERS B
6
10
-10
-6
-2 2 X (fm)
6
10
Fig. 3. Comparison of the HFB density and the sharp surface liquid drop shape (thick dashed line) at the minima. they are not. This reduces the m o m e n t o f inertia in the second m i n i m u m and gives a larger rotational energy correction than one would have obtained in the second m i n i m u m in the absence o f pairing correlations. The particle n u m b e r correction provides an additional 1.7 MeV, caused again by the vanishing of the pairing correlations in the first m i n i m u m and the sizeable pairing energy found in the second m i n i m u m ~1. When we a d d these corrections to the HFB energies along the constrained path we obtain the full line o f fig. 1. N o w the second m i n i m u m appears at an excitation energy o f 2 MeV with respect to the first minimum, in better agreement with the Strutinsky result. The cranked Strutinsky calculations differ slightly from the the results given previously [ 16 ], because the calculations have been extended to more grid points in the three dimensional deformation space. The values of u2 have been extended from 3.75 to 4.75 in steps of 0.05; u4 was varied from 0.50 to 1.36 in steps o f 0.02; and Unk was varied from - 1.0 to 0.20 in steps of 0.2. In fig. 3, we show a two dimensional projection o f the density for the nuclear shapes found at the two minima. We compare these shapes with the liquid drop shapes found at these two m i n i m a in the cranked Strutinsky calculations. The shapes found in the Gogny calculations differ slightly from the sharp surface liquid drop shapes found in the cranked Strutinsky calculations. The elongations are very similar, but the extension in the p direction is somewhat larger in the Strutinsky calculations. As can be seen the den*~l The effect of the zero point corrections is qualitative, a more quantitative estimate could be obtained by projection before the variation.
10 F e b r u a r y 1994
sity is not uniform in the nuclear interior in the Gogny calculations, whereas a uniform density is assumed in the liquid drop calculations. F o r the very extended shape, the curvature at the ends (at z ,,~ 10 fm) is reduced substantially relative to pure quadrupole shapes, and at the same time there is no necking in. The similarity o f the H a r t r e e - F o c k and the liquid drop shapes indicate that the necking degree o f freed o m is very useful for the parameterization o f extended nuclear shapes. In conclusion, we have performed the first constrained HFB calculations at high spin with the density dependent Gogny force to study shape isomers. Our results are in semi-quantitative agreement with earlier calculation [16] based on W o o d - S a x o n plus Strutinsky shell corrections. The calculations reported here were carried out in part at NERSC in Livermore. The research o f R.C. is supported by the US Dept. of Energy, Nuclear Physics Division under contract W31-109-Eng-38. The research of J.L.E. and L.M.R. is supported in part by D G I C y T , Spain under project PB91-0006. The collaboration has been greatly facilitated by a N A T O collaborative research grant 921182.
References [1] S.M. Polikanov et al., Zh. Exsp. Teor. Fiz. 42 (1962) 1016. [2] V.M. Strutinsky, Arkiv for Fysik 36 (1966) 629. [3] K. Neergaard and V. Paskevich, Phys. Lett. B 59 (1975) 218. [4] J. Dudek et al., Phys. Lett. B 112 (1982) 1. [5] J. Dudek and W. Nazarewicz, Phys. Rev. C 31 (1985) 298. [6] P.J. Twin et al., Phys. Rev. Lett. 57 (1986) 811. [7] R.R. Chasman Phys. Lett. B 219 (1989) 227. [8] E.F. Moore et al., Phys. Rev. Lett. 63 (1989) 360. [ 9 ] C.F. Tsang and S.G. Nilsson Nucl. Phys. A 140 (1970) 275. [10] M. Girod, J.P. Delaroche and J.F. Berger, Phys. Rev. C 38 (1988) 1, 519. [11] P. Bonche et al., Nucl. Phys. A 500 (1989) 308. [12] J. Dudek, T. Werner and L.L. Reidinger, Phys. Lett. B 211 (1988) 2, 52. [ 13] T.R. Werner and J. Dudek, Atomic Data Nuclear Data Tables 50 (1992) 179. [ 14 ] A. Galindo-Uribarri et al., Phys. Rev. Lett. 71 ( 1993 ) 231. 25
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[ 15 ] S. Aberg, Lund Inst. of Technology Report Lund-Mph-
92/09. [16] R.R. Chasman, Phys. Lett. B 302 (1993) 134. [17] H. Flocard et al., Nucl. Phys. A 557 (1993) 559c. [18] J.L. Egido and L.M. Robledo, Phys. Rev. Lett. 70 (1993) 2876. [19] D. Gogny, in Nuclear Selfconsistent Fields, eds. G. Ripka and M. Pomeuf (North-Holland, Amsterdam, 1975).
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[20] J.L. Egido, P. Ring and H.J. Mang, A 334 (1980) 1. [21 ] J.F. Berger, M. Girod and D. Gogny Nucl. Phys. A 428 (1984) 23c. [22] J.L. Egido and P. Ring, Phys. Lett. B 95 (1980) 331. [23] J.L. Egido, H.J. Mang and P. Ring, Nucl. Phys. A 341 (1980) 229.