3. Fusion barrier distribution

Nuclear fusion is related to the transmission of the incident wave through a potential barrier, resulting from nuclear attraction plus Coulomb repulsion. However, the meaning of the fusion barrier depends on the description of the collision. Coupled-channel calculations include static barriers, corresponding to frozen densities of the projectile and the target. Its most dramatic consequence is the enhancement of the total fusion reaction cross section σfus at Coulomb barrier energies Vb, in some cases by several orders of magnitude. One of the possible ways to describe the effect of coupling channels is a division of the fusion barrier into several, the so-called fusion barrier distribution Dfus and given by [20, 29]

$$D\_{fus}(E) = \frac{d^2F(E)}{dE^2} \tag{23}$$

where F Eð Þ is related to the total fusion reaction cross section through [29]

$$F(E) = E \sigma\_{\hat{f}us}(E) \tag{24}$$

The experimental determination of the fusion reaction barrier distribution has led to significant progress in the understanding of fusion reaction. This comes about because, as already mentioned, the fusion reaction barrier distribution gives information on the coupling channels appearing in the collision. However, we note from Eq. (24) that, since fusion reaction barrier distribution should be extracted from the values of the total fusion reaction cross section, it is the subject to experimental as well as numerical uncertainties [29–31]:

$$D\_{fus}(E) \approx \frac{F(E + \Delta E) + F(E - \Delta E) - 2F(E)}{\Delta E^2} \tag{25}$$

where ΔE is the energy interval between measurements of the total fusion reaction cross section. From Eq. (25), one finds that the statistical error associated with the fusion reaction barrier distribution is approximately given by [30]

$$
\delta D\_{\rm fus}^{\rm stat}(E) \approx \frac{\sqrt{\left[\delta F(E + \Delta E)\right]^2 + \left[\delta F(E - \Delta E)\right]^2 + 4\left[\delta F(E)\right]^2}}{\left(\Delta E\right)^2} \tag{26}
$$

where δF Eð Þ means the uncertainty in the measurement of the product of the energy by the total fusion reaction cross section at a given value of the collision energy. Then the uncertainties can approximately be written as [30]

$$
\delta D\_{fus}^{stat}(E) \approx \frac{\sqrt{\mathsf{G}} \delta F(E)}{\left(\Delta E\right)^2} \tag{27}
$$
