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Shape-coexistence in the mass-70 region
Prog. Part. Nucl. Phys., Vol. 28, pp. 129-130, 1992.
0146-6410/92 $15.00 @ 1992 Pergamon Press Ltd
Printed in Great Britain. All tights reserved.
Shape-Coexistence in the Mass-70 Region E. BENDER*, K. W. SCHMID*, A. PETROVICVf, F. GR(~VlMER~ and A. FAESSLER* * Institutfllr Theoretische Physik der Universittlt Tabingen, Aufder MorgensteUe 14, W-7400 Tabingen, Germany t Institutefor Physics and Nuclear Engineering, Bucharest, Romania ~iInstitutfar Theoretische Physik H der Universitllt Bocham, Universitatsstrafle 150, W-4360Bochum, Germany
We used microscopic nuclear structure wavefunctions of the EXCITED VAMPIR and EXCITED FED VAMPIR models to analyse the experimental situation found in the nuclei around mass number 70. The calculated results agree reasonably well with experiment. They support strongly the predicted shape coexistence structures in the nuclei of this mass region. The doubly even nuclei in the mass 70 region reveal a couple of interesting features unique in the nuclear mass table. For instance these nuclei show an amazing great number of excited 0+-states at low excitation energies. The quadrupole moment of the first excited 2+-state changes rapidly with the mass number. And these nuclei display a rich band structure (Hamilton, 1989). Qualitatively these properties can be explained with help of the Nilsson-model by a competition of different configurations corresponding to various shapes. But for a detailed theoretical understanding one needs a completely microscopic model. In the models of the VAMPIR family used in this work the calculation of the wavefunctions is based on the variational principle (Schmid et al., 1987a, 1987b, 1989). The VAMPIR approach uses one symmetry-projected Hartree-Fock-Bogoliubov determinant to describe an y-rast state of a given symmetry, i.e. mass-number, z-component of isospin and spin-parity. In the EXCITED VAMPIR model also excited states of a given symmetry are described by one determinant. Here the y-rast solution found by a VAMPIR calculation is taken out of the variational space. In this restricted variational space the optimal symmetry projected determinant is searched for the first excited state. Then the lowest two solutions are excluded from the variational space and the third determinant is searched for the second excited state. This procedure is continued until one has found m different symmetry-projected determinants for the m lowest states. Finally the residual interaction between these states is diagonalized. The EXCITED FED VAMPIR uses several determinants to describe one state. In this approach the first determinant describing a state is obtained by an EXCITED VAMPIR calculation. It is not taken out of the variational space, but one looks for a second determinant correlating the first one. Then a third determinant is seached correlating the first two solutions and so on until one has found n determinants for the description of one state. Then this correlated state is taken out of the variational space and the next state is obtained in the same way. Again finally the residual interaction is diagonalized. This procedure accounts for the dominant correlations on top of the projected mean field solution irrespective where in energy they occur. In all these models states
129
130
E. Bender et al.
with totally different structures can be described simultaneously. Thus the models are well suited to give proof of shape coexistence phenomena. For details of the nuraereal calculation see Petrovici et
al. (1988, 1989, 1990, 1991). ~,, ~
O.0tB
. . . .
ft~l ~;ne
,
,o a e
(z:)
: '~neory
0.016i
>,
0.012 0.010 O.OOS
°¢
0.006
O.QOZ 0,000 -.002
r/~
Fig. 1. The calculated transition charge density for the first 2+-state in 7°Ge.
The EXCITED VAMPIR approach yielded in all considered even Ge and Se isotopes some low-lying 0+-states as observed by experiment. Also the measured quadrupole moments and other low spin properties could be well reproduced (Petrovici et al., 1988). Energy spectra of eSGe and r2Se calculated with the EXCITED VAMPIR and the EXCITED FED VAMPIR agree up to high spins with the complicated experimental band structure (Petrovici et al., 1989 and 1990, respectively). But the excitation energies and decay schemes allow only a rough comparison with experiment. To get a more quantitative comparison with experiment also the transitioncharge densities of some even mass Ge isotopes were determined within the FED VAMPIR approach (Petrovici et al., 1991). Fig. 1 shows the calculated and measured transition charge density for the first 2+-state in r°Ge. The agreement is very good for this collective transition. In all calculations the wavefunctions were found to be a complicated mixture of several configurations corresponding to different deformations. The results give a strong indication for shape coexistence in the mass-70
region.
REFERENCES Hamilton, J. H. (1989). Structures of Nuclei far from Stability. In: Treatise on Heavy Ion Science, (D. A. Bromley, ed.), Vol. 8., pp. 3-98. Plenum Press, New York. Petrovici, A. , K. W. Schmid, F. Griimmer, A. Faessler and T. Horibata (1988). Microscopic description of low-lying states in even Ge and Se nuclei. Nucl. Phys. A483, 317-347. Petrovici, A. , K. W. Schmid, F. Grfimmer and A. Faessler (1989). Shape-coexistence at high spins in the nuclei eSGe and r~Se. Nucl. Phys. A504, 277-299. Petrovici, A. , K. W. Schmid, F. Griimmer and A. Faessler (1990). Some new aspects of the shape coexistence in the A = 70 mass region. Nucl. Phys. A517, 108-124. Petrovici, A., K. W. Schmid, F. Griimmer and A. F~essler (1991). Calculation of charge and transition charge densities in some even mass Ge isotopes from microscopic nuclear structure wavefunctions. Z. Phys. 339, 71-79. Schmid, K. W., F. Griinuner and A. Faessler (1987a). Complex mean fields and unnatural parity pairing in the Hartree-Fock-Bogoliubov problem with symmetry-projection before the variation. Ann. Phys. 180, 1-73. Schmid, K. W. and F. Grt$mmer (1987b). Large-scale nuclear structure studies. Rep. Prog. Phys. 50, 731-781. Schmid, K. W., R.-R. Zheng, F. Gr/immer and A. Faessler (1989). Beyond symmetry-projected quasiparticle mean fields: A new variational procedure for nuclear structure calculations. Nucl. Phys. A499, 63-92.
0146-6410/92 $15.00 @ 1992 Pergamon Press Ltd
Printed in Great Britain. All tights reserved.
Shape-Coexistence in the Mass-70 Region E. BENDER*, K. W. SCHMID*, A. PETROVICVf, F. GR(~VlMER~ and A. FAESSLER* * Institutfllr Theoretische Physik der Universittlt Tabingen, Aufder MorgensteUe 14, W-7400 Tabingen, Germany t Institutefor Physics and Nuclear Engineering, Bucharest, Romania ~iInstitutfar Theoretische Physik H der Universitllt Bocham, Universitatsstrafle 150, W-4360Bochum, Germany
We used microscopic nuclear structure wavefunctions of the EXCITED VAMPIR and EXCITED FED VAMPIR models to analyse the experimental situation found in the nuclei around mass number 70. The calculated results agree reasonably well with experiment. They support strongly the predicted shape coexistence structures in the nuclei of this mass region. The doubly even nuclei in the mass 70 region reveal a couple of interesting features unique in the nuclear mass table. For instance these nuclei show an amazing great number of excited 0+-states at low excitation energies. The quadrupole moment of the first excited 2+-state changes rapidly with the mass number. And these nuclei display a rich band structure (Hamilton, 1989). Qualitatively these properties can be explained with help of the Nilsson-model by a competition of different configurations corresponding to various shapes. But for a detailed theoretical understanding one needs a completely microscopic model. In the models of the VAMPIR family used in this work the calculation of the wavefunctions is based on the variational principle (Schmid et al., 1987a, 1987b, 1989). The VAMPIR approach uses one symmetry-projected Hartree-Fock-Bogoliubov determinant to describe an y-rast state of a given symmetry, i.e. mass-number, z-component of isospin and spin-parity. In the EXCITED VAMPIR model also excited states of a given symmetry are described by one determinant. Here the y-rast solution found by a VAMPIR calculation is taken out of the variational space. In this restricted variational space the optimal symmetry projected determinant is searched for the first excited state. Then the lowest two solutions are excluded from the variational space and the third determinant is searched for the second excited state. This procedure is continued until one has found m different symmetry-projected determinants for the m lowest states. Finally the residual interaction between these states is diagonalized. The EXCITED FED VAMPIR uses several determinants to describe one state. In this approach the first determinant describing a state is obtained by an EXCITED VAMPIR calculation. It is not taken out of the variational space, but one looks for a second determinant correlating the first one. Then a third determinant is seached correlating the first two solutions and so on until one has found n determinants for the description of one state. Then this correlated state is taken out of the variational space and the next state is obtained in the same way. Again finally the residual interaction is diagonalized. This procedure accounts for the dominant correlations on top of the projected mean field solution irrespective where in energy they occur. In all these models states
129
130
E. Bender et al.
with totally different structures can be described simultaneously. Thus the models are well suited to give proof of shape coexistence phenomena. For details of the nuraereal calculation see Petrovici et
al. (1988, 1989, 1990, 1991). ~,, ~
O.0tB
. . . .
ft~l ~;ne
,
,o a e
(z:)
: '~neory
0.016i
>,
0.012 0.010 O.OOS
°¢
0.006
O.QOZ 0,000 -.002
r/~
Fig. 1. The calculated transition charge density for the first 2+-state in 7°Ge.
The EXCITED VAMPIR approach yielded in all considered even Ge and Se isotopes some low-lying 0+-states as observed by experiment. Also the measured quadrupole moments and other low spin properties could be well reproduced (Petrovici et al., 1988). Energy spectra of eSGe and r2Se calculated with the EXCITED VAMPIR and the EXCITED FED VAMPIR agree up to high spins with the complicated experimental band structure (Petrovici et al., 1989 and 1990, respectively). But the excitation energies and decay schemes allow only a rough comparison with experiment. To get a more quantitative comparison with experiment also the transitioncharge densities of some even mass Ge isotopes were determined within the FED VAMPIR approach (Petrovici et al., 1991). Fig. 1 shows the calculated and measured transition charge density for the first 2+-state in r°Ge. The agreement is very good for this collective transition. In all calculations the wavefunctions were found to be a complicated mixture of several configurations corresponding to different deformations. The results give a strong indication for shape coexistence in the mass-70
region.
REFERENCES Hamilton, J. H. (1989). Structures of Nuclei far from Stability. In: Treatise on Heavy Ion Science, (D. A. Bromley, ed.), Vol. 8., pp. 3-98. Plenum Press, New York. Petrovici, A. , K. W. Schmid, F. Griimmer, A. Faessler and T. Horibata (1988). Microscopic description of low-lying states in even Ge and Se nuclei. Nucl. Phys. A483, 317-347. Petrovici, A. , K. W. Schmid, F. Grfimmer and A. Faessler (1989). Shape-coexistence at high spins in the nuclei eSGe and r~Se. Nucl. Phys. A504, 277-299. Petrovici, A. , K. W. Schmid, F. Griimmer and A. Faessler (1990). Some new aspects of the shape coexistence in the A = 70 mass region. Nucl. Phys. A517, 108-124. Petrovici, A., K. W. Schmid, F. Griimmer and A. F~essler (1991). Calculation of charge and transition charge densities in some even mass Ge isotopes from microscopic nuclear structure wavefunctions. Z. Phys. 339, 71-79. Schmid, K. W., F. Griinuner and A. Faessler (1987a). Complex mean fields and unnatural parity pairing in the Hartree-Fock-Bogoliubov problem with symmetry-projection before the variation. Ann. Phys. 180, 1-73. Schmid, K. W. and F. Grt$mmer (1987b). Large-scale nuclear structure studies. Rep. Prog. Phys. 50, 731-781. Schmid, K. W., R.-R. Zheng, F. Gr/immer and A. Faessler (1989). Beyond symmetry-projected quasiparticle mean fields: A new variational procedure for nuclear structure calculations. Nucl. Phys. A499, 63-92.