1. Ch. Stoyanov Two-Phonon Mixed-Symmetry States in the Domain N=52
Nuclear structure calculations in a large domain of excitation energies
Two-Phonon Mixed-Symmetry States in the Domain N=52
Ch. Stoyanov
Institute for Nuclear Research and Nuclear Energy
Sofia, Bulgaria

2. Nuclear structure calculations in a large domain of excitation energies
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)

3. U(5) limit of IBM-2
1 July 2016
Ch. Stoyanov

4. E2 and M1transitions connecting one phonon states
1 July 2016
Ch. Stoyanov

5. M1 Transitions conneting two-phonon states
1 July 2016
Ch. Stoyanov

6. General view
11 July 2015
Ch. Stoyanov

7. Nuclear structure calculations in a large domain of excitation energies
The model Hamiltonian

8. Quasiparticle RPA (collective effects)
Nuclear structure calculations in a large domain of excitation energies
Quasiparticle RPA (collective effects)

9. Quasiparticle RPA (2) (quasiboson approximation)
Ch. Stoyanov

10. Quasiparticle RPA (3) (collective effects)
Nuclear structure calculations in a large domain of excitation energies
Nuclear structure calculations in a large domain of excitation energies
Quasiparticle RPA (3) (collective effects) 14

11. mixed-symmetry states
Nuclear structure calculations in a large domain of excitation energies
Applications
Even-even nuclei
mixed-symmetry states

12. Mixed symmetry states Experiment
Nuclear structure calculations in a large domain of excitation energies
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
1 July 2016
Ch. Stoyanov 23

13. Mixed symmetry states Experiment
Nuclear structure calculations in a large domain of excitation energies
Mixed symmetry states Experiment
MS have been populated by means of many nuclear reactions as
Inelastic scattering of
Electrons
Photons
β decay
Coulomb excitation
1 July 2016
Ch. Stoyanov 24

14. Nuclear structure calculations in a large domain of excitation energies
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)

15. Test of the Structure
In order to test the isospin nature of 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

16. Nuclear structure calculations in a large domain of excitation energies
The dependence of M1 and E2 transitions on the ratio G(2)/k0(2) in 136Ba.

17. Nuclear structure calculations in a large domain of excitation energies
Structure of the first RPA phonons (only the largest components are given) and corresponding B(2+) ratios for 136Ba
B(2+)

18. Nuclear structure calculations in a large domain of excitation energies
11 July 2015
Ch. Stoyanov 30

19. 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.

20. B(E2; g.s.→ 2+) strength distributions in 94Mo.
Nuclear structure calculations in a large domain of excitation energies
B(E2; g.s.→ 2+) strength distributions in 94Mo.

21. B(M1;2+k →2+1 ) strength distributions in 94Mo
Nuclear structure calculations in a large domain of excitation energies
B(M1;2+k →2+1 ) strength distributions in 94Mo

22. 96Ru New experimental information
Hennig et al.
Phys. Rev. C (2015)
Properties of the
two-phohon mixed-symmetry quintuplet
2+1(sm) ⃰ 2+2(ms)
Ch. Stoyanov

23. Contribution of main components in the structure of low-lying QRPA 2+ states in 96Ru.Jπ
E(MeV)
Structure
B(E2,g.st.→2+)
(W.u.)
B(M1) ( μ2N)
2+2→2+1
2+ 1
0.999
0.79(2d5/2)2n (1g9/2)2p 96
2+ 2
2.276
(2d5/2)2n (1g9/2)2p
3.8
1.37
Ch. Stoyanov

24. 96 Ru isospin nature of 2+ states
Jπ B (2+) isoscalar (symmetric) isovector (mixed symmetry)
Ch. Stoyanov

25. State Jπ E(MeV) EXP. QPM Structure,% 2+1 2+2 2+3
2+1
0.832
0.775
88%[2+1]QRPA
+5%{[2+1]QRPA ⃰ [2+1]QRPA} + …
2+2
1.932
1.826
80% {[2+1]QRPA ⃰ [2+1]QRPA} + ….
2+3
2.283
2.164
90% [2+2 ]QRPA + ..
Ch. Stoyanov

26. State E (MeV) Transition E2 [Wu] M1 [μN] strength Jπ EXP. QPM
Nuclear structure calculations in a large domain of excitation energies
State
E (MeV)
Transition E2
[Wu] M1
[μN]
strength Jπ
EXP.
QPM
Jπi → Jπf σλ
B(σλ)exp
B(σλ)QPM
g e f f = 0.8 g f ree
B(σλ)IBM
2+1
0.832
0.775
2+1 →0+1
18.1(5) 16
18.4
2+2
1.932
1.826
2+2 →2+1
2+2 →0+1
0.05(2)
……..
28(9)
0.11
0.77 30 24
2+3
2.283
2.164
2+3 →2+1
2+3 →0+1
0.69(14)
1.36(19)
0.63
0.75
0.69
2.53
Ch. Stoyanov

27. Distribution of two-phonon component
{[2+1(sm)]QRPA ⃰ [2+2(ms)]QRPA}
Jπ contribution E[MeV]
% MeV
% MeV
% MeV
Ch. Stoyanov

28. Forth and fifth quadrupole excitations
Jπ State
E (MeV)
Transition
strength
EXP.
QPM
Jπi → Jπf σλ
B(σλ)exp
B(σλ)QPM
g e f f = 0.8 g free
B(σλ)IBM
2+4
2.980
2+4→0+1
2+4→2+1
2+4→2+2 E2 M1
0.01
2.9
0.06
2+5
2.740
3.338
2+5 →0+1
2+5 →2+1
2+5 →2+2
0.006(12)
0.58(15)
0.17(3)
0.16
1.1
0.15
1.92
0.17
Ch. Stoyanov

29. 3+ excited states structureJπ
E(MeV)
EXP.
QPM
Structure,%
3+1
2.852
2.644
9% {[2+1]QRPA ⃰ [2+2]QRPA} + ….
3+2
2.898
3.164
76% {[2+1]QRPA ⃰ [2+2]QRPA} + ….
Ch. Stoyanov

30. 3+ excited states Jπ State E (MeV) Transition strength EXP. QPM
strength
EXP.
QPM
Jπi → Jπf σλ
B(σλ)exp
B(σλ)QPM
g e f f = 0.8 g free
B(σλ)IBM
3+1
2.852
2.644
3+1→2+1
3+1→2+2 E2 M1
< 0.01
0.008(1)
< 5.58
0.09
0.016
0.009
14.7
3+2
2.898
3.164
3+2→2+1
3+2→2+2
< 0.28
0.02(4)
0.078(14)
0.66
0.07
0.27
3.17
0.56
Ch. Stoyanov

31. 4+ excited states structureJπ
E(MeV)
EXP.
QPM
Structure,%
4+1
1.518
1.585
63% {[2+1]QRPA ⃰ [2+1]QRPA} + ….
4+2
2.462
2.207
61%[4+1]QRPA
+13%{[2+1]QRPA ⃰ [2+1]QRPA} +
2.3% {[2+1]QRPA ⃰ [2+2]QRPA} +…
4+5
3.300
86% {[2+1]QRPA ⃰ [2+2]QRPA} + ….
Ch. Stoyanov

32. 4+ excited states Jπ State E (MeV) Transition strength EXP. QPM
strength
EXP.
QPM
Jπi → Jπf σλ
B(σλ)exp
B(σλ)QPM
g e f f = 0.8 g f ree
B(σλ)IBM
4+1
1.518
1.585
4+1→2+1 E2
22.6(17) 15
25.6
4+2
2.462
2.207
4+2→4+1
4+2→2+1 M1
0.90(18)
1.52(19)
0.07
5.5
1.13
1.44
4+5
3.300
1.6
1.8
Ch. Stoyanov

33. 1+ excited states structureJπ
E(MeV)
EXP.
QPM
Structure,%
1+1
3.154
3.118
93%{[2+1]QRPA ⃰ [2+2]QRPA} + ..
Ch. Stoyanov

34. 1+ excited states Jπ State E (MeV) Transition strength EXP. QPM
strength
EXP.
QPM
Jπi → Jπf σλ
B(σλ)exp
B(σλ)QPM
g e f f = 0.8 g f ree
B(σλ)IBM
1+1
3.154
3.192
1+1 →0+1 M1
0.17(6)
0.13
Ch. Stoyanov

35. Nuclear structure calculations in a large domain of excitation energies
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.

36. Thank You for Your attention!!!
Nuclear structure calculations in a large domain of excitation energies
Thank You for Your attention!!!