# Recent Advances in Microscopic Approaches to Nuclear Matter and Symmetry Energy

## Abstract

**:**

## 1. Introduction

_{V}coefficient is therefore the binding energy per nucleon in infinite symmetric matter, approximately 15–16 MeV. Clearly, the (repulsive) Coulomb term favors a nucleus with a large neutron excess. On the other hand, the destabilizing effect of the symmetry term increases with larger values of N − Z = A − 2Z. Fits to empirical nuclear binding energies determine the coefficients of the mass formula, resulting in a value of about 23 MeV for a

_{sym}. We note that this is not the same as the symmetry energy at the equilibrium density of infinite nuclear matter.

## 2. The Symmetry Energy

#### 2.1. Empirical Facts

_{n}, and the proton density, ρ

_{p}. It is more convenient to refer to the total density ρ = ρ

_{n}+ ρ

_{p}and the asymmetry (or neutron excess) parameter $\alpha =\frac{{\rho}_{n}-{\rho}_{p}}{\rho}$. Clearly, α = 0 corresponds to symmetric matter and α = 1 to neutron matter. In terms of α and the average Fermi momentum, k

_{F}, related to the total density in the usual way,

_{sym}at nuclear matter saturation density (ρ

_{0}) is 30 MeV, with theoretical predictions spreading approximately between 26 and 35 MeV.

_{i}appearing in the potential energy part is found to be between 0.4 and 1.0. Recent measurements of elliptic flows in

^{197}Au +

^{197}Au reactions at GSI at 400–800 MeV/nucleon favor a potential energy term with γ

_{i}equal to 0.9 ± 0.4. Giant dipole resonance excitation in fusion reactions [33] is also sensitive to the symmetry energy, since the latter is responsible for isospin equilibration in isospin-asymmetric collisions. Recent efforts to set more stringent constraints on the symmetry energy will be reviewed in the next section.

_{n.m.}), that is, the neutron matter pressure. As to be expected on physical grounds, the neutron skin thickness,

^{−}/π

^{+}ratio, the K

^{+}/K

^{0}ratio, neutron/proton differential transverse flow or nucleon elliptic flow [35].

_{0}the incompressibility of symmetric nuclear matter. Attempts to extract the parameter K

_{asy}from the giant monopole resonances of neutron-rich nuclei have been made, but the extracted value carries very large uncertainties.

_{n/p}, are different from each other and satisfy the approximate relation:

_{sym}, being proportional to the gradient between the single-neutron and the single-proton potentials, should be comparable with the Lane potential [37], namely the isovector part of the nuclear optical potential. This suggests that optical potential analyses (in isospin unsaturated nuclei) can be an alternative way to constrain the symmetry energy.

#### 2.2. Experimental Constraints on the Symmetry Energy

#### Heavy ion collisions:

^{124}Sn +

^{124}Sn and

^{112}Sn +

^{112}Sn at 50 MeV per nucleon have been measured, as well as transverse collective flows of hydrogen, helium isotopes and other fragments at 35 MeV per nucleon in

^{70}Zn +

^{70}Zn,

^{64}Zn +

^{64}Zn and

^{64}Ni +

^{64}Ni reactions.

#### Nuclear binding energies:

#### Neutron skin measurements with hadronic or electroweak probes:

^{208}Pb was extracted through polarized proton elastic scattering data. The proton optical potential was parametrized and constrained via

^{52}Ni measurements and then fit to

^{204}

^{,}

^{206}

^{,}

^{208}Pb data by adjusting the neutron densities.

^{208}Pb is expected to be re-measured at the Jefferson Laboratory in the PREXII experiment planned for the near future. Parity-violating electron scattering at low momentum transfer is especially suitable to probe neutron densities, as the Z

^{0}boson couples primarily to neutrons. A much higher level of accuracy can be achieved with electroweak probes than with hadronic scattering. With the success of this program, reliable empirical information on neutron skins will be able to provide more stringent constraints on the density dependence of the symmetry energy.

^{48}Ca have also been proposed and are expected to take place in the near future with the CREX experiment [39,40]. Being much lighter than

^{208}Pb,

^{48}Ca is considerably more sensitive to surface and shell effects, which makes it possible to extract useful structure information. Furthermore, ab initio calculations are feasible for calcium isotopes, thus CREX experiments have the potential to test models of three-nucleon forces and density functional theories.

#### Electric dipole strength function:

^{208}Pb and the electric dipole strength, providing an alternative way to measure the skin and, thus, extract information on the slope of the symmetry energy.

#### 2.3. The Slope of the Symmetry Energy and the Radii of Neutron Stars

^{4}≤ ∊ ≤ 10

^{6}g·cm

^{−3}. As density increases, charge neutrality requires matter to become more neutron rich. In this density range (about 10

^{7}< ∊ < 10

^{11}g·cm

^{−3}), neutron-rich nuclei appear, mostly light metals, while electrons become relativistic. This is the outer crust. Above densities of approximately 10

^{11}g·cm

^{−3}, free neutrons begin to form a continuum of states. The inner crust is a compressed solid with a fluid of neutrons that drip out and populate free states outside the nuclei, which have become neutron-saturated. Densities in the inner crust range between 10

^{11}and 10

^{14}g·cm

^{−3}. At densities equal to approximately 1/2 of the saturation density, clusters begin to merge into a continuum. In this phase, matter is a uniform fluid of neutrons, protons and leptons. Above a few times nuclear matter density, the actual composition of stellar matter is not known. Strange baryons can appear when the nucleon chemical potential is of the order of their rest mass. Meson production can also take place. At even higher densities, transitions to other phases are speculated, such as a deconfined, rather than hadronic, phase. The critical density for such a transition cannot be predicted reliably, because it lies in a range where QCD is non-perturbative [42].

_{⊙}, to date the best mass determination.

_{⊙}[43]. The observation of a heavier star was confirmed, namely J1614-2230, with a mass of 1.97 ± 0.04 M

_{⊙}[44], one of the highest yet measured with this certainty. Furthermore, a more recent paper has reported the observation of a pulsar with even heavier mass [45], 2.01 ± 0.04 M

_{⊙}. Clearly, these observations represent a challenge for the softest EoS.

_{⊙}[46]. The smallest reliably estimated neutron star mass is the companion of the binary pulsar, J1756-2251, which has a mass of 1.18 ± 0.02 M

_{⊙}[47].

_{∞}, which is related to the actual stellar radius by R

_{∞}= R(1 − 2GM/Rc

^{2})

^{−1/2}. Estimates are usually based on thermal emissions of cooling stars, including redshifts, and the properties of sources with bursts or thermonuclear explosions at the surface. A major problem associated with the determination of radii is that the distance from the source is not well known, hence the need for additional assumptions. Much more stringent constraints could be imposed on the EoS if mass and radius could be determined independently of each other.

^{208}Pb (we recall that the correlation coefficient is equal to one for a perfect correlation). The strong correlation between L and the neutron skin thickness was mentioned earlier in the discussion following Equations (9) and (10) and can be seen clearly from the table.

^{208}Pb deteriorates for the heaviest stars, which have central densities much higher than those probed by ordinary nuclei. In summary, the larger the predicted skin of a heavy nucleus, the larger the predicted stellar radius of (low-mass) neutron stars. Therefore, constraints from both neutron skins and neutron star radii would be extremely useful for a better understanding of the slope of the symmetry energy.

## 3. Our Microscopic Approach to Isospin-Asymmetric Nuclear Matter

#### 3.1. Brief Review of the Model

^{2}are used. Thus,

^{∗}, is defined as m

^{∗}= m + U

_{S}, with U

_{S}an attractive scalar potential (this will be derived below). It can be shown that both the description of a single nucleon via Equation (17) and the evaluation of the Z-diagram, Figure 2, generate a density-dependent repulsive effect on the energy per particle in symmetric nuclear matter, which provides the saturating mechanism missing from conventional BHF calculations.

_{ij}is the in-medium reaction matrix (ij = nn, pp, or np), and the asterisk signifies that medium effects are applied to those quantities. Thus, the NN potential, ${v}_{ij}^{*}$, is constructed in terms of effective Dirac states (in-medium spinors) as in Equation (

_{p}17). In Equation (18), $\overrightarrow{q}$, ${\overrightarrow{q}}^{\prime}$ and $\overrightarrow{K}$ are the initial, final and intermediate relative momenta, and ${E}_{i}^{*}=\sqrt{{({m}_{i}^{*})}^{2}+{K}^{2}}$. The momenta of the two interacting particles in the nuclear matter rest frame have been expressed in terms of their relative momentum and the center-of-mass momentum, $\overrightarrow{P}$. The energy of the two-particle system is:

_{ij}, prevents scattering to occupied nn, pp or np states.

#### 3.2. Microscopic Predictions of the EoS and Related Quantities

^{−3}and −16.14 MeV, respectively, and the compression modulus is 252 MeV. The increased stiffness featured by the DBHF EoS at the higher densities originates from the strongly density-dependent repulsion characteristic of the Dirac–Brueckner–Hartree–Fock method.

_{n}predicted with the three potentials at large α.

_{ρ}= f

_{ρ}/g

_{ρ}, whereas a stronger value of κ

_{ρ}= 6.6 was determined from partial-wave analyses [65]. In other words, a larger value of the ρ tensor coupling as compared to its vector coupling is well supported by evidence. This fact is reflected in meson exchange models where, typically, the ratio κ

_{ρ}is about six. Therefore, a Lagrangian density with only a vector coupling for the ρ may miss the most important part of how this meson couples to the nucleon.

## 4. A Different Approach: Chiral Interactions

^{3}LO) [71] along with 3BF at next-to-next-to-leading order (N

^{2}LO). High-precision NN potentials at N

^{3}LO have been developed for Λ = 450, 500 and 600 MeV [71].

^{3}LO, such calculation for nuclear matter is not feasible at this time. Thus, the combination of 2BF at N

^{3}LO and 3BF at N

^{2}LO is presently state-of-the-art. We note further that four-body forces also appear at this order [72], namely the fourth order in the chiral expansion, but are left out because they are expected to be small [73].

^{2}LO) are: the long-range two-pion exchange graph; the medium-range one-pion exchange diagram; and the short-range contact term. The corresponding diagrams are shown in Figure 11.

_{1},

_{3},

_{4}, which are fixed in the NN system [71]. Two are generated by the one-pion exchange diagram and depend on the low-energy constant c

_{D}. Finally, the short-range component depends on the constant c

_{E}. In pure neutron matter, the contributions proportional to the low-energy constants c

_{4}, c

_{D}and c

_{E}vanish [74].

_{1},

_{3},

_{4}, which are already present in the two-pion-exchange 2BF. Concerning the low-energy constants c

_{D}and c

_{E}appearing in the N

^{2}LO 3BF, a very important aspect of our calculations is that they are completely determined from the three-nucleon system. Specifically, they are constrained to reproduce the A = 3 binding energies and the triton Gamow–Teller matrix elements. Their values are given in Table 2 [75,76]. The procedure [75] is based on the consistency of 2BF, 3BF and currents, as required by chiral EFT.

^{−3}.

## 5. Summary and Conclusions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) The baryon density and (

**b**) the mass-energy density profile for a neutron star with a mass of 1.4 solar masses.

**Figure 3.**Our Dirac–Brueckner–Hartree–Fock (DBHF) predictions for the equation of state (EoS) of symmetric matter (solid red) and neutron matter (dashed black).

**Figure 4.**Pressure in symmetric matter from the Idaho DBHF calculation. The shaded area corresponds to the region of pressure consistent with the flow data analyzed in [62].

**Figure 5.**DBHF prediction for the symmetry energy (solid red) compared with various phenomenological parametrizations (dashed black). See the text for details.

**Figure 6.**Pressure in neutron (red curve) and baryon-lepton (green curve) matter from the Idaho DBHF calculation. The shaded area corresponds to the region of pressure consistent with flow data and the inclusion of strong density dependence in the asymmetry terms [62].

**Figure 7.**Momentum dependence of the single-nucleon potential in isospin asymmetric matter, U

_{i}(i = n, p), predicted with Bonn A (

**a**), Bonn B (

**b**) and Bonn C (

**c**). The total density is equal to 0.185 fm

^{−3}, and the isospin asymmetry parameter is equal to 0.4. The momentum is given in units of the (average) Fermi momentum, which is equal to 1.4 fm

^{−1}.

**Figure 8.**(

**a**) Neutron and (

**b**) proton single-particle potentials as functions of the asymmetry parameter at fixed average density and momentum equal to the average Fermi momentum, which is equal to 1.4 fm

^{−1}.

**Figure 10.**Predictions of the symmetry energy vs. density for various parameterizations of the Skyrme model.

**Figure 11.**Leading three-body forces at N

^{2}LO: (

**a**) one-pion exchange; (

**b**) two-pion exchange; (

**c**) contact term. See the text for more details.

**Figure 12.**Energy/particle in symmetric nuclear matter as a function of density obtained with 2BF and 3BF as explained in the text. The solid (red), dashed (green) and dotted (blue) curves correspond to Λ = 450, 500 and 600 MeV, respectively.

**Figure 13.**As in Figure 12, but for neutron matter.

**Figure 14.**As in Figure 12, for the symmetry energy.

**Table 1.**Correlation between “observable” A and the neutron skin thickness in

^{208}Pb. P is the pressure, and R

_{x}is the radius of a neutron star with mass equal to x solar masses.

Physical property A | Correlation coefficient between A and the neutron skin thickness in ^{208} Pb |
---|---|

L | 0.9952 |

${P}_{\rho}{}_{{}_{0}}$ | 0.9882 |

${P}_{2{\rho}_{0}}$ | 0.8016 |

R_{0.6} | 0.9953 |

R_{0.8} | 0.9931 |

R_{1.0} | 0.9866 |

R_{1.4} | 0.9486 |

R_{1.6} | 0.8361 |

**Table 2.**Values of n and low-energy constants used for each type of cutoff in the regulator function as given in Equation (30). The low-energy constants of the dimension-two πN Lagrangian, c

_{1},

_{3},

_{4}, are given in units of GeV

^{−1}.

Λ (MeV) | n | c_{1} | c_{3} | c_{4} | c_{D} | c_{E} |
---|---|---|---|---|---|---|

450 | 3 | −0.81 | −3.40 | 3.40 | −0.24 | −0.11 |

500 | 2 | −0.81 | −3.20 | 5.40 | 0.0 | −0.18 |

600 | 2 | −0.81 | −3.20 | 5.40 | −0.19 | −0.833 |

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**MDPI and ACS Style**

Sammarruca, F.
Recent Advances in Microscopic Approaches to Nuclear Matter and Symmetry Energy. *Symmetry* **2014**, *6*, 851-879.
https://doi.org/10.3390/sym6040851

**AMA Style**

Sammarruca F.
Recent Advances in Microscopic Approaches to Nuclear Matter and Symmetry Energy. *Symmetry*. 2014; 6(4):851-879.
https://doi.org/10.3390/sym6040851

**Chicago/Turabian Style**

Sammarruca, Francesca.
2014. "Recent Advances in Microscopic Approaches to Nuclear Matter and Symmetry Energy" *Symmetry* 6, no. 4: 851-879.
https://doi.org/10.3390/sym6040851