Lorentz polarization

The parameters were then further refined by four successive least-squares procedures, as described by Hughes (1941). Only hk() data were used. The form factor for zinc was taken to be 2-4 times the average of the form factors for magnesium and aluminum. The values of the form factor for zinc used in making the average was corrected for the anomalous dispersion expected for copper Kot radiation. The customary Lorentz, polarization, temperature, and absorption factors were used. A preliminary combined scale, temperature, and absorption factor was evaluated graph-... 

Figure 8.3. LORENTZ-polarization corrected WAXS curve of poly(3-dodecylthiophene) before and after background subtraction (from PROSA et al. [109]). The authors define q in the way that is identical to the definition of s in this book...

Hall and Pass used a rather different method (8), whereby the intensity was measured on a 25pm lattice of points covering the entire area of a reflection. From this an intensity contour map of the reflection was created and its total intensity determined by measuring the area within each contour line, multiplying this area by the intensity difference between adjacent contours, and summing these products for all contour areas within the boundary of the reflection. The measured intensity was corrected by application of the value at the centre of the reflection of the Lorentz-polarization factor. 

A polycrystalline thin film does not have any preferred orientation (Figure 6.4 (c)). In such a case, the diffraction from the crystal is not a spot but a so-called Debye-Scherrer ring. Therefore, the sample does not have to be inclined to obtain the diffraction pattern. Conventional 2 0-6 scans move the scattering vector H in the radial direction along the film surface normal. Thus, these scans give sufficient information when the film is polycrystalline. The obtained diffracted intensity must be corrected in terms of the absorption and the Lorentz polarization. These two terms and the obtained diffracted intensity have the following relation ... 

The factors that are included when calculating the intensity of a powder diffraction peak in a Bragg-Brentano geometry for a pure sample, composed of three-dimensional crystallites with a parallelepiped form, are the structure factor Fhkl 2=l/ TS )l2, the multiplicity factor, mm, the Lorentz polarization factor, LP(0), the absorption factor, A, the temperature factor, D(0), and the particle-size broadening factor, Bp(0). Then, the line intensity of a powder x-ray diffraction pattern is given by [20-22,24-26]... 

Correct the experimental data for Lorentz, polarization, absorption, and other factors, and the results to relative values of the structure amplitudes F hkl). 

The Lorentz and polarization contributions to the scattered intensity are nearly always combined together in a single Lorentz-polarization factor, which in the case when no monochromator is employed is given as ... 

Figure 2.47. Lorentz-polarization factor as a function of Bragg angle the solid line represents calculation using Eq. 2.70 (no monochromator), and the dash-dotted line is calculated assuming graphite monochromator and Cu Ka radiation with K = 0.5 (Eq. 2.71).

Intensity gain due to Lorentz-polarization factor (see Chapter 2, section 2.10.4) is partially offset by the requirement of reduced divergence slit opening (see sections 3.5,3 and 3.6.3), provided all other things remain constant, including the brightness of the incident beam. 

This in turn is combined with the polarization factor + cos 26) of Sec. 4-2 to give the combined Lorentz-polarization factor which, with a constant factor of i omitted, is given by... 

In this case, A = 1.542 A (Cu Ka) and a = 3.615 A (lattice parameter of copper). Therefore, multiplication of the integers in column 3 by A /4o = 0.0455 gives the values of sin 6 listed in column 4. In this and similar calculations, three-figure accuracy is ample. Column 6 Needed to determine the Lorentz-polarization factor and (sin 0)/A. 

The agreement obtained here between observed and calculated intensities is satisfactory. Note how the value of the multiplicity p exerts a strong control over the line intensity. The values of F and of the Lorentz-polarization factor vary smoothly with 6, but the values ofp, and therefore of /, vary quite irregularly. 

The use of a monochromator produces a change in the relative intensities of the beams diffracted by the specimen. Equation (4-19), for example, was derived for the completely unpolarized incident beam obtained from the x-ray tube. Any beam diffracted by a crystal, however, becomes partially polarized by the diffraction process itself, which means that the beam from a crystal monochromator is partially polarized before it reaches the specimen. Under these circumstances, the usual polarization factor (1 - - cos 26)12, which is included in Eqs. (4-19) through (4-21), must be replaced by the factor (1 + cos 2a cos 20)/(l -I- cos 2a), where 2a is the diffraction angle in the monochromator (Fig. 6-16). Since the denominator in this expression is independent of 6, it may be omitted the combined Lorentz-polarization factor for crystal-monochromated radiation is therefore (1 + cos 2a cos 20)/sin 6 cos 6. This factor may be substituted into Eqs. (4-19) and (4-20), although a monochromator is not often used with a Debye-Scherrer camera, or into Eq. (4-21), when a monochromator is used with a diffractometer (Sec. 7-13). But note that Eq. (4-20) does not apply to the focusing cameras of the next section. 

We have already seen that the intensity of a superlattice line from an ordered solid solution is much lower than that of a fundamental line. Will it ever be so low that the line cannot be detected We can make an approximate estimate by ignoring the variation in multiplicity factor and Lorentz-polarization factor from line to line, and assuming that the relative integrated intensities of a superlattice and fundamental line are given by their relative F values. For fully ordered AuCus, for example, we find from Eqs. (13-1) that... 

Usually not taken into account is the variation of the structure factor due to the variation of the diffusion factor (X-ray), or of the Lorentz-polarization factor. 

Depending on which Rietveld program has been used, it might be necessary to remove the effect of the Lorentz-polarization Lp) factor from each observed peak intensity ... 

The term X /sin 20 is referred to as the Lorentz factor, the Lorentz-polarization factor, which is the product of the two previous expressions, is also commonly used. 

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