Short-range order theory

The integral under the heat capacity curve is an energy (or enthalpy as the case may be) and is more or less independent of the details of the model. The quasi-chemical treatment improved the heat capacity curve, making it sharper and narrower than the mean-field result, but it still remained finite at the critical point. Further improvements were made by Bethe with a second approximation, and by Kirkwood (1938). Figure A2.5.21 compares the various theoretical calculations [6]. These modifications lead to somewhat lower values of the critical temperature, which could be related to a flattening of the coexistence curve. Moreover, and perhaps more important, they show that a short-range order persists to higher temperatures, as it must because of the preference for unlike pairs the excess heat capacity shows a discontinuity, but it does not drop to zero as mean-field theories predict. Unfortunately these improvements are still analytic and in the vicinity of the critical point still yield a parabolic coexistence curve and a finite heat capacity just as the mean-field treatments do. 

To summarize we have reproduced the intricate structural properties of the Fe-Co, Fe-Ni and the Fe-Cu alloys by means of LMTO-ASA-CPA theory. We conclude that the phase diagram of especially the Fe-Ni alloys is heavily influenced by short range order effects. The general trend of a bcc-fcc phase transition at lower Fe concentrations is in accordance with simple band Ailing effects from canonical band theory. Due to this the structural stability of the Fe-Co alloys may be understood from VGA and canonical band calculations, since the common band model is appropriate below the Fermi energy for this system. However, for the Fe-Ni and the Fe-Cu system this simple picture breaks down. 

TABLE I.a Comparison of Observed Activity Coefficients of Cu and Au with Values Calculated from Observed Short-Range Order Parameters by the First-Order Quasi-Chemical Theory... 

It is simplest to consider these factors as they are reflected in the entropy of the solution, because it is easy to subtract from the measured entropy of solution the configurational contribution. For the latter, one may use the ideal entropy of mixing, — In, since the correction arising from usual deviation of a solution (not a superlattice) from randomness is usually less than — 0.1 cal/deg-g atom. (In special cases, where the degree of short-range order is known from x-ray diffuse scattering, one may adequately calculate this correction from quasi-chemical theory.) Consequently, the excess entropy of solution, AS6, is a convenient measure of the sum of the nonconfigurational factors in the solution. 

This observation is expected from theory, as the observed thickness distributions are exactly the functions by which one-dimensional short-range order is theoretically described in early literature models (Zernike and Prins [116] J. J. Hermans [128]). From the transformed experimental data we can determine, whether the principal thickness distributions are symmetrical or asymmetrical, whether they should be modeled by Gaussians, gamma distributions, truncated exponentials, or other analytical functions. Finally only a model that describes the arrangement of domains is missing - i.e., how the higher thickness distributions are computed from two principal thickness distributions (cf. Sect. 8.7). Experimental data are fitted by means of such models. Unsuitable models are sorted out by insufficient quality of the fit. Fit quality is assessed by means of the tools of nonlinear regression (Chap. 11). 

No investigation of a solid, such as the electrode in its interface with the electrolyte, can be considered complete without information on the physical structure of that solid, i.e. the arrangement of the atoms in the material with respect to each other. STM provides some information of this kind, with respect to the 2-dimensional array of the surface atoms, but what of the 3-dimensional structure of the electrode surface or the structure of a thick layer on an electrode, such as an under-potential deposited (upd) metal At the beginning of this chapter, electrocapillarity was employed to test and prove the theories of the double layer, a role it fulfilled admirably within its limitations as a somewhat indirect probe. The question arises, is it possible to see the double layer, to determine the location of the ions in solution with respect to the electrode, and to probe the double layer as the techniques above have probed adsorption Can the crystal structure of a upd metal layer be determined In essence, a technique is required that is able to investigate long- and short-range order in matter. 

The importance of interactions amongst point defects, at even fairly low defect concentrations, was recognized several years ago. Although one has to take into account the actual defect structure and modifications of short-range order to be able to describe the properties of solids fully, it has been found useful to represent all the processes involved in the intrinsic defect equilibria in a crystal (with a low concentration of defects), as well as its equilibrium with its external environment, by a set of coupled quasichemical reactions. These equilibrium reactions are then handled by the law of mass action. The free energy and equilibrium constants for each process can be obtained if we know the enthalpies and entropies of the reactions from theory or... 

The ratio Eg(T)/Eg(0) is compared with experimentally determined values in Figure 5 where it can be seen that the theory reproduces the weak temperature dependence of the gap. We expect that an accurate theory with fluctuations included would remove the discontinuity shown in Figure 4 due to short range order in the pairing amplitude in the neighbourhood of Tc and such an effect can account for the trends reviewed by Deutcher in which the gap does not disappear at Tc. 

At the Curie point, the Weiss theory predicts that for T > Te, in the absence of an external field, the spin order vanishes completely. Actually there is considerable short-range order just above Te, as has been verified by neutron diffraction experiments (405,675). It is the problem of short-range order that is tackled by the more exact quantum statistics mentioned in connection with equation 94. At very high temperatures (T Te), there is no short-range order, and the experimental curve approaches the Curie-Weiss curve asymptotically. Theory shows that the possibility of short-range ordering lowers the... 

Calculation of the multiple scattering by means of a real-space cluster approach is considerably more flexible than band-structure methods. Since this technique does not rely on crystal periodicity, it can readily be applied to interpret data for materials of arbitrary atomic arrangements. The sensitivity to higher order correlations has been shown. Fujikawa et al. (94,96) favor short-range-order multiple-scattering XANES theory, in which atoms are not divided into shells but the scattered waves are classified into a direct term and a fully multiple-scattering term. 

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