Fayans Energy Density Functional

• Fayans EDF is a nuclear energy density functional based on the Theory of Finite Fermi Systems, characterized by a rational (Padé-like) density dependence and the preservation of the bare nucleon mass.
• It features a nonlinear surface term and a unique pairing functional with explicit density and gradient dependence, which drives a rearrangement mechanism that improves predictions of charge radii and odd–even staggering.
• Advanced parameterizations such as DF3 variants and FaNDF⁰ link detailed optimization strategies with enhanced performance in describing spherical, deformed, and superheavy nuclei.

Searching arXiv for recent and foundational Fayans EDF papers to support the article. The Fayans energy density functional (EDF) is a nonrelativistic local nuclear EDF rooted in the self-consistent Theory of Finite Fermi Systems (TFFS) and formulated to describe finite nuclei with a more sophisticated density dependence than standard Skyrme functionals. Its characteristic ingredients are a rational or Padé-like density dependence in the bulk channel, the use of the bare nucleon mass $m^*=m$, a nonlinear surface term with gradient dependence in the denominator, and a pairing functional with explicit density and density-gradient dependence. Within nuclear density functional theory, these features have been linked to accurate descriptions of charge radii, differential charge radii, odd–even staggering, single-particle spectra, and, in the localized FaNDF$^0$ form, subtle isotope-shift systematics such as the near equality of the charge radii of $^{40}\mathrm{Ca}$ and $^{48}\mathrm{Ca}$ and the “bell shape” of the Ca chain (Saperstein et al., 2015, Reinhard et al., 2017, Naito et al., 19 Jun 2026).

1. TFFS origin and defining theoretical ideas

The Fayans EDF was developed as a practical realization of the self-consistent TFFS. In that framework, the quasiparticle self-energy $\Sigma$ depends on both energy $\varepsilon$ and momentum $k$, and the effective mass factorizes into momentum and energy components, $m^*=m_k^*m_E^*$. Fayans and collaborators argued that the complicated energy dependence of $\Sigma$, together with the associated $Z$-factor, can be mimicked in an EDF by a more elaborate density dependence while retaining the bare mass. This is the conceptual basis of the characteristic Fayans choice $^0$0, justified by the TFFS cancellation $^0$1 (Saperstein et al., 2015).

A standard schematic expression for the Fayans in-volume term is

$^0$2

where the denominator distinguishes the Fayans form from the polynomial density dependence of Skyrme–Hartree–Fock, recovered when $^0$3. The same logic underlies the localized FaNDF$^0$4 form, which was constructed to make the original finite-range Fayans EDF practical in solvers developed for Skyrme-like functionals (Saperstein et al., 2015, Tolokonnikov et al., 2016).

This TFFS lineage is central to the interpretation of Fayans results. In the spherical-nucleus literature, the denominator-type density dependence is repeatedly connected with improved charge radii and better single-particle spacings relative to Skyrme fits with $^0$5, while the bare-mass prescription avoids the expanded spectra that often appear in such fits (Gnezdilov et al., 2014, Saperstein et al., 2015).

2. Functional structure and local densities

In Fayans implementations, the total energy is written as an integral over a local energy density built from normal and anomalous densities. A generic decomposition used in the FaNDF$^0$6 pairing analysis is

$^0$7

while deformed FaNDF$^0$8 calculations employ

$^0$9

The particle–hole sector contains central, surface, spin–orbit, and Coulomb terms; the pairing sector is local and zero range in the forms used in the cited studies (Naito et al., 19 Jun 2026, Tolokonnikov et al., 2016).

Two notational systems are common. In FaNDF$^{40}\mathrm{Ca}$0-like forms one writes $^{40}\mathrm{Ca}$1, $^{40}\mathrm{Ca}$2, and $^{40}\mathrm{Ca}$3 with $^{40}\mathrm{Ca}$4. In the Ca-radius studies, normalized isoscalar and isovector densities are written as

$^{40}\mathrm{Ca}$5

The Wigner–Seitz radius and Fermi-energy scale are introduced as

$^{40}\mathrm{Ca}$6

and the local anomalous density in spherical HFB is

$^{40}\mathrm{Ca}$7

These definitions are not merely formal: they determine how density dependence and density gradients enter both the particle–hole and pairing sectors, and therefore how pairing feeds back into normal mean fields (Naito et al., 19 Jun 2026).

In localized FaNDF$^{40}\mathrm{Ca}$8, the volume energy density takes the form

$^{40}\mathrm{Ca}$9

with

$^{48}\mathrm{Ca}$0

and the surface term has a denominator containing both $^{48}\mathrm{Ca}$1 and gradient contributions. This nonlinear surface structure is one of the canonical signatures of the Fayans formalism (Tolokonnikov et al., 2016, Inakura et al., 2024).

3. Pairing functional and the rearrangement mechanism

The pairing sector is the most distinctive part of the Fayans EDF in discussions of radii systematics. In the FaNDF$^{48}\mathrm{Ca}$2 pairing analysis of Ca isotopes, the pairing energy density is

$^{48}\mathrm{Ca}$3

with $^{48}\mathrm{Ca}$4, $^{48}\mathrm{Ca}$5, $^{48}\mathrm{Ca}$6, and $^{48}\mathrm{Ca}$7. The corresponding local pair potential is

$^{48}\mathrm{Ca}$8

Unlike standard mixed, surface, or volume pairing, this form depends explicitly on both density and density gradients in the normal channel (Naito et al., 19 Jun 2026).

Because the pairing coupling depends on $^{48}\mathrm{Ca}$9 and $\Sigma$0, it generates a rearrangement contribution to the normal mean field: $\Sigma$1 For $\Sigma$2 and $\Sigma$3, the $\Sigma$4 piece is positive everywhere, while $\Sigma$5 and $\Sigma$6 are typically repulsive in the interior and weakly attractive near the surface. This produces an isoscalar repulsive rearrangement mechanism that pushes nucleon densities outward in open-shell nuclei and modifies proton and neutron radii in tandem (Naito et al., 19 Jun 2026).

The Ca chain provides the clearest demonstration. The strong density- and gradient-dependent pairings reproduce the observed “bell shape” between $\Sigma$7 and $\Sigma$8: the charge radius grows from $\Sigma$9 to mid-shell $\varepsilon$0 and then decreases toward $\varepsilon$1. For the pairing set $\varepsilon$2, the rearrangement potential in $\varepsilon$3 shows a strong repulsive peak around $\varepsilon$4 fm and a small attractive pocket around $\varepsilon$5 fm. Including $\varepsilon$6 increases the neutron pairing gap from $\varepsilon$7 MeV to $\varepsilon$8 MeV, changes the neutron $\varepsilon$9 quasiparticle energy and occupation from $k$0 MeV and $k$1 to $k$2 MeV and $k$3, raises the proton rms radius from $k$4 fm to $k$5 fm, and raises the neutron rms radius from $k$6 fm to $k$7 fm, while leaving the neutron skin nearly unchanged (Naito et al., 19 Jun 2026).

This mechanism is also the basis of the radial decomposition analysis of Ca radii. In spherical HFB with $k$8, the proton shell remains closed, so $k$9 and the isotope-shift pattern is almost entirely radial. The Fayans pairing rearrangement acts through the proton central field $m^*=m_k^*m_E^*$0, reshapes proton $m^*=m_k^*m_E^*$1-shell wave functions, and produces the mid-shell enhancement of $m^*=m_k^*m_E^*$2. The same analysis also showed that this mechanism overshoots on the proton-rich side: experiment gives $m^*=m_k^*m_E^*$3, whereas FaNDF$m^*=m_k^*m_E^*$4 gives $m^*=m_k^*m_E^*$5; omitting the Fayans pairing rearrangement term changes the result to approximately $m^*=m_k^*m_E^*$6 (Inakura et al., 2024).

4. Parameterizations, calibration strategies, and model families

Several parameter families exist within the Fayans framework, reflecting different emphases in optimization and application.

Parameterization	Distinguishing feature	Typical use
DF3, DF3-a, DF3-b	TFFS-based spherical EDFs with different spin–orbit and tensor-sector parameters	Charge radii, $m^*=m_k^*m_E^*$7 excitations, single-particle spectra
FaNDF$m^*=m_k^*m_E^*$8	Localized Fayans EDF with rational density dependence and nonlinear surface term	Spherical and deformed HFB, Ca radii, superheavy $m^*=m_k^*m_E^*$9
Fy(std), Fy($\Sigma$0), Fy($\Sigma$1)	Optimized Fayans variants with progressively stronger differential-radius constraints	Global radii, odd–even staggering, isotope shifts
13D and 14D extended Fayans	14D frees the isovector pairing strength $\Sigma$2	Sensitivity analysis and global calibration of spherical even-even nuclei

The DF3 family differs mainly in the spin–orbit and effective tensor sector. The values quoted for the four spin–orbit parameters are, for DF3, $\Sigma$3, $\Sigma$4, $\Sigma$5, $\Sigma$6; for DF3-a, $\Sigma$7, $\Sigma$8, $\Sigma$9, $Z$0; and for DF3-b, $Z$1, $Z$2, $Z$3, $Z$4. DF3-b was optimized to reproduce 35 measured spin–orbit differences and gives the smallest rms deviation for those splittings (Gnezdilov et al., 2014).

Modern optimization work has concentrated on the FaNDF$Z$5-like local form. In the 2017 radius-focused study, adding differential charge radii to the fit drove the surface and pairing gradient couplings upward by orders of magnitude. In particular, the pairing gradient strength changed from $Z$6 in Fy(std) to $Z$7 in Fy($Z$8) and $Z$9 in Fy($^0$00); the surface gradient strength likewise increased from $^0$01 to $^0$02 and $^0$03, respectively. This is a direct quantitative indication that differential radii constrain the gradient sector very strongly (Reinhard et al., 2017).

The extended Fayans study compared a 13-parameter model with a 14-parameter model that frees the isovector pairing strength $^0$04. The heterogeneous calibration dataset contained 194 observables from 69 spherical even-even nuclei, including binding energies, diffraction radii, surface thicknesses, charge radii, spin–orbit splittings, differential radii, and neutron and proton three-point energy differences. The optimization used the derivative-free POUNDerS algorithm, and the 14D extension improved the overall quality of the model by about 30%. In the best fit, $^0$05, giving $^0$06 and $^0$07, so the fit preferred stronger proton pairing than neutron pairing (Reinhard et al., 2024).

A separate calibration strategy was used in the 2026 Ca-pairing analysis. There, 25 pairing parameterizations were generated on the grid $^0$08, with $^0$09, and for each pair the overall neutron pairing strength $^0$10 was adjusted to reproduce the empirical neutron gap in $^0$11, $^0$12 MeV. Acceptable sets also had to satisfy gap criteria in $^0$13 and $^0$14 and vanish in doubly magic $^0$15, $^0$16, and $^0$17, leaving five “starred” pairings (Naito et al., 19 Jun 2026).

5. Performance for spherical observables

Charge radii have long been a strong point of the Fayans EDF. Using DF3-a, the systematic accuracy in radii reaches $^0$18 fm across many nuclei, including deformed cases treated approximately. In the Pb chain, the isotopic trend is reproduced essentially perfectly. By contrast, HFB-17 agrees in heavier Pb isotopes but fails for $^0$19 with deviations up to $^0$20 fm because of spurious stable deformation, while SLy4 radii are systematically too large by $^0$21 fm (Saperstein et al., 2015).

The Ca chain is the canonical differential-radius benchmark. The FaNDF$^0$22 particle–hole sector by itself already yields $^0$23, close to the experimental $^0$24, thereby reproducing the small difference between the magic endpoints. Once pairing is included, only Fayans-like pairings with both strong density and gradient dependence reproduce the bell shape in $^0$25. The set $^0$26 gives a very good match, whereas $^0$27 overshoots the $^0$28 and $^0$29 shifts, and volume-like pairing $^0$30 yields near-flat $^0$31 and fails to generate the bell (Naito et al., 19 Jun 2026).

The Fayans EDF also performs well for collective and spectroscopic observables in spherical systems. In semi-magic Sn isotopes, DF3-a plus self-consistent TFFS/QRPA gives $^0$32 energies and $^0$33 values in very good agreement with data; surface pairing yields $^0$34 energies typically $^0$35 keV lower than volume pairing and is on average closer to experiment. For single-particle spectra in seven doubly magic nuclei, the rms deviation over 105 levels is $^0$36 MeV for DF3-a and $^0$37 MeV for DF3 and DF3-b, compared with $^0$38 MeV for HFB-17 (Saperstein et al., 2015).

Particle–phonon coupling (PC) further refines this picture when treated with both pole and tadpole terms. In $^0$39, including PC with the tadpole term improves the rms deviation from $^0$40 MeV to $^0$41 MeV for DF3-a. The tadpole contribution is always positive, the pole term is usually negative, and neglecting the tadpole can overestimate PC effects and even give the wrong sign. In light nuclei such as $^0$42, the simplified tadpole approximation overestimates the correction by about $^0$43, so PC slightly worsens the agreement there (Saperstein et al., 2015, Gnezdilov et al., 2014).

For odd–even mass differences in semi-magic nuclei, the direct Dyson-equation treatment of PC on top of DF3-a improves the Pb chain markedly. The combined rms deviation for $^0$44 and $^0$45 falls from $^0$46 MeV at the mean-field level to $^0$47 MeV when the $^0$48 and $^0$49 phonons are included. In Sn, the situation is more selective: the addition mode improves from $^0$50 MeV to $^0$51 MeV, but the removal mode worsens from $^0$52 MeV to $^0$53 MeV because of the peculiar intruder $^0$54 hole topology (Saperstein et al., 2017).

6. Deformed nuclei, heavy systems, and superheavy applications

Although the Fayans EDF was first calibrated mainly on spherical nuclei, it has also been applied to deformed systems. Early axial HFB calculations with FaNDF$^0$55 in the uranium and lead chains showed that the uranium isotopes have ground-state deformations and deformation energies broadly similar to HFB-17 and HFB-27, while the light Pb isotopes remain spherical for all $^0$56, in agreement with experimental trends in charge radii and magnetic moments and unlike HFB-17 and HFB-27, which predicted strong deformations in many neutron-deficient Pb isotopes (Tolokonnikov et al., 2014).

A broader study of deformed actinides used FaNDF$^0$57 with volume pairing in axial HFBTHO. For the uranium chain, the two-neutron drip line was found at $^0$58 with $^0$59; for thorium and plutonium the corresponding values were $^0$60 and $^0$61. Across the even-$^0$62 elements from Pb to Fm, FaNDF$^0$63 drip-line predictions were close to SLy4 and HFB-17/HFB-27 and systematically lower than SkM$^0$64, which often predicted much more neutron-rich drip points (Tolokonnikov et al., 2015).

Recent octupole-deformation surveys extend the Fayans program further into heavy deformed nuclei. Using Fy(std) and Fy($^0$65) in constrained HFB calculations, the actinide octupole island was found to be centered near $^0$66 and to occupy essentially the same region as in UNEDF0-based Skyrme studies. Significant octupole deformation was characterized by $^0$67, the strongest octupole minima occur around $^0$68 in Th, U, and Pu, and the octupole correlation energy ranges from about $^0$69 MeV to $^0$70 MeV, with Fy($^0$71) generally giving slightly stronger octupole minima than Fy(std) (Danneaux et al., 23 Mar 2026).

The Fayans framework has also been tested in superheavy $^0$72-decay systematics. Using a modified HFBTHO solver with FaNDF$^0$73, the rms deviations in $^0$74 across six superheavy decay chains were $^0$75 MeV for surface pairing and $^0$76 MeV for volume pairing, compared with $^0$77 MeV for SLy4, $^0$78 MeV for SkM$^0$79, and $^0$80 MeV for a macro–micro model. When those theoretical $^0$81 values were propagated to half-lives, the Fayans results outperformed SLy4 and SkM$^0$82 with both Parkhomenko–Sobiczewski and Royer–Zhang mappings, although they remained less accurate than the macro–micro model (Tolokonnikov et al., 2016).

7. Limitations, tensions, and current extensions

The modern Fayans literature is explicit about the framework’s unresolved tensions. In the Ca-pairing study, no single pairing parameterization simultaneously described empirical gaps and isotope shifts with equal quality across Ca, Sn, and Pb. Large $^0$83 values reproduce the Ca bell shape but generate too-strong arches in $^0$84 and too-rapid growth of pairing gaps in heavier nuclei, while more moderate sets improve Sn and Pb but miss Ca. The authors therefore argued that the standard Fayans EDF may require a more general form, either through refitting the particle–hole sector with modern optimization methods, through isovector pairing extensions, or through finite-range pairing within DFT (Naito et al., 19 Jun 2026).

A related issue is that the same Fayans pairing rearrangement mechanism that succeeds in $^0$85 also produces the wrong-sign enhancement on the proton-rich side. The detailed orbital decomposition for Ca showed that the mid-shell parabola and the failure for $^0$86 have a parallel origin. This suggests that future refinements of the Fayans pairing channel must constrain both regions simultaneously rather than fitting only the $^0$87 bell shape (Inakura et al., 2024).

Optimization studies have begun to address this lack of flexibility. The 14D extended Fayans model introduces an explicit isovector pairing parameter and reduces the total objective function by about 30% relative to the 13D model, while also reducing correlations between pairing and surface parameters. At the same time, some parameters, notably $^0$88, become weakly constrained in the 14D fit, indicating that the isovector volume denominator is effectively saturated in the relevant density domain. The calibration is still restricted to spherical even-even nuclei and uses BCS pairing in the optimization loop, so a deformed HFB-level extension remains an open step (Reinhard et al., 2024).

Further limitations concern explicit correlation effects. In the spherical PC studies, the simplified tadpole approximation is excellent in heavy nuclei but overestimates the tadpole magnitude by about $^0$89 in $^0$90. This does not challenge the TFFS-based PC framework itself, but it does limit quantitative accuracy in light nuclei unless the full variation equations, including in-volume corrections, are solved (Saperstein et al., 2015, Gnezdilov et al., 2014).

Taken together, these developments define the present status of the Fayans EDF. Its characteristic rational density dependence, bare-mass prescription, nonlinear surface term, and density- and gradient-dependent pairing have established it as a distinct branch of nuclear EDF theory, particularly strong in radii systematics and in observables sensitive to pairing-induced rearrangement. At the same time, the current literature points toward a more general Fayans functional with improved isovector pairing, reoptimized particle–hole terms, and broader deformed-HFB calibrations as the natural next stage of the framework’s development (Reinhard et al., 2024, Naito et al., 19 Jun 2026).