Relative Lorentz factor

Figure 14.12 Relative Lorentz factor. The image depicts a perpendicular section of a capillary of diameter L being illuminated by a beam of width W. Only crystallites falling in the light grey inner circle are rotated completely (27t) within the beam. Crystallites outside this region but still within the beam path only experience a limited rotation co, thus reducing the effective single crystal Lorentz factor to be applied to them.

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.

Since ip depends on space-time coordinates, the relative phase factor of ip at two different points would be completely arbitrary and accordingly, a must also be a function of space-time. To preserve invariance it is necessary to compensate the variation of the phase a (a ) by introducing the electromagnetic potentials (T4.5). In similar vein the gravitational field appears as the compensating gauge field under Lorentz invariant local isotopic gauge transformation [150]. 

In the case of a static field, the macroscopic relative permittivity e° has to be used in (82) for the cavity field factor, while the optical relative permittivity extrapolated to infinite wavelength e can be applied to estimate the static polarizability a(0 0) in (84). In this way the Onsager-Lorentz factor for a pure dipolar liquid is obtained (87). 

The microphysics in the field generation and particle acceleration described here is clearly beyond the reach of the magneto hydrodynamic approximation. A parameter study utilizing a PIC code working from first principles is necessary to fully understand the interdependence between the relative bulk Lorentz factors of the colliding plasma shells, the power law index of the non-thermal electron population, eb and in a broader sense the detailed evolution and structure in collisionless shocks. 

On the right-hand side of relation (6.3) are the L, P and A factors. The Lorentz factor, L, takes account of the relative time each reflection spends in the reflecting position. It depends on the precise diffraction geometry used. For example, in the rotation method... 

Note that we have restricted ourselves to proper Lorentz transformations with A°o > 1/ cf. Eq. (3.18). Application of the Lorentz back and forth transformations to the f-coordinate yields exactly the same result, of course. The Lorentz factor 7 depends only on the relative velocity v between IS and IS is strictly greater than 1, and tends to unity and features a smooth and vanishing derivative with respect to v for v 0,... 

In some instances, it is possible to replace the above integrations by multiplying the intensity with appropriate factors analogous to Lorentz factors. Absolute intensity measurements is necessary if the intensity from a single sample is to be of any use. However, relative integrated intensities such as the one given in Table I can be of immense value in comparing the data from samples within a set... 

Of course, in Eq. (6) the contraction form factor p is valid only in the arm that is parallel to the velocity vector. Equation (6) was interpreted by Lorentz and Fitz-Gerald as a real contraction [17]. It is important to see that in Eq. (6) the hidden parameter p is only one possible solution for the contradiction, but the result of the M-M experiment allows numerous other solutions based on the inner properties and features of the light. The M-M experiment destroyed the world picture of classical physics, and it required a new physical system of paradigms. Thus, for example, the applicability of Galilean relativity principle was rendered invalid. 

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

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... 

The overall intensities of the peaks are related to the abundance of each phase in the sample. For each phase, the relative intensities may be determined by the calculation of structure factors, multiplicity factors, preferred orientation, and Lorentz/polarization factors. The latter two are normally tabled as a function of the scattering angle. The atomic arrangement within the cell also influences individual peak intensities via structure factor. 

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