Thomas-Fermi Solver: Mean-Field Applications

• Thomas-Fermi Solver is a semiclassical mean-field method that replaces discrete quantum sums with local density functionals, enabling efficient evaluation of bulk nuclear properties.
• The solver employs a self-consistent iterative scheme to compute local nucleon densities and meson fields, ensuring convergence of energy functionals in finite nuclei and non-uniform matter.
• Applications include modeling finite nuclei and astrophysical systems such as Wigner-Seitz cells in neutron-star crusts, with outcomes validated against experimental measurements.

The Thomas-Fermi solver is a computational and theoretical framework designed to obtain semiclassical, mean-field approximations to the properties of many-fermion systems, particularly those governed by short-range or Coulombic interactions. In modern implementations, especially in nuclear physics and astrophysics, the Thomas-Fermi approach is often paired with self-consistency and relativistic mean-field models to describe the spatial distribution of nucleons in finite nuclei and non-uniform matter, such as that found in supernovae and neutron-star crusts. The method replaces discrete, shell-resolved quantum sums with local density and energy functionals, enabling efficient and accurate calculations of bulk nuclear properties, Wigner-Seitz cells, and energy densities under the influence of slowly varying mean fields.

1. Self-Consistent Thomas–Fermi Approximation: Core Formulation

In the self-consistent Thomas-Fermi (STF) approximation as used for finite nuclei or nuclear matter, nucleons are treated as locally homogeneous Fermi gases in spatially varying meson mean fields. The local scalar and vector nucleon densities, essential for building energy functionals, are given by

$$n_i^s(\mathbf{r}) = \frac{1}{\pi^2} \int_0^{k_{Fi}(\mathbf{r})} \frac{M^*}{\sqrt{k^2 + {M^*}^2}} k^2 dk,$$

$$n_i(\mathbf{r}) = \frac{(k_{Fi}(\mathbf{r}))^3}{3\pi^2},$$

where $M^* = M + g_\sigma \sigma$ is the effective nucleon mass, $k_{Fi}(\mathbf{r})$ is the local Fermi momentum, and $i = p, n$ for protons and neutrons, respectively.

The total energy for a nucleus is obtained by integrating the contributions of kinetic energy, mean-field (meson exchange and self-interactions), and Coulomb energy: $$E_{\text{total}} = \int d^3r \left[ \sum_{i=p,n} \frac{1}{\pi^2} \int_0^{k_{Fi}(\mathbf{r})} k^2 \sqrt{k^2 + {M^*}^2} dk - \frac{1}{2}g_\sigma \sigma (n_p^s + n_n^s) + \frac{1}{2}g_\omega \omega (n_p + n_n) + \dots \right],$$ with $g_\sigma$, $g_\omega$, etc., denoting meson-nucleon coupling constants, and additional terms coming from nonlinear meson self-interactions and the Coulomb field.

The meson fields $(\sigma, \omega, \rho)$ and electrostatic potential $A(\mathbf{r})$ themselves are determined from coupled, nonlinear differential equations sourced by the local densities.

2. Self-Consistent Iterative Scheme and Solver Workflow

The STF solver uses an iterative, self-consistent procedure for determining ground state solutions within finite nuclei or Wigner-Seitz (WS) cells:

1. Initialization: Begin with trial meson field profiles ($$\sigma(\mathbf{r}),\ \omega(\mathbf{r}),\ \rho(\mathbf{r}),\ A(\mathbf{r})$$).
2. Compute Local Densities: Using the local Fermi momenta (obtained from the chemical potentials $\mu_p,\ \mu_n$), evaluate $n_p(\mathbf{r})$ and $n_n(\mathbf{r})$.
3. Update Meson and Field Equations: Solve the field equations,

$$-\nabla^2 \sigma + m_\sigma^2 \sigma + g_2 \sigma^2 + g_3 \sigma^3 = -g_\sigma (n_p^s + n_n^s),$$

and similarly for $\omega,\ \rho,\ A$, with source terms from the densities.

1. Normalize and Enforce Constraints: Imposing chemical potential consistency, global charge neutrality, or total baryon number dictates $\mu_p,\ \mu_n$.
2. Convergence Check: Iterate the above steps until all fields and densities converge within prescribed tolerances.

In WS cell and non-uniform matter applications, the radius $R_C$ of the cell is treated as a variational parameter; the solver minimizes the free energy per baryon

$F = \frac{F_{\text{cell}}}{N_B},$

by adjusting $R_C$ to locate the physical (thermodynamically favored) configuration.

3. Comparison: STF versus Relativistic Mean-Field (RMF) Approaches

The relativistic mean-field method solves the Dirac equation for nucleons in the background of meson fields, generating quantal single-particle wavefunctions $\phi_\alpha$ whose densities capture shell effects and quantum oscillations: $$\psi_i = \sum_\alpha A_\alpha \phi_\alpha,\quad \text{where}\ \phi_\alpha \ \text{are Dirac eigenfunctions under the same meson fields}.$$ Density expressions then involve explicit sums over discrete levels. The core distinctions are:

Feature	STF Approximation	RMF Approach
Quantum Shell Effects	Neglected (densities are smooth)	Retained (densities exhibit oscillations)
Computational Complexity	Low (local density functionals and ODEs)	High (solving many coupled Dirac eqs)
Surface/Fine Structure	Approximated via density gradients	Direct, fully quantal
Applicability	Bulk properties, heavy/large systems, WS cells	Spectroscopy, single-particle detail

Both methods use identical underlying nuclear interactions and parameter sets, enabling direct comparison of average energies, charge radii, and neutron skin thickness. However, the STF systematically smooths out quantum interference terms, resulting in modest but systematic deviations from RMF in single-particle-sensitive observables.

4. Energy Functional and Observables

In both methods, energies, densities, and chemical potentials are computed using the same (or analogous) formulas but with different evaluations of the spatial densities. In the STF approximation, all densities are calculated from integrals over momentum of the local Fermi gas: $$n_{i}(\mathbf{r}) = \frac{(k_{Fi}(\mathbf{r}))^3}{3\pi^2}$$ where $$k_{Fi}(\mathbf{r}) = [(\mu_i - g_\omega \omega(\mathbf{r}) \mp g_\rho\rho(\mathbf{r})/2 - eA(\mathbf{r}))^2 - {M^*}^2 ]^{1/2}$$. The total binding energy, charge radius, and neutron skin are post-processed directly from these local densities.

5. Applications and Validation with Experimental Data

The implementation described in (Li et al., 31 Jul 2025) evaluates finite nuclei, WS cell structures in nonuniform matter, and bulk nuclear matter, validating theoretical predictions against experimental ground-state properties.

• Charge radii and energy per nucleon: Comparison to AME2020 and electron scattering measurements. The STF approximation tends to closely track experimental values in heavy nuclei but may slightly underpredict shell-sensitive observables.
• Neutron-skin thickness ($\Delta R_{np}$): Correlated with symmetry energy slope in the model; comparison to PREX-2/CREX.
• WS cells in neutron stars/supernovae: The STF solver enables efficient exploration of thermodynamic and compositional properties by minimizing the free energy with respect to R_C for given temperature (T), baryon density ($\rho_B$), and proton fraction (Y_p), aiding realistic equation of state (EOS) modeling.

6. Advantages and Limitations

• Advantages of STF:
  • Efficient—no need to solve large sets of Dirac equations.
  • Direct connection to mean-field energy functionals.
  • Capable of describing nonuniform phases and extended bulk matter (nuclear pasta, neutron-star crust).
  • Sufficient for computation of global observables in heavy nuclei and astrophysical contexts.
• Limitations:
  • Cannot capture shell effects or single-particle spectra.
  • Surface and gradient energies are approximate; detailed spectroscopic or fine-structure predictions require RMF.
  • For light nuclei or where quantum interference is significant, STF is less reliable than RMF.

7. Summary and Future Directions

The Thomas-Fermi solver, particularly in its self-consistent relativistic implementation, remains a central tool for the paper of nuclear matter, finite nuclei, and astrophysical systems. Its strengths lie in its ability to efficiently capture average properties of large systems while being sufficiently flexible to model nonuniform matter, WS cells, and global trends. Its limitations are well-understood and systematically linked to neglect of quantum shell structure, which can be addressed—if needed—via hybrid or RMF-based approaches. Contemporary work (Li et al., 31 Jul 2025, Zhang et al., 2014) confirms its practical utility for EOS construction once validated against experimental mass, radius, and skin thickness measurements, and highlights the importance of consistent parameterization and benchmarking. Further improvements could focus on better incorporation of gradient corrections or hybrid matching to RMF methods at boundaries where quantal effects become prominent.