Lattice gradient

The term eq is referred to as the lattice gradient. It arises from the ionic charges... 

For an ion to move through the lattice, there must be an empty equivalent vacancy or interstitial site available, and it must possess sufficient energy to overcome the potential barrier between the two sites. Ionic conductivity, or the transport of charge by mobile ions, is a diffusion and activated process. From Fick s Law, J = —D dn/dx), for diffusion of a species in a concentration gradient, the diffusion coefficient D is given by... 

The other major defects in solids occupy much more volume in the lattice of a crystal and are refeiTed to as line defects. There are two types of line defects, the edge and screw defects which are also known as dislocations. These play an important part, primarily, in the plastic non-Hookeian extension of metals under a tensile stress. This process causes the translation of dislocations in the direction of the plastic extension. Dislocations become mobile in solids at elevated temperamres due to the diffusive place exchange of atoms with vacancies at the core, a process described as dislocation climb. The direction of climb is such that the vacancies move along any stress gradient, such as that around an inclusion of oxide in a metal, or when a metal is placed under compression. 

Why is this relevant to the diffusion of zinc in copper Imagine two adjacent lattice planes in the brass with two slightly different zinc concentrations, as shown in exaggerated form in Fig. 18.5. Let us denote these two planes as A and B. Now for a zinc atom to diffuse from A to B, down the concentration gradient, it has to squeeze between the copper atoms (a simplified statement - but we shall elaborate on it in a moment). This is another way of saying the zinc atom has to overcome an energy barrier... 

The first and second derivatives in the gradient and Laplacian term of the functional (1) at the point r = i,j k)h on the lattice are calculated according to the following formulas [24]... 

For such a condition of equilibrium to be reached, the atoms must acquire sufficient energy to permit their displacement at an appreciable rate. In the case of metal lattices, this energy can be provided by a suitable rise in temperature. In the application of coatings the diffusion process is arrested at a suitable stage when there is a considerable solute concentration gradient between the surface and the required depth of penetration. 

In an ideal ionic crystal, all ions are held rigidly in the lattice sites, where they perform only thermal vibratory motion. Transfer of an ion between sites under the effect of electrostatic fields (migration) or concentration gradients (diffusion) is not possible in such a crystal. Initially, therefore, the phenomenon of ionic conduction in solid ionic crystals was not understood. 

While these results are qualitatively eorreet, in an exaet treatment of the model the critical interaction is generally not at a = 2, but lies at a different value, which will depend on the geometry of the lattice and not only on the number of nearest neighbors. Also, this treatment says nothing about the structure of the interface. It can be extended by introducing gradient terms, which allow the calculation of density profiles for the two solvents [1,5]. Since there are more accurate methods available for this purpose we refer to the cited literature. 

When the two phases separate the distribution of the solvent molecules is inhomogeneous at the interface this gives rise to an additional contribution to the free energy, which Henderson and Schmickler treated in the square gradient approximation [36]. Using simple trial functions, they calculated the density profiles at the interface for a number of system parameters. The results show the same qualitative behavior as those obtained by Monte Carlo simulations for the lattice gas the lower the interfacial tension, the wider is the interfacial region in which the two solvents mix (see Table 3). 

SFC-NMR is available from 200 to 800 MHz, and is suitable for all common NMR-detected nuclei. SFC/SFE-NMR requires dedicated probe-heads for high pressure (up to 350 bar) and elevated temperature (up to 100 °C). SFC-NMR is carried out with conventional packed columns, using modifier, pressure and temperature gradients. The resolution of 1H NMR spectra obtained in SFE-NMR and SFC-NMR coupling under continuous-flow conditions approaches that of conventionally recorded NMR spectra. However, due to the supercritical measuring conditions, the 111 spin-lattice relaxation times 7) are doubled. 

The tortuosity factor appears as a squared term because it decreases the concentration gradient and increases the diffusive path length. Using a cubic lattice model and inquiring how many steps a diffusing molecule needs to take to get around an obstacle, 0 was derived to be... 

The smoothing terms have a thermodynamic basis, because they are related to surface gradients in chemical potential, and they are based on linear rate equations. The magnitude of the smoothing terms vary with different powers of a characteristic length, so that at large scales, the EW term should predominate, while at small scales, diffusion becomes important. The literature also contains non-linear models, with terms that may represent the lattice potential or account for step growth or diffusion bias, for example. 

In the literature [55], typical energies involved in the nuclear quadrupole moments -crystalline electric field gradient interactions range up to A E 2x 10-25 J. The measured AE seems to confirm the hypothesis that the excess specific heat of the metallized wafer is due to boron doping of the Ge lattice. 

Displacements of lattice members are determined by energy factors and concentration gradients. To a considerable extent, diffusion in solids is related to the existence of vacancies. The "concentration" of defects, N0, (sites of higher energy) can be expressed in terms of a Boltzmann distribution as... 

For nuclei that have perfect cubic site symmetry (e.g., those in an ideal rock salt, diamond, or ZB lattice) the EFG is zero by symmetry. However, defects, either charged or uncharged, can lead to non-zero EFG values in nominally cubic lattices. The gradient resulting from a defect having a point charge (e.g., a substitutional defect not isovalent with the host lattice) is not simply the quantity calculated from simple electrostatics, however. It is effectively amplified by factors up to 100 or more by the Sternheimer antishielding factor [25],... 

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