Particle damping constant

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

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) ... 

This relaxation time—which, to be specific, we have written in the Landau-Lifshitz representation—has the anticipated behavior the smaller the precession damping constant (the higher the quality factor of the oscillations), the slower does the particle magnetic moment approach its equilibrium position. For ferromagnet or ferrite nanoparticles the typical values of the material parameters are Is Is < 103G, Vm 10 18cm3, and a 0.1. Substituting them in formula (4.28) aty 2 x 108 rad/Oe s and room temperature, one obtains xD 10-9 s. 

As the radius of the sphere is increased, the n>ro band of the bulk mode appears in the spectrum. Eor example, for ZnS particles this occurs at r > 2 p,m. As the radius increases further, (i) both the surface- and bulk-mode absorption bands broaden and split due to the appearance of higher order surface modes, (ii) the band maxima shift toward lower frequencies, and (iii) the ratio of the intensities of the surface to bulk modes decreases [293]. As can be deduced from Eq. (3.35), an increase in the dielectric constant of the surrounding medium will also cause the surface modes to shift toward the red, as does an increase in the particle dimensions. The explanation is the same as for plane-parallel films (Section 3.3.1). The presence of a dispersion in particle size causes the two absorption bands to broaden, the fine sfiucture to disappear, and the a>io band to shift to lower frequencies. An analogous effect can be observed if the damping constant y of the sphere material increases. 

The preceding discussion has shown that both elastic and inelastic collisions cause spectral line broadening. The elastic collisions may additionally cause a line shift which depends on the potential curves E. (R) and E (R). This can be quantitatively seen from a model introduced by LINDHOLM [3.6], which treats the excited atom A as a damped oscillator which suffers collisions with particles B (atoms or molecules). In this model inelastic collisions damp the amplitude of the oscillation. This is described by introducing a damping constant such that the sum of radiative and col-lisional damping is represented by y = y + y qi From the derivation in Sect.3.1 one obtains for the line broadened by inelastic collisions a Lorentzian profile with halfwidth (3.38)... 

The diffusion constant should be small enough to damp out inertial motion. In the presence of a force the diffusion is biased in the direction of the force. When the friction constant is very high, the diffusion constant is very small and the force bias is attenuated— the motion of the system is strongly overdamped. The distance that a particle moves in a short time 8t is proportional to... 

Theoretical predictions relating to the orientation and deformation of fluid particles in shear and hyperbolic flow fields are restricted to low Reynolds numbers and small deformations (B7, C8, T3, TIO). The fluid particle may be considered initially spherical with radius ciq. If the surrounding fluid is initially at rest, but at time t = 0, the fluid is impulsively given a constant velocity gradient G, the particle undergoes damped shape oscillations, finally deforming into an ellipsoid (C8, TIO) with axes in the ratio where... 

We shall call this a quasilinear Fokker-Planck equation, to indicate that it has the form (1.1) with constant B but nonlinear It is clear that this equation can only be correct if F(X) varies so slowly that it is practically constant over a distance in which the velocity is damped. On the other hand, the Rayleigh equation (4.6) involves only the velocity and cannot accommodate a spatial inhomogeneity. It is therefore necessary, if F does not vary sufficiently slowly for (7.1) to hold, to describe the particle by the joint probability distribution P(X, V, t). We construct the bivariate Fokker-Planck equation for it. 

Suppose we idealize the polymeric chain as bang a linear array of Z masses m, each separated by a distance a, coupled by springs of force constant a. Let there be N such macromolecules per c.c. and let / be the frictional damping factor associated with the motion of each particle. 

In atomic-molecular media the damping of plasmon states is due to the interaction of plasmon waves with electrons, lattice vibrations, and impurities. The electron-plasmon interaction is a long-range one. With absorption of a plasmon, the momentum q is transferred to the electron, resulting in a decay of the collective state into a single-particle one. The latter process is identical with absorption of a photon with the same energy. Wolff102 (see also Ref. 103) has shown that in this case the lifetime can be expressed in terms of two optical constants the absorption coefficient k and the refractive index nT, namely,... 

Here, z0 10-10 -10 13s is the inverse attempt frequency that depends on the damping of the magnetic moments by the phonons. The superparamagnetic blocking occurs when r equals the measuring time of each experimental point, te, therefore TB = all / kB ln(/(, / r0 ), where a is a constant that depends on the width of the particle size distribution. 

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