One-dimensional solids

The smectic A phase is a liquid in two dimensions, i.e. in tire layer planes, but behaves elastically as a solid in the remaining direction. However, tme long-range order in tliis one-dimensional solid is suppressed by logaritlimic growth of tliennal layer fluctuations, an effect known as tire Landau-Peierls instability [H, 12 and 13]... 

Figure 6.15. Potential energy of a one-dimensional solid with lattice distance a between the atoms.

The theory assumes that the nuclei stay fixed on their lattice sites surrounded by the inner or core electrons whilst the outer or valence electrons travel freely through the solid. If we ignore the cores then the quantum mechanical description of the outer electrons becomes very simple. Taking just one of these electrons the problem becomes the well-known one of the particle in a box. We start by considering an electron in a one-dimensional solid. 

Besides magnetic perturbations and electron-lattice interactions, there are other instabilities in solids which have to be considered. For example, one-dimensional solids cannot be metallic since a periodic lattice distortion (Peierls distortion) destroys the Fermi surface in such a system. The perturbation of the electron states results in charge-density waves (CDW), involving a periodicity in electron density in phase with the lattice distortion. Blue molybdenum bronzes, K0.3M0O3, show such features (see Section 4.9 for details). In two- or three-dimensional solids, however, one observes Fermi surface nesting due to the presence of parallel Fermi surface planes perturbed by periodic lattice distortions. Certain molybdenum bronzes exhibit this behaviour. 

A closely related method is that of Boley (B8), who was concerned with aerodynamic ablation of a one-dimensional solid slab. The domain is extended to some fixed boundary, such as X(0), to which an unknown temperature is applied such that the conditions at the moving boundary are satisfied. This leads to two functional equations for the unknown boundary position and the fictitious boundary temperature, and would, therefore, appear to be more complicated for iterative solution than the Kolodner method. Boley considers two problems, the first of which is the ablation of a slab of finite thickness subjected on both faces to mixed boundary conditions (Newton s law of cooling). The one-dimensional heat equation is once again... 

D. Jerome and L. G. Caron (Eds.), Low Dimensional Conductors and Superconductors, NATO Advanced Study Institute, Series B, Vol. 155, (Plenum, New York 1987) J.T. Devreese (Ed.), Highly Conducting One-Dimensional Solids, (Plenum, New York/London 1979) G. Griiner, Density Waves in Solids (Addison-Wesley, London, 1994)... 

Devreese, J. T., Evrard, R. P., Van Doren, V. E. (Eds.) Highly Conducting One-Dimensional Solids Plenum Press, New York 1979... 

S. Barisic and A. Bjelis, in Theoretical Aspects of Band Structures and Electronic Properties of Pseudo-One-Dimensional Solids, H. Kanimura, ed., Reidel, Dordrecht, 1985. 

Fig. 4-6 Nomenclature for one-dimensional solids suddenly sub ected to convection environment at T (a) infinite plate of thickness 2L (0) infinite cylinder of radius...

There is a fair amount of unity, as seen in Section VI, in the mechanisms of phase transitions of the organic conductors and the oxide superconductors. Correlations, be they inter- or intramolecular, play an important role. Quantum and thermal fluctuations are important in the quasi-one-dimensional solids and, to a lesser extent, in the quasi-two-dimensional conductors. There is a striking richness of phases and unusual phenomena in the organic conductors. These are explored in the following chapters of this book. 

Conduction with Heat Source Application of the law of conservation of energy to a one-dimensional solid, with the heat flux given by (5-1) and volumetric source term S (W/m3), results in the following equations for steady-state conduction in a flat plate of thickness 2R (b = 1), a cylinder of diameter 2R (b = 2), and a sphere of diameter 2R (b = 3). The parameter b is a measure of the curvature. The thermal conductivity is constant, and there is convection at the surface, with heat-transfer coefficient h and fluid temperature I. ... 

To couple the intrinsic coke burning kinetics described in Section II.B with gas and solid flow models, the simplest approach is provided by the one-dimensional solid dispersion model based on following assumptions ... 

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