Harmonic amplitudes solid systems

To this date, no stable simulation methods are known which are successful at obtaining quantum dynamical properties of arbitrary many-particle systems over long times. However, significant progress has been made recently in the special case where a low-dimensional nonlinear system is coupled to a dissipative bath of harmonic oscillators. The system-bath model can often provide a realistic description of the effects of common condensed phase environments on the observable dynamics of the microscopic system of interest. A typical example is that of an impurity in a crystalline solid, where the harmonic bath arises naturally from the small-amplitude lattice vibrations. The harmonic picture is often relevant even in situations where the motion of individual solvent atoms is very anhaimonic in such cases validity of the linear response approximation can lead to Gaussian behavior of appropriate effective modes by virtue of the central limit theorem. ... 

Visco-elastic fluids like pectin gels, behave like elastic solids and viscous liquids, and can only be clearly characterized by means of an oscillation test. In these tests the substance of interest is subjected to a harmonically oscillating shear deformation. This deformation y is given by a sine function, [ y = Yo sin ( t) ] by yo the deformation amplitude, and the angular velocity. The response of the system is an oscillating shear stress x with the same angular velocity . 

Figure 12. The dependence of the activation energy R on the amplitude A of the harmonic driving force F(t) =A cos (1.2r) as determined [141] by electronic experiment (filled circles), numerical sumulations (open circles) and analytical calculation (solid line), based on (28) for an overdamped duffing oscillator U q) = —q2/2 +

In the first part of this chapter we studied the radial vibrations of a solid or hollow sphere. This problem was considered an extension to the dynamic situation of the quasi-static problem of the response of a viscoelastic sphere under a step input in pressure. Let us consider now the simple case of a transverse harmonic excitation in which separation of variables can be used to solve the motion equation. Let us assume a slab of a viscoelastic material between two parallel rigid plates separated by a distance h, in which a sinusoidal motion is imposed on the lower plate. In this case we deal with a transverse wave, and the viscoelastic modulus to be used is, of course, the shear modulus. As shown in Figure 16.7, let us consider a Cartesian coordinate system associated with the material, with its X2 axis perpendicular to the shearing plane, its xx axis parallel to the direction of the shearing displacement, and its origin in the center of the lower plate. Under steady-state conditions, each part of the viscoelastic slab will undergo an oscillatory motion with a displacement i(x2, t) in the direction of the Xx axis whose amplitude depends on the distance from the origin X2-... 

In the harmonic method and its extensions it is always assumed that the amplitudes of the molecular vibrations about their equilibrium positions and orientations remain small. It will be illustrated by several examples in Sect. 2.3 that this is often not realistic for molecular crystals. In the plastic phases there is not even a well-defined equilibrium orientation of the molecules, but also in the ordered phases the librational amplitudes may become substantial. The motions in such systems have been studied by classical methods, in particular the Molecular Dynamics method and the Monte Carlo method [73]. The advantage of these methods is that they can also be applied to study liquids, and the melting of solids, and other systems (glasses, solutions, mixed crystals) which have lost translational periodicity. Large amplitude motions in molecular crystals can also be studied quantum mechanically, however, by the methods described below. 

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