Fine structure in the atomic energy levels is due to the interaction of the three quantum numbers n, l, and s (for Dirac atoms).
Fine structure in the atomic energy levels is due to the interaction of the two quantum numbers n and l (for Schroedinger (or Klein Gordon) atoms).
It is noteworthy that the atomic energy levels have fine structure even when the atomic particle has no spin. This has been proved to about 1 part per million in pionic atoms.
The quantum numbers arise from the separation of the wave equations in spherical coordinates into radial and azimuthal components. See for example Schiff Quantum Mechanics (second edition) pages 321 and 337.
Bob S