Low-energy nuclear spectroscopy in a microscopic multiphonon approach

1. Low-energy nuclear spectroscopy in a microscopic multiphonon approach Ch. Stoyanov Institute for Nuclear Research and Nuclear Energy Sofia, Bulgaria

2. The model Hamiltonian

3. Quasiparticle RPA(collective effects)

4. 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

5. 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

6. Quasiparticle RPA (3)(collective effects)

7. Harmonic vibrations To avoid Pauli principle problem

8. Landau-Migdal form of the Skyrme interaction Nguyen Van Giai, Sagawa, H., Phys. Lett B106 (1981) 379 11 July 2015 9 Ch. Stoyanov

9. Gauss integration formula with abscissas and weights {rk, wk}. cutoff radius R Introducing the coefficient and the p-h matrix elements 11 July 2015 11 Ch. Stoyanov

10. Applications • Even-even nuclei • mixed-symmetry states

11. Microscopic description of mixed-symmetry states in nearly spherical nuclei • 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)

12. Ch. Stoyanov U(5) limit of IBM-2

13. Ch. Stoyanov General view

14. 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)

15. Mixed symmetry states Experiment • N. Pietralla et al., Phys. Rev. C 58, 796 (1998), • N. Pietralla et al., Phys. Rev. Lett. 83, 1303 (1999) • inelastic hadron scattering cross sections • measurements of the electron conversion coefficients in β decay 11 July 2015 18 Ch. Stoyanov

16. Mixed symmetry states Experiment • MS have been populated by means of many nuclear reactions as • Inelastic scattering of • Electrons • Photons • β decay • Coulomb excitation 11 July 2015 19 Ch. Stoyanov

17. Review papers • N. Pietralla, P. von Brentano, • and A. F. Lisetskiy, • Prog. Part. Nucl. Phys. 60, 225 (2008). • N Lo Iudice, V Yu Ponomarev, Ch Stoyanov, A V Sushkov, V V Voronov • J. Phys. G: Nucl. Part. Phys. 39 (2012) 043101

18. 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

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

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

21. 11 July 2015 25 Ch. Stoyanov

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

23. 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.

24. Explanation of the method used (3) • The normalization condition for states reads

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

26. B(E2; g.s.→2+) strength distributions in 94Mo.

27. B(M1;2+k →2+1 ) strength distributions in 94Mo

28. 96Ru 2+1 QRPA

29. 96Ru 2+2 QRPA

30. Ch. Stoyanov

31. Ch. Stoyanov

32. 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.

33. Experimental results

34. Fermi energy as a function of the mass number

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

36. N=84: Experimental results

37. 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).

38. Comparison to the experiment

39. Ch. Stoyanov 204Hg 2+

40. Ch. Stoyanov 96RuNew experimental information • Hennig et al. • Phys. Rev. C 92 064317 (2015) • Properties of the • two-phohon mixed-symmetry quintuplet • 2+1(sm)⃰ 2+2(ms)

41. Ch. Stoyanov Distribution of two-phonon component • {[2+1(sm)]QRPA ⃰ [2+2(ms)]QRPA} • Jπ contribution E[MeV] • 2+3 2.47% 2.16 MeV • 2+4 21 % 2.98 MeV • 2+5 69%3.34 MeV

42. Ch. Stoyanov Forth and fifth quadrupole excitations

43. Conclusions • There are two modes in the low-lying quadrupole excitations – isoscalar and isovector one. • The properties of these two modes are close to IBM-2 symmetric and mixed-symmetry states. • The coupling of the modes leads to variety of excited states. There are well pronounced regularities of E2 and M1 transitions connecting the states.

44. Thank You for Your attention!!!