Site occupation parameters

I Site-occupatioH parameters. A complete description of a system requires detailed knowledge about the occupancy of all available sites by all die participating species. However, with K constituents and N sites, this leads to different configurations, which is computationally unrealistic. A usefid approximation is to use average site-occupation parameters, for equivalent crystallographic sites. 

To complete the picture it is useful to define distribution variables, Xj, which are atomic fiactions normalised to a mole of atoms as related to site-occupation parameters. The x and y variables are related geometrically through the equation ... 

These were largely due to inappropriate use of the method being applied. For the Rietveld based techniques, these errors included (i) entry of incorrect crystal structure information, including atom coordinates and thermal vibration parameters, (ii) incorrect space group notation, (iii) incorrect atom site occupation parameters, (iv) allowing structure parameters (especially thermal parameters) to refine to physically unrealistic values, and (v) not completing the refinement. 

Yak] Yakel, H.L., Brynestad, J., Non-random Site-occupation Parameters in (Cr,Fe)23Ce Phases , Scr. Metall, 16(4), 453-454 (1982) (Crys. Structure, Thermodyn., Calculation, Experimental, 6)... 

The output from the Le Bail/Rietveld refinements were combined to show the variation of several important structure-property indicators (unit cell, thermal, phase fraction and occupancy parameters) versus time/temperature profiles (Fig. 4.3). Given the fact that thermal B-factors and site occupancy parameters are... 

The density of lattice vacancies can be determined, usually accurately, by the site occupation parameter. Thus if the local atomic potential is bifurcated and atoms are locally displaced, it may appear that partial occupation of the displaced site describes the system quite well. However, in such cases there is usually strong coupling between the occupation parameter and the DW factor during the process of refinement. Also, local displacement of an atom is usually accompanied by a lattice relaxation around the displaced atom and a much increased DW factor of the displaced site. Thus, in practice, it is very difficult to describe anharmonic local displacements through partial occupation of two closely spaced sites. 

The atomic scattering factors are the Fourier transforms of the spherical atomic electron distributions. They are considered as known from quantum-chemical calculations. The site occupation parameters may assume values different from unity if the structure is disordered. The Debye-Waller factors allow for the atomic thermal motions. They are functions of the atomic displacement parameters W. Omitting the atom index n and representing the Miller indices and lengths of the reciprocal lattice vectors by and a, respectively ... 

There results nine adjustable parameters per atom, i.e. three positional coordinates and six displacement parameters, and sometimes an additional site occupation parameter, although the point symmetry of an atom in a special position may reduce the number of adjustable parameters. The model value of the net intensity I h, k, /), Bragg peak minus background, is... 

Crl, Grr, A r and A r are the potential parameters of the constituents A and B of the alloy, S r r r, is the structure matrix in the most localized representation, tir are local site-occupation variables which randomly takes value 1 or 0 according to whether the site is occupied by an atom of type A or not, with probabilities proportional to the concentrations of the constituents. According to the prescription of the augmented space formalism, the effective non-random Hamiltonian H in augmented space is then... 

Site Localized moment Spin proportion Occupation parameter ... 

Figure 3. The effect of degree of polymerization on surface coverage (fractional site occupancy) at various polymer concentrations. The solid lines represent the present model and the symbols correspond to the theory of Scheutjens and Fleer. The parameter values are the same as in Figure 2.

Figure 5.7 Molar volumes and cell edges in binary olivine mixtures (from Akimoto et al., 1976 modified). Note conspicuous deviations from linearity in cell edges of (Mg,Ni)2Si04 mixture, satisfactorily reproduced by site-occupancy calculations based on parameters in table 5.7.

Table 5.8 Measured (m) structural parameters for mixture Mgi 4Nio eSi04 compared with results of calculations (c) based on equation 5.24 and observed site occupancies (Ottonello et al., 1989)...

Essentially, sublattice models originate from the concepts of Temkin (1945) who proposed that two separate sublattices exist in a solid-state crystal for cations and anions. The configurational entropy is then governed by the site occupation of the various cations and anions on their respective sublattices. When the valence of the cations and anions on the sublattices are equal, and electroneutrality is maintained, the model parameters can be represented as described in Section 5.4.2. However, when the valence of the cations and anions varies, the situation becomes more complex and some additional restrictions need to be made. These can be expressed by considering equivalent fractions (/) which, for a sublattice phase with the formula (/, . .. )(M"", . ..), are given by... 

Fig. 2.14 Relationship between the oxidation parameter z and the Fe site occupancy per 32...

Virtually all of the reported structural data on titanium alloy hydrides and deuterides indicate that the solute atoms occupy tetrahedral interstitial sites in the metal lattice. Neutron diffraction data obtained for deuterium in Ti/34 atom % Zr and in Ti/34 atom % Nb (17) indicate tetrahedral site occupancy in the bcc /3-phase. Similarly, data reported for deuterium in Ti/19 atom % V and in Ti/67 atom % Nb (18) indicate tetrahedral site occupancy in the fee 7-phase. Crystallographic examination of the 7-phase Ti-Nb-H system (19) reveals that increasing niobium content linearly increases the lattice parameter of the fee 7-phase for Nb contents ranging from 0 to 70.2 atom %. Vanadium, on the other hand, exerts the opposite effect (6) at H/M = 1.85, the 7-phase lattice parameter decreases with increasing vanadium contents. 

Order parameter, rj Magnetization, M Density difference, pt - pg Site occupation... 

Cation Site Distribution, Thin-film EDS analysis can also be used to quantitatively determine the site occupancy of atoms in a known crystal structure. Atom Location by Channeling Enhanced Microanalysis (ALCHEMI) is a technique which utilises electronchanneling enhanced X-ray emission for specific atoms in a crystal when appropriately oriented relative to the incident beam [43]. The method involves no adjustable parameters, can be used on relatively small areas of sample and provides fractional occupancies of atom positions [44] Unlike X-ray diffraction which has had limited success with adjacent elements in the periodic table [e.g. 45], ALCHEMI can provide site occupancies for adjacent elements and is relatively insensitive to sample thickness or the precise electron beam orientation [44] ... 

The upper limit for tetrahedral Fe3+ I.S. in silicates is shown to be 0.25 mm/sec., whereas the lower limit for octahedral Fe3+ is 0.29 mm/sec. The correlations point to inconsistencies in Mossbauer spectral parameters and cation site occupancy assignments for clintonite, yoderite and sapphirine. New Mossbauer spectral data obtained for these minerals demonstrate that clintonites from skarn deposits contain tetrahedral Fe3+ and octahedral Fe3+ and Fe2+, with relative enrichment of Fe3+ in tetrahedral sites only octahedral Fe2+ and Fe3+ occur in sapphirines from granulite facies rocks and five-coordinated Fe3+ predominates over octahedral Fe3+ ions in yoderites from high grade metamorphic rocks. 

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