Radius, atomic Subject

Returning now to our main topic of crystalline solids, we consider a primitive cell with h atoms centred at positions q. Some of these atoms may be of the same type, and we denote the number of type t atoms in the cell h. . In the atomic-sphere approximation each atom is surrounded by a sphere of suitably chosen radius S subject to the constraint... 

Some metals are soluble as atomic species in molten silicates, the most quantitative studies having been made with Ca0-Si02-Al203(37, 26, 27 mole per cent respectively). The results at 1800 K gave solubilities of 0.055, 0.16, 0.001 and 0.101 for the pure metals Cu, Ag, Au and Pb. When these metal solubilities were compared for metal alloys which produced 1 mm Hg pressure of each of these elements at this temperature, it was found drat the solubility decreases as the atomic radius increases, i.e. when die difference in vapour pressure of die pure metals is removed by alloy formation. If the solution was subjected to a temperature cycle of about 20 K around the control temperamre, the copper solution precipitated copper particles which grew with time. Thus the liquid metal drops, once precipitated, remained stable thereafter. 

Hg is much more dense than Cd, because the decrease in atomic radius that occurs between Z = 58 and Z = 71 (the lanthanide contraction) causes the atoms following the rare earths to he smaller than might have been expected for their atomic masses and atomic numbers. Zn and Cd have densities that are not too dissimilar because the radius of Cd is subject only to a smaller d-block contraction. 

The role of d-type orbitals is not addressed in Table 1. This subject has been addressed for second-row atoms in previous reviews21,22 which contain many references to this subject. It is very difficult to define the energy and radius of the outer-sphere d-type orbitals (nd) in isolated atoms since they are not occupied in the ground state. Rather than make use of some excited state definition, we prefer to postpone a discussion of this subject until after an inspection of the calculated results on the molecular compounds. 

An introductory example to this subject is the well-known diagrams developed by Darken and Gurry (1953) for solid solution prediction. In such a diagram (as shown in Fig. 2.14) all elements may be included. The two coordinates represent the atomic size, generally the radius corresponding to the coordination number (CN) 12, and the electronegativity of the elements. 

The atomic radius of an element is considered to be half the interatomic distance between identical (singly bonded) atoms. This may apply to iron, say, in its metallic state, in which case the quantity may be regarded as the metallic radius of the iron atom, or to a molecule such as Cl2. The difference between the two examples is sufficient to demonstrate that some degree of caution is necessary when comparing the atomic radii of different elements. It is best to limit such comparisons to elements with similar types of bonding, metals for example. Even that restriction is subject to the drawback that the metallic elements have at least three different crystalline arrangements with possibly different coordination numbers (the number of nearest neighbours for any one atom). 

The first attempt to calculate realistic wave functions for electrons in metals is that of Wigner and Seitz (1933). These authors pointed out that space in a body-or face-centred cubic crystal could be divided into polyhedra surrounding each atom, that these polyhedra could be replaced without large error by spheres of radius r0, so that for the lowest state one has to find spherically symmetrical solutions of the Schrodinger equation (6) subject to the boundary condition that... 

The main problem in the use of the hard-sphere potential is the selection of values of the van der Waals radii of the atoms. Much has been written on this subject, and the difficulty lies primarily in the fact that the concept of a van der Waals radius is itself a nebulous and inexact one. Also, since the radii are assigned to spherical atoms, no consideration is taken of the angle at which the two atoms approach each other. Thus, it is not surprising to find large ranges of values in the literature for the... 

The covalent radii of transition elements are subject to two additional effects that influence the values of ionic radii also. A large covalent radius for a given atom is favored by both a low oxidation number and a high coordination number. These two effects are independent neither of each other nor of bond order effects however, an adequate unified treatment of the interrelationships between bond number, coordination number, oxidation number, and bond distances for compounds of the transition metals is best postponed to a more advanced text. 

The Sn 5 s and 5p radial functions, from a nonrelativistic calculation for the free 5sz5pz atom, are plotted in Fig. 7. Roughly 8% of the 5s charge extends outside the Wigner-Seitz radius, rws, for / —Sn the 5s orbital, with much of its density in a region in which Zen is about equal to the valence, is actually somewhat in the interior of the atom. It is not unlike the d orbitals of transition metals, which, as earlier noted, maintain much of their atomic quality in a metal. Thus it is quite plausible that the valence s character in Sn is much like the free atom 5 s, except for a renormalization within the Wigner-Seitz cell. The much more extended 5p component, on the other hand, is not subject to simple renormalization the p character near the bottom of the band takes on a form more like the dot-dash curve of Fig. 7. It nevertheless appears useful to account for charge terms of a pseudo P component and a renormalized s. 

Chemical reaction occurs between reactants in their valence state, which is different from the ground state. It requires excitation by the environment, to the point where a valence electron is decoupled from the atomic or molecular core and set free to establish new liaisons, particularly with other itinerant electrons, likewise decoupled from their cores [114]. The energy required to promote atoms into their valence state has been studied before [24] in terms of the simplest conceivable model of environmental pressure, namely uniform isotropic compression. This was simulated by an atomic Hartree-Fock procedure, subject to the boundary condition that confines all electron density to within an impenetrable sphere of adjustable finite radius. 

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