Fayans-like Pairing Interaction

• Fayans-like pairing interaction is a nuclear energy density functional model that modulates pairing strength using local density and gradient dependence to enhance low-density surface pairing.
• It incorporates isovector and gradient terms to refine predictions for odd-even staggering, quadrupole excitations, and charge radii in diverse nuclei.
• The approach bridges standard zero-range methods and finite-range pairing schemes, while posing challenges in ultraviolet convergence and numerical stability.

Searching arXiv for the specified Fayans-pairing papers and closely related work. arxiv_search(query="Fayans pairing interaction density dependence pairing correlation unstable nuclei (Changizi et al., 2015)", max_results=5, sort_by="relevance") arxiv_search(query="The first quadrupole excitations in spherical nuclei and nuclear pairing (Tolokonnikov et al., 2012) Fayans EDF", max_results=5, sort_by="relevance") arxiv_search(query="Extended Fayans energy density functional optimization and analysis isovector pairing (Reinhard et al., 2024)", max_results=5, sort_by="relevance") Fayans-like pairing interaction denotes a class of nuclear pairing energy-density functionals in which the effective strength of the $T=1$ superfluid channel is not taken as purely volume-like and density independent, but is modulated by the local isoscalar density and, in the modern Fayans formulation, by density gradients; in extended variants, an isovector term allows systematically different proton and neutron pairing fields. In practical nuclear density functional theory, this places Fayans-like pairing between standard zero-range density-dependent delta interactions and explicitly finite-range pairing schemes: it remains local and contact-like in its basic form, yet it is constructed to enhance pairing in the low-density surface region and to feed back into surface-sensitive observables such as odd–even staggering, low-lying quadrupole excitations, differential charge radii, and drip-line systematics [(Tolokonnikov et al., 2011); (Reinhard et al., 2024); (Naito et al., 19 Jun 2026)].

1. Definition and formal structure

In the Fayans framework, the ground-state energy is written as a functional of normal densities $\rho_q(\mathbf r)$ and anomalous densities $\nu_q(\mathbf r)$ or $\breve\rho_q(\mathbf r)$. In the self-consistent Theory of Finite Fermi Systems (TFFS), the effective pairing kernel is generated from second functional derivatives of the EDF,

$$\mathcal{F}^{\xi}=\frac{\delta^2 \mathcal{E}}{\delta \nu^2}, \qquad \mathcal{F}^{\omega\xi}=\frac{\delta^2 \mathcal{E}}{\delta \rho\,\delta \nu},$$

so that density dependence in pairing is equivalent to a nonzero mixed derivative $\mathcal{F}^{\omega\xi}$. In historical TFFS language, surface pairing corresponds to different strengths inside and outside the nucleus, $\gamma_{\rm in}$ and $\gamma_{\rm ex}$, with $$\left|\gamma_{\rm in}/\gamma_{\rm ex}\right|\simeq 10$$, whereas volume pairing corresponds to $\gamma_{\rm in}=\gamma_{\rm ex}$ and $\rho_q(\mathbf r)$0 (Tolokonnikov et al., 2012).

A widely used modern Fayans pairing EDF is local and contact-like but explicitly dependent on the isoscalar density and its gradient. In the 2024 extended Fayans model, the pairing energy density for nucleon species $\rho_q(\mathbf r)$1 is

$\rho_q(\mathbf r)$2

with $\rho_q(\mathbf r)$3, $\rho_q(\mathbf r)$4, $\rho_q(\mathbf r)$5, $\rho_q(\mathbf r)$6, and $\rho_q(\mathbf r)$7. Functional differentiation yields local pairing fields

$\rho_q(\mathbf r)$8

Thus the Fayans pairing field is strictly local and proportional to the local anomalous density, but with a strength controlled by density, density gradient, and, in the 14D model, isospin through $\rho_q(\mathbf r)$9 (Reinhard et al., 2024).

An older two-parameter Fayans-like form used in DF3-a-based TFFS calculations suppresses the explicit gradient term and keeps

$\nu_q(\mathbf r)$0

with $\nu_q(\mathbf r)$1 for volume pairing and $\nu_q(\mathbf r)$2 for surface pairing. This reduced form already produces a surface-peaked gap $\nu_q(\mathbf r)$3 and nonzero mixed residual interaction terms when density dependence is retained (Tolokonnikov et al., 2011).

2. Relation to standard volume, surface, and mixed pairing prescriptions

A simplified realization of Fayans-like physics is provided by density-dependent zero-range pairing of the form

$\nu_q(\mathbf r)$4

with $\nu_q(\mathbf r)$5 and $\nu_q(\mathbf r)$6 for volume pairing, $\nu_q(\mathbf r)$7 for surface pairing, and $\nu_q(\mathbf r)$8 for mixed pairing. In coordinate space the local pairing field becomes

$\nu_q(\mathbf r)$9

where $\breve\rho_q(\mathbf r)$0 is the anomalous density. This form was used in global Skyrme-HFB calculations with SLy4 to assess how density dependence modifies pairing gaps, odd–even staggering, and drip-line trends (Changizi et al., 2015).

Setting the Fayans gradient coupling to zero recovers the structure of the standard density-dependent delta interaction. In the 2026 analysis of charge radii, the Fayans pairing EDF was written as

$\breve\rho_q(\mathbf r)$1

with $\breve\rho_q(\mathbf r)$2, $\breve\rho_q(\mathbf r)$3, $\breve\rho_q(\mathbf r)$4, and $\breve\rho_q(\mathbf r)$5. The authors explicitly noted that $\breve\rho_q(\mathbf r)$6 reduces the functional to a standard density-dependent delta interaction, with $\breve\rho_q(\mathbf r)$7 corresponding to volume-, mixed-, and surface-type pairing (Naito et al., 19 Jun 2026).

The comparison between formulations can be summarized compactly.

Formulation	Pairing strength modulation	Distinctive feature
Volume pairing	density independent	$\breve\rho_q(\mathbf r)$8
Surface or mixed DDDI	linear or simple density dependence	low-density enhancement
Fayans-like pairing	density and gradient dependence; optionally isovector	strong surface localization and rearrangement feedback

This suggests that “Fayans-like” is not identical to any single contact-force parametrization. The common feature is surface-dominated, density-dependent pairing; the full Fayans form adds gradient dependence, and the extended variant adds isovector splitting between proton and neutron pairing fields (Changizi et al., 2015, Reinhard et al., 2024).

3. Microscopic mechanisms

The first mechanism is surface enhancement of the pairing field. In neutron-rich systems, a substantial part of the anomalous density resides at low density near the surface or in the exterior region. In the zero-range density-dependent ansatz,

$\breve\rho_q(\mathbf r)$9

the $$\mathcal{F}^{\xi}=\frac{\delta^2 \mathcal{E}}{\delta \nu^2}, \qquad \mathcal{F}^{\omega\xi}=\frac{\delta^2 \mathcal{E}}{\delta \rho\,\delta \nu},$$0 surface choice increases the local pairing strength as $$\mathcal{F}^{\xi}=\frac{\delta^2 \mathcal{E}}{\delta \nu^2}, \qquad \mathcal{F}^{\omega\xi}=\frac{\delta^2 \mathcal{E}}{\delta \rho\,\delta \nu},$$1 decreases, so the local gap is amplified where $$\mathcal{F}^{\xi}=\frac{\delta^2 \mathcal{E}}{\delta \nu^2}, \qquad \mathcal{F}^{\omega\xi}=\frac{\delta^2 \mathcal{E}}{\delta \rho\,\delta \nu},$$2 is large and the level density near threshold is high. The result is stronger mixing of weakly bound and continuum states and enhanced di-neutron correlations (Changizi et al., 2015).

The second mechanism is enhancement of anomalous transition amplitudes. In QRPA-like TFFS calculations of the first $$\mathcal{F}^{\xi}=\frac{\delta^2 \mathcal{E}}{\delta \nu^2}, \qquad \mathcal{F}^{\omega\xi}=\frac{\delta^2 \mathcal{E}}{\delta \rho\,\delta \nu},$$3 states, the response matrix $$\mathcal{F}^{\xi}=\frac{\delta^2 \mathcal{E}}{\delta \nu^2}, \qquad \mathcal{F}^{\omega\xi}=\frac{\delta^2 \mathcal{E}}{\delta \rho\,\delta \nu},$$4 contains normal propagators $$\mathcal{F}^{\xi}=\frac{\delta^2 \mathcal{E}}{\delta \nu^2}, \qquad \mathcal{F}^{\omega\xi}=\frac{\delta^2 \mathcal{E}}{\delta \rho\,\delta \nu},$$5 and anomalous Gor’kov functions $$\mathcal{F}^{\xi}=\frac{\delta^2 \mathcal{E}}{\delta \nu^2}, \qquad \mathcal{F}^{\omega\xi}=\frac{\delta^2 \mathcal{E}}{\delta \rho\,\delta \nu},$$6,

$$\mathcal{F}^{\xi}=\frac{\delta^2 \mathcal{E}}{\delta \nu^2}, \qquad \mathcal{F}^{\omega\xi}=\frac{\delta^2 \mathcal{E}}{\delta \rho\,\delta \nu},$$7

The pole structure of the effective field defines the excitation energy $$\mathcal{F}^{\xi}=\frac{\delta^2 \mathcal{E}}{\delta \nu^2}, \qquad \mathcal{F}^{\omega\xi}=\frac{\delta^2 \mathcal{E}}{\delta \rho\,\delta \nu},$$8, and a perturbative estimate gives

$$\mathcal{F}^{\xi}=\frac{\delta^2 \mathcal{E}}{\delta \nu^2}, \qquad \mathcal{F}^{\omega\xi}=\frac{\delta^2 \mathcal{E}}{\delta \rho\,\delta \nu},$$9

Because surface pairing produces anomalous amplitudes $\mathcal{F}^{\omega\xi}$0 that are much stronger at the nuclear surface, it lowers $\mathcal{F}^{\omega\xi}$1 relative to volume pairing (Tolokonnikov et al., 2012).

The third mechanism, specific to the Fayans functional, is the pairing-induced rearrangement potential in the particle–hole channel. Since the pairing EDF depends on $\mathcal{F}^{\omega\xi}$2 and $\mathcal{F}^{\omega\xi}$3, the functional derivative with respect to the density generates

$\mathcal{F}^{\omega\xi}$4

For the gradient-dependent Fayans form, this contains a density-dependent term, a $\mathcal{F}^{\omega\xi}$5 term, and a $\mathcal{F}^{\omega\xi}$6 term. In open-shell nuclei, the interior contribution is repulsive, with a sizable peak near the surface, while gradient terms may become weakly attractive outside. The repulsive rearrangement potential shifts density from the interior toward the exterior and increases rms charge radii. The 2026 calcium study concluded that this effect “cannot simply be mocked up by a refit of the pairing strength” (Naito et al., 19 Jun 2026).

A frequent misconception is that pairing acts only in the particle–particle channel and therefore modifies only gaps. In the Fayans case, the mixed derivative $\mathcal{F}^{\omega\xi}$7 and the rearrangement potential imply direct feedback from pairing to the normal mean field, making charge radii, surface profiles, and collective transition densities pairing-sensitive observables rather than passive by-products [(Tolokonnikov et al., 2012); (Naito et al., 19 Jun 2026)].

4. Phenomenology in collective states, odd–even staggering, and charge radii

For low-lying quadrupole excitations in spherical even-even nuclei, surface pairing systematically lowers $\mathcal{F}^{\omega\xi}$8 relative to volume pairing. In the TFFS calculations for tin and lead isotopes, the $\mathcal{F}^{\omega\xi}$9 energies are higher by about $\gamma_{\rm in}$0 for volume pairing than for surface pairing. In tin, the rms deviations from experiment were $\gamma_{\rm in}$1 for surface pairing and $\gamma_{\rm in}$2 for volume pairing; in lead they were $\gamma_{\rm in}$3 and $\gamma_{\rm in}$4, respectively. The effect on $\gamma_{\rm in}$5 is less regular because interference between normal and anomalous amplitudes makes the pairing dependence non-monotonic (Tolokonnikov et al., 2012).

In older DF3-a calculations for first $\gamma_{\rm in}$6 states and quadrupole moments of odd nuclei, the same qualitative pattern appeared: volume pairing raises $\gamma_{\rm in}$7 by $\gamma_{\rm in}$8–$\gamma_{\rm in}$9 relative to surface pairing, while quadrupole moments are often more sensitive to the single-particle energy $\gamma_{\rm ex}$0 through the Bogolyubov factor

$\gamma_{\rm ex}$1

This makes predictions for odd nuclei strongly dependent on near-Fermi single-particle structure, especially for high-$\gamma_{\rm ex}$2 states (Tolokonnikov et al., 2011).

For odd–even staggering in masses, all three standard zero-range prescriptions—volume, mixed, and surface—reproduce empirical $\gamma_{\rm ex}$3 reasonably well near the $\gamma_{\rm ex}$4-stability line in global Skyrme-HFB calculations. The analysis also extracted a residual $\gamma_{\rm ex}$5 interaction from neighboring $\gamma_{\rm ex}$6 values, with averages $\gamma_{\rm ex}$7 from neutron gaps and $\gamma_{\rm ex}$8 from proton gaps, and no visible shell dependence (Changizi et al., 2015).

For charge radii, the signature of Fayans-like pairing is stronger. The generalized Fayans pairing functional with a gradient term reproduced the odd–even staggering of charge radii in semi-magic chains far better than functionals without such a term. In the 2017 global study, adding $\gamma_{\rm ex}$9 observables to the fit drove the pairing gradient coupling $$\left|\gamma_{\rm in}/\gamma_{\rm ex}\right|\simeq 10$$0 from $$\left|\gamma_{\rm in}/\gamma_{\rm ex}\right|\simeq 10$$1 in Fy(std) to $$\left|\gamma_{\rm in}/\gamma_{\rm ex}\right|\simeq 10$$2 in Fy($$\left|\gamma_{\rm in}/\gamma_{\rm ex}\right|\simeq 10$$3) and $$\left|\gamma_{\rm in}/\gamma_{\rm ex}\right|\simeq 10$$4 in Fy($$\left|\gamma_{\rm in}/\gamma_{\rm ex}\right|\simeq 10$$5), showing that differential radii constrain the gradient term very strongly (Reinhard et al., 2017).

The calcium chain is the paradigmatic case. FaNDF0 reproduces the parabolic behavior of $$\left|\gamma_{\rm in}/\gamma_{\rm ex}\right|\simeq 10$$6 in $$\left|\gamma_{\rm in}/\gamma_{\rm ex}\right|\simeq 10$$7Ca, whereas standard Skyrme plus usual pairing gives nearly flat radii. A comparative decomposition showed that the proton-orbital shifts required for the calcium parabola are driven primarily by the Fayans pairing rearrangement term. However, the same mechanism produces the wrong sign for $$\left|\gamma_{\rm in}/\gamma_{\rm ex}\right|\simeq 10$$8 in $$\left|\gamma_{\rm in}/\gamma_{\rm ex}\right|\simeq 10$$9, because the enhancement below $\gamma_{\rm in}=\gamma_{\rm ex}$0 has an origin parallel to the parabolic behavior in $\gamma_{\rm in}=\gamma_{\rm ex}$1 (Inakura et al., 2024).

5. Drip-line and neutron-rich behavior

Systematic Skyrme-HFB calculations over the whole nuclear chart show that near the $\gamma_{\rm in}=\gamma_{\rm ex}$2-stability line, volume, mixed, and surface prescriptions behave comparably in both $\gamma_{\rm in}=\gamma_{\rm ex}$3 and $\gamma_{\rm in}=\gamma_{\rm ex}$4. Far from stability, the differences become large. In semi-magic chains such as He, O, Ca, Ni, Sn, and Pb, surface pairing produces larger $\gamma_{\rm in}=\gamma_{\rm ex}$5 and a larger separation between $\gamma_{\rm in}=\gamma_{\rm ex}$6 and $\gamma_{\rm in}=\gamma_{\rm ex}$7 near and beyond the drip line, whereas volume and mixed pairing typically yield $\gamma_{\rm in}=\gamma_{\rm ex}$8 and both go to zero promptly beyond the drip line, except for some nickel cases (Changizi et al., 2015).

The nickel isotopes illustrate the mechanism quantitatively. For $\gamma_{\rm in}=\gamma_{\rm ex}$9Ni with $\rho_q(\mathbf r)$00, the reported $\rho_q(\mathbf r)$01 values were $\rho_q(\mathbf r)$02 for volume pairing, $\rho_q(\mathbf r)$03 for mixed pairing, and $\rho_q(\mathbf r)$04 for surface pairing. For $\rho_q(\mathbf r)$05Ni they were $\rho_q(\mathbf r)$06, $\rho_q(\mathbf r)$07, and $\rho_q(\mathbf r)$08, respectively. When $\rho_q(\mathbf r)$09 was increased to $\rho_q(\mathbf r)$10, the surface gaps grew further, reaching $\rho_q(\mathbf r)$11 in $\rho_q(\mathbf r)$12Ni and $\rho_q(\mathbf r)$13 in $\rho_q(\mathbf r)$14Ni. The loosely bound $\rho_q(\mathbf r)$15 neutron orbital also became much more fragmented with surface pairing, with $\rho_q(\mathbf r)$16 in $\rho_q(\mathbf r)$17Ni versus $\rho_q(\mathbf r)$18–$\rho_q(\mathbf r)$19 for volume and mixed pairing (Changizi et al., 2015).

These results have consequences for drip-line indicators. Surface pairing drives the neutron chemical potential more negative, flattens the two-neutron separation-energy trend, and yields smoother driplines defined by $\rho_q(\mathbf r)$20 or $\rho_q(\mathbf r)$21. A plausible implication is that surface-dominated pairing smears shell closures in neutron-rich regions more efficiently than volume or mixed pairing, but this same sensitivity makes extrapolations dependent on model-space choices (Changizi et al., 2015).

The phenomenology aligns with the central Fayans idea that pairing should be stronger at low density. At the same time, the zero-range linear-density ansatz used in these drip-line calculations is only Fayans-like, not a full Fayans pairing functional: it omits gradient terms and finite-range regulation, which likely contributes to the pronounced cutoff and basis dependence near threshold (Changizi et al., 2015).

6. Calibration, sensitivities, and modern extensions

Modern Fayans EDF optimization confirms that the pairing channel is statistically identifiable and that isovector pairing improves the fit. In the 13D model without explicit isovector pairing, the pairing parameters were $\rho_q(\mathbf r)$22, $\rho_q(\mathbf r)$23, and $\rho_q(\mathbf r)$24. In the 14D model, adding

$\rho_q(\mathbf r)$25

changed the corresponding values to $\rho_q(\mathbf r)$26, $\rho_q(\mathbf r)$27, and $\rho_q(\mathbf r)$28. This single isovector term improved the total objective function by about $\rho_q(\mathbf r)$29 and reduced parameter correlations (Reinhard et al., 2024).

The fitted density-independent proton and neutron contact strengths in the 14D model become

$\rho_q(\mathbf r)$30

so enabling isovector pairing increases proton pairing strength while leaving neutron strength nearly unchanged. The spectral gaps

$\rho_q(\mathbf r)$31

show that going from 13D to 14D raises proton gaps and lowers neutron gaps somewhat, improving odd–even mass staggering and differential charge radii (Reinhard et al., 2024).

The principal limitation of zero-range Fayans-like pairing is numerical pathology in large model spaces. Contact pairing already requires regularization; adding gradient-density dependence aggravates ultraviolet and continuum sensitivity. In coordinate-space HFB, this shows up as strong dependence on pairing cutoff, box size, basis geometry, and marginal occupations. These deficiencies are particularly severe for Fayans-type gradient terms (Lalit et al., 11 Nov 2025).

A finite-range remedy has therefore been proposed by folding the anomalous density with a Gaussian kernel of range $\rho_q(\mathbf r)$32. In that construction,

$\rho_q(\mathbf r)$33

so the density and gradient modulation of the Fayans kernel is retained, but the pairing operator becomes nonlocal and ultraviolet convergent. The 2025 study concluded that a folding radius of about $\rho_q(\mathbf r)$34 offers the best compromise between quality and stability, and substantially reduces pathological behavior in different numerical applications (Lalit et al., 11 Nov 2025).

The resulting picture is two-sided. Fayans-like pairing is supported by its ability to describe low-lying collectivity, odd–even staggering, and especially differential charge radii through explicit surface and rearrangement physics. Yet the same surface focus can overenhance pairing in light nuclei, exaggerate arches in heavier chains, or become strongly cutoff dependent when implemented as a pure zero-range gradient functional. Current developments therefore move toward more faithful realizations: explicit isovector dependence, smoother regularization, and finite-range pairing while preserving the defining density- and gradient-sensitive character of the Fayans pairing channel (Reinhard et al., 2017, Reinhard et al., 2024, Lalit et al., 11 Nov 2025).