Lattice points/sites

Suppose the geometry of a compartmentalized system is described by a lattice of integral or fractal dimension of given size and shape, and characterized by N discrete lattice points (sites) embedded in a Euclidean space of dimension d = de and local connectivity or valency v. At time t = 0, assume that the diffusing coreactant A is positioned at a certain site j with unit probability. For f > 0 the probability distribution function p(f) governing the fate of the diffusing particle is determined by the stochastic master equation... 

Each lattice point extraneous to the sites connected by graphs also contributes a factor of two on summing over spin states. Hence all lattice points contribute a factor of 2 whether they are coimected or not, and... 

FIQ. 1 Sketch of the BFM of polymer chains on the three-dimensional simple cubic lattice. Each repeat unit or effective monomer occupies eight lattice points. Elementary motions consist of random moves of the repeat unit by one lattice spacing in one lattice direction. These moves are accepted only if they satisfy the constraints that no lattice site is occupied more than once (excluded volume interaction) and that the bonds belong to a prescribed set of bonds. This set is chosen such that the model cannot lead to any moves where bonds should intersect, and thus it automatically satisfies entanglement constraints [51],... 

One of the first attempts to calculate the thermodynamic properties of an atomic solid assumed that the solid consists of an array of spheres occupying the lattice points in the crystal. Each atom is rattling around in a hole at the lattice site. Adding energy (usually as heat) increases the motion of the atom, giving it more kinetic energy. The heat capacity, which we know is a measure of the ability of the solid to absorb this heat, varies with temperature and with the substance.8 Figure 10.11, for example, shows how the heat capacity Cy.m for the atomic solids Ag and C(diamond) vary with temperature.dd ee The heat capacity starts at a value of zero at zero Kelvin, then increases rapidly with temperature, and levels out at a value of 3R (24.94 J-K -mol-1). The... 

The order-disorder transition of a binary alloy (e.g. CuZn) provides another instructive example. The body-centred lattice of this material may be described as two interpenetrating lattices, A and B. In the disordered high-temperature phase each of the sub-lattices is equally populated by Zn and Cu atoms, in that each lattice point is equally likely to be occupied by either a Zn or a Cu atom. At zero temperature each of the sub-lattices is entirely occupied by either Zn or Cu atoms. In terms of fractional occupation numbers for A sites, an appropriate order parameter may be defined as... 

The importance of dislocations becomes evident when we consider the strain on the microstructure of a simple crystal. The atoms or ions in a crystal are in symmetric energy wells and so vibrate around their lattice site. When we track across a crystal plane, the potential energy increases and decreases in a regular fashion with the minima at the lattice points... 

Working with clusters defined by r lattice points reduces the number of configurations by a further factor of r. A multi-site correlation parameter ( ) can then be constructed from the sum of all such values and used to define the overall configuration. 

Careful measurements of the structure factors of vanadium (Ohba et al. 1981) and chromium (Ohba et al. 1982) up to sin 6/2 = 1.72 A / using AgKa radiation and small spherical crystals ( 0.2 mm diameter), have been reported. The bcc structure of these metals leads to pairs of reflections such as (330/441), (431/510), at identical values of sin 6/2, which have the same intensity for a structure with one spherical atom per lattice point. This is no longer true when the t2g and eg orbitals of the cubic site are no longer equally occupied. This is easiest seen as follows. 

As mentioned above, the non-stoichiometric compounds originate from the existence of point defects in crystals. Let us consider a crystal consisting of mono-atoms. In ideal crystals of elements, atoms occupy the lattice points regularly. In real crystals, on the other hand, various kinds of point defects can exist in thermodynamic equilibrium. First, we shall consider vacancies , which are empty regular lattice points. Consider a crystal composed of one element which has N atoms sited on regular lattice points and vacancies,... 

Another type of lattice defect for elements is interstitial atoms, in which an atom is transferred from a regular lattice point to an interstitial position, normally unoccupied by an atom. Consider a crystal which has N atoms sited on regular lattice points and N, atoms sited on interstitial lattice points (the number of interstitial lattice points is A, which is fixed by the crystal structure under consideration), by a similar calculation, the free energy increment from the ideal crystal is expressed as... 

This notation means that the number of vacancies is equal to the number of interstitial atoms. Therefore it is assumed that an interstitial atom is produced by removing an atom from a regular lattice site, placing it at the surface in a regular lattice site, and then transferring it to an interstitial lattice point. 

Schottky type in which equal numbers of M, and X are created (Fig. 1.9(c)), in this case, the composition is also stoichiometric, and anti-Schottky type, in which equal numbers of metal atoms and anions on respective regular sites move to respective interstitial lattice points leaving vacancies and X . 

Consider a crystal Mj X which contains both metal vacancies and interstitial metal atoms in low concentration, i.e. M occupies lattice points in N lattice sites of metal, X occupies N in N lattice sites (generally, N, N, but in this calculation we assume = Nf and, moreover, interstitial M occupies in Na, where a is a constant which is fixed by crystal structure. If the conditions N N — N ), (N — N ), N are satisfied, it is not necessary to take the interaction energy between defects, as mentioned below, into consideration. The free energy of the crystal may be written as... 

Let us consider a crystal similar to that discussed in Sections 1,3.3 and 1.3.4, which, in this case, shows a larger deviation from stoichiometry. It is appropriate to assume that there are no interstitial atoms in this case, because the Frenkel type defect has a tendency to decrease deviation. Consider a crystal in which M occupies sites in N lattice points and X occupies sites in N lattice points. It is necessary to take the vacancy-vacancy interaction energy into consideration, because the concentration of vacancies is higher. The method of calculation of free energy (enthalpy) related to is shown in Fig. 1.12. The total free energy of the crystal may be written... 

Consider a crystal of composition M0.5X, the metal vacancies are regularly arranged among the lattice sites at lower temperatures, shown in Fig. 1.18 as a basic model of a vacancy-ordered structure with a two-dimensional lattice (in this figure, the anion atoms are omitted for clarity). This structure is realized if the composition of the crystal is Mq 5X, and metal atoms M fully occupy the B-sites and metal vacancies fully occupy the A-sites, this only occurs at absolute zero temperature (perfect order). The occupation probabilities, p and Pg, denote the ratio of number of metal atoms on the A-sites (ma) to the number of lattice points of the A-sites ( 1V) and the ratio of number of metal atoms on the B-sites (Ug) to the number of lattice points of the B-sites (ilV), respectively. Thus p and pg can be expressed as... 

Consider a metal hydride MH in which there are X lattice points for metals and sN for hydrogens, and hydrogen atoms occupy Nh in sN lattice points (sN = + Ny, where A/y is the number of vacant hydrogen sites). 

Exercise. For the random walk on a two-dimensional square lattice, either with discrete or continuous time, show that every lattice point is reached with probability 1, but on the average after an infinite time. In three dimensions, however, the probability of reaching a given site is less than unity there is a positive probability for disappearing into infinity. 

There are some apparent exceptions. In the ammonium halides, which have the rock salt lattice, the ammonium ions, which do not themselves have centers of symmetry, lie at lattice points which would, were the ammonium ions not present, be centers of inversion. Depending on the particular halide and the temperature, either the ammonium ions are freely rotating, so that the time-averaged symmetry is centric, or else ammonium ions at different sites have different orientations so that, although any particular site is not centrosymmetric, the crystal as a whole appears to be centrosymmetric. In this brief discussion we shall consider only cases in which the crystals are completely ordered and contain no rotating molecules. 

We introduce a variable for the particles (1 G 0, H,N,A, B where 0 represents a vacant site, A represents a NH particle and B a NH2 particle. The state of a lattice point 07 consists of the state of the catalyst (activated or unactivated) and its coverage with a particle. This leads to the following possible states ... 

If h > k and k 0 then the super-lattice is unsymmetrically disposed with respect to the original lattice. If the lattice is turned over and superimposed on itself so that the super-lattice points coincide, then we have a coincidence site lattice (in which a fraction 1 /(h2 + hk + k2) of the original lattice points coincide). Coincidence site lattices can also be found in three dimensions, particularly for cubic lattices. 

The fee unit cell can be thought of as having holes in which other atoms or ions can be placed. For example, the Na + ions occupy octahedral holes in the fee CF lattice (connecting the six CF ions surrounding a Na+ by lines defines an octahedron). The fee lattice also has tetrahedral holes if we divide the unit cell into 8 smaller cubes, the centers of these little cubes are surrounded by 4 lattice points which define a tetrahedron. The diamond allotropic form of carbon has a fee structure with C atoms in 4 of these tetrahedral sites each C atom is surrounded by (covalently bound to) 4 other C s. 

The simple CSL model is directly applicable to the cubic crystal class. The lower symmetry of the other crystal classes necessitates the more sophisticated formalism known as the constrained coincidence site lattice, or CCSL (Chen and King, 1988). In this book we treat only cubic systems. Interestingly, whenever an even value is obtained for E in a cubic system, it will always be found that an additional lattice point lies in the center of the CSL unit cell. The true area ratio is then half the apparent value. This operation can always be applied in succession until an odd value is obtained thus, E is always odd in the cubic system. A rigorous mathematical proof of this would require that we invoke what is known as O-lattice theory (Bollman, 1967). The O-lattice takes into account all equivalence points between two neighboring crystal lattices. It includes as a subset not only coinciding lattice points (the CSL) but also all nonlattice sites of identical internal coordinates. However, expanding on that topic would take us well beyond the scope of this book. The interested reader is referred to Bhadeshia (1987) or Bollman (1970). 

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