One-dimensional behaviour

Fig. 2.14 The crystal Structure of 1,4-dibromo naphthalene. The crystal consist of stacks of molecules, closely spaced with parallel molecular planes. This yields a strong one-dimensional behaviour in energy transport, i.e. linear excitons. The crystal is monoclinic, with Z= 8 molecules in the unit cell. The lattice constants are a = 27.45 k, b= 16.62 A, c = 4.09 A. The angle between the c axis and the ab plane s p = 91°55. ...

Ringland, J. Turner, J. S. One-dimensional behaviour in a model of the Belousov-Zhabotinskii reaction, Phys. Letters (in press). 

The discussion so far has been dominated by one-dimensional behaviour, reflecting the most convenient and customary materials testing methods. However, any engineering application will be for a three-dimensional body, subject to multi-axial stresses. It is now feasible to implement non-linear viscoelastic models in numerical schemes to perform analyses of structures, and this is often the motivation for generalising a viscoelastic theory to two or three dimensions. 

At the TS the energy along the reaction path is a maximum, but it is a minimum in the perpendicular direction(s). A one-dimensional cut through the (0,0) and (1,1) comers for path A in Figure 15.30 thus corresponds to Figure 15.28. A similar cut through the (0,1) and (1,0) comers will display a normal (as opposed to inverted) parabolic behaviour. 

Moholkar VS, Pandit AB (2001) Numerical investigations in the behaviour of one-dimensional bubbly flow in hydrodynamic cavitation. Chem Eng Sci 56 1411-1418... 

We have already seen (p. 2) that the individual electrons of an atom can be symbolised by wave functions, and some physical analogy can be drawn between the behaviour of such a wave-like electron and the standing waves that can be generated in a string fastened at both ends—the electron in a (one-dimensional) box analogy. The first three possible modes of vibration will thus be (Fig. 12.1) ... 

The way in which tunnelling affects the energy levels of a system is illustrated well by the behaviour of a particle in a one-dimensional box with a central potential barrier. 

B.D. Lambourn J.E, Hartley, "The Calculation of Hydrodynamic Behaviour of Plane One Dimensional Explosive/Metal... 

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. 

In previous chapters we have dealt only with systems which have one or two independent concentrations. This has been sufficient for a wide range of intricate behaviour. Even with just a single independent concentration (one variable), reactions may show multiple stationary states and travelling waves. Oscillations are, however, not possible. To understand the latter point we can think in terms of the phase plane or, more correctly for a one-dimensional system, the phase line (Fig. 13.1(a)). As the concentration varies in time, so the system moves along this line. Stationary-state solutions are points on the line the arrows indicate the direction of motion along the line, as time increases, towards stable states and away from unstable ones. Figure 13.1(b) shows this motion and phase line behaviour represented in terms of some potential, with stable states a minima and an unstable (saddle) solution as a maximum. 

Squared Fluctuation Functions F2(f, /), Expressed in Terms of Autocorrelation Functions p(/,/) and the Limiting Forms Taken by the Scaled rms Fluctuation Functions (/(/,/) at Short and Long Times, for the Functions /Shown and the Following Systems . (A) Harmonically bound particle. (B) One-dimensional particle in a box. (C) Plane rotor. (D) Spherical rotor. The limiting behaviour at short times is given in terms of e = 1 — p. 

We now consider some models of polymer structure and ascertain their usefulness as representative volume elements. The Takayanagi48) series and parallel models are widely used as descriptive devices for viscoelastic behaviour but it is not correct to use them as RVE s for the following reasons. First, they assume homogeneous stress and displacement throughout each phase. Second, they are one-dimensional only, which means that the modulus derived from them depends upon the directions of the surface tractions. If we want to make up models such as the Takayanagi ones in three dimensions then we shall have a composite brick wall with two or more elements in each of which the stress is non-uniform. 

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