Nuclear Matter Near Saturation

Near nuclear saturation density, uniform matter can be organized by a local expansion in compression and neutron–proton asymmetry. Its accuracy depends on the omitted terms and the underlying nuclear calculation. The expansion defines the binding energy, incompressibility, symmetry energy, and its slope without pretending that a few coefficients determine all higher-density physics. Saturation is the minimum of symmetric-matter energy per baryon, so its pressure vanishes there; neutron-rich matter acquires pressure and composition dependence primarily through the symmetry-energy sector.

Required background. Hilbert Stress–Energy from Metric Variation supplies the energy density and pressure entering relativistic matter stress.

Helpful background. The Einstein Equation supplies stellar gravity; Relativistic Matter and Compact Objects gives the equation-of-state context; and Computational Relativity supplies applications of tabulated dense matter.

Saturation as a thermodynamic minimum

Consider cold uniform nucleonic matter, without Coulomb or lepton contributions for the moment. Let $n=n_n+n_p$ be baryon density and

$$\delta=\frac{n_n-n_p}{n}=1-2Y_p$$

the isospin asymmetry. Write the zero-temperature energy per baryon, excluding or including a declared common nucleon rest mass, as

$$e(n,\delta)=e_{\rm sat}(n)+S(n)\delta^2+O(\delta^4).$$

Charge symmetry removes odd powers to a good approximation. Symmetric nuclear matter has $\delta=0$ and saturates at $n=n_0$, defined by

$$\left.\frac{d e_{\rm sat}}{dn}\right|_{n_0}=0.$$

Since cold pressure at fixed asymmetry is $p=n^2(\partial e/\partial n)_\delta$, symmetric matter has $p(n_0,0)=0$. Representative empirical scales are $n_0\simeq0.16\ {\rm fm}^{-3}$ and binding near $-16\ {\rm MeV}$ per baryon relative to separated nucleons; these are scales, not precision measurements. Lattimer and Prakash define the symmetric-matter and symmetry-energy conventions in Lattimer and Prakash 2016, §2.4, Eqs. (14)–(18).

Compression and symmetry coefficients

Introduce

$$x=\frac{n-n_0}{3n_0}.$$

The local expansions are conventionally written

$$e_{\rm sat}(n)=e_0+\frac12K_0x^2+\frac16Q_0x^3+\cdots,$$ $$S(n)=J+Lx+\frac12K_{\rm sym}x^2+\cdots.$$

The coefficients are derivatives at $n_0$:

$$K_0=9n_0^2e_{\rm sat}''(n_0), \qquad J=S(n_0), \qquad L=3n_0S'(n_0).$$

The missing linear term in $e_{\rm sat}$ is the saturation condition, not a fitted accident. $K_0$ controls the leading curvature under symmetric compression. $J$ is the symmetry energy at saturation, and $L$ controls its leading density slope and an important part of neutron-rich pressure near $n_0$.

For nearly pure neutron matter, $\delta\simeq1$, the quadratic approximation gives

$$p(n_0,1)\simeq n_0^2S'(n_0)=\frac{n_0L}{3},$$

up to higher-asymmetry and many-body terms. This relation explains why neutron-matter calculations and neutron-skin observables inform $L$, while also showing the assumptions behind that mapping. Drischler, Holt, and Wellenhofer discuss chiral interactions, many-body truncation, and the density domain of such constraints in Drischler, Holt, and Wellenhofer 2021, §§ 2–4.

Beta equilibrium and composition

In cold catalyzed matter without trapped neutrinos, weak equilibrium requires

$$\mu_n-\mu_p=\mu_e=\mu_\mu$$

when muons are present, together with charge neutrality $n_p=n_e+n_\mu$. In the quadratic approximation,

$$\mu_n-\mu_p\simeq4S(n)\delta.$$

To derive the factor four, vary $n_n$ and $n_p$ at fixed total $n$: $dn_n=-dn_p=(n/2)d\delta$. The first law gives $d(ne)=(n/2)(\mu_n-\mu_p)d\delta$, whereas the quadratic functional gives $d(ne)=2nS\delta\,d\delta$. Equating them gives the stated relation. It neglects the neutron–proton rest-mass difference and charge-symmetry breaking consistently with the common-rest-mass convention above.

The lepton Fermi energies then determine $Y_p(n)$. The proton fraction is not freely selectable once the nuclear functional and equilibrium conditions are specified. Pressure of beta-equilibrated stellar matter must include leptons; it is not simply the fixed-$\delta$ nuclear pressure evaluated while differentiating an arbitrary $Y_p(n)$. Minimize the total energy per baryon over composition, then differentiate that minimum; the composition derivative vanishes by the equilibrium condition. With trapped neutrinos, the relation becomes $\mu_n-\mu_p=\mu_e-\mu_{\nu_e}$ and a lepton fraction, not zero neutrino chemical potential, is the appropriate constraint.

Composition thresholds can introduce muons, hyperons, condensates, or quark degrees of freedom. The saturation expansion alone does not predict those appearances. Effective masses and momentum dependence also affect thermal response and transport even if the cold pressure curve is similar. Oertel and collaborators connect saturation coefficients, composition, and finite-temperature models in Oertel et al. 2017, §§ II–III.

Empirical meaning and limitations

The coefficients are correlated inference products, not independent knobs measured at one point. Nuclear masses, giant resonances, neutron skins, heavy-ion reactions, and microscopic neutron-matter calculations probe different combinations and density ranges. Extracted values depend on the energy-density functional, many-body truncation, regulator, and treatment of surface and Coulomb effects.

The expansion is local around $n_0$. At $2n_0$ or higher, omitted skewness, kurtosis, composition changes, and phase transitions can be as important as $K_0$ or $L$. A polynomial that fits the neighborhood of $n_0$ can become unstable or acausal when extrapolated. High-density extensions must instead enforce thermodynamic consistency, stability, and causality while propagating the low-density uncertainty.

Dimensional checks help prevent convention errors. $K_0$, $J$, $L$, and $Q_0$ have energy units; $x$ and $\delta$ are dimensionless. If rest mass is added to $e$, pressure derivatives are unchanged but total energy density and relativistic enthalpy are not. The rest-mass convention must therefore be declared before coupling to stellar structure.

Exercises

Using the expansions above, show that symmetric-matter pressure vanishes at $n_0$ and that the leading symmetry pressure of pure neutron matter is $n_0L/3$.

Solution

For symmetric matter, $de_{\rm sat}/dn=(K_0x+O(x^2))/(3n_0)$, which vanishes at $x=0$; hence $p=n^2de/dn=0$. For $\delta=1$, the symmetry part has $dS/dn=L/(3n_0)+O(x)$. Evaluated at $n_0$,

$$p_{\rm sym}=n_0^2\frac{L}{3n_0}=\frac{n_0L}{3}.$$

Higher powers of $\delta$ and density modify this leading local relation.

References

• Drischler, Christian, Jeremy W. Holt, and Christian Wellenhofer. “Chiral Effective Field Theory and the High-Density Nuclear Equation of State.” Annual Review of Nuclear and Particle Science 71, 403–432 (2021). Open preprint.
• Lattimer, James M., and Madappa Prakash. “The Equation of State of Hot, Dense Matter and Neutron Stars.” Physics Reports 621, 127–164 (2016). Open preprint.
• Oertel, Micaela, Matthias Hempel, Thomas Klähn, and Stefan Typel. “Equations of State for Supernovae and Compact Stars.” Reviews of Modern Physics 89, 015007 (2017). DOI.