Preferred ray directions

Transit time for the lateral shift 10-6 Lateral shift in planar waveguides 10-7 Preferred ray directions... 

The geometric optics analysis of the excitation of the planar waveguide by sources assumes that power may enter any bound ray whose direction belongs to the continuum of directions defined by Eq. (I-5a). However, as a consequence of diffraction, power enters certain preferred directions only, and the smaller V, the smaller the number of preferred directions. If V is sufficiently small, then there is only one preferred direction and all bound rays propagate with the same transit time. As V increases, the number of preferred ray directions becomes sufficiently large that it can be accurately approximated by a continuum. 

Thus the modal and ray transit times are equal only when tj - 1. This condition is satisfied only by those rays belonging to modes well above cutoff, i.e. when Vp U, or, equivalently, when 0 < 0c- Hence is inaccurate for arbitrary values of 9. This inaccuracy arises because the ray transit time ignores diffraction effects, which were discussed in Chapter 10. The step-profile planar waveguide is a special case, however, because all diffraction effects can be accounted for exactly by including the lateral shift at each reflection, together with recognizing the preferred ray directions. TWs was carried out in Section 10-6, and for rays, or local plane waves, whose electric field is polarized in the y-direction in Fig. 10-2, leads to the modified ray transit time of Eq. (10-13). If we use Table 36-1 to express 0, and 0(.in terms of U, Vand Wand substitute rj for TE modes from Table 12-2, we find that Eqs. (10-13) and (12-8) are identical since 0 = 0. It is readily verified that the same conclusion holds for TM modes and local plane waves whose magnetic field is polarized in the y-direction of Fig. 10-2. 

The most conventional investigations on the adsorption of both modifier and substrate looked for the effect of pH on the amount of adsorbed tartrate and MAA [200], The combined use of different techniques such as IR, UV, x-ray photoelectron spectroscopy (XPS), electron microscopy (EM), and electron diffraction allowed an in-depth study of adsorbed tartrate in the case of Ni catalysts [101], Using these techniques, the general consensus was that under optimized conditions a corrosive modification of the nickel surface occurs and that the tartrate molecule is chemically bonded to Ni via the two carbonyl groups. There were two suggestions as to the exact nature of the modified catalyst Sachtler [195] proposed adsorbed nickel tartrate as chiral active site, whereas Japanese [101] and Russian [201] groups preferred a direct adsorption of the tartrate on modified sites of the Ni surface. 

It shall be assumed in this chapter that molecular arrangement in the bulk of solid explosives, and all amorphous and liquid explosives, has no preferred orientation direction. The diffraction patterns in this case are isotropic around the primary X-ray beam, and the vector quantity, x, can be replaced by its scalar magnitude. It is customary to speak of diffraction profiles, rather than patterns, when isotropy obtains and the diffraction profiles are derived by integration of the (circularly-symmetric) diffraction pattern over the azimuthal component of the scattering angle. 

Instrumental Quantitative Analysis. Methods such as x-ray spectroscopy, oaes, and naa do not necessarily require pretreatment of samples to soluble forms. Only reUable and verified standards are needed. Other instmmental methods that can be used to determine a wide range of chromium concentrations are atomic absorption spectroscopy (aas), flame photometry, icap-aes, and direct current plasma—atomic emission spectroscopy (dcp-aes). These methods caimot distinguish the oxidation states of chromium, and speciation at trace levels usually requires a previous wet-chemical separation. However, the instmmental methods are preferred over (3)-diphenylcarbazide for trace chromium concentrations, because of the difficulty of oxidizing very small quantities of Cr(III). 

Whereas imaging agents that can serve to detect molecular and cellular processes must in some way directly or indirectly participate in these processes or interact with components participating in the processes, the development of X-ray contrast agents led in the opposite direction Molecules displaying the least possible interaction with body constituents and processes were preferred because of advantages in tolerability [2]. 

The initial results of an early directed evolution study are all the more significant, because no X-ray data or homology models were available then to serve as a possible guide [89]. In a model study using whole E. coU cells containing the CHMO from Adnetohacter sp. NCIM B9871,4-hydroxy-cydohexanone (3 5) was used as the substrate. The WT leads to the preferential formation of the primary product (i )-36, which spontaneously rearranges to the thermodynamically more stable lactone (R)-37. The enantiomeric excess of this desymmetrization is only 9%, and the sense of enantioselectivity (R) is opposite to the usually observed (S)-preference displayed by simple 4-alkyl-substituted cydohexanone derivatives (see Scheme 2.10) [84—87]. 

In the case of selective oxidation catalysis, the use of spectroscopy has provided critical Information about surface and solid state mechanisms. As Is well known( ), some of the most effective catalysts for selective oxidation of olefins are those based on bismuth molybdates. The Industrial significance of these catalysts stems from their unique ability to oxidize propylene and ammonia to acrylonitrile at high selectivity. Several key features of the surface mechanism of this catalytic process have recently been descrlbed(3-A). However, an understanding of the solid state transformations which occur on the catalyst surface or within the catalyst bulk under reaction conditions can only be deduced Indirectly by traditional probe molecule approaches. Direct Insights Into catalyst dynamics require the use of techniques which can probe the solid directly, preferably under reaction conditions. We have, therefore, examined several catalytlcally Important surface and solid state processes of bismuth molybdate based catalysts using multiple spectroscopic techniques Including Raman and Infrared spectroscopies, x-ray and neutron diffraction, and photoelectron spectroscopy. 

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