Lorentz Corrections

The scattered intensity measured from the isotropic three-dimensional object can be transfonned to the onedimensional mtensity fiinction/j(<3 ) by means of the Lorentz correction [15]... 

Equation (8.59) defines the ID interference function of a layer stack material. G (s) is one-dimensional, because p has been chosen in such a way that it extinguishes the decay of the Porod law. Its application is restricted to a layer system, because misorientation has been extinguished by Lorentz correction. If the intensity were isotropic but the scattering entities were no layer stacks, one would first project the isotropic intensity on a line and then proceed with a Porod analysis based on p = 2. For the computation of multidimensional anisotropic interference functions one would choose p = 2 in any case, and misorientation would be kept in the state as it is found. If one did not intend to keep the state of misorientation, one would first desmear the anisotropic scattering data from the orientation distribution of the scattering entities (Sect. 9.7). 

D Intensity. As already mentioned (cf. p. 126 and Fig. 8.12), the isotropic scattering of a layer-stack structure is easily desmeared from the random orientation of its entities by Lorentz correction (Eq. 8.44). For materials with microfibrillar structure this is more difficult. Fortunately microfibrils are, in general, found in highly oriented fiber materials where they are oriented in fiber direction. In this case the one-dimensional intensity in fiber direction,... 

Warning. For isotropic materials the ID projection /, and the Lorentz correction yield different ID intensities. Both are related by... 

If the structural entities are lamellae, Eq. (8.80) describes an ensemble of perfectly oriented but uncorrelated layers. Inversion of the Lorentz correction yields the scattering curve of the isotropic material I (5) = I (s) / (2ns2). On the other hand, a scattering pattern of highly oriented lamellae or cylinders is readily converted into the ID scattering intensity /, (53) by ID projection onto the fiber direction (p. 136, Eq. (8.56)). The model for the ID intensity, Eq. (8.80), has three parameters Ap, dc, and <7C. For the nonlinear regression it is important to transform to a parameter set with little parameter-parameter correlation Ap, dc, and oc/dc. When applied to raw scattering data, additionally the deviation of the real from the ideal two-phase system must be considered in an extended model function (cf. p. 124). 

Reflection intensity in the SAED negatives was measured with a microdensitometer. The refinement of the structure analysis was performed by the least square method over the intensity data (25 reflections) thus obtained. A PPX single-crystal is a mosaic crystal which gives an "N-pattem". Therefore we used the 1/d hko as the Lorentz correction factor [28], where d hko is the (hkO) spacing of the crystal. In this case, the reliability factor R was 31%, and the isotropic temperature factor B was 0.076nm. The molecular conformation of the P-form took after that of the P-form since R was minimized with this conformation benzene rings are perpendicular to the trans-zigzag plane of -CH2-CH2-. 

Before this information can be used, the data set has to undergo some routine corrections, this process is known as data reduction. The Lorentz correction, L, relates to the geometry of the collection mode the polarization correction, p, allows for the fact that the nonpolarized X-ray beam, may become partly polarized on reflection, and an absorption correction is often applied to data, particularly for inorganic structures, because the heavier atoms absorb some of the X-ray beam, rather than just scatter it. Corrections can also be made for anomalous dispersion, which affects the scattering power of an atom when the wavelength of the incident X-ray is close to its absorption edge. These corrections are applied to the scattering factor, 4 of the atom. 

Fig. 4 Effect of nanoclay loading on neat SEBS a Lorentz -corrected SAXS profiles (vertically shifted for better clarity) showing effect of nanoclay arrows indicate peak positions, b Lengths corresponding to first- and second- order scattering vector positions along with the 2D SAXS patterns for each sample of clay-loaded nanocomposites...

N is the number density in the units of number of molecules per cm. F is the Lorentz correction factor defined by Equation 6. If the solute is in dilute concentration, Equation 9 can be written as... 

Celia, Lee and Hughes ( ) have studied the Lorentz correction for fibres in some detail they consider that the correction 2 sin 8cos0 is most appropriate for the equatorial scatter from a parallel bundle of fibres. 

The effect of the Lorentz correction is illustrated in Figure 3. Here the equatorial trace of a viscose rayon specimen is shown uncorrected (LOR 0), corrected in the normal way for fibres (LOR 1), and for powders (LOR 2). Table I shows the results of profile analysis on the viscose rayon specimen using the different... 

Resolved parameters for an untreated viscose rayon with different Lorentz corrections... 

Lorentz corrections. As can be seen, there are significant changes in the resolved parameters, particularly the peak positions and peak heights the effect on peak width is less marked. The standard fibre correction (LOR 1) gives the most reasonable results, the correction (LOR 2), suggested for the equatorial trace of a fibre specimen, is not considered to be realistic. 

Treatment of Experimental Values. The experimental values are corrected for air scattering, polarization, but absorbtion - geometric (Lorentz) corrections are not made. After the variable 20 is transformed into s = 2 sin 6, the experimental curves are normalized, in electronic units, by adjustment to a theoretical curve. Theoretical curves (total scattering power, summing up coherent and incoherent scatterings) are calculated from the stoichiometric composition of polymers. 

When a non-centrosymmetric solvent is used, there is still hyper-Rayleigh scattering at zero solute concentration. The intercept is then determined by the number density of the pure solvent and the hyperpolarizability of the solvent. This provides a means of internal calibration, without the need for local field correction factors at optical frequencies. No dc field correction factors are necessary, since in HRS, unlike in EFISHG, no dc field is applied. By comparing intercept and slope, a hyperpolarizability value can be deduced for the solute from the one for the solvent. This is referred to as the internal reference method. Alternatively, or when the solvent is centrosymmetric, slopes can be compared directly. One slope is then for a reference molecule with an accurately known hyperpolarizability the other slope is for the unknown, with the hyperpolarizability to be determined. This is referred to as the external reference method. If the same solvent is used, then no field correction factor is necessary. When another solvent needs to be used, the different refractive index calls for a local field correction factor at optical frequencies. The usual Lorentz correction factors can be used. 

The Lorentz and polarization corrections,often called Lp, are geometrical corrections made necessary by the nature of the X-ray experiment. The Lorentz factor takes into account the different lengths of time that the various Bragg reflections are in the diffracting position. This correction factor differs for each type of detector geometry. For example, the Lorentz correction for a standard four-circle diffractometer... 

For purposes of display, the intensity of each calculated reflection was assumed to be distributed about its mean position according to Gaussian distributions in 20 and p. The breadths of these distributions represent the expected peak broadening due to finite crystallite sizes, the crystal mosaic, and paracrystal distortions, which are not available a priori from the model calculations. Additionally, we require an estimate for p1/2 for the Lorentz correction of meridional reflection intensities. The peak broadening distributions were selected of the form... 

Figure 23. SAXS traces (Lorentz corrected intensities vs q) of /J-C191H385COOH recorded during cooling an already partially crystallized sample at 0.3 °C/min from 132 to 112 °C. Bragg spacings corresponding to the observed peak positions, recording temperatures and diffraction orders of the bilayer phase are marked (from ref 137 by permission of American Chemical Society).

Lorentz corrections applied to powder diffraction data are slightly different to those applied to single-crystal data. Whereas the single-crystal correction only consists of a rotational factor, the powder correction contains an additional statistical factor. " This corrects for the likelihood of a crystallite being in diffraction position. This factor has a simple (sin0) dependence, and is found in the common Lorentz correction... 

Combining this with the statistical factor leads to the general formulation of the Lorentz correction for powder samples rotated within the beam ... 

Figure 14.11 Two dimensional single crystal Lorentz correction for an ideally aligned detector. Note the zero values in the central valley. These cause divergent intensities as they are multiplied with the inverse of the Lorentz correction. Therefore, the intensities in that region have no meaning. The central valley is parallel to the sample rotation axis.

Lorentz Correction for Highly Collimated Beams. The rotational correction should be used if the powder sample is rotated within the beam in the single crystal sense, i.e. all crystallites should complete their rotation within the beam. Should the beam be collimated to dimensions below those of the sample containment then this further reduces the rotational impact on the cumulative Lorentz factor. A term Rl can be introduced to quantify the rotational Lorentz factor from 0 for no rotational element to 1 for full rotation of all crystallites within the beam (Figure 14.12). The introduction of this factor leads to ... 

Now this rotation needs to be put into relation to the entire illuminated area. The normalization takes the form of the average rotational angle of the entire illuminated area relative to the full 2n rotation of the standard Lorentz correction ... 

Spend time on the measurement of the WAXS intensity curve. If noise can be detected by the eye, data are insufficient for further analysis. Correct the raw data for varying absorption as a function of scattering angle depending on the geometry of the beamline setup (Sect. 7.6, p. 76). Measure and eliminate instrumental broadening (cf. Sect. 8.2.5.3). Carry out polarization correction, i.e., divide each intensity by the polarization factor (cf. Sect. 2.2.2 and ( [6], p. 99)). Transform the data to scattering vector representation (20 —> s). Carry out the Lorentz correction ... 

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