EFwT Metric: Threshold in Sub-Barrier Fusion

• EFwT metric is defined as the center-of-mass energy at which short-range nuclear forces switch off, marking the onset of deep sub-barrier fusion hindrance.
• It unifies empirical observations—such as a dramatic increase in the logarithmic slope of capture cross sections—with theoretical criteria like the interaction 'turn-off' energy.
• By integrating nuclear potential scaling and quantum-diffusion dynamics, the EFwT metric provides a compact prescription for predicting fusion outcomes in heavy-ion reactions.

The Energy and Forces within Threshold (EFwT) metric is a quantitative prescription for characterizing the onset of deep sub-barrier fusion hindrance in heavy-ion reactions. Defined as the center-of-mass energy at which the short-range nuclear forces cease to contribute to the fusion process, EFwT unifies the empirically observed hindrance threshold with a theoretically motivated interaction "turn-off" energy. This metric provides a physical link between the disappearance of nuclear friction, the structure of the ion-ion potential, and the observed dramatic reduction in fusion (capture) cross sections far below the Coulomb barrier (Sargsyan et al., 2012).

1. Definition of the Sub-Barrier Fusion Hindrance Threshold

The deep sub-barrier fusion hindrance threshold, denoted $E_{\rm th}$ (or $E_s$ in some literature), refers to the energy below which measured fusion (capture) cross sections $\sigma_{\rm cap}(E)$ decline much more steeply than predicted by conventional coupled-channels models or standard barrier-penetration estimates such as the Hill-Wheeler formula. Mathematically, the logarithmic slope

$S(E) = \frac{d\ln\sigma_{\rm cap}}{dE}$

rises significantly above the usual barrier-penetration value $S_{\rm standard}\approx 2\pi/(\hbar\omega_b)$, where $\omega_b$ is the barrier curvature at its apex. Physically, $E_{\rm th}$ marks the regime in which the outer classical turning point of the relative motion, $R_{\rm ex}(E)$, moves outside the radius $R_{\rm int}$ where the nuclear forces and friction become active. For $R_{\rm ex} > R_{\rm int}$, only the long-range Coulomb repulsion and centrifugal barrier contribute, leading to diminished coupling to internal nuclear degrees of freedom and suppressed capture probability.

2. Interaction "Turn-Off" Energy and Underlying Potentials

The interaction "turn-off" energy, $E_s$0, is defined as the center-of-mass energy for which the external turning point satisfies $E_s$1, where the interaction radius is given by

$E_s$2

with $E_s$3 denoting the top-of-barrier radius (typically for $E_s$4). For $E_s$5, the short-range nuclear potential $E_s$6 is negligible, and the system is governed by the Coulomb and centrifugal terms alone:

$E_s$7

In practice, since $E_s$8 is small at $E_s$9, a good approximation is

$\sigma_{\rm cap}(E)$0

where $\sigma_{\rm cap}(E)$1 are the atomic numbers of the colliding nuclei.

3. Empirical Correspondence and Scaling Properties

Across a broad range of medium-light and heavy-ion systems, measurements reveal that $\sigma_{\rm cap}(E)$2 within uncertainties. Both energies exhibit a strong linear dependence on the scaling parameter

$\sigma_{\rm cap}(E)$3

where $\sigma_{\rm cap}(E)$4 are mass numbers. Empirical fits yield

$\sigma_{\rm cap}(E)$5

with $\sigma_{\rm cap}(E)$6. This proportionality offers a compact parametrization for predicting the hindrance threshold in various systems and highlights the centrality of electrostatic and mass factors.

4. Quantum-Diffusion Framework for Capture Dynamics

The quantum-diffusion approach models the relative motion of two nuclei in terms of a quantum-corrected Langevin or Fokker-Planck equation:

• The coordinate evolves as $\sigma_{\rm cap}(E)$7.
• The momentum obeys

$\sigma_{\rm cap}(E)$8

where $\sigma_{\rm cap}(E)$9 is the reduced mass, $S(E) = \frac{d\ln\sigma_{\rm cap}}{dE}$0 includes nuclear, Coulomb, and centrifugal contributions, $S(E) = \frac{d\ln\sigma_{\rm cap}}{dE}$1 is the non-Markovian friction kernel, and $S(E) = \frac{d\ln\sigma_{\rm cap}}{dE}$2 is a quantum Langevin force with statistics governed by the fluctuation-dissipation theorem.

Capture probability for partial wave $S(E) = \frac{d\ln\sigma_{\rm cap}}{dE}$3 is given by

$S(E) = \frac{d\ln\sigma_{\rm cap}}{dE}$4

with $S(E) = \frac{d\ln\sigma_{\rm cap}}{dE}$5 the distribution of barrier heights from coupling and $S(E) = \frac{d\ln\sigma_{\rm cap}}{dE}$6 a transmission factor (e.g., Hill–Wheeler, WKB). The total capture cross section is then

$S(E) = \frac{d\ln\sigma_{\rm cap}}{dE}$7

5. Behavior of Cross Sections and Slope Transitions

Just below the Coulomb barrier, $S(E) = \frac{d\ln\sigma_{\rm cap}}{dE}$8 can be approximated by the Hill–Wheeler formula, giving a nearly constant logarithmic slope $S(E) = \frac{d\ln\sigma_{\rm cap}}{dE}$9. However, for energies $S_{\rm standard}\approx 2\pi/(\hbar\omega_b)$0, with $S_{\rm standard}\approx 2\pi/(\hbar\omega_b)$1 and $S_{\rm standard}\approx 2\pi/(\hbar\omega_b)$2 over most of the forbidden region, the effective barrier curvature is reduced, the barrier becomes broader, and $S_{\rm standard}\approx 2\pi/(\hbar\omega_b)$3 increases sharply. This sudden change in slope—fusion hindrance—is the phenomenological indicator of the EFwT regime.

6. Diagnostic Role of Quasielastic Barrier Distributions

Quasielastic barrier distributions $S_{\rm standard}\approx 2\pi/(\hbar\omega_b)$4 provide direct experimental access to the transition at $S_{\rm standard}\approx 2\pi/(\hbar\omega_b)$5:

$S_{\rm standard}\approx 2\pi/(\hbar\omega_b)$6

with $S_{\rm standard}\approx 2\pi/(\hbar\omega_b)$7 calculated from backward-angle quasielastic and Rutherford differential cross sections. Changes in the width or a discontinuity (kink) in $S_{\rm standard}\approx 2\pi/(\hbar\omega_b)$8 signal the onset of sub-barrier hindrance and thus indicate the EFwT energy.

7. The EFwT Metric: Unified Prescription

The Energy-and-Forces-within-Threshold metric packages the experimental and theoretical criteria into a single construct:

$S_{\rm standard}\approx 2\pi/(\hbar\omega_b)$9

• $\omega_b$0 is the experimentally observed hindrance threshold.
• $\omega_b$1 is the energy at which the classical turning point coincides with $\omega_b$2.
• $\omega_b$3 and $\omega_b$4 are the nuclear and Coulomb potentials, respectively.

The EFwT value thus marks the energy at which the short-range nuclear interaction (and accompanying dissipative processes) effectively switch off, and fusion ceases to follow the standard barrier-penetration trend. Practically, computation of EFwT involves evaluating $\omega_b$5 and $\omega_b$6 for the chosen system, offsetting the radius by $\omega_b$7, and calculating the total potential at this interaction cutoff. The empirical concordance between EFwT and the observed $\omega_b$8 affirms the quantum-diffusion picture of hindrance as a consequence of the absence of nuclear forces beyond $\omega_b$9. EFwT consequently provides an operational criterion bridging theoretical dynamics and experimental measurement in sub-barrier fusion studies (Sargsyan et al., 2012).