Transfer matrix calculations

Transfer matrix calculations of the adsorbate chemical potential have been done for up to four sites (ontop, bridge, hollow, etc.) or four states per unit cell, and for 2-, 3-, and 4-body interactions up to fifth neighbor on primitive lattices. Here the various states can correspond to quite different physical systems. Thus a 3-state, 1-site system may be a two-component adsorbate, e.g., atoms and their diatomic molecules on the surface, for which the occupations on a site are no particles, an atom, or a molecule. On the other hand, the three states could correspond to a molecular species with two bond orientations, perpendicular and tilted, with respect to the surface. An -state system could also be an ( - 1) layer system with ontop stacking. The construction of the transfer matrices and associated numerical procedures are essentially the same for these systems, and such calculations are done routinely [33]. If there are two or more non-reacting (but interacting) species on the surface then the partial coverages depend on the chemical potentials specified for each species. 

The finite-size scaling theory combined with transfer matrix calculations had been, since the development of the phenomenological renormalization in 1976 by Nightingale [70], one of the most powerful tools to study critical phenomena in two-dimensional lattice models. For these models the partition function and all the physical quantities of the system (free energy, correlation length, response functions, etc) can be written as a function of the eigenvalues of the transfer matrix [71]. In particular, the free energy takes the form... 

For a layer thickness corresponding to the screening length, S aa k, scaling arguments predict a rather abrupt desorption transition [84]. This is in accord with previous transfer-matrix calculations for a semiflexible polymer bound by short-ranged (square-well) potentials [88-91]. Setting S in Eq. (42), we obtain an expression for the adsorption threshold... 

The zone transfer matrix calculated in the Laplace domain for a single zone in the one-dimensional model exactly expresses its behavior as a component of any system built from such a zone, not only as part of a TAP reactor. We now study what happens if we were to consider concentric situations in two and three dimensions instead. Let d denote the dimension (d= 1, 2, or 3). Then the spatial PDE, Eq. (5.123), takes the following form for radial situations... 

The procedure of transfer matrix calculation of dielectric mirrors is described in detail in, e.g., [243]. An approach similar to the above is used for 2D and 3D structures, but the obtained expressions are much more involved and must be solved numerically. In [244] a detailed approach is described and the diagrams of photonic bandgap structures for 2D and 3D photonic crystals are given. An alternative method is presented in [241],... 

For ideal chains, one has (p = v= 1/2, and thus we recover the prediction from the transfer-matrix calculations, Eq. (7). For nonideal chains, the crossover exponent

different from the swelling exponent v. However, extensive Monte Carlo eomputer simulations point to a value for (p very close to v, such that the adsorption exponent v/q> appearing in Eq. (11) is very close to unity for polymers embedded in three-dimensional space [31]. 

Figure 7.7. N+1 layers representation of a EC cell between polarizers for transfer matrix calculation. Insert shows the incident beam direction.

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. 

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. 

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. 

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]

Fig. 30. Phase diagram of a model for Si/W(110) in the temperature versus 9 plane. Experimentally determined interactions J Jj,are used. Full dots are from Monte Carlo calculations, while triangles are based on transfer matrix finite size scaling using strip widths of 8 and 12. The point labelled L indicates approximate location of Lifshitz point. The dotted line indicates the transition region between the (5 x l)and(6 x 1) phases. (From...

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 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 " . ... 

Different quantum chemical approaches can be invoked to calculate the electronic couplings. In many cases one can reliably estimate electron-transfer matrix elements on the basis of a one-electron approximation [27-29]. 

Each WCP dimer model consists of two purine bases (G and A) and two pyrimidine bases (C and T). According to the calculations, the two highest-lying orbitals HOMO and HOMO-1 of each duplex are mainly locahzed on the purine nucleobases, whereas the two occupied MOs following at lower energies, HOMO-2 and HOMO-3, are locahzed on pyrimidine nucleobases. Therefore, the purine-purine electronic coupling provides the dominant contribution to the hole transfer matrix elements, irrespective whether the bases belong to the same or to opposite strands. 

There is a general statement [17] that spin-orbit interaction in ID systems with Aharonov-Bohm geometry produces additional reduction factors in the Fourier expansion of thermodynamic or transport quantities. This statement holds for spin-orbit Hamiltonians for which the transfer matrix is factorized into spin-orbit and spatial parts. In a pure ID case the spin-orbit interaction is represented by the Hamiltonian //= a so)pxaz, which is the product of spin-dependent and spatial operators, and thus it satisfies the above described requirements. However, as was shown by direct calculation in Ref. [4], spin-orbit interaction of electrons in ID quantum wires formed in 2DEG by an in-plane confinement potential can not be reduced to the Hamiltonian H s. Instead, a violation of left-right symmetry of ID electron transport, characterized by a dispersion asymmetry parameter Aa, appears. We show now that in quantum wires with broken chiral symmetry the spin-orbit interaction enhances persistent current. 

Stewart WE, Prober R. Matrix calculation of multicomponent mass transfer in isothermal systems. Ind Eng Chem Fundam 1964 3 224-235. 

Rates of non-adiabatic intramolecular electron transfer were calculated in Ref. [331] using a self-consistent perturbation method for the calculation of electron-transfer matrix elements based on Lippman-Schwinger equation for the effective scattering matrix. Iteration of this perturbation equation provides the data that show the competition between the through-bond and through-space coupling in bridge structures. 

R. J. Cave and M. D. Newton, Generalization of the Mulliken-Hush treatment for the calculation of electron transfer matrix elements, Chem. Phys. Lett., 249 (1996) 15-19. 

---