The lattice concept

A study of the external symmetry of crystals naturally leads to the idea that a single crystal is a three-dimensional periodic structure i.e., it is built of a basic structural unit that is repeated with regular periodicity in three-dimensional space. Such an infinite periodic structure can be conveniently and completely described in terms of a lattice (or space lattice), which consists of a set of points (mathematical points that are dimensionless) that have identical environments. 

An infinitely extended, linear, and regular system of points is called a row, and it is completely described by its repeat distance a. A planar regular array of points is called a net, which can be specified by two repeat distances a and b and the angle y between them. The analogous system of regularly distributed points in three dimensions constitute a lattice (or space lattice), which is described by a set of three non-coplanar vectors (a, b, c) or six parameters the repeat intervals a, b, c and the angles a, p,y between the vectors. As far as possible, the angles are chosen to be obtuse, particularly 90° or 120°, and lattices in one, two, and three dimensions are illustrated in Fig. 9.2.1. 

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The presence of a defect in the lattice (impurity, surface, vacancy...) breaks the symmetry and induces perturbations of the electronic structure in its vicinity. Thus it is convenient to introduce the concept of local density of states (LDOS) at site i ... 

More recently, D. Emin [24] developed an alternative analysis of activated hopping by introducing the concept of coincidence. The tunneling of an electron from one site to the next occurs when the energy state of the second site coincides with that of the first one. Such a coincidence is insured by the thermal deformations of the lattice. By comparing the lifetime of such a coincidence and the electron transit time, one can identify two classes of hopping processes. If the coincidence lime is much laigcr than the transit lime, the jump is adiabatic the electron has lime to follow the lattice deformations. In the reverse case, the jump is non-adia-batic. 

Palladium hydride is a unique model system for fundamental studies of electrochemical intercalation. It is precisely in work on cold fusion that a balanced materials science approach based on the concepts of crystal chemistry, crystallography, and solid-state chemistry was developed in order to characterize the intercalation products. Very striking examples were obtained in attempts to understand the nature of the sporadic manifestations of nuclear reactions, true or imaginary. In the case of palladium, the elfects of intercalation on the state of grain boundaries, the orientation of the crystals, reversible and irreversible deformations of the lattice, and the like have been demonstrated. 

Girault and Schiffrin [4] proposed an alternative model, which questioned the concept of the ion-free inner layer at the ITIES. They suggested that the interfacial region is not molecularly sharp, but consist of a mixed solvent region with a continuous change in the solvent properties [Fig. 1(b)]. Interfacial solvent mixing should lead to the mixed solvation of ions at the ITIES, which influences the surface excess of water [4]. Existence of the mixed solvent layer has been supported by theoretical calculations for the lattice-gas model of the liquid-liquid interface [23], which suggest that the thickness of this layer depends on the miscibility of the two solvents [23]. However, for solvents of experimental interest, the interfacial thickness approaches the sum of solvent radii, which is comparable with the inner-layer thickness in the MVN model. 

On the other hand, one strength of the approach is the availability of algorithms (such as the slithering snake algorithm) by which undercooled polymer melts can be equilibrated at relatively low temperatures. This allows the static properties of the model to be established over a particularly wide parameter range. Furthermore, the lattice structure allows many questions to be answered in a well-defined, unique way, and conceptional problems of the approach can be identified and eliminated. Last but not least, the lattice structure allows the formulation of very efficient algorithms for many properties. 

Such an approach is conceptually different from the continuum description of momentum transport in a fluid in terms of the NS equations. It can be demonstrated, however, that, with a proper choice of the lattice (viz. its symmetry properties), with the collision rules, and with the proper redistribution of particle mass over the (discrete) velocity directions, the NS equations are obeyed at least in the incompressible limit. It is all about translating the above characteristic LB features into the physical concepts momentum, density, and viscosity. The collision rules can be translated into the common variable viscosity, since colliding particles lead to viscous behavior indeed. The reader interested in more details is referred to Succi (2001). 

Disordered structures belonging to the class (i) are interesting because, in some cases, they may be characterized by disorder which does not induce changes of the lattice dimensions and of the crystallinity, and a unit cell may still be defined. These particular disordered forms are generally not considered as mesomorphic modifications. A general concept is that in these cases the order-disorder phenomena can be described with reference to two ideal structures, limit-ordered and limit-disordered models, that is, ideal fully ordered or fully disordered models. 

A description which in some simple cases could be considered alternative to those exemplified in Table 3.2 is based on the lattice complex concept. Listing the symbols of the lattice complexes occupied by the different atoms in a structure (for instance, symbol P for the point 0, 0, 0 and its equivalent points), provides in fact... 

Fig. 4. Illustrative model of paths between two trap sites embedded in a three-dimensional cubic lattice. The dashed 24-link line has 7 unnecessary kinks which reduce its contribution to the path sum, but there are many of them (Table 2) note that the kinks in the figure are two-dimensional but the count in Eq. 17 is three-dimensional. The paths corresponding to terms in Eq. 14 may in general cross over themselves and backtrack, but may not visit the initial or final sites twice. The latter condition does not arise directly from Eq. 13 but rather from the irreversibility concept underlying the theory of the rate constant...

According to the model, a perturbation at one site is transmitted to all the other sites, but the key point is that the propagation occurs via all the other molecules as a collective process as if all the molecules were connected by a network of springs. It can be seen that the model stresses the concept, already discussed above, that chemical processes at high pressure cannot be simply considered mono- or bimolecular processes. The response function X representing the collective excitations of molecules in the lattice may be viewed as an effective mechanical susceptibility of a reaction cavity subjected to the mechanical perturbation produced by a chemical reaction. It can be related to measurable properties such as elastic constants, phonon frequencies, and Debye-Waller factors and therefore can in principle be obtained from the knowledge of the crystal structure of the system of interest. A perturbation of chemical nature introduced at one site in the crystal (product molecules of a reactive process, ionized or excited host molecules, etc.) acts on all the surrounding molecules with a distribution of forces in the reaction cavity that can be described as a chemical pressure. 

Some of the earliest concepts came from Japan, where Matsuchita developed the Li/(CF) battery that was used, for example, in fishing floats. Lithium fluoride and carbon are the final reaction products, but the cell potential of 2.8—3.0 V suggests a different electrochemical reaction. It was proposed that lithium initially intercalates the carbon monofluoride lattice and subsequently the lithium fluoride is formedF Li + (CF)n — L CF)n C + LiF. Although much work... 

A system of adsorbed particles is often treated as a two-dimensional gas covering the adsorbent surface. Such an approach is quite justified and fruitful, as long as we are dealing with physical adsorption when the influence of the adsorbent on the adsorbate can be regarded as a weak perturbation. In case of chemical adsorption (the most frequent in catalysis), the concept of a two-dimensional gas becomes untenable. In this case the adsorbed particles and the lattice of the adsorbent form a single quantum-mechanical system and must be regarded as a whole. In such a treatment the electrons of the crystal lattice are direct participants of the chemical processes on the surface of the crystal in some cases they even regulate these processes. 

We shall proceed from a concept which in a certain sense is contrary to that of the two-dimensional gas. We shall treat the chemisorbed particles as impurities of the crystal surface, in other words, as structural defects disturbing the strictly periodic structure of the surface. In such an approach, which we first developed in 1948 (I), the chemisorbed particles and the lattice of the adsorbent are treated as a single quantum-mechanical system, and the chemisorbed particles are automatically included in the electronic system of the lattice. We observe that this by no means denotes that the adsorbed particles are rigidly localized they retain to a greater or lesser degree the ability to move ( creep ) over the surface. 

The concept of different forms of chemisorption which differ in the character of the bond between the adsorbed particle and the adsorbent lattice is the first important result of the electron theory. The possibility of different bonding types in chemisorption is due to the ability of the chemisorbed particles to form bonds to which either free electrons or free holes of the lattice can contribute. In other words, it is due to the ability of the chemisorbed particle to generate a free electron or a free hole and to give them up to the lattice. 

It was therefore appropriate that the first attempt to produce lattice stabilities for non-allotropic elements dealt with Cu, Ag and Zn (Kaufman 1959b). It is also significant that, because of the unfamiliarity of the lattice stability concept, this paper did not appear as a mainstream publication although the work on Ti and Zr (Kaufman 1959a) was published virtually at the same time. It was also realised that Ae reliability of metastable melting points derived by extrapolation were best... 

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