Phenomenological descriptions

If atom A undergoes a radiative transition between levels Ei and Ek during the collision time, the frequency 

In a gas mixture of atoms A and B, the mutual distance R(A, B) shows random fluctuations with a distribution around a mean value R that depends on pressure and temperature. According to (3.47), the fluorescence yields a corresponding frequency distribution around a most probable value coikiRm), which may be shifted against the frequency coq of the unperturbed atom A. The shift Aco = coo — coik depends on how differently the two energy levels Ei and Ek are shifted at a distance / m(A, B) where the emission probability has a maximum. The intensity profile f(o)) of the collision-broadened and shifted emission line can be obtained from 

From (3.48) it can be seen that the intensity profile of the collision-broadened line reflects the difference of the potential curves 

Let V R) be the interaction potential between the ground-state atom A and its collision partner B. The probability that B has a distance between R and R- - dR is proportional to AitR dR and (in thermal equilibrium) to the Boltzmann factor oxp[—V(R)/kT]. The number N(R) of collision partners B with distance R from A is therefore 

Frequently, different spherical model potentials V R) are substituted in (3.50), such as the Lennard-Jones potential 

To obtain rate expressions for photoelectrochemical reactions, all the steps mentioned above as well as the potential distribution under illumination should be taken into account. 

In the early stages of study of photoelectrochemical kinetics, two extreme models were presented. One is the model of Bockris and Uosaki,93 108,109 in which the charge transfer step was considered to be the rate-determining step and the other is Butler s model,110 in which the semiconductor/electrolyte interface was considered as a Schottky barrier, i.e., the electrochemical kinetics were neglected and all the potential drop occurred only within the semiconductor. 

In this section, these two models are explained first, then the effects of other terms will be considered. 

Bockris and Uosaki93 108,109 calculated photocurrent-potential relations mainly for the hydrogen evolution reaction (HER) at p-type semiconductors with the assumption that the following is the ratedetermining step  

Ne(x) dx9 is equal to the number of photons absorbed betweeen x and x + dx9 TVph(x) dx, which is given by 

The outgoing electron is shown as a spherical wave, whose wavelength Ae=2Jt/k depends on the energy of the electron (fay—Z K) according to the formula 

The scattered wave from the near neighbours overlaps the outgoing wave from the absorption site, the K shell, and can interfere constructively or destructively depending on the total phase shift experienced by the electron (see figure A4.2). This phase shift depends, in turn, on the distance between atoms and Ae (i.e. on fi(o—EK) and the propagation of the electron between the absorbing site and the scattering atom. 

Since the effect is dependent only on the presence of an excitable atom and an ordered local environment of that atom, the sample can be disordered (e.g. amorphous, solution) or ordered (e.g. crystalline) the only constraint being that sufficient material be present and the concentration of the excitable atom be enough for a reasonable signal-to-noise ratio. 

In summary, information concerning the electronic structure at and/or in the immediate environment about the primary absorbing atom can be obtained by the accurate measurement of  

An energy-dispersive detector can be used to isolate the fluorescent radiation from the Compton and Rayleigh components. 

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Hamiltonian, but in practice one often begins with a phenomenological set of equations. The set of macrovariables are chosen to include the order parameter and all otlier slow variables to which it couples. Such slow variables are typically obtained from the consideration of the conservation laws and broken synnnetries of the system. The remaining degrees of freedom are assumed to vary on a much faster timescale and enter the phenomenological description as random themial noise. The resulting coupled nonlinear stochastic differential equations for such a chosen relevant set of macrovariables are collectively referred to as the Langevin field theory description. 

In the process of establishing the kinetic scheme, the rate studies determine the effects of several possible variables, which may include the temperature, pressure, reactant concentrations, ionic strength, solvent, and surface effects. This part of the kinetic investigation constitutes the phenomenological description of the system. 

As the density of a gas increases, free rotation of the molecules is gradually transformed into rotational diffusion of the molecular orientation. After unfreezing , rotational motion in molecular crystals also transforms into rotational diffusion. Although a phenomenological description of rotational diffusion with the Debye theory [1] is universal, the gas-like and solid-like mechanisms are different in essence. In a dense gas the change of molecular orientation results from a sequence of short free rotations interrupted by collisions [2], In contrast, reorientation in solids results from jumps between various directions defined by a crystal structure, and in these orientational sites libration occurs during intervals between jumps. We consider these mechanisms to be competing models of molecular rotation in liquids. The only way to discriminate between them is to compare the theory with experiment, which is mainly spectroscopic. 

A phenomenological description of the differential cross-section for emission of photoelectrons into solid angle O in the lab frame can be written, assuming random molecular orientation and an axis of cylindrical symmetry defined by the photon polarization, as... 

The relative contribution of each driving force (X) generated by component j to the flux of solute i (./,) is expressed by coefficients Li in this phenomenological description of parallel transport processes. 

In reaction rate studies one s goal is a phenomenological description of a system in terms of a limited number of empirical constants. Such descriptions permit one to predict the time-dependent behavior of similar systems. In these studies the usual procedure is to try to isolate the effects of the different variables and to investigate each independently. For example, one encloses the reacting system in a thermostat in order to maintain it at a constant temperature. 

Linear response theory10 provides a link between the phenomenological description of the kinetics in term of reaction rate constants and the microscopic dynamics of the system [33]. All information needed to calculate the reaction rate constants is contained in the time correlation function... 

We should note that the use of the Lipari-Szabo analysis implies that relaxation data are available at multiple magnetic fields. It provides a phenomenological description of the rotational motion that can be very useful for comparing systems with similar structure. Nevertheless, one should be aware of the limits of this approach and avoid direct comparison of local or global rotational correlation times for structurally very different compounds. 

Since we don t usually know enough about pore structure and other matters to assess the relative importance of these modes, we fall back on the phenomenological description of the rate of diffusion in terms of Fick s (first) law. According to this, for steady-state diffusion in one dimension (coordinate x) of species A, the molar flux, NA, in, say, mol m-2 (cross-sectional area of diffusion medium) s-1, through a particle is... 

The solution of this equation is a precession of the vector M around H0 with an angular velocity equilibrium value. In order to explain how the spins approach their equilibrium distribution we must take account of the interactions of the spins among themselves and with the other degrees of freedom of the system. The simplest phenomenological description of the approach to equilibrium of an assembly of magnetic moments, placed in a constant field H0, is given by the equations... 

Most characterisation of non-linear responses of materials with De < 1 have concerned the application of a shear rate and the shear stress has been monitored. The ratio at any particular rate has defined the apparent viscosity. When these values are plotted against one another we produce flow curves. The reason for the popularity of this approach is partly historic and is related to the type of characterisation tool that was available when rheology was developing as a subject. As a consequence there are many expressions relating shear stress, viscosity and shear rate. There is also a plethora of interpretations for meaning behind the parameters in the modelling equations. There are a number that are commonly used as phenomenological descriptions of the flow behaviour. 

This follows the form of phenomenological descriptions of polymer systems such as the BKZ model10 which is encouraging. Very good fits to experiments have been found using this approach. In order to take this idea forward the most convenient method is to use the memory function ... 

A PHENOMENOLOGICAL DESCRIPTION OF THE MASS-DEPENDENT INSTRUMENTAL BIAS... 

The loss of alkenes from aliphatic onium ions via onium reaction comprises scission of the C-X bond and concomitant transfer of a hydrogen from the leaving alkyl moiety to the heteroatom, and a merely phenomenological description of this reaction has already been included in the preceding schemes. 

A phenomenological description of the dynamic structure factor at this Q-value by KWW functions ... 

Fig. 5.1 Mode number dependence of the relaxation times Tj and T2 (solid lines). The dashed-dotted line shows the relaxation time ip in the Rouse model (Eq. 3.12). The horizontal dashed line displays the value of r. The dashed and the dotted lines represent the relaxation time when the influence of the chain stiffness is considered mode description of the chain statistics iq (dashed, Eq. 5.11) and bending force model tp (dotted, Eq. 5.7). The behaviour of the relaxation time used in the phenomenological description is also shown for the lowest modes (see text). (Reprinted with permission from [217]. Copyright 1999 American Institute of Physics)...

Hence, the two main foci of this review are (i) the current understanding of the underlying elementary processes and (ii) a phenomenological description of the resulting macroscopic transport phenomena. Because the first aspect comprises proton conduction mechanisms other than the mechanisms of parasitic transport , this review may also be considered an... 

Now, our purpose is to simulate these different kinetic behaviors. A phenomenological description was tried with some simple models noticing that the starts were autocatalytic in zone A. The best agreements were obtained with a model composed of two successive reactions, the first step being autocatalytic ... 

Apart from this phenomenological description and that of Jameson, we have seen that the theoretical approach of Cowley and White was capable of reproducing this behaviour (Section III.A.2). It would therefore be interesting to similarly investigate the geometrical dependence which will now be considered. (ii)... 

As the alternative, a phenomenological description of turbulent mixing gives good results for many situations. An apparent diffusivity is defined so that a diffusion-type equation may be used, and the magnitude of this parameter is then found from experiment. The dispersion models lend themselves to relatively simple mathematical formulations, analogous to the classical methods for heat conduction and diffusion. 

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