Vacancy atom fraction

As we have already stated, Ny so that the atomic fraction of vacancies is ... 

The effect of atomic motion in the solid state on nuclear resonance line width is illustrated by the behavior of Na resonance from NaCl as a function of temperature 97). In Fig. 9 is shown the variation of the Na line width with temperature for pure NaCl and NaCl doped with an atomic fraction concentration of 6 X 10 of CdCU. As discussed in Section II,A,2 the low-temperature, rigid-lattice line width will narrow when the frequency of motion of the nuclei under observation equals the line width expressed in sec.-. The number of vacancies present should be equal to the concentration of divalent impurities and the jump frequency of Na+ is the product of the atomic vacancy concentration and the vacancy jump frequency... 

Figure 4.44 Diffusivity as a function of temperature for KCl with 10 atom fraction divalent cation impurities. Insert plot shows the variation of vacancies with temperature. Adapted from W. D. Kingery, H. K. Bowen, and D. R. Uhhnann, Introduction to Ceramics. Copyright 1976 by John Wiley Sons, Inc. This material is used by permission of John Wiley Sons, Inc.

In self-diffusion by the vacancy mechanism, a lattice atom moves from a normal lattice site to a vacancy. As shown in Figure 4, the atom must move from the normal lattice site in a to the saddle point position in b to reach the vacancy at c. The energy at the saddle point is greater than that at the equilibrium lattice sites, and the atoms must be sufficiently activated in order to move to b and then to c. The fraction of the lattice atoms activated to the saddle point is related to the Gibbs free energy change between positions a and b. The atom fraction of activated atoms, Xm, is expressed by... 

In order to simplify the task, we should accept several assumptions (i) that the predominant atomic defects are vacancies in the metal and oxygen sublattices (ii) that metal M is nonvolatile in the conditions under investigation (iii) that the concentration of different defects in the crystal lattice is small enough, so that the atomic fraction ofatoms in regular positions can be taken as [M ] w 1, [Oq] w 1 and interaction of defects can also be neglected and (iv) that the vacancies are completely ionized. Accordingly, the disordering processes can be presented as ... 

An appreciable N solubility in Th02 at 2550 and 2660X has been demonstrated metallogra-phically with the atom fraction values of x = 0.002 and 0.03, respectively [2]. Incorporation of N and of Th atoms in the ThOa lattice must be accompanied by creation of 0 ion vacancies in the nonmetal sublattice. Substoichiometry of pure ThOa is insignificant, ThO 993, above 2800 K [8]. 

In the expression for both fluxes (the last term), we take into account the input of vacancy gradients due to the Gibbs-Thomson efiect [8]. We note that the diffusion of the B-flux is directed out of the shell and the difiusion of the A-flux is directed into it. Concentrations of B and A inside the compound can slightly change in the narrow homogeneity ranges for atomic fractions of B and A inside the compound, respectively (Acb = — Aca). We assume that 9Ca dCB... 

At subcritical potentials a single oxidation process with participation of only electronegative component takes place on the surface of the alloy. The surface layer is saturated with nonequilibrium defects (mainly vacancies), maintains morphological stability and represents a diffusion zone in which the atomic fraction of the noble component gradually increases as we approach the interface with the solution [6-9], According to the volume-diffusion model [10, 11], the formation of such zone is limited by the time-dependent interdiffusion of alloy components for the vacancy mechanism. 

Since the alloy is a solution of substitution, the vacancies created are common to both components and the alloy is in fact a ternary solution within the meaning of the stmcture elements A1 atoms, A2 atoms, and vacancies whose fractions of sites are, respectively, xi, X2, and xy in a plan parallel with the AIG layer to a X-coordinate y in metal. The positive direction of diffusion will be chosen as the one of A1 atom diffusion (Figure 16.2)... 

Here AH , d/ffj represent the monovacancy energies of R and M atoms in the pure metals R and M, and Fr and are the atomic volumes of these metals. The quantities/m and/r are the atomic fractions of M and R atoms surrounding a given M site in RM . All these quantities are listed by Miedema for practically all metals, so that A Hu can be calculated for given metal combinations RM as a function of concentration n. In fig. 5 the vacancy energies at Ni sites in various LaNi intermetallics have been calculated and plotted as a function of Ni concentration. Note that d/ffj in La-rich compounds is considerably less than in pure Ni metal. The initial concentration independence of results from the fact that in intermetallic compounds the Ni atoms try to surround themselves with an optimal number of the larger La atoms. The broken line represents the results for a solid solution, realized more or less in amorphous La-Ni alloys. 

The random-walk model of diffusion needs to be modified if it is to accurately represent the mechanism of the diffusion. One important change regards the number of point defects present. It has already been pointed out that vacancy diffusion in, for example, a metal crystal cannot occur without an existing population of vacancies. Because of this the random-walk jump probability must be modified to take vacancy numbers into account. In this case, the probability that a vacancy is available to a diffusing atom can be approximated by the number of vacant sites present in the crystal, d], expressed as a fraction, that is... 

Suppose that vacancy diffusion is the principal mechanism involved in atom transport. An expression for the fraction of vacancies in a pure crystal is [Eq. (2.6)]... 

The XANES region of the Pt Lm and Ln absorption edges can be used to determine the fractional d-electron occupancy of the Pt atoms in the catalyst sample by a so-called white line analysis. Figure 2 shows the XAS spectrum collected at both Pt Lm and Lii absorption edges of Na2Pt(OH)e. The sharp features at the absorption edges are called white lines after the white line observed in early photographic film based XAS measurements. Mansour and coworkers have shown that comparison of the white line intensities of a sample with those of a reference metal foil provides a measure of the fractional d-electron vacancy, f, of the absorber atoms in the sample. is defined as follows ... 

It is found that the atomic arrangement, or a vacancy network, in a depleted zone in a refractory metal or a dilute alloy of a refractory metal, created by bombardment of an ion can be reconstructed on an atomic scale from which the shape and size of the zone, the radial distribution function of the vacancies, and the fraction of monovacancies and vacancy clusters can be calculated. For example, Wei Seidman108 studied structures of depleted zones in tungsten produced by the bombardment of 30 keV ions of different masses, W+, Mo+ and Cr+. They find the average diameters of the depleted zones created by these ions to be 18,25 and 42 A, respectively. The fractions of isolated monovacancies are, respectively, 0.13,0.19and0.28,andthe fractions of vacancies with more than six nearest neighbor vacancies (or vacancy clusters) are, respect-... 

In addition to stress, the other important influence on solid state kinetics (again differing from fluids) stems from the periodicity found within crystals. Crystallography defines positions in a crystal, which may be occupied by atoms (molecules) or not. If they are not occupied, they are called vacancies. In this way, a new species is defined which has attributes of the other familiar chemical species of which the crystal is composed. In normal unoccupied sublattices (properly defined interstitial lattices), the fraction of vacant sites is close to one. The motion of the atomic structure elements and the vacant lattice sites of the crystal are complementary (as is the motion of electrons and electron holes in the valence band of a semiconducting crystal). 

Chemical solid state processes are dependent upon the mobility of the individual atomic structure elements. In a solid which is in thermal equilibrium, this mobility is normally attained by the exchange of atoms (ions) with vacant lattice sites (i.e., vacancies). Vacancies are point defects which exist in well defined concentrations in thermal equilibrium, as do other kinds of point defects such as interstitial atoms. We refer to them as irregular structure elements. Kinetic parameters such as rate constants and transport coefficients are thus directly related to the number and kind of irregular structure elements (point defects) or, in more general terms, to atomic disorder. A quantitative kinetic theory therefore requires a quantitative understanding of the behavior of point defects as a function of the (local) thermodynamic parameters of the system (such as T, P, and composition, i.e., the fraction of chemical components). This understanding is provided by statistical thermodynamics and has been cast in a useful form for application to solid state chemical kinetics as the so-called point defect thermodynamics. 

Correlation diminishes the effectiveness of atomic jumps in diffusional random motion. For example, when an atom has just moved through site exchange with a vacancy, the probability of reversing this jump is much higher than that of making a further vacancy exchange step in one of the other possible jump directions. Indeed, if z is the coordination number of equivalent atoms in the lattice, the fraction of ineffective jumps is approximately 2/z (for sufficiently diluted vacancies as carriers) [C. A. Sholl (1992)]. 

This fraction is determined by the step-dance between a specified vacancy and the (tagged) atom during their encounter, which does not end before the atom-vacancy pair has definitely separated. Normally, a new and independently moving vacancy comes along much later and begins the next encounter with the tagged atom. 

This important result modifies the description of encounters between tagged atoms and vacancies. If the fraction 7Vv is sufficiently low, then the encounters of a specified atom with different vacancies are independent of each other. In this case, the correlation factor depends only on the properties of a single encounter... 

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