Phase differences

These constants are dependent upon pressure, temperature and also the composition of the hydrocarbon fluid, as the various components within the system will interact with each other. K values can be found in gas engineering data books. The basic separation process is similar for oil and gas production, though the relative amounts of each phase differ. 

This region has been divided into two subphases, L and S. The L phase differs from the L2 phase in the direction of tilt. Molecules tilt toward their nearest neighbors in L2 and toward next nearest neighbors in L (a smectic F phase). The S phase comprises the higher-ir and lower-T part of L2. This phase is characterized by smectic H or a tilted herringbone structure and there are two molecules (of different orientation) in the unit cell. Another phase having a different tilt direction, L, can appear between the L2 and L 2 phases. A new phase has been identified in the L 2 domain. It is probably a smectic L structure of different azimuthal tilt than L2 [185]. 

Depending on the relative phase difference between these temis, one may observe various experimental spectra, as illustrated in figure Bl.5.14. This type of behaviour, while potentially a source of confiision, is familiar for other types of nonlinear spectroscopy, such as CARS (coherent anti-Stokes Raman scattering) [30. 31] and can be readily incorporated mto modelling of measured spectral features. 

For molecules having dimensions comparable with the wavelength, phase differences will occur between waves scattered from different regions of the molecule. These phase differences result in an angular dependence of the scattered intensity. The reduction may be expressed in temis of a particle interference factor P(2Q) such that... 

This transition is usually second order [18,19 and 20]. The SmC phase differs from the SmA phase by a tilt of the director with respect to the layers. Thus, an appropriate order parameter contains the polar (0) and azimuthal ((]i) angles of the director ... 

In order to write the previously obtained equations in the nearly nonrelativistic limit, we introduce phase differences s, that remain finite in the limit c —> oo. Then... 

We must describe the light scattered with interference in terms of phase differences that develop as the waves pass through a molecule consisting of multiple scattering sites. 

We must find a way to describe these phase differences in terms of the distances traveled through the array of scattering sites, since this is how the size of the molecule enters the theory. 

For Rayleigh scattering, 5j - 6, = 0-there are no phase differences-and each of the cosine terms in Eq. (10.71) equals unity. In this case, which corresponds to i Rayleigh Ee right-hand side of Eq. (10.71) equals E n, and we can write... 

Fig. 10. The rotary actuator (a) side view where SAW = surface acoustic wave and (b) view of the poled pie2oelectric ceramic ring showing poled segments and how temporal and spatial phase differences are estabUshed. Courtesy of Shinsei Kogyo Co.

Interference of Waves. The coherent scattering property of x-rays is used in x-ray diffraction appHcations. Two waves traveling in the same direction with identical wavelengths, X, and equal ampHtudes (the intensity of a wave is equal to the square of its ampHtude) can interfere with each other so that the resultant wave can have anywhere from zero ampHtude to two times the ampHtude of one of the initial waves. This principle is illustrated in Figure 1. The resultant ampHtude is a function of the phase difference between the two initial waves. 

In Figure la, the two waves have a zero phase difference and the resultant ampHtude is twice that of each of the initial waves. In this case the waves are in phase with one another and the interference is "constmctive."... 

In Figure lb, the two waves have a phase difference of 1/4 of the wavelength, X/4, and the resultant ampHtude is the square root of two times that... 

A whole science, called metallography, is devoted to this. The oldest method is to cut the alloy in half, polish the cut faces, etch them in acid to colour the phases differently, and look at them in the light microscope. But you don t even need a microscope to see some grains. Look at any galvanised steel fire-escape or cast brass door knob and you will see the grains, etched by acid rain or the salts from people s hands. 

How do we find phase differences between diffracted spots from intensity changes following heavy-metal substitution We first use the intensity differences to deduce the positions of the heavy atoms in the crystal unit cell. Fourier summations of these intensity differences give maps of the vectors between the heavy atoms, the so-called Patterson maps (Figure 18.9). From these vector maps it is relatively easy to deduce the atomic arrangement of the heavy atoms, so long as there are not too many of them. From the positions of the heavy metals in the unit cell, one can calculate the amplitudes and phases of their contribution to the diffracted beams of the protein crystals containing heavy metals. 

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