Transfer Matrix Methods

Lattice models have been studied in mean field approximation, by transfer matrix methods and Monte Carlo simulations. Much interest has focused on the occurrence of a microemulsion. Its location in the phase diagram between the oil-rich and the water-rich phases, its structure and its wetting properties have been explored [76]. Lattice models reproduce the reduction of the surface tension upon adsorption of the amphiphiles and the progression of phase equilibria upon increasmg the amphiphile concentration. Spatially periodic (lamellar) phases are also describable by lattice models. Flowever, the structure of the lattice can interfere with the properties of the periodic structures. 

A natural question is just how big does Mq have to be to see this ordered phase for M > Mq. It was shown in Ref 189 that Mq <27, a very large upper bound. A direct computation on the Bethe lattice (see Fig. 2) with q neighbors [190,191] gives Mq = [q/ q — 2)f, which would suggest Mq 4 for the square lattice. By transfer matrix methods and by Pirogov-Sinai theory asymptotically (M 1) exact formulas were derived [190,191] for the transition lines between the gas and the crystal phase (M 3.1962/z)... 

This review is structured as follows. In the next section we present the theory for adsorbates that remain in quasi-equilibrium throughout the desorption process, in which case a few macroscopic variables, namely the partial coverages 0, and their rate equations are needed. We introduce the lattice gas model and discuss results ranging from non-interacting adsorbates to systems with multiple interactions, treated essentially exactly with the transfer matrix method, in Sec. II. Examples of the accuracy possible in the modehng of experimental data using this theory, from our own work, are presented for such diverse systems as multilayers of alkali metals on metals, competitive desorption of tellurium from tungsten, and dissociative... 

With the availabihty of computers, the transfer matrix method [14] emerged as an alternative and powerful technique for the study of cooperative phenomena of adsorbates resulting from interactions [15-17]. Quantities are calculated exactly on a semi-infinite lattice. Coupled with finite-size scaling towards the infinite lattice, the technique has proved popular for the determination of phase diagrams and critical-point properties of adsorbates [18-23] and magnetic spin systems [24—26], and further references therein. Application to other aspects of adsorbates, e.g., the calculation of desorption rates and heats of adsorption, has been more recent [27-30]. Sufficient accuracy can usually be obtained for the latter without scaling and essentially exact results are possible. In the following, we summarize the elementary but important aspects of the method to emphasize the ease of application. Further details can be found in the above references. 

To introduce the transfer matrix method we repeat some well-known facts for a 1-D lattice gas of sites with nearest neighbor interactions [31]. Its grand canonical partition function is given by... 

The transfer matrix method extends rather straightforwardly to more than one dimension, systems with multiple interactions, more than one adsorption site per unit cell, and more than one species, by enlarging the basis in which the transfer matrix is defined. 

As an example of a multilayer system we reproduce, in Fig. 3, experimental TPD spectra of Cs/Ru(0001) [34,35] and theoretical spectra [36] calculated from Eq. (4) with 6, T) calculated by the transfer matrix method with M = 6 on a hexagonal lattice. In the lattice gas Hamiltonian we have short-ranged repulsions in the first layer to reproduce the (V X a/3) and p 2 x 2) structures in addition to a long-ranged mean field repulsion. Second and third layers have attractive interactions to account for condensation in layer-by-layer growth. The calculations not only successfully account for the gross features of the TPD spectra but also explain a subtle feature of delayed desorption between third and second layers. As well, the lattice gas parameters obtained by this fit reproduce the bulk sublimation energy of cesium in the third layer. 

Additional applications of the transfer matrix method to adsorption and desorption kinetics deal with other molecules on low index metal surfaces [40-46], multilayers [47-49], multi-site stepped surfaces [50], and co-adsorbates [51-55]. A similar approach has been used to study electrochemical systems. 

The calculation of the cladding mode effective index can be accomplished by an extension of the model for doubly clad fibers28 or by following the transfer matrix method (TMM) proposed by Anemogiannis et al.26 and successively widely adopted for the analysis of coated LPGs29 30. 

An efficient formalism for the calculation of eigenmodes of the multilayer is known as the transfer matrix method . We will briefly outline its fundamentals. 

The transfer matrix method is known to be often unstable. If this is the case for the matrix A, other methods can be alternatively used to calculate the Bloch modes and their propagation constants " . ... 

The effort to solve Eqs.(l) evidently depends on the refractive index profile. For isotropic media in a one-dimensional refractive index profile the modes are either transversal-electric (TE) or transversal-magnetic (TM), thus the problem to be solved is a scalar one. If additionally the profile consists of individual layers with constant refractive index, Eq.(l) simplifies to the Flelmholtz-equation, and the solution functions are well known. Thus, by taking into account the relevant boundary conditions at interfaces, semi-analytical approaches like the Transfer-Matrix-Method (TMM) can be used. For two-dimensional refractive index profiles, different approaches can be... 

The enumeration of Kekule structures for rectangle-shaped benzenoids is treated. Combinatorial formulas for K (the Kekule structure count) are derived by several methods. The oblate rectangles, Rj(m, n), with fixed values of m are treated most extensively and used to exemplify different procedures based on the method of fragmentation (chopping, summation), a fully computerized method (fitting of polynominal coefficients), application of the John-Sachs theorem, and the transfer-matrix method. For Rj(m, n) with fixed values of n the relevant recurrence relations are accounted for, and general explicit combinatorial K formulas are reported. Finally a class of multiple coronoids, the perforated oblate rectangles, is considered in order to exemplify a perfectly explicit combinatorial K formula, an expression for arbitraty values of the parameters m and n. 

The transfer-matrix method is a powerful tool for studying Kekule structures and their numbers [29-33]. In this approach one studies the manner in which a Kekule... 

In this connection the transfer-matrix method (Sect. 2.7) shows its superiority to the other methods, which have been treated here. By an elaborate expansion of the eigenvectors and eigenvalues of T it was arrived at a general formula for K R (m, n), which is suitable for producing the explicit formulas of K Rj(m, n) with fixed values of n. It reads... 

In Ref. 30 the values of Awd of Sec. 6.1 for w ranging from one to twelve and D from zero to (w — b)/2 (with Aw(-b-D) = Awd) have been obtain by a dimer-covering-counting transfer-matrix method as follows. 

For co = 0, this equation reduces to (14), whereas dm/dz = 0 reproduces (23). Mathematically, Eq. (24) is a well-known random-potential eigenvalue problem, which can be solved numerically or by transfer-matrix methods [5, 153],... 

It is possible to include phase transformers in scalar diffraction theory. The calculations are lengthy, however, and we refer the reader to Anan ev (1992) and Martin and Bowen (1993) for details. An alternative approach exists that is equivalent to the transfer matrix method of geometrical optics, although the results are justifiable in terms of diffraction theory (Anan ev, 1992 Martin and Bowen, 1993). The formalism is discussed, for example, in Hecht and Zajac (1979, pp. 171-175) and we will briefly outline the necessary results. 

A Buckminsterfullerene. Vukicevic and Randic174 examined the 12,500 Kekule structures of buckminsterfullerene (this number was first obtained by Klein et al 15 using the transfer-matrix method) and were able to classify them according to the different innate degree of freedom that they possess. The innate degree of freedom or, for short, the degree offreedom (df) of a Kekule... 

In the majority of numerical calculations of the anomalous frequency behavior of such composites (in particular, near the percolation threshold pc) under the action of an alternating current, lattice (discrete) models have been used, which were studied in terms of the transfer-matrix method [91,92] combined with the Frank-Lobb algorithm [93], Numerical calculations and the theoretical analysis of the properties of composites performed in Refs. 91-109 have allowed significant progress in the understanding of this phenomenon however, the dielectric properties of composites with fractal structures virtually have not been considered in the literature. 

Fig. 6.30. Phase diagram for oxygen as obtained using the transfer matrix method in conjunction with effective Hamiltonian parameters from first-principles calculations (adapted from Liu et at.

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