Collision damping

More recently, Gilbert (1955) [6,7] proposed an equation describing the dynamic behavior of M which incorporated the collision damping incurred by the precessional motion, in an effective damping field term. He assumed that the damping field is... 

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)... 

Judging by these results the angular momentum relaxation in a dense medium has the form of damped oscillations of frequency jRo = (Rctc/to)i and decay decrement 1/(2tc). This conclusion is quantitatively verified by computer experiments [45, 54, 55]. Most of them were concerned with calculations of the autocorrelation function of the translational velocity v(t). However the relation between v(t) and the force F t) acting during collisions is the same as that between e> = J/I and M. Therefore, the results are qualitatively similar. In Fig. 1.8 we show the correlation functions of the velocity and force for the liquid state density. Oscillations are clearly seen, which point to a regular character of collisions and non-Markovian nature of velocity changes. 

This damping function s time scale parameter x is assumed to characterize the average time between collisions and thus should be inversely proportional to the collision frequency. Its magnitude is also related to the effectiveness with which collisions cause the dipole function to deviate from its unhindered rotational motion (i.e., related to the collision strength). In effect, the exponential damping causes the time correlation function <% I Eq ... 

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... 

A. Ben-Reuven. Radiatively damped collisions of ultracold atoms. In L. Frommhold and J. W. Keto, eds., Spectral Line Shapes 6, p. 206, Am. 

The oldest and best known example of a Markov process in physics is the Brownian motion.510 A heavy particle is immersed in a fluid of light molecules, which collide with it in a random fashion. As a consequence the velocity of the heavy particle varies by a large number of small, and supposedly uncorrelated jumps. To facilitate the discussion we treat the motion as if it were one-dimensional. When the velocity has a certain value V, there will be on the average more collisions in front than from behind. Hence the probability for a certain change AV of the velocity in the next At depends on V, but not on earlier values of the velocity. Thus the velocity of the heavy particle is a Markov process. When the whole system is in equilibrium the process is stationary and its autocorrelation time is the time in which an initial velocity is damped out. This process is studied in detail in VIII.4. 

We begin with an innocuous case. Consider a pendulum suspended in air and consequently subject to damping accompanied by a Langevin force. This force is, of course, the same as the one in equation (1.1) for the Brownian particle, because the collisions of the air molecules are the same. They depend on the instantaneous value of V, but they are insensitive to the fact that there is a mechanical force acting on the particle as well. Hence for small amplitudes the motion is governed by the linear equation (1.10). For larger amplitudes the equation becomes nonlinear ... 

It may also be mentioned here that in specific molecular actions a particularly marked influence of like molecules upon one another is often to be observed. This is encountered in various ways in spectroscopy, in the extinction of the polarization of mercury resonance radiation with increasing vapour pressure, in the damping of fluorescence in concentrated solutions, and in various chemical reactions. As an example of the latter the decomposition of acetaldehyde (p. 70) may be quoted, where collisions between two molecules of the aldehyde are much more effective than collisions of aldehyde molecules with those of other gases. 

Dislocations move when they are exposed to a stress field. At stresses lower than the critical shear stress, the conservative motion is quasi-viscous and is based on thermal activation that overcomes the obstacles which tend to pin the individual dislocations. At very high stresses, > t7crit, the dislocation velocity is limited by the (transverse) sound velocity. Damping processes are collisions with lattice phonons. 

In view of the calculations considered in Section V and in other publications (VIG), these interactions, giving rise to FIR absorption and to low-frequency Debye loss, resemble interactions pertinent to strongly polar nonassociated liquids. However, if we compare water with a nonassociated liquid (e.g., CH3F), then we shall find that in the latter (i) the R-band is absent (ii) the number mvjb of the reorientation cycles is much less, so that the reduced collision frequency y is substantially greater thus, molecular rotation is more damped and chaotic and (iii) the fitted form factor/is greater. 

In Fig. 62a we demonstrate by solid lines the absorption dependence (463) in the R-band calculated for liquid H2O (a) and liquid D20 (b) at temperature 22.2°C the ordinates are fitted in such a way that the magnitude of the Astr(v) peak is equal to 1. The chosen dimensionless collision frequency Y is chosen 0.6 for H20 and 1.32 for D20, with correspondingly the lifetimes xstr 0.071 and 0.036 ps. These curves exhibit a damped resonance, since the fitted lifetime xstr is very short they resemble the dashed curves g(vstr) in Fig. 60d and 61d. 

It is also expected that reactive collisions may diminish the effects of collisional damping of the z-oscillation. An unreactive collision removes energy from the z-mode oscillation so that the ion contributes more signal current at its original cyclotron frequency whereas a reactive collision removes an ion from a reactant population giving a true indication of the loss from the original population. The loss rate from the reactant population for ions of z-oscillation, Az, is proportional to the density of reactant ions of amplitude Az. Thus, for very reactive ions, no change in sensitivity due to collisional relaxation is expected. 

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