Phase relations different atoms

The curious phase relations between phosphorus, sulfur and their binai compounds are worth noting. Because both P4 and Sg are stable molecules the phase diagram, if studied below 100°, shows only solid solutions with a simple eutectic at 10° (75 atom % P). By contrast, when the mixtures are heated above 200° the elements react and an entirely different phase diagram is obtained however, as only the most stable compounds P4S3, P4S7 and P4S10... 

In the Fe2P-type structure there are four different groups of equipoints. The distribution of P and Fe atoms in different groups of positions is reported. A number of isostructural binary compounds are known. To the same structure, however, ternary or even more complex phases may be related if different atomic species are distributed in the different sites. This structure can be considered as an example of more complex structures built up by linked triangular prisms of Fe atoms. 

The determination of absolute configurations in these investigations was based on a qualitative use of the anomalous scattering effects—on simple observations of which of two opposite reflections, hkl or hkl, was the stronger. It is, however, possible in principle to make quantitative use of the differences between Fhkl and Fjm to determine phase angles directly. The difference between Fhkl and depends on the phase relations of the waves from the anomalously scattering atom and from the rest of the molecule when these two waves are nearly in phase,... 

In addition to Fig. 10 in Fig. 12 some data are given regarding phase relations and stability areas with different rare earth cations. The lowest x-value for trivalent cations is 0.33 for Ca2+ the lowest value is 0.3 [123-127, 169, 187], It should be emphasised that all ass do not include the pure a-Si3N4. The atomic positions for different cations M are listed in Table 5, showing that the Si- and N-positions do not differ significantly from that in pure a. This is... 

Both of these quantities contain an arbitrary constant, the zero from which the potentials are measured, but differences of either the electrostatic potential or of the electrochemical potential, between two phases, are definite. The thermionic work function, x, the work required to extract electrons from the highest energy level within the phase, to a state of rest just outside the phase, is also definite and the relation between the three definite quantities fa, V, and x is given by (3.1), where is the electrochemical potential of electrons very widely separated from all other charges. The internal electric potential , and other expressions relating to the electrical part of the potential inside a phase containing dense matter, are undefined, and so are the differences of these quantities between two phases of different composition. This indefiniteness arises from the impossibility of separating the electrostatic part of the forces between particles, from the chemical, or more complex interactions between electrons and atomic nuclei, when both types of force are present. 

Since there is an exact correspondence of atoms at the boundaries of the blocks, the shear structures can be represented as the ordered overgrowth of a particular phase upon itself. There are certain cases where this is also true for oxides and oxyhydroxides of different structure and composition, which may rise to a nonstoichiometric compound if disorder exists, or to a series of phases related by a general formula if there is order. 

The question at this stage is How does one derive the atomic arrangement in a crystal, such as that of sodium chloride or potassium chloride, from the intensities in their respective diffraction patterns The answer is that the diffracted X-ray beams, which have amplitudes represented by the square roots of their measured intensities, must be recombined in a manner similar to that achieved by a lens in an optical microscope. This recombination is done by a mathematical calculation called a Fourier synthesis. The recombination cannot be done directly because the phase relations among the different diffracted X-ray beams usually cannot be measured. If the phases, however, can be estimated by one of the methods described in Chapter 8, an approximate image of the arrangement of atoms in the crystal can be obtained. 

Since the ordering process can start at non-equivalent atomic sites, the boundary can result in a discontinuity, dividing regions with a different phase relation. Such boundaries are therefore called APBs (Figure 13.6). 

The arrangement of the atoms in the NaTl structure and its relation to some other simple cubic structures is illustrated in Fig. 1 and Table 1. In Fig. la a unit cell is shown with four different lattice sites A to D. The A2 structure results if all four sites are occupied by the same kind of atoms or statistically by different atoms. For AB compounds the CsCl structure results for the configurations Aa, Ab, Bq and Bp (cf. Fig. lb). The NaTl structure follows from Aa, Bg, Ac and Bp as displayed in Fig. Ic. For ternary Zintl phases A2BC one finds that either the B and C atoms are statistically distributed on the B and D sites given in Fig. la (B32A structure) or they are ordered and the XA type of structure results 5 (cf. Table 1). 

The intensity of the FT peak increases showing that the atoms of the first shell either have a larger backscattering amplitude or are in increasing number. At the end of the process, the characteristic FT of metallic copper is obtained. Figure 11 (a, b, c, d) shows the filtered back-transformed spectra of the first shell. These curves exhibit a continuous decrease of the amplitude of the oscillations with the appearance of a beat node at about 250 eV (Fig. 11c) directly related to the splitting of the Fourier transform. This beat node evidences that two different atoms with a k difference in their phase shifts contribute to the EXAFS oscillations. A direct explanation involves the O and S atoms in the first shell. This is consistent with the EXAFS characteristics drawn from two samples used as standards the... 

For phase-boundary controlled reactions, the situation is somewhat different. Diffusion of species is fast but the reaction is slow so that the diffusing species pile up. That is, the reaction to rearrange the structure is slow in relation to the arrival of the diffusing ions or atoms. Thus, a phase-boundary (difference in structure) focus exists which controls the overall rate of solid state reaction. This rate may be described by ... 

Here the structure factor signifies the vectorial sum of the waves scattered by the single atoms which show amplitude f and phase y. Every atom contributes a scattered wave to the whole diffraction effect, the amplitude of which is proportional to the so-called form factor. The phase is thus defined by the position of the atom in the elementary cell, whilst the form factor is a characteristic constant for every sort of atom which represents a measure of its scattering power. Hence no special differences exist in the positions of the diffracted beams, which in both X-ray and electron diffraction cases satisfy the geometric relations between lattice constant and X-ray or material wavelengths, according to the Bragg equation. However, there are definitely differences in their intensities. 

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