The anomalous X-ray scattering from an atom is described by a modification of the scattering factor defined above. The atomic scattering factor becomes a complex quantity and is written
where 
is the Thomson scattering factor and 
and 
are respectively
the real and imaginary anomalous scattering correction terms.
The effect of anomalous scattering can most simply be described by treating an electron as a classical
dipole oscillator driven by a force effected by the incident X-ray and with damping corresponding to
the bound state in which the electron exists. Consider a bound electron placed at the origin having a
damping constant 
and a natural resonant frequency 
. An X-ray of frequency 
is incident on the electron with an electric field vector at time 
at the electron given
by 
.
The electron is forced into motion by the presence of the oscillating electric field and at time , has
a displacement 
, perpendicular to the X-rays direction of travel, described by the equation of motion
The steady state solution to this equation will have the form 
and in fact turns out to be
At a large distance 
from an oscillating dipole the magnitude of the instantaneous electric field at time
observed in the forward direction is given by [33]
From Eq. 
we obtain for 
,
Substituting this into Eq. and taking the maximum amplitude of 
gives
The analogous expression for a free electron acting as a dipole oscillator may be obtained by letting 
.
Including the scattering angle dependence we recover Eq. for scattering in the free electron limit.
Now by using the definition for the atomic scattering factor above and applying it to our present case,
we obtain a result for the scattering factor of a single bound electron with resonant frequency , and
damping constant for incident X-rays of frequency ,
It can be seen that the scattering factor is a complex quantity due to the effects of the damping term. This
means that an additional phase shift has been introduced into the scattered beam. We may
split this expression into real and imaginary parts by first multiplying top and bottom
by 
and separating out the terms. We obtain
which is equivalent to Eq. for the single bound electron case; the first term on the left hand
side includes the Thompson scattering term and the real part of the anomalous scattering correction. Again we notice
that as 
and 
, 
.
Before the extension to the many electron atom we must first make several assumptions. Firstly the anomalous
scattering effects observed are all due to core electrons and their charge distributions are therefore small
compared to the wavelength of the incident radiation. We therefore assume that the anomalous scattering
factor correction terms are constant with respect to the scattering angle. This has been shown
by Hazell [45] to be the case, at least to within a fraction of an electron unit of 
, and is generally
accepted as such. To a first approximation we also notice from Eq. that for very small values
of r (the distance of the classical electron from the nucleus) the term 
will be close to unity
an almost independent of 
.
Secondly, in showing classically that the first term of Eq. does indeed correspond
to 
it is necessary to assume that the damping constant is small.
We now let the atom contain N oscillators, each represented by an oscillator strength 
and each having
natural frequency 
. The oscillator strengths quantify the contribution of each electron to the
scattering factor and are assumed to comply to the
Thomas-Reiche-Kühn [110] [70] [96]
sum rule 
. (In the quantum mechanical treatment of anomalous scattering the values
are proportional to the transition probabilities between initial and final energy states). With the added
assumption that 
and that only the inner core electron states 
contribute to
anomalous scattering, we may rewrite the real part
of Eq. as
where 
equals the Thompson scattering component for forward scattering.
