Frenkel equation

The variational derivation of TDHF equations (Frenkel, 1934) relates immediately to the stationary state approach and the SCF method. The quantities needed for substitution in the variation principle (12.2.2) are (5V V ) and (69 H 9 ) the first follows from first principles (for example from Slater s rules, noting that d/8r behaves like a 1-electron operator) as... 

If attractive forces are present, then according to an equation by Frenkel (see Ref. 2), the average time of stay t of the molecule on the surface will be... 

This may be based on Eq. XVI-2 [232] or on related equations with film thickness given by some version of the Frenkel-Halsey-Hill equation (Eq. XVII-79) [233,234],... 

The multilayer region. The Frenkel-Halsey-Hiil (FHH) equation... 

By analogy with similar materials in which free elecU ons and electron holes are formed, NiO is called a p-type compound having vacant site Schottky defects, and ZnO is an n-type compound having interstitial Frenkel defects. The concentrations of these defects and their relation to the oxygen pressure in the suiTounding atmosphere can be calculated, for a dilute solution of defects by the application of a mass action equation. The two reactions shown above are represented by the equations... 

In Eq. (13), medium resistance to bubble compression-decompression depends on viscosity r, and is described by the second member in the right-hand part of the equation. It should be mentioned at this point that bubble growth in a Newtonian liquid was originally examined by the Soviet physicist Y. I. Frenkel [29], in a rarely cited work published in 1946. 

FHH (Frenkel-Halsey-Hill) theory is valid for multi molecules adsorption model of the flat surfrtce material. When this model is applied for the surface fractal in the range of capillary condensation, in other words, in the state of interface which was controlled by the surface tension between liquid and gas, the modified FHH equation can be expressed as Eq. (3). 

In this case, we use 6 as a small fraction since the actual number of defects is small in relation to the overall number of ions actually present. For the F-Center, the brackets enclose the complex consisting of an electron captured at an anion vacancy. Note that these equations encompass all of the mechanisms that we have postulated for each of the individual reactions. That is, we show the presence of vacancies in the Schottlqr case and interstitial cations for the Frenkel case involving either the cation or anion. The latter, involving an interstitlcd anion is called, by convention, the "Anti-Frenkel" case. The defect reaction involving the "F-Center" is also given. 

Write a series of equations for the Frenkel defect, similar to those given for the Schottky defect, i.e.- Equations 3.7.2 to 3.7.3. 

We now proceed as we did for the stoichiometric case, namely to develop defect- concentration equations for the non-stoichiometric case. Consider the effect of Anti-Frenkel defect production. From Table 2-1, we get Kaf with its associated equation, kAF In Table 2-2, we use Kxi for X-interstitial sites. Combining these, we get ... 

Equations (2) and (3) relate intermolecular interactions to measurable solution thermodynamic properties. Several features of these two relations are worth noting. The first is the test-particle method, an implementation of the potential distribution theorem now widely used in molecular simulations (Frenkel and Smit, 1996). In the test-particle method, the excess chemical potential of a solute is evaluated by generating an ensemble of microscopic configurations for the solvent molecules alone. The solute is then superposed onto each configuration and the solute-solvent interaction potential energy calculated to give the probability distribution, Po(AU/kT), illustrated in Figure 3. The excess... 

A brief sketch of the derivation of the WHAM equations follows we note that a detailed explanation is available in the book by Frenkel and Smit [14], Consider the canonical reweighing (3.5). Our goal will be to combine the histograms pi(U) from several runs at different temperatures T to predict the distribution of potential energies at a new temperature T. Individually, each run enables us to reweight its histogram to obtain the distribution at T... 

A similar equation can be written for Frenkel defects on the anion positions ... 

The approximations to use depend upon the pressure regime and the values of the equilibrium constants. This oxide is an insulator under normal conditions, and so, in the middle region of the diagram, Frenkel equilibrium is dominant, that is, Kf > Ke and the electroneutrality equation is approximated by... 

The addition of either donors or acceptors will, however, upset the charge balance, and these must be included in the electroneutrality equation. Consider donor doping by a trivalent ion D3+ due to reaction with D2X3 to introduce D defects, once again assuming that Frenkel defects are not important. The original electroneutrality Eq. (7.12) ... 

The configurational entropy change, A.Sf, due to the distribution of the defects over the available positions, can be determined by using the Boltzmann equation. The number of ways of distributing the nP vacancies that have been created over the N available positions in the atom array affected by Frenkel defects is... 

Results of the ideal solution approach were found to be identical with those arrived at on the basis of a simple quasichemical method. Each defect and the various species occupying normal lattice positions may be considered as a separate species to which is assigned a chemical potential , p, and at equilibrium these are related through a set of stoichiometric equations corresponding to the chemical reactions which form the defects. For example, for Frenkel disorder the equation will be... 

In tightly bound (Frenkel) excitons, the observed peaks do not respond to the hy-drogenic equation (4.39), because the excitation is localized in the close proximity of a single atom. Thus, the exciton radius is comparable to the interatomic spacing and, consequently, we cannot consider a continuous medium with a relative dielectric constant as we did in the case of Mott-Wannier excitons. 

From classic thermodynamics alone, it is impossible to predict numeric values for heat capacities these quantities are determined experimentally from calorimetric measurements. With the aid of statistical thermodynamics, however, it is possible to calculate heat capacities from spectroscopic data instead of from direct calorimetric measurements. Even with spectroscopic information, however, it is convenient to replace the complex statistical thermodynamic equations that describe the dependence of heat capacity on temperature with empirical equations of simple form [15]. Many expressions have been used for the molar heat capacity, and they have been discussed in detail by Frenkel et al. [4]. We will use the expression... 

Some results of the calculations of Frenkel et al. [4] for the coefficients in this equation are summarized in Table 4.9 and are illustrated in Figure 4.2. 

Figure 4.2. Variation of heat capacity with temperature as calculated from the equations of Frenkel et al. [4]. The differences observed between isotopic species and the way heat capacity depends on molecular size and structure can be described thermodynamically, but they must be explained by the methods of quantum-statistical thermodynamics. The right-hand scale is for H2 and D2 the left-hand scale is for the other compounds.

Equation 4.75 finds its application in the region of intrinsic disorder (a similar equation can be developed for Frenkel defects), where Schottky and Frenkel defects are dominant with respect to point impurities and nonstoichiometry. 

The significance of empirical equations 4.78, 4.79, 4.80, and 4.81 has not yet been clarified in a satisfactory fashion. The enthalpy of Frenkel processes in metals can be related to the Debye temperature of the solid through... 

To overcome the problems which arise from a limited size range and the different adsorption properties of different adsorbates, one may use the entire adsorption isotherm obtained with just one probe molecule instead. There are two approaches both leading to a Frenkel-Halsey-Hill type equation ... 

Now that we know how to write defect equations, let s look at Frenkel and Schottky defects in more detail. 

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