Neighbour, definition

According to the Frank-Kasper definition, the coordination number is unambiguously 12 in the hexagonal close-packed metals and assumes the value 14 in a body-centred cubic metal. Generally in several complex metallic structures this definition yields reasonable values such as 14, even when the nearest-neighbour definition would give 1 or 2. 

We define an "i-th nearest neighbour complex to be a pair of oppositely charged defects on lattice sites which are i-th nearest neighbours, such that neither of the defects has another defect of opposite charge at the i-th nearest neighbour distance, Rit or closer. This corresponds to what is called the unlike partners only definition. A different definition is that the defects be Rt apart and that neither of them has another defect of either charge at a distance less than or equal to R. This is the like and unlike partners definition. For ionic defects the difference is small at the lowest concentrations the definition to be used depends to some extent on the problem at hand. We shall consider only the first definition. It is required to find the concentration of such complexes in terms of the defect distribution functions. It should be clear that what is required is merely a particular case of the specialized distribution functions of Section IV-D and that the answer involves pair, triplet, and higher correlation functions. In fact this is not the procedure usually employed, as we shall now see. 

The quantity G( a, ) is defined as the probability of finding a defect of type / on a site a distance R( from a particular a defect so that they constitute an i-th nearest neighbour complex ( unlike partners only definition). Let E(t Rt) be the probability that a defect of type t does not have a defect partner at a distance less than or equal to Rt except on one site at a distance Rt. From the definitions made it follows that... 

Coordination number and packing geometry. On the basis of the previous definition of the coordination number (as the number of first neighbours), and of the corresponding coordination geometries (as listed in Table 3.4) some additional remarks may be useful for the particular case of packing of hard sphere atoms. 

To conclude, kinetic measurements and structural analysis of the copolymers have allowed a quantitative and self-consistent description of the reaction of RCI Li species on PMMA taking into account PMMA chain reactivity through the simplified model of the nearest neighbouring group effects. Two main features are particularly relevant the definite influence of tacticity, and the independance of the reaction process on the total charge of the copolymer. In this sense, the R-C Li/PMMA systems are closed to the PMMA basic hydrolysis in presence of excess base. (29,31). 

The first satisfactory definition of crystal radius was given by Tosi (1964) In an ideal ionic crystal where every valence electron is supposed to remain localised on its parent ion, to each ion it can be associated a limit at which the wave function vanishes. The radial extension of the ion along the connection with its first neighbour can be considered as a measure of its dimension in the crystal (crystal radius). This concept is clearly displayed in figure 1.7A, in which the radial electron density distribution curves are shown for Na and Cl ions in NaCl. The nucleus of Cl is located at the origin on the abscissa axis and the nucleus of Na is positioned at the interionic distance experimentally observed for neighboring ions in NaCl. The superimposed radial density functions define an electron density minimum that limits the dimensions or crystal radii of the two ions. We also note that the radial distribution functions for the two ions in the crystal (continuous lines) are not identical to the radial distribution functions for the free ions (dashed lines). 

Chemical structures may be well explained and predicted by means of a set of differently applied constant atomic radii roughly constant for the many cases of a few main types of bond. Thus Pauling (1) used covalent, metallic, van der Waals and ionic radii, aU Structural Radii r chosen to add up to observed Structural Distances D where identical atoms are nearest neighbours D = 2r. But in ionic crystals identical ions are never nearest neighbours hence a special problem arises which forty years ago seemed to have been definitively solved by Goldschmidt (2) and Pauling (/). 

In the case of molecules which do not dissociate, the electromotive force must be due to the orientation of the molecules in the surface layer, which molecules must have a definite electric moment. In the few cases where the electromotive force can be accurately compared with the surface concentration such as n-but3rric, ri-valeric and n-caproic acids, the E.M.F. is almost approximately proportional to the surface concentration F, as would be the case if the moment of a molecule were independent of the proximity of its neighbours. 

The distance matrix D(G) of a graph G is another important graph-invariant. Its entries dy, called distances, are equal to the number of edges connecting the vertices i and j on the shortest path between them. Thus, all dy are integers, including du = 1 for nearest neighbours, and, by definition, d = 0. The distance matrix can be derived readily from the adjacency matrix ... 

For the case 5=1 and D = 1 the results of the stochastic model are in good agreement with the CA model y = 0.262). This is understandable because the different definition of the reaction which leads to a difference in the blocking of activated sites cannot play significant role because all sites are activated. The diffusion rate of D = 10 leads nearly to the same reactivity as if we define the reaction between the nearest-neighbour particles. If the diffusion rate is considerably lowered (D = 0.1), the behaviour of the system changes completely because of the decrease of the reaction probability. This leads to the disappearance of the kinetic phase transition at y because different types of particles may reside on the surface as the nearest neighbours without reaction, a case which does not occur at all in the CA approach. 

Definitions. We consider a lattice with coordination number To each lattice site is given a lattice vector l. The state of the site l is represented by the lattice variable oi which may depend on the state of the catalyst site (e.g., promoted or not) and on its coverage with a particle. Here we deal only with the simple case in which all sites are identical and therefore cr depends only on its coverage (the other case is explained in Section 9.1.4. Therefore ai G 0,A,B,... where 0 represents a vacant site, A is a site which is occupied by an A particle and so on. Next we want to define a variable aij7l which is unity for the case that l and n are the nearest neighbour sites on the lattice and zero otherwise. Sometimes we will use the abbreviation oi = A, a l = A, on = v and o n = v. The states of the neighbourhood (z sites) of site l are denoted by o f. 

These difficulties seem to stem from the definition of the overlap parameter r. This parameter is based on the next-neighbour p7t,p7t overlap value in benzene according to equation 5 ... 

Before commencing a course in practical organic chemistry, the student should have a definite list of preparations to follow. These should be arranged in increasing order of difficulty, and in such a way that, as far as possible, each preparation leads naturally to the next. Where several students are working in the laboratory, the best results are obtained when each works through a different fist and compares notes with his neighbour. 

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