Angular-Momentum Projected Potential Energy Surfaces Based on a Combined Method. Jianzhong Gu. (China Institute of Atomic Energy, Beijing, China)

Transcription

1 Angular-Momentum Projected Potential Energy Surfaces Based on a Combined Method Jianzhong Gu (China Institute of Atomic Energy, Beijing, China) 2011 KLFTP-BLTP Joint Workshop on Nuclear Physics (Sep. 6-8, 2011, ITP, Beijing)

2 The work was done in collaboration with Wenhua Zou (CIAE), Yuan Tian (CIAE), Bangbao Peng (CIAE) Jiangming Yao (Southwest U), Zhongyu Ma (CIAE), and Shuifa Shen (East China Institute of Technolgy).

3 Outline The combined method and its justification 80,82,84 Zr Hg isotopes Summary

4 There are many nuclear models and theories, their connections are not so clear. Collecting their merits together one may better understand the nuclear many-body problem.

5 The angular momentum projected potential energy surfaces ( AMPPESs) method The Hamiltonian of the PSM does not contain the Coulomb interaction of protons which is indispensable for the potential energy surface (PES). To remedy this shortcoming of the PSM and compute the AMPPESs we combine the PSM with the QCRHB-NL3+separable- Gogny-D1S-force-theory. The quadrupole constrained RHB theory, in which the relativistic mean-field (RMF) Lagrangian is described by the NL3 effective interaction and the pairing correlations by a separable Gogny D1S force (QCRHB-NL3+separable-Gogny-D1S-forcetheory), We first calculate the ground-state PES based on the QCRHB-NL3 +separable-gogny-d1s-force-theory. Then we calculate the PES with a given angular momentum in the framework of the PSM. Finally, the energy difference between the PSM calculated PES with a non-zero angular momentum and that with zero spin is added to the ground-state PES, and a new PES is then formed, which, roughly speaking, has a given angular momentum. Certainly, since angular momentum projection has not yet been performed to the ground-state PES, anything added to the top of it is also unprojected. Those new PESs together with the ground-state PES constitute a group of the PESs with (approximate) given angular momenta. We would say that the ground-state PES serves as a kind of the band head of the PES group.

6 We would furthermore give some justifications for our combined method of calculation of the AMPPESs as follows. (a) For a great variety of many-body systems (including the nucleus), it is possible to describe the excitation spectra in terms of elementary modes of excitation representing the different, approximately independent, fluctuations about equilibrium( A. Bohr and B. R. Mottelson, Nuclear Structure (World Scientific, Singapore, 1998), Vol. 2). This implies a separation of scale between the excitations of many-body systems and their ground-state energies and justifies therefore our combined method where nuclear ground states are treated with the RHB, and nuclear excitations are described by the PSM. (b) The Hamiltonian used in the PSM is rather schematic for nuclear excitations, however, it takes into account the most important long-range correlations (the quadrupole-quadrupole correlation) and the most important short-range correlations (the pairing forces) (see, for example, P. Ring and P. Schuck, The nuclear many body problem, Spinger-Verlag (1980)). In this sense, the PSM is a shell-model like approach.

7 80,82,84 Zr near the N=Z line, abundant and exotic nuclear structure due to large parts of protons and neutrons in pfg orbitals and the intruder of the 1g 9/2.

8 A=80 mass region,shapes, shape coexistence, shape transitions and decay out of the super-deformed bands.

9 The decay out could be rather fragmented since the energy difference between the SD states and ND (spherical) states is as high as 6-8 MeV for 82 Zr and 84 Zr nuclei at high spins. Nevertheless, for 80 Zr nucleus, there is no decay out of the SD band since the barrier is so thick. Decay out in 84 Zr C. J. Chiara et al., Phys. Rev. C 73 (2006) (R).

10 The quadrupole constrained RHB theory, in which the relativistic mean-field (RMF) Lagrangian is described by the NL3 effective interaction and the pairing correlations by a separable Gogny D1S force (QCRHB-NL3+separable- Gogny-D1S-force-theory), Phys. Rev. C 82(2010) The quadrupole constrained relativistic mean-field framework with PC-PK1 parameter set. The pairing correlation is considered through a standard BCS method with a density-independent delta pairing force (AMP-QCPC-PK1+BCS approach). P. W. Zhao, Z. P. Li, J. M. Yao et al., New parametrization for the nuclear covariant energy density functional with point-coupling interaction, arxiv: v1 [nucl-th]; Phys. Rev. C 82 (2010) Hartree-Fock-Bogoliubov (HFB) axial mean field calculations based on the D1S Gogny interaction (HFB-full-Gogny-D1S), Eur. Phys. J. A 33 (2007) 237.

11 From Fig.2 one can see that the AMP-QCPC-PK1+BCS approach yields the equilibrium shapes which are consistent with the experimental data. However, the QCHFB-full-Gogny- D1S approach predicts the spherical equilibriumshapes for the three nuclei, which are inconsistent with the experimental data. So it is not a good approach to study the ground-state PESs. The experimental data: a strongly prolate shape > +0.4 for 80 Zr, approx 0.3 for 82 Zr and approx 0.2 for 84 Zr.

12 We recalculated the AMPPESs of 80,82,84 Zr nuclei by replacing the QCRHB- NL3+separable-Gogny-D1S-force-theory by the AMP-QCPC-PK1+BCS approach, and found that the two kinds of AMPPESs have a few common features:the strong shape mixing in 82 Zr and the decay out of the SD bands in nuclei 82,84 Zr although at low spins they are different from each other. The common features imply that the strong shape mixing and the decay out of the SD bands are not so sensitive to the choice of the band heads.

13 Experiments: for A=190 mass region, I=8-12 hbar at decay out points. Calculations: the barrier gets thin and low at such spins, and the decay out suddenly happens. Information for the excitation energies and spins at decay out points can be obtained, which is the most wanted for experiments. Bandheads here are taken from Eur. Phys. J. A 33 (2007) 237. A HFB approach based on the D1S Gogny force. Jianzhong Gu, Bangbao Peng, Wenhua Zou and Shuifa Shen, Nucl. Phys. A 834 (2010) 87c.

14 Feeding and Decay Process a beautiful double cycle between disorder and order A nucleus Suddenly changes its shape at low spins.

15 Over 300 SD bands in total up to now! Around 200 SD bands in these mass regions Total number of paths around 40 for 152Dy, while around 100 for 192Hg. 133Nd, path completely clear, 59Cu almost clear.

16 The intensities of the E2 gamma transitions within a SD band show a remarkable feature: The intra-band E2 transitions follow the band down with practically constant intensity. At some point, the transition intensity starts to drop sharply. This phenomenon is referred to as the decay out of a SD band. It is due to the mixing of the SD state and the normally deformed (ND) states with equal (similar) spin. The barrier separating the first and second minima of the deformation potential depends on and decreases with decreasing spin. Decay out of the SD band sets in at a certain spin value for which penetration through the barrier is competitive with the E2 decay within the SD band. The decay mechanism for the rapid depopulation?

17 E. Vigezzi, R. A. Broglia and T. Dossing, Nucl. Phys. A 520 (1990) 179c; Phys. Lett. B 249 (1990) 163. The theoretical description of the mixing between SD and ND states uses a statistical model for the ND states first proposed by Vigezzi et al. The ND states to which the SD state is coupled, lie several MeV above the ground state. The spectrum of these states is expected to be rather complex. The ND states can be described in terms of random--matrix theory, more precisely, by the Gaussian Orthogonal Ensemble (GOE) of random matrices. The results of this approach have been used to analyze experimental data. The formula actually used by Vigezzi et al. is not really derived from the statistical model. It is rather based on physically plausible and intuitive reasoning.

18 The Basic Picture

19 J. Z. Gu and H. A. Weidenmueller, Nucl. Phys. A 660 (1999) 197. exactly treated the model analytically and numerically The Hamiltonian H of the system is a matrix of dimension K+1 and has the form (j,l=1, K)

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21 Using the supersymmetry approach developed in Phys. Rep. 129 (1985) 367

22 A link among the most relevant observables for the decay out process has been established. Valid for A=150, 190 mass regions.

23 Comparison with the approach by Vigezzi et al J. Z. Gu and H. A. Weidenmueller, Nucl. Phys. A 660 (1999) 197. J. Z. Gu, Int. J. Mod. Phys. E 17 (Supplement ) (2008) 292. H. A. Weidenmueller et al., Rev. Mod. Phys. 81 (2009) 539.

24 Chaoticity dependence of the decay out intensity Aberg once concluded that the enhancement of the decay out of the SD band is due to the onset of Chaos (S. Aberg, Phys. Rev. Lett. 82 (1999) 299.) How does the degree of the chaoticity affect the decay out intensity?

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27 We conclude that the decay-out intensity less depends on the degree of chaoticity of the normal deformed states, putting Aberg s conclusion into question!

28 Overview theoretical activities of the decay out problem Decay out of a SD band continues to receive considerable theoretical attention. Sargeant et al. derived the formulae for the energy average and variance of the intraband decay intensity. They are strictly valid when the ND states are well overlapped. A. Sargeant, M. Hussein, M. Pato et al., Phys. Rev. C 65 (2002)

29 Stafford et al. calculated the decay out intensity based on a so-called two-level model (C. Stafford and B. Barrett, Phys. Rev. C60 (1999) ) where only one ND state is involved in the decay out process. This approach could be valid when the coupling between the SD state and ND states is rather weak, namely spreading width is small. Very recently, however, this approach was used to analyze the data in the 190 mass region (D. Cardamone, B. Barrett and C. Stafford, Phys. Lett. B 661 (2008) 233). However, in the 190 mass region, the decay from the SD to the normal states is spread over many different available paths. This means the SD state are coupled to many ND states. Therefore it is difficult to understand how the single ND state model is able to account for the data in the 190 mass region. In addition, we notice that the decay out intensity based on this model depends on the same ratios as those appearing in the Vigezzi model. The decay out intensity, therefore, is independent of the value of the fine structure constant, which is not physically plausible. The two-level model was generalized by Dzyublik and Utyuzh (Phys. Rev. C 68 (2003) ) a few years ago. They considered infinite equidistant ND states in their calculations.

30 Shimizu et al. studied the decay out problem by using the cranked Nilsson-Strutinsky model (Nucl. Phys. A 682 (2001) 464c; 696 (2001) 85). This model allows one to calculate the action for the superfluid tunnelling through the potential barrier separating the SD and ND potential wells. It predicts the dependence of the action on the spin of the state for which decay out of the SD band occurs. The action is related to the spreading width. Nevertheless, the large overestimation of the spreading width by this model has not been understood. A cluster model was suggested to study the decay out process by Adamian et al.( Phys. Rev. C 67 (2003) ; 69 (2004) ), in which a collective Hamiltonian depends only on a special degree of freedom (mass asymmetry coordinate) and determines the contribution of each cluster component to the total wave function of a nucleus.

31 Table 1 Tunneling width Γ tunn (in units of ev) J 188 Hg 190 Hg 192 Hg 194 Hg 196 Hg E-5 1.7E-7 2.1E-9 6.7E E E-4 5.7E-4 9.1E-7 2.3E E E-3 1.5E-6 3.8E-8 1.8E E-6 1.4E-8 The tunneling width could be identical to the spreading width, which shares the same order of magnitude as those predicted by the GW model (for instance, R. Kruecken et al., Phys. Rev. C 64 (2001) ).

32 Summary: (a) We proposed a new method to compute angular momentum projected nuclear potential energy surfaces; (b) The equilibrium shapes, shape coexistence, shape transitions and decay out of super-deformed bands for 80,82,84 Zr nuclei were studied; (c) The decay out for Hg isotopes is isospin dependent, the barrier gets thin and low at low spins, and the decay out suddenly happens, not the degree of the chaoticity, putting Aberg's conclusion into question (Phys. Rev. Lett. 82 (1999) 299). A challenge : describe the evolution of nuclear structure in a wide range of deformation!

33 There is no one size fits all nuclear theory. If you think all of the extant models and theories are useful, and if you want to develop a unified theory for the nuclear many-body problem you have to consider the relationship of the extant models and theories, which could be tough. Let us first bring them together, we may learn more and better.

34 Large amplitude collective motions (LACM) such as fission and shape coexistence. Microscopic understanding of the LACM is a long-standing fundamental subject of nuclear structure physics.

35 Shape coexistence stems from the competition between the pairing and deformation. An unstable HB minimum could serve as a starting point of time evolution of a manybody system.