Lateral lattice expansion

The theory later developed by Spiering and co-workers [24, 196] takes as its basis changes of volume, shape, and elasticity of the lattice as the main factors influencing the cooperative interactions. This model of lattice expansion and elastic interactions has been developed further and is described in detail by Spiering in Chap. 28. 

Our development has assumed temperature independent force constants. In real liquids, however, there is a small temperature dependence of frequencies and force constants due to anharmonicities, lattice expansion, etc. The incorporation of these effects into the theory is treated in later sections. 

We note first that not all amorphous substances actually exhibit a negative a in the experimentally probed temperature range. In such cases, it is likely that the contraction coming from those interactions in these materials is simply weaker than the regular, anharmonic lattice thermal expansion. Other contributions to the Griineisen parameter will be discussed later as well. 

Expansion (7.58) becomes obvious if we consider the lattice symmetry. Two nearest neighbors belonging to the same sublattice are situated on the diagonals of the opposite vertices and their contributions to the crystalline field should be equal, so that the odd harmonics vanish. Coefficients /34 and j86 can be calculated using the AAP method. Yamamoto et al. [1977] found /34 = 5.43, p6 = 7.18 these numbers were later revised by Smith [1990] so as to give the best fit to the libration spectrum (/34 = 6.50, /36 = -5.33). The intermolecular potential is expanded in a series in rotational tetrahedral functions uv, which can be expressed in terms of Wigner functions at 7 = 3 ... 

Introduction of the metal layer leads, for all the stages, to the same increase in the distance between the adjacent carbon planes, as may be seen from the identity periods for potassium-graphite (see Table I). In addition the carbon planes next to the metal layers always have identical positions. Thus entry of the alkali metal into the lattice is linked not only with an expansion lattice but also with a lateral displacement of the carbon planes. Metal atoms are then able to arrange themselves so that they lie over and under the carbon hexagons and are surrounded symmetrically by twelve carbon atoms. Figure 8 shows the sequence of... 

Values of (SiO /Al O ) were taken from solid state NMR data (presented later). The difference between lines A and B (indicating unit cell expansion) is due to the incorporation of Fe in the lattice framework. However if all the Fe in the solids were to occupy lattice positions the unit cell parameters would have been larger than those observed (line A) and would have increased with Fe content. The observed intermediate position of line A leads to the conclusion that only a fraction of Fe is in the faujasite lattice the balance occupying nonframework positions. Thus while the observed a... 

When diamond films are deposited chi non-diamond substrates, stresses may be generated in the films due to lattice mismatch and/or differences in thermal expansion coefiBcients between diamond and the substrate materials.In addition, lateral variations in the grain size, density, or impurities incorporated during growth may also lead to stresses, which may be either tensile or compressive. The stresses are known to generally build up with increasing film thickness, and will influence diamond-substrate adhesion and properties of diamond films. ... 

These results, in turn, then formed the basis for the non-jellium part of the jel-lium model in [73]. Berkowitz [52-54] and later Zhu and Philpott [55] and Spohr [56] followed the approach by Steele [93] and Fourier-expanded the lattice sum of all (pairwise) interactions between the atoms in the solid and one molecule or ion in the liquid. Only the lowest order corrugation terms are kept in the expansion but in principle the summation can be extended to any desired accuracy. The procedure is adequate as long as there is no substantial coupling between liquid and metal motions that could influence the liquid structure and relaxation phenomena. Spohr [56] used a corrugated Morse potential for the oxygen-metal interactions and an exponentially repulsive potential for the hydrogen metal interactions in the form... 

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