Extended Double-Folding Model

• The Extended Double-Folding Model is a semi-microscopic approach that computes effective nuclear potentials by double folding projectile and target densities with effective NN interactions.
• It augments baseline Hartree–Fock or M3Y schemes with corrections such as density dependence, rearrangement effects, local momentum modifications, and complex absorption terms.
• It finds practical applications in heavy-ion scattering, fusion, alpha decay, and cluster spectroscopy, where renormalization and modern density inputs enhance modeling precision.

The extended double-folding model is a family of semi-microscopic and microscopic constructions for nucleus–nucleus or $\alpha$-core interactions in which the effective potential is obtained by folding projectile and target densities with an effective nucleon–nucleon kernel, and then augmenting that baseline prescription with ingredients absent from simpler Hartree–Fock-level or M3Y-only schemes. In the current literature, “extended” does not denote a single universally standardized formalism. It may instead refer to density dependence, rearrangement effects required by the Hugenholtz–van Hove theorem, local momentum dependence, RMF-derived R3Y kernels, chiral EFT interactions, dispersive or derivative-generated imaginary parts, modern EDF density inputs, or constrained renormalization factors used in spectroscopy, scattering, fusion, and $\alpha$-decay calculations (Khoa et al., 2016, Durant et al., 2020, Mohr, 2017, Yahya et al., 2021).

1. Terminological scope and representative variants

The common core of all variants is the double convolution of two nuclear densities with an effective $NN$ interaction. What changes from one “extended” model to another is the status of the interaction kernel, the treatment of exchange and medium effects, the density input, and the way the folded potential is embedded into a reaction or structure framework. The literature therefore uses the phrase in a plural rather than singular sense.

Extension axis	Representative modification	Representative application
In-medium mean field	$F_0(\rho)\to F_0(\rho)+\Delta F_0(\rho)$, with local $g(k)$	Rainbow scattering in $^{12}\mathrm C+^{12}\mathrm C$, $^{16}\mathrm O+^{12}\mathrm C$
Density-dependent $\alpha$-daughter folding	$F(\rho_1,\rho_2)$ in the folding kernel; RMF-derived R3Y forces	$\alpha$-decay of $\alpha$0
Chiral EFT kernel replacement	Local chiral EFT interactions at LO, NLO, N$\alpha$1LO or N$\alpha$2LO-only implementations	Heavy-ion scattering/fusion; $\alpha$3-cluster structure
Semi-microscopic renormalization	$\alpha$4, $\alpha$5, or $\alpha$6	$\alpha$7 cluster bands and $\alpha$8 scattering
Complex folded optical potential	Dispersive $\alpha$9 or derivative-generated imaginary term	Heavy-ion elastic scattering; $NN$0He-induced fusion
Density modernization	M3Y folding with EDF matter densities	$NN$1 elastic scattering

A recurrent misconception is that “extended” must imply a new formal double-folding theory. In several cases the formal direct-plus-exchange structure is left intact, while the extension is instead the microscopic origin of the interaction, the consistency of the in-medium mean field, or the quality of the density and absorptive input (Durant et al., 2017, Heo et al., 2024).

2. Core formalism and standard points of extension

The baseline folded potential is usually decomposed as

$NN$2

with a direct term

$NN$3

and a localized exchange term of the form

$NN$4

The local relative momentum is determined self-consistently from

$NN$5

This structure is explicit in heavy-ion chiral-EFT folding calculations and in $NN$6-core folding from chiral EFT. In those implementations, the nonlocal exchange term is rendered local through a local-momentum prescription together with a density-matrix expansion or realistic localization approximation. The heavy-ion chiral-EFT work retains only the leading density-matrix-expansion term, while the $NN$7-cluster study follows the localization strategy associated with Khoa and collaborators (Durant et al., 2017, Bai et al., 2021).

A second, simpler branch of the formalism replaces the nonlocal exchange integral by a zero-range pseudopotential $NN$8. That approximation is used in the Po $NN$9-decay calculations with M3Y and R3Y kernels and in the $F_0(\rho)\to F_0(\rho)+\Delta F_0(\rho)$0He-fusion SRTM implementation. In these cases the nuclear part of the effective interaction remains finite-range in its direct Yukawa terms, but exchange is collapsed into a contact term (Yahya et al., 2021, Mohammad et al., 11 Dec 2025).

The density input is equally variable. The cited literature uses Gaussian $F_0(\rho)\to F_0(\rho)+\Delta F_0(\rho)$1-particle densities, two-parameter Fermi forms, phenomenological densities from the São Paulo group, empirical electron-scattering densities in Fourier-Bessel or sum-of-Gaussians form, TALYS densities, and self-consistent EDF densities. Accordingly, the formalism is often only partially microscopic: the interaction may be derived from RMF or chiral EFT, while the densities remain empirical or parametrized (Bai et al., 2021, Heo et al., 2024).

3. In-medium extensions: density dependence, rearrangement, and local momentum

One major sense of “extended” is the explicit treatment of medium effects. In the Po $F_0(\rho)\to F_0(\rho)+\Delta F_0(\rho)$2-decay study, the nuclear potential is written as

$F_0(\rho)\to F_0(\rho)+\Delta F_0(\rho)$3

with density-independent DD0 corresponding to $F_0(\rho)\to F_0(\rho)+\Delta F_0(\rho)$4, and density dependence introduced through

$F_0(\rho)\to F_0(\rho)+\Delta F_0(\rho)$5

The DDM3Y1 parameter sets used are quoted explicitly for Reid and Paris forms, and the calculations show systematic improvement when density dependence is included rather than omitted (Yahya et al., 2021).

A more formal in-medium extension is the rearrangement-consistent heavy-ion DFM based on CDM3Y3 and CDM3Y6. There the effective interaction is modified as

$F_0(\rho)\to F_0(\rho)+\Delta F_0(\rho)$6

where $F_0(\rho)\to F_0(\rho)+\Delta F_0(\rho)$7 encodes the rearrangement term derived from the Hugenholtz–van Hove theorem, and $F_0(\rho)\to F_0(\rho)+\Delta F_0(\rho)$8 is a local momentum-dependent factor fitted to the empirical energy dependence of the nucleon optical potential. In this formulation the rearrangement contribution is repulsive, grows with overlap density, and affects both direct and exchange folding terms. Its impact is strongest at small internuclear distances, where the local density is highest under the frozen density approximation (Khoa et al., 2016).

The numerical consequences are substantial. For $F_0(\rho)\to F_0(\rho)+\Delta F_0(\rho)$9 at 240 MeV and $g(k)$0 at 200 MeV, the rearrangement contribution reaches about $g(k)$1–$g(k)$2 of the potential strength at the smallest radii. In optical-model fits, the renormalization factor of the real folded potential moves from $g(k)$3–$g(k)$4 to $g(k)$5–$g(k)$6 for $g(k)$7, and from $g(k)$8–$g(k)$9 to $^{12}\mathrm C+^{12}\mathrm C$0–$^{12}\mathrm C+^{12}\mathrm C$1 for $^{12}\mathrm C+^{12}\mathrm C$2. In that literature, the extension is therefore not merely lexical: it directly cures the standard folded potential’s tendency to be too deep in the interior (Khoa et al., 2016).

A related but more limited medium-sensitive construction appears in the density-dependent double-folding complex potential used within SRTM. There the authors label the interaction “DDM3Y-Reid,” but do not print an explicit density-overlap multiplier $^{12}\mathrm C+^{12}\mathrm C$3. The only explicit density dependence is through the folding over projectile and target matter densities, while the imaginary term is generated phenomenologically from the radial derivative of the real folded potential. This suggests a semi-microscopic rather than fully documented DDM3Y implementation (Mohammad et al., 11 Dec 2025).

4. Modern microscopic kernels and upgraded density inputs

Another major extension axis is the replacement of traditional M3Y-like kernels by interactions with a more explicit microscopic origin. One route proceeds through RMF-derived R3Y interactions. In the Po $^{12}\mathrm C+^{12}\mathrm C$4-decay calculations, the effective force is tied to RMF meson exchange, with $^{12}\mathrm C+^{12}\mathrm C$5, $^{12}\mathrm C+^{12}\mathrm C$6, and $^{12}\mathrm C+^{12}\mathrm C$7 contributions plus a zero-range exchange term, and four parameterizations—R3Y-L1, R3Y-W, R3Y-Z, and R3Y-HS—are tested against M3Y-Paris and M3Y-Reid. In that setting the “extended” character is the substitution of RMF-derived Yukawa kernels for conventional phenomenological M3Y ones, together with optional density dependence (Yahya et al., 2021).

A second route is the use of local chiral EFT interactions. Heavy-ion double-folding potentials have been built from soft local chiral EFT forces order by order at LO, NLO, and N$^{12}\mathrm C+^{12}\mathrm C$8LO, with coordinate-space cutoffs $^{12}\mathrm C+^{12}\mathrm C$9 fm and spectral-function regularization $^{16}\mathrm O+^{12}\mathrm C$0 MeV. The key finding is that sufficiently soft interactions can generate realistic folded potentials at Hartree–Fock level, whereas harder local interactions become repulsive unless missing many-body correlations are restored (Durant et al., 2017).

The chiral-EFT extension is carried further in heavy-ion optical potentials with dispersive imaginary parts and in $^{16}\mathrm O+^{12}\mathrm C$1-cluster semi-microscopic models. For $^{16}\mathrm O+^{12}\mathrm C$2, $^{16}\mathrm O+^{12}\mathrm C$3, and $^{16}\mathrm O+^{12}\mathrm C$4, the real part is built from local chiral N$^{16}\mathrm O+^{12}\mathrm C$5LO $^{16}\mathrm O+^{12}\mathrm C$6 interactions with cutoffs $^{16}\mathrm O+^{12}\mathrm C$7 fm, using only two-body forces. For $^{16}\mathrm O+^{12}\mathrm C$8-cluster structure above double shell closures, the $^{16}\mathrm O+^{12}\mathrm C$9-core potentials are constructed from local chiral N$\alpha$0LO interactions with $\alpha$1 fm, again omitting three-nucleon forces for simplicity (Durant et al., 2020, Bai et al., 2021).

Density modernization forms a third branch of extension. The $\alpha$2 study retains a traditional M3Y kernel but replaces phenomenological densities by four EDF density sets—SLy4, KIDS0, QHD, and QMC. The matter densities agree closely in the surface region and differ mainly in the interior; correspondingly, the folded potentials are nearly identical outside the overlap region but differ in their interior depth, with the QMC density producing the largest deviation. For that system, the folded potential is also weakly energy dependent because the only energy dependence enters through the coefficient of the zero-range exchange term (Heo et al., 2024).

5. Renormalization, localization, and complex-potential closures

Extended folding models almost never stop at the bare convolution. They are typically closed by constrained renormalization, semiclassical quantization, or an explicit complex optical-potential construction.

In $\alpha$3-cluster spectroscopy, the bound-state potential is often taken as

$\alpha$4

with a smooth angular-momentum dependence

$\alpha$5

when a constant $\alpha$6 compresses the rotational spectrum. For $\alpha$7, the fitted variation of $\alpha$8 stays below $\alpha$9 across the band; for $F(\rho_1,\rho_2)$0, below $F(\rho_1,\rho_2)$1. In $F(\rho_1,\rho_2)$2 scattering the real part is further refined to $F(\rho_1,\rho_2)$3, with $F(\rho_1,\rho_2)$4 constrained to remain very close to unity. This near-unity width scaling is used as a small correction to the radial extent of the folded real potential (Mohr, 2017).

In $F(\rho_1,\rho_2)$5-decay the closure is instead semiclassical. The effective barrier is

$F(\rho_1,\rho_2)$6

with the Langer replacement $F(\rho_1,\rho_2)$7. The quantization factor $F(\rho_1,\rho_2)$8 is fixed by the Bohr–Sommerfeld condition together with the Wildermuth rule rather than fitted freely, and the half-life is then obtained from

$F(\rho_1,\rho_2)$9

This procedure embeds the folded nuclear attraction into a WKB tunneling problem that also contains finite-size Coulomb and centrifugal barriers (Yahya et al., 2021).

For optical-potential applications, the extension often means adding absorption. In the dispersive chiral-EFT heavy-ion model, the complex folded potential is

$\alpha$0

and the imaginary part is generated from the energy dependence of the exchange term through

$\alpha$1

After factorizing $\alpha$2, the radial shape of the absorptive part is inherited from the exchange term itself rather than from an independent fitted Woods–Saxon geometry (Durant et al., 2020).

A different semi-microscopic complexification appears in SRTM, where

$\alpha$3

The real part is folded, but the imaginary part is not independently folded from a separate $\alpha$4 interaction; it is generated from the surface derivative of the real folded potential. This remains a semi-microscopic prescription because $\alpha$5, $\alpha$6, and the radius parameter $\alpha$7 are still adjustable (Mohammad et al., 11 Dec 2025).

6. Applications, performance, and limitations

The most systematic $\alpha$8-decay benchmark in the cited literature concerns $\alpha$9 Po isotopes, $\alpha$00. For density-independent DD0 calculations with $\alpha$01, the root-mean-square deviations in $\alpha$02 are $\alpha$03 for M3Y-Paris and $\alpha$04 for M3Y-Reid, versus $\alpha$05 for R3Y-HS, $\alpha$06 for R3Y-L1, $\alpha$07 for R3Y-W, and $\alpha$08 for R3Y-Z. With an empirical preformation factor, the DD0 R3Y deviations become $\alpha$09, $\alpha$10, $\alpha$11, and $\alpha$12, and after adding DDM3Y1 density dependence they improve further to $\alpha$13, $\alpha$14, $\alpha$15, and $\alpha$16, with R3Y-Z marginally best. The maximum half-life occurs at $\alpha$17, i.e. $\alpha$18, while the minimum occurs at $\alpha$19, associated with daughter $\alpha$20 and the $\alpha$21 shell effect (Yahya et al., 2021).

In refractive heavy-ion scattering, the rearrangement-consistent extended DFM is most notable for its interior corrections. At 240 MeV in $\alpha$22, the first Airy minimum moves from about $\alpha$23 with the deeper HF folded potential to about $\alpha$24 with the HF+RT potential, compared with the observed value near $\alpha$25. At 200 MeV in $\alpha$26, the predicted first Airy minimum moves from about $\alpha$27 to about $\alpha$28, and the renormalization factor of the real part becomes close to unity. This is the clearest evidence in the cited literature that the rearrangement term is not a marginal correction but a structural ingredient of the folded interior potential (Khoa et al., 2016).

The chiral-EFT heavy-ion program shows a different kind of success. Double-folded real parts from local chiral interactions, combined with a dispersively constrained imaginary part, describe elastic scattering in $\alpha$29, $\alpha$30, and $\alpha$31 up to about $\alpha$32 MeV and also reproduce low-energy fusion $\alpha$33-factors when realistic electron-scattering densities are used. A central conclusion is that density choice is comparatively more important for fusion than cutoff variation, and that the dispersive construction removes the earlier ad hoc proportional-absorption parameter $\alpha$34 from the final optical potential (Durant et al., 2020).

In cluster spectroscopy, the DDM3Y-based extended folding model for $\alpha$35 gives reasonable excitation energies, reduced widths, intercluster separations, and $\alpha$36 values, while the associated $\alpha$37 scattering analysis indicates that the neutron distribution of $\alpha$38 is more radially extended than the charge distribution, since a folded potential built from the empirical charge density must be enlarged by about $\alpha$39 in width whereas the TALYS density requires essentially no width correction. The chiral-EFT $\alpha$40-cluster program extends this line of work across $\alpha$41, $\alpha$42, $\alpha$43, $\alpha$44, and $\alpha$45, with fitted $\alpha$46 values generally close to $\alpha$47, and with the $\alpha$48 $\alpha$49 mismatch interpreted as evidence that this state is likely shell-model dominated rather than a good $\alpha$50-cluster state (Mohr, 2017, Bai et al., 2021).

The $\alpha$51He-induced fusion study illustrates both the utility and the limits of semi-microscopic extensions. Replacing the complex square-well input of earlier SRTM work by a density-based folded real part plus derivative-generated absorption improves the physical plausibility of the model and yields good agreement for several tabulated points, especially in $\alpha$52. However, the low-energy discrepancies are not uniformly small: for $\alpha$53 at $\alpha$54 keV the quoted experimental cross section is $\alpha$55 mb while the calculation gives $\alpha$56 mb, and for $\alpha$57 at $\alpha$58 keV the quoted experimental value is $\alpha$59 mb versus a calculated $\alpha$60 mb (Mohammad et al., 11 Dec 2025).

The main limitations are consistent across the literature. Exchange is often treated by a zero-range $\alpha$61 term or localized only at leading order in the density-matrix expansion; three-nucleon forces are omitted in the chiral-EFT implementations; densities are frequently empirical or phenomenological rather than generated from the same Hamiltonian as the interaction; the imaginary part remains Woods–Saxon, proportional, dispersive, or derivative-generated rather than fully microscopic; frozen-density prescriptions may be inappropriate at lower energies where adiabatic overlap is more natural; deformation and explicit coupled-channel effects are often absent; and formal uncertainty quantification remains limited. The phrase “extended double-folding model” should therefore be understood as denoting a hierarchy of improvements over the conventional folding prescription, not a uniformly self-consistent ab initio optical-potential theory (Durant et al., 2017, Heo et al., 2024).