Equilibrium, thermal relaxation

B1.15.2.2 THERMAL EQUILIBRIUM, MAGNETIC RELAXATION AND LORENTZIAN UNESHAPE... 

The speed and time by which the perturbed TLS systems relax to the equilibrium thermal populations depend on the TLS characteristics and in particular on the coupling energy of tunnelling states. 

Provided a Boltzmann distribution is maintained the absorption intensity is proportional to the square of the frequency. It is important, however, that there be collisions in the gas which maintain thermal equilibrium if there is insufficient thermal relaxation, the population difference is reduced by absorption of radiation. Consequently case (ii) is approached and ultimately, from (6.290), there is no net observable absorption of radiation. 

The aimealing kinetics of the light-induced defects are shown in Fig. 6.29. Several hours at 130 °C are needed to anneal the defects completely, but only a few minutes at 200 C. The relaxation is nonexponential, and in the initial measurements of the decay the results were analyzed in terms of a distribution of time constants, Eq. (6.78) (Stutzmann, Jackson and Tsai 1986). The distribution is centered close to 1 eV with a width of about 0.2 eV. Subsequently it was found that the decay fits a stretched exponential, as is shown in Fig. 6.29. The parameters of the decay-the dispersion, p, and the temperature dependence of the decay time, t - are similar to those found for the thermal relaxation data and so are consistent with the same mechanism of hydrogen diffusion. The data are included in Fig. 6.23 which describes the general relation between x and D,. The annealing is therefore the process of relaxation to the equilibrium state with a low defect density. 

When y is very small, the thermal relaxation in the well is not fast relative to the escape rate, and the assumption that the distribution within the well can be represented by the equilibrium Boltzmann distribution no longer holds. On the other hand we can make use of the fact that the total energy E varies on a time scale much longer than either x or u (it is conserved for y = 0). Thus changing variables from (x, v) to ( , ) and eliminating the fast phase variable (f> leads to a Smoluchowski (diffusion) equation for E. [Kramers gave the equivalent equation in terms of the action variable J( ).]... 

We assume that V, the operator that couples systems L and R to each other, mixes only / and r states, that is, = Vr,r — 0- We are interested in the transition between these two subsystems, induced by V. We assume that (1) the coupling V is weak coupling in a sense explained below, and (2) the relaxation process that brings each subsystem by itself (in the absence of the other) into thermal equilibrium is much faster that the transition induced by V between them. Note that assumption (2), which implies a separation of timescales between the L 7 transition and the thermal relaxation within the L and R subsystems, is consistent with assumption (1). 

Below we will use the timescale separation between the (fast) thermal relaxation within the L and R subsystems and the (slow) transition between them in one additional way We will assume that relative equilibrium within each subsystem is maintained, that is. 

Real locus of the system in pressure-composition space during an isochoric step from the /f-1th to the kth point on the recorded isotherm (Eqns 7.10, 7.11). The excursion of the pressure in the hydrogenator to pfys,o) occurs when the valve S (Fig. 7.4) is opened instantaneously. The system then approaches equilibrium according to the kinetics of the sample and the thermal relaxation time of the sample/cell sub-system. The pressure excursion is lessened if S is opened slowly, so that absorption commences while Psys is still rising. The slope of the isochore is constant only if the compressibility is constant. 

Under conditions of local thermodynamic equilibrium (LTE) thermal relaxation processes will be rapid and maintain the population of the lower level. If, however, one were able to depopulate the upper level n compared with its LTE population, the absorption signals would increase. The technique of doubleresonance spectrometry exploits this by application of an intense MMW or laser field that excites a transition, e.g. n q, where is a higher state (Figure 1.3). This depletes the population of n and permits greater absorption of a second MMW field, exciting the transitions m n ox q r. 

In general, the intensity of an ESR spectrum increases with an increase in the microwave power R When the applied power level is sufficiently low, thermal relaxation processes can, to a good approximation, maintain the Boltzmann equilibrium between spin levels. When the power level exceeds that amount, the ESR spectrum broadens and its intensity begins to decrease and eventually disappears. This phenomenon is called the power saturation or saturation broadening effect and depends... 

In the electronic ground states of molecules collision-induced transitions represent, for most experimental situations, the dominant mechanism for the redistribution of energy, because the radiative processes are generally too slow. In cases where a nonequilibrium distribution has been produced (for example, by chemical reactions or by optical pumping), these collisions try to restore thermal equilibrium. The relaxation time of the system is determined by the absolute values of collision cross sections. 

Thermally stimulated current (TSC) spectroscopy is used to characterize the relaxation processes and structural transitions occurring in samples that have been polarized at a temperature greater than the temperature where molecular motion in the sample is enhanced and subsequently quenched so that the high mobility state is frozen. On heating the sample at a controlled rate, depolarization of the polymer electrets (molecular or ionic dipoles, trapped electrons, mobile ions) occurs and the oriented dipoles, frozen in the quenched sample, relax to a state of thermal equilibrium. This relaxation process is observed as a depolarization current, which is typically of the order of picoamperes, and is referred to as the thermally stimulated current. 

Fig.4 shows the structure of a typical partly dispersed shock wave. For strong shocks such as this (ugo/afo = 1.5) very large departures from equilibrium occur just downstream of the discontinuity. In the relaxation zone the droplet temperature rises very rapidly to the saturation value on a time scale Xd and this is followed by velocity and thermal relaxation on time scales Xv and Xt respectively. 

---