EXAFS :== extended X-ray absorption fine structure.
At some Xray energies, almost all absorption is due to photoelectric effect, an (inner) electron being promoted to a higher energy
state, often escaping the atom entirely.
The 'scattered state' that contains the outgoing photoelectron has
to overlap the core state. That is where the matrix element
comes from, that determines the rate of photoelectric absorption
of an X-ray. The key here, is that the outgoing states are
perturbed by the cores of any neighboring atoms, which are significant
parts of the Hamiltonian of that outgoing electron.
It is the extended electron wave, in all its not-exactly-a-point-particle
extent, which carries away the X-ray's quantum of energy,
and not just a small local orbital (like a bound electron in
the 1S shell).
The result is that some energies and directions are unlikely (because
the outgoing electron wave backscatters), which means the
matrix element is small, because of those other, distant, atoms.
EXAFS with polarized X-rays (synchrotron radiation) can give
directional information on crystalline samples with orientation,
because of the different neighbor positions as a function of
the crystal orientation with respect to the outgoing-electron
direction. With liquids or polycrystalline samples, an average
over many directions is seen instead.
And with a dilute gas of atoms, a 'pure' absorption of the Xrays
by a free atom is observed. The cases are all different, because
the outgoing electrons are waves that may overlap multiple neighbors.
The absorption depends on those outgoing-electron wavefunctions, so
it is modulated by the absorbing atom's neighbors.