Coupling Approximations

We now show that the elaborate formalism we have developed is a convenient starting point for approximations. The type of approximations one makes depends, of course, on the nature of the problem. In our case, which is a classical fluid with short-range forces, we want to rewrite G as a sum of pieces, one of which leads to the Boltzmann-Enskog memory function. We can do this by applying the identity 

The term will lead to a generalized Boltzmann-Enskog approximation. This 

Our next task is to deduce a tractable representation of the G KG term. Notice that when we put G KG into the memory function expression (79), at each end of the interaction K one has 

One can argue that this quantity has its largest values in the region near r4 — r4. Since this quantity multiplies K, we need to treat 1 (44 22 ) accurately for r4 = r4 . Since K is proportional to T, an accurate treatment of K requires an understanding of the behavior of T(44 22 ) for f4 near r4-. An analysis of T in the low-density limit shows that T can be written in the form 

Equation (100) can be taken as a definition of 7 and we will assume that T has this structure for all densities. Returning to (99) we now want to extract factors of G in the same way as (75), 

---

If there are no reactions, the conservation of the total quantity of each species dictates that the time dependence of is given by minus the divergence of the flux ps vs), where (vs) is the drift velocity of the species s. The latter is proportional to the average force acting locally on species s, which is the thermodynamic force, equal to minus the gradient of the thermodynamic potential. In the local coupling approximation the mobility appears as a proportionality constant M. For spontaneous processes near equilibrium it is important that a noise term T] t) is retained [146]. Thus dynamic equations of the form... 

Vector representation of momenta and vector coupling approximations 7.1.2.1 Angular momenta and magnetic moments... 

A second approximation neglects coupling between the spin of an electron and its orbital momentum but assumes that coupling between orbital momenta is strong and that between spin momenta relatively weak but appreciable. This represents the opposite extreme to the 77-coupling approximation. It is known as the Russell-Saunders coupling approximation and serves as a useful basis for describing most states of most atoms and is the only one we shall consider in detail. 

However complex the atom, we can use the Russell-Saunders coupling approximation (or jj coupling, if necessary) to derive the states that arise from any configuration. The four general selection mles that apply to transitions between these states are as follows. 

Assume that the Russell-Saunders coupling approximation applies to both configurations. Answer. The ground electron configuration of zirconium (Z = 40) is (see Table 7.1)... 

How would the components of a electronic state be described in the case (c) coupling approximation ... 

Working in the same weak-coupling approximation, it takes little effort to produce the expression for the rate constant in the asymmetric case, by simply replacing J in (2.42)-(2.44) by the energy bias . 

Uncoupled solutions for current and electric field give simple and explicit descriptions of the response of piezoelectric solids to shock compression, but the neglect of the influence of the electric field on mechanical behavior (i.e., the electromechanical coupling effects) is a troublesome inconsistency. A first step toward an improved solution is a weak-coupling approximation in which it is recognized that the effects of coupling may be relatively small in certain materials and it is assumed that electromechanical effects can be treated as a perturbation on the uncoupled solution. 

The determination of piezoelectric constants from current pulses is based on interpretation of wave shapes in the weak-coupling approximation. It is of interest to use the wave shapes to evaluate the degree of approximation involved in the various models of piezoelectric response. Such an evaluation is shown in Fig. 4.5, in which normalized current-time wave forms calculated from various models are shown for x-cut quartz and z-cut lithium niobate. In both cases the differences between the fully coupled and weakly coupled solutions are observed to be about 1%, which is within the accuracy limits of the calculations. Hence, for both quartz and lithium niobate, weakly coupled solutions appear adequate for interpretation of observed current-time waveforms. On the other hand, the adequacy of the uncoupled solution is significantly different for the two materials. For x-cut quartz the maximum error of about 1%-1.5% for the nonlinear-uncoupled solution is suitable for all but the most precise interpretation. For z-cut lithium niobate the maximum error of about 8% for the nonlinear-uncoupled solution is greater than that considered acceptable for most cases. The linear-uncoupled solution is seriously in error in each case as it neglects both strain and coupling. 

Finally we shall derive the equation used by Bixon and Jortner. Suppose that an intramolecular vibrational mode, say Qi, plays a very important role in electron transfer. To this mode, we can apply the strong-coupling approximation (or the short-time approximation). From Eq. (3.40), we have... 

Linear coupling approximation, geometric phase theory, 3... 

Using the long time-weak coupling approximation and the hypothesis of random phases for the thermostat, Bloch and Wangsness find, after taking the trace over the heat bath, the following equations for the reduced density matrix a ... 

J.-P. Bouchaud, L. F. Cugliandolo, J. Kurchan, andM. Mezard. Mode-coupling approximations,... 

In contrast, when a stress acts in the 2-3 plane, as in Figure 5.86b, the matrix plays a crucial load-bearing role. Fibers and matrix now couple approximately in series, and the whole tensile force is assumed to be carried fully by both the fibers and matrix. The tensile forces in the fiber and matrix, a/2 and ct 2, are therefore equal to each other and to the overall stress in the composite, CT2I... 

The practical utility of the coupled assumptions of unit Lewis number and a single, over-all reaction step in complex reaction processes is unknown. The use of relations derivable from Equation 18, in conjunction with an experimental study, should help in providing some insight concerning the utility of these coupled approximations for representative combustible gas mixtures. 

This equation is, of course, well known and often called the Pauli equation. We recognize on the right-hand side the familiar gain and loss terms. The transition probabilities which appear in the Pauli equation correspond to the Born approximation for one-photon processes. For further reference let us summarize the main properties of this weakly coupled approximation. 

Continua. The wavefunctions of scattering and bound states have been calculated numerically in the close coupled approximation [358]. Converged partial wave expansions of the elastic scattering solutions have been calculated for pairs of angular momenta 71/2 = 00, 02, 22, 10, 30, 12, 11, and 13 at several hundred energy points. Rotationally inelastic... 

The distortionless enhancement by polarization transfer (DEPT) sequence 253) for the J-coupled heteronuclear spin system (J is 13C- H coupling, approximately 135 to 170 Hz) is shown in Fig. 48. The pulse sequence is based on a resolvable spin-spin coupling between two nuclei, one of them (lH) being the polarization source for the... 

This model is based on the weak coupling approximation and it takes into account neither reorganization nor vibration assistance in the sense of Section 2.5. Although the term vibration-assisted tunneling is also applied to the Skinner-Trommsdorf model, this assistance signifies only that vibrations supply the energy needed to provide a resonance. 

Sinha, P.K., Chaudhury, P. and Ghosh, A.S. (1997). Ps-H scattering using three-state positronium close-coupling approximation. J. Phys. B At. Mol. Opt. Phys. 30 4643-4652. 

---