1. Shalom Shlomo Cyclotron Institute Texas A&M University
EQUATION OF STATE OF SYMMETRIC AND ASYMMETRIC NUCLEAR MATTER AT VARIOUS DENSITIES AND TEMPERATURES
Shalom Shlomo
Cyclotron Institute
Texas A&M University

2. Outline
1. Introduction
Equation of State: nuclear matter incompressibility coefficient K
Giant Resonances: Compression Modes, Isovector Giant Dipole
2. Energy Density Functional
3. Hartree-Fock based Random Phase Approximation (RPA)
4. Hot Nuclear Matter at Low densities
Heavy Ion Collisions: Freeze-Out And Fragmentation
Determination of Temperature and Density
5. Medium Modifications of Cluster Properties
6. Symmetry Energy of Low Density Nuclear Matter
7. Summary And Conclusions

3. Equation of state and nuclear matter compressibility
The nuclear matter (N=Z and no Coulomb interaction) incompressibility coefficient, K, is a very important physical quantity in the study of nuclei, supernova collapse, neutron stars, and heavy-ion collisions, since it is directly related to the curvature of the nuclear matter (NM) equation of state (EOS), E = E(ρ).
E/A [MeV]
ρ = 0.16 fm-3
ρ [fm-3]
E/A = -16 MeV

4. Macroscopic picture of giant resonance
L = 0
L = 1
L = 2

5. Important task: Develop a modern EDF with enhanced
Important task: Develop a modern EDF with enhanced predictive power for properties of rare nuclei.
We start from EDF obtained from Skyrme N-N interaction.
The effective Skyrme interaction has been used in mean-field models for several decades and many different parameterizations of the interaction have been realized to better reproduce nuclear masses, radii, and various other data. Today, there are more experimental data of nuclei far from the stability line. It is time to improve the parameters of Skyrme interactions We fit our mean-field results to an extensive set of experimental data and obtain the parameters of the Skyrme type effective interaction for nuclei at and far from the stability line.

6. Map of the existing nuclei
Map of the existing nuclei. The black squares in the central zone are stable nuclei, the broken inner lines show the status of known unstable nuclei as of 1986 and the outer lines are the assessed proton and neutron drip lines (Hansen 1991).

7. The total energy
Where

9. (2p1/2) - (2p3/2) = 1.83 MeV (proton).
Fitted data
- The binding energies for 14 nuclei ranging from normal to the exotic (proton or neutron) ones: 16O, 24O, 34Si, 40Ca, 48Ca, 48Ni, 56Ni, 68Ni, 78Ni, 88Sr, 90Zr, 100Sn, 132Sn, and 208Pb.
- Charge rms radii for 7 nuclei: 16O, 40Ca, 48Ca, 56Ni, 88Sr, 90Zr, 208Pb.
The spin-orbit splittings for the 2p proton and neutron orbits for 56Ni (2p1/2) - (2p3/2) = 1.88 MeV (neutron)
(2p1/2) - (2p3/2) = 1.83 MeV (proton).
- Rms radii for the valence neutron:
in the 1d5/2 orbit for 17O
in the 1f7/2 orbit for 41Ca
- The breathing mode energy for 4 nuclei: 90Zr (17.81 MeV), 116Sn (15.9 MeV), 144Sm (15.25 MeV), and 208Pb (14.18 MeV).

10. Constraints 1. The critical density Landau stability condition:
Example:
2. The Landau parameter
should be positive at
3. The quantity
must be positive for densities up to
4. The IVGDR enhancement factor

11. The values of the Skyrme parameters
0.1667
(0.0144)
(0.0163) α
126.00
(4.39)
(3.33)
W0 (MeV fm5)
1.3910
(0.0767)
(0.0882) x3
-1.000
(0.0270)
(0.0179) x2
(0.0298)
(0.0165) x1
0.8460
(0.0579)
( ) x0
(641.99)
(680.73)
t3 (MeV fm3(1+α))
(14.26)
(27.31)
t2 (MeV fm5)
457.97
(27.63)
(16.67)
t1 (MeV fm5)
(115.32)
(140.63)
t0 (MeV fm3)
SLy7
KDE
KDE0
Parameter

12. Fully self-consistent HF-RPA results for ISGMR centroid energy (in MeV) with the Skyrme interaction SK255, SGII and KDE0 and compared with the RRPA results using the NL3 interaction. Note the corresponding values of the nuclear matter incompressibility, K, and the symmetry energy , J, coefficients. ω1-ω2 is the range of excitation energy. The experimental data are from TAMU.
33.0
26.8
37.4
J (MeV)
229
215
255
272
K (MeV)
13.8
13.6
14.4
13.96±0.30
10-35
14.3
14.2
0-60
208Pb
15.5
15.2
16.2
15.40±0.40
15.3
16.1
144Sm
16.6
16.4
17.3
15.85±0.20
17.1
116Sn
18.0
17.9
18.9
17.81±0.30
18.7
90Zr
KDE0
SGII
SK255
NL3
Expt.
ω1-ω2
Nucleus

13. Hot Nuclear Matter
In an intermediate energy heavy ion collision a dense and hot nuclear system is created which then breaks into fragments. Assuming a thermal and chemical equilibrium at freeze out, the temperature T of the disassembling hot system is then determined from the ratios of the yields of the emitted fragments using the method first introduced by Saha. The dependence of excitation energy on T, i.e., the caloric curve, was found to show irregularities which may be interpreted as a possible signal for liquid gas phase transition. In the analysis of the experimental data of fragment yields we have considered the effects of:
The Coulomb interaction
Post emission decay
Flow due to compression
The medium on the binding energies of clusters

14. Introducing the average density ρs of clusters s:
We have for the nucleon
where χ=V’/V0. The spin-degeneracy factor 2 is included. The relative yield of fragments s is given by
This expression is very close to that of Albergo et al except for the term (1+κ)/χ and the Wigner-Seitz energy EC(s).

15. The Albergo et al relation is modified by
with
The fragment yields must be selected such that
where A = N + Z and n and p are integer numbers.
For isotopes, ΔEC = 0. For isotone fragments, ΔT can be as much as 50%. For R=(Y(16O)/Y(12C))/(Y(6Li)/Y(d)), ΔB = 5.69 MeV and ΔEC = MeV.

16. Medium Effects Formation of clusters at sub-saturation densities:
For the nuclei embedded in nuclear matter, an effective in-medium Schrödinger equation can be derived
This equation contains the effects of the medium in the single-nucleon quasiparticle shifts as well as in the Pauli blocking terms.

17. The in medium Fermi distribution function
contains the effective chemical potential which is determined by the total proton or neutron density, calculated in the quasiparticle approximation,
It describes the occupation of the phase space neglecting any correlations in the medium.

19. The EoS can be evaluated as in the non-interacting case except that the number densities of clusters must be calculated with the quasiparticle energies,
In the cluster-quasiparticle approximation, the EoS reads:
for the total proton and neutron density, respectively.
This result is an improvement of the NSE and allows for the smooth transition from the low-density limit up to the region of saturation density.

20. FIG. 1: Comparisons of the scaled internal symmetry energy Esym(n)/Esym(n0) as a function of the scaled total density n/n0 for different approaches and the experiment. Left panel: The symmetry energies for the commonly used MDI parameterization for T = 0 and different asy-stiffnesses, controlled by the parameter x (dotted, dot-dashed and dashed lines); for the QS model including light clusters for temperature T = 1 MeV (solid line), and for the RMF model at T = 0 including heavy clusters (long-dashed line). Right panel: The internal scaled symmetry energy in an expanded low density region. Shown are again the MDI curves and the QS results for T = 1, 4, and 8 MeV compared to the experimental data with the NSE entropy (solid circles) and the results of the self-consistent calculation (open circles).

21. SUMMARY AND CONCLUSIONS
1) Fully self-consistent calculations of the compression modes (ISGMR and ISGDR) using modern energy density functionals (Skyrme forces) lead to → K∞ = 240 ± 20 MeV, with sensitivity to symmetry energy.
2 ) Accounting for post-emission decay allows one to obtain consistent values of temperature of a disassembling source from the “double-ratio” method.
3) Although , at low densities, the temperature calculated from given yields changes only modestly if medium effects are taken into account, larger discrepancies are observed when the nucleon densities are determined from measured yields,
4) Due to clusterization at low density nuclear matter, the symmetry energy is much larger than that predicted by mean field approximation