Lattices, space coordinates

As the dynamics of the system are removed from the model, it is no longer necessary to allow the molecules to live in a continuous space. Instead, the use of lattices - discrete sets of coordinates on to which the molecules are restricted - is popular. Digital computers are of course much more efficient with discrete space than with continuum space. The use of a lattice implies that one removes all properties that occur on shorter length scales than the lattice spacing from the model. This is no problem if the main interest is in phenomena that are larger than this length scale. 

Unless one is willing to become involved in many intricacies, a lattice model with united atoms (segments) features segments which are all of equal size. The price we have to pay for this is that there is no unique way to convert from lattice units to real space coordinates. We will discuss this point in the Result sections in more detail. 

The first model of porous space as a 2D lattice of interconnected pores with a variation of randomness and branchness was offered by Fatt [220], He used a network of resistors as an analog PS. Further, similar approaches were applied in a number of publications (see, e.g., Refs. [221-223]). Later Ksenjheck [224] used a 3D variant of such a model (simple cubic lattice with coordination number 6, formed from crossed cylindrical capillaries of different radii) for modeling MP with randomized psd. The plausible results were obtained in these works, but the quantitative consent with the experiment has not been achieved. 

Since the variation of any physical property in a three dimensional crystal is a periodic function of the three space coordinates, it can be expanded into a Fourier series and the determination of the structure is equivalent to the determination of the complex Fourier coefficients. The coefficients are indexed with the vectors of the reciprocal lattice (one-to-one relationship). In principle the expansion contains an infinite number of coefficients. However, the series is convergent and determination of more and more coefficients (corresponding to all reciprocal lattice points within a sphere, whose radius is given by the length of a reciprocal lattice vector) results in a determination of the stmcture with better and better spatial resolution. Both the amplitude and the phase of the complex number must be determined for any Fourier coefficient. The amplitudes are determined from diffraction... 

Both the components of E and the elements of the electric field gradient as given by Eqs. (8.30) and (8.32) are with respect to the reciprocal-lattice coordinate system. A transformation is required if the values in the direct-space coordinate systems are needed. To obtain the elements of the traceless V tensor, the quantity — (47t/3)pe(r) = — (47r/3K) F(H) exp ( — 27tiHT) must be subtracted from each of the diagonal elements VEU. 

If Tfl were to occur in a BO2 layer which also provides six coordination, the lattice spacing within the layer would be 452 pm, much too large to match to any available A cations. 

By absorbing excited radiation the electrons are raised from the ground state to the excited state. These transitions take place so rapidly that no displacement of the atomic nuclei occurs (Franck-Condon principle). The space coordinate thus remains unchanged and the transitions can be represented by vertical lines. Because the excited system is not immediately in a state of equilibrium after absorption of energy, it first moves towards the lowest vibrational level with loss of energy to the lattice... 

Tables 3.1, 3.2 and 3.3 compiled by Povarennykh (1963) specify the initial data accepted for the calculation of hardness from formulae (3.5) and (3.6). As the ratio WJWa increases, the coefficient a decreases (Table 3.1). For compounds with ratios inverse to those given in the table, i.e., for compounds having a so-called antistructure, the coefficient a will be exactly the same, e.g., 1/2 and 2/1. In both cases, x — 80. The link attenuation coefficient / varies over a relatively narrow range, usually between 0.7 and 1.0 (Table 3.2). This coefficient requires the state of lattice linkage to be considered in each case, and like coefficient a it depends on the type of compound involved. For various types of compounds, the values of the coefficient / may be lower taking as an example minerals in the pyrite and skutterudite group, they are as follows for compounds 2/2—0.60, for 3/3—0.48 and for 4/4—0.39. The values of the coefficient y grow proportionally with coordination number (Table 3.3). The constancy of the coefficient y depends on the constancy of the coordination number which is influenced by the valence ratio of electropositive and electronegative atoms. Lattice spacings, state of chemical bonds and electron-shell structure, and for complex compounds, also the degree of action of the remain-...

Three additional concepts may be introduced by means of Fig. 2A. First, all measurements of the diffracted beams are made relative to the incident x-ray beam so that the diffracted beam leaves the crystal at an angle 20, as illustrated. Thus, while the diffraction condition is determined by the angle between the incident and diffracted beams relative to a lattice plane, the measured position of the diffracted beam is determined relative to the incident beam. Second, there is a reciprocal relation between the spacing between lattice planes and the positions of the diffraction spots small lattice spacings give diffraction spots with large values of 20. This leads to two types of space. The crystal coordinate system is in real space, whereas the diffrac-... 

When the direct space coordinates are allowed to take on all integral values m, n, p, the vector termini xmnp define the direct lattice [Eq. 

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