Cluster-Bethe-lattice method

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. 

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