The Transfer Matrix

If Kl(a) or Ki (b) denotes the number of Kekuld structures for a polyphenanthrene strip with the length L and with local structures a orb at the terminal end, then 

FIGURE 5.2 A part of the polyphenanthrene strip and the corresponding local structures. 

the standard matrix multiplication produces the number of Kekule structures. For example, the number of Kekule structures for the polyphenanthrene strip of length 6 is 987. 

Babic and A. Graovac, Enumeration of Kekule structures in one-dimensional polymers, Croat. Chem. Acta 59 (1986) 731.744. 

Lukovits, S. Nikolic, and N. Trinajstic, Resistance-distance matrix A computional algorithm and its application, Int. J. Quantum Chem. 90 (2002) 166-176. A.T. Balaban, Highly discriminating distance-based topological index, Chem. Phys. Lett. 89 (1989) 399-404. 

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We suppose that a measurement signal is a mix of r (unknown) independent sources Sj with variance cf. A detector give p> r measurement signals /W/t obtained with several frequencies. The relations between sources and measurement signals are supposed to be linear, but the transfer matrix T is unknown. If we get n >p>r samples m i) of measurement signals, mix of n... 

This is an important general result which relates the free energy per particle to the largest eigenvalue of the transfer matrix, and the problem reduces to detennining this eigenvalue. 

By a deft application of the transfer matrix teclmique, Onsager showed that the free energy is given by... 

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... 

To avoid numerical differentiation (which is inherently unstable) one uses the fact that an eigenvalue can be expressed as Ai = v Tvf where are the corresponding normalized left and right eigenvectors. Differentiation of the eigenvalue with respect to any parameter is then equivalent to the differentiation of the transfer matrix, and one finds... 

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. 

For the extension to two dimensions we consider a square lattice with nearest-neighbor interactions on a strip with sites in one direction and M sites in the second so that, with cyclic boundary conditions in the second dimension as well, we get a toroidal lattice with of microstates. The occupation numbers at site i in the 1-D case now become a set = ( ,i, /25 5 /m) of occupation numbers of M sites along the second dimension, and the transfer matrix elements are generalized to... 

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. 

Our last example of the utility of the transfer matrix approach concerns the thermodynamics and desorption kinetics of H/Rh(311) obtained from... 

FIG. 5 Schematic of site parameters and interactions employed for the hollow-bridge site model of Te on W(IOO). Also depicted are the six hollow sites (squares) and adjacent bridge sites (small open circles) allowed in one strip in the construction of the transfer matrix. (Reprinted from Ref. 37 with permission from Elsevier Science.)... 

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. 

For the equihbrium properties and for the kinetics under quasi-equilibrium conditions for the adsorbate, the transfer matrix technique is a convenient and accurate method to obtain not only the chemical potentials, as a function of coverage and temperature, but all other thermodynamic information, e.g., multiparticle correlators. We emphasize the economy of the computational effort required for the application of the technique. In particular, because it is based on an analytic method it does not suffer from the limitations of time and accuracy inherent in statistical methods such as Monte Carlo simulations. The task of variation of Hamiltonian parameters in the process of fitting a set of experimental data (thermodynamic and... 

In the general case there will be n roots which are the eigenvalues of the transfer matrix K. Each of the eigenvalues defines a particular phase of the time course of the contents in the n compartments of the model. The eigenvalues are the hybrid transfer constants which appear in the exponents of the exponential function. For example, for the ith compartment we obtain the general solution ... 

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. 

The transfer matrix formalism enables us to find the modal field profile in the case of an arbitrary arrangement of annular concentric dielectric rings. However, we are especially interested in structures that can confine the modal energy near a predetermined radial distance, i.e. within the defect. 

This system was modelled in terms of the lattice gas with interactions shown in Fig. Ib. The phase diagram was first calculated by the transfer matrix finite size scaling technique for various choices of the interaction parameters [Pg.122]

The diagram above shows an interactive MIMO system, where the controlled variables, outlet flow temperature and concentration, both depend on the manipulated variables. In order to design a decentralized control, a pairing of variables should be decided. A look at the state Eq.(23) suggests the assignment of the control of the temperature to the cooling flow and the concentration control to the reactor inlet flow. In this case, the internal variable Tj may be used to implement a cascade control of the reactor temperature. Nevertheless, a detailed study of the elements of the transfer matrix may recommend another option (see, for instance, [1]). 

Exercise 2. Compute the transfer matrix in Eq.(32) assuming the reactor data corresponding to the smallest one in Table 1. Choose a gain around the value of Ko in Figure 4 for both, Gn(s) and G22(s). Apply this control to the reactor and plot the step responses to changes in the references. Is there something you did not expect What about the system interaction See the next Section. 

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

Here, A(AC) is the transfer matrix describing the translation by A within the section s. The mode field amplitudes p(C) and q(C) can be alternatively expressed in terms of the amplitudes of forward and backward propagating modes and b. 

The Bloeh mode ean be defined as a wave eorresponding to the eigenmode of the transfer matrix of one period of the strueture. Let A is the transfer matrix deseribing wave transition from the left to the right of one period, calculated by sueeessive applications of Eq. (13) to each section and Eq. (13) to each interface between sections within the period. The Bloch mode then has to satisfy the condition... 

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... 

If those matrices U are made unitary and are related to their mean phase shift, stacking gives the transfer matrix of the full network. 

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