Coordination number effective

In 1961 Bauer [226], within the framework of the ionic model, demonstrated that in structures of the Ti02 type the minimum energy of a crystal lattice is reached at 

How we can define Nc if ligands are situated at different distances For the first time this question was addressed by Frank and Casper [227] who defined the effective coordination number (Nc ) as the number of planes in the Voronoi polyhedron. This approach assumes that all atoms have identical volume, all planes in a polyhedron are equivalent, i.e. there are no distinctions in the character of the bonds formed by different ligands. The metal structures generally conform to these criteria [228] but application of this approach to binary inorganic and complex compounds [229,230] is not straightforward and requires some accounting for different atom sizes. 

Simultaneously, a similar formula was proposed by Brunner [238], and later by Beck [239], In [237] Nc was calculated in terms of the covalent theory. Since the overlap integrals (5) of the bonds are directly proportional to the bond energies (see Sect. 2.3), then 

In 1978 Carter [240] proposed to calculate Nc under the general formula  

Besides the distortion of the coordination polyhedron, deviations of Nc from integer values can be caused by point defects (vacancy and interstitial) in a crystal structure or in the first coordination sphere of an amorphous solid. These values of Nc are defined experimentally by XRD (see Chap. 7) and by optical methods. Thus, Wemple [242] estimated by spectroscopy that in the structure of AS2S3 the N = 3.4 0.2 whereas Eq. 5.7 gives 3.7. Optical estimates of Nc of Se andTe are 2.8 and 3.0, respectively, whereas the calculated values are 2.8 and 3.1. When the composition of chalcogenide glasses (Ge-S, Ge-Se, As-Se, Ge-As-Se) is altered, there occur phase transformations with changing structural, mechanical and electric properties 

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Saa = -esB = e was chosen, and an effective coordination number Zcir = 2.5 occurred ) Thus one finds that the coefficient in the relation T N ,r is somewhat smaller than expected from Eq. (3), and in addition there is a "chain end"-correction. However, an integral equation theoryyielded Tc cc Vn, and th simulations showed that this result clearly is incorrect. More recent improved versions of integral equation theories now yield Tc N as well but still the theoretical understanding of the prefactor in this relation is limited. 

Calculate effective coordination numbers (e.c.n.) with the formula given on page 3 for ... 

R. Hoppe, Effective coordination numbers and mean Active ionic radii. Z Kristallogr. 150 (1979) 23. 

For a near-neighbor shell of atoms (Nt) whose interatomic vector with the absorber makes some angle 0, relative to the plane of polarization, one can relate the effective coordination number (Nf) and the true coordination number through57... 

For a discussion on the effective coordination number , its relation to atomic size, bond strength, Madelung constant, etc., see also Simon (1983). For a computation of the heats of formation based on the so-called effective coordination, see a formula by Kubaschewski and Evans (1958), and for a discussion on its application and limits, see Borzone etal. (1993). 

The effective coordination number should certainly be taken to have a value less than 6 for p- or d-functions, which are not spherically symmetric, giving smaller critical values, since substantial overlap between neighbours will occur only in certain directions. Experimental evidence that this is so is discussed later. 

Fig. 2. The reduced effective coordination number, x/o, versus ax for the face-centered...

Mechanisms for such reactions have been viewed by the effective coordination number of the transition state. Bond rupture processes proceed via five-coordinate transition states for which idealized square-pyramidal (SP) and trigonal-bipyramidal (TBP) geometries with the dangling ligand axial or equatorial have been considered. Twist mechanisms are considered to proceed via six-coordinate transition states with idealized trigonal prismatic (TP) geometry. Twists of the chelate rings about the real or pseudo C3 axis in the case of the cis or trans isomer, respectively, and about the... 

Nephelauxetic effects (1 — f ), mean lanthanide-ligand distances (R (Ln-O) and effective coordination numbers of lanthanides (CNeff) in Pr(III) and Nd(III) ODA complexes determined from baricentres (a) of spectral bands ODA = oxydiacetate, OOCCH2OCH2COO- [184],... 

Results of best fits to SEXAFS modulations, recorded at normal and grazing incidence, from MgO(100)-Ag at 1 ML equivalent cover e [43]. Also listed are the theoretically exjjccted effective coordination numbers (N j,eo) for the adsorption geometry discussed in the text. 

Results of best fits to SEXAFS modulations, recorded at normal incidence, from Al203(0001)lxl-Cu and Al203(0001)(V31xV31)R 9°-Cu at 0.5 ML equivalent coverage of Cu [110]. N is the effective coordination number of a shell of scatterers, R is its distance, and Aa2 the mean square relative displacement. 

The effective coordination numbers resulting from single shell fits to K T-edge SEXAFS data recorded from ZnO(000 I)2x2-K [132]. Also listed are the calculated effective coordination numbers for the three-fold hollow, bridge and atop adsorption sites. 

For a rough estimate of A(r ) we take the isomer shift difference between the AuF complexes and AuFj, AS= 3.4 mm/s, as representative for the one between Au(V) and Au(III). It is obvious from the above discussion that for this pair of gold compounds the difference in effective coordination number is smaller than for any other pair of the stud-... 

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