How tightly is the nuclear symmetry energy constrained by a unitary Fermi gas?
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摘要
We examine critically how tightly the density dependence of nuclear symmetry energy E_(sym)(q) is constrained by the universal equation of state of the unitary Fermi gas EUG(q) considering currently known uncertainties of higher order parameters describing the density dependence of the equation of state of isospin asymmetric nuclear matter. We found that E_(UG)(q) does provide a useful lower boundary for the E_(sym)(q). However, it doesnot tightly constrain the correlation between the magnitude E_(sym)(q_0) and slope L unless the curvature K_(sym)of the symmetry energy at saturation density q_0 is more precisely known. The large uncertainty in the skewness parameters affects the E_(sym)(q_0) versus L correlation by the same almost as significantly as the uncertainty in K_(sym).
We examine critically how tightly the density dependence of nuclear symmetry energy E_(sym)(q) is constrained by the universal equation of state of the unitary Fermi gas EUG(q) considering currently known uncertainties of higher order parameters describing the density dependence of the equation of state of isospin asymmetric nuclear matter. We found that E_(UG)(q) does provide a useful lower boundary for the E_(sym)(q). However, it doesnot tightly constrain the correlation between the magnitude E_(sym)(q_0) and slope L unless the curvature K_(sym)of the symmetry energy at saturation density q_0 is more precisely known. The large uncertainty in the skewness parameters affects the E_(sym)(q_0) versus L correlation by the same almost as significantly as the uncertainty in K_(sym).
We examine critically how tightly the density dependence of nuclear symmetry energy E_(sym)(q) is constrained by the universal equation of state of the unitary Fermi gas EUG(q) considering currently known uncertainties of higher order parameters describing the density dependence of the equation of state of isospin asymmetric nuclear matter. We found that E_(UG)(q) does provide a useful lower boundary for the E_(sym)(q). However, it doesnot tightly constrain the correlation between the magnitude E_(sym)(q_0) and slope L unless the curvature K_(sym)of the symmetry energy at saturation density q_0 is more precisely known. The large uncertainty in the skewness parameters affects the E_(sym)(q_0) versus L correlation by the same almost as significantly as the uncertainty in K_(sym).
引文
1.B.A.Li,A.Ramos,G.Verde,I.Vidana(eds.),Topical issue on nuclear symmetry energy.Eur.Phys.J.A 50(2),9(2014)
2.B.A.Li,C.M.Ko,W.Bauer,Isospin physics in heavy-ion collisions at intermediate energies.Int.J.Mod.Phys.E 7,147(1998).https://doi.org/10.1142/S0218301398000087
3.B.A.Li,W.Udo Schro¨der(eds.),Isospin Physics in Heavy-Ion Collisions at Intermediate Energies(Nova Science Publishers Inc,New York,2001)
4.V.Baran,M.Colonna,V.Greco et al.,Reaction dynamics with exotic nuclei.Phys.Rep.410,335-466(2005).https://doi.org/10.1016/j.physrep.2004.12.004
5.B.A.Li,L.W.Chen,C.M.Ko,Recent progress and new challenges in isospin physics with heavy-ion reactions.Phys.Rep.464,113-281(2008).https://doi.org/10.1016/j.physrep.2008.04.005
6.W.G.Lynch,M.B.Tsang,Y.Zhang et al.,Probing the symmetry energy with heavy ions.Prog.Nucl.Part.Phys.62,427-432(2009).https://doi.org/10.1016/j.ppnp.2009.01.001
7.P.Danielewicz,J.Lee,Symmetry energy I:semi-infinite matter.Nucl.Phys.A 818,36-96(2009).https://doi.org/10.1016/j.nucl physa.2008.11.007
8.W.Trautmann,H.H.Wolter,Elliptic flow and the symmetry energy at supra-saturation density.Int.J.Mod.Phys.E 21,1230003(2012).https://doi.org/10.1142/S0218301312300032
9.M.B.Tsang,J.R.Stone,F.Camera et al.,Constraints on the symmetry energy and neutron skins from experiments and theory.Phys.Rev.C 86,015803(2012).https://doi.org/10.1103/Phys Rev C.86.015803
10.C.J.Horowitz,E.F.Brown,Y.Kim et al.,A way forward in the study of the symmetry energy:experiment,theory,and observation.J.Phys.G 41,093001(2014).https://doi.org/10.1088/0954-3899/41/9/093001
11.M.Baldo,G.F.Burgio,The nuclear symmetry energy.Prog.Part.Nucl.Phys.91,203-258(2016).https://doi.org/10.1016/j.ppnp.2016.06.006
12.B.A.Li,Nuclear symmetry energy extracted from laboratory experiments.Nuclear Physics News(2017)(to appear).ar Xiv:1701.03564
13.A.W.Steiner,M.Prakash,J.M.Lattimer et al.,Isospin asymmetry in nuclei and neutron stars.Phys.Rep.411,325-375(2005).https://doi.org/10.1016/j.physrep.2005.02.004
14.J.M.Lattimer,The nuclear equation of state and neutron star masses.Annu.Rev.Nucl.Part.Sci.62,485-515(2012).https://doi.org/10.1146/annurev-nucl-102711-095018
15.W.G.Newton,J.Hooker,M.Gearheart et al.,Constraints on the symmetry energy from observational probes of the neutron star crust.Eur.Phys.J.A 50,41(2014).https://doi.org/10.1140/epja/i2014-14041-x
16.K.Iida,K.Oyamatsu,Symmetry energy,unstable nuclei and neutron star crusts.Eur.Phys.J.A 50,42(2014).https://doi.org/10.1140/epja/i2014-14042-9
17.J.M.Pearson,N.Chamel,A.F.Fantina,Symmetry energy:nuclear masses and neutron stars.Eur.Phys.J.A 50,43(2014).https://doi.org/10.1140/epja/i2014-14043-8
18.F.J.Fattoyev,W.G.Newton,B.A.Li,Probing the high-density behavior of symmetry energy with gravitational waves.Eur.Phys.J.A 50,45(2014).https://doi.org/10.1140/epja/i2014-14045-6
19.T.Fischer,M.Hempel,I.Sagert et al.,Symmetry energy impact in simulations of core-collapse supernovae.Eur.Phys.J.A 50,46(2014).https://doi.org/10.1140/epja/i2014-14046-5
20.D.Blaschke,D.E.Alvarez-Castillo,T.Klahn,Universal symmetry energy contribution to the neutron star equation of state.ar Xiv:1604.08575
21.B.A.Li,X.Han,Constraining the neutron-proton effective mass splitting using empirical constraints on the density dependence of nuclear symmetry energy around normal density.Phys.Lett.B727,276-281(2013).https://doi.org/10.1016/j.physletb.2013.10.006
22.M.Oertel,M.Hempel,T.Kla¨hn et al.,Equations of state for supernovae and compact stars.Rev.Mod.Phys.89,015007(2017).https://doi.org/10.1103/Rev Mod Phys.89.015007
23.M.W.Zwierlein,Ch.18 in Novel Superfluids vol.2,ed.by K.-H.Bennemann,J.B.Ketterson(Oxford University Press,Oxford,2015)
24.M.J.H.Ku,A.T.Sommer,L.W.Cheuk et al.,Revealing the superfluid lambda transition in the universal thermodynamics of a unitary Fermi gas.Science 335,563-567(2012).https://doi.org/10.1126/science.1214987
25.G.Zu¨rn,T.Lompe,A.N.Wenz et al.,Precise characterization of6Li feshbach resonances using trap-sideband-resolved RF spectroscopy of weakly bound molecules.Phys.Rev.Lett.110,135301(2013).https://doi.org/10.1103/Phys Rev Lett.110.135301
26.M.G.Endres,D.B.Kaplan,J.W.Lee et al.,Lattice Monte Carlo calculations for unitary fermions in a finite box.Phys.Rev.A 87,023615(2013).https://doi.org/10.1103/Phys Rev A.87.023615
27.E.E.Kolomeitsev,J.M.Lattimer,A.Ohnishi,I.Tews.ar Xiv:1611.07133v1(2016)
28.I.Tews,J.M.Lattimer,A.Ohnishi,E.E.Kolomeitsev,Astrophys.J.848,105(2017)
29.B.J.Cai,L.W.Chen,Constraints on the skewness coefficient of symmetric nuclear matter within the nonlinear relativistic mean field model.Nucl.Sci.Tech.28(to be published)
30.L.W.Chen,B.J.Cai,C.M.Ko et al.,Higher-order effects on the incompressibility of isospin asymmetric nuclear matter.Phys.Rev.C 80,014322(2009).https://doi.org/10.1103/Phys Rev C.80.014322
31.M.Dutra,O.Lourenco,J.S.Sa Martins,Skyrme interaction and nuclear matter constraints.Phys.Rev.C 85,035201(2012).https://doi.org/10.1103/Phys Rev C.85.035201
32.M.Dutra,O.Lourenco,S.S.Avancini et al.,Relativistic meanfield hadronic models under nuclear matter constraints.Phys.Rev.C 90,055203(2014).https://doi.org/10.1103/Phys Rev C.90.055203
33.F.Fattoyev,W.G.Newton,J.Xu et al.,Generic constraints on the relativistic mean-field and Skyrme-Hartree-Fock models from the pure neutron matter equation of state.Phys.Rev.C 86,025804(2012).https://doi.org/10.1103/Phys Rev C.86.025804
34.G.Colo`,U.Garg,H.Sagawa,Symmetry energy from the nuclear collective motion:constraints from dipole,quadrupole,monopole and spin-dipole resonances.Eur.Phys.J.A 50,26(2014).https://doi.org/10.1140/epja/i2014-14026-9
35.U.Garg,Y.Lui,Private communications
36.M.D.Cozma,Feasability of constraining the curvature parameter of the symmetry energy using elliptic flow data.Eur.Phys.J.A(2017).ar Xiv:1706.01300
37.B.J.Cai,B.A.Li,Isospin quartic term in the kinetic energy of neutron-rich nucleonic matter.Phys.Rev.C 92,011601(2015).https://doi.org/10.1103/Phys Rev C.92.011601
38.B.A.Brown,A.Schwenk,Constraints on Skyrme equations of state from properties of doubly magic nuclei and ab initio calculations of low-density neutron matter.Phys.Rev.C 89,011307(2014).https://doi.org/10.1103/Phys Rev C.89.011307.[Erratum:Phys.Rev.C 91,049902(2015)doi:10.1103/Phys Rev C.91.049902]
39.J.R.Stone,N.J.Stone,S.A.Moszkowski,Incompressibility in finite nuclei and nuclear matter.Phys.Rev.C 89,044316(2014).https://doi.org/10.1103/Phys Rev C.89.044316
40.S.Shlomo,V.M.Kolomietz,G.Colo,Deducing the nuclearmatter incompressibility coefficient from data on isoscalar compression modes.Eur.Phys.J.A 30,23-30(2006).https://doi.org/10.1140/epja/i2006-10100-3
41.J.Piekarewicz,Do we understand the incompressibility of neutron-rich matter?J.Phys.G 37,064038(2010).https://doi.org/10.1088/0954-3899/37/6/064038
42.E.Khan,J.Margueron,I.Vidana,Constraining the nuclear equation of state at subsaturation densities.Phys.Rev.Lett.109,092501(2012).https://doi.org/10.1103/Phys Rev Lett.109.092501
43.M.Meixner,J.P.Olson,G.Mathews,et al.,The NDL equation of state for supernova simulations.ar Xiv:1303.0064(2013)
44.L.W.Chen,Higher order bulk characteristic parameters of asymmetric nuclear matter.Sci.China Phys.Mech.Astron.54,s124-s129(2011).https://doi.org/10.1007/s11433-011-4415-9
45.M.Farine,J.M.Pearson,F.Tondeur,Nuclear-matter incompressibility from fits of generalized Skyrme force to breathingmode energies.Nucl.Phys.A 615,135-161(1997).https://doi.org/10.1016/S0375-9474(96)00453-8
46.T.Kla¨hn,D.Blaschke,S.Typel et al.,Constraints on the highdensity nuclear equation of state from the phenomenology of compact stars and heavy-ion collisions.Phys.Rev.C 74,035802(2006).https://doi.org/10.1103/Phys Rev C.74.035802
47.K.A.Maslov,E.E.Kolomeitsev,D.N.Voskresensky,Relativistic mean-field models with scaled hadron masses and couplings:hyperons and maximum neutron star mass.Nucl.Phys.A 950,64-109(2016).https://doi.org/10.1016/j.nuclphysa.2016.03.011
48.B.J.Cai,L.W.Chen,Constraints on the skewness coefficient of symmetric nuclear matter within the nonlinear relativistic mean field model.ar Xiv:1402.4242v1(2014)
49.N.B.Zhang et al.,in preparation(2017)
50.T.Li,U.Garg,Y.Liu et al.,Isotopic dependence of the giant monopole resonance in the Even-A112à124Sn isotopes and the asymmetry term in nuclear incompressibility.Phys.Rev.Lett.99,162503(2007).https://doi.org/10.1103/Phys Rev Lett.99.162503
51.J.Piekarewicz,Why is the equation of state for tin so soft?Phys.Rev.C 76,031301(2007).https://doi.org/10.1103/Phys Rev C.76.031301
We would like to thank Umesh Garg for helpful communications.
2.B.A.Li,C.M.Ko,W.Bauer,Isospin physics in heavy-ion collisions at intermediate energies.Int.J.Mod.Phys.E 7,147(1998).https://doi.org/10.1142/S0218301398000087
3.B.A.Li,W.Udo Schro¨der(eds.),Isospin Physics in Heavy-Ion Collisions at Intermediate Energies(Nova Science Publishers Inc,New York,2001)
4.V.Baran,M.Colonna,V.Greco et al.,Reaction dynamics with exotic nuclei.Phys.Rep.410,335-466(2005).https://doi.org/10.1016/j.physrep.2004.12.004
5.B.A.Li,L.W.Chen,C.M.Ko,Recent progress and new challenges in isospin physics with heavy-ion reactions.Phys.Rep.464,113-281(2008).https://doi.org/10.1016/j.physrep.2008.04.005
6.W.G.Lynch,M.B.Tsang,Y.Zhang et al.,Probing the symmetry energy with heavy ions.Prog.Nucl.Part.Phys.62,427-432(2009).https://doi.org/10.1016/j.ppnp.2009.01.001
7.P.Danielewicz,J.Lee,Symmetry energy I:semi-infinite matter.Nucl.Phys.A 818,36-96(2009).https://doi.org/10.1016/j.nucl physa.2008.11.007
8.W.Trautmann,H.H.Wolter,Elliptic flow and the symmetry energy at supra-saturation density.Int.J.Mod.Phys.E 21,1230003(2012).https://doi.org/10.1142/S0218301312300032
9.M.B.Tsang,J.R.Stone,F.Camera et al.,Constraints on the symmetry energy and neutron skins from experiments and theory.Phys.Rev.C 86,015803(2012).https://doi.org/10.1103/Phys Rev C.86.015803
10.C.J.Horowitz,E.F.Brown,Y.Kim et al.,A way forward in the study of the symmetry energy:experiment,theory,and observation.J.Phys.G 41,093001(2014).https://doi.org/10.1088/0954-3899/41/9/093001
11.M.Baldo,G.F.Burgio,The nuclear symmetry energy.Prog.Part.Nucl.Phys.91,203-258(2016).https://doi.org/10.1016/j.ppnp.2016.06.006
12.B.A.Li,Nuclear symmetry energy extracted from laboratory experiments.Nuclear Physics News(2017)(to appear).ar Xiv:1701.03564
13.A.W.Steiner,M.Prakash,J.M.Lattimer et al.,Isospin asymmetry in nuclei and neutron stars.Phys.Rep.411,325-375(2005).https://doi.org/10.1016/j.physrep.2005.02.004
14.J.M.Lattimer,The nuclear equation of state and neutron star masses.Annu.Rev.Nucl.Part.Sci.62,485-515(2012).https://doi.org/10.1146/annurev-nucl-102711-095018
15.W.G.Newton,J.Hooker,M.Gearheart et al.,Constraints on the symmetry energy from observational probes of the neutron star crust.Eur.Phys.J.A 50,41(2014).https://doi.org/10.1140/epja/i2014-14041-x
16.K.Iida,K.Oyamatsu,Symmetry energy,unstable nuclei and neutron star crusts.Eur.Phys.J.A 50,42(2014).https://doi.org/10.1140/epja/i2014-14042-9
17.J.M.Pearson,N.Chamel,A.F.Fantina,Symmetry energy:nuclear masses and neutron stars.Eur.Phys.J.A 50,43(2014).https://doi.org/10.1140/epja/i2014-14043-8
18.F.J.Fattoyev,W.G.Newton,B.A.Li,Probing the high-density behavior of symmetry energy with gravitational waves.Eur.Phys.J.A 50,45(2014).https://doi.org/10.1140/epja/i2014-14045-6
19.T.Fischer,M.Hempel,I.Sagert et al.,Symmetry energy impact in simulations of core-collapse supernovae.Eur.Phys.J.A 50,46(2014).https://doi.org/10.1140/epja/i2014-14046-5
20.D.Blaschke,D.E.Alvarez-Castillo,T.Klahn,Universal symmetry energy contribution to the neutron star equation of state.ar Xiv:1604.08575
21.B.A.Li,X.Han,Constraining the neutron-proton effective mass splitting using empirical constraints on the density dependence of nuclear symmetry energy around normal density.Phys.Lett.B727,276-281(2013).https://doi.org/10.1016/j.physletb.2013.10.006
22.M.Oertel,M.Hempel,T.Kla¨hn et al.,Equations of state for supernovae and compact stars.Rev.Mod.Phys.89,015007(2017).https://doi.org/10.1103/Rev Mod Phys.89.015007
23.M.W.Zwierlein,Ch.18 in Novel Superfluids vol.2,ed.by K.-H.Bennemann,J.B.Ketterson(Oxford University Press,Oxford,2015)
24.M.J.H.Ku,A.T.Sommer,L.W.Cheuk et al.,Revealing the superfluid lambda transition in the universal thermodynamics of a unitary Fermi gas.Science 335,563-567(2012).https://doi.org/10.1126/science.1214987
25.G.Zu¨rn,T.Lompe,A.N.Wenz et al.,Precise characterization of6Li feshbach resonances using trap-sideband-resolved RF spectroscopy of weakly bound molecules.Phys.Rev.Lett.110,135301(2013).https://doi.org/10.1103/Phys Rev Lett.110.135301
26.M.G.Endres,D.B.Kaplan,J.W.Lee et al.,Lattice Monte Carlo calculations for unitary fermions in a finite box.Phys.Rev.A 87,023615(2013).https://doi.org/10.1103/Phys Rev A.87.023615
27.E.E.Kolomeitsev,J.M.Lattimer,A.Ohnishi,I.Tews.ar Xiv:1611.07133v1(2016)
28.I.Tews,J.M.Lattimer,A.Ohnishi,E.E.Kolomeitsev,Astrophys.J.848,105(2017)
29.B.J.Cai,L.W.Chen,Constraints on the skewness coefficient of symmetric nuclear matter within the nonlinear relativistic mean field model.Nucl.Sci.Tech.28(to be published)
30.L.W.Chen,B.J.Cai,C.M.Ko et al.,Higher-order effects on the incompressibility of isospin asymmetric nuclear matter.Phys.Rev.C 80,014322(2009).https://doi.org/10.1103/Phys Rev C.80.014322
31.M.Dutra,O.Lourenco,J.S.Sa Martins,Skyrme interaction and nuclear matter constraints.Phys.Rev.C 85,035201(2012).https://doi.org/10.1103/Phys Rev C.85.035201
32.M.Dutra,O.Lourenco,S.S.Avancini et al.,Relativistic meanfield hadronic models under nuclear matter constraints.Phys.Rev.C 90,055203(2014).https://doi.org/10.1103/Phys Rev C.90.055203
33.F.Fattoyev,W.G.Newton,J.Xu et al.,Generic constraints on the relativistic mean-field and Skyrme-Hartree-Fock models from the pure neutron matter equation of state.Phys.Rev.C 86,025804(2012).https://doi.org/10.1103/Phys Rev C.86.025804
34.G.Colo`,U.Garg,H.Sagawa,Symmetry energy from the nuclear collective motion:constraints from dipole,quadrupole,monopole and spin-dipole resonances.Eur.Phys.J.A 50,26(2014).https://doi.org/10.1140/epja/i2014-14026-9
35.U.Garg,Y.Lui,Private communications
36.M.D.Cozma,Feasability of constraining the curvature parameter of the symmetry energy using elliptic flow data.Eur.Phys.J.A(2017).ar Xiv:1706.01300
37.B.J.Cai,B.A.Li,Isospin quartic term in the kinetic energy of neutron-rich nucleonic matter.Phys.Rev.C 92,011601(2015).https://doi.org/10.1103/Phys Rev C.92.011601
38.B.A.Brown,A.Schwenk,Constraints on Skyrme equations of state from properties of doubly magic nuclei and ab initio calculations of low-density neutron matter.Phys.Rev.C 89,011307(2014).https://doi.org/10.1103/Phys Rev C.89.011307.[Erratum:Phys.Rev.C 91,049902(2015)doi:10.1103/Phys Rev C.91.049902]
39.J.R.Stone,N.J.Stone,S.A.Moszkowski,Incompressibility in finite nuclei and nuclear matter.Phys.Rev.C 89,044316(2014).https://doi.org/10.1103/Phys Rev C.89.044316
40.S.Shlomo,V.M.Kolomietz,G.Colo,Deducing the nuclearmatter incompressibility coefficient from data on isoscalar compression modes.Eur.Phys.J.A 30,23-30(2006).https://doi.org/10.1140/epja/i2006-10100-3
41.J.Piekarewicz,Do we understand the incompressibility of neutron-rich matter?J.Phys.G 37,064038(2010).https://doi.org/10.1088/0954-3899/37/6/064038
42.E.Khan,J.Margueron,I.Vidana,Constraining the nuclear equation of state at subsaturation densities.Phys.Rev.Lett.109,092501(2012).https://doi.org/10.1103/Phys Rev Lett.109.092501
43.M.Meixner,J.P.Olson,G.Mathews,et al.,The NDL equation of state for supernova simulations.ar Xiv:1303.0064(2013)
44.L.W.Chen,Higher order bulk characteristic parameters of asymmetric nuclear matter.Sci.China Phys.Mech.Astron.54,s124-s129(2011).https://doi.org/10.1007/s11433-011-4415-9
45.M.Farine,J.M.Pearson,F.Tondeur,Nuclear-matter incompressibility from fits of generalized Skyrme force to breathingmode energies.Nucl.Phys.A 615,135-161(1997).https://doi.org/10.1016/S0375-9474(96)00453-8
46.T.Kla¨hn,D.Blaschke,S.Typel et al.,Constraints on the highdensity nuclear equation of state from the phenomenology of compact stars and heavy-ion collisions.Phys.Rev.C 74,035802(2006).https://doi.org/10.1103/Phys Rev C.74.035802
47.K.A.Maslov,E.E.Kolomeitsev,D.N.Voskresensky,Relativistic mean-field models with scaled hadron masses and couplings:hyperons and maximum neutron star mass.Nucl.Phys.A 950,64-109(2016).https://doi.org/10.1016/j.nuclphysa.2016.03.011
48.B.J.Cai,L.W.Chen,Constraints on the skewness coefficient of symmetric nuclear matter within the nonlinear relativistic mean field model.ar Xiv:1402.4242v1(2014)
49.N.B.Zhang et al.,in preparation(2017)
50.T.Li,U.Garg,Y.Liu et al.,Isotopic dependence of the giant monopole resonance in the Even-A112à124Sn isotopes and the asymmetry term in nuclear incompressibility.Phys.Rev.Lett.99,162503(2007).https://doi.org/10.1103/Phys Rev Lett.99.162503
51.J.Piekarewicz,Why is the equation of state for tin so soft?Phys.Rev.C 76,031301(2007).https://doi.org/10.1103/Phys Rev C.76.031301
We would like to thank Umesh Garg for helpful communications.