Hi,
Fine structure (oscillations) primarily arises from the fact that the photoelectron (as a result of the core electron removal) gets scattered by all neighboring atoms.
Here (in thumbnail) is a sample of X-ray lines in molybdenum between 0.5 and 0.6 Angstroms (24 KeV to 21 KeV).. This is taken from the NIST table on atomic transition energies.
There are two ways a particular atomic transition can be split:
1) Fine structure. I have drawn a box around the 3p3/2 and 3p1/2 to the 1s1/2 transitions. These are split by j = l + 1/2 and j = l - 1/2 fine structure in the 3p initial state; about 44 eV energy difference out of 21,800 eV.
2) The ionization state of atom. This table shows only two ionization states (# of electrons missing), Mo LXII and Mo LXI (LXI and LXII represent the ionization state), but there are many more. Go explore.