First neighbors

It turns out that for some systems the GPM yields the pair interactions, particularly those between first neighbors, which do not correspond to experimental phase diagrams. It is the purpose of the present work to show some of these cases and make a comparison with results obtained by other methods, particularly by the CWIS. 

In order to check the consistency and mutual relations of ECIs calculated by various methods, as well as to compare them with experimental data, we have performed calculations for several alloy systems, as diverse as Cu-Nl, Al-Li, Al-Ni, Ni-Pt and Pt-Rh. Here we present the results for Al-Ni, Pt-Rh and Ni-Pt alloys in some detail, because the pair interactions between the first neighbors are dominant in these alloys which makes the interpretation relatively simple. On the other hand, the pair interactions between more distant neighbors and also triplet interactions are important for Al-Li and Cu-Ni. The equilibrium atomic radii, bulk moduli and electronegativities of A1 and Ni are rather different, while Pt and Rh are quite similar in this respect. The Ni and Pt atoms differ mainly by their size. 

To conclude this section it may be worthwhile to remark that, although the L-J-D method is generally applied to a face-centered cubic lattice (z = 12), it is equally valid for the cavities in a clathrate (z = 20—28), as long as one restricts oneself to first-neighbor interactions. 

It will be observed that entropies of dilution (as indicated by i) are highly variable from one system to another. This is contrary to the theory developed from consideration of lattice arrangements, according to which pi should be approximately 1/2 and nearly independent of the system. For polystyrene in methyl ethyl ketone, the entropy of dilution is nearly zero i.e., this solvent is a poor one not because of an adverse energy of interaction but because of the low entropy. First neighbor interactions apparently contribute to the entropy as well as to the energy, a point which was emphasized in Chapter XII. It will be noted also that cyclic solvents almost without exception are more favorable from the standpoint of the entropy than acyclic ones. 

Parameter expressing the first neighbor interaction free energy, divided by kTj for solvent with polymer (Chap. XII et seq.). 

Fig. 3. Formal potentials of oligo(dihexylferrocenylene)s 13 (O) (60,76), those calculated from the first neighboring site interaction (A) with u, = 15 kJ mol-1 and u2 = 4.5 kJmol-1, and those calculated from both first and second neighboring-site interaction (V) with u, = 15kJmol u2 = 4.5 kJ mol 1, and oxr = -3.8 kJ mol-1 (77).

In the mesomorphic form of iPP, chains in the ordered 3/1 helical conformation are arranged in small domains where only the first neighboring chains... 

Figure 6.13 Rh K-edge EXAFS spectra, uncorrected Fourier transforms according to (6-10) and isolated EXAFS contribution from the first neighbor shell of Rh metal (top), Rh20 ( (middle) and RhCI3 (bottom). The first shell contributions clearly reflect the different backscattering properties of Rh, O and Cl atoms. Note the high number of coordination shells that are visible in Rh bulk metal (from van Zon et al. [35]).

As noted earlier, the diffraction of X-rays, unlike the diffraction of neutrons, is primarily sensitive to the distribution of 00 separations. Although many of the early studies 9> of amorphous solid water included electron or X-ray diffraction measurements, the nature of the samples prepared and the restricted angular range of the measurements reported combine to prevent extraction of detailed structural information. The most complete of the early X-ray studies is by Bon-dot 26>. Only scanty description is given of the conditions of deposition but it appears likely his sample of amorphous solid water had little or no contamination with crystalline ice. He found a liquid-like distribution of 00 separations at 83 K, with the first neighbor peak centered at 2.77 A. If the pair correlation function is decomposed into a superposition of Gaussian peaks, the area of the near neighbor peak is found to correspond to 4.23 molecules, and to have a root mean square width of 0.50 A. 

Correlation functions hoo(R) for the amorphous deposit prepared and studied at 77 K are shown in Fig. 7b together with the curve for polycrystalline ice Ih. As in the crystalline phase, the nearest-neighbor oxygen-oxygen correlations in H20(as) occur in an exceptionally narrow band centered at 2.76 A, with rms-deviation 0.114 A. The distance ratio for second and first neighbors indicates tetrahedral coordination on the average, but the second neighbor peak near... 

To approach the complexity of a real tridimensional structure, let us first consider the case of a monodimensional array of alternating positive and negative charges, each at distance r from its first neighbor. We will assume for the sake of simplicity that the dispersive potential is negligible. The total potential is therefore... 

Consider a diatomic chain in which the atoms, of distinct masses mi and m2, are positioned at distance a (figure 3.6). The repetition distance of the chain is 2a, and the Brillouin zone falls between —irlla and i lla. If only the interactions between first neighbors are significant, the equation of motion for atom r at position ja is given by... 

For a pair of Schottky defects with, respectively, x and y first neighbors and v and v" vibrational frequencies, vibrational term / has the form... 

In the case of a Frenkel defect pair, if v, is the frequency of interstitial atom i and v is the vibrational frequency of the Z first neighbors surrounding it, we have... 

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