Lattice arbitrary origin

Figure 1.2. Illustration of an arbitrary origin of a lattice. The original and the alternative lattices are shown using solid and dash-dotted lines, respectively. The unit cells (shaded) have identical shapes.

As a unit of length, let us choose a distance d which is characteristic of the lattice. An ion / will be at distance from an arbitrary origin, chosen in place of an ion A and at... 

Figure 3.5. A bi-dimensional lattice of points is shown which can be built on the basis of the translation units a (the shortest one) and the corresponding unit cell. The origin of the cell is arbitrary (for inst. (a) or (b)) it contains 1 point. A more symmetric cell (c) may be built with the edges A and B, however it is double primitive centred rectangular, containing two equivalent points.

Figure 4.8 a shows an ab section of lattice with an arbitrary lattice point O chosen as the origin of the reciprocal lattice I am about to define. This point is thus the origin for both the real and reciprocal lattices. Each + in the figure is a real lattice point. 

The increase of AE with chain length must originate from the difference between the monomer lattice constant and the length of oligomer/polymer repeat unit. Anticipating proportionality between 5AE/5n and lattice mismatch as suggested by studies under pressure (see 5.2), the variation of AE with n can be calculated for arbitrary conversion X from the low conversion value (8AE/8n)x=o ... 

First of all, consider a crystal cell, the origin of which is located at an arbitrary point M of the crystal lattice. Inside this cell, an arbitrary point N can be defined by... 

Various results of graph theory enabled Kasteleyn to evaluate the number of configurations of the system in a rather simple manner (the origin of the chain is fixed once and for all in an arbitrary way). The calculation is simpler when the lattice is periodic. 

This is equivalent to translating (without rotating) the original coordinate system in such a way that for the two polymers under consideration, the smallest coordinate of any of their beads is unity (in lattice units). While the choice of unity is somewhat arbitrary, the translation of the coordinate system is done to avoid large numbers when computing the linking number. 

Basis vectors ai, and 33 define a parallelepiped called the unit cell, which is primitive, because it contains one lattice point. All cells that are obtained by translation of this unit cell, the origin cell, through the application of all vectors g in Eq. [1], fill the space completely. Then, the entire lattice can be subdivided into cells and every vector g can be used to label a cell with respect to the origin cell, or 0-cell. Actually, the definition of a unit cell is arbitrary, and many (an infinite number) different possible choices exist, because all cells containing the same number of lattice points are equivalent. The actual shape of a unit cell depends on the lattice type. 

The total energy of the dislocation is a function of misfit distribution f(j ) or, equivalently, pix), and it is invariant with respect to arbitrary translation of p x) and f(j ). To regain the lattice discreteness, the integration of the /-energy in Equation (8.2) was discretized and replaced by a lattice sum in the original P-N formulation... 

A two dimensional crystal lattice (Figure 6.1) may be used to illustrate the main principles for characterization of a crystal. The crystal planes are defined by choosing an origin in an arbitrary lattice point and require that the plane passes infinitely through many lattice points. Crystal directions are straight lines passing infinitely through many points. The crystal axes are chosen as three of the crystal directions. 

The theory of the lattice dynamics of covalent semiconductors has been developed in the late sixties and early seventies [l]. The essence of this theory is the description of the density distribution of the valence electrons between the ions at their arbitrary and instantaneous positions. Originally this part of the electron distribution due to deviations of the ions from their equilibrium positions is derived from linear response theory. Subsequently phonon dispersion curves can be obtained from this dielectric screening method. From 1972 the present authors started their efforts to calculate phonon frequencies using this method. A presentation of linear response and dielectric screening theory is given in these proceedings in the paper by J.T. 

Because B is an arbitrary parameter, the free-volume fraction cannot be determined directly from these equations. As noted previously, Doolittle " " was concerned with the definition of free volume and in Figure 20 is shown a possible definition for the free-volume fraction based on the concept of an occupied volume. When one considers B to be unity, the value of the free-volume fraction at falls between 0.013 and 0.034 for many polymers, and the universal value originally proposed by Williams, Landel and Ferrywas taken to be 0.025 (see equation 23 also the discussion about the free-volume fraction derived from the Gibbs-DiMarzio " lattice model). 

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