# Calculating Scattering matrix of a nuclear fusion reaction using Fortran

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?

• What are $\nu,\ W_f,\ k?$ And assuming you know them, what is it that is blocking you from directly implementing them in Fortran? Dec 27, 2023 at 6:29
• k is the wave number Wf is the wood saxon potential when the radial separation is less than r . The issue is when i solve for the Schrodinger wave the wave function obtained is not as same as the one as the model states. Hence it is causing an error Dec 28, 2023 at 8:43

The formulas you used are sometimes called absorptive (or absorption) cross section. You can find the details calculation of it in good reaction textbooks like "Nuclear Reactions for Astrophysics" of Thompson-Nunes or "Theory of Nuclear Reactions" of Fröbrich-Lipperheide. Basically the biggest tasks in your code is to numerically solve the Schroedinger equation with scattering boundary condition to find the wave function $$u_l(r)$$. I would suggest the the Numerov method for that. For more information on the numerical methods, you can check out standard computational physics textbooks.