Lattice molecule

To represent the elasticity and dispersion forces of the surface, an approach similar to that of Eqs. (3) and (4) can be taken. The waU molecules can be assumed to be smeared out. And after performing the necessary integration over the surface and over layers of molecules within the surface, a 10-4 or 9-3 version of the potential can be obtained [54,55], Discrete representation of a hexagonal lattice of wall molecules is also possible by the Steele potential [56], The potential is essentially one dimensional, depending on the distance from the wall, but with periodic variations according to lateral displacement from the lattice molecules. Such a representation, however, has not been developed in the cylindrical pore... 

Our procedure is to set up a site balance in terms of lattice molecules, i.e.- AgBr ... 

The X-ray diffraction analysis of these complexes (4 -Ua--i -Ue) revealed atom positions and connectivities of one molecule of alkane in the coordination sphere of the uranium(III) center and a second molecule of cycloalkane co-crystallized in the lattice. Molecules 4 -Ua—4 "-Ue are isostructural and isomorphous. The molecular... 

At about this time, J. Frenkel published a most seminal theoretical paper [J. Frenkel (1926)]. He suggested that in a similar way as (neutral) water dissociates to a very small extent into protons and hydroxyl ions, a perfect lattice molecule of a crystal (such as AgBr, which crystallizes in the B1-structure) will dissociate its regular structure elements, AgAg, into silver ions which are activated to occupy vacant sites in the interstitial sublattice, V). (The notation is explained in the list of symbols.) They leave behind empty regular silver ion sites (silver vacancies) symbolized here by V Ag. This dissociation process can be represented in a more chemical language (Kroeger-Vink notation) in Eqn. (1.1)... 

If z corresponds to the number of lattice sites comprised by the formula unit of a compound (e.g. AB204), we call M a lattice molecule . At equilibrium, M has a constant chemical potential 1/°, which we may set equal to zero by definition. Equation (2.21) then reduces to... 

While Eqn. (2.24) justifies the introduction of virtual chemical potentials of SE s including vacancies, it also assumes that the lattice molecule M, according to Eqn. (2.23), is in equilibrium with all the vacancies V, x = 1,..., K. 

The first reaction is a site exchange reaction and so does not alter the number of lattice sites. The second reaction describes the formation of a complete lattice molecule M. An example of the first type of reaction (exchange reaction, Eqn. (2.59)) is the so-called Frenkel defect formation reaction in AX (e.g., in silver halides, see Fig. 1-2)... 

The equilibrium and electroneutrality conditions lead to an equation with the same form as Eqn. (2.62) where Nf = Nv- = Nv-. In contrast to Eqn. (2.61), however, the sites of a new lattice molecule AX have been added to the crystal which, for example, has consequences for the pressure dependence of Nf. ... 

Point defects which are not in equilibrium react either with each other or they react with components or lattice molecules at sites of repeatable growth (normally sur-... 

By modeling r in this way, we have tacitly assumed that the atomic oxygen has free access to the vacancy sinks which are the internal sites of repeatable growth for the lattice molecules. Properly spaced dislocations with fast oxygen diffusion could be a prototype of those sinks. 

For each [A]-lattice molecule we have mA-n0 cation lattice sites. Thus, mA-n0-nB-m0 is the number of cation vacancies which nB ions of type B bring into the cation sublattice when [B] dissolves in [A], We can now formulate Vp [B] in terms of the individual SE potentials as... 

Therefore, if A(a = 0, the cation flux changes its density at the AX/AY interface. This means that this interface (by application of a sufficiently strong electric field) acts either as an A sink or as an A source depending on the direction of the A flux. In the first case, metallic A will be precipitated at the AX/AY interface. Since AtA = A/e> the difference in electric current, A7e, will supply the necessary electrons for the (internal) reduction of the A cations. In the second case, the AX/AY interface operates as an A source and the lattice molecules AX or AY will be decomposed. Consequently, either X(Y) atoms or X2(Y2) molecules are formed and the corresponding reactions read... 

Particle irradiation effects in halides and especially in alkali halides have been intensively studied. One reason is that salt mines can be used to store radioactive waste. Alkali halides in thermal equilibrium are Schottky-type disordered materials. Defects in NaCl which form under electron bombardment at low temperature are neutral anion vacancies (Vx) and a corresponding number of anion interstitials (Xf). Even at liquid nitrogen temperature, these primary radiation defects are still somewhat mobile. Thus, they can either recombine (Xf+Vx = Xx) or form clusters. First, clusters will form according to /i-Xf = X j. Also, Xf and Xf j may be trapped at impurities. Later, vacancies will cluster as well. If X is trapped by a vacancy pair [VA Vx] (which is, in other words, an empty site of a lattice molecule, i.e., the smallest possible pore ) we have the smallest possible halogen molecule bubble . Further clustering of these defects may lead to dislocation loops. In contrast, aggregates of only anion vacancies are equivalent to small metal colloid particles. 

If local stresses exceed the forces of cohesion between atoms or lattice molecules, the crystal cracks. Micro- and macrocracks have a pronounced influence on the course of chemical reactions. We mention three different examples of technical importance for illustration. 1) The spallation of metal oxide layers during the high temperature corrosion of metals, 2) hydrogen embrittlement of steel, and 3) transformation hardening of ceramic materials based on energy consuming phase transformations in the dilated zone of an advancing crack tip. 

Let us consider a homogeneously, but not hydrostatically, stressed solid which is deformed in the elastic regime and whose structure elements are altogether immobile. If we now isothermally and reversibly add lattice molecules to its different surfaces (with no shear stresses) from the same reservoir, the energy changes are different. This means that the chemical potential of the solid is not single valued, or, in other words, a non-hydrostatically stressed solid with only immobile components does not have a unique measurable chemical potential [J. W. Gibbs (1878)]. 

We have pointed out before that during creep, demixing of solid solutions is to be expected. Creep in compounds, however, occurs in such a way that the rate is determined by the slowest constituent since complete lattice molecules have to be displaced and the various constituent fluxes are therefore coupled. If extra fast diffusion paths operate for one (or several) of the components in the compound crystal, the coupling is cancelled. Therefore, if creep takes place in an oxide semiconductor surrounded by oxygen gas, it is not necessarily the slow oxygen diffusion that determines the creep rate. Rather, the much faster cations may determine it if oxygen can be supplied to or taken away from the external surfaces via dislocation pipes. 

Despite potential complications, simple simulations that hold the lattice molecules stationary have yielded results in very good agreement with experimental observations [13,37]. Even simpler calculations which only locate empty regions in the crystal lattice that could accommodate reactant... 

Except for the very early stages of surface layers, it is justified during practically the entire observable course of metal oxidation to assume that the concentration n. of ionic and electronic defects is small compared to the concentrations N, Ng or Njyj of the lattice constituents A and B or the lattice molecules. Consequently the particle currents will not contribute appreciably to a time-variation of defect concentrations, but almost exclusively to the layer formation. For defects we may thus write... 

Fig. 5. Sequence of layers of an oxidation system with several reaction product layers. N is the number of lattice molecules per cm at the layer.

From this it follows that the quantitative ratio of lattice molecules in the two phases is determined to within a numerical factor by the ratio of their Tammann layer formation constants k the ratio of the partial layer thicknesses is ... 

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