Linked cluster expansion on trees.

The linked cluster expansion has been shown to be highly efficient in calculating equilibrium and nonequilibrium properties of a variety of 1D and 2D classical and quantum lattice models. In this article, we extend the linked cluster method to the Cayley tree and its boundaryless cousin the Bethe lattice. We aim to (a) develop the linked cluster expansion for these lattices, a novel application, and (b) to further understand the surprising convergence efficiency of the linked cluster method, as well as its limitations. We obtain several key results. First, we show that for nearest-neighbor Hamiltonians of a specific form, all finite treelike clusters can be mapped to one dimensional finite chains. We then show that the qualitative distinction between the Cayley tree and Bethe lattice appears due to differing lattice constants that is a result of the Bethe lattice being boundaryless. We use these results to obtain the explicit closed-form formula for the zero-field susceptibility for the entire disordered phase up to the critical point for Bethe lattices of arbitrary degree; remarkably, only 1D chainlike clusters contribute. We also obtain the exact zero field partition function for the Ising model on both trees with only the two smallest clusters, similar to the 1D chain. Finally, these results achieve a direct comparison between an infinite lattice with a nonnegligible boundary and one without any boundary, allowing us to show that the linked cluster expansion eliminates boundary terms at each order of the expansion, answering the question about its surprising convergence efficiency. We conclude with some ramifications of these results, and possible generalizations and applications.

Promoting effects of water on the NH3-SCR reaction over Cu-SAPO-34 catalysts: transient and permanent influences on Cu species.

Cu-SAPO-34 catalysts with varied Cu loadings were synthesized through ion exchange to study the influence of water on the NH3-SCR reaction. The catalytic activities were evaluated by selective catalytic reduction of NO under a reactant feed in the presence/absence of water. Transient experiments were designed to study the response of NO conversion to the presence of water. H2-TPR and DFT calculations were performed to study the reducibility of Cu species. NH3-TPD and XPS were conducted to reveal the migration of Cu species. The results show that water could remarkably improve NO reduction activities and the promoting effect is more significant on the catalyst with low Cu loading. Both transient and permanent influences were found in this promoting phenomenon. For the transient influence, water has been proved to accelerate the re-oxidation half-cycle. Moreover, water can enhance the promoting effect of the SCR feed on the migration of Cu species. These unanchored Cu ions migrate to defect sites to form active sites, which lead to a permanent influence of water.