In trying to understand the significance of the anomalous scattering correction terms it is convenient to consider dispersion in the refractive index of a plane sheet of atoms when a plane wave of radiation is incident on it.
A concise derivation for the field produced by a plane of oscillating charges has been
given by Feynman [33]. Consider a plane containing 
charges per unit area each with charge 
,
oscillating in phase with one another. Their individual displacements about their rest positions at time 
is 
where 
is the maximum displacement.
We wish to obtain an expression for the total electric field 
at a point 
a large distance 
from
the origin 
of the plane, along the direction normal to the plane. For any charge 
in the plane a distance 
from , the field at due solely to that charge is given by Eq. 
and may be written in full as
All charges a distance from lie on a circle of radius 
, such that 
. The field
at due to all charges lying within a narrow shell of area 
is given by summing
over the field produced by the 
charges within the shell. The total field at is
thus obtained by integrating over all such shells from 
to 
, i.e.
which is in fact equivalent to the velocity of the charges at time 
multiplied by the
factor 
.
Let us now consider a thin plate of thickness 
having a refractive index 
and containing 
oscillators per unit volume. As the plane wave of
incident radiation, described in front of the plate by 
, travels through
the plate it will apparently travel with a speed 
and take an additional
time 
to travel the distance through the plate compared to through vacuum.
The wave behind the plate may then be expressed as
If the plate is thin then the addition phase change 
will be small then we may expand
the second exponential term into 
. If we
truncate this series after the second term we can rewrite the above equation as
which is in effect the initial wave 
minus a contribution to the field behind the plate from the oscillating charges in the material, 
.
If we now assume that the equations of motion of the charges in the material have the solution given
in Eq. then by differentiating to obtain the velocity at time and substituting the result
for the velocity part of Eq. we obtain a second expression for the field due to the oscillating
charges behind the plate,
We can now equate the two results for and by substituting 
for in the above
equation we arrive at an equation for the refractive index of the material in terms of the dispersive
properties of its constituent particles which in the case of N equal electrons per unit volume will be
We note that the refractive index is now a complex quantity. The physical meaning of a complex refractive
index may be realised by substituting 
for in Eq. .
The resulting expression is
The final exponential term in this equation represents an attenuation or absorption of the primary beam over the
distance . The presence of an imaginary part to the refractive index thus implies that absorption of
the primary beam occurs the material. An expression for the linear coefficient of absorption 
of
the material may be found by squaring the above equation so that we deal with intensities instead
of amplitudes. By doing this we include a factor of two in the absorption exponent and we can thus
define as
By substituting the imaginary part of Eq. for 
in the last expression and
rearranging we arrive at an equation for 
, the imaginary part of the anomalous scattering
correction, in terms of the linear absorption coefficient , i.e.
This is one form of the Optical theorem, but it is more commonly written in terms of the atomic cross
section 
. This relationship is important as it allows direct experimental measurement
of for a scatterer via measurement of its atomic absorption cross-section. In terms
of the incident X-ray energy, 
, and the Optical theorem may be written
The real and imaginary parts of the atomic scattering factor
are related via a set of dispersion formulae known as the Kramers-Kronig relations [71][73].
The real part of the anomalous scattering correction 
may be obtained from by
In fact this is not strictly true and an additional correction term arising from QED considerations must be added
to Eq. . Cromer & Libermann [25] showed the additional term to be 
where 
is the total energy of the atom. Eqs. and are used in
Chapter to establish and from experimental X-ray fluorescence
data.
