Absorption and the complex refractive index.



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Absorption and the complex refractive index.

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. gif 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. gif then by differentiating to obtain the velocity at time and substituting the result for the velocity part of Eq. gif 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. gif. 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. gif 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. gif. Cromer & Libermann [25] showed the additional term to be where is the total energy of the atom. Eqs. gif and gif are used in Chapter gif to establish and from experimental X-ray fluorescence data.



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Next: Diffraction from a Up: Anomalous scattering. Previous: Classical description of



Gwyndaf Evans
Fri Oct 7 15:42:16 MET 1994