I am trying to find out the $S$-matrix elements for the reaction: $${}^{19}\textrm{F} + {}^{208}\textrm{Pb}. $$
The model followed is Direct reaction model where the optical potential is:
$$ V_{op}(r) = V_R(r) + iW_R .$$
A wave function was obtained as:
$$ \psi(r) = \frac{1}{kr}\sum(2\ell+1)i^\ell u_\ell(r)P_\ell(\cos\theta) $$
From this wave function transmission coefficient was calculated to be:
$$ T_\ell = \frac{-8}{\hbar\nu}\int_0^\infty |u_\ell(r)|^2 W(r) dr $$
Which can be separated into 2 terms, which are the fusion and direct reaction components:
$$ T_\ell^F = \frac{-8}{\hbar\nu}\int_0^\infty |u_\ell(r)|^2 W_f(r) dr $$ $$ T_\ell^D = \frac{-8}{\hbar\nu}\int_0^\infty |u_\ell(r)|^2 W_D(r) dr $$
The spin distribution of fusion is given as:
$$ \sigma_f(\ell) = \frac{\pi}{k^2}(2\ell + 1)T_\ell^F $$
And the fusion cross section is obtained as:
$$ \sigma_f = \sum\sigma_f(\ell) $$
Under this model how to calculate the scattering matrix elements and hence calculate the fusion cross-section in Fortran?
