In cooperation with the Iranian Nuclear Society

Document Type : Research Paper

Authors

1 Department of Physics, Ilam University, P.O.Box: 69315-516, Ilam - Iran

2 Department of Physics, University of Tabriz, P.O.Box: 51664, Tabriz - Iran

Abstract

In this paper, the Hamiltonian of Bohr-Mottelson model was solved to determine the energy levels and also energy surfaces of 190Hg nucleus in the Z(5) critical point. The Morse and Harmonic oscillator potentials are used for radial and angular parts of Hamiltonian, respectively. The asymptotic iteration method is used to solve radial equation and the constants of model are extracted in comparison with experimental data. The results are compared with the predictions of previous studies which solved Bohr-Hamiltonian in this critical point with using infinite well potential for radial part and also the predictions of O(6) dynamical limit of interacting boson model. Significant improvements are yield with using Morse potential in determination of energy levels of excited energy bands. Also, the results of this potential show more corresponding with the predictions of O(6) dynamical model.

Highlights

1. A. Bohr, B.R. Mottelson, Nuclear Structure, II, Benjamin, New York, )1975).

 

2. R.K. Sheline, Vibrational states in deformed even-even nuclei, Rev. Mod. Phys, 32, 1 (1960).

 

3. W. Greiner, J.A. Maruhn, Nuclear Models, Springer, Berlin, (1996).

 

4. F. Iachello, Analytic description of critical point nuclei in a spherical-axially deformed shape phase transition, Phys. Rev. Lett, 87, 052502 (2001).

 

5. Dennis Bonatsos, et. al, Z(5): critical point symmetry for the prolate to oblate nuclear shape phase transition, Phys. Lett. B. 588, 172 (2004).

 

6.  I. Boztosun, D. Bonatsos, I. Inci, Analytical solutions of the Bohr Hamiltonian with the Morse potential, Phys. Rev. C. 77, 044302 (2008).

 

7. G. Chen, The exact solutions of the Schrödinger equation with the Morse potential via Laplace transforms, Phys. Lett. A. 326, 55 (2004).

 

8. I. Inci, I. Boztosun, D. Bonatsos, The Bohr Hamiltonian Solution with the Morse Potential for the y-unstable and the Rotational Cases, AIP Conference Proceedings, 1072, 219 (2008).

 

9. I. Inci, I. Boztosun, D. Bonatsos, Electric quadrupole transitions of the Bohr Hamiltonian with the Morse potential, Phys. Rev. C. 84, 024309 (2011).

 

10. I. Inci, Exactly separable Bohr Hamiltonian with the Morse potential for triaxial nuclei, Int. J. Mod. Phys. E, 23, 1450053 (2014).

 

11. I. Inci, Investigation of γ-unstable odd-even nuclei in the Collective model framework with the Morse potential, Nuclear Phys. A, 991, 121611 (2019).

 

12. L. Fortunato, A. Vitturi, Analytically solvable potentials for γ-unstable nuclei, J. Phys. G, Nucl. Part. Phys. 29, 1341 (2003).

 

13. J.P. Elliott, Collective motion in the nuclear shell model. I. Classification schemes for states of mixed configurations, Proc. Roy. Soc. Lond. Ser. A. 245, 128 (1958).

 

14. F. Iachello, A. Arima, The Interacting Boson Model, (Cambridge U. Press, Cambridge, 1987).

 

15. R.F. Casten, D.D. Warner, The interacting boson approximate, Rev. Mod. Phys. 60, 389 (1988).

 

16. R.F. Casten, N.V. Zamfir, Empirical Realization of a Critical Point Description in Atomic Nuclei, Phys. Rev. Lett. 87, 052503 (2001).

 

17. G. Thiamova, D.J. Rowe, J.L. Wood, Coupled-SU(3) models of rotational states in nuclei, Nuclear Phys. A, 780, 112 (2006).

 

18. H. Fathi, et al., Investigation of shape phase transition in the U(5) - SO(6) transitional region by catastrophe theory and critical exponents of some quantities, Int. J. Mod. Phys. E 23, 1450045 (2014).

 

19. A.E.L. Dieperink, O. Scholten, F. Iachello, Classical Limit of the Interacting-Boson Model, Phys. Rev. Lett. 44, 1747 (1980).

 

20. D.J. Rowe, A computationally tractable version of the collective model, Nucl. Phys. A,735, 372 (2004).

 

21. H. Ciftci, R.L. Hall, N. Saad, Asymptotic iteration method for eigenvalue problems, J. Phys. A: Math. Gen. 36, 11807 (2003).

 

22. Nasser Saad, Richard L Hall, Hakan Ciftci, Criterion for polynomial solutions to a class of linear differential equations of second order, J. Phys. A: Math. Gen. 39, 13445 (2006).

 

23. C.L. Pekeris, The Rotation-Vibration Coupling in Diatomic Molecules, Phys. Rev. 45, 98 (1934).

 

24. C. Berkdemir, J. Han, Any l-state solutions of the Morse potential through the Pekeris approximation and Nikiforov–Uvarov method, Chem. Phys. Lett. 409, 203 (2005).

 

25. M.A. Caprio, Effects of β−γ coupling in transitional nuclei and the validity of the approximate separation of variables, Phys. Rev. C 72, 054323 (2005).

 

26. L. Fortunato, A. Vitturi, New analytic solutions of the collective Bohr Hamiltonian for a β-soft, γ-soft axial rotor, J. Phys. G: Nucl. Part. Phys. 30, 627 (2004).

 

27. Balraj Singh, Jun Chen, Nuclear Data Sheets A = 200, Nucl. Data Sheets, 169, 1 (2020).

28. M. Seidi, H. Sabri, The 116Te nucleus as candidate for U(5) dynamicalsymmetry, Acta Phys. Pol. B, 51, 2139 (2020).

 

29. D. Bonatsos, et al, Exactly separable version of the Bohr Hamiltonian with the Davidson potential, Phys. Rev. C, 76, 064312 (2007)

 

30. Z. Podoly´ak, Prolate-oblate shape transition in heavy neutron-rich nuclei, Journal of Physics: Conference Series, 381, 012052 (2012).

 

31. J. Jolie, A. Linnemann, Prolate-oblate phase transition in the Hf-Hg mass region, Phys. Rev. C 68 031301(R)(2007).

Keywords

1. A. Bohr, B.R. Mottelson, Nuclear Structure, II, Benjamin, New York, )1975).
 
2. R.K. Sheline, Vibrational states in deformed even-even nuclei, Rev. Mod. Phys, 32, 1 (1960).
 
3. W. Greiner, J.A. Maruhn, Nuclear Models, Springer, Berlin, (1996).
 
4. F. Iachello, Analytic description of critical point nuclei in a spherical-axially deformed shape phase transition, Phys. Rev. Lett, 87, 052502 (2001).
 
5. Dennis Bonatsos, et. al, Z(5): critical point symmetry for the prolate to oblate nuclear shape phase transition, Phys. Lett. B. 588, 172 (2004).
 
6.  I. Boztosun, D. Bonatsos, I. Inci, Analytical solutions of the Bohr Hamiltonian with the Morse potential, Phys. Rev. C. 77, 044302 (2008).
 
7. G. Chen, The exact solutions of the Schrödinger equation with the Morse potential via Laplace transforms, Phys. Lett. A. 326, 55 (2004).
 
8. I. Inci, I. Boztosun, D. Bonatsos, The Bohr Hamiltonian Solution with the Morse Potential for the y-unstable and the Rotational Cases, AIP Conference Proceedings, 1072, 219 (2008).
 
9. I. Inci, I. Boztosun, D. Bonatsos, Electric quadrupole transitions of the Bohr Hamiltonian with the Morse potential, Phys. Rev. C. 84, 024309 (2011).
 
10. I. Inci, Exactly separable Bohr Hamiltonian with the Morse potential for triaxial nuclei, Int. J. Mod. Phys. E, 23, 1450053 (2014).
 
11. I. Inci, Investigation of γ-unstable odd-even nuclei in the Collective model framework with the Morse potential, Nuclear Phys. A, 991, 121611 (2019).
 
12. L. Fortunato, A. Vitturi, Analytically solvable potentials for γ-unstable nuclei, J. Phys. G, Nucl. Part. Phys. 29, 1341 (2003).
 
13. J.P. Elliott, Collective motion in the nuclear shell model. I. Classification schemes for states of mixed configurations, Proc. Roy. Soc. Lond. Ser. A. 245, 128 (1958).
 
14. F. Iachello, A. Arima, The Interacting Boson Model, (Cambridge U. Press, Cambridge, 1987).
 
15. R.F. Casten, D.D. Warner, The interacting boson approximate, Rev. Mod. Phys. 60, 389 (1988).
 
16. R.F. Casten, N.V. Zamfir, Empirical Realization of a Critical Point Description in Atomic Nuclei, Phys. Rev. Lett. 87, 052503 (2001).
 
17. G. Thiamova, D.J. Rowe, J.L. Wood, Coupled-SU(3) models of rotational states in nuclei, Nuclear Phys. A, 780, 112 (2006).
 
18. H. Fathi, et al., Investigation of shape phase transition in the U(5) - SO(6) transitional region by catastrophe theory and critical exponents of some quantities, Int. J. Mod. Phys. E 23, 1450045 (2014).
 
19. A.E.L. Dieperink, O. Scholten, F. Iachello, Classical Limit of the Interacting-Boson Model, Phys. Rev. Lett. 44, 1747 (1980).
 
20. D.J. Rowe, A computationally tractable version of the collective model, Nucl. Phys. A,735, 372 (2004).
 
21. H. Ciftci, R.L. Hall, N. Saad, Asymptotic iteration method for eigenvalue problems, J. Phys. A: Math. Gen. 36, 11807 (2003).
 
22. Nasser Saad, Richard L Hall, Hakan Ciftci, Criterion for polynomial solutions to a class of linear differential equations of second order, J. Phys. A: Math. Gen. 39, 13445 (2006).
 
23. C.L. Pekeris, The Rotation-Vibration Coupling in Diatomic Molecules, Phys. Rev. 45, 98 (1934).
 
24. C. Berkdemir, J. Han, Any l-state solutions of the Morse potential through the Pekeris approximation and Nikiforov–Uvarov method, Chem. Phys. Lett. 409, 203 (2005).
 
25. M.A. Caprio, Effects of β−γ coupling in transitional nuclei and the validity of the approximate separation of variables, Phys. Rev. C 72, 054323 (2005).
 
26. L. Fortunato, A. Vitturi, New analytic solutions of the collective Bohr Hamiltonian for a β-soft, γ-soft axial rotor, J. Phys. G: Nucl. Part. Phys. 30, 627 (2004).
 
27. Balraj Singh, Jun Chen, Nuclear Data Sheets A = 200, Nucl. Data Sheets, 169, 1 (2020).
28. M. Seidi, H. Sabri, The 116Te nucleus as candidate for U(5) dynamicalsymmetry, Acta Phys. Pol. B, 51, 2139 (2020).
 
29. D. Bonatsos, et al, Exactly separable version of the Bohr Hamiltonian with the Davidson potential, Phys. Rev. C, 76, 064312 (2007)
 
30. Z. Podoly´ak, Prolate-oblate shape transition in heavy neutron-rich nuclei, Journal of Physics: Conference Series, 381, 012052 (2012).
 
31. J. Jolie, A. Linnemann, Prolate-oblate phase transition in the Hf-Hg mass region, Phys. Rev. C 68 031301(R)(2007).