Relativistic Thomas–Fermi Equation

• The relativistic Thomas–Fermi equation is a semiclassical model that combines Fermi statistics, relativistic kinematics, and mean-field interactions to describe electron screening and nuclear structure.
• It is applied in diverse contexts, from compressed atoms and white dwarf matter to finite nuclei and subnuclear matter, with distinct formulations for atomic versus nuclear systems.
• Recent advances include generalizations incorporating deformed phase-space measures, thermal effects, and non-extensive corrections that refine its predictive power across astrophysical and nuclear applications.

Searching arXiv for recent and foundational papers on the relativistic Thomas–Fermi equation and related formulations. The relativistic Thomas–Fermi equation denotes a class of semiclassical self-consistent equations in which Fermi statistics, electrostatic or mean-field interactions, and relativistic kinematics are combined without solving a fully microscopic many-body wave equation. In the literature represented here, the term has two principal usages. In atomic and astrophysical applications it can refer to a specific nonlinear screening equation for electrons in a compressed atom or Wigner–Seitz cell, derived from a relativistic Fermi-energy condition and Poisson’s equation (Rotondo et al., 2010). In nuclear-structure and dense-matter applications it more often denotes a relativistic Thomas–Fermi approximation: a coupled set of RMF field equations, local density relations, and chemical-equilibrium conditions for finite nuclei, hot nuclei, hypernuclei, nuclear pasta, and subnuclear matter (Li et al., 31 Jul 2025). A recurring source of confusion is therefore terminological rather than physical: the atomic formulation is often a single radial differential equation, whereas the nuclear formulation is typically an integral-differential mean-field system rather than one closed ODE (1411.1584).

1. Scope, terminology, and nonrelativistic baseline

The standard nonrelativistic Thomas–Fermi equation for a neutral atom is

$$\varphi''(x)=x^{-1/2}\,\varphi(x)^{3/2}, \qquad \varphi(0)=1,\qquad \varphi(\infty)=0,$$

or equivalently $y''(x)=x^{-1/2}y^{3/2}(x)$ in the notation of the Emden–Fowler form with $p=3/2$ (Alizzi et al., 2023, Fernández et al., 2021). This equation describes the screening of the Coulomb potential inside heavy neutral atoms and supplies the historical baseline from which relativistic variants depart (Oulne, 2015).

A useful way to organize the subject is to distinguish the physical regimes in which “relativistic Thomas–Fermi” is used.

Context	Mathematical object	Representative papers
Neutral atoms	Nonrelativistic TF boundary-value problem $\varphi''=x^{-1/2}\varphi^{3/2}$	(Alizzi et al., 2023, Fernández et al., 2021, Oulne, 2015)
Compressed atoms / white dwarfs	Relativistic screening equation in a Wigner–Seitz cell	(Rotondo et al., 2010)
Finite nuclei / nonuniform matter	Self-consistent relativistic Thomas–Fermi approximation with RMF fields	(Li et al., 31 Jul 2025, 1411.1584, Zhang et al., 2015)

This classification matters because it prevents the misleading impression that there is a single canonical “the” relativistic Thomas–Fermi equation. The atomic and white-dwarf literature centers on relativistic electron screening in a Coulomb field, while the nuclear literature uses the same Thomas–Fermi label for a covariant semiclassical limit of meson-exchange mean-field theory. The nonrelativistic atomic equation also remains analytically important because several later works use it as the benchmark from which relativistic or generalized variants are assessed (Esposito et al., 2020).

2. Relativistic screening in compressed atoms and white-dwarf matter

A concrete relativistic Thomas–Fermi equation is developed within the relativistic Feynman–Metropolis–Teller theory for compressed atoms in a Wigner–Seitz cell (Rotondo et al., 2010). The starting point is the relativistic electron equilibrium condition

$$E_e^F = \sqrt{c^2 (P_e^F)^2+m_e^2 c^4} - m_e c^2 - e V(r) = {\rm constant} > 0,$$

which replaces the nonrelativistic kinetic term by the full relativistic dispersion relation. The local electron density is then

$$n_e(r) = \frac{(P_e^{F})^3}{3\pi^2 \hbar^3} = \frac {1}{3\pi^2 \hbar^3 c^3}\left[\hat V^2(r)+ 2m_e c^2 \hat V(r)\right]^{3/2}, \qquad \hat V = e V + E_e^F,$$

and Poisson’s equation reads

$\nabla^2 V(r)= -4\pi e\left[n_p(r)-n_e(r)\right].$

After introducing

$$x= \frac{r}{\lambda_{\pi}}, \qquad x_c= \frac{R_c}{\lambda_{\pi}}, \qquad \frac{\chi}{r} = \frac{\hat V(r)}{\hbar c},$$

the paper obtains the relativistic Thomas–Fermi equation

$$\frac{1}{3x}\frac {d^2\chi(x)}{d x^2} = -\frac{\alpha}{\Delta^3}\theta( x_c- x) + \frac {4\alpha}{9\pi}\left[\frac {\chi^2(x)}{ x^2} +2\frac{m_e}{m_\pi}\frac{\chi(x)}{x}\right]^{3/2}.$$

The finite-size proton source is encoded by the step-function term, and the electron density by the relativistic $[\cdots]^{3/2}$ source (Rotondo et al., 2010).

Finite nuclear size is not a secondary correction in this framework. The proton density is taken as a finite uniform distribution with radius

$y''(x)=x^{-1/2}y^{3/2}(x)$0

because the point-nucleus approximation would generate a non-integrable singularity at the origin in the relativistic regime (Rotondo et al., 2010). The boundary conditions

$y''(x)=x^{-1/2}y^{3/2}(x)$1

enforce regularity at the center and global neutrality at the cell boundary.

The same paper embeds the cell problem into general relativity. The stellar equilibrium condition becomes

$y''(x)=x^{-1/2}y^{3/2}(x)$2

equivalently the redshifted Wigner–Seitz chemical potential is constant throughout the star (Rotondo et al., 2010). This relation links the microscopic relativistic Thomas–Fermi cell calculation to macroscopic TOV equilibrium. Within that unified framework, the paper reports inverse beta-decay threshold densities for $y''(x)=x^{-1/2}y^{3/2}(x)$3He, $y''(x)=x^{-1/2}y^{3/2}(x)$4C, $y''(x)=x^{-1/2}y^{3/2}(x)$5O, and $y''(x)=x^{-1/2}y^{3/2}(x)$6Fe, and gives maximum masses $y''(x)=x^{-1/2}y^{3/2}(x)$7 for $y''(x)=x^{-1/2}y^{3/2}(x)$8C, $y''(x)=x^{-1/2}y^{3/2}(x)$9 for $p=3/2$0O, and $p=3/2$1 for $p=3/2$2Fe (Rotondo et al., 2010). These results are lower than the corresponding Chandrasekhar-type benchmarks quoted in the same work, because Coulomb interactions, finite nuclear size, weak interactions, and GR are all treated self-consistently.

3. Deformed phase-space and Planck-scale modifications

A distinct relativistic Thomas–Fermi construction appears in the reformulation based on a linear-quadratic generalized uncertainty principle (Barman et al., 2020). Here the central modification is not a new interaction term in the Hamiltonian but a deformed phase-space measure. Retaining terms up to second order in the deformation parameter $p=3/2$3, the measure is written as

$p=3/2$4

This altered density of states is inserted directly into the relativistic Thomas–Fermi electron density.

The relativistic number density is then expressed as

$p=3/2$5

where the single-particle energy contains the relativistic combination

$p=3/2$6

To handle the integral, the paper introduces the Jüttner transformation

$p=3/2$7

followed by

$p=3/2$8

so that Sommerfeld-type expansion methods can be applied (Barman et al., 2020).

The resulting relativistic density contains a leading relativistic Thomas–Fermi term, a correction linear in $p=3/2$9, and a correction quadratic in $\varphi''=x^{-1/2}\varphi^{3/2}$0. Substituting that density into Poisson’s equation,

$\varphi''=x^{-1/2}\varphi^{3/2}$1

produces a generalized relativistic Thomas–Fermi equation with explicit Planck-scale corrections (Barman et al., 2020). The displayed final expression is lengthy and organized in terms of powers of

$\varphi''=x^{-1/2}\varphi^{3/2}$2

together with deformation terms proportional to $\varphi''=x^{-1/2}\varphi^{3/2}$3. The paper then introduces dimensionless variables and states the relativistic TF equation in the presence of linear-quadratic GUP as a source equation plus GUP corrections. In the limit $\varphi''=x^{-1/2}\varphi^{3/2}$4, the equation reduces to the non-relativistic TF equation, and for $\varphi''=x^{-1/2}\varphi^{3/2}$5 the GUP corrections vanish. In the absence of deformation or thermal effects, the paper gives the simplified form

$\varphi''=x^{-1/2}\varphi^{3/2}$6

The conceptual significance of this formulation is methodological. The relativistic correction enters through the relativistic dispersion relation, while the quantum-gravity-inspired correction enters through the modified phase-space volume element rather than by perturbing the Hamiltonian (Barman et al., 2020). This separates two kinds of generalization that are often conflated: relativization of the kinetic term and deformation of the density of states.

4. Relativistic Thomas–Fermi approximation in finite nuclei

In nuclear structure, the relativistic Thomas–Fermi equation is usually not a single ODE but a semiclassical, self-consistent limit of covariant density functional theory (Li et al., 31 Jul 2025). The self-consistent Thomas–Fermi approximation assumes that meson mean fields vary slowly in space, so nucleons can be treated as a locally uniform relativistic Fermi gas moving in position-dependent scalar and vector fields. This avoids solving the full Dirac single-particle problem while retaining relativistic dynamics.

The framework is built from the same covariant Lagrangian used in RMF calculations, with nucleons coupled to the isoscalar scalar $\varphi''=x^{-1/2}\varphi^{3/2}$7, isoscalar vector $\varphi''=x^{-1/2}\varphi^{3/2}$8, isovector vector $\varphi''=x^{-1/2}\varphi^{3/2}$9, and electromagnetic field $$E_e^F = \sqrt{c^2 (P_e^F)^2+m_e^2 c^4} - m_e c^2 - e V(r) = {\rm constant} > 0,$$0 (Li et al., 31 Jul 2025). In the STF approximation, local scalar and vector densities are computed from local Fermi momenta, and the effective mass is

$$E_e^F = \sqrt{c^2 (P_e^F)^2+m_e^2 c^4} - m_e c^2 - e V(r) = {\rm constant} > 0,$$1

The proton and neutron chemical potentials are spatially constant because of baryon-number conservation, and the total STF energy is obtained by integrating a local functional containing the nucleon kinetic contribution together with $$E_e^F = \sqrt{c^2 (P_e^F)^2+m_e^2 c^4} - m_e c^2 - e V(r) = {\rm constant} > 0,$$2, $$E_e^F = \sqrt{c^2 (P_e^F)^2+m_e^2 c^4} - m_e c^2 - e V(r) = {\rm constant} > 0,$$3, $$E_e^F = \sqrt{c^2 (P_e^F)^2+m_e^2 c^4} - m_e c^2 - e V(r) = {\rm constant} > 0,$$4, Coulomb, nonlinear, and coupling terms (Li et al., 31 Jul 2025).

The numerical procedure is explicitly iterative: specify $$E_e^F = \sqrt{c^2 (P_e^F)^2+m_e^2 c^4} - m_e c^2 - e V(r) = {\rm constant} > 0,$$5 and $$E_e^F = \sqrt{c^2 (P_e^F)^2+m_e^2 c^4} - m_e c^2 - e V(r) = {\rm constant} > 0,$$6, guess $$E_e^F = \sqrt{c^2 (P_e^F)^2+m_e^2 c^4} - m_e c^2 - e V(r) = {\rm constant} > 0,$$7, $$E_e^F = \sqrt{c^2 (P_e^F)^2+m_e^2 c^4} - m_e c^2 - e V(r) = {\rm constant} > 0,$$8, $$E_e^F = \sqrt{c^2 (P_e^F)^2+m_e^2 c^4} - m_e c^2 - e V(r) = {\rm constant} > 0,$$9, and $$n_e(r) = \frac{(P_e^{F})^3}{3\pi^2 \hbar^3} = \frac {1}{3\pi^2 \hbar^3 c^3}\left[\hat V^2(r)+ 2m_e c^2 \hat V(r)\right]^{3/2}, \qquad \hat V = e V + E_e^F,$$0, determine $$n_e(r) = \frac{(P_e^{F})^3}{3\pi^2 \hbar^3} = \frac {1}{3\pi^2 \hbar^3 c^3}\left[\hat V^2(r)+ 2m_e c^2 \hat V(r)\right]^{3/2}, \qquad \hat V = e V + E_e^F,$$1 and $$n_e(r) = \frac{(P_e^{F})^3}{3\pi^2 \hbar^3} = \frac {1}{3\pi^2 \hbar^3 c^3}\left[\hat V^2(r)+ 2m_e c^2 \hat V(r)\right]^{3/2}, \qquad \hat V = e V + E_e^F,$$2 under particle-number constraints, compute local densities from the Thomas–Fermi formulas, solve the meson-field equations, and repeat until convergence (Li et al., 31 Jul 2025). For the finite nuclei considered there, spherical symmetry is assumed. The paper also states practical simplifications for heavy nuclei: center-of-mass corrections and pairing correlations are neglected in STF, and nucleons are treated semiclassically.

A detailed comparison with RMF using the same interaction shows that the same parameter set does not necessarily give the same finite-nucleus result in STF and RMF, because RMF solves the Dirac equation explicitly for single-particle states whereas STF uses local Fermi-gas momentum integration (Li et al., 31 Jul 2025). The benchmark set comprises 19 doubly magic and semi-magic nuclei. RMF generally reproduces experimental binding energies better than STF, especially for parameter sets fitted to finite nuclei. In STF, TM1m*, TM1m, and SFHo tend to give binding energies below experiment, TM1e and TM1 give results closer to RMF and experiment, and NL3 tends to overbind (Li et al., 31 Jul 2025). For charge radii, STF with TM1m* and TM1m reproduces the data well, SFHo tends to give slightly smaller radii, and TM1e, TM1, and NL3 yield STF radii systematically larger than RMF values.

The density profiles illustrate the semiclassical character of the approximation. For $$n_e(r) = \frac{(P_e^{F})^3}{3\pi^2 \hbar^3} = \frac {1}{3\pi^2 \hbar^3 c^3}\left[\hat V^2(r)+ 2m_e c^2 \hat V(r)\right]^{3/2}, \qquad \hat V = e V + E_e^F,$$3Ca and $$n_e(r) = \frac{(P_e^{F})^3}{3\pi^2 \hbar^3} = \frac {1}{3\pi^2 \hbar^3 c^3}\left[\hat V^2(r)+ 2m_e c^2 \hat V(r)\right]^{3/2}, \qquad \hat V = e V + E_e^F,$$4Pb, RMF profiles show oscillatory shell structure, while STF yields smooth distributions with lower central density, slower falloff at intermediate radii, and a more rapid tail decrease (Li et al., 31 Jul 2025). In heavy nuclei such as $$n_e(r) = \frac{(P_e^{F})^3}{3\pi^2 \hbar^3} = \frac {1}{3\pi^2 \hbar^3 c^3}\left[\hat V^2(r)+ 2m_e c^2 \hat V(r)\right]^{3/2}, \qquad \hat V = e V + E_e^F,$$5Pb, STF proton densities can show an inward indentation at the center due to Coulomb repulsion. None of the models studied simultaneously matches both CREX and PREX-2 neutron-skin constraints within uncertainties, and STF predictions for neutron skins are systematically smaller than RMF results (Li et al., 31 Jul 2025).

5. Hot nuclei, hypernuclei, nuclear pasta, and subnuclear matter

The relativistic Thomas–Fermi formalism is especially prominent in nonuniform matter and finite-temperature applications. At subnuclear density, matter in supernovae and neutron-star crusts is modeled as a lattice of heavy nuclei surrounded by dripped nucleons, with a Wigner–Seitz cell introduced to simplify the geometry (Zhang et al., 2014, Zhang et al., 2015). In the self-consistent relativistic STF treatment, the nucleon distributions are not imposed by a parameterized ansatz but obtained from the coupled RMF field equations and local Fermi–Dirac occupation probabilities. The thermodynamically favored state is found by minimizing the free energy with respect to the cell radius, and the relevant phases include droplets, bubbles, and uniform matter (Zhang et al., 2014).

A central comparison in this literature is between self-consistent STF and the parameterized Thomas–Fermi approximation used in the Shen EOS (Zhang et al., 2015, Zhang et al., 2014). In PTF, the surface energy is estimated with a phenomenological parameter $$n_e(r) = \frac{(P_e^{F})^3}{3\pi^2 \hbar^3} = \frac {1}{3\pi^2 \hbar^3 c^3}\left[\hat V^2(r)+ 2m_e c^2 \hat V(r)\right]^{3/2}, \qquad \hat V = e V + E_e^F,$$6, and the nucleon density profile is assumed a priori. In STF, by contrast, the density profile and surface energy emerge self-consistently from the RMF equations. The numerical differences in free energy are generally modest, but the surface and Coulomb contributions are systematically sensitive to the treatment of the surface term. In the example quoted for $$n_e(r) = \frac{(P_e^{F})^3}{3\pi^2 \hbar^3} = \frac {1}{3\pi^2 \hbar^3 c^3}\left[\hat V^2(r)+ 2m_e c^2 \hat V(r)\right]^{3/2}, \qquad \hat V = e V + E_e^F,$$7, $$n_e(r) = \frac{(P_e^{F})^3}{3\pi^2 \hbar^3} = \frac {1}{3\pi^2 \hbar^3 c^3}\left[\hat V^2(r)+ 2m_e c^2 \hat V(r)\right]^{3/2}, \qquad \hat V = e V + E_e^F,$$8 MeV, PTF with $$n_e(r) = \frac{(P_e^{F})^3}{3\pi^2 \hbar^3} = \frac {1}{3\pi^2 \hbar^3 c^3}\left[\hat V^2(r)+ 2m_e c^2 \hat V(r)\right]^{3/2}, \qquad \hat V = e V + E_e^F,$$9 gives smaller surface energies and lower free energies than STF, while $\nabla^2 V(r)= -4\pi e\left[n_p(r)-n_e(r)\right].$0 makes PTF results very close to STF (Zhang et al., 2015).

Finite-temperature nuclei are handled by a subtraction procedure in which one solves two coupled mean-field problems, a nucleus-plus-gas phase $\nabla^2 V(r)= -4\pi e\left[n_p(r)-n_e(r)\right].$1 and a gas-only phase $\nabla^2 V(r)= -4\pi e\left[n_p(r)-n_e(r)\right].$2, and defines the isolated nucleus by subtraction (1411.1584). The thermodynamic potential is

$\nabla^2 V(r)= -4\pi e\left[n_p(r)-n_e(r)\right].$3

and the physical densities satisfy

$\nabla^2 V(r)= -4\pi e\left[n_p(r)-n_e(r)\right].$4

This construction removes dependence on the box size and makes it possible to study hot nuclei in equilibrium with their vapor. In that setting, the symmetry energy coefficient of finite nuclei is found to be significantly affected by Coulomb polarization, and the extracted symmetry energy depends strongly on whether Coulomb is subtracted after the calculation or switched off before solving (1411.1584).

The same subtraction formalism is extended to hot single-$\nabla^2 V(r)= -4\pi e\left[n_p(r)-n_e(r)\right].$5 hypernuclei in a relativistic Thomas–Fermi approximation (Hu et al., 2016). The baryons are $\nabla^2 V(r)= -4\pi e\left[n_p(r)-n_e(r)\right].$6, $\nabla^2 V(r)= -4\pi e\left[n_p(r)-n_e(r)\right].$7, and $\nabla^2 V(r)= -4\pi e\left[n_p(r)-n_e(r)\right].$8, the microscopic input is an RMF Lagrangian with $\nabla^2 V(r)= -4\pi e\left[n_p(r)-n_e(r)\right].$9, $$x= \frac{r}{\lambda_{\pi}}, \qquad x_c= \frac{R_c}{\lambda_{\pi}}, \qquad \frac{\chi}{r} = \frac{\hat V(r)}{\hbar c},$$0, $$x= \frac{r}{\lambda_{\pi}}, \qquad x_c= \frac{R_c}{\lambda_{\pi}}, \qquad \frac{\chi}{r} = \frac{\hat V(r)}{\hbar c},$$1, and Coulomb fields, and the thermodynamic potential of the isolated hypernucleus is again defined by $$x= \frac{r}{\lambda_{\pi}}, \qquad x_c= \frac{R_c}{\lambda_{\pi}}, \qquad \frac{\chi}{r} = \frac{\hat V(r)}{\hbar c},$$2 plus Coulomb energy. The paper reports that the $$x= \frac{r}{\lambda_{\pi}}, \qquad x_c= \frac{R_c}{\lambda_{\pi}}, \qquad \frac{\chi}{r} = \frac{\hat V(r)}{\hbar c},$$3 central density is very sensitive to temperature, the $$x= \frac{r}{\lambda_{\pi}}, \qquad x_c= \frac{R_c}{\lambda_{\pi}}, \qquad \frac{\chi}{r} = \frac{\hat V(r)}{\hbar c},$$4 radius becomes very large at high temperature, the level density parameters are almost constant at low temperature, and the single-$$x= \frac{r}{\lambda_{\pi}}, \qquad x_c= \frac{R_c}{\lambda_{\pi}}, \qquad \frac{\chi}{r} = \frac{\hat V(r)}{\hbar c},$$5 binding energies are largely reduced with increasing temperature (Hu et al., 2016). In this usage, the “relativistic Thomas–Fermi equation” is explicitly the coupled set of self-consistent integral-differential equations rather than a single nonlinear screening ODE.

Nuclear pasta provides a further extension. In the relativistic Thomas–Fermi treatment of protoneutron-star matter, the WS cell may take spherical droplet, cylindrical rod, slab, cylindrical tube, or spherical bubble geometry (Furtado et al., 2021). Pairing is added through a BCS gap equation, either non-self-consistently or self-consistently. The paper states that the uniform matter energy is always higher than the most stable pasta phase in the density range where pasta exists, pairing changes transition densities only slightly, and pairing strongly suppresses the specific heat at low temperature (Furtado et al., 2021). This suggests that, in crust physics, the chief value of the relativistic Thomas–Fermi formalism lies in its ability to scan large ranges of density, proton fraction, geometry, and thermodynamic state at manageable computational cost.

6. Analytical structure, classical correspondences, and common misconceptions

The relativistic literature remains closely connected to the analytical structure of the classical Thomas–Fermi equation. The nonrelativistic equation possesses a homology symmetry under

$$x= \frac{r}{\lambda_{\pi}}, \qquad x_c= \frac{R_c}{\lambda_{\pi}}, \qquad \frac{\chi}{r} = \frac{\hat V(r)}{\hbar c},$$6

and this symmetry allows a reduction of the second-order ODE to a first-order ODE in homology-invariant variables (Alizzi et al., 2023). Majorana’s specific transformation,

$$x= \frac{r}{\lambda_{\pi}}, \qquad x_c= \frac{R_c}{\lambda_{\pi}}, \qquad \frac{\chi}{r} = \frac{\hat V(r)}{\hbar c},$$7

reduces the Thomas–Fermi equation to

$$x= \frac{r}{\lambda_{\pi}}, \qquad x_c= \frac{R_c}{\lambda_{\pi}}, \qquad \frac{\chi}{r} = \frac{\hat V(r)}{\hbar c},$$8

The paper explicitly does not treat a relativistic Thomas–Fermi equation, but it argues that the reduction mechanism is tied to scaling symmetry and is therefore conceptually informative for possible relativistic generalizations if they possess analogous scaling properties (Alizzi et al., 2023).

Other classical works in the record are likewise important by contrast. The variational and series treatment for neutral atoms provides improved approximations to the standard equation and explicitly states that it does not derive or study a relativistic Thomas–Fermi equation (Oulne, 2015). The analysis of Majorana’s series solution likewise concerns the nonrelativistic neutral-atom problem and not a relativistic generalization, although it shows how a transformed first-order equation can yield extremely accurate control of the slope at the origin (Fernández et al., 2021). A useful corrective to a common misconception is therefore straightforward: not every paper with “Thomas–Fermi” in the title contributes to the relativistic equation proper.

A different route to generalization appears in the study of a generalized Thomas–Fermi equation with relativistic, non-extensive, and thermal effects (Esposito et al., 2020). There the starting point is

$$x= \frac{r}{\lambda_{\pi}}, \qquad x_c= \frac{R_c}{\lambda_{\pi}}, \qquad \frac{\chi}{r} = \frac{\hat V(r)}{\hbar c},$$9

with $$\frac{1}{3x}\frac {d^2\chi(x)}{d x^2} = -\frac{\alpha}{\Delta^3}\theta( x_c- x) + \frac {4\alpha}{9\pi}\left[\frac {\chi^2(x)}{ x^2} +2\frac{m_e}{m_\pi}\frac{\chi(x)}{x}\right]^{3/2}.$$0 encoding relativistic effects, $$\frac{1}{3x}\frac {d^2\chi(x)}{d x^2} = -\frac{\alpha}{\Delta^3}\theta( x_c- x) + \frac {4\alpha}{9\pi}\left[\frac {\chi^2(x)}{ x^2} +2\frac{m_e}{m_\pi}\frac{\chi(x)}{x}\right]^{3/2}.$$1 non-extensive effects, and $$\frac{1}{3x}\frac {d^2\chi(x)}{d x^2} = -\frac{\alpha}{\Delta^3}\theta( x_c- x) + \frac {4\alpha}{9\pi}\left[\frac {\chi^2(x)}{ x^2} +2\frac{m_e}{m_\pi}\frac{\chi(x)}{x}\right]^{3/2}.$$2 thermal effects (Esposito et al., 2020). The paper concludes that the standard Sommerfeld inverse-power tail is no longer universally valid once $$\frac{1}{3x}\frac {d^2\chi(x)}{d x^2} = -\frac{\alpha}{\Delta^3}\theta( x_c- x) + \frac {4\alpha}{9\pi}\left[\frac {\chi^2(x)}{ x^2} +2\frac{m_e}{m_\pi}\frac{\chi(x)}{x}\right]^{3/2}.$$3 and $$\frac{1}{3x}\frac {d^2\chi(x)}{d x^2} = -\frac{\alpha}{\Delta^3}\theta( x_c- x) + \frac {4\alpha}{9\pi}\left[\frac {\chi^2(x)}{ x^2} +2\frac{m_e}{m_\pi}\frac{\chi(x)}{x}\right]^{3/2}.$$4 are present; for sufficiently small parameters it can survive only up to a finite cutoff scale $$\frac{1}{3x}\frac {d^2\chi(x)}{d x^2} = -\frac{\alpha}{\Delta^3}\theta( x_c- x) + \frac {4\alpha}{9\pi}\left[\frac {\chi^2(x)}{ x^2} +2\frac{m_e}{m_\pi}\frac{\chi(x)}{x}\right]^{3/2}.$$5, estimated from

$$\frac{1}{3x}\frac {d^2\chi(x)}{d x^2} = -\frac{\alpha}{\Delta^3}\theta( x_c- x) + \frac {4\alpha}{9\pi}\left[\frac {\chi^2(x)}{ x^2} +2\frac{m_e}{m_\pi}\frac{\chi(x)}{x}\right]^{3/2}.$$6

This suggests that “relativistic Thomas–Fermi equation” may also designate a broader family of generalized screening equations in which relativity is only one ingredient among several nonclassical deformations.

Taken together, these strands show that the relativistic Thomas–Fermi equation is best understood as a methodological family rather than a single formula. In compressed-atom and white-dwarf physics it is a relativistic screening equation tied to Wigner–Seitz-cell neutrality and, in GR, to the condition $$\frac{1}{3x}\frac {d^2\chi(x)}{d x^2} = -\frac{\alpha}{\Delta^3}\theta( x_c- x) + \frac {4\alpha}{9\pi}\left[\frac {\chi^2(x)}{ x^2} +2\frac{m_e}{m_\pi}\frac{\chi(x)}{x}\right]^{3/2}.$$7 (Rotondo et al., 2010). In nuclear structure and dense matter it is a self-consistent semiclassical RMF scheme for locally uniform relativistic Fermi gases in spatially varying meson and Coulomb fields (Li et al., 31 Jul 2025, 1411.1584). In generalized formulations it can absorb modified density-of-states effects or thermal and non-extensive corrections (Barman et al., 2020, Esposito et al., 2020). The common invariant across these usages is the Thomas–Fermi strategy itself: local phase-space filling plus self-consistent fields, pushed into regimes where relativistic kinematics and covariant mean fields can no longer be ignored.