Effects of n-p Mass Splitting on Symmetry Energy

Effects of n-p Mass Splitting on Symmetry Energy Li Ou1,2 Zhuxia Li3 Bao-An Li1 1 Department of Physics and Astronomy Texas A&M University-Commerce, USA 2 College of Physics and Technology Guangxi Normal University, P. R. China 3 China Institute of Atomic Energy, P. R. China Jun. 19th 211 NuSYM11@Smith College

Outline 1 Introduction 2 Approach 3 Results and discussion 4 Summary

Motivation Symmetry energy (ρ, T ) = E(ρ, T, δ = 1) E(ρ, T, δ = ). The multifaceted influence of the nuclear symmetry energy 1 Nuclear structure for nuclei far away from β-stability line (cold); Mechanism for rare isotope reaction (hot); Astrophysics & Cosmology issues (cold and hot). (ρ) a lot works, great progress; (T ) few attention, more unclear. 1 A.W. Steiner, M. Prakash, J.M. Lattimer and P.J. Ellis, Phys. Rep. 411, 325 (25).

Some works on (T ) Theoretical Jun Xu et.al, Phys. Rev. C 77, 1432 (28) Self-consistent Thermal model+mdi Ch. C. Moustakidis, Phys. Rev. C 76, 2585 (27) Experimental S. Kowalski et.al PRC 75, 1461(27). J. B. Natowitz,et.al PRL 14, 2251 (21). D. V. Shetty et al., arxiv:nucl-ex/6632. A. Le Fèvre et al., Phys. Rev. Lett. 94, 16271 (25) W. Trautmann et al., arxiv: nucl-ex/6327. Taken from Jun Xu et.al, Phys. Rev. C 75, 1467 (27) Low ρ, Low T : a few data, High ρ: none.

Effective mass splitting Taken from W. Zuo, et.al, PRC 74 (26) 14317. (D)BHF: m n > m p m * /m m * /m m * /m 2. SLy7 n, p = Skz3 SkT5 1.5 n =1 1. p =1.5. SkMP SIII SkM 1.5 1..5. SKXm 1.5 1..5...5 1. 1.5 SkP..5 1. 1.5..5 1. 1.5 2. v7 SHF: interaction dependence. m n > m p? m n = m p? or m n < m p? Experimental information are needed!

Question Motivation Obtain (ρ, T ); Study the relationship between (ρ, T ) & m n/m p

Approach Thermodynamics model with Skyrme energy density functional. Skyrme energy density functional H = 2 2m [τ p(r) + τ n (r)] + 1 4 t [(2 + x )ρ 2 (2x + 1)(ρ 2 p + ρ 2 n)] + 1 24 t 3ρ α [(2 + x 3 )ρ 2 (2x 3 + 1)(ρ 2 p + ρ 2 n)] + 1 8 [t 1(2 + x 1 ) + t 2 (2 + x 2 )]τρ + 1 8 [t 2(2x 2 + 1) t 1 (2x 1 + 1)](τ p ρ p + τ n ρ n ), nucleon density ρ q = 2 n q (p) 4πp2 dp, and kinetic energy density τ q = 2 n q (p) 4πp2 3 p 2 2 dp. 3

phase-space distribution function obeys Dirac-Fermi distribution, n q = 1 1 + exp[β(ε q µ q )]. ε q = p2 2m + U q is the single particle energy for proton or neutron. q The EFM m q is expressed as 2 2m q = 2 2m q + 1 8 [t 1(2 + x 1 ) + t 2 (2 + x 2 )]ρ + 1 8 [t 2(2x 2 + 1) t 1 (2x 1 + 1)]ρ q. The potential energy U q for proton or neutron reads U q (r) = 1 2 t [(2 + x )ρ (1 + 2x )ρ q ] + 1 24 t 3[(2 + x 3 )(2 + α)ρ α+1 (1 + 2x 3 )[2ρ α ρ q + αρ α 1 (ρ 2 n + ρ 2 p)]] + 1 8 [t 1(2 + x 1 ) + t 2 (2 + x 2 )]τ + 1 8 [t 2(2x 2 + 1) t 1 (2x 1 + 1)]τ q.

Introducing effective chemical potential µ q = µ q U q, then n q = 1 1 + exp[β(p 2 /2m q µ q)]. For fixed density ρ, temperature T, and isospin asymmetry δ, the effective chemical potential µ q can be determined numerically by a self-consistency iteration scheme, and then we get EOS. (ρ, T ) = E(ρ, T, δ = 1) E(ρ, T, δ = ).

(ρ, T ) R m = m 1 m for neutron 4 3 2 1 (d) 3 2 1 3 2 1 (a) (b) (c) SLy7 R m =.87 1>m * p >m* n SkMP R m =1.13 (e) Skz3 R =.91 m 1>m * p >m* n (f) SIII R m =1.19 (g) (h) (i) SkT5 R m =1. m * p =m* n =1 SkM R m =1.23 SKXm SkP v7 R =1.41 m R =1.55 m R =1.98 m..5 m * m * n 1. 1.5 p n >1>m* p m * n >1>m* p..5 1. 1.5..5 1. 1.5 2. / / /

(ρ, T ) R m = m 1 m for neutron 4 3 2 1 (d) 3 2 1 3 2 1 (a) (b) (c) SLy7 R m =.87 1>m * p >m* n SkMP R m =1.13 (e) Skz3 R =.91 m 1>m * p >m* n (f) SIII R m =1.19 (g) (h) (i) MeV 5 MeV 1 MeV 2 MeV SkT5 R =1. m m * p =m* =1 n SkM R m =1.23 SKXm SkP v7 R =1.41 m R =1.55 m R =1.98 m..5 m * m * n 1. 1.5 p n >1>m* p m * n >1>m* p..5 1. 1.5..5 1. 1.5 2. / / / Density dependence of has no obvious relation to EFM splitting; Temperature dependence of has relation to EFM splitting.

5 4 3 2 1 4 3 2 1 v7 R m =2.1 v1 R m =1.5..5 1. / T= MeV T= 5 MeV T=1 MeV T=2 MeV v8 R m =1.45 v15 R m =1. v9 R m =1.2 v11 R m =.95..5 1...5 1. 1.5 / /

Relationship between m n/m p and (ρ, T ) (ρ, T ) = (ρ, T ) (ρ, ) = ( 2 2mρ + b 3 + b 4 )(τ T 1 τ Rm )(1 R τ ) R m = m 1 m R τ = τ T 1 τ 1 τ T τ E 5 4 3 2 1-1 isosvector part isoscalar part ver R m Skz-1 1.84 Skz2 1.13 Skz3.91-2..5 1. 1.5 2. / R 6 4 2 1.2 1..8.6.4 Skz-1 R m =1.84 T=2 MeV R m R R m / R (a1) MeV 2 MeV (b1) Skz2 R m =1.13 (a2) Skz3 R m =.91..5 1...5 1...5 1. 1.5 / / / (b2) (a3) (b3)

How effective mass influence R τ? τ q = 2 n q = n q (p) 4πp2 3 p 2 2 dp. 1 1 + exp[β(ε q µ q )] ε q = p2 2m q + U q

n q n q 1.2 (a) Skz-1 (b) Skz-1.3.8 1.5 MeV = MeV =1.4. (c).8.4. 2 4 p (MeV/c) Skz2.3 (d) Skz2 1.5 2 4 6 p (MeV/c)

n q n q 1.2 (a) Skz-1 (b) Skz-1.3.8 1.5 MeV = MeV =1.4 2 MeV =. (c).8.4. 2 4 p (MeV/c) Skz2.3 (d) Skz2 1.5 2 4 6 p (MeV/c) Deviations of nucleon distributions: in SNM: τ T τ

n q n q 1.2 (a) Skz-1 (b) Skz-1.3.8 1.5 MeV = MeV =1.4 2 MeV = 2 MeV =1. (c).8.4. 2 4 p (MeV/c) Skz2.3 (d) Skz2 1.5 2 4 6 p (MeV/c) Deviations of nucleon distributions: in SNM: τ T τ in PNM:τ T 1 τ 1 R τ = τ 1 T τ 1 τ T τ (ρ, T ) (1 Rm R τ ) Low ρ: Deviations of n n (p) in PNM is weaker than n q (p) in SNM, R τ < 1. (T ) < (T = ). High ρ: Skz2, same as low ρ; Skz-1, n n (p) in PNM deviate stronger than n q (p) in SNM, R τ > 1. (T ) > (T = ).

Summary 1 The work involves two open issues: (ρ, T ), m n/m p, and their correlation; 2 There is not obvious correlation between n-p effective mass splitting and density dependence of symmetry energy; 3 The behavior of temperature dependence of symmetry energy has relation to n-p effective mass splitting; 4 For all Skyrme interaction, symmetry energy at low density becomes softer with temperature increase. For some interactions with m n > m p TrTDSE may occur, which makes symmetry energy at high density stiffer. 5 No only itself, temperature and density dependence of symmetry energy is a matter of significance for extracting n-p EFM splitting.

Thanks

Density dependence of n-p effect mass The catalogue of 93 sets of Skyrme interaction based on m /m. Group symmetric matter neutron matter I m p = m n >1 BSk1 3, MSk2, MSk4 6, SVII, SKXce, vx BSk1, MSk2, MSk4 6, V15 II m p = m n =1 MSk1, MSk3, SkP, SkSC1 4, SkT1 6 MSk1, MSk3, SkSC1 4, SkT1 6 BSk4 17, Es, FitB, Gs, RATP, Rs, SGI, III m p = m n <1 IV m p < m n <1 V m p < 1 < m n VI 1 > m p > m n SGII,SI VI, SIII, SKRA, SkIx, SkM1, SkM, SkM, SkMP, SkO, SkT7 9, SKX, SKXm, Skzx, SLyx, Zs SkT8, SkT9 BSk14 17 Gs, RATP, Rs, SGI, SGII, SII SV, SIII, SKRA, SkM, SkM1, SkMP, SkM, SkO, SkT7, Skz1, Skz2, Zs BSk2 5,1 13, Es, FitB, SI, SkP, SKX, SKXce, SKXm, Skz-1, Skz, SVI, SVII, v7, v75, v8, v9, v1 BSk6 9, SkIx, Skz3, Skz4, SLyx VII m p > m n > 1 v11

4 3 2 1 3 2 1 3 2 BSK1 R m =1. BSK4 R m =1.8 T= MeV T=2 MeV 1 BSK7 R =.93 m..5 1. 1.5 / 4 BSK1 3 R =1.15 m 2 1 3 2 1 3 2 1 BSK13 R m =1.14 BSK16 R m =1.1..5 1. 1.5 / T= MeV T=2 MeV BSK2 R m =1.27 BSK5 R m =1.11 BSK8 R m =.93 BSK3 R m =1.35 BSK6 R m =.94 BSK9 R m =.89..5 1. 1.5..5 1. 1.5 2. / / BSK11 R m =1.14 BSK14 R m =1.2 BSK17 R m =1.2 BSK12 R m =1.14 BSK15 R m =1.4..5 1. 1.5..5 1. 1.5 2. / / 5 SkI1 4 R =.89 m 3 2 1 SkI4 4 R =.84 m 3 2 1..5 1. / 5 Skz-1 4 R =1.84 m 3 2 1 Skz2 4 R =1.13 m 3 2 1..5 1. / SkI2 R m =.87 T= MeV T=2 MeV SkI5 R =.78 m SkI3 R m =.78 SkI6 R m =.82..5 1...5 1. 1.5 / / Skz R m =1.66 T= MeV T= 5 MeV T=1 MeV T=2 MeV Skz3 R =.91 m Skz1 R m =1.3 Skz4 R m =.79..5 1...5 1. 1.5 / /

Hugenholtz Van Hove (HVH) theorem: m n m p = 2δ m / ( du sym m 2 k F dk 1 + 2 m ) du kf 2 k F dk kf = 2δ m / [ ( )] du sym m 2 k F dk 1 + 2 1 kf Chang Xu, et al, PRC 82, 5467(21) m