Static contour

The origin of the rotational structure of the isotropic Q-branch (Av = 0, Aj = 0) is connected with the dependence of the vibrational transition frequency shift on rotational quantum number j [121, 126] 

As soon as condition (3.2) is satisfied the static contour 6q(o)) arises, which is already insensitive to broadening of individual rotational components, but has not yet transformed as a whole. The static contour is almost indifferent to further increase in gas density until 

The shape of the static contour is readily calculated from Eq. (2.13) with the correlation function 

As discussed in [91], the shape of a static spectrum determines significantly the spectral transformation as frequency exchange increases. In particular, spectral narrowing will take place only if the second moment of the spectrum is finite. In our case 

Consequently, when the gas density increases, the increase of the frequency exchange rate causes the narrowing of the isotropic scattering spectrum. 

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The intensity at the periphery of the line ( Ageneral rule (2.62) [20, 104]. However, the most valuable advantage of general formula (3.34) is its ability to describe continuously the spectral transformation from a static contour to that narrowed by motion (Fig. 3.1). In the process of the spectrum s transformation its maximum is gradually shifted, the asymmetry disappears and it takes the form established by perturbation theory. 

The results obtained allow one to follow the collapse of a static contour and its further narrowing by collisions. The limiting cases are the simplest to describe. In particular, from (6.45) it can be easily obtained that as 1/tj -> 0... 

Each term of this sum corresponds to a static contour of the corresponding branch, except the Q-branch, which by virtue of (6.29) is a -function. In the opposite limiting case (as 1 /xj —> oo)... 

Convection. In these mixers an impeller operates within a static shell and particles are moved from one location to another within the bulk. bbon Tjpe. Spiral or other blade styles transfer materials from one end to the other or from both ends to the center for discharge (Fig. 37b). This mixer can be used for dry materials or pastes of heavy consistency. It can be jacketed for heating or cooling. Blades can be smoothly contoured and highly pohshed when cleanliness is an important process requirement. 

Cavitation Loosely regarded as related to water hammer and hydrauhc transients because it may cause similar vibration and equipment damage, cavitation is the phenomenon of collapse of vapor bubbles in flowing liquid. These bubbles may be formed anywhere the local liquid pressure drops below the vapor pressure, or they may be injected into the hquid, as when steam is sparged into water. Local low-pressure zones may be produced by local velocity increases (in accordance with the Bernouhi equation see the preceding Conservation Equations subsection) as in eddies or vortices, or near bound-aiy contours by rapid vibration of a boundaiy by separation of liquid during water hammer or by an overaU reduction in static pressure, as due to pressure drop in the suction line of a pump. 

Fig. 17. Contour plots for a

The quasi-classical description of the Q-branch becomes valid as soon as its rotational structure is washed out. There is no doubt that at this point its contour is close to a static one, and, consequently, asymmetric to a large extent. It is also established [136] that after narrowing of the contour its shape in the liquid is Lorentzian even in the far wings where the intensity is four orders less than in the centre (see Fig. 3.3). In this case it is more convenient to compare observed contours with calculated ones by their characteristic parameters. These are the half width at half height Aa)i/2 and the shift of the spectrum maximum ftW—< > = 5a>+A, which is usually assumed to be a sum of the rotational shift 5larger scale A determined by vibrational dephasing. 

Figure 3. Static deformation density in Si-O-Si bridge planes Si,-0,-Si2 and Si,-O,o-Sij. Contours as in Figure 1.

Fig. 3. Contour plot of the proton-detected local field (PDLF) spectrum of a static 5CB sample. The traces taken parallel to for each carbon site show distinct doublets that are related to individual C-H couplings. (Reproduced by permission of American Chemical Society.)...

FIG. 11.3 Comparison of the ab-initio local density functional deformation density for Si with the experimental static model deformation density. Contour interval is 0.025 e A-3. Negative contours are dashed lines. Source Lu and Zunger (1992), Lu et al. (1993). 

Figure 1.65 CFD simulations giving 3-D mass contour plots in the cross-channel structure for a design without and one with two static mixing elements. The completeness of mixing can be judged from the cross-sectional mass distribution at the outlet [71] (by courtesy of Elsevier Ltd.).

The velocity contour plots show a higher velocity, a higher velocity gradient and rapid change of the direchon of the velocity components in the proximity of the static mixing elements compared with the rest of the flow in the channel (see Figure 1.66) [71]. 

Table 1.5 Measures for mixing efficiency calculated from mass contour plots yielded by CFD simulation - benchmarking cross-shaped mixers with and without static mixing elements (SME) [71].

Here Lw is the contour length of the chain. In an effort to generalize Eq. (14) for non-theta solutions, Reed et al. [46] have provided an ad hoc treatment by combining theories of Odijk [35], Odijk and Houwaart [36], Skolnick and Fixman [37], and Gupta and Forsman [49]. Here they append an additional contribution f30 to A arising from short-ranged non-electro-static interactions, so that f3 of Eq. (11) is given by... 

Fig. 15. Comparison between the static, MAS and MQMAS 23Na NMR spectra of multisite sodium salts. Contour levels in the 2D plots were taken at 70%, 35%, 16%, 8%, and 4% of the maximum spectral intensities asterisks indicate the spinning sidebands. (Reproduced, with permission, from Medek et a/.209)...

Fig. 9. Polymorphism in p-nitrophenol static deformation density in the plane of the phenyl rings for the a- and the (3-forms (contours at 0.1 eA-3). Intramolecular and lone-pair regions exhibit many differences. Relief maps of the Laplacians in the inteimolecular hydrogen bond region are also shown (range -250 to 250 eA-5). In the a-fonn, H(l) bonds not only with 0(3) but also with 0(2) and N(l) of the neighboring nitro group (reproduced with permission from Kulkami et al. [61]).

Fig. 15. (a) Intramolecular hydrogen bonds in urea crystal with displacement ellipsoids at 50% probability, (b) Static deformation density obtained from the multipolar analysis of the experimental data corrected for the thermal diffuse scattering. Theoretical deformation density obtained using (c) the Hartree-Fock method (d) the DFT method by generalized gradient approximation (contours at 0.0675 eA-3) (reproduced with permission from Zavodnik et al. [69]). 

Figure 3. ORTEP view (a) and theoretical static deformation density of H3PO4 in the 0=P-0(H) plane (b). Contours interval 0.1 e A 3 (—) positive contours, ( ) negative contours zero contour omitted.

Figure 4. Model static deformation densities of H3PO4 in the 0=P-0(H) plane from simulated structure factors with U = 0 (a), at 75K (b), at 150K (c), at 300K (d). Contours as in Figure 3.

Figure 9. Experimental static maps in the plane of the tyrosine (a) and of the phenylalanine residue, (b) in leu-enkephalin. Contours as in Figure 6.

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