Gianluca Colò. Density Functional Theory for isovector observables: from nuclear excitations to neutron stars. Università degli Studi and INFN, MIlano

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1 Density Functional Theory for isovector observables: from nuclear excitations to neutron stars Gianluca Colò NSMAT2016, Tohoku University, Sendai 1

2 Outline Energy density functionals (EDFs): a short introduction - unique way to connect observables and nuclear EoS. Incompressibility of symmetric nuclear matter from the monopole resonance. EoS of asymmetric matter: parameters deduced from nuclear collective motion and global status. Neutron stars: from their mass to EoS. Composition of the outer crust. Conclusions. NSMAT2016, Tohoku University, Sendai 2

3 Energy Density Functionals (EDFs) E[ ] E = Ψ Ĥ Ψ = Φ Ĥ Φ = ρˆ eff Φ Slater determinant ρˆ 1-body density matrix H eff = T + V eff. If V eff or E[] are well designed, the resulting g.s. (minimum) energy can fit experiment at best. Hartree-Fock or Kohn-Sham. Within a time-dependent theory (TDHF), one can describe harmonic oscillations around the minimum. The restoring force is:. The linearization of the equation of the motion leads to Random Phase Approximation. NSMAT2016, Tohoku University, Sendai 3

4 Skyrme vs. relativistic functionals attraction Skyrme effective force short-range repulsion In the relativistic (that is, covariant) models the nucleons are described as Dirac particles that exchange effective mesons. There are also point coupling versions! NSMAT2016, Tohoku University, Sendai 4

5 Modern functionals and their performance They are as fundamental as other models because of the Kohn- Hohenberg theorem. They differ among one another (only) because of the ansatz about density dependence. They are applicable to almost the whole isotope chart and (!) to highly excited states, as well as to nuclear and neutron matter. Typical number of parameters 10 at most. Error on masses of the order of 1 MeV (can go further down). Trends of charge radii and deformations fairly well reproduced. They also describe well states such as giant resonances, rotational bands. ( ) Where is the neutron drip line located? Cf. e.g. J. Erler et al., Nature 486, 509 (2012) NSMAT2016, Tohoku University, Sendai 5

6 Symmetric and asymmetric matter EoS E/A called EoS. We consider UNIFORM matter. Nuclear matter EOS Symmetric matter EOS Symmetry energy S Knowledge aroundρ 0 SATURATION POINT of SNM We can introduce neutron matter S can also be defined as: S = E/A(neutron matter) E/A(symmetric matter) NSMAT2016, Tohoku University, Sendai 6

7 Incompressibility of symmetric matter K The relationship between K and the energy of the GMR has been discussed for decades. Cf. J.P. Blaizot, Phys. Rep. 64, 171 (1980). There was some consensus to deduce this quantity by using functionals, but for a while it looked like nonrelativistic and relativistic functionals were providing different results for some reason E GMR GC, N. Van Giai, NPA 731, 15 (2004) Skyrme RMF K NSMAT2016, Tohoku University, Sendai 7

8 The model dependence is actually density dependence PRC 70, (2004) = 1/6 The concept by Blaizot is too simplistic but the model dependence in the extraction of the incompressibility has nothing to do with relativity or not. It is related to the density dependence of the functional. The figures refer to Skyrme calculations: = 1/3 The incompressibility from Skyrme can be made consistent with that from RMF, and viceversa: NSMAT2016, Tohoku University, Sendai 8

9 Open question: asymmetric matter EOS Symmetric matter EOS Symmetry energy S There is a small uncertainty on J, a considerable uncertainty on L and basically no constraint on K sym. The correlation between L and the neutron skin thickness is well established. B.A. Brown and S. Typel, Phys. Rev. C 64, (2001) NSMAT2016, Tohoku University, Sendai 9

10 Isovector collective modes Neutrons and protons oscillate in opposition of phase. Promising observables to extract the properties of the symmetry energy. Problems: the nucleus is not a homogeneous system, it has a shell structure, and there is isoscalar/isovector mixing. NSMAT2016, Tohoku University, Sendai 10

11 Correlations involving nuclear collective modes: review of several attempts MEASURABLE QUANTITY A IVGDR PRC 77, (R) (2008) PDR PRC 81, (R) (2010) J = 32.3 ± 1.3; L = 64.8 ± 15.7 Dipole polarizability PRC 88, (2013) J = 31 ± 2; L = 43 ± 16 EoS PARAMETER B Points correspond to the results of calculations with Skyrme or RMF but is every correlation of this type really 1-dimensional and sound? IVGQR PRC 87, (2013) J = 32 ± 1; L = 37 ± 18 Anti-analog dipole PRC 94, (2016) J = 33.3 ± 2.1; L = 98.8 ± 23.6 ALL NUMBERS IN MeV J and/or L? NSMAT2016, Tohoku University, Sendai 11

12 A sound correlation The droplet model provides an expression for the dipole polarizability: Under the hypothesis that (i) JA -1/3 /Q is small; (ii) the density has a Fermi profile; (iii) J/Q is linearly correlated with L, M. Warda et al., Phys. Rev. C 80, (2009) X. Roca-Maza et al.. Phys. Rev. C 88, (2013) NSMAT2016, Tohoku University, Sendai 12

13 L or neutron skin from dipole polarizability We could put L instead of the skin thickness on the x-axis. Advantages: It includes the whole dipole response The correlation with the symmetry energy parameters can be understood Measurements are available in more than one nucleus: (p,p ) at small angle mainly (cf. A. Tamii / T. Aumann) NSMAT2016, Tohoku University, Sendai 13

14 Main result for symmetry energy parameters X. Roca-Maza et al., PRC 92, (2015) If one selects (red points) the functionals that reproduce experimental data (1) in all nuclei, then one obtains Cf. the paper for OTHER NUCLEI like 48 Ca and 90 Zr. (1) Quasi-deuteron contribution subtracted NSMAT2016, Tohoku University, Sendai 14

15 The isovector giant quadrupole resonance (IVGQR) S. Henshaw et al., PRL 93, (2011). HIγS (10 7 γ/s, ΔE/E 2-3%) High intensity polarized photon beam on 209 Bi Scattering parallel and perpendicular to the polarization plane Three-parameter fit of the IVGQR energy, width and strength NSMAT2016, Tohoku University, Sendai 15

16 Consistency between dipole and quadrupole Reminder: values to be represented in the (J,L) plane. There is a region in which J and L extracted either from (IV) dipole or quadrupole are compatible. From E of ISGQR and IVGQR These values are also compatible with our previous studies. Discrepancy with AGDR only. NSMAT2016, Tohoku University, Sendai 16

17 Comparison with the global status J.M. Lattimer, J. Lim, Ap. J. 771, 51 (2013) J = 31.6 ± 2.7 MeV L = 58.9 ± 31.6 MeV B.A. Li, 2016 Consistency in the (J,L) plane! C.H. Horowitz et al., J. Phys. G 41 (2014) NSMAT2016, Tohoku University, Sendai 17

18 Neutron skin in 208 Pb PDR ΔR np = ± fm Dipole polarizability ΔR np = 0.16 ± 0.03 fm IVGQR ΔR np = 0.14 ± 0.03 fm AGDR < ΔR np < fm The value extracted from dipole polarizability corresponds to the most comprehensive analysis Values from IVGQR and PDR compatible Value from AGDR? NSMAT2016, Tohoku University, Sendai 18

19 Symmetry energy and n-stars (I) Ultimately, the energy balance is dominated by the energy of β-equilibrated matter vs. gravitational energy. The stiffer the energy of neutron matter grows with density, the larger is the mass. J. Stone et al., PRC68, (2003), L. Trippa, M.Sc. Thesis 2007, X. Roca Maza et al., PRC 86, (2012). N-stars: despite the complex structure, simple assumption of uniform matter can be effective to assess the proper nuclear physics input. A. Steiner, PRC 77, (2008). NSMAT2016, Tohoku University, Sendai 19

20 Symmetry energy and the composition of the outer crust Outer crust: the composition is determined by masses of n-rich nuclei The stiffer is the symmetry energy, the more exotic is the nuclear composition. Strong magnetic field: Electrons affected for B around B c ( G) Nuclei affected for B larger than G D. Basilico et al., PRC 92, (2015) X. Roca-Maza et al., PRC 78, (2008) NSMAT2016, Tohoku University, Sendai 20

21 Model dependence a first study C. Mondal et al., PRC 93, (2016) Different families of models built with the same protocol. FSV like FSU (ρ-ω coupling) TSV with σ-ω coupling DDME and SAMi-J introduced in the previous discussion KDE0 is of Skyrme type Difference in skin much larger than difference in L! NSMAT2016, Tohoku University, Sendai 21

22 Why? Different densities distributions! Despite the care in choosing the same observables when fitting EDFs, details differ and matter. Saturation density and density inside 208 Pb are not exactly the same. Also the density that characterises the skin is not always the same! NSMAT2016, Tohoku University, Sendai 22

23 Conclusions EDFs provide a unique framework to study ground-state and excited states of nuclei together with nuclear matter and neutron stars. Connections between observables and equation of state are possible. Many works have been devoted to pin down the values of the symmetry energy. Convergence around normal saturation density is coming but with open questions yet. Connection with neutron stars has been elucidated. To improve EDFs: constrain their density dependence. NSMAT2016, Tohoku University, Sendai 23

24 Co-workers M. Brenna, X. Roca-Maza (University of Milano and INFN, Italy) M. Centelles, X. Viñas (University of Barcelona, Spain) N. Paar, D. Vretenar (University of Zagreb, Croatia) J. Piekarewicz (Florida State University, USA) C. Mondal, B.K. Agrawal (SINP, Kolkata, India) S.K. Singh, S.K. Patra (Institute of Physics, Bhubhaneshwar, India) L. Cao (NCEPU, Beijing, P.R. China) H. Sagawa (University of Aizu and RIKEN, Japan) NSMAT2016, Tohoku University, Sendai 24

25 Backup slides NSMAT2016, Tohoku University, Sendai 25

26 Applications to uniform matter (EoS) Nuclear matter is an idealized UNIFORM system of neutrons and protons having constant density. The Coulomb interaction among protons must be taken out! It is analogous to the uniform electron gas for condensed matter physcists. WE STICK TO T = 0! NSMAT2016, Tohoku University, Sendai 26

27 Examples NSMAT2016, Tohoku University, Sendai 27

28 QHO model and the relationship with S Schematic RPA: Bohr-Mottelson formula: pot We assume: (i) simple density profile; (ii) relationship with S shell gap E(ISGQR) = 61 A -1/3, Fermi energy = 37 MeV, S(0.1) = 24 MeV E(IVGQR) = 135 A -1/3 NSMAT2016, Tohoku University, Sendai 28

29 Systematically varied families of functionals: SAMi e DDME X. Roca-Maza, G.C., H. Sagawa, Phys. Rev. C 86, (R) (2012). D. Vretenar, T. Nikšić, P. Ring, Phys. Rev. C68, (2002). All sets have comparable quality. Fits on exp. data (binding energies, radii etc.) are repeated each time by fixing only either m* (SAMim) or J (SAMi-J or DDME-x). NSMAT2016, Tohoku University, Sendai 29

30 Model dependence Interestingly, experiment lies in the region where the model dependence is minimal. NSMAT2016, Tohoku University, Sendai 30

31 The anti-analog giant dipole resonance The AGDR is the analogous state of the GDR, in the same way as the IAS is the analogous of the g.s. Anti-? Perhaps misleading. We expect E1 transitions between AGDR and IAS in the same way as between GDR and g.s. C. Ligang, X. Roca-Maza, GC, H. Sagawa, Phys. Rev. C 92, (2015); Phys. Rev. C94, (2016). In this respect, we expect sensitivity to the symmetry energy but the argument should be refined. NSMAT2016, Tohoku University, Sendai 31

32 Why is E(AGDR)-E(IAS) correlated with L? Z N Using sum rules and schematic RPA, as above: By taking the difference, the Lane potential and the Coulomb energy disappear. The shell gap is related to the isoscalar effective mass. One is left with quantities that can be treated within the DM, to conclude that NSMAT2016, Tohoku University, Sendai 32

33 Improved theoretical analysis (i) Exact form for the twobody spin-orbit residual force; The functional has a twoparameter form that is not associated with an interaction (i) exact Coulomb exchange; No Slater approximation (ii) resonance centroids are extracted by careful choice of energy interval. Model dependence is clear here Families of forces created with the same protocol and with different J NSMAT2016, Tohoku University, Sendai 33

34 Constraints from the experimental data Errors are not in the plots. By including experimental errors one arrives at Should experiment be re-done? NSMAT2016, Tohoku University, Sendai 34

35 Attempts to extract S from HI reactions Courtesy: B. Tsang If x is the asymmetry β, we speak of isospin diffusion and a stiff symmetry energy causes small diffusion (too much energy cost!), whereas a soft symmetry energy causes a rather good degree of equilibration. Time evolution of the one-body distribution function f ( r, p, t) t Vlasov Boltzman n Langevin { h( f ), f ( r, p) } = K ( f ) + δk ( r, p, ) f ( r, p) t NSMAT2016, Tohoku University, Sendai 35