Lorentz equations corrections

If the craze layer extends with complete lateral constraint, the strain in the craze is related to the change in its density. From a relationship between density and refractive index, an equation between strain in the craze and its refractive index can be derived. Although it is usual to start with the Lorenz-Lorentz equation, this may not be the correct relationship for a material having the structure of the craze (9). For the present purposes a linear relationship is assumed. The error introduced is at most 10% and only a few percent for the stretched craze with a high void content. 

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). 

The Wess-Zumino term in Eq. (11) guarantees the correct quantization of the soliton as a spin 1/2 object. Here we neglect the breaking of Lorentz symmetries, irrelevant to our discussion. The Euler-Lagrangian equations of motion for the classical, time independent, chiral field Uo(r) are highly non-linear partial differential equations. To simplify these equations Skyrme adopted the hedgehog ansatz which, suitably generalized for the three flavor case, reads [40] ... 

To obtain the absorbances at 910 and 967 cm. 1, it was necessary to correct the observed band intensities for the overlapping of adjacent bands. The band at 910 cm."1 for the vinyl group was corrected for the absorbance from the wing of the 967-cm."1 frarw-vinylene band,. and the latter band was corrected for the vinyl band at 995 cm. 1. The Lorentz band shape equation was used to calculate the absorbance in the wings, and in the thicker specimens, successive approximations were necessary. This treatment gave the four equations below, which yielded the concentrations of trans and vinyl groups for the emulsion and sodium polybutadienes listed in Table I. Implicit in these equations is the assumption that the absorptivities are independent of concentration. 

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... 

For each Bragg reflection, the raw data normally consist of the Miller indices (h,k,l), the integrated intensity I(hkl), and its standard deviation [ a[I) ]. In Equation 7.2 (earlier), the relationship between the measured intensity / [hkl] and the required structure factor amplitude F[hkl) is shown. This conversion of I hkl) to F hkl) involves the application of corrections for X-ray background intensity, Lorentz and polarization factors, absorption effects, and radiation damage. This process is known as data reduction.The corrections for photographic and diffractometer data are slightly different, but the principles behind the application of these corrections are the same for both. 

Liquid Phase Calculations of the Linear Response. The data in Table 5 for the isotropic polarizability, derived formally via the Lorentz-Lorenz equation (1) from the measured refractive index, shows that the assumption that individual molecular properties are largely retained at high frequency in the liquid is very reasonable. While the specific susceptibilities for the gas and liquid phases differ, once the correction for the polarization of the surface of a spherical cavity, which is the essential feature of the Lorentz-Lorenz equation, has been applied, it is clear that the average molecular polarizabilities in the gas and liquid have values which always agree within 5 or 10%. 

Now, a theorem in Riemannian geometry tells us that locally any metric (7) with the correct signature can be rewritten as (6) by an appropriate change of coordinates. At different points we use different transformations of coordinates, but always end up with the Lorentz metric in the new coordinates. So the equation (8), when written in terms of the coordinates for which the metric looks like (7), must describe the trajectory in the gravitational field. This is the geodesic equation (sum over / , 7)... 

Although empirical and lacking a theoretical basis, (2) is useful for interpolation purposes. Lorentz (1909) himself pointed out possible causes for the slight limitations of (1), and Bottcher (1952) has discussed an appropriate correction nevertheless, in practice the inconstancy of (1) with temperature is usually within the experimental error and the equation may safely be used as written above. 

The remaining factors in equation (6.1) have to be calculated for each reflection separately, i.e. LPA, the Lorentz, polarisation and absorption corrections respectively. The following relationship will now be considered in more detail ... 

This relationship as such is not well obeyed for most compounds if the static or low-frequency relative permittivity is used, as can be judged from Table 11.1. The relationship can be correctly interpreted by using the relative permittivity due to electronic polarisation in the equation. With this in mind, substitution of the relationship given in Equation (11.10) into the Clausius-Mossotti equation yields the Lorentz-Lorenz equation ... 

In the above equations, /3 and y" represent the microscopic coefficients at site n which are averaged over molecular orientations 0 and 0 and summed over all sites n. The terms F(a>i) are the local field corrections for a wave of frequency Generally, one utilizes the Lorentz approximation for the local field, where ... 

A change between different coordinate systems can be described by a Lorentz transformation, which may mix space and time coordinates. The postulate that physical laws should be identical in all coordinate systems is equivalent to the requirement that equations describing the physics must be invariant (unchanged) to a Lorentz transformation. Considering the time-dependent Schrbdinger equation (8.1), it is clear that it is not Lorentz invariant since the derivative with respect to space coordinates is of second order, but the time derivative is only first order. The fundamental structure of the Schrbdinger equation is therefore not relativisticaUy correct. 

The intensity distributions calculated for the atomic models (using equation 8) are shown in fig. 6, together with the diffractometer traces overlaid for direct comparison. The calculated data are for a distribution of chain lengths centered on M=10, and have been corrected for the Lorentz and polarization effects. The intensity agreement is very good, especially for the peaks at d v sA, and... 

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