Lorentz correction factor

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-. 

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... 

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. 

In this text, we will use the Lorentz correction factor, which arises from the assumption that the molecule is located in a spherical cavity ... 

The used S5mbols are K, scale factor n, number of Bragg peaks A, correction factor for absorption P, polarization factor Jk, multiplicity factor Lk, Lorentz factor Ok, preferred orientation correction Fk squared structure factor for the kth reflection, including the Debye-Waller factor profile function describing the profile of the k h reflection. 

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. 

Munera [57] took into account both earth rotation and orbital motion, as a function of the local latitude and longitude. Prediction of the variation of speed difference as function of time of day are given in Munera [57] for the locations of Miller s experiment. The qualitative shape of the variations is of the same sort observed by Miller in the 1930s. However, the magnitudes are not correct because solar motion was not included.2 Selleri [58] allowed for small violations of Lorentz invariance a correction factor around 10-3 reproduces Miller s observations. Also independently, Allais [59] revisited Miller s work. He argues that Miller s seasonal variations are strong proof for a local anisotropy of space. 

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. 

In this section, a simple description of the dielectric polarization process is provided, and later to describe dielectric relaxation processes, the polarization mechanisms of materials produced by macroscopic static electric fields are analyzed. The relation between the macroscopic electric response and microscopic properties such as electronic, ionic, orientational, and hopping charge polarizabilities is very complex and is out of the scope of this book. This problem was successfully treated by Lorentz. He established that a remarkable improvement of the obtained results can be obtained at all frequencies by proposing the existence of a local field, which diverges from the macroscopic electric field by a correction factor, the Lorentz local-field factor [27],... 

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... 

Data reduction Conversion of intensities, I(hkl), to structure amplitudes. F hkl), by application of various correction factors including Lorentz factors, polarization factors, and absorption factors. 

Lorentz factor A factor that is used to correct diffraction intensities that allows for the varying time that different Bragg reflections (as reciprocal lattice points with finite sizes) take to pass through the sphere of reflection (Ewald sphere). The value of this correction factor depends on the scattering angle and the geometry of the measurement of the Bragg reflection. 

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 ... 

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 ... 

Figure 14.13 Relative Lorentz factor. The image depicts the surface of the correction factor for a primary beam that is collimated to below the sample size. Should the primary beam be larger than the sample, the factor is unity.

The other two correction factors account for the effects of induced dipoles in the medium through electronic polarization, and are known as the Lorentz-Lorenz correction factors. The correction for the optical field at frequency 2fti is given by ... 

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 ... 

Structure factors were computed for all crystallographic planes possible, and the values thus obtained (with the exception of those for the meridional reflections) were then corrected for multiplicity and for the Lorentz polarization factor, in the usual maimer,in order to obtain the calculated intensities for all reflections. The meridional reflections were not used, as the Lorentz factor does not apply to them, and, in general, the meridional intensities found experimentally are unreliable. 

To determine the coordinates of these atoms, the intensities of about 300 structurally independent reflections of the hkO and Okf type were measured and anal3 ed. The reflection intensity was measured by the ionization method. The measurements were made on a spherical crystal of 0.51-mm diameter. The spherical shape was obtained by rolling [6]. Corrections were made for the absorption and for the Lorentz-polarization factor. 

Fig. 3.6 The effective electric field acting on a molecule in a polarizable medium (shaded rectangles) is Eiq fErtj d where is the field in the medium and / is the local-field correction factor, hi the cavity-field model (A) Ei c is the field that would be present if the molecule were replaced by an empty cavity (Ecav), in the Lorentz model (B) Ei c is the sum of E av and the reaction field (Ereaa) resulting from polarization of the medium by induced dipoles within the molecule (P)...

The factor (n + 2)13 is called the Lorentz correction. Liptay [19] gives expressions that include the molecular radius, dipole moment and polarizability explicitly. More elaborate expressions for /also have been derived for cylindrical or elipsoidal cavities that are closer to actual molecular shapes [20, 21]. 

Figure 3.7 shows the local-field correction factors given by Eqs. (3.35) and (3.36). The Lorentz correction is somewhat larger and may tend to overestimate the contribution of the reaction field, because the cavity-field expression agrees better with experiment in some cases (Fig. 4.5). 

Fig. 3.7 The cavity-field (/mv. solid curve) and Lorentz (/j, dashed curve) correction factors for the local electric field acting on a spherical molecule in a homogeneous medium, as a function of the refractive index of the medium...

Fig. 4.5 Dipole strength of the long-wavelength absorption band of bacteriochlorophyll-a, calculated by Eq. (4.16a) from absorption spectra measured in solvents with various refractive indices. Three treatments of the local-field correction factor (/) were used down triangles, f= 1.0 (no correction) filled circles, f is the cavity-field factor empty circles, f is the Lorentz factor. The dashed lines are least-squares fits to the data. Spectra measured by Connolly et al. [148] were converted to dipole strengths as described by Alden et al. [4] and Knox and Spring [5]...

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