Metastable states descriptions

If 5v //v /coex is not small, the simple description Eq. (14) in terms of bulk and surface terms no longer holds. But one can find AF from Eq. (5) by looking for a marginally stable non-uniform spherically symmetric solution v /(p) which leads to an extremum of Eq. (5) and satisfies the boundary condition v /(p oo) = v(/ . Near the spinodal curve i = v /sp = Vcoex /a/3 (at this stability limit of the metastable states both and S(0) diverge) one finds "... 

Finally, we have considered an example of metastable state without potential barrier exponential approximation with the MFPT of the point d for kT = 1 (Fig. 13). It is seen that even for such an example the exponential approximation [with the mean decay time (6.5)] gives an adequate description of the probability evolution and that this approximation works better for larger noise intensity. 

The radiative transitions of the previous descriptions have all been spontaneous Relaxation from the excited state to the ground state and emission of photons occur without external aid. In contrast, a stimulated emission occurs when the half-life of the excited state is relatively long, and relaxation can occur only through the aid of a stimulating photon. In stimulated emission, the emitted photon has the same direction as, and is in phase with, the stimulating photon. The example of Cr +-doped AI2O3 that we utilized earlier for our description of the color of ruby works equally well for a description of stimulated emission. Recall that the presence of chromium in alumina alters the electronic structure, creating a metastable state between the valence and conduction bands. Absorption of a blue-violet photon results in the excitation of an electron from... 

Yet this description is not entirely correct, because even when the system is at a site in Da, and even inside Aai there is still a probability, however small, for a giant fluctuation to occur, which takes it across into Dc. It will then move to the neighborhood of c until a similar giant fluctuation takes it back to Da. Thus a mesostate represented by a probability peak in Aa does not survive forever its probability is slowly depleted in favor of a peak near (f)c. Although a is a stable solution of the macroscopic equation, the related mesostate is not strictly stable but merely long-lived, and may be called metastable. Indeed, a metastable state in thermodynamics, such as supersaturated vapor, also exists because it is stable with respect to small fluctuations, but an improbable giant fluctuation may carry it into a macro-scopically different thermodynamic state. 

A relaxation process will occur when a compound state of the system with large amplitude of a sparse subsystem component evolves so that the continuum component grows with time. We then say that the dynamic component of this state s wave function decays with time. Familiar examples of such relaxation processes are the a decay of nuclei, the radiative decay of atoms, atomic and molecular autoionization processes, and molecular predissociation. In all these cases a compound state of the physical system decays into a true continuum or into a quasicontinuum, the choice of the description of the dissipative subsystem depending solely on what boundary conditions are applied at large distances from the atom or molecule. The general theory of quantum mechanics leads to the conclusion that there is a set of features common to all compound states of a wide class of systems. For example, the shapes of many resonances are nearly the same, and the rates of decay of many different kinds of metastable states are of the same functional form. 

A pseudo potential approach was adopted by Hickman et al. [259] to calculate the excited metastable states of a He atom under liquid He. The density functional approach developed by Dupont-Roc et al. [260] was applied subsequently [261] for the description of the nature of the cavity formed around an alkali atom in the excited state of non-zero angular momentum. The resulting form of the cavity differs very much from the spherical shape. A similar approach was adopted by De Toffol et al. [262] to find qualitatively the first excited states of Na and Cs in liquid He. Earlier work in this direction was given in detail in Ref. [263]. 

ADAS is centred on generalized collisional-radiative (GCR) theory. The theory recognizes relaxation time-scales of atomic processes and how these relate to plasma relaxation times, metastable states, secondary collisions etc. Attention to these aspects - rigorously specified in generalized collisional-radiative theory - allow an atomic description suitable for modeling and analyzing spectral emission from most static and dynamic plasmas in the fusion and astrophysical domains [3,4]. 

Such effects make quantitative studies of these re Kctions very difficult but by no means less interesting. Another difficulty associated with all studies of the reactions of solids is the dependence of the reaction rate on the previous history of the solid. A major contribution to such erratic behavior is the property of solids of being able to exist for long periods of time in metastable states of physical stress. This introduces into the description of solids an additional set of thermodynamic variables which are not necessarily at the disposal of the experimenter orSey observable. 

Kramers paper spurred an enormous amount of research on the theory of activated rate processes, especially in the physics community, as evidenced in numerous textbooks see, for example, Refs. 13 and 14. However, as noted by Landauer (15) in his subjective description of the history of noise activated escape from metastable states, up till the end of the seventies, the physical chemistry community largely ignored the theory of rates introduced by Kramers. The first experimental measurements of viscosity effects on activated rate processes were performed on the isomerization of frans-stilbene to c/s-stil-bene by Fischer and co-workers in 1968 (16). These authors were not aware of Kramers work and interpreted their results in terms of the free volume necessary for isomerization to occur. Since then, experimental work has proliferated see, for example, the recent textbook (17). 

The sensitivity of fronts to the dynamics of small perturbations about the unstable or metastable states has been studied by Brunet and Derrida [61] for pulled fronts and Kessler et al. [227] for pulled and pushed fronts. The mean-field description of reacting and diffusing systems ceases to be valid for low values of the particle density p, values that correspond to less than one particle. This fact can be incorporated into the RD equation by introducing a cutoff for the reaction term. Such a cutoff strongly affects the front velocity. Throughout this section we consider for simplicity that space and time have been rescaled such that D = r =. ... 

To summarize the observations of Sect. 6.3, one must remember that the history of a metastable sample is stored in its internal variables, i.e., for description, one must use the irreversible thermodynamics of Sect. 2.4, instead of the equilibrium thermodynamics. The internal variables that characterize the metastable state must be uniquely coupled to the history to be discovered. The evaluation of the thermal history of a glass is simple if the initial cooling of the sample is the only contributing factor to the history. In this case, a simple hysteresis determination, matched empirically to a reference, may be sufficient for the task (see Sect 6.3.1, Fig. 6.6). 

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