Symmetry energy in Skyrme models

J.R.Stone University of Oxford/University of Tennessee

Tony Hilton Royle Skyrme! 1922 - 1987!

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 0.01 – 3.0) ρ0

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

Energy per particle in the Skyrme model

Parameters: t0 ,t1,t2 ,t31,t32 ,t33,t4 ,t5 , x0 , x1, x2 , x31, x32 , x33, x4 , x5 ,σ1,σ 2 ,σ 3,δ ,γ

More for finite nuclei (spin-orbit term etc)

y=Z/A

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?

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, 24210-150 Boa Viagem, Niter´oi RJ, Brazil

J. R. S.

P. D. Stevenson Department of Physics, University of Surrey, Guildford, GU2 7XH UK

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 ρ0 ~0.16 fm-3

Farine, Pearson and Tondeur, Nucl. Phys. A615, 135, (1997).

ref later

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)

B. Pure neutron matter (PNM)

PNM1: Pressure in low density dilute neutron gas:

Schwenk and Pethick, PRL 79, 055801 (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)

Constraints involving both symmetric and pure neutron matter

Symmetry energy and its density dependence:

x =ρ − ρ0

3ρ0

y= ZA

Stone and Reinhard, Prog.Part.Nucl.Phys.58, 587 (2007)

Chen et al., PRC 80, 014322 (2009)

Reduction of the symmetry energy at half of the saturation density

Slope of the symmetry energy at the saturation density

Symmetry energy at saturation density

Danielewicz, Nucl.Phys. A727, 233 (2003)

Isospin incompressibility: Analysis of GMR data

Stone et al. 2011

Relativistic field models, parabolic equation of state, HIC, neutron star

Piekarewicz, PRC76, 064310 (2007)

MIX3: -700 ≤ Kτ ,v ≤ −370 MeV

B. Tsang et al

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 J0737-3039

0 0.5 1 1.5 2 2.5 3!/!

0

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Mn*/M

GSkI

GSkII

KDE0v1

MSL0

NRAPR

SQMC650

SQMC700

SQMC750

SkO’

SV-sym32

0 0.5 1 1.5 2 2.5 3!/!

0

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Mp*/M

Density dependence of the effective mass in BEM matter

neutrons protons

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

Landau parameters in PNM

Density dependence of the symmetry energy All selected parameter sets predict INCREASING symmetry

energy with increasing density:

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?

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)

Low mass neutron star - The double pulsar J0737-3039 : Podsiadlowski et al. Mon.Not.R.Astron.Soc. 361, 1243 (2005)

Prediction of 0.019 – 0.024 Msolar loss in the progenitor mass

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

R = (0.86±0.01)A1/3 + (0.47±0.05) fm

Centelles et al., PRL 102, 122502 (2009) neutron skin thickness (antiprotons)

+ data on charge radii from Fricke et al.

S = (0.9±0.15)I + (-0.03±0.02) fm

Values of KA, calculated with charge and matter radii of 112-124 Sn

Blaizot 1980,1995; Treiner et al. 1981

isospin incompressibility

ratio of the surface to volume terms

Li et al., 2007, 2010

Stone et al.,2011

Blaizot 1980,1995; Treiner et al. 1981

converges only in the “scaling” approximation:

In this approximation: (i) Kvol = K0 = Kn.m.

(ii) Kcourv and higher order terms are small

The scaling approximation is related to the cubic weighted sum rule and the energy of GMR is determined as

E3 = (m3 / m1)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 KA would be uniquely determined

In real world this is not the case so a selection has to be made.

Two step fitting process:

MESH – minimum sought on a fine mesh of K0 and Kτ

MINUIT – CERN minimization package – calculation of errors including correlations

Protocol:

K0 and Kτ for fixed c and Kcoul K0 and Kτ varying (fixed) values of c and Kcoul Estimation of Kτ,v from a known value of Kτ

Data:

Sn and Cd isotopes (Li 2007, Garg 2011) Available data on 90Zr, 92Mo, Sn, Cd, 144,148Sm and 208Pb

112-124 Sn 106,110-116 Cd Sn + Cd

Results of the MESH fit – c, K0, Kτ (Kcoul from a model) Li et al., PRL 99, 162503 (2009) Sagawa PRC 76, 034327 (2007) Garg 2011, Zakopane Proceedings)

Sn + Cd

Sn -514 ± 157 MeV Cd -690 ± 225 MeV

Sn+Cd -634 ± 127 MeV All data -632 ± 77 MeV

Total isospin incompressibility Kτ

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:

Kτ ,v + Kτ ,sA−1/3 = Kτ

Kτ ,sA−1/3

Kτ ,v

≤ 0.5 Kτ ,v ≤ Kτ

−700 ≤ Kτ ,v ≤ −370 MeV

Incompressibility of infinite matter K0:

Sn -206 ± 10 MeV Cd -210 ± 11 MeV

Sn+Cd -256 ± 10 MeV (variable c) All data -280 ± 40 MeV

Compare with:

K0 = 300 ± 25 MeV

Kτ = 320 ± 180 MeV

Sharma et al., PRC 38, 2562 (1988)

Danielewicz 2002 FOPI, PLB 612, 173 (2005)

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, 014310 (2005) Cao et al., PRC 73, 014313 (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

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!