The Three-Dimensional Case

For illustrative purposes, we will present some results for the case N = 3. In this case, we use the three Euler angles to describe the rotation matrix W in the (z,x,z) convention 

The influence of W is essentially attributable to the displacement parameters. According to Equations 4.69, there are a total of 15 parameters Three direct geometrical displacements defined by the vector ki23 = col (feu3i 123)- three reciprocal displacements specified in Equation 4.70 

Given values for all these displacements together with the frequencies factors = co /h and pj, = oi)J,/h (p = 1,2,3) (the latter, for instance, may be determined from spectroscopic data), the coefficients and 

Aj23 and ajGri normalized according to the procedure given in Section 8.1 (see Equations 8.14 and 8.15. The latter quantities are quite convenient because the numerical values have an order of magnitude of 1. Next, starting from 8p,v and Op,v, we obtain Cj, v, gft.v, and dj,, via Equations C1-C8. With all of these coefficients and the initial value of I3, Equation 4.98, the system of recurrence Equations 4.96 and 4.97 may be solved. 

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It is worth mentioning here several things for later use. Scheme (33) with the boundary conditions (45) is in common usage for step-shaped regions G, whose sides are parallel to the coordinate axes. In the case of an arbitrary domain this scheme is of accuracy 0( /ip + r Vh). Scheme (9)-(10) cannot be formally generalized for the three-dimensional case, since the instability is revealed in the resulting scheme. 

The wave function for this system is a function of the N position vectors (ri, r2,. .., r v, i). Thus, although the N particles are moving in three-dimensional space, the wave function is 3iV-dimensional. The physical interpretation of the wave function is analogous to that for the three-dimensional case. The quantity... 

The vector product X x Y is somewhat more complicated in matrix notation. In the three-dimensional case, an antisymmetric (or skew symmetric) matrix can be constructed from the elements of the vector AT in the form... 

Equation (9.3) has been derived for one-dimensional diffusion and supported by molecular dynamics simulation in the three-dimensional case (Powles, 1985 Tsurumi and Takayasu, 1986 Rappaport, 1984). For the partially diffusion-controlled recombination reaction we again refer to Figure 9.1, where the inner (Collins-Kimball) boundary condition is now given as... 

The phenomenon of nucleation considered is not limited to metal deposition. The same principles apply to the formation of layers of certain organic adsorbates, and the formation of oxide and similar films. We consider the kinetics of the growth of two-dimensional layers in greater detail. While the three-dimensional case is just as important, the mathematical treatment is more complicated, and the analytical results that have been obtained are based on fairly rough approximations details can be found in Ref. 3. 

Baer, M. Adiabatic and diabatic representations for atom-diatom collisions Treatment of the three-dimensional case, Chem.Phys., 15 (1976), 49-57... 

Let us consider the three-dimensional case and work within the Parrinel-lo-Rahman framework. A rather general three-dimensional b matrix of Eq. [31] will be considered ... 

Although they did not obtain a closed-form analytic expression for the three-dimensional case, they dealt with a trasformed one-matrix for the single Slater determinant constructed from plane waves, and rewrote the energy in terms of this transformed matrix. The conditions on the transformation were not imposed through the Jacobian but rather through the equations ... 

From the second of these derivatives, evaluate 0C as predicted by this model. Use this value of 6C and the first of these derivatives to evaluate the relationship between Tc and the two-dimensional a and b constants. How does this result compare with the three-dimensional case The van der Waals constant b is four times the volume of a hard -sphere molecule. What is the relationship between the two dimensional b value and the area of a haid-disk molecule ... 

For the three-dimensional case the system, [Eqs. (14) and (15)], applies also if in the first equation djjdq is replaced by div j (j being a vector) and in the second equation dnjdq is replaced by grad n. We have then... 

A series of curves drawn for the same molecular weights of Fig. 5 is presented in Fig. 6. Notice that the modulus function falls off much more rapidly in the three-dimensional case, reflecting the sharper distribution of relaxation times. (The broken line indicates the slope-1/2 associated with the linear chain behavior.)... 

In the three-dimensional case [114] the saturation concentration is smaller than for d = 1 and 2 the maximum value is f/o = 1 -02 (for vp = 266 sites in the recombination sphere), and its dependence on vp is also weaker. Both the magnitude of U0 and this trend are in complete agreement with analytical theory presented in Section 7.1. 

The simulation was performed for the three-dimensional case with imposition of the periodic boundary conditions on the cube in which the defects are being created. The initial distribution function of genetic defects was chosen in the form... 

More generally, it may be shown for the three-dimensional case that whenever and 4>v differ in any one transformation property, then the overlap between them vanishes. Thus if acquires the factor eia under a rotation through a = 2 /5 radians and is "even with respect to inversion it is... 

In the three-dimensional case, the computation of the exchange matrix element can be carried out by the asymptotic method [15]. Without dwelling upon the details of the calculation, we quote the final result... 

Equation (23) is derived for describing the kinetics of electron tunneling in the three-dimensional case (i.e. in the bulk of dielectrics). At present, electron tunneling is studied experimentally not only in the bulk but also on the surface of dielectrics (see Chap. 7, Sect. 6). Thus, it is also of interest to analyze the kinetics of electron tunneling in the two-dimensional case. In the latter case, the analog of eqn. (23) is [37]... 

Once again, we note that direct, step-by-step consideration of the three-dimensional case has not yet been possible, and the question of the character of growth of field in the three-dimensional case remains open. 

The paper also proves that the fluid region where turbulent motion occurs expels the external magnetic field like a diamagnetic body with low magnetic permeability /i. In the two-dimensional case considered in this paper, fi turns out to be of order (Rem)-1. However, the assertion regarding the diamagnetic properties of the turbulent motion is valid for the three-dimensional case as well. Here we find p (Rem)-1 (see review ). The next article in this collection is devoted to further development of the results obtained in this paper. 

A more complicated situation emerges in motion along nonintersecting surfaces with variable curvatures. If the distance between these surfaces remains finite everywhere, then the field lines do not expand infinitely in the directions normal to the surfaces. In the absence of dissipation this means that there is no unbounded growth of the normal field component. However, introduction of the finite conductivity yields an equation for the normal component which is not decoupled it contains the contribution of the Laplacian of the remaining components. At the same time, it is possible for all other components to increase exponentially with an increment which depends on the conductivity and vanishes for infinite conductivity. The authors called this mechanism of field amplification a slow dynamo, in contrast to the fast dynamo feasible in the three-dimensional case, i.e., the mechanism related only to infinite expansion of the field lines as, for example, in motion with magnetic field loop doubling. In a fast dynamo the characteristic time of the field increase must be of the same order as the characteristic period of the motion s fundamental scale. 

Finally, taking into account the new normalization of the eigenfunctions for the three-dimensional case (the normalization factor is (/i )"3/2), we obtain... 

Example 13.2. It is instructive to relate the film pressure to a three-dimensional pressure (Fig. 13.5). On the barrier of length l the film exerts a force nl. In the three-dimensional case we estimate the force from the pressure P which acts upon a surface Id, where d is the thickness of the monolayer. This force is Pld. If the forces are set equal, we obtain P = Tt/d. Typical values for a monolayer in the Li phase arc d 1 nm and 7r = 10-3 N/m. Then we estimate a three-dimensional pressure of P = 106 N/m2 = 10 atm. 

The mechanical characteristics of thin films on liquids are described in a similar way to the three-dimensional case. The surface elasticity [618] is defined as... 

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