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]

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.

Fri Oct 7 15:42:16 MET 1994