valence shell excitations in even-even spherical nuclei within microscopic model

1. valence shell excitations in even-even spherical nucleiwithin microscopic model Ch. Stoyanov Institute for Nuclear Research and Nuclear Energy Sofia, Bulgaria

2. The model Hamiltonian

3. Woods-Saxon potential

4. Spin-orbital term

5. Coulomb potential

6. Constant pairing

7. Separable force and multipole expansion

8. Central forces

9. Spherical case Nguyen Van Giai, Ch. Stoyanov, V. V. Voronov, Phys. Rev. C 57 1204 (1998) Contribution of F0(r):

10. Landau-Migdal form of the Skyrme interaction Nguyen Van Giai, Sagawa, H., Phys. Lett B106 (1981) 379

11. Gauss integration formula with abscissas and weights {rk, wk}. cutoff radius R Introducing the coefficient and the p-h matrix elements

12. Quasiparticle RPA(collective effects)

13. Quasiparticle RPA (2)(quasiboson approximation) • Jm denote a single-particle level of the average field for neutrons (or protons) • The neutron […]λμ means coupling to the total momentum λ with projection μ: • The quantity is Clebsch-Gordon coefficient • Bogoliubov linear transformation

14. Phonon properties • Phonons are not only collective • Collective  many amplitudes • Non-collective  a few amplitudes • Pure quasi-particle state  only one amplitude • Diverse Momentum and Parity Jπ spin-multipole phonons • The interaction could include any kind of correlations (particle-particle channel) LARGE PHONON SPACE

15. Quasiparticle RPA (3)(collective effects)

16. Harmonic vibrations To avoid Pauli principle problem

17. Microscopic description of mixed-symmetry states in nearly spherical nuclei

18. Introduction • Low-lying isovector excitations are naturally predicted in the algebraic IBM-2 as mixed symmetry states. Their main signatures are relatively weak E2 and strong M1 transition to symmetric states. • T. Otsuka , A.Arima, and Iachello, Nucl .Phys. A309, 1 (1978) • P. van Isacker, K.Heyde, J.Jolie et al., Ann. Phys. 171, 253 (1986)

19. Definitions • The low-lying states of isovector nature were considered in a geometrical model as proton-neutron surface vibrations. • is in-phase (isoscalar) vibration of protons and neutrons. • is out-of-phase (isovector) vibration of protons and neutrons. • A.Faessler, R. Nojarov, Phys. Lett., B166, 367 (1986) • R. Nojarov, A. Faessler, J. Phys. G, 13, 337 (1987)

20. Review paper • N. Pietralla, P. von Brentano, • and A. F. Lisetskiy, • Prog. Part. Nucl. Phys. 60, 225 (2008).

21. Microscopic calculations • Within the nuclear shell modelA. F. Lisetskiy, N. Pietralla, C. Fransen, R. V. Jolos, P. von Brentano, Nucl. Phys. A677, 1000 (2000) • Within the quasi-particle-phonon model (QPM)N. Lo Iudice and Ch. Stoyanov, Phys. Rev. C 62, 047302 (2000) N. Lo Iudice and Ch. Stoyanov, Phys. Rev. C 65, 064304 (2002)

22. Definition • In order to test the isospin nature of 2+states the following ratio is computed: • This ratio probes: • The isoscalar ((2+)<1) and • The isovector (B(2+)>1)properties of the 2+ state under consideration

23. The dependence of M1 and E2 transitions on the ratio G(2)/k0(2) in 136Ba.

24. B(2+) Structure of the first RPA phonons (only the largest components are given) and corresponding B(2+) ratios for 136Ba

25. The values of B(2+) for 144Nd

26. Explanation of the method used • The quasi-particle Hamiltonian is diagonalized using the variational principle with a trial wave function of total spin JM Where ψ0 represents the phonon vacuum state and R, P and T are unknown amplitudes; ν labels the specific excited state.

27. Energies and structure of selected low-lying excited states in 94Mo. Only the dominant components are presented.

28. 94Mo level scheme./low-lying transitions/

29. E2 transitions connecting some excite states in 94Mo calculated within QPM.

30. M1 transitions connecting some excite states in 94Mo calculated within QPM.

31. 92 Zr

33. 92 Zr Contribution of N and Z in the 2+ QRPA phonons

34. E2 and M1 transitions connecting excited st. in 92 Zr

35. QPM, EXP and SM g-fact. of low-lying excited st. in 92 Zr

36. The N=80 isotones N. Pietralla et al., Phys. Rev. C 58, 796 (1998). G. Rainovski, N. Pietralla et al., Phys. Rev. Lett. 96, 122501 (2006). T. Ahn, N. Pietralla, G. Rainovski et al., Phys. Rev. C 75, 014313 (2007). K. Sieja et al., Phys. Rev. C, v. 80 (2009) 054311.

37. Experimental results

38. Fermi energy as a function of the mass number

39. Results on QRPA level

40. QPM Results for N=80 isotones 134Xe 134Xe 136Ba 138Ce 138Ce

41. N=84: theoretical description N. Pietralla et al., Phys. Rev. C 58, 796 (1998). G. Rainovski, N. Pietralla et al., Phys. Rev. Lett. 96, 122501 (2006). T. Ahn, N. Pietralla et al.,Phys. Rev. C 75, 014313 (2007).

42. N=84: theoretical description

43. Comparison to the experiment

44. Recent experimental results Sn • PRL 98, 172501 (2007) • PRL 99, 162501 (2007) • PRL 101, 012502 (2008) • LoI • Phys. Lett. B 695, 110 (2011).

45. Experimental and theoretical B(E2) values for the Sn isotopes reported from Ref.[5]. The dashed and solid curves represent the results from shell model calculations using different cores (for details see Ref.[5]).

46. Calculations • A. Ansari, Phys. Lett. B 623, 37 (2005). • A. Ansari and P. Ring, Phys. Rev. C 74, 054313 (2006). • J. Terasaki, Nucl. Phys. A 746, 583c (2004). • N. Lo Iudice, Ch. Stoyanov, • and D. Tarpanov PRC 84, 044314 (2011)

47. Selected proton s. p. states around the Fermi energy

48. Selected neutron s. p. states around the Fermi energy

49. Selected neutron s. p. states around the Fermi energy

50. Experimental values of B(E2, g.s.-->2+1) and calculated neutron gaps in tin isotopic chain