Contour projection length

The Maier-Saupe theory is now adapted to take concentration and flexibility into account via the strength of the anisotropic potential. This is done by power-law relations. First the contour projection length L(T) is introduced. It is defined as the projection of a polymer chain along the direction of the first segment. Next it is assumed that the strength of the potential is given by... 

Lch being the contour length of the chain. The projection length L of the chain at a tensile stress [Pg.490]

Fig. 4. Tentative allocation of probe binding sites within the three-dimensional structure of Ca -ATPase derived from vanadate-induced E2-type crystals. The top picture is the projection view of the Ca -ATPase down the x-axis, revealing the pear-shaped contours of ATPase molecules. The maximum length of the cytoplasmic domain to the tip of the lobe is =r65A. In the middle and bottom pictures the same structure is viewed down the x-axis, revealing the gap between the bridge and the bilayer surface and the connections between ATPase molecules in neighboring dimer chains. The proposed binding sites for lAEDANS and FITC are indicated. The bottom right picture is the same structure viewed down the y-axis. Adapted from Taylor et al. [90].

Average projection of the end-to-end vector on the tangent to the chain contour at a chain end in the limit of infinite chain length. 

Fig. 1 [Reproduced from ref. (15)]. Electron density distribution of NaCI at 290° K in a plane (x, y, 0), after Schoknecht (17). Note that this is not the projection of the electron density on this plane. The length of the diametrical lines indicates the total error of the result. The contour lines with 0.1 and 0.2 el/A3 published by Witte and Wolfel (14), contrary to Fig. 1, have quadrilateral-like shapes. The authors emphasize that this is caused by the deviation of some reflections from the straight Gaussian lines . They found the deviations larger than those recorded by Schoknecht. Higher precision in the measured values of the integral intensities is necessary to decide the issue unequivocally...

Two or more vertices within a defined small space indicate the location of an atom in the structure (see Figure 4.9c). Atoms are numbered, and the neighborhood relationship among them is kept. Any atom with only a single neighbor is considered a possible terminal atom. A linear projection of its previous bond up to a distance similar to the length of that bond is made (see Figure 4.9d). If no on-pixels are encountered, the atom is considered a default carbon atom in the structure. Otherwise, the contour determination process is done over the newly found on-pixel. In this way, the detected chemical symbols are submitted to the OCR module. 

The deterioration of the grating (shown in Fig. 91) with successive pulses cannot be attributed to thermal effects, since in an experiment using a mask and projection technique (1 pm slit pattern and multiple pulses with 200 mj cm-2) sharp contours remained. Additionally, the thermal diffusion length for the laser pulse corresponds only to 30-40 nm for commercial polymers such as poly(ethylene terephthalate). 

Fig. 11. Features of a difference electron density projection for TMA (3) H O PA. The difference electron density which represents the enclathrated picric acid molecules is contoured at levels of 1, 2, 3 e A In TMA H2O PA the molecules lie in the (III) planes and the hydrogen bonds do not lie in the plane of the projection but are directed out of this plane. For TMA H O the present diagram serves as a slightly distorted representation of the arrangement in the layers of the framework TMA and HjO molecules, as viewed along a. The hydrogen bonds have the following lengths, averaged over corresponding bonds in TMA H O % PA and the five independent layers in TMA HjO (the values in parentheses are the RMS deviations from the means) a 2.641 (15) A b 2.632 (52) c 2.874 (44) d 2.850 (52) e 2.592 (29). (Taken from Ref. )...

Let us repeat derivation of Equation (3) by projection of the random walk on a periodic lattice (see Figure 2). As before, we suppose the chain molecule consists of N chemically identical units each of length a. Its contour length is Na. Owing to its flexibility the chain forms a coil. What is the average end-to-end distance of the coil (r ) ... 

In Fig. 3, the lowest contour levels indicate the lowest positive density in the reconstruction. These levels indicate the maximal dimensions of the monomer in projection. The maximal length is 10 nm, the maximal width occurs asymmetrically at one side of the particle and is about 7.5 nm. Both dimensions are somewhat smaller than those found for the top views of the isolated monomers (9-11 nm), The latter, however, contain contributions from the detergent Dodecylmaltoside, suggesting that in our crystals the detergent has partially been removed. The area covered by one monomer (58 nm leads to a volume of about 350 nm if a thickness of 6 nm is assumed. The molecular mass of a monomer would then be 175 40 kDa, assuming a specific volume of 2 + 0.5 nm /kDa. This mass seems sufficient to include a monomeric complex of apoproteins of CP47, Dl,... 

Fig. 11.19 shows a plan view (horizontal projection) of the site. In this projection the contours of the rooms and the main drift are marked by continuous lines (Rooms 2 and 3) and dashed lines (Rooms 9s and 9n). 24 AE sensors are installed in boreholes of up to 30 m length which were drilled from the drift (gray area) because the rooms are not accessible anymore. 

The central region of approximately constant film thickness remains, but the relative change In film thickness along Its length Is much larger. This feature may be observed In Isometric projections of film shape, but It Is not particularly evident on the contour map due to the contour positions. The film thickness ratio (H, Is shown, as a... 

The distances from the source to various time-integrated concentrations or ground deposition predicted by Equations (5) through (7) depend upon a constant wind velocity. This condition seldom prevails for any length of time, and it certainly varies with distance as the cloud moves downwind. On the other hand, areas enclosed by contours of equal concentration or contamination are far less dependent upon the constant velocity assun tlon. As a matter of fact, the areas enclosed by these Isopleths have proven to be much more predictable than distances In both Project Prairie Qrass(3) and Hanford s Green Glow experiments... 

Let us apply these ideas to the case of flexible polymers in solution. We assume the chains to be composed of N monomers of dimension a, giving a contour length, Lq = Na. In the absence of interactions, the polymer takes on a random walk, or Gaussian, conformation in order to maximize its entropy. This results in a random coil of projected size, Rq (N 1). Let us consider a... 

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