a) Cluster model:

Some nuclei permit an approximate description of their structure in terms of cluster. This is the case of ⁷Li, which is sometimes modeled as two inert clusters ⁴He+³H. Other examples are ⁸B$\rightarrow$⁸B+n, ¹¹Be$\rightarrow$¹⁰Be+n, etc.

For such systems, Coulomb and nuclear excitations can be interpreted and calculated in terms of the interactions of each cluster and the target. For example, consider that the projectile is composed of two fragments, denoted 1 and 2. Then:

$$U_{\alpha\alpha}(\mathbf{R})=\int d\mathbf{r}\phi_{\alpha}^{*}(\mathbf{r})U(\mathbf{r},\mathbf{R})\phi_{\alpha}(\mathbf{r})$$ (16)

$$U_{\alpha\alpha}'(\mathbf{R})=\int d\mathbf{r}\phi_{\alpha}^{*}(\mathbf{r})U(\mathbf{r},\mathbf{R})\phi_{\alpha'}(\mathbf{r})$$ (17)

where

U(r,R)=U₁(R₁)+U₂(R₂) (18)

with

$$\mathbf{R}_{1}=\mathbf{R}+\frac{m_{2}}{m_{1}+m_{2}}_{1}\math... ...\,\mathbf{R}_{2}=\mathbf{R}-\frac{m_{1}}{m_{1}+m_{2}}\mathbf{r}$$ (19)

Previous to the solution of the coupled equations, FRESCO has to evaluate the coupling potentials $U_{\alpha\alpha}$ and $U_{\alpha\alpha'}$. This in turn require the internal wavefunctions $\phi_{\alpha}(\mathbf{r})$ and $\phi_{\alpha'}(\mathbf{r})$.