P. Marevic 1,2, R.D. Lasseri 2, J.-P. Ebran 1, E. Khan 2, T. Niksic 3, D.Vretenar 3

Transcription

1 -Nuclear Clustering in the Energy Density Functional Approach- P. Marevic 1,2, R.D. Lasseri 2, J.-P. Ebran 1, E. Khan 2, T. Niksic 3, D.Vretenar 3 1 CEA,DAM,DIF 2 IPNO 3 University of Zagreb 1

2 Introduction Nuclear systems as a mixture of 4 types of fermion r p T=0 MeV Fermi liquid proton neutron BEC BCS pairing Mott line Clustering a 2-component Fermi gas O O a a a t h a h a t t a a h QPT crossover 3-component Fermi gas r n subsaturation density saturation density These features leave fingerprints in finite nuclei 2

3 Nuclei at high spin/energy Introduction Emergence of ordered sub-structures inside nuclei proton neutron a low energy resonance deformation clustering skin halo molecular orbital Nuclear EDFs provide a unified and consistent description of these various properties 3

4 Introduction Outline Theoretical framework : EDF Pair correlations Nuclear clustering 4

5 - Theoretical framework : EDF- 5

6 EDF Nuclear many-body problem : strategies Many-body approaches as implementation of different strategies to apprehend the many-body problem Traditional starting point of a many-body approach : HF product state + + Fermi correlations E E RHF missing correlations E GS How to incorporate such correlations starting with a HF product state? 6

7 EDF Nuclear many-body problem : strategies Vertical Philosophy Excit. 2p2h non collective excitations Excit. 1p1h physics accounted through numerous rearrangements of nucleons within the orbitals given by the reference vacuum conservation of symmetries Reference vacuum i j = C 0 + C 1p-1h i + C 2p-2h j + Fermi correlations Major contribution Dynamical correlations in all nuclei HF E E RHF E GS 7

8 EDF Nuclear many-body problem : strategies Vertical Philosophy Excit. 2p2h non collective excitations Excit. 1p1h physics accounted through numerous rearrangements of nucleons within the orbitals given by the reference vacuum conservation of symmetries Reference vacuum i j = C 0 + C 1p-1h i + C 2p-2h j + comparable contributions Fermi correlations Dynamical correlations in all nuclei HF Nondynamical correlations open shell systems E HF E RHF E GS 8

9 EDF Vertical Philosophy Nuclear many-body problem : strategies Horizontal philosophy Excit. 2p2h non collective excitations broken symmetries Excit. 1p1h physics accounted through numerous rearrangements of nucleons within the orbitals given by the reference vacuum (0,0 ) (q 0,j 0 ) (q 0,j 1 ) (q 1,j 0 ) conservation of symmetries Reference vacuum i j = C 0 + C 1p-1h i + C 2p-2h j + Dynamical correlations comparable contributions in all nuclei HF Fermi correlations Collective excitations Physics accounted for through spontaneous symmetry breaking + fluctuations of the order parameter dynamical correlations transferred in E = dq f(q) H eff (, ) ( q, arg(q)) Nondynamical correlations open shell systems E RHF E HF E RHF E UHF q E GS E GS Arg(q) 9

10 EDF Vertical Philosophy Nuclear many-body problem : strategies Horizontal philosophy Excit. 2p2h non collective excitations broken symmetries Excit. 1p1h physics accounted through numerous rearrangements of nucleons within the orbitals given by the reference vacuum (0,0 ) (q 0,j 0 ) (q 0,j 1 ) (q 1,j 0 ) conservation of symmetries Reference vacuum i j = C 0 + C 1p-1h i + C 2p-2h j + Dynamical correlations comparable contributions in all nuclei HF Fermi correlations Collective excitations Physics accounted for through spontaneous symmetry breaking + fluctuations of the order parameter dynamical correlations transferred in E = dq f(q) H eff (, ) ( q, arg(q)) Initial wave function Nondynamical correlations open shell systems E RHF E HF E RHF E UHF q Optimized wave function with {q (1) } E GS E GS Arg(q) 10

11 EDF Vertical Philosophy Nuclear many-body problem : strategies Horizontal philosophy Excit. 2p2h non collective excitations broken symmetries Excit. 1p1h physics accounted through numerous rearrangements of nucleons within the orbitals given by the reference vacuum (0,0 ) (q 0,j 0 ) (q 0,j 1 ) (q 1,j 0 ) conservation of symmetries Reference vacuum i j = C 0 + C 1p-1h i + C 2p-2h j + Dynamical correlations comparable contributions in all nuclei HF Fermi correlations Collective excitations Physics accounted for through spontaneous symmetry breaking + fluctuations of the order parameter dynamical correlations transferred in E = dq f(q) H eff (, ) ( q, arg(q)) Nondynamical correlations open shell systems E RHF E HF E RHF E UHF q E GS E GS Arg(q) 11

12 EDF Vertical Philosophy Nuclear many-body problem : strategies Horizontal philosophy Excit. 2p2h non collective excitations broken symmetries Excit. 1p1h physics accounted through numerous rearrangements of nucleons within the orbitals given by the reference vacuum (0,0 ) (q 0,j 0 ) (q 0,j 1 ) (q 1,j 0 ) conservation of symmetries Reference vacuum i j = C 0 + C 1p-1h i + C 2p-2h j + Dynamical correlations comparable contributions in all nuclei HF Fermi correlations Collective excitations Physics accounted for through spontaneous symmetry breaking + fluctuations of the order parameter dynamical correlations transferred in E = dq f(q) H eff (, ) ( q, arg(q)) Nondynamical correlations open shell systems E RHF E HF E RHF E UHF q E GS E GS Arg(q) 12

13 EDF Vertical Philosophy Nuclear many-body problem : strategies Horizontal philosophy Excit. 2p2h non collective excitations broken symmetries Excit. 1p1h physics accounted through numerous rearrangements of nucleons within the orbitals given by the reference vacuum (0,0 ) (q 0,j 0 ) (q 0,j 1 ) (q 1,j 0 ) conservation of symmetries Reference vacuum i j = C 0 + C 1p-1h i + C 2p-2h j + Dynamical correlations comparable contributions in all nuclei HF Fermi correlations Collective excitations Physics accounted for through spontaneous symmetry breaking + fluctuations of the order parameter dynamical correlations transferred in E = dq f(q) H eff (, ) ( q, arg(q)) Nondynamical correlations open shell systems E RHF E HF E RHF E UHF q E GS E GS Arg(q) 13

14 EDF Vertical Philosophy Nuclear many-body problem : strategies Horizontal philosophy Excit. 2p2h non collective excitations broken symmetries Excit. 1p1h physics accounted through numerous rearrangements of nucleons within the orbitals given by the reference vacuum (0,0 ) (q 0,j 0 ) (q 0,j 1 ) (q 1,j 0 ) conservation of symmetries Reference vacuum i j = C 0 + C 1p-1h i + C 2p-2h j + Dynamical correlations comparable contributions in all nuclei HF Fermi correlations Collective excitations Physics accounted for through spontaneous symmetry breaking + fluctuations of the order parameter dynamical correlations transferred in E = dq f(q) H eff (, ) ( q, arg(q)) Nondynamical correlations open shell systems E RHF E HF E RHF E GS Novel many-body approaches combining both philosophies P. Arthuis, J. Ripoche, T. Duguet, D. Lacroix, J.-P. Ebran J.Ripoche et al PRC 95, (2017) T. Duguet et al EPJA 51, 162 (2015) E proj E GS E UHF Arg(q) q 14

15 - Pair correlations - 15

16 Pair correlation How to describe a pair of nucleons in the medium? Reduced density matrix Form of the A-body wave function All the information of a many-body system is contained in its Nth-order density matrix BCS assumption: all the pairs of fermions occupy the same pair wave function : Only keep information about p- cluster embedded in the medium composed by the other N-p particles : 2-RDM : eigenfunctions provide an in-medium pair wave function 16

17 Pair correlation Covariant HFB with projection on good particle number prior to variation 2-neutron distribution properties r n rms d nn R.D. Lasseri et al, in prep 17

18 Pair correlation 2-neutron distribution properties R.D. Lasseri et al, in prep 18

19 - Nuclear Clustering - 19

20 Nuclear clustering How do es clustering show up in nuclear EDFs? First indicator: mass density at the MF level Marevic et al, submitted to PRC

21 Nuclear clustering How do es clustering show up in nuclear EDFs? Localization measure Conditional probability Reinhard et al, Phys. Rev. C 83, (2011) 0.5 : signals a nearly homogeneous Fermi gas 1 : localized a-like state (in N=Z systems) 16 O 4 He 20 Ne r C 21

22 Nuclear clustering How do es clustering show up in nuclear EDFs? Beyond MF level Marevic et al, submitted to PRC 22

23 Nuclear clustering How do es clustering show up in nuclear EDFs? Beyond MF level Marevic et al, submitted to PRC 23

24 Nuclear clustering How do es clustering show up in nuclear EDFs? rotation-vibration bands Bijker (2016) Marevic et al, in prep Work in progress Ebran et al, in prep 24

25 Nuclear clustering Deepe r understanding about nuclear clustering Guidance : the localization parameter particle number depth of the confining potential average inter-particle distance density α~1 crystal localization quantum liquid Ebran, Khan, Niksic & Vretenar, Nature 487, (2012) - PRC 87, (2013) 25

26 Nuclear clustering Deepe r understanding about nuclear clustering Influence of the particle number particle number Clustering is more likely to be found in light systems Ebran, Khan, Niksic & Vretenar, PRC 87, (2013) - PRC 89, (R) (2014) 26

27 Nuclear clustering Deepe r understanding about nuclear clustering Influence of the density average inter-particle distance density Ebran, Khan, Niksic & Vretenar, PRC 89,031303(R) (2014) Girod & Schuck, PRL 111, (2013) 27

28 Nuclear clustering Deepe r understanding about nuclear clustering Influence of the confining potential Deeper confining potential depth of the confining potential Electrons in quantum dots Yannouleas and Landman, Rep.Prog.Phys. 70, 2067 (2007) Neutral bosons in rotating trap Nucleons in 40 Ca 28

29 Nuclear clustering Deepe r understanding about nuclear clustering Influence of the confining potential 29

30 Nuclear clustering Deepe r understanding about nuclear clustering Influence of the confining potential depth of the confining potential 30

31 Nuclear clustering Deepe r understanding about nuclear clustering Influence of the confining potential Harmonic oscillator case Nazarewicz et. al., AIP Conf. Proc. 259, 30 (1992) 31

32 Nuclear clustering Deepe r understanding about nuclear clustering Influence of the neutron excess 2-center cluster in a N=Z system Neutron rich isotope melting Covalent bonding p s p * Be case p*3/2 d5/2 p1/2 p3/2 8 p1/2 s1/2 6 N=8 magic number breaking p3/2 s1/2 32

33 Nuclear clustering Deepe r understanding about nuclear clustering Influence of the neutron excess Be 12 Be 14 Be 33

34 Nuclear clustering Deepe r understanding about nuclear clustering Influence of the neutron excess 34

35 Nuclear clustering Deepe r understanding about nuclear clustering Influence of the neutron excess 35

36 Nuclear clustering Deepe r understanding about nuclear clustering Influence of the neutron excess 36

37 Conclusion & Perspectives Nuclear EDFs frame the various nuclear properties in a unified and consistent way Key feature : breaking/restoration of symmetries to efficiently account for nondynamical correlations Di-neutron like configuration at the surface of superfluid nuclei Clustering in ground and excited states of nuclei : impact of the average density and the depth of the confining potential Covalent bonding in neutron rich systems Particle number projection, conditional probability distribution, form factor, quarteting under development Thank you for your attention 37

38 EDF Nuclear many-body problem : strategies Good description of the ground state of ~50 nuclei 38

39 EDF Nuclear many-body problem : strategies Breaking U(1) : good description of the ground state of ~300 nuclei 39

40 EDF Nuclear many-body problem : strategies Breaking U(1) and O(3) : good description of the ground state of all nuclei 40