Showing papers on "Cluster expansion published in 1970"

Nature of the Rigid‐Rod Mesophase

Abstract: The thermodynamic phases of the rigid‐rod gas are re‐examined. Introduced in its present form by Zwanzig, the model postulates elongated molecules of parallelepiped shape with only three allowed orientations (mutually perpendicular). We have added to the virial series the cluster integrals represented by eight‐point bicolored graphs. It is demonstrated that the Pade approximants generated by the truncated cluster expansion provide a much more stable sequence of parameters characterizing the isotropic–anisotropic phase transition. It is similarly shown that there are no stable phases predicted for which the orientational distribution is not axially symmetric. The relative volume change accompanying the transition is still predicted to be larger than that observed in the isotropic–nematic liquid crystal transition.

Cluster-expansion procedures for the correlated charge form factor

Abstract: A factor-cluster or Van Kampen-type expansion is proposed for the evaluation of expectation values with respect to Jastrow wave functions, with special attention to the correlated form factor for elastic electron scattering from light nuclei. The advantages of this expansion over other means (familiar iwamoto-Yamada-type expansion, single- or multiple-pair approximation) for evaluating the Jastrow form factor are displayed.

Theory of Phenomenological Coefficients in Solid‐State Diffusion. II. Cluster Expansion for Ionic Crystals

Abstract: Previously derived time correlation formulas for phenomenological coefficients of solid‐state diffusion are studied. The model is that the system evolves by the master equation of a Markoff process and is characterized by known defect‐configuration dependent jump probabilities. To study the effect of defect interactions on matter transport the cluster and diagram expansion methods of equilibrium theory are used to convert the expressions to series expansions in powers of defect concentrations, the migration mechanisms being vacancy and interstitial. The coefficients of the expansion are cluster summations dependent on concentration through equilibrium correlation functions. To obtain convergent summations for ionic crystals a further chain summation step like that of Mayer ionic solution theory, as well as an additional bond summation of a different kind, are necessary. Formal results are given for ionic conducting crystals whose defect composition is arbitrary except that there are no macroscopic amounts...

Entropy principle for the derivation of a new cluster expansion.

Abstract: A product expansion for the s‐particle distribution function (s arbitrary) is derived by means of a variational principle involving the entropy associated with the (s + 1)‐particle distribution function.

General Method for Handling the Hard Core with Shell-Model Wave Functions

Abstract: The left-unitary operators that introduce the correlations of the hard core in an uncorrelated bound-state wave function of a finite number $A$ of particles are explicitly determined up to a unitary transformation. Left unitarity is required in order to keep orthonormality. Using a shell-model basis, transformed according to the correlation operator, it is possible to diagonalize the Hamiltonian in a subspace of the full Hilbert space with the technique of Bloch and Horowitz. Ground as well as excited states are treated on the same footing. In practice, the diagonalization is accomplished using the original single-particle basis and transforming back the Hamiltonian. This results in an $A$-body operator for which a cluster expansion is possible. In this cluster expansion, there are three- and four-body terms easy to evaluate, because they factorize. A particularly interesting application can be made to few-body systems, evaluating the integrals numerically. The unique approximation is then connected with the cut of the basis. The method developed seems comparatively simple and is free of the mathematical problems of the hard core.