I checked the paper. It describes how figure 9 was computed by referring the reader to older papers:
the spherically symmetric powder pattern scatter-
ing intensity was calculated from the atom-atom pair
distribution functions. The functions for all of the atom
pairs are available elsewhere40. Standard techniques
as discussed by Wagner49 and invoked previously22,26
were used. The atomic scattering functions are from
Alexander50.
I didn't check these references, though. I assume the calculation is based on the Debye scattering formula. It's a standard way of computing a diffraction pattern from atomic positions, derived by P. Debye in 1915. This formula was derived by spherical averaging of the intensity – that's what "spherically symmetric" refers to.
The Debye formula is applied to a finite set of atoms. If you performed your simulation in periodic boundary conditions (PBC), you might need to first generate a larger system to capture longer-range interatomic distances.
Then, instead of writing your own code, you might search what programs are already available. One of them is mine: https://github.com/wojdyr/debyer (written 15 years ago and not maintained anymore).