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Cranked Hartree-Fock calculations of high spin states in 162Yb with the skyrme interaction
¢'olume 79B, number 4,5
PHYSICS LETTERS
4 December 1978
CRANKED HARTREE-FOCK CALCULATIONS OF HIGH SPIN STATES IN 162yb WITH THE SKYRME INTERACTION ¢: J. FLECKNER and U. MOSEL [nstitut far Theoretische Physik, Universitfft Giessen, 6300 Giessen, West Germany and Department of Physics, University o f Washington, Seattle, WA, USA and
P.G. ZINT Institut ffir Theoretisehe Physik, Universitfft Giessen, 6300 Giessen, West Germany Received 8 August 1978
The single-particle structure of the yrast line, the moment of inertia Q and the deformations/3 and ~ are discussed as a function of I in the spin region I ~ 40h-70//and compared with recent experimental results.
New techniques for measurements of continuum gamma-ray spectra have recently made the study of nuclear structure at angular momenta above 30h feasible. In this region of very high spin the pairing correlations in the nucleus have probably broken down so that the H a r t r e e - F o c k (HF) picture of independent particles seems to be an appropriate description of the nucleonic motion, in contrast to the H a r t r e e - F o c k Bogolyubov (HFB) approach for lower spins. As an approximation to the projection of the nuclear wavefunction onto good angular momentum a cranking approach was chosen [1 ]. The cranking hamiltonian H - c o I x is minimized in the subspace of Slater determinants, where H describes the hamiltonian of the system, c~ is the cranking frequency and I x the x-component of the total spin I. Denoting by q~,j a solution of the cranking hamiltonian to a given co the total spin of the system is determined by the equation [ I ( I + 1)] 1/2 = (Ocollxl(aw)" Due to the cranking part in the hamiltonian in the rotating case time reversal symmetry is broken, so that the remaining symmetries are parity and cR-invariance, Work supported by BMFT, GSI Darmstadt and US Department of Energy.
i.e. the invariance of the nucleus under a rotation of 7r around the rotation axis. The calculations were performed in a deformed cylinder-symmetric harmonic oscillator basis and were confirmed by HF calculations for this system in a rotating triaxial harmonic oscillator basis [2]. At each basis deformation the lowest 352 proton and 352 neutron states were used that include most of the N = 8 and N = 9 oscillator shells. This basis is large enough to make the results independent of the basis parameters. In order to include the effects of the core nucleons all single particle states were taken into account in the HF calculations. The numerical method used is similar to that given in ref. [3]. As a measure for convergence of the HF procedure the matrix elements of the HF harniltonian between occupied and unoccupied states were employed. As n u c l e o n - n u c l e o n interaction the Skyrme interaction with the parameter set V was used [4]. Giving good results for ground state properties of even-even nuclei [5], it is more fundamental than the usually applied schematic q u a d r u p o l e - q u a d r u p o l e forces. The breaking of the time reversal invariance due to the cranking term causes the appearance of additional densities in the HF hamiltonian [3,6]. We have found, however, that only the terms containing the current density, 343
Volume 79B, n u m b e r 4,5
70
i
PHYSICS LETTERS
I
I
4 December 1978
I
b6
60
b
b6 30-
s
bs
L
5o-
i
1 - -
f
b~
.jb2
/
./
/
b3 5-
40
0,4
I
0,6
~
0,8
I
1,0
U]
]5 w/MeV Fig. h Tile total spin I of the nucleus as a function of the cranking frequency co. Shown are the six bands which constitute the yrast line in this spin region. Thick lines refer to optiinal states, thin lines to particle-hole excitations. The horizontal lines give the transition points from one band to the other as can be inferred from fig. 2.
15
10
I
j = (1/2i)(V - V')p(r,
r')lr=r',
I 2000
I 3000 I(I+1)
contribute significantly so that we have neglected all other additional densities in obtaining the final results. The j-dependent terms are essential to restore the local galilean invariance once the nucleus is rotated. The calculations were performed with different but fixed values of co. After each iteration normally the N lowest neutron and Z lowest proton states were occupied (optimal states). Not all states on the yrast line, however, are such optimal states so that the different bands which arise from the different occupation numbers had to be continued to higher and lower spins with this fixed occupation. Fig. 1 shows that the total spin I is a multivalued function of the cranking frequency co corresponding to different initial configurations. The horizontal lines indicate the band crossings on the yrast line which lead to large changes in co. The thick lines indicate the optimal states, the thin lines particle hole states. The six yrast bands which were found in this spin region are labeled b 1 to b 6. In fig. 2 the lowest bands which resulted from the calculations are shown. Due to the plot o f energy versus I(I + 1) a constant moment o f inertia would show up as a straight line. The common structure of all the 344
--
[ 4000 ~--
t:ig. 2. Calculated yrast bands bl to b 6 and averaged yrast line Ydyn with a m o m e n t of inertia Qdyn = 77.5]/2 MeV-1. The excitation energy is given with respect to an HFB calculation for the ground state [12].
six bands is to start with a small slope near the crossing with the lower band and then to increase the slope gradually until they intersect with the higher band. In order to investigate how many nucleons actually do participate in the nuclear rotation, we have analyzed the single-particle contributions to the total angular momentum. There are basically two perhaps related - mechanisms for an increase o f l with co: first through changes of the sp-occupation numbers and second through a continuous change of the wave functions. For example, at I ~ 401i about 2 8 / / c a n be assigned to ph-excitations of only 5 nucleons whereas the remaining 12h are generated out of a different mixture of the wave functions compared with the ground state. Between most of the six bands only small changes of the occupation of the states occur. However, these give rise to large changes in L The most important states which become occupied are a proton i11/2 s t a t e and a n e u t r o n i13/2 state.
Volume 79B, number 4,5 L
PHYSICS LETTERS L
I
]'germ I
[
]'star
!&\
150
~0
-kay,
t
:0
100 o 14
T
>~
13
o
la2yb
o r~
50
-
o
o o
0 0
I 03
I i 1 [ 0,2 0,3 0,/~ 0,5 h 2t~2/MeV2 Fig. 3. Experimental (circles) [7] and theoretical results (lines) for various moments of inertia as explained in the text. The numbers in the figure denote the corresponding spins L It is interesting to derive collective parameters from this Hartree-Fock calculation. The shape of the yrast line indicates that the moment of inertia Q of the nucleus is not constant with respect to I. Because it has to be expected that the residual interaction will mix the different bands shown in fig. 2 and thus smoothen the transition from one to the other we have approximated the six bands by a straight line labeled Ydyn' The "dynamical" moment of inertia obtained from this line in Qdyn = 77"5h2 MeV-1, valid for the spin region I = 4 0 h - 7 0 h in 162yb. Also calculated are the "geometrical" moment of inertia ~ e o m which is that of a homogenous mass distribution with an ellipsoidal shape of the same deformation and the "static" inertia value Qstat defined as the expectation value o f y 2 + z 2 over the matter distribution generated by the Hartree-Fock wavefunctions. In fig. 3 all three moments of inertia are plotted together with experimental results obtained by Simon et al. [7] for this same nucleus. The moments of inertia 9dyn and ~stat agree within 10%. This agreement holds here even though the HF hamiltonian contains an effective mass. In Nilssonmodel calculations the momentum-dependent terms in the hamiltonian lead to an enhancement of Qdyn by a factor of meff/m over 9stat [8,9]. In the present case, however, this effect is compensated by the galilean in-
4 December 1978
variance of the hamiltonian [10] that is restored by the/'-dependent terms. The remaining difference between ~stat and 9dyn is probably due to shell effects. The difference in rotational energy for these moments of inertia is less than 1.5 MeV over the angular momentum range I = 40h-70fi. It is seen that the calculated values for the moment of inertia are all larger than the ones extracted from the experiment. These latter ones are even slightly smaller than the moment of inertia ~sph of a rigid sphere with a diffuse surface [7]. Since the nucleus 162yb is probably highly deformed at the high angular momenta treated here, the discrepancy between theory and experiment is not easy to explain. Some possibilities are: (i) that pairing correlations, that are known to decrease the moment of inertia still survive at these high spins; this seems unlikely in view of existing calculations [ 11 ] ; (ii) that there may be steeper bands above the yrast line, perhaps connected with rotations around a smaller axis of the nuclear shape. These bands will lead to higher transition energies and, therefore, smaller moments of inertia in the analysis of ref. [7]. More probably, however, is that the sharp-cut-off approximation is too unreliable for a determination of the spin of the yrast states and that in addition the evaporated particles may lead to a more complex angular momentum population of the evaporation residue 162yb perhaps even populating higher spins than in the primary nucleus. The deformation behaviour was determined in terms of the parameters/3 and 3'. The overall behaviour of the nucleus 162yb is like in ref. [6], but the changes in deformation are not so pronounced. At I = 40h the nucleus is already triaxial with a negative 3' value of about - 4 0 ° (note the different definition of the sign of 3' in ref. [8] ). For I = 60h 7 gets close to - 6 0 ° (band b 5 of the yrast line),/3 = 0.24, then the transition to the band b 6 reduces 3` to - 3 0 ° and increases ~3to 0.3 (centrifugal stretching). The nucleus is rather hard against changes in/3, whereas in ref. [8] 13changes from 0.15 for I = 60h to ~3= 0.5 for I = 70h occur, in these calculations/3 changes only from 0.2 to 0.3.
References [1] R. Beck, H.J. Mang and P. Ring, Z. Phys. 231 (1970) 26. [2] P.G. Zint, Z. Phys. A286 (1978) 281. [3] K.H. Passler and U. Mosel, Nucl. Phys. A257 (1976) 242. 345
Volume 79B, number 4,5
PHYSICS LETTERS
[4] T.H.R. Skyrme, Nucl. Phys. 9 (1959) 615; M. Beiner, H. Flocard, N. van Giai and P. Quentin, Nucl. Phys. A238 (1975) 29. [5] D. Vautherin and D.M. Brink, Phys. Rev. C5 (1971) 626. [6] K.H. Passler, Nucl. Phys. A257 (1976) 253; Y.M. Engel, D.M. Brink, K. Goeke, S.1. Krieger and D. Vautherin, Nucl. Phys. A249 (1975) 215. [7] R.S. Simon, M.V. Banaschik, R.M. Diamond, J.O. Newton and F.S. Stephens, Nucl. Phys. A290 (1977) 253. [8] G. Andersson et al., Nucl. Phys. A268 (1976) 205.
346
4 December 1978
[9] K. Neergard, V.V. Pashkevich and S. Frauendorf, Nucl. Phys. A262 (1976) 61. [10] A. Bohr and B.R. Mottelson, Nuclear structure, Vol. II (Benjamin, Reading, MA, 1975); A.B. Migdal, Nucl. Phys. 13 (1959) 655. [11] A. Faessler, K.R. Sandhya Devi, F. Griimmer, K.W. Schmid and R.R. Hilton, Nucl. Phys. A256 (1976) 106. [12] 1. Fleckner, U. Mosel, P. Ring and H.J. Mang, to be published.
PHYSICS LETTERS
4 December 1978
CRANKED HARTREE-FOCK CALCULATIONS OF HIGH SPIN STATES IN 162yb WITH THE SKYRME INTERACTION ¢: J. FLECKNER and U. MOSEL [nstitut far Theoretische Physik, Universitfft Giessen, 6300 Giessen, West Germany and Department of Physics, University o f Washington, Seattle, WA, USA and
P.G. ZINT Institut ffir Theoretisehe Physik, Universitfft Giessen, 6300 Giessen, West Germany Received 8 August 1978
The single-particle structure of the yrast line, the moment of inertia Q and the deformations/3 and ~ are discussed as a function of I in the spin region I ~ 40h-70//and compared with recent experimental results.
New techniques for measurements of continuum gamma-ray spectra have recently made the study of nuclear structure at angular momenta above 30h feasible. In this region of very high spin the pairing correlations in the nucleus have probably broken down so that the H a r t r e e - F o c k (HF) picture of independent particles seems to be an appropriate description of the nucleonic motion, in contrast to the H a r t r e e - F o c k Bogolyubov (HFB) approach for lower spins. As an approximation to the projection of the nuclear wavefunction onto good angular momentum a cranking approach was chosen [1 ]. The cranking hamiltonian H - c o I x is minimized in the subspace of Slater determinants, where H describes the hamiltonian of the system, c~ is the cranking frequency and I x the x-component of the total spin I. Denoting by q~,j a solution of the cranking hamiltonian to a given co the total spin of the system is determined by the equation [ I ( I + 1)] 1/2 = (Ocollxl(aw)" Due to the cranking part in the hamiltonian in the rotating case time reversal symmetry is broken, so that the remaining symmetries are parity and cR-invariance, Work supported by BMFT, GSI Darmstadt and US Department of Energy.
i.e. the invariance of the nucleus under a rotation of 7r around the rotation axis. The calculations were performed in a deformed cylinder-symmetric harmonic oscillator basis and were confirmed by HF calculations for this system in a rotating triaxial harmonic oscillator basis [2]. At each basis deformation the lowest 352 proton and 352 neutron states were used that include most of the N = 8 and N = 9 oscillator shells. This basis is large enough to make the results independent of the basis parameters. In order to include the effects of the core nucleons all single particle states were taken into account in the HF calculations. The numerical method used is similar to that given in ref. [3]. As a measure for convergence of the HF procedure the matrix elements of the HF harniltonian between occupied and unoccupied states were employed. As n u c l e o n - n u c l e o n interaction the Skyrme interaction with the parameter set V was used [4]. Giving good results for ground state properties of even-even nuclei [5], it is more fundamental than the usually applied schematic q u a d r u p o l e - q u a d r u p o l e forces. The breaking of the time reversal invariance due to the cranking term causes the appearance of additional densities in the HF hamiltonian [3,6]. We have found, however, that only the terms containing the current density, 343
Volume 79B, n u m b e r 4,5
70
i
PHYSICS LETTERS
I
I
4 December 1978
I
b6
60
b
b6 30-
s
bs
L
5o-
i
1 - -
f
b~
.jb2
/
./
/
b3 5-
40
0,4
I
0,6
~
0,8
I
1,0
U]
]5 w/MeV Fig. h Tile total spin I of the nucleus as a function of the cranking frequency co. Shown are the six bands which constitute the yrast line in this spin region. Thick lines refer to optiinal states, thin lines to particle-hole excitations. The horizontal lines give the transition points from one band to the other as can be inferred from fig. 2.
15
10
I
j = (1/2i)(V - V')p(r,
r')lr=r',
I 2000
I 3000 I(I+1)
contribute significantly so that we have neglected all other additional densities in obtaining the final results. The j-dependent terms are essential to restore the local galilean invariance once the nucleus is rotated. The calculations were performed with different but fixed values of co. After each iteration normally the N lowest neutron and Z lowest proton states were occupied (optimal states). Not all states on the yrast line, however, are such optimal states so that the different bands which arise from the different occupation numbers had to be continued to higher and lower spins with this fixed occupation. Fig. 1 shows that the total spin I is a multivalued function of the cranking frequency co corresponding to different initial configurations. The horizontal lines indicate the band crossings on the yrast line which lead to large changes in co. The thick lines indicate the optimal states, the thin lines particle hole states. The six yrast bands which were found in this spin region are labeled b 1 to b 6. In fig. 2 the lowest bands which resulted from the calculations are shown. Due to the plot o f energy versus I(I + 1) a constant moment o f inertia would show up as a straight line. The common structure of all the 344
--
[ 4000 ~--
t:ig. 2. Calculated yrast bands bl to b 6 and averaged yrast line Ydyn with a m o m e n t of inertia Qdyn = 77.5]/2 MeV-1. The excitation energy is given with respect to an HFB calculation for the ground state [12].
six bands is to start with a small slope near the crossing with the lower band and then to increase the slope gradually until they intersect with the higher band. In order to investigate how many nucleons actually do participate in the nuclear rotation, we have analyzed the single-particle contributions to the total angular momentum. There are basically two perhaps related - mechanisms for an increase o f l with co: first through changes of the sp-occupation numbers and second through a continuous change of the wave functions. For example, at I ~ 401i about 2 8 / / c a n be assigned to ph-excitations of only 5 nucleons whereas the remaining 12h are generated out of a different mixture of the wave functions compared with the ground state. Between most of the six bands only small changes of the occupation of the states occur. However, these give rise to large changes in L The most important states which become occupied are a proton i11/2 s t a t e and a n e u t r o n i13/2 state.
Volume 79B, number 4,5 L
PHYSICS LETTERS L
I
]'germ I
[
]'star
!&\
150
~0
-kay,
t
:0
100 o 14
T
>~
13
o
la2yb
o r~
50
-
o
o o
0 0
I 03
I i 1 [ 0,2 0,3 0,/~ 0,5 h 2t~2/MeV2 Fig. 3. Experimental (circles) [7] and theoretical results (lines) for various moments of inertia as explained in the text. The numbers in the figure denote the corresponding spins L It is interesting to derive collective parameters from this Hartree-Fock calculation. The shape of the yrast line indicates that the moment of inertia Q of the nucleus is not constant with respect to I. Because it has to be expected that the residual interaction will mix the different bands shown in fig. 2 and thus smoothen the transition from one to the other we have approximated the six bands by a straight line labeled Ydyn' The "dynamical" moment of inertia obtained from this line in Qdyn = 77"5h2 MeV-1, valid for the spin region I = 4 0 h - 7 0 h in 162yb. Also calculated are the "geometrical" moment of inertia ~ e o m which is that of a homogenous mass distribution with an ellipsoidal shape of the same deformation and the "static" inertia value Qstat defined as the expectation value o f y 2 + z 2 over the matter distribution generated by the Hartree-Fock wavefunctions. In fig. 3 all three moments of inertia are plotted together with experimental results obtained by Simon et al. [7] for this same nucleus. The moments of inertia 9dyn and ~stat agree within 10%. This agreement holds here even though the HF hamiltonian contains an effective mass. In Nilssonmodel calculations the momentum-dependent terms in the hamiltonian lead to an enhancement of Qdyn by a factor of meff/m over 9stat [8,9]. In the present case, however, this effect is compensated by the galilean in-
4 December 1978
variance of the hamiltonian [10] that is restored by the/'-dependent terms. The remaining difference between ~stat and 9dyn is probably due to shell effects. The difference in rotational energy for these moments of inertia is less than 1.5 MeV over the angular momentum range I = 40h-70fi. It is seen that the calculated values for the moment of inertia are all larger than the ones extracted from the experiment. These latter ones are even slightly smaller than the moment of inertia ~sph of a rigid sphere with a diffuse surface [7]. Since the nucleus 162yb is probably highly deformed at the high angular momenta treated here, the discrepancy between theory and experiment is not easy to explain. Some possibilities are: (i) that pairing correlations, that are known to decrease the moment of inertia still survive at these high spins; this seems unlikely in view of existing calculations [ 11 ] ; (ii) that there may be steeper bands above the yrast line, perhaps connected with rotations around a smaller axis of the nuclear shape. These bands will lead to higher transition energies and, therefore, smaller moments of inertia in the analysis of ref. [7]. More probably, however, is that the sharp-cut-off approximation is too unreliable for a determination of the spin of the yrast states and that in addition the evaporated particles may lead to a more complex angular momentum population of the evaporation residue 162yb perhaps even populating higher spins than in the primary nucleus. The deformation behaviour was determined in terms of the parameters/3 and 3'. The overall behaviour of the nucleus 162yb is like in ref. [6], but the changes in deformation are not so pronounced. At I = 40h the nucleus is already triaxial with a negative 3' value of about - 4 0 ° (note the different definition of the sign of 3' in ref. [8] ). For I = 60h 7 gets close to - 6 0 ° (band b 5 of the yrast line),/3 = 0.24, then the transition to the band b 6 reduces 3` to - 3 0 ° and increases ~3to 0.3 (centrifugal stretching). The nucleus is rather hard against changes in/3, whereas in ref. [8] 13changes from 0.15 for I = 60h to ~3= 0.5 for I = 70h occur, in these calculations/3 changes only from 0.2 to 0.3.
References [1] R. Beck, H.J. Mang and P. Ring, Z. Phys. 231 (1970) 26. [2] P.G. Zint, Z. Phys. A286 (1978) 281. [3] K.H. Passler and U. Mosel, Nucl. Phys. A257 (1976) 242. 345
Volume 79B, number 4,5
PHYSICS LETTERS
[4] T.H.R. Skyrme, Nucl. Phys. 9 (1959) 615; M. Beiner, H. Flocard, N. van Giai and P. Quentin, Nucl. Phys. A238 (1975) 29. [5] D. Vautherin and D.M. Brink, Phys. Rev. C5 (1971) 626. [6] K.H. Passler, Nucl. Phys. A257 (1976) 253; Y.M. Engel, D.M. Brink, K. Goeke, S.1. Krieger and D. Vautherin, Nucl. Phys. A249 (1975) 215. [7] R.S. Simon, M.V. Banaschik, R.M. Diamond, J.O. Newton and F.S. Stephens, Nucl. Phys. A290 (1977) 253. [8] G. Andersson et al., Nucl. Phys. A268 (1976) 205.
346
4 December 1978
[9] K. Neergard, V.V. Pashkevich and S. Frauendorf, Nucl. Phys. A262 (1976) 61. [10] A. Bohr and B.R. Mottelson, Nuclear structure, Vol. II (Benjamin, Reading, MA, 1975); A.B. Migdal, Nucl. Phys. 13 (1959) 655. [11] A. Faessler, K.R. Sandhya Devi, F. Griimmer, K.W. Schmid and R.R. Hilton, Nucl. Phys. A256 (1976) 106. [12] 1. Fleckner, U. Mosel, P. Ring and H.J. Mang, to be published.