Symmetry energy in Skyrme models

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1 Tony Hilton Royle Skyrme! ! Symmetry energy in Skyrme models J.R.Stone University of Oxford/University of Tennessee

2 Skyrme model: density dependent NN force for finite nuclei and for nuclear matter Low momentum expansion of the two-body interaction validity range (density range ) 0 Nuclear matter: idealized infinite medium made of interacting nucleons uniform density, no surface and Coulomb effects Close to: matter in the interior of heavy nuclei stellar matter matter created in heavy ion collisions Excellent laboratory for testing of nuclear models

3 Energy per particle in the Skyrme model y=z/a Parameters: t 0,t 1,t 2,t 31,t 32,t 33,t 4,t 5, x 0, x 1, x 2, x 31, x 32, x 33, x 4, x 5,σ 1,σ 2,σ 3,δ,γ More for finite nuclei (spin-orbit term etc)

4 Determination of the Skyrme parameters: Calculation of ground state properties of finite nuclei: masses, radii, moments, single-particle energies, fission barriers, energies of giant resonances etc and comparison to experimental data More pieces of information than parameters but the parameters are highly correlated In principle infinite number of parameter sets fits the data well Currently more than 237 sets can be found in the literature (almost) impossible to find the best set by looking at finite nuclei How about nuclear matter?

5 The Skyrme Interaction and Nuclear Matter Constraints M. Dutra, O. Lourenco, J. S. S. Martins, and A. Delfino Departamento de Fısica - Universidade Federal Fluminense, Av. Litorˆanea s/n, Boa Viagem, Niter oi RJ, Brazil J. R. S. P. D. Stevenson Department of Physics, University of Surrey, Guildford, GU2 7XH UK

6 11 macroscopic (bulk) constraints A. Symmetric nuclear matter (SNM) (equal number of protons and neutrons) minimal requirement saturation energy E/A = ~16 MeV at density 0 ~0.16 fm -3 ref later Farine, Pearson and Tondeur, Nucl. Phys. A615, 135, (1997).

7 SM3: Density dependence of pressure in SNM Extracted from measurement of particle flow in heavy ion collision (HIC) Danielewicz et al, Science 298, 1592 (2002) SM4: The same as above but including emission of kaons Lynch et al., Prog.Part,Nucl.Phys. 62, 427 (2009)

8 B. Pure neutron matter (PNM) PNM1: Pressure in low density dilute neutron gas: Schwenk and Pethick, PRL 79, (2009) Epelbaum et al., Eur. Phys. J. A40, 199 (2009) PNM2: Pressure in high density PNM extracted from measurement of particle flow in HIC Danielewicz et al, Science 298, 1592 (2002)

9 Constraints involving both symmetric and pure neutron matter Symmetry energy and its density dependence: x = ρ ρ 0 3ρ 0 y= Z A

10 Symmetry energy at saturation density Stone and Reinhard, Prog.Part.Nucl.Phys.58, 587 (2007) Slope of the symmetry energy at the saturation density Chen et al., PRC 80, (2009) Reduction of the symmetry energy at half of the saturation density Danielewicz, Nucl.Phys. A727, 233 (2003)

11 Isospin incompressibility: Analysis of GMR data MIX3: -700 K τ,v 370 MeV Stone et al Relativistic field models, parabolic equation of state, HIC, neutron star Piekarewicz, PRC76, (2007)

12 B. Tsang et al

13 5 microscopic constraints: Applied to parameterizations that successfully passed the macroscopic constraints (i) Density dependence of the effective mass in BEM (ii) Landau parameters and the critical density for transition to spin-ordered nuclear matter (iii) Density dependence of the symmetry energy (iv) (Gravitational mass and radius of high mass neutron stars) (v) Correlation of gravitational and baryonic mass of double pulsar J

14 Density dependence of the effective mass in BEM matter GSkI GSkII KDE0v1 MSL0 NRAPR SQMC650 SQMC700 SQMC750 SkO SV-sym32 M n */M M p */M neutrons 0.4 protons !/! !/! 0

15 Landau parameters in SNM > -(2l+1)

16 Landau parameters in PNM

17 Density dependence of the symmetry energy All selected parameter sets predict INCREASING symmetry energy with increasing density:

18 Comments on the symmetry energy in nuclear matter: S(ρ) = (E / A) PNM (ρ) (E / A) SNM (ρ) Should we seek models of PNM at high densities instead of the symmetry energy?

19 Maximum mass of a neutron star calculated with a Skyrme model corresponds to central density well beyond the validity range of the Skyrme interaction (left panel) Need to add another high density model e.g. quark-meson coupling (right panel) Stone et al., Nucl.Phys. A792, 341 (2007)

20 Low mass neutron star - The double pulsar J : Podsiadlowski et al. Mon.Not.R.Astron.Soc. 361, 1243 (2005) Prediction of M solar loss in the progenitor mass

21 Constraints extracted from Giant Monopole Resonance data: Li et al, 2007 and 2010, Texas A&M and ND + Osaka groups Brissaud et al.,nucl.phys. 191, 145 (1972) elastic alpha-scattering at 166 MeV + data from elastic proton scattering Centelles et al., PRL 102, (2009) neutron skin thickness (antiprotons) + data on charge radii from Fricke et al. R = (0.86±0.01)A 1/3 + (0.47±0.05) fm S = (0.9±0.15)I + (-0.03±0.02) fm

22 Values of K A, calculated with charge and matter radii of Sn

23 Blaizot 1980,1995; Treiner et al isospin incompressibility Li et al., 2007, 2010 Stone et al.,2011 ratio of the surface to volume terms

24 Blaizot 1980,1995; Treiner et al converges only in the scaling approximation: In this approximation: (i) K vol = K0 = K n.m. (ii) K courv and higher order terms are small

25 The scaling approximation is related to the cubic weighted sum rule and the energy of GMR is determined as E 3 = (m 3 / m 1 ) 1/2 moments of the strength function In general, If the strength function is a delta function, then all energies will be the same and K A would be uniquely determined In real world this is not the case so a selection has to be made.

26 Two step fitting process: MESH minimum sought on a fine mesh of K 0 and K MINUIT CERN minimization package calculation of errors including correlations Protocol: K 0 and K for fixed c and K coul K 0 and K varying (fixed) values of c and K coul Estimation of K τ,v from a known value of K τ Data: Sn and Cd isotopes (Li 2007, Garg 2011) Available data on 90 Zr, 92 Mo, Sn, Cd, 144,148 Sm and 208 Pb

27 Results of the MESH fit c, K 0, K (K coul from a model) Li et al., PRL 99, (2009) Garg 2011, Zakopane Proceedings) Sagawa PRC 76, (2007) Sn 106, Cd Sn + Cd

28 Sn + Cd

29 Total isospin incompressibility K τ Sn Cd -514 ± 157 MeV -690 ± 225 MeV Sn+Cd -634 ± 127 MeV All data -632 ± 77 MeV Limits on the volume part of the isospin incompressibility Why important? GMR data contain both volume and surface terms Skyrme models calculate only the volume term: K τ,v + K τ,s A 1/3 = K τ K τ,s A 1/3 K τ,v 0.5 K τ,v K τ 700 K τ,v 370 MeV

30 Incompressibility of infinite matter K 0 : Sn Cd -206 ± 10 MeV -210 ± 11 MeV Sn+Cd -256 ± 10 MeV (variable c) All data -280 ± 40 MeV Compare with: K 0 = 300 ± 25 MeV K τ = 320 ± 180 MeV Sharma et al., PRC 38, 2562 (1988) Danielewicz 2002 FOPI, PLB 612, 173 (2005)

31 Final result: Macroscopic constraints: 18 out of 237 passed GSkI, GSkII, KDE0v1, LNS, MSL0,NRAPR, Ska25s20, Ska35s20, SkO, SkT1, SkT2, SkT3, SkT1*, SkT3*, SQMC650, SQMC700, SQMC750 and SV -sym32, Microscopic and observational constraints: only 5 out of 18 passed KDE0v1, LNS, NRAPR, SQMC700, SQMC750 Agrawal, Shlomo and Au, PRC72, (2005) Cao et al., PRC 73, (2006) BHF+3BF Steiner et al., Phys.Rep. 411, 325 (2005) variational Akmal et al Guichon et al., Nucl.Phys. A772, 1 (2006) Quark-Meson Coupling

32 Purpose of this work: Guidance in selection and improvement of Skyrme parameterizations: Observation we made: It is essential to pay the utmost attention to applicability of our models in physical situations: It is not good enough to be amazed when a dog speaks we have to care not only what it says. but also what it really means! Skyrme Sk5346 is really good!