Diffraction volume, correction

Absorption and Variation of the Diffracting Volume Correction. During the measurement of a pole figure in reflection mode the specimen is irradiated under various angles. The paths of the incident and diffracted beams through the specimen, li x) and ld x) vary according to the depth x of the point where diffraction occurs. The absorption coefficient A x) is. 

H. Lipson. Absorption corrections. In International Tables for X-ray Crystallography. Volume II. Mathematical Tables. Section 5. Physics of Diffraction Methods. (Eds., Kasper, J. S., and Lonsdale, K.) pp. 291-312. Kynoch Press Birmingham (1959). 

To take into account the disparity in electrons, and hence scatting power, for the various atoms in the unit cell, the atomic number Zj has been introduced as a means of defining amplitudes for the component waves. The total diffracted wave for the entire crystal will be the product of the equation above with the total number of unit cells in the crystal.1 For a single unit cell, like that in Figures 5.12a and 5.12b, an atom s contribution to the total structure factor of one unit cell can also be illustrated in vector terms as in Figure 5.13. To put the expression in the correct units, it is necessary to multiply the summation by a constant, the volume of the unit cell. Here, that is simply V = a x b x c. Thus... 

In calculating the value of R for a particular diffraction line, various factors should be kept in mind. The unit cell volume v is calculated from the measured lattice parameters, which are a function of carbon and alloy content. When the martensite doublets are unresolved, the structure factor and multiplicity of the martensite are calculated on the basis of a body-centered cubic cell this procedure, in effect, adds together the integrated intensities of the two lines of the doublet, which is exactly what is done experimentally when the integrated intensity of an unresolved doublet is measured. For greatest accuracy in the calculation of F, the atomic scattering factor f should be corrected for anomalous scattering by an amount A/ (see Sec. 13-4), particularly when Co Ka radiation is used. The value of the temperature factor can be taken from the curve of Fig. 4-20. 

The thermal expansion of solids depends on their structure symmetry, and may be either isotropic or anisotropic. For example, graphite has a layered structure, and its expansion in the direction perpendicular to the layers is quite different from that in the layers. For isotropic materials, ay w 3 a . However, in anisotropic solid materials the total volume expansion is distributed unequally among the three crystallographic axes and, as a rule, cannot be correctly measured by most dilatometric techniques. The true thermal expansion in such case should be studied using in situ X-ray diffraction (XRD) to determine any temperature dependence of the lattice parameters. 

Figure 6.2 The analytical calculation of a crystal sample correction involves the integration over the whole illuminated volume of the crystal. dV is a volume element, ts and tp are the path lengths through the crystal sample of the secondary (diffracted) and primary (incident) beams. In macromolecular crystallography this calculation is never done because the crystal is bathed in a blob of mother liquor, which may well decrease or swell during the experiment. A better method is to use short wavelengths (e.g. 0.9 A or in future 0.33 A) - see figure 6.3.

Here, d//dA denotes the spectral intensity distribution of the incident X-ray beam V is the volume of sample illuminated V0 is the sample unit cell volume 0 is the Bragg angle for the reflection h P is the polarisation factor A is an absorption correction for the sample in its capillaiy and D is a detector sensitivity and obliquity factor. Quantities such as P, A and D vary with any or all of A, 0 and x, the position of the diffracted beam on the detector the spectral intensity distribution is, in general, not precisely known in advance and the detector may suffer from spatial distortion and non-uniformity. Thus equation (7.17) may be written as... 

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