12 Nuclear Matter Properties at High Densities…
151
E sym,0 = 31.6 ± 2.7 MeV and L = 59 ± 16 MeV, despite the quite large variations
in the individual measurements. However, L and E sym,0 cannot be individually determined in most experiments, often a larger value of L is compensated by a smaller
E sym,o or vice versa.The authors of [7] have extracted experimental constraints from
nuclear physics and astrophysical measurements in the E sym,0 –L plane, which give a
consensus region of 40 MeV < L(ρ 0 ) < 60 MeV and 30 MeV < E sym,0 < 32 MeV
with 68 % confidence, which is in mutual agreement with the results quoted in [6].
However, in another publication [8] it was argued, while using isobaric analog states
and isovector skins on neutron-rich nuclei, that both symmetry parameters may be
larger than the commonly adopted values.
The symmetry energy at higher densities ρ > ρ 0 can be accessed by the determination of the mass and the radii of neutron stars [9] or by investigating observables in
heavy-ion collisions which are related to the early, high-density phase of the reactions
and its isospin content. Theoretical model calculations predict that for a short time
period of 20 fm/c densities up to 3 ρ 0 are reached in the central zone of a heavy-ion
collision even at moderate energies ≈ 1 GeV/nucleon [2]. At lower energies around
200 MeV/nucleon up to 2ρ/ρ 0 may still be reached. However, one should be aware
that the highest density reached during a heavy-ion collision is not necessarily equivalent to the density which is probed by a certain observable. This question has to be
addressed when extracting constraints on the density dependence of the symmetry
energy from experimental data.
A multitude of observables have been proposed to be sensitive to the symmetry
energy (for a review see [2]): ratio of multiplicities or spectra of isospin partners (e.g.,
π
−
/π
+ , n/p or t/
3 He) and the comparison of their flows: The ratio of positively and
negatively charged pions measured close to or below the production threshold in the
NN system (E beam,thr = 280 MeV) is one of the observables discussed. It is predicted
to be sensitive to the density dependence of the symmetry energy. Indeed, model predictions obtained with the transport code IBUU4 [11] could only reproduce existing
experimental data on pion production around E beam = 400 MeV/nucleon in various collisions systems [12] when a rather soft density dependence of the symmetry
energy was applied. Incorporation of in-medium effects in addition to the symmetry
energy, like, e.g., pion potentials, s-wave production of pions, and the properties of
intermediate Delta resonances, may lead to different and even opposite conclusions
[13, 14], while describing the experimental data equally well. It is still not settled
how the symmetry energy influences the pion production ration. An experimental
way out is to measure double ratios of pion production, i.e., compare pion production in a neutron-rich and proton-rich system having the same Z. Here, some input
parameters to the models will drop out. Such experiments have been accomplished
at the Riken facility with the BIGRIPS magnet and the SPIRIT TPC, recently.
Other observables sensitive to the symmetry energy at supra-normal densities are
collective flows. At energies below 1 GeV/nucleon the reaction dynamics is largely
determined by the nuclear mean field. The resulting pressure produces a collective
motion of the compressed material whose strength will be influenced by the symme-
151
E sym,0 = 31.6 ± 2.7 MeV and L = 59 ± 16 MeV, despite the quite large variations
in the individual measurements. However, L and E sym,0 cannot be individually determined in most experiments, often a larger value of L is compensated by a smaller
E sym,o or vice versa.The authors of [7] have extracted experimental constraints from
nuclear physics and astrophysical measurements in the E sym,0 –L plane, which give a
consensus region of 40 MeV < L(ρ 0 ) < 60 MeV and 30 MeV < E sym,0 < 32 MeV
with 68 % confidence, which is in mutual agreement with the results quoted in [6].
However, in another publication [8] it was argued, while using isobaric analog states
and isovector skins on neutron-rich nuclei, that both symmetry parameters may be
larger than the commonly adopted values.
The symmetry energy at higher densities ρ > ρ 0 can be accessed by the determination of the mass and the radii of neutron stars [9] or by investigating observables in
heavy-ion collisions which are related to the early, high-density phase of the reactions
and its isospin content. Theoretical model calculations predict that for a short time
period of 20 fm/c densities up to 3 ρ 0 are reached in the central zone of a heavy-ion
collision even at moderate energies ≈ 1 GeV/nucleon [2]. At lower energies around
200 MeV/nucleon up to 2ρ/ρ 0 may still be reached. However, one should be aware
that the highest density reached during a heavy-ion collision is not necessarily equivalent to the density which is probed by a certain observable. This question has to be
addressed when extracting constraints on the density dependence of the symmetry
energy from experimental data.
A multitude of observables have been proposed to be sensitive to the symmetry
energy (for a review see [2]): ratio of multiplicities or spectra of isospin partners (e.g.,
π
−
/π
+ , n/p or t/
3 He) and the comparison of their flows: The ratio of positively and
negatively charged pions measured close to or below the production threshold in the
NN system (E beam,thr = 280 MeV) is one of the observables discussed. It is predicted
to be sensitive to the density dependence of the symmetry energy. Indeed, model predictions obtained with the transport code IBUU4 [11] could only reproduce existing
experimental data on pion production around E beam = 400 MeV/nucleon in various collisions systems [12] when a rather soft density dependence of the symmetry
energy was applied. Incorporation of in-medium effects in addition to the symmetry
energy, like, e.g., pion potentials, s-wave production of pions, and the properties of
intermediate Delta resonances, may lead to different and even opposite conclusions
[13, 14], while describing the experimental data equally well. It is still not settled
how the symmetry energy influences the pion production ration. An experimental
way out is to measure double ratios of pion production, i.e., compare pion production in a neutron-rich and proton-rich system having the same Z. Here, some input
parameters to the models will drop out. Such experiments have been accomplished
at the Riken facility with the BIGRIPS magnet and the SPIRIT TPC, recently.
Other observables sensitive to the symmetry energy at supra-normal densities are
collective flows. At energies below 1 GeV/nucleon the reaction dynamics is largely
determined by the nuclear mean field. The resulting pressure produces a collective
motion of the compressed material whose strength will be influenced by the symme-