Section 2.5.1.4. Integrated intensity for powder sample

The kinematical theory of diffraction and a non-absorbing crystal with a `frozen' lattice are assumed. Corrections for thermal vibrations, absorption, extinction, etc. are discussed in Subsection 2.5.1.5. The total diffracted power, , for a Bragg reflection of a powder sample can then be written (Buras & Gerward, 1975; Kalman, 1979) where h is the diffraction vector, the classical electron radius, the intensity per unit energy range of the incident beam evaluated at the energy of the diffraction peak, V the irradiated sample volume, the number of unit cells per unit volume, j the multiplicity factor, F the structure factor, and the polarization factor. The latter is given by where P(E) is the degree of polarization of the incident beam. The definition of P(E) is where and are the parallel and normal components of with respect to the plane defined by the incident- and diffracted-beam directions.

Generally, has to be calculated from equations (2.5.1.4) and (2.5.1.5). However, the following special cases are sometimes of interest: Equation (2.5.1.6a) can often be used in connection with Bremsstrahlung from an X-ray tube. The primary X-ray beam can be treated as unpolarized for all photon energies when there is an angle of 45° between the plane defined by the primary and the diffracted beams and the plane defined by the primary beam and the electron beam of the X-ray tube. In standard configurations, the corresponding angle is 0° or 90° and equation (2.5.1.6a) is generally not correct. However, for it is correct to within 2.5% for all photon energies (Olsen, Buras, Jensen, Alstrup, Gerward & Selsmark, 1978).

Equations (2.5.1.6b) and (2.5.1.6c) are generally acceptable approximations for synchrotron radiation. Equation (2.5.1.6b) is used when the scattering plane is horizontal and (2.5.1.6c) when the scattering plane is vertical.

The diffraction directions appear as generatrices of a circular cone of semi-apex angle about the direction of incidence. Equation (2.5.1.3) represents the total power associated with this cone. Generally, only a small fraction of this power is recorded by the detector. Thus, the useful quantity is the power per unit length of the diffraction circle on the receiving surface, . At a distance r from the sample, the circumference of the diffraction circle is and one has (constants omitted) The peak areas in an XED powder spectrum are directly proportional to the of equation (2.5.1.7).

Quantitative structural analysis requires the knowledge of and P(E). As mentioned above, these quantities are not known with sufficient accuracy for Bremsstrahlung. For synchrotron radiation they can be calculated, but they will nevertheless contribute to the total uncertainty in the analysis. Accordingly, XED is used rather for identification of a known or assumed structure than for a full structure determination.