SUBJECTS lattice theory

Approximation of the orientation factor Zoriem the partition function according to Eq. (9) of the text provides the key to the earlier version of the lattice theory of rodlike particles not subject to orientation-dependent interactions. Although approximate, this formulation offers advantages of simplicity that for most purposes outweigh the incident errors. The latter are generally small (see Table 1). 

The main topics in lattice theories, which are relevant for the polymer subject are avoided random walk, lattice percolation [3] and lattice spin models. In this work we shall put the emphasis on the numerical investigation of the systems in the framework of lattice percolation methodologies and avoided random walks on square and cubic lattices. 

Dislocation theory as a portion of the subject of solid-state physics is somewhat beyond the scope of this book, but it is desirable to examine the subject briefly in terms of its implications in surface chemistry. Perhaps the most elementary type of defect is that of an extra or interstitial atom—Frenkel defect [110]—or a missing atom or vacancy—Schottky defect [111]. Such point defects play an important role in the treatment of diffusion and electrical conductivities in solids and the solubility of a salt in the host lattice of another or different valence type [112]. Point defects have a thermodynamic basis for their existence in terms of the energy and entropy of their formation, the situation is similar to the formation of isolated holes and erratic atoms on a surface. Dislocations, on the other hand, may be viewed as an organized concentration of point defects they are lattice defects and play an important role in the mechanism of the plastic deformation of solids. Lattice defects or dislocations are not thermodynamic in the sense of the point defects their formation is intimately connected with the mechanism of nucleation and crystal growth (see Section IX-4), and they constitute an important source of surface imperfection. 

In a review of the subject, Ubbelohde [3] points out that there is only a relatively small amount of data available concerning the properties of solids and also of the (product) liquids in the immediate vicinity of the melting point. In an early theory of melting, Lindemann [4] considered that when the amplitude of the vibrational displacements of the atoms of a particular solid increased with temperature to the point of attainment of a particular fraction (possibly 10%) of the lattice spacing, their mutual influences resulted in a loss of stability. The Lennard-Jones—Devonshire [5] theory considers the energy requirement for interchange of lattice constituents between occupation of site and interstitial positions. Subsequent developments of both these models, and, indeed, the numerous contributions in the field, are discussed in Ubbelohde s book [3]. 

The theory of melting continues to be the subject of recent publications, including consideration of vacancy concentrations near the melting point [8,9], lattice vibrations and expansions [8,10—12], Meanwhile, the phenomenon also continues to be the subject of experimental investigations Coker et al. [13], from studies of the fusion of tetra-n-amyl ammonium thiocyanate, identify the greatest structural change as that which... 

Crystal-field theory (CFT) was constructed as the first theoretical model to account for these spectral differences. Its central idea is simple in the extreme. In free atoms and ions, all electrons, but for our interests particularly the outer or non-core electrons, are subject to three main energetic constraints a) they possess kinetic energy, b) they are attracted to the nucleus and c) they repel one another. (We shall put that a little more exactly, and symbolically, later). Within the environment of other ions, as for example within the lattice of a crystal, those electrons are expected to be subject also to one further constraint. Namely, they will be affected by the non-spherical electric field established by the surrounding ions. That electric field was called the crystalline field , but we now simply call it the crystal field . Since we are almost exclusively concerned with the spectral and other properties of positively charged transition-metal ions surrounded by anions of the lattice, the effect of the crystal field is to repel the electrons. 

The classical theory for electronic conduction in solids was developed by Drude in 1900. This theory has since been reinterpreted to explain why all contributions to the conductivity are made by electrons which can be excited into unoccupied states (Pauli principle) and why electrons moving through a perfectly periodic lattice are not scattered (wave-particle duality in quantum mechanics). Because of the wavelike character of an electron in quantum mechanics, the electron is subject to diffraction by the periodic array, yielding diffraction maxima in certain crystalline directions and diffraction minima in other directions. Although the periodic lattice does not scattei the elections, it nevertheless modifies the mobility of the electrons. The cyclotron resonance technique is used in making detailed investigations in this field. 

Oxidic surfaces in particular develop acid or basic properties which are important in catalysis. We will approach this subject first by taking as a starting point the ionic bond model [2]. The lattice is considered to consist of cations and anions held together by electrostatic interactions. Later we will discuss a more balanced theory that also accounts for covalent bonding aspects. 

We will not deal here with the subject of EPR spectroscopy of the solid state. In this field of investigation a kind of delta-like approach such as that recently proposed to deal with molecular dynamics in the liquid state has developed naturally. According to the European Molecular Liquid Group (EMLG), the symbol A symbolizes the cooperative efforts of computer simulation, experiment, and theory. Knak Jensen and Hansen, for instance, carried out a computer simulation of the dynamics of N identical spins placed in a rigid simple cubic lattice subject to an external magnetic field Bq. a further example of numerical study is the paper of Sur and Lowe. Free-induction decay measurements,on the other hand, represent the experimental comer of this ideal triangle, the theoretical comer of which is, of course, expressed by the theoretical papers mentioned above. 

The question whether the dispersion bands observed with feebly damped waves by Colley, Obolensky, Romanoff, Potapenko, and others (Handbuch der Physiky 15, 514 et seq. (Berlin, 1927)) in the region of wave-lengths of a few decimetres correspond to intramolecular vibrations might be determined by comparison of the spectra of the vapours with those of the liquids. As is well known, the kinetic theory of liquids has recently exhibited a decided tendency to follow the theory of crystal lattices more closely in many respects for a comprehensive account of some of the most important papers on this subject see K. Jellinek, Lehrbuch der physikalischen ChemiCy 1, 824 et seq, especially pp. 828, 831 (Stuttgart, 1928). 

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