Mean field studies of odd mass nuclei and quasiparticle excitations. Luis M. Robledo Universidad Autónoma de Madrid Spain

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1 Mean field studies of odd mass nuclei and quasiparticle excitations Luis M. Robledo Universidad Autónoma de Madrid Spain

2 Odd nuclei and multiquasiparticle excitations(motivation) Nuclei with odd number of protons and/or neutrons constitute ¾ of all possible. However, they are not as extensively studied with microscopic methods as the even-even ones. The reason is pairing: it couples nucleons together to form Cooper pairs coupled to spin zero. The equation that governs the mean field dynamics is the HFB equation. The HFB equation for even-even nuclei involve time-even fields and densities facilitating the development of computer codes. There is a double degeneracy (Krammers) and orbitals are occupied pairwise. Time reversal symmetry is preserved. In time-odd systems (including odd-a, 2qp excitations, 4qp excitations, etc ) time reversal symmetry is broken New, time-odd fields and densities are required (new computer codes) The effective interaction in those channels is poorly characterized (new terms and new observables to fix parameters) Many more excited states are accessible to the HFB description (more complexity)

3 The mean field : Hartree- Fock- Bogoliubov The nuclear mean field has to encompass at the same time long range correlations typical of the Hartree- Fock method and short range correlations leading to the formation of Cooper pairs and superfluidity, well handled by the BCS theory This is done with the help of the Bogoliubov quasiparticles HO basis W orthogonal

4 The mean field : Hartree- Fock- Bogoliubov The key ingredient is the generalized density matrix And the same quantity but in the quasiparticle basis Even systems They are related through the W Bogoliubov amplitude Any mean value can be expressed in terms of those elementary contractions using Wick s theorem

5 The mean field: Hartree- Fock- Bogoliubov The HFB equation is obtained by looking a the absolute minimum of the HFB energy as a function of the truly independent parameters (Thouless parametrization) Trivial algebra leads to with That leads to the HFB equation And the gradient

6 The mean field: Hartree- Fock- Bogoliubov The HFB equation and the gradient expression for blocked odd-a states, 2qp excitations, etc are the same but replacing the generalized density matrix by the corresponding one Where is defined the traditional way but replacing They can be written as With the swapping matrix The swapping matrix can be re-absorbed in W by introducing explaining in a natural way the swap U and V columns in the Bogoliubov amplitudes recipe used in solving the HFB equation for 1qp, 2qp, etc systems. It allows to extend the gradient method to the 1qp, 2qp, etc cases (advantageous for handling many constraints)

7 Computer codes We have written two computer codes to solve the HFB equations for general n-quasiparticle excitations and the finite range Gogny force HFBtri (L.M. Robledo, unpublished) Triaxial code, preserves reflection symmetry. K mixing is allowed, what leads to practical difficulties in solving the problem (see below) Atb (Robledo, Bertsch, Bernard, PRC86, ) Axial symmetry preserved but reflection symmetry is not imposed. K is a good quantum number. Much less computer power demanding than HFBtri. All time-odd fields are taken into account in the two codes

8 The mean field : Hartree- Fock- Bogoliubov Fully paired even systems: ρ has doubly degenerated eigenvalues Odd (number parity) systems (1qp excitation): Vacuum of Two quasiparticle excitations

9 The blocking strategy The solution of the HFB equation follows the following strategy Solve HFB (even number parity, time reversal invariant) for the target N and Z values Choose the quasiparticles to block (usually the 10 with the lowest qp energy) Swap the appropriate U and V columns in the Bogoliubov amplitudes and start the iterative solution of the HFB equation computing all time-odd fields Problem Orthogonality is not preserved by the iterative process Initial quasiparticles are orthogonal even if they have the same quantum numbers However, orthogonality is lost in the iterative process and usually, no matter the initial quasiparticle is, the final solution is the same and corresponds to the lowest energy This is the most prominent advantage of preserving axial symmetry: K is a good quantum number and quasiparticles with different K values are orthogonal by construction. The orthogonality problem only matters within quasiparticles with the same K

10 The orthogonality constraint The orthogonality issue In odd mass systems, or two- four- etc quasiparticle states it is common to consider several excited states. Most of them are orthogonal to the others because of symmetry considerations like the K quantum number or parity. When the symmetries are not preserved or the quantum numbers are the same the states are not necessarily orthogonal and the solution of the HFB equation based on the minimization of the energy usually ends up in the lowest energy solution. For instance, in even-even nuclei is very difficult to reach 2qp K=0 + solutions if orthogonality is not addressed in the proper way (always converge to the ground state) It is very difficult to obtain solutions different from the ground state with triaxial, codes Another typical situation is when two different solutions of the HFB equations have a non-zero overlap meaning, according to the rules of QM, that they are not true excited states and a re-orthogonalization is required (modifying excitation energies and other properties)

11 The orthogonality constraint To minimize the energy to obtain imposing orthogonality to use Lagrange multipliers Gradient with The gradient is the product of a singular matrix A -1 times a tiny number det A To handle this situation the SVD of A is very handy C, D are orthogonal matrices and σ is diagonal.

12 Gogny force The Gogny force is a popular choice (also Skyrme, BCPM, relativistic, etc) Two body kinetic energy is always subtracted to correct for COM motion Parameters fixed by imposing some nuclear matter properties and a few values from finite nuclei (binding energies, spin-orbit splitting and some radius information. D1S: surface energy fine tuned to reproduce fission barriers (1989) D1N: Realistic neutron matter equation of state reproduced (2008) D1M: Realistic neutron matter + Binding energies of essentially all nuclei with beyond mean field effects (2009) Pairing and time-odd fields are taken from the interaction itself

13 Odd A super-heavy Shell structure is very important for the stability of Super Heavy (SH) nuclei and in particular the position of the proton 7/2[633] and neutron 11/2[725] orbitals J. Dobaczewski et al. Nucl Phys A Most of the levels in a ±1 MeV window are reproduced Specific excitation energies off by 0.5 MeV (poor spectroscopic quality)

14 Odd A super-heavy Increasing the spin-orbit strength in Gogny D1S improves the agreement But increases binding energy by 100 MeV A careful refitting of the EDF is required

15 Odd A super-heavy Increasing pairing strength improves the agreement for the excitation energy, but worsens other quantities like moments of inertia, etc Present day functionals do not have spectroscopic quality

16 Odd A super-heavy Extensive calculations in Es (Z=99) isotopes and N=151 isotones reveal again a reasonable agreement with experiment but we are still far from spectroscopic quality as typical excitation energy differences are around kev In addition, these are pure HFB results and beyond mean field effects (symmetry restoration and configuration mixing) could alter the final picture. More theory effort required.

17 Non collective K=0 2qp states: 160 Gd Two 0+ states at MeV and MeV observed. Are they β vibrations? 160 Gd GS is predicted prolate with an additional oblate minimum at 5 MeV (Gogny D1S) β vibration energy (GCM quadrupole) 3.59 MeV Several 2qp K=0 + states with excitation energies below 3 MeV (Orthogonality constraint is crucial in this calculation) (5/2[523]) 2 Neutron at 1.07 MeV (1.9 Pert) (3/2[411]) 2 Proton at 1.19 MeV (2.26 Pert) (5/2[532]) 2 Proton at 1.31 MeV (2.20 Pert) 5/2[532] 5/2[413] Proton at 1.92 MeV (2.36 Pert) The blocking of 2qp almost destroys pairing correlations in the corresponding channel The calculation and measurement of B(E2) and B(E0) is required before any conclusion can be reached

18 Non collective 2qp states: 178 Hf Several high K 2qp excitations are known in 178 Hf (Mullins PLB393, 279 (1997)) K π E exp (MeV) E th (MeV) Configuration E th pert (MeV) Neutron 7/2[514] 9/2[624] Proton 7/2[404] 9/2[514] Proton 5/2[402] 7/2[404] 2.30 Self-consistent two quasiparticle excitations with Gogny D1S and orthogonality constrain The agreement with experiment is good indicating that the single particle levels are at the right position and pairing strength is also good.

19 Non collective 4qp high K isomer states: 178 Hf More interesting are the 4qp (2qp(prot) 2qp(neut)) states like the long lived K=16 + isomer with a half life of 31 yr. Our predictions with Gogny D1S for the excitation energies are: 1.67 MeV for the 16 + and 2.16 MeV for the They come out a bit too low but still are satisfactory.

20 Non collective high K isomer states: 254No Several isomeric 2qp (3 + and 8 - ) and 4qp (16 + ) states observed in 254No Relevant for the understanding of shell effects in SH nuclei Nature 442, 896 (2006)

21 Non collective high K isomer states: 254No In PLB690, 19 (2010) Clark et al studied the decay pattern involving an additional K=10 + isomeric rotational band state

22 Non collective high K isomer states: 254No High K orbitals near the Fermi level Protons: 7/2[514] 7/2[633] 9/2[624] Neutrons: 7/2[624] 9/2[734] 11/2[725]

23 Non collective high K isomer states: 254No 3 + Proton 2qp excitation K=7/2[514] K=-1/2[521] 8 - Neutron 2qp excitation K=9/2[734] K=7/2[624] 10 + Neutron 2qp excitation K=9/2[734] K=11/2[725] 8 - Proton 2qp excitation K=7/2[514] K=9/2[624] 16 + Proton Neutron 4qp Next is the calculation of transition strengths using AMP wave functions

24 Decay out of high-k and other observables The decay out of high-k isomers is a big challenge as The role of fragmentation is difficult to asses: intemediate triaxial states can play a key role Validity of the rotational formula connecting static intrinsic deformations with electromagnetic strengths is at stake These two problems call out for a symmetry restoration framework We are identifying K in the intrinsic frame with J in the laboratory: This assumption has to be carefully tested by restoring angular momentum quantum numbers

25 Odd-odd systems and the Gallagher-Moszkowski rule Gallagher-Moszkowski (GM) rule: In odd-odd systems with an unpaired proton K p and neutron K n a doublet is obtained with J=K n +K p and J= K n - K p. The configuration with the lowest energy is that with parallel intrinsic spins. Typical example: 174 Lu (Z=71) Results for 173 Lu and 173 Yb also given The inversion observed in 173 Lu explains why the doublet is not the GS GM rule is violated GM rule is a consequence of the properties of the spin-spin neutron-proton nuclear interaction LMR, R Bernard and G. Bertsch Phys Rev C89, (2014)

26 Odd-odd systems and the Gallagher-Moszkowski rule Analyzing the agreement of calculations with experimental data on the ordering of the doublets provides a handle on a poorly-determined part of the interaction In the Gogny EDF two terms contribute to the splitting: Brink-Boecker central term Density dependent term The region around Z=71 (Lu) is well known experimentally and the GM rule is fulfilled in more than 95% of the cases Perturbative calculation Calculated GM doublet splittings Lu isotopes ( ) Positive ΔE: agree with GM BB: Brink-Boeker DD: Density dependent

27 Odd-odd systems and the Gallagher-Moszkowski rule GM fails % of the cases BB contribution correct DD contribution incorrect The spin-spin neutron-proton density dependent interaction is wrong

28 Future developments Due to polarization, beyond mean field effects are expected to be more important than in the even-even case Parity restoration is in the impending horizon: not only important for static octupole deformed but also for all nuclei through dynamic octupole correlations ( see J. Phys. G: Nucl. Part. Phys. 42 (2015) in the even-even case) Particle number projection to come next: relevant because pairing correlations are highly quenched in the presence of blocked configurations and dynamical pairing is competing with static one Angular momentum projection is still far away but the preservation of axial symmetry make it feasible. Large variability of moments of inertia is expected Particle-vibration coupling: can be addressed by considering a generator coordinator like wave function to couple to the collective degrees of freedom Pffafian techniques to compute multi-quasiparticle overlaps very handy: PRC79, (2009), PRL108, (2012)

29 Future developments Very preliminary results on parity projection (time-odd fields not considered in the hamiltonian overlap) Very significant change in excitation energies and ordering of levels!

30 Summary and conclusions Solving the HFB for 1qp, 2qp, etc configurations with the Gogny force provides reasonable results for odd-a systems and noncollective excitations (including K isomers) However, none of the variants of Gogny (and other Skyrme or relativistic functionals) do have spectroscopic quality yet (at the level of a few tens of kev). Difficulties with odd-odd nuclei imply missing time-odd components in the effective (density dependent) interaction Orthogonality constraint is fundamental to have access to many more excited states within the formalism The role of beyond mean field correlations, including symmetry restoration and particle-vibration coupling has still to be elucidated Very preliminary results point to a relevant role of parity projection in a RVAP framework.