Damping constants

The damping constant a and tire spring constant K can be written compactly in tenns of tire atomic and field parameters as... 

Example The differential equation My" + Ay + ky = 0 represents the vibration of a linear system of mass M, spring constant k, and damping constant A. If A < 2 VkM. the roots of the characteristic equation... 

The system is still comprised of the inertia force due to the mass and the spring force, but a new force is introduced. This force is referred to as the damping force and is proportional to the damping constant, or the coefficient of viscous damping, c. The damping force is also proportional to the velocity of the body and, as it is applied, it opposes the motion at each instant. 

Note that for undamped free vibration, the damping constant, c, is zero and, therefore, ji, is zero. 

With damped vibration, the damping constant, c, is not equal to zero and the solution of the equation gets quite complex assuming the function, X =Xo sin(ft)/ — ). In this equation, cj) is the phase angle, or the number of degrees that the external force, Fo sin(ft)/), is ahead of the displacement, Xo sin(ft)/ — cj>). Using vector concepts, the... 

In the absence of damping (and in units where ( b = 1), the invariant manifolds bisect the angles between the coordinate axes. The presence of damping destroys this symmetry. As the damping constant increases, the unstable manifold rotates toward the Agu-axis, the stable manifold toward the A<7u-axis. In the limit of infinite damping the invariant manifolds coincide with... 

Figure 5. Reaction probabilities for a given instance of the noise as a function of the total integration time Tint for different values of the anharmonic coupling constant k. The solid lines represent the forward and backward reaction probabilities calculated using the moving dividing surface and the dashed lines correspond to the results obtained from the standard fixed dividing surface. In the top panel the dotted lines display the analytic estimates provided by Eq. (52). The results were obtained from 15,000 barrier ensemble trajectories subject to the same noise sequence evolved on the reactive potential (48) with barrier frequency to, = 0.75, transverse frequency co-y = 1.5, a damping constant y = 0.2, and temperature k%T = 1. (From Ref. 39.)...

The damping constant T and the frequency shift 8w are expressed through the coupling constants m for the interaction of the oscillator with the thermal bath and through the frequency characteristics of the latter.86 The frequency shift will be neglected in what follows for the sake of simplicity. 

Metal Temperature (K) Viscous Damping Constant Reference... 

Finally, we can recover the classical damping constant from Eq. (A3.30) by writing... 

According to the theory proposed by Horton and McGie,[io] the particle damping constant is determined according to ... 

If, however, the transition is of a pure displacive nature, the fluctuation amplitude of the order parameter is critical and is by no means temperature-independent. Since the soft mode is an under-damped lattice vibration (at least outside the close vicinity of Tc), defined by its frequency a>s and damping constant Tj, the spectral density is a Lorentzian centred at s and the... 

In the presence of both order-disorder and displacive, as in the KDP family, the two dynamic concepts have somehow to be merged. It could well be that the damping constant Zs becomes somewhat critical too (at least in the over-damped regime of the soft mode), because of the bihnear coupling of r/ and p. It would, however, lead too far to discuss this here in more detail. The corresponding theory of NMR spin-lattice relaxation for the phase transitions in the KDP family has been worked out by Blinc et al. [19]. Calculation of the spectral density is here based on a collective coordinate representation of the hydrogen bond fluctuations connected with a soft lattice mode. Excellent and comprehensive reviews of the theoretical concepts, as well as of the experimental verifications can be found in [20,21]. 

By introducing standard parameters, the natural frequency fo (or the natural circular frequency too = 2tt ) and the damping constant y. 

While using Eq. (10.36) to calculate the damping constant, the dimensionless parameter Co can be chosen from this chart. (After Nagaya, 1984.)... 

In the region between to, and to, which for SiC is between about 800 and 1000 cm-1, the reflectance is high not because of large k but because of small n. If n = 0, the normal incidence reflectance is nearly 100% only for the undamped oscillator (y = 0) is the reflectance actually 100%, but solids like SiC approach this rather closely. If the damping constant y in (9.20) is set equal to zero, the real part of the dielectric function becomes... 

An elementary treatment of the free-electron motion (see, e.g., Kittel, 1962, pp. 107-109) shows that the damping constant is related to the average time t between collisions by y = 1 /t. Collision times may be determined by impurities and imperfections at low temperatures but at ordinary temperatures are usually dominated by interaction of the electrons with lattice vibrations electron-phonon scattering. For most metals at room temperature y is much less than oip. Plasma frequencies of metals are in the visible and ultraviolet hu>p ranges from about 3 to 20 eV. Therefore, a good approximation to the Drude dielectric functions at visible and ultraviolet frequencies is... 

The dielectric function of a metal can be decomposed into a free-electron term and an interband, or bound-electron term, as was done for silver in Fig. 9.12. This separation of terms is important in the mean free path limitation because only the free-electron term is modified. For metals such as gold and copper there is a large interband contribution near the Frohlich mode frequency, but for metals such as silver and aluminum the free-electron term dominates. A good discussion of the mean free path limitation has been given by Kreibig (1974), who applied his results to interpreting absorption by small silver particles. The basic idea is simple the damping constant in the Drude theory, which is the inverse of the collision time for conduction electrons, is increased because of additional collisions with the boundary of the particle. Under the assumption that the electrons are diffusely reflected at the boundary, y can be written... 

There are several interesting observations that can be made about (12.32). Integrated absorption is independent of the damping constant y the only bulk parameter that affects it is the plasma frequency. If the particles are in air, then integrated absorption is independent of the shape this is true not only for a single oriented ellipsoid but also for a collection of randomly oriented ellipsoids. It is instructive to rewrite (12.32) using (12.29) ... 

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