Coulomb and Proximity Potential Model

• Coulomb and Proximity Potential Model is a semi-classical framework that represents nuclear interactions through a repulsive Coulomb term, an attractive proximity term, and an optional centrifugal barrier.
• It is applied in heavy-ion fusion using the Wong formalism and in radioactive decay via WKB methods to evaluate barrier penetration and compare with experimental cross sections and half-lives.
• Different proximity potential variants (e.g., Prox.77, Prox.81, Guo2013) critically influence model predictions, highlighting its versatility and sensitivity across various nuclear processes.

The Coulomb and Proximity Potential Model (CPPM) is a semi-classical barrier-penetration framework in which the interaction between two nuclear fragments is represented by a repulsive Coulomb term, a short-range attractive proximity term, and, when required, a centrifugal term. In the literature represented here, CPPM is used in two broad settings: heavy-ion fusion, where the barrier is inserted into a one-dimensional barrier penetration model such as the Wong formalism, and radioactive decay, where tunneling through the barrier is treated by WKB methods for $\alpha$ decay, cluster decay, proton radioactivity, and two-proton radioactivity (Santhosh et al., 2013, Santhosh et al., 2012, Zhu et al., 2022).

1. Core definition and physical content

In its common form, CPPM writes the interaction barrier as

$V(r)=V_C(r)+V_P(z)+V_\ell(r),$

or explicitly,

$$V=\frac{Z_1 Z_2 e^2}{r}+V_p(z)+\frac{\hbar^2 \ell(\ell+1)}{2\mu r^2}.$$

Here $r$ is the center-to-center distance, $z$ is the near-surface separation, $Z_1$ and $Z_2$ are the fragment charges, $\mu$ is the reduced mass, and $\ell$ is the orbital angular momentum. The Coulomb and centrifugal terms are repulsive, whereas the proximity term is attractive. Fusion or decay is then interpreted as penetration through the resulting barrier (Santhosh et al., 2013).

The nuclear term is usually written in Blocki-type form,

$$V_p(z)=4\pi \gamma b\left(\frac{C_1 C_2}{C_1+C_2}\right)\Phi\!\left(\frac{z}{b}\right),$$

with surface energy coefficient

$V(r)=V_C(r)+V_P(z)+V_\ell(r),$0

Süssmann central radii

$V(r)=V_C(r)+V_P(z)+V_\ell(r),$1

and sharp radii

$V(r)=V_C(r)+V_P(z)+V_\ell(r),$2

Equivalent radius corrections also appear in the form $V(r)=V_C(r)+V_P(z)+V_\ell(r),$3, which is algebraically the same relation. These ingredients encode the near-surface geometry of the two fragments and make the attraction depend primarily on surface separation rather than on a fully microscopic many-body interaction (Santhosh et al., 2013, Zhu et al., 2022).

A commonly used universal function is the Prox.81 piecewise form

$V(r)=V_C(r)+V_P(z)+V_\ell(r),$4

Other studies instead use Prox.77-type universal functions, Woods-Saxon-like forms, Bass-family expressions, or logistic forms such as Guo2013. This suggests that CPPM is best understood as a family of related barrier models defined by a common decomposition of the interaction, rather than as a single immutable potential (Liu et al., 2024).

2. Barrier-penetration formalisms and computational workflow

For radioactive decay, CPPM is typically coupled to a one-dimensional WKB penetrability,

$V(r)=V_C(r)+V_P(z)+V_\ell(r),$5

with turning points determined by

$V(r)=V_C(r)+V_P(z)+V_\ell(r),$6

The decay constant is then written either as

$V(r)=V_C(r)+V_P(z)+V_\ell(r),$7

or, when explicit preformation is introduced,

$V(r)=V_C(r)+V_P(z)+V_\ell(r),$8

leading to

$V(r)=V_C(r)+V_P(z)+V_\ell(r),$9

In older cluster-decay applications, the overlap region $$V=\frac{Z_1 Z_2 e^2}{r}+V_p(z)+\frac{\hbar^2 \ell(\ell+1)}{2\mu r^2}.$$0 is described by a power-law interpolation $$V=\frac{Z_1 Z_2 e^2}{r}+V_p(z)+\frac{\hbar^2 \ell(\ell+1)}{2\mu r^2}.$$1, smoothly matched to the external barrier, so that both overlap and post-touching regions enter the tunneling action (Santhosh et al., 2012, Mageed et al., 2016).

A distinct implementation appears in heavy-ion fusion. There the barrier generated by Coulomb plus proximity plus centrifugal terms is inserted into a one-dimensional barrier penetration model, specifically the Wong formalism. The partial-wave transmission coefficient is taken in Hill-Wheeler form,

$$V=\frac{Z_1 Z_2 e^2}{r}+V_p(z)+\frac{\hbar^2 \ell(\ell+1)}{2\mu r^2}.$$2

and the total fusion cross section is

$$V=\frac{Z_1 Z_2 e^2}{r}+V_p(z)+\frac{\hbar^2 \ell(\ell+1)}{2\mu r^2}.$$3

Replacing the sum by an integral gives Wong’s expression

$$V=\frac{Z_1 Z_2 e^2}{r}+V_p(z)+\frac{\hbar^2 \ell(\ell+1)}{2\mu r^2}.$$4

which reduces at relatively large $$V=\frac{Z_1 Z_2 e^2}{r}+V_p(z)+\frac{\hbar^2 \ell(\ell+1)}{2\mu r^2}.$$5 to

$$V=\frac{Z_1 Z_2 e^2}{r}+V_p(z)+\frac{\hbar^2 \ell(\ell+1)}{2\mu r^2}.$$6

In that application, the barrier height $$V=\frac{Z_1 Z_2 e^2}{r}+V_p(z)+\frac{\hbar^2 \ell(\ell+1)}{2\mu r^2}.$$7 and barrier radius $$V=\frac{Z_1 Z_2 e^2}{r}+V_p(z)+\frac{\hbar^2 \ell(\ell+1)}{2\mu r^2}.$$8 are obtained from the maximum of the total potential curve, rather than from a closed analytic formula (Santhosh et al., 2013).

Two practical workflows therefore coexist within the CPPM literature. In one, the model is a full barrier-tunneling treatment with external and, in some studies, overlap-region contributions. In the other, the internal part is absorbed into an explicit preformation factor and the WKB integral starts at the touching configuration $$V=\frac{Z_1 Z_2 e^2}{r}+V_p(z)+\frac{\hbar^2 \ell(\ell+1)}{2\mu r^2}.$$9 (Liu et al., 2024). The distinction is methodological rather than terminological: both are called CPPM in the cited work.

3. Heavy-ion fusion implementation

In fusion studies, CPPM is used as a bare interaction model for generating the entrance-channel barrier. A representative case is the calculation of total fusion cross sections for

$r$0

with the explicit aim of comparing a weakly bound projectile, $r$1, against a tightly bound projectile, $r$2 (Santhosh et al., 2013).

The calculated cross sections, obtained from the Coulomb and proximity barrier together with one-dimensional barrier penetration, were compared both with experiment and with coupled-channel calculations using CCFULL. For $r$3, the tabulated theory values range from $r$4 mb at $r$5 MeV to $r$6 mb at $r$7 MeV, against experimental values from $r$8 mb to $r$9 mb. For $z$0, the theory values range from $z$1 mb at $z$2 MeV to $z$3 mb at $z$4 MeV, compared with $z$5 mb to $z$6 mb experimentally. For $z$7, the model gives $z$8 mb at $z$9 MeV and $Z_1$0 mb at $Z_1$1 MeV, compared with $Z_1$2 mb and $Z_1$3 mb (Santhosh et al., 2013).

The same study reports that with and without couplings there is almost no difference above the barrier in the CCFULL calculations. Reduced cross sections,

$Z_1$4

for $Z_1$5 and $Z_1$6 are described as “rather similar.” The stated interpretation is that breakup does not significantly affect total fusion cross sections above the barrier, static weak-binding effects are not strong above the barrier, and channel couplings are also not very important above the barrier (Santhosh et al., 2013).

Within this domain, CPPM is therefore a static bare-potential model. It does not explicitly include breakup channels, transfer channels, continuum coupling, or dynamic polarization effects. Its reported success is strongest in the near-barrier to above-barrier regime, especially above the Coulomb barrier (Santhosh et al., 2013).

4. Radioactivity applications

CPPM has been applied to cluster decay, $Z_1$7 decay, proton radioactivity, and two-proton radioactivity. In all of these cases, the emitted object and daughter nucleus are treated as a two-body system moving in an effective barrier; what changes from application to application is the treatment of preformation, the centrifugal term, and the choice of proximity potential (Mageed et al., 2016, Deng et al., 2019, Zhu et al., 2022).

For cluster decay of trans-lead nuclei, CPPM has been used for parent isotopic chains from Fr to Cm, with emitted clusters ranging from $Z_1$8 to $Z_1$9. In one implementation, the model includes both the overlap region and the separated region, evaluates WKB penetrability through the full barrier, and estimates the assault frequency through an empirical vibration energy,

$Z_2$0

These studies report good agreement with available experimental data and emphasize that decays leading to $Z_2$1 and neighboring $Z_2$2 daughters are especially favored, with the role of neutron shell closure described as more crucial than proton shell closure (Santhosh et al., 2012). A related application to even-even 5d transition-metal nuclei, Hf, W, Os, Pt, and Hg with $Z_2$3, also uses CPPM to calculate $Z_2$4- and heavier-cluster half-lives and reports that the calculated $Z_2$5-decay half-lives are in good agreement with experimental data, especially with the CPPM results (Mageed et al., 2016).

For proton radioactivity of spherical proton emitters, CPPM is formulated with a uniformly charged-sphere Coulomb potential and a Langer-modified centrifugal term,

$Z_2$6

and the half-life is written through $Z_2$7. In this context, the sensitivity to the inner turning point is decisive: if the total potential does not intersect $Z_2$8 on the inner side, $Z_2$9 cannot be obtained and the WKB calculation fails (Deng et al., 2019).

For two-proton radioactivity, CPPM is extended by treating the emitted two protons as a correlated cluster, effectively a $\mu$0He-like object. The total barrier is

$\mu$1

the penetrability is computed by WKB, and the decay constant is

$\mu$2

The preformation probability is taken as

$\mu$3

with candidate selection based on

$\mu$4

The model belongs explicitly to the category of correlated $\mu$5 emission rather than full three-body radioactivity (Zhu et al., 2022).

For $\mu$6-decay of superheavy nuclei, a temperature-dependent CPPM has been studied with 22 proximity-potential models. The practical barrier is

$\mu$7

because the centrifugal term is neglected. The half-life is written as

$\mu$8

with a shell-sensitive phenomenological preformation factor,

$\mu$9

using $\ell$0, $\ell$1, $\ell$2, and $\ell$3. Temperature is introduced phenomenologically through

$\ell$4

together with a $\ell$5- and $\ell$6-dependent diffuseness

$\ell$7

(Qi et al., 2 Apr 2025).

5. Proximity-potential variants and comparative performance

A central development in the recent CPPM literature is the systematic replacement of the nuclear term $\ell$8 by different proximity-potential prescriptions. The cited studies examine Prox.77 and modified Prox.77 forms, Prox.81, Prox.00, Prox.00 DP, Prox.2010, Dutt2011, Bass73, Bass77, Bass80, CW76, BW91, AW95, Ngô80, Denisov, Denisov DP, and Guo2013 (Deng et al., 2019, Liu et al., 2024, Qi et al., 2 Apr 2025).

Domain	Variants compared	Reported best result
Proton radioactivity of spherical emitters	28 proximity versions	Guo2013, $\ell$9 (Deng et al., 2019)
Cluster radioactivity in trans-lead nuclei	28 proximity versions	Prox.77-12 and Prox.81, both $$V_p(z)=4\pi \gamma b\left(\frac{C_1 C_2}{C_1+C_2}\right)\Phi\!\left(\frac{z}{b}\right),$$0 (Liu et al., 2024)
$$V_p(z)=4\pi \gamma b\left(\frac{C_1 C_2}{C_1+C_2}\right)\Phi\!\left(\frac{z}{b}\right),$$1 decay of superheavy nuclei	22 proximity models with temperature effects	Prox.77-3 T-DEP, $$V_p(z)=4\pi \gamma b\left(\frac{C_1 C_2}{C_1+C_2}\right)\Phi\!\left(\frac{z}{b}\right),$$2 (Qi et al., 2 Apr 2025)

The proton-radioactivity comparison is especially stringent. Out of 28 proximity-potential versions, 21 are reported to be unsuitable because the depth of the total interaction potential remains above the proton radioactivity energy, so the classical inner turning point cannot be found. Only seven versions are usable in that CPPM framework: Bass73, CW76, Denisov, Denisov DP, Guo2013, Prox.00 DP, and Prox.2010. Among them, Guo2013 gives the lowest rms deviation,

$$V_p(z)=4\pi \gamma b\left(\frac{C_1 C_2}{C_1+C_2}\right)\Phi\!\left(\frac{z}{b}\right),$$3

compared with $$V_p(z)=4\pi \gamma b\left(\frac{C_1 C_2}{C_1+C_2}\right)\Phi\!\left(\frac{z}{b}\right),$$4 for Denisov, $$V_p(z)=4\pi \gamma b\left(\frac{C_1 C_2}{C_1+C_2}\right)\Phi\!\left(\frac{z}{b}\right),$$5 for Bass73, $$V_p(z)=4\pi \gamma b\left(\frac{C_1 C_2}{C_1+C_2}\right)\Phi\!\left(\frac{z}{b}\right),$$6 for CW76, $$V_p(z)=4\pi \gamma b\left(\frac{C_1 C_2}{C_1+C_2}\right)\Phi\!\left(\frac{z}{b}\right),$$7 for Denisov DP, $$V_p(z)=4\pi \gamma b\left(\frac{C_1 C_2}{C_1+C_2}\right)\Phi\!\left(\frac{z}{b}\right),$$8 for Prox.00 DP, and $$V_p(z)=4\pi \gamma b\left(\frac{C_1 C_2}{C_1+C_2}\right)\Phi\!\left(\frac{z}{b}\right),$$9 for Prox.2010 (Deng et al., 2019).

The cluster-radioactivity comparison leads to a different ranking. In the trans-lead benchmark of 26 known cluster decays, the best-performing potentials are Prox.77-12 and Prox.81, both with

$V(r)=V_C(r)+V_P(z)+V_\ell(r),$00

Other relatively strong performers include Prox.77-1 with $V(r)=V_C(r)+V_P(z)+V_\ell(r),$01, Prox.77-2 with $V(r)=V_C(r)+V_P(z)+V_\ell(r),$02, Prox.77-8 with $V(r)=V_C(r)+V_P(z)+V_\ell(r),$03, Prox.77-13 with $V(r)=V_C(r)+V_P(z)+V_\ell(r),$04, Prox.77-11 with $V(r)=V_C(r)+V_P(z)+V_\ell(r),$05, Bass80 with $V(r)=V_C(r)+V_P(z)+V_\ell(r),$06, and Ngô80 with $V(r)=V_C(r)+V_P(z)+V_\ell(r),$07. Poorer performers include Prox.00 with $V(r)=V_C(r)+V_P(z)+V_\ell(r),$08, AW95 with $V(r)=V_C(r)+V_P(z)+V_\ell(r),$09, Denisov with $V(r)=V_C(r)+V_P(z)+V_\ell(r),$10, BW91 with $V(r)=V_C(r)+V_P(z)+V_\ell(r),$11, CW76 with $V(r)=V_C(r)+V_P(z)+V_\ell(r),$12, Prox.2010 with $V(r)=V_C(r)+V_P(z)+V_\ell(r),$13, Bass73 with $V(r)=V_C(r)+V_P(z)+V_\ell(r),$14, Guo2013 with $V(r)=V_C(r)+V_P(z)+V_\ell(r),$15, Dutt2011 with $V(r)=V_C(r)+V_P(z)+V_\ell(r),$16, Prox.00 DP with $V(r)=V_C(r)+V_P(z)+V_\ell(r),$17, and Denisov DP with $V(r)=V_C(r)+V_P(z)+V_\ell(r),$18 (Liu et al., 2024).

For superheavy $V(r)=V_C(r)+V_P(z)+V_\ell(r),$19 decay, the best CPPM result is Prox.77-3 in temperature-dependent form,

$V(r)=V_C(r)+V_P(z)+V_\ell(r),$20

improving upon

$V(r)=V_C(r)+V_P(z)+V_\ell(r),$21

The same study also reports improvement for Prox.77-6, $V(r)=V_C(r)+V_P(z)+V_\ell(r),$22, and Prox.77-7, $V(r)=V_C(r)+V_P(z)+V_\ell(r),$23, whereas models such as Bass77, AW95, Ngô80, and Guo2013 remain comparatively poor. Ni’s empirical formula gives $V(r)=V_C(r)+V_P(z)+V_\ell(r),$24, slightly smaller than the best CPPM value in that benchmark (Qi et al., 2 Apr 2025).

A recurring lesson is that success is application-dependent. Prox.81 is unsuitable in the proton-radioactivity benchmark because it can fail the turning-point requirement, yet it is one of the two best performers in the cluster-radioactivity benchmark and is also used successfully in heavy-ion fusion and two-proton radioactivity studies (Deng et al., 2019, Liu et al., 2024, Santhosh et al., 2013, Zhu et al., 2022). This suggests that the short-distance barrier geometry required for one observable need not transfer unchanged to another.

6. Assumptions, limitations, and interpretive issues

CPPM is consistently presented as a physically transparent but approximate model. Its most common assumptions are spherical geometry, one-dimensional barrier penetration, and a phenomenological treatment of cluster formation or internal dynamics. Depending on the application, deformation, multidimensional couplings, breakup, transfer, continuum effects, configuration mixing, and explicit three-body asymptotics are omitted (Santhosh et al., 2013, Zhu et al., 2022, Santhosh et al., 2012).

In fusion, the model is explicitly a static bare-potential description and does not include breakup channels, transfer channels, continuum coupling, or dynamic polarization effects. Its strongest validity in the cited work is above the barrier, where the Coulomb-plus-proximity barrier reproduces measured total fusion cross sections reasonably well and where coupled-channel corrections were reported to make almost no difference (Santhosh et al., 2013).

In decay problems, preformation is not handled uniformly across the literature. Some CPPM studies absorb the internal barrier or overlap-region physics into the WKB action by introducing an overlap potential for $V(r)=V_C(r)+V_P(z)+V_\ell(r),$25 (Santhosh et al., 2012). Others instead factorize the decay constant as $V(r)=V_C(r)+V_P(z)+V_\ell(r),$26 or $V(r)=V_C(r)+V_P(z)+V_\ell(r),$27 and interpret $V(r)=V_C(r)+V_P(z)+V_\ell(r),$28 or $V(r)=V_C(r)+V_P(z)+V_\ell(r),$29 as a preformation probability at the nuclear surface (Liu et al., 2024, Zhu et al., 2022). This difference is substantive: it changes which part of the decay physics is assigned to barrier penetrability and which part is assigned to an external phenomenological factor.

Several papers explicitly identify missing physics as a source of discrepancies. In two-proton radioactivity, deformation effects, three-body asymptotic behavior, and configuration mixing are suggested as likely reasons for mismatches, especially for $V(r)=V_C(r)+V_P(z)+V_\ell(r),$30Kr (Zhu et al., 2022). In cluster radioactivity, the strong sensitivity of half-lives to the touching-region potential means that modest changes in radii or universal functions can shift the WKB action enough to alter $V(r)=V_C(r)+V_P(z)+V_\ell(r),$31 by many orders of magnitude (Liu et al., 2024). In superheavy $V(r)=V_C(r)+V_P(z)+V_\ell(r),$32 decay, temperature corrections to radius, surface energy, and diffuseness are introduced phenomenologically rather than from a microscopic thermal derivation (Qi et al., 2 Apr 2025).

A common misconception is that “the” CPPM implies a unique nuclear potential. The benchmark studies contradict that view directly: the nuclear term may be taken from 22 or 28 distinct proximity-potential prescriptions, and the ranking of those prescriptions depends strongly on whether the observable is fusion, cluster radioactivity, proton radioactivity, two-proton radioactivity, or superheavy $V(r)=V_C(r)+V_P(z)+V_\ell(r),$33 decay (Deng et al., 2019, Liu et al., 2024, Qi et al., 2 Apr 2025). Another misconception is that success in a bare-barrier calculation eliminates the relevance of couplings or structure effects. The cited results are more specific: above-barrier fusion may be described reasonably well without explicit breakup or channel-coupling dynamics, but the same papers do not claim such sufficiency for all energies or all observables (Santhosh et al., 2013).

Taken together, the cited works define CPPM as a broad semi-classical methodology centered on the near-surface nuclear barrier. Its durable value lies in the combination of a parameterized short-range proximity attraction, standard Coulomb geometry, and analytically tractable tunneling formalisms. Its limitations arise from the same source: the model is intentionally reduced, and its predictive quality depends sensitively on how the effective barrier is parameterized for the specific nuclear process under study.