Equilibrium spacing

Pseudomorphism received methodical study from about 1905. A micro-section taken across the interface between a substrate and an electrodeposit shows the grain boundaries of the former continue across the interface into the deposit (Fig. 12.5). As grain boundaries are internal faces of metal crystals, when they continue into the deposit the latter is displaying the form of the substrate. Hothersall s 1935 paper contains numerous excellent illustrations with substrates and deposits chosen from six different metals, crystallising in different lattice systems and with different equilibrium spacing. Grain boundary continuation and hence pseudomorphism is evident despite the differences. 

Non-epitaxial electrodeposition occurs when the substrate is a semiconductor. The metallic deposit cannot form strong bonds with the substrate lattice, and the stability conferred by co-ordination across the interface would be much less than that lost by straining the lattices. The case is the converse of the metal-metal interface the stable arrangement is that in which each lattice maintains its equilibrium spacing, and there is consequently no epitaxy. The bonding between the met lic lattice of the electrodeposit and the ionic or covalent lattice of the substrate arises only from secondary or van der Waals forces. The force of adhesion is not more than a tenth of that to a metal substrate, and may be much less. 

The simple theory of electronegativity fails in this discussion because it is based merely on electron transfer energies and determines only the approximate number of electrons transferred, and it does not consider the interactions which take place after transfer. Several calculations in the alkali halides of the cohesive energy (24), the elastic constants (24), the equilibrium spacing (24), and the NMR chemical shift 17, 18, 22) and its pressure dependence (15) have assumed complete ionicity. Because these calculations based on complete ionicity agree remarkably well with the experimental data, we are not surprised that the electronegativity concept of covalency fails completely for the alkali iodide isomer shifts. 

Our starting point in this discussion is the resemblance between Figure 6.8a and Figure 10.6b. The former illustrates the work of cohesion and the latter the interaction between two blocks of material. Suppose we identify by d0 the equilibrium spacing between molecules of a bulk sample of the material under consideration. Then the cohesion process represented by Figure 6.8a can be viewed as one in which two blocks of material are separated from d = d0 to d = 00. In terms of Equation (63),... 

Fig. 6. Schematic energy levels of a solid as a function of interatomic distance where the vertical line represents the equilibrium spacing (68). A band of states obeying Fermi distribution is required by the Pauli principle. High electron velocities and equivalent temperatures exist in conductors even when the...

The interest in lipid bilayers is due to their relevance to biological membranes [1], They exhibit a richness of structures due to the interplay between many different inter- and intrabilayer forces. Among all the multilamellar bilayer structures, probably the most pertinent to biological membranes are the lamellar ones. Their equilibrium spacing is considered to be the result of a balance between attractive and repulsive forces. While the former forces are just the usual van der Waals interactions, the latter are composed of double layer forces (for charged bilayers) [2], hydration forces (due to the structuring of water near interfaces) [3] and repulsive forces generated by the thermal undulation of the membranes [4]. 

Let us briefly highlight the effect of equilibrium space charges on the transport in our prototype oxide (with just O and h as defects).244 The situation becomes particularly clear if we assume equal mobilities and equal bulk conductivities. Since in the case of chemical diffusion space charge splitting... 

For nonpolar materials the Hamaker constant can be calculated from the surface tension or surface free energy of the material, y, and the equilibrium spacing, d, between molecules in the material as follows [11] ... 

If Ca > Cr in Equation 4, then attractive forces can result. Forces measurements on DNA double helices spontaneously assembled by polycations provide evidence for attractive hydration forces (12, 13). The binding of several polycations result in spontaneous condensation of DNA. Packaging of genetic material in vivo and compaction of DNA for use in nonviral gene therapy exploit this attraction. The equilibrium spacing between helix surfaces in the condensed state varies between... 

Relatively few direct comparisons have been made between results of Xoc calculations and those based on the more elaborate potentials. For many properties of simple systems (atomic ionization potentials , equilibrium spacings, vibrational frequencies and binding energies of first-row diatomics" , clusters of a simple metal, K" , the adsorption of O2 on Ag clusters ), the differences among the results provided by the various potentials are far from overwhelming. This is not surprising in view of the vast body of reasonable results which have been obtained with the Xa. method. 

Our fundamental assertion concerning the geometry of pile-ups is that they reflect the equilibrium spacing of the various dislocations which are participants in such a pile-up. From a discrete viewpoint, what one imagines is an equilibrium between whatever applied stress is present and the mutual interactions of the dislocations. In simple terms, using the geometry depicted schematically in fig. 11.12, we argue that each dislocation satisfies an equilibrium equation of the form... 

The problem may not be as severe for the/-shell metals particularly in the 4f series. The / electrons (in most cases n — 3 electrons in the Fn column for 4 < < 17) are so well localized that they may be treated as core electrons. The corresponding values were calculated from the ionization potentials of the atoms (see Problem 16-2,c), and are listed in the Solid State Table. Then the bonding properties of the /-shell metals can be treated exactly as the simple metals (or as beginning transition series if the effects of d states are sufficiently large to make that necessary) and effects of the /-shell electrons (such as the magnetic properties discussed in Section 20-F) can be treated separately. As an example, the equilibrium spacing of the rare earths is discussed in Problem 20-2, in which any d-state effects are ignored. This is a rather crude approximation, since with three non-/ electrons there is always some occupation of /-like bands. 

The equilibrium value of r is obtained by minimizing (1). This leads to the formula for U at the equilibrium spacing at 0 °K... 

The above leads us to conclude that In pair structures the molecules are more tightly bound than In the stack ones. It Is reasonable to believe that the close-spaced substituents In the stack structures Introduce an extra repulsive term In the Inter-molecular Interaction as compared to that of the pair structures. We assume that at the ground-state equilibrium spacing the same repulsive term Is operative In the exclmer. This assumption Is supported by the similarity of the excitation spectra for exclmer emissions for the two types of structure. Further, because of the similarity of uis for pair and stack structures. 

Fig. 6.18 Potential curves and an emission transition for excimers in a crystal. Absorption occurs at the equilibrium spacing in the ground state and leads to the excited state S-. From there, a radiationless relaxation into the excimer state with the smaller equilibrium spacing tex takes place. The excimer emission...

The stress, a, on a solid as a function of the interatomic spacing, r, can be estimated from the interactions given in Section S4.1.7. A typical form for such a curve is sketched in Figure S4.7a. This can be understood as follows. The stress is zero at the equilibrium spacing, r . When the atoms are stretched, the stress will increase rapidly. In the case when the atoms are very far apart, there will be no interatomic force and thus the stress will become zero. This reasoning indicates that the curve must pass though a maximum, at some value This... 

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