---
title: 'EFwT Metric: Threshold in Sub-Barrier Fusion'
url: https://www.emergentmind.com/topics/energy-and-forces-within-threshold-efwt-metric
type: topic
---

# EFwT Metric: Threshold in Sub-Barrier Fusion

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 [1211.2312].

## 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, $E_{\rm off}$, is defined as the center-of-mass energy for which the external turning point satisfies $R_{\rm ex}(E_{\rm off})=R_{\rm int}$,
where the interaction radius is given by
$$
R_{\rm int} = R_b + \Delta R, \quad \Delta R \approx 1.1\,\rm fm,
$$
with $R_b$ denoting the top-of-barrier radius (typically for $J=0$). For $R>R_{\rm int}$, the short-range nuclear potential $V_N(R)$ is negligible, and the system is governed by the Coulomb and centrifugal terms alone:
$$
E_{\rm off} = V(R_{\rm int}) = V_C(R_{\rm int}) + V_N(R_{\rm int}).
$$
In practice, since $V_N$ is small at $R_{\rm int}$, a good approximation is
$$
E_{\rm off} \approx \frac{Z_1 Z_2 e^2}{R_b + 1.1\,\rm fm},
$$
where $Z_{1,2}$ 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 $E_{\rm th} \simeq E_{\rm off}$ within uncertainties. Both energies exhibit a strong linear dependence on the scaling parameter
$$
X = Z_1Z_2 \sqrt{\frac{A_1 A_2}{A_1 + A_2}},
$$
where $A_{1,2}$ are mass numbers. Empirical fits yield
$$
E_{\rm th}(X) \approx E_{\rm off}(X) = \alpha X + \beta,
$$
with $\beta \approx 0$. 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 $dR/dt = P/\mu$.
- The momentum obeys
  $$
  \frac{dP}{dt} = -\frac{dV}{dR} - \int_0^t d\tau \,\gamma(t-\tau) \,\frac{P(\tau)}{\mu} + \xi(t),
  $$
  where $\mu$ is the reduced mass, $V(R)$ includes nuclear, Coulomb, and centrifugal contributions, $\gamma(t-\tau)$ is the non-Markovian friction kernel, and $\xi(t)$ is a quantum Langevin force with statistics governed by the fluctuation-dissipation theorem.
  
Capture probability for partial wave $J$ is given by
$$
P_{\rm cap}(E,J) = \int d\mathcal{B} \, f_J(\mathcal{B}) T(E,\mathcal{B}),
$$
with $f_J(\mathcal{B})$ the distribution of barrier heights from coupling and $T(E,\mathcal{B})$ a transmission factor (e.g., Hill–Wheeler, WKB). The total capture cross section is then
$$
\sigma_{\rm cap}(E) = \frac{\pi}{k^2} \sum_{J=0}^{J_{\max}} (2J+1) P_{\rm cap}(E,J), \quad k = \sqrt{2\mu E}/\hbar.
$$

## 5. Behavior of Cross Sections and Slope Transitions

Just below the Coulomb barrier, $P_{\rm cap}(E,0)$ can be approximated by the Hill–Wheeler formula, giving a nearly constant logarithmic slope $S(E)\approx 2\pi / (\hbar\omega_b)$. However, for energies $E < E_{\rm th}$, with $R_{\rm ex} > R_{\rm int}$ and $V_N \approx 0$ over most of the forbidden region, the effective barrier curvature is reduced, the barrier becomes broader, and $S(E)$ 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 $D_{\rm qel}(E)$ provide direct experimental access to the transition at $E_{\rm th}$:
$$
D_{\rm qel}(E) = -\frac{dP_{\rm qe}(E, J=0)}{dE} = \frac{dP_{\rm cap}(E, J=0)}{dE},
$$
with $P_{\rm qe}(E,0)$ calculated from backward-angle quasielastic and Rutherford differential cross sections. Changes in the width or a discontinuity (kink) in $D_{\rm qel}(E)$ 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:
$$
\boxed{
\mathrm{EFwT} = E_{\rm th} \simeq E_{\rm off} = V(R_{\rm int}) = V_N(R_b + \Delta R) + \frac{Z_1 Z_2 e^2}{R_b + \Delta R}
}
$$
- $E_{\rm th}$ is the experimentally observed hindrance threshold.
- $E_{\rm off}$ is the energy at which the classical turning point coincides with $R_{\rm int} = R_b + 1.1\,\rm fm$.
- $V_N$ and $V_C$ 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 $R_b$ and $V(R)$ for the chosen system, offsetting the radius by $1.1\,\rm fm$, and calculating the total potential at this interaction cutoff. The empirical concordance between EFwT and the observed $E_s$ affirms the quantum-diffusion picture of hindrance as a consequence of the absence of nuclear forces beyond $R_{\rm int}$. EFwT consequently provides an operational criterion bridging theoretical dynamics and experimental measurement in sub-barrier fusion studies [1211.2312].

Source: https://www.emergentmind.com/topics/energy-and-forces-within-threshold-efwt-metric