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The interacting boson model and the structure of 182Hg
Volume 149B, number 4,5
PHYSICS LETTERS
20 December 1984
THE INTERACTING BOSON MODEL AND THE STRUCTURE OF 182Hg
A.F. BARFIELD and B.R. BARRETT 1 Department of Physics, Bldg. 81, University of Arizona, Tucson, AZ 85721, USA Received 10 August 1984
It is d e m o n s t r a t e d that the interacting boson model is completely consistent with newly observed energy levels in 182Hg. This is accomplished by making a slight change in only one o f the parameter values previously used.
Recent in-beam studies by Ma et al. [ 1] have identified some of the energy levels of 182Hg for the first time. In ref. [1] Ma et al. emphasize that our previous interacting boson model (IBM) calculations [2] predict that the deformed band in 182Hg will rise in energy compared to the same band in 184Hg, while their observations indicate that the energies of the deformed levels in 182Hg drop compared with 184Hg. The purpose of this paper is to make two points regarding our application of the IBM to the mercury isotopes. First, as clearly stated in ref. [2], the calculated deformed-band energies for 182Hg were higher than the corresponding energy levels in 184Hg because of "the assumption that the effective middle of the shell is at 184Hg" for the mercury isotopes. The deformedband energies in 182Hg could easily have been lower than those for 184Hg, had a slightly different choice of the IBM parameters been used in the calculations. Hence, it was not a prediction of the IBM, but an assumption on our part which cased the deformed-band energies in 182Hg to rise. The second purpose of this paper is to report a new IBM calculation for 182Hg, now that energy levels are known for this nuclide. A new neutron-proton IBM calculation (the so-called IBM-2) [3], with configuration mixing [4], has been performed for 182Hg. This calculation is the same as the previous calculation [2] except for a change in the value of the z Alexander yon H u m b o l d t Senior US Scientist Awardee
1983-84. 0370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
strength parameter r(3) for the Qrr" Qv interaction for the deformed band, from -0.120 MeV to -0.122 MeV. In short, an IBM-2 calculation with one protonhole boson (NTr = 1) was performed, in order to describe the nearly spherical ground-state (gs) band. A second IBM-2 calculation was then carried out with three proton-hole bosons (N~r = 3), representing the excitation of a proton boson across the Z = 82 shell gap, so as to describe the deformed band levels. The results of these two IBM-2 calculations were then mixed, as described in earlier work [2,4]. The results of these calculations are shown in fig. 1. It should be emphasized that the only difference between these
Experiment gsband
100D
Theory
Deformed band Obsr'd. Pred.
Before mixing_ N~r=l NT=3
After mixinq N~r=l N~r=3
1010 4" - 935 6"
1051 4*
941 6"
614 4o6_33__4"
629 4"
638 4" 526 2"
,.. 50C
=.,
955 8°
352 2"
t.~
0
0"
4_2J___2" 3~_8__.0÷
413 2" 429 2" 342 0"
~3 2" 3~
0
0
0÷
0*
0°
Fig. 1. Comparison between experimental energy levels [ 1] and theoretical levels calculated with the IBM-2 [2-4] for 182Hg" 277
Volume 149B, number 4,5
PHYSICS LETTERS
calculations and the previous ones is a slight change in the value of g(3) in the Qr" Qv interaction. All other parameter values are identical with those reported in ref. [2]. In fig. 1, the experimental results of Ma et al. [ 1] for the nearly spherical gs band and the deformed band are given on the left-hand side of the figure. It should be noted that the deformed-band levels indicated by dashed lines are predicted values based on the observed energies of the 6 + to 12 + levels, using the rotational energy formula E = E 0 + A J ( J + 1) + BJ2(J + 1) 2 withA = 15 keV andB = -11.5 eV [1]. As such, these predicted energy values for the 0 + to 4 + levels represent unperturbed values, i,e., they do not contain the effects of mixing with the nearly spherical gs band levels. Consequently, these predicted values should be compared with the IBM-2 results for N,r = 3 before mixing. These theoretical results are given in the middle of fig. 1 and are in excellent agreement (within a few keV) with the experimentally predicted unperturbed values (dashed lines). The theoretical results after mixing the nearly spherical gs band and the deformed band are shown on the right-hand side of fig. 1. The strong mixing between the 2 + states should be noted. Also, the energy of the deformed 4 + state has been increased slightly, compared with its unperturbed value. At first, this might seem to be a strange result; however, it must be remembered that all energies are measured with respect to the energy of the 0 + gs, which has been lowered by mixing with the deformed 0 + state. So, in this calculation, it is seen that the 0 + gs has been lowered in energy 9 keV more than the deformed 4 + state. This causes an apparent upward shift in the energy of the latter state. Unfortunately, this is in the wrong direction with regard to the experimental value. However, the overall agreement between the known experimentai energies and the theoretical values after mixing is extremely good, the differences being 24 keV or less. If we compare the theoretical values of E(02) + = 368 keV and E(2~) = 526 keV for 182Hg with the experimental values ofE(0~) = 375 keV and E(2~) = 535 keV for 184Hg [5], it would appear that the deformed band has reached or is close to its minimum energy at 182Hg. The effect of changing g(3) from -0.120 MeV to -0.122 MeV not only causes the deformed band to 278
20 December 1984
move down in energy for 182Hgcompared with 184Hg, but also makes some small changes in the eigenstates after mixing. The influence of these changes in the eigenstates on the calculated values of the electromagnetic properties is such that the new values for 182Hg continue the trends obtained previously for 184 -202Hg [2] ,instead of exhibiting an abrupt change from the latter trends. In conclusion, it has been demonstrated that the IBM is completely consistent with the newly observed energy levels in 182Hg, without making changes of any kind in the calculational techniques or in the calculated results for other mercury isotopes and with only a slight change in one of the IBM-2 parameter values for 182Hg. As an empirical model, the IBM-2 with configuration mixing [4] is consistent with all relevant experimental data for the mercury isotopes known at this time. One of us (B.R.B.) would like to thank Professor J.P. Elliot for his hospitality and that of the School of Mathematical and Physical Sciences at the University of Sussex where this manuscript was written; the Science and Engineering Research Council for a Visiting Fellowship; and Professor H.A. Weidenmilller for his hospitality and that of the Max-Planck Institute for Nuclear Physics, Heidelberg, where this work originated. One of us (A.F.B.)would like to thank Dr. G.A. Leander for valuable discussions about 182Hg and for his hospitality at Oak Ridge National Laboratory. We would also like to thank Oak Ridge Associated Universities for travel support for AFB under Department of Energy Contract No. DE-AC05760R00033. This material is based upon work supported in part by the National Science Foundation under Grant No. PHY-81-00141.
References [1] W.C. Ma et al., Phys. Lett. 139B (1984) 276. [2] A.F. Barfield, B.R. Barrett, K.A. Sage and P.D. Dural, Z. Phys. A311 (1983) 205. [3] A. Arima, T. Otsuka, F. Iachello and I. Talmi, Phys. Lett. 66B (1977) 205; 76B (1978) 139. [4] P.D. Dural and B.R. Barrett, Phys. Lett. 100B (1981) 223; Nucl. Phys. A376 (1982) 213. [5 ] C.M. Lederer and V.S. Shirley, eds., Table of isotopes, 7th Ed. (Wiley,New York, 1978).
PHYSICS LETTERS
20 December 1984
THE INTERACTING BOSON MODEL AND THE STRUCTURE OF 182Hg
A.F. BARFIELD and B.R. BARRETT 1 Department of Physics, Bldg. 81, University of Arizona, Tucson, AZ 85721, USA Received 10 August 1984
It is d e m o n s t r a t e d that the interacting boson model is completely consistent with newly observed energy levels in 182Hg. This is accomplished by making a slight change in only one o f the parameter values previously used.
Recent in-beam studies by Ma et al. [ 1] have identified some of the energy levels of 182Hg for the first time. In ref. [1] Ma et al. emphasize that our previous interacting boson model (IBM) calculations [2] predict that the deformed band in 182Hg will rise in energy compared to the same band in 184Hg, while their observations indicate that the energies of the deformed levels in 182Hg drop compared with 184Hg. The purpose of this paper is to make two points regarding our application of the IBM to the mercury isotopes. First, as clearly stated in ref. [2], the calculated deformed-band energies for 182Hg were higher than the corresponding energy levels in 184Hg because of "the assumption that the effective middle of the shell is at 184Hg" for the mercury isotopes. The deformedband energies in 182Hg could easily have been lower than those for 184Hg, had a slightly different choice of the IBM parameters been used in the calculations. Hence, it was not a prediction of the IBM, but an assumption on our part which cased the deformed-band energies in 182Hg to rise. The second purpose of this paper is to report a new IBM calculation for 182Hg, now that energy levels are known for this nuclide. A new neutron-proton IBM calculation (the so-called IBM-2) [3], with configuration mixing [4], has been performed for 182Hg. This calculation is the same as the previous calculation [2] except for a change in the value of the z Alexander yon H u m b o l d t Senior US Scientist Awardee
1983-84. 0370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
strength parameter r(3) for the Qrr" Qv interaction for the deformed band, from -0.120 MeV to -0.122 MeV. In short, an IBM-2 calculation with one protonhole boson (NTr = 1) was performed, in order to describe the nearly spherical ground-state (gs) band. A second IBM-2 calculation was then carried out with three proton-hole bosons (N~r = 3), representing the excitation of a proton boson across the Z = 82 shell gap, so as to describe the deformed band levels. The results of these two IBM-2 calculations were then mixed, as described in earlier work [2,4]. The results of these calculations are shown in fig. 1. It should be emphasized that the only difference between these
Experiment gsband
100D
Theory
Deformed band Obsr'd. Pred.
Before mixing_ N~r=l NT=3
After mixinq N~r=l N~r=3
1010 4" - 935 6"
1051 4*
941 6"
614 4o6_33__4"
629 4"
638 4" 526 2"
,.. 50C
=.,
955 8°
352 2"
t.~
0
0"
4_2J___2" 3~_8__.0÷
413 2" 429 2" 342 0"
~3 2" 3~
0
0
0÷
0*
0°
Fig. 1. Comparison between experimental energy levels [ 1] and theoretical levels calculated with the IBM-2 [2-4] for 182Hg" 277
Volume 149B, number 4,5
PHYSICS LETTERS
calculations and the previous ones is a slight change in the value of g(3) in the Qr" Qv interaction. All other parameter values are identical with those reported in ref. [2]. In fig. 1, the experimental results of Ma et al. [ 1] for the nearly spherical gs band and the deformed band are given on the left-hand side of the figure. It should be noted that the deformed-band levels indicated by dashed lines are predicted values based on the observed energies of the 6 + to 12 + levels, using the rotational energy formula E = E 0 + A J ( J + 1) + BJ2(J + 1) 2 withA = 15 keV andB = -11.5 eV [1]. As such, these predicted energy values for the 0 + to 4 + levels represent unperturbed values, i,e., they do not contain the effects of mixing with the nearly spherical gs band levels. Consequently, these predicted values should be compared with the IBM-2 results for N,r = 3 before mixing. These theoretical results are given in the middle of fig. 1 and are in excellent agreement (within a few keV) with the experimentally predicted unperturbed values (dashed lines). The theoretical results after mixing the nearly spherical gs band and the deformed band are shown on the right-hand side of fig. 1. The strong mixing between the 2 + states should be noted. Also, the energy of the deformed 4 + state has been increased slightly, compared with its unperturbed value. At first, this might seem to be a strange result; however, it must be remembered that all energies are measured with respect to the energy of the 0 + gs, which has been lowered by mixing with the deformed 0 + state. So, in this calculation, it is seen that the 0 + gs has been lowered in energy 9 keV more than the deformed 4 + state. This causes an apparent upward shift in the energy of the latter state. Unfortunately, this is in the wrong direction with regard to the experimental value. However, the overall agreement between the known experimentai energies and the theoretical values after mixing is extremely good, the differences being 24 keV or less. If we compare the theoretical values of E(02) + = 368 keV and E(2~) = 526 keV for 182Hg with the experimental values ofE(0~) = 375 keV and E(2~) = 535 keV for 184Hg [5], it would appear that the deformed band has reached or is close to its minimum energy at 182Hg. The effect of changing g(3) from -0.120 MeV to -0.122 MeV not only causes the deformed band to 278
20 December 1984
move down in energy for 182Hgcompared with 184Hg, but also makes some small changes in the eigenstates after mixing. The influence of these changes in the eigenstates on the calculated values of the electromagnetic properties is such that the new values for 182Hg continue the trends obtained previously for 184 -202Hg [2] ,instead of exhibiting an abrupt change from the latter trends. In conclusion, it has been demonstrated that the IBM is completely consistent with the newly observed energy levels in 182Hg, without making changes of any kind in the calculational techniques or in the calculated results for other mercury isotopes and with only a slight change in one of the IBM-2 parameter values for 182Hg. As an empirical model, the IBM-2 with configuration mixing [4] is consistent with all relevant experimental data for the mercury isotopes known at this time. One of us (B.R.B.) would like to thank Professor J.P. Elliot for his hospitality and that of the School of Mathematical and Physical Sciences at the University of Sussex where this manuscript was written; the Science and Engineering Research Council for a Visiting Fellowship; and Professor H.A. Weidenmilller for his hospitality and that of the Max-Planck Institute for Nuclear Physics, Heidelberg, where this work originated. One of us (A.F.B.)would like to thank Dr. G.A. Leander for valuable discussions about 182Hg and for his hospitality at Oak Ridge National Laboratory. We would also like to thank Oak Ridge Associated Universities for travel support for AFB under Department of Energy Contract No. DE-AC05760R00033. This material is based upon work supported in part by the National Science Foundation under Grant No. PHY-81-00141.
References [1] W.C. Ma et al., Phys. Lett. 139B (1984) 276. [2] A.F. Barfield, B.R. Barrett, K.A. Sage and P.D. Dural, Z. Phys. A311 (1983) 205. [3] A. Arima, T. Otsuka, F. Iachello and I. Talmi, Phys. Lett. 66B (1977) 205; 76B (1978) 139. [4] P.D. Dural and B.R. Barrett, Phys. Lett. 100B (1981) 223; Nucl. Phys. A376 (1982) 213. [5 ] C.M. Lederer and V.S. Shirley, eds., Table of isotopes, 7th Ed. (Wiley,New York, 1978).