Direction cosine method

We will not pursue the analysis in detail any further. It does illustrate the power of spherical tensor methods, and one can only shudder at the possibility of developing the theory in a cartesian coordinate system, with direction cosines. We list the final values of the molecular parameters for 14N35C1 in table 10.12. The values of the hyperfine constants may be interpreted semi-empirically in the following way. The outmost pair of electrons occupy a 3 /rT molecular orbital and the Fermi contact constants, given in table 10.12, may be compared with the atomic values [144] of 1811 and 5723 MHz for the nitrogen and chlorine atoms respectively one concludes that the s electron character... 

In some crystalline systems where there are no reflecting planes normal to the desired crystallographic direction one can still obtain the orientation for this direction providing the crystalline structure is defined. This method has been developed by Wilchinsky (77). He shows that by measuring the orientation functions associated with various known [hkl] planes, one can calculate the value of the desired orientation function. This method should not be confused with the direct application of the direction cosines relationship as discussed earlier. 

It is necessary to define precisely what is meant by a molecule-fixed component of any angular momentum that includes R (e.g., R, N, O, J, and F). Brown and Howard do this using spherical tensor methods the following derivation is in terms of direction cosines. The direction cosine operator, a, is a 3 x 3 matrix, the components of which are defined by... 

Differential and Moment Methods. The radiative transfer equation can be approximated using the moments of radiation intensity instead of intensity itself. A moment is defined as the integral of intensity multiplied by a power of a direction cosine over a predetermined solid angle division. For example,... 

Coll and Harris (1970) have described another method of dealing with the librations of a crystal of linear molecules by use of direction cosine displacement coordinates. These authors eliminate the 2-component (AJ at the beginning of the treatment and thereby avoid the redundancy problem. 

AP = p — Pq, where = sin 0 and Pq = sin 0j, is a method of expressing the angle between the specular and scatter directions in direction cosine space along the surface for scatter in the PLIN. This is very useful normalization when scatter results only from surface microroughness, and the grating equation ... 

In microporous materials where Knudsen diffusion prevails, De cannot be calculated by solving Fick s law. The use of a discrete particle simulation method such as dynamic MC is appropriate in such cases (Coppens and Malek, 2003 Zalc et al., 2003, 2004). In the Knudsen regime, relatively few gas molecules collide with each other compared with the number of collisions between molecules and pore walls. One of the fundamental assumptions of the Knudsen diffusion is that the direction in which a molecule rebounds from a pore wall is independent of the direction in which it approaches the wall, and is governed by the cosine law the probability d.v that a molecule leaves the surface in the solid angle dm forming an angle 0 with the normal to the surface is... 

The value of / is derived via comparison with model calculations. Model calculations can be carried out with Monte Carlo methods The particles start at random positions outside the cavity with a randomly chosen direction. The angular distribution of the particle directions does, however, not necessarily have to be uniform. Each particle is followed if it enters the slit and as long as it is inside the cavity. Upon each wall collision a fraction s of the particle sticks to the wall and a fraction r = 1 — / is re-emitted with a cosine distribution with respect to the surface normal. When only a negligible part of the particle is left, e. g. 10-3, the next particle is started outside the cavity. In general, the trajectories of more than 106 particles have to be calculated to reach good statistics. For convenience, s = / is chosen in the calculation (this is equivalent to 7 = 0). As said before, the normalized profiles depend on the surface loss probability (3 only, so that this choice has no influence on the profile. 

White, P. S., and Woolfson, M. M. The application of phase relationships to complex structures. VII. Magic integers. Acta Cryst. A31, 367-372 (1975). Hauptman, H. On the identity and estimation of those cosine invariants, cos(V>i + 2 + V a -I- which are probably negative. Acta Cryst. A30, 472-476 (1974). Patterson, A. L. A direct method for the determination of the components of interatomic distances in crystals. Z. Krist. A90, 517-542 (1935). 

Most problems in crystallography " are not linear therefore, it is not possible to solve for the parameters directly. For example, the structure factor F hkl) is not a linear function of x, y, z, and B, because it contains cosine functions (Figure 6.21, Chapter 6). If, however,we have an initial set of parameters that are significantly better than a random guess, we can use the method of least squares to improve the parameters. The use of a Taylor s series allows certain functions, such as cosines and sines, to be expanded as a convergent power series so that the higher powers of X, the parameter, make successively smaller and smaller contributions to the total function. If/(x) is a differentiable function for x = xq -i-Ax, then the Taylor s series expression is ... 

The pressure difference is directly proportional to the cosine of the contact angle. For a nonwetting fluid, 6 approaches 90°, and AP approaches zero. The implication of immobilizing a nonwetting or poorly wetting fluid through solvent-exchange or other methods is that the Immobilized liquid membrane would have little resistance to small transmembrane pressures. 

An alternate method of deriving the vectors S( is illustrated in the case of the bending coordinate as follows. The scalar product of unit vectors directed outwards from the central atom gives the cosine of the angle ... 

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