Isovector and spin-isospin nuclear matter symmetry energy (Skyrme forces):

Isovector and spin-isospin nuclear matter symmetry energy (Skyrme forces): dependences on density and temperature Fábio L. Braghin Int. Centr Cond Matter Physics-University of Brasilia, Brasilia, Brasil (and IF-USP, São Paulo, Brasil) Financial Support: IBEM/Ministry Scie.-Tech-Brazil FAP-DF (Brasilia) - Brazil FAPESP (São Paulo) F.L.Braghin, IWND SE-09, Shanghai, August, 2009

Drop liquid (macroscopic) models NUCLEAR FORCE dependences on isospin in A(s,t) => (spin, isospin) = (s,t) F.L.Braghin, IWND SE-09, Shanghai, August, 2009

Spin-Isospin symmetry energy Essentially restoring force of Gamow Teller (GT) resonances Probe for pionic degree of freedom, eg. in relativistic models Lagrangian (pseudo-scalar coupling): Related to an eventual instability of pion condensation (early 70, 80’s), higher r Eventually to DCC F.L.Braghin, IWND SE-09, Shanghai, August, 2009

Relating the symmetry energy coefficient GENERAL AND response function (static) IDEA For a small external perturbation in binding energy: Standard form: F.L.Braghin, IWND SE-09, Shanghai, August, 2009

Dynamical Polarizabilities with Skyrme forces Linear response on time dependent Hartree Fock for Skyrme forces such as: Behavior of small density fluctuations induced by small amplitude external perturbation that separates densities of neutrons and protons (s,t= 0,1) neutrons/protons spin up/down (s,t=1,1) F.L.Braghin, IWND SE-09, Shanghai, August, 2009

Writing only the static limit for any densities of protons and neutrons Same standard form of polarizability v = b, c, d Asymmetry parameters Temperature dependence of A Form factors for Densities of states Particle densities Densities of momentA F.L.Braghin, IWND SE-09, Shanghai, August, 2009 NPA(2000); PRC(2005)

Each of the densities (form factors)for rp = rn F.L.Braghin, IWND SE-09, Shanghai, August, 2009

Density dependence of the spin-isospin symmetry energy: FLB,IJMPE(2003) Very different predictions N Kaiser, CPT, (2006) (strongly dependent on each d.o.f./effect Mesons exchanges / D F.L.Braghin, IWND SE-09, Shanghai, August, 2009

Temperature dependence of n –p symmetry energy: A0,1 F.L.Braghin, IWND SE-09, Shanghai, August, 2009

0.1 r0 (NN09) Exp (diff r and T*) Shetty et al 2006; Le Fevre et al (2005) Trautmann et al (2006) (INDRA; ALADIN) Kowalski et al(2006) .05r SLyb,b=0.5 (x) SLyb,b=0 ___ SGII,b=0.5 (+) SGII,b=0 ..... Samaddar et al: down triangles (2007) Lie-Wen Chen et al: up triangles (2007) Moustakidis: dashed (2008) Different momentum Dependent interactions, Clusters,.. F.L.Braghin, IWND SE-09, Shanghai, August, 2009

Full solid symbols , x: 1.33 r0 , b = 0 Empty symbols , +: 1.33 r0 , b = 0.25 • -Onsi et al (a) SkSC4 • - Lyon group SLy4 • - Onsi et al (b)SkSC6 • Sagawa/Giai • SGII F.L.Braghin, IWND SE-09, Shanghai, August, 2009 FLB, PRC(2009)

0.1 r0 Spin-isospin: ++ SGII (b=.5) ... SGII (b=0) xx SLy4 (b=.5) ___ SLy4 (b=0) F.L.Braghin, IWND SE-09, Shanghai, August, 2009

1.33 r0 Smoother variation at higher densities SGII SkSC4 SLy4 F.L.Braghin, IWND SE-09, Shanghai, August, 2009

Momentum dependence of the polarizability Limit of equal densities protons-neutrons Depending on the Skyrme force F.L.Braghin, IWND SE-09, Shanghai, August, 2009

Momentum dependence 0.1 r0 2 r0 Instability for n-matter F.L.Braghin, IWND SE-09, Shanghai, August, 2009

Spin-isospin: Momentum dependence at r0 T=0,1,2,4 Momentum dependence does not favor spin-isospin instabilities for several Skyrme forces (but yes for neutron-matter instabilities) (not yet extensive) F.L.Braghin, IWND SE-09, Shanghai, August, 2009

Considering always: In static (or ~ dynamical) framework Eventually assuming: A = A (b,r, T,q,..) F.L.Braghin, IWND SE-09, Shanghai, August, 2009

Simultaneous dependence of symmetry energy on different variables Standard expression Variation of P Or More generally: Looking for data to plug with e.o.s. For finite nuclei； Densities-> N,Z/Vn Vp F.L.Braghin, IWND SE-09, Shanghai, August, 2009 PRC(2005)

Summary • Calculating symmetry energies from nuclear polarizabilities • Dependences on T, n-p, momentum exchanged depend strongly on density • Ｈｏｗ　ｃan spin-isospin sym.energ interaction　（ｔｏｇｅｔｈｅr with(0,1)) ｂｅ　ｐｒｏｂｅｄ　by pions in 　ｈｉｃ? • Differential equation available：ｇｉｖｅ　ｍｅ　eos　ａｎｄ　ｋｎｏｗｗｗｈｉｃｈ symmetryｅｎｅｒｇｙ isｃｏｍｐａｔｉｂｌｅ．． • - So far no surface or finite size effect taken into account. Thank you. 謝謝 F.L.Braghin, IWND SE-09, Shanghai, August, 2009

For The same Variation of A with a (or b) Example of ~ opposite behavior: De-Samaddar-nucl-th/0708.2183 F.L.Braghin, IWND SE-09, Shanghai, August, 2009

For: Heiselberg, Hjorth Jensen and Differential equation becomes: Stable Density As function Of n-p For the eos F.L.Braghin, IWND SE-09, Shanghai, August, 2009

Neutron matter: No assumption or consideration on dynamical and nuclear microscopical effects (forces, processes) - These will provide more precise expressions F.L.Braghin, IWND SE-09, Shanghai, August, 2009

Based onF.L.B., Phys. Rev. C (2009); Phys. Rev. C 71, 064303 (2005); Int. Journ. Mod. Phys. E12, 755 (2003); Nucl. Phys. A 665, 13 (2000); Nucl. Phys. A 696, 413 (2001)+ A 709, 487 (2002). Proceedings Brazilian Meetings on Nuc Phys 2007, 2008. F.L.Braghin, IWND SE-09, Shanghai, August, 2009

Mostly quadratic terms in d r Scalar channel: incompressibility K and A00 have nearly the same behavior with b, a F.L.Braghin, IWND SE-09, Shanghai, August, 2009

Isovector and spin-isospin nuclear matter symmetry energy (Skyrme forces):