Cluster-Bethe-Lattice

In the cluster-Bethe-lattice method, a central cluster is treated exactly and the surrounding medium is replaced by a successively branching network containing no closed paths. 

We choose here instead an analytic formulation based upon the simple molecular lattice, which will immediately be generalized. This will bring the most important concepts to light and provide an interpretation of the spectrum. It can also provide the starting point for a quantitative application of the cluster-Bethe-lattice method use of the Bethe lattice should improve accuracy and reduce the computation required in comparison to the direct cluster technique. 

Figure 8.04. Comparison of densities of states for amorphous (solid line) and crystalline rhombohedral (dashed line) As. Experimental data are from XPS of Ley et al. (1973), and theoretical curves are from Kelly (1980) (recursion method) and from Pollard and Joannopoulos (1978) (cluster-Bethe-lattice method, CBLM) (Combined Figure from Elliott, 1984).

AP atom probe CBLM cluster Bethe lattice method... 

Percolation theory describes [32] the random growth of molecular clusters on a d-dimensional lattice. It was suggested to possibly give a better description of gelation than the classical statistical methods (which in fact are equivalent to percolation on a Bethe lattice or Caley tree, Fig. 7a) since the mean-field assumptions (unlimited mobility and accessibility of all groups) are avoided [16,33]. In contrast, immobility of all clusters is implied, which is unrealistic because of the translational diffusion of small clusters. An important fundamental feature of percolation is the existence of a critical value pc of p (bond formation probability in random bond percolation) beyond which the probability of finding a percolating cluster, i.e. a cluster which spans the whole sample, is non-zero. 

The model in Fig. 3.2 is sufficient to predict the general features of N E), but much more detailed calculations are needed to obtain an accurate density of states distribution. Present theories are not yet as accurate as the corresponding results for the crystalline band structure. The lack of structural periodicity complicates the calculations, which are instead based on specific structural models containing a cluster of atoms. A small cluster gives a tractable numerical computation, but a large fraction of the atoms are at the edge of the cluster and so are not properly representative of the real structure. Large clusters reduce the problem of surface atoms, but rapidly become intractable to calculate. There are various ways to terminate a cluster which ease the problem. For example, a periodic array of clusters can be constructed or a cluster can be terminated with a Bethe lattice. Both approaches are chosen for their ease of calculation, but correspond to structures which deviate from the actual a-Si H network. 

The central result of percolation theory is the existence of the critical probability pc, the percolation threshold at which a cluster of infinite size appears. In a one-dimensional lattice, the percolation threshold is obviously equal to one. For higher dimensions, pc will be smaller than one. To illustrate this central result, we consider the Bethe lattice (also called the Cayley tree). 

Figure 3.15 Evolution of the probability P(p) that a site belongs to the infinite cluster as a function of the occupation probability p for the Bethe lattice with three branches. The percolation threshold is pc = 0.5.

As noted previously, the critical probability for the Bethe lattice is (equivalent to Pc defined in Chapter 4) = /(( — 1). For lattices below this critical probability (e < Cc), the root of Equation 9-26 is e = e. The accessible porosity, from Equation 9-25, is zero, which indicates that a lattice spanning cluster is not present. For lattices above the critical probability (e > e ) Equation 9-26 can be solved to find e. Results for coordination numbers 3 and 4 are ... 

In Figure 9.14b, the fraction accessible porosity is plotted versus the total porosity for Bethe lattices of coordination numbers 3 and 7. For all coordination numbers, a is zero for porosities less than the critical value. The critical porosity is indicated for each coordination number by the intercept of the curve with the x-axis. Above the critical porosity, rises sharply. In this transition region, the infinite, lattice-spanning cluster is growing and incorporating pores and smaller pore clusters that are isolated at lower porosities. At high porosities, 0a becomes equal to unity, indicating that all the pores are members of the infinite cluster. 

Even below the critical porosity, pore clusters exist. The clusters are finite in size and do not span the lattice. In the transition region—as the infinite cluster is incorporating more of these finite clusters—finite clusters still exist. For the Bethe lattice, the mean size of these finite clusters, S, depends on the lattice-filling probability (i.e., porosity) [45] ... 

Figure 9.14 Fraction accessible porosity and mean cluster size for a Bethe lattice. Fraction accessible porosity and mean cluster size for Bethe lattices with f = 3 (solid lines) and 7 (dashed lines). The fraction of accessible porosity (b), or the fraction of porosity that is part of an infinite cluster, is plotted versus the total porosity. The mean cluster size (a) exhibits a singularity at the critical porosity.

Equation (2.215) deals with the level shifts for a cluster with metal atoms assumed to be uncoupled from the rest of the surface. Equation (2.238) is the formula to be solved for the embedded cluster. is the Green s function of the indented lattice. For the Bethe lattice it becomes equal to gz(E). The first-order correction to q.(2.215) becomes ... 

The first theory that attempted to derive the divergences in cluster mass and average radius accompanying gelation is that of Flory [52] and Stockmayer [53]. In their model, bonds are formed at random between adjacent nodes on an infinite Cayley tree or Bethe lattice (see Figure 47.7). The Flory-Stockmayer (FS) model is qualitatively successful because it correctly describes the emergence of an infinite cluster at some critical extent of reaction and... 

Although there are probably other universality classes, this transition was successfully modeled by bond percolation [6]. Generally, bond percolation on a lattice has each bond (line connecting two neighboring lattice sites) present randomly with probability p and absent with probability 1-p. Clusters are groups of sites connected by present bonds. For p > Pc zn infinite cluster is formed. Percolation theory (in a Bethe lattice approximation) was invented by Flory (1941) to describe gelation for three-functional polymers. 

Disaster approaches for cluster (a) with e = 3 growing on a Cayley tree (or Bethe lattice), because lattice contains no loops, so its density increases without limit (b) as the radius grows. Additional bonds forming at the periphery indicated by thick lines. 

By contrast, since Bethe lattices are tree structures-containing no loops or closed paths within them-they are more easily analyzed than other, less regular, lattice structures. Analytical expressions have been derived for the percolation probability, cluster size distribution function [42], and effective conductivity [43] of Bethe lattices. The properties of these special Bethe lattices are quantitatively similar to regular... 

Effective transport coefficients Lattice representations of space provide a convenient means for representing porous materials. As shown in the previous subsection, some important material properties (critical porosity, accessible porosity, cluster size) can be predicted given a suitable lattice model for the structure. In order to determine the rate of solute transport in the structure, h(9l) must be evaluated to find the effective diffusion coefficient. For diffusion on a Bethe lattice, analytical expressions for the effective diffusion coefficient are available [43,44]. For a Bethe lattice with coordination number the effective diffusion coefficient is found from ... 

Curiously, at the upper critical dimension d = 6, equ.(2) gives zero for D nti-red Unfortunately the reason given in proof in reference [9] to support this result, is also not correct. At the upper critical dimension, as the loops are irrelevant in the cluster structure, the problem can be replaced by a Bethe lattice problem. In this case, there effectively exists only one bond to reconnect a cluster of size s, but this reconnecting bond (anti-red bond) is highly degenerated. Hence to confirm or to infirm the above assumption it is important to examine more in details the case of the Bethe lattice. This is the purpose of the following part. 

The disconnection probability, per fra enting bond, b5 s(Pc) of a cluster of size s from a cluster of size s in a Bethe lattice with coordination z, has been recently given by Edwards et al. [8],... 

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