Crystalline solids correlation

The structure of low density gas phases and crystalline solid states is easy to describe, because the correlation functions for them are easy to obtain. In low density gases, intermolecular interactions are negligible and there are no correlations between the particles. g(r) is 1 for all r. At the other extreme, in crystalline solids correlations are very strong but, due to the lattice symmetry, the structure and correlation functions can be easily determined. Correlation functions are particularly important for dense disordered systems such as liquids and glasses, and hence for polymers, for which the crystalline state is rather the exception than the norm. Different models used in classical liquid state theory are essentially different approximations to calculate the correlation functions. All thermodynamic properties can be expressed in terms of the structure correlation functions and interaction potentials [37,46]. 

Calculations for Ceo in the LDA approximation [62, 60] yield a narrow band (- 0.4 0.6 eV bandwidth) solid, with a HOMO-LUMO-derived direct band gap of - 1.5 eV at the X point of the fee Brillouin zone. The narrow energy bands and the molecular nature of the electronic structure of fullerenes are indicative of a highly correlated electron system. Since the HOMO and LUMO levels both have the same odd parity, electric dipole transitions between these levels are symmetry forbidden in the free Ceo moleeule. In the crystalline solid, transitions between the direct bandgap states at the T and X points in the cubic Brillouin zone arc also forbidden, but are allowed at the lower symmetry points in the Brillouin zone. The allowed electric dipole... 

A general presentation and discussion of the origin of structure of crystalline solids and of the structural stability of compounds and solid solutions was given by Villars (1995) and Pettifor (1995). For an introduction to the electronic structure of extended systems, see Hoffmann (1987, 1988). In this chapter a brief sampling of some useful semi-empirical correlations and, respectively, of methods of classifying (predicting) phase and structure formation will be summarized. 

The emphasis of the present chapter is on the correlation of the physical properties and structures of oxide melts. Since long-range order is destroyed in the process of fusion, the meaning of structure is necessarily different for the crystalline solid and its melt. For the latter, structural information is often only obtainable at the present by indirect means such as the comparison of certain properties at a particular temperature. Here, a meaningful interpretation may become doubtful because of the lack of a corresponding temperature. For instance, if the melting points of two oxides differ by 1000°C, on what basis can a property of their respective melts be compared For such reasons, some of the conclusions regarding structure discussed below must be considered as qualitative and treated with reservations. 

Metal-insulator transitions in both crystalline and non-crystalline materials are often associated with the existence of magnetic moments. Moments on atoms in a solid are of course an effect of correlation, that is of interaction between electrons, and their full discussion is deferred until Chapter 3. But even within the approximation of non-interacting electrons in crystalline solids, metal-insulator transitions can occur. These will now be discussed. 

An electron diffraction study of [Zn(acac)2] has recently been reported. The Zn—O distances of 1.942 0.006 A correlate reasonably well with those determined from an X-ray structural study of the crystalline solid (Zn—O, 1.999 A) and are shorter than those observed in the solid state structure of the monohydrate (Zn—O, 2.02 A).744... 

It will be useful now to review some elementary facts regarding the structure of liquids at equilibrium. When a crystalline solid melts to form a liquid, the long range order of the crystal is destroyed. However, a residue of local order persists in the liquid state with a range of several molecular diameters. The local order characteristic of the liquid state is described in terms of a pair correlation function, g-i(R)> defined as the ratio of the average molecular density, p(R), at a distance R from an arbitrary molecule to the mean bulk density, p, of the liquid... 

Cr Cub, Cubv d E G HT Iso Isore l LamN LaniSm/col Lamsm/dis LC LT M N/N Rp Rh Rsi SmA Crystalline solid Spheroidic (micellar) cubic phase Bicontinuous cubic phase Layer periodicity Crystalline E phase Glassy state High temperature phase Isotropic liquid Re-entrant isotropic phase Molecular length Laminated nematic phase Correlated laminated smectic phase Non-correlated laminated smectic phase Liquid crystal/Liquid crystalline Low temperature phase Unknown mesophase Nematic phase/Chiral nematic Phase Perfluoroalkyl chain Alkyl chain Carbosilane chain Smectic A phase (nontilted smectic phase)... 

Currently the problems involved in calculating the electronic band structures of molecular crystals and other crystalline solids centre around the various ways of solving the Schrodinger equation so as to yield acceptable one-electron solutions for a many-body situation. Fundamentally, one is faced with an appropriate choice of potential and of coping with exchange interactions and electron correlation. The various computational approaches and the many approximations and assumptions that necessarily have to be made are described in detail in the references cited earlier. 

There is little doubt that the basic species formed when almost any compound MnX2 is dissolved in water204,205 is [Mn(H20)6]2+. It is seen in various crystalline solids, and the solution data give no real hint that we need to suspect the other coordination polyhedra, either higher or lower, are present in the aqueous solutions. So, [Mn(H20)6]2+ it is—mainly on the basis205 of the optical spectra and correlation with the crystalline solids. In this respect, we note that XAFS studies on aqueous solutions derive an Mn—OH2 distance of 2.18 A, in good agreement with the observed distances in solids (Table 1). 

Now, when spin-spin correlations are not neglected, there should exist a temperature below which a phase transition to a more organized state will occur. If this state is characterized by order on an infinite length scale (i.e. the length scale of the correlations is essentially the dimension of the sample) this is known as long-range order (LRO). The transition that occurs is analogous to the condensation of a gas to form a crystalline solid. We are not concerned here with a detailed discussion of the voluminous literature on phase transitions of this type (also known as critical phenomena). 

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