The Cellular Method

The cellular method is of little direct practical importance today. Its significance is historical (as a precursor of the APW and related methods) and conceptual (as a convenient framework for the understanding of the cohesion of solids). 

Orthogonalized Plane Wave, Augmented Plane Wave, 

Historically speaking, the orthogonalized plane wave (OPW) method should have been discussed before the pseudopotential method (Section 3.6). The work of Phillips and Kleinman, which is 

As was remarked in earlier sections, the use of plane waves as basis functions is inadequate because of the strong potential near the ion core. This induces rapid oscillations in the wave function which cannot be described by any reasonable number of plane waves. One way of looking at these oscillations is to consider them as necessitated by the condition that the wave function must be orthogonal to all the bound states which lie lower in energy. 

Plane waves made orthogonal to these bound states by the addition of suitable linear combinations of the bound states are thus suitable basis functions. They are still plane waves outside the core, but now have the right character in the core to provide a reasonably convergent expansion of the true wave function. 

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Here the last expression was found by taking the arithmetic mean between the two forms, Eq. 11.45 and Eq. 11.74. Formulas of this type have actually been used in the cellular method" for treating crystals, but our own experience from work on atoms is that the orbital energies sk seem to be the quantities in the HF scheme which are most easily influenced by numerical uncertainties and errors. Even if Eqs. 11.74 and 11.75 are practically simpler to handle than Eq. 11.45, they are probably less numerically reliable. Further investigations on this point are desired. 

Shockley, W. (1937) The empty lattice test of the cellular method in solids. Phys. Rev. 52,... 

Cellular Method. The cellular method of Wigner and Seitz (1933) assumes that the solid is divided into "cells" with an ionic core at the origin the atomic wavefunction is of the type... 

We shall now test the accuracy of our cellular method , i.e., the replacing of the cell-polyhedra by spheres. We shall do this by estimating the error in case of a simple cubic lattice - the cellular method is evidently much more accurate for face or body centered lattices. The correction which we shall obtain gives, at the same time, the difference between a simple cubic and a centered lattice from the point of view of thermal utilization. 

There are two exhaustive reviews of the band structure of rare earth metals by Dimmock (1971) and Freeman (1972). The first article gives a complete discussion of the band calculation results and the second article enlphasizes the magnetic properties as delineated by the electronic properties. Earlier calculations on the energy bands and Fermi surface of rare-earth-like metals Sc and Y by the cellular method have been reviewed by Cracknell (1971). Whenever feasible we will avoid duplicating the materials contained in these articles. 

Various other approaches to the description of energy bands in amorphous materials (particularly Si, Ge) have been proposed, including the cellular method, the use of a smeared version of the density of states of a crystalline phase, the study of finite clusters, and the isotropic Penn model. Probably there is something to be learned from all of these approaches to what remains a difficult problem. Much of the current interest in these materials has centered on the existence of a band gap, which seems to be implied by experimental results for suitably prepared amorphous Si and Ge. 

Other methods for detennining the energy band structure include cellular methods. Green fiinction approaches and augmented plane waves [2, 3]. The choice of which method to use is often dictated by die particular system of interest. Details in applying these methods to condensed matter phases can be found elsewhere (see section B3.2). 

The nomenclature of cellular polymers is not standardized classifications have been made according to the properties of the base polymer (22), the methods of manufacture, the cellular stmcture, or some combination of these. The most comprehensive classification of cellular plastics, proposed in 1958 (23), has not been adopted and is not consistent with some of the common names for the more important commercial products. 

Stabilization of the Cellular State. The increase in surface area corresponding to the formation of many ceUs in the plastic phase is accompanied by an increase in the free energy of the system hence the foamed state is inherently unstable. Methods of stabilizing this foamed state can be classified as chemical, eg, the polymerization of a fluid resin into a three-dimensional thermoset polymer, or physical, eg, the cooling of an expanded thermoplastic polymer to a temperature below its second-order transition temperature or its crystalline melting point to prevent polymer flow. 

Poly(vinylchloride). Cellular poly(vinyl chloride) is prepared by many methods (108), some of which utili2e decompression processes. In all reported processes the stabili2ation process used for thermoplastics is to cool the cellular state to a temperature below its second-order transition temperature before the resia can flow and cause coUapse of the foam. 

Urea.—Forma.IdehydeResins. Cellular urea—formaldehyde resins can be prepared in the following manner an aqueous solution containing surfactant and catalyst is made into a low density, fine-celled foam by dispersing air into it mechanically. A second aqueous solution consisting of partially cured urea—formaldehyde resin is then mixed into the foam by mechanical agitation. The catalyst in the initial foam causes the dispersed resin to cure in the cellular state. The resultant hardened foam is dried at elevated temperatures. Densities as low as 8 kg/m can be obtained by this method (117). 

Several countries have developed their own standard test methods for cellular plastics, and the International Organization for Standards (ISO) Technical Committee on Plastics TC-61 has been developing international standards. Information concerning the test methods for any particular country or the ISO procedures can be obtained in the United States from the American National Standards Institute. The most complete set of test procedures for cellular plastics, and the most used of any in the world, is that developed by the ASTM these procedures are pubUshed in new editions each year (128). There have been several reviews of ASTM methods and others pertinent to cellular plastics (32,59,129—131). 

In order to perform effectively as an insulant a material must restrict heat flow by any (and preferably) all three methods of heat transfer. Most insulating materials adequately reduce conduction and convection elements by the cellular structure of the material. The radiation component is decreased by absorption into the body of the insulant and is further reduced by the application of bright foil outer facing to the product. 

The synthesis of porphyrins from bilanones is free of all symmetry restraints.77 The oxo function is necessary to stabilize the bilane system by its electron-withdrawing effect. The synthesis of porphyrins from the parent bilane without the oxo function and with /3-substituents is possible,54 but the method gives rise to preparative problems due to the sensitivity of these compounds to oxidation, to electrophiles and to acids. Nature circumvents these problems in the cellular environment by exclusion of oxygen, when porphyrinogens, the precursors of porphyrins, are produced from bilanes55 in the course of their biosynthesis. 

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