Equilibrium Pair Correlation Function Research Articles

Theory of self -diffusion in liquids

A new theory of self-diffusion in simple liquids is outlined. Dynamical superposition is assumed and then it is shown that with no further approximations the self-diffusion coefficient can be computed if the functional form of the equilibrium pair correlation function $g(r)$ is known. The result is formally equivalent to that found in the theory of dilute gases with the potential of mean force, $\ensuremath{\omega}(r)=\ensuremath{-}\mathrm{ln}g(r)$, taking the place of the intermolecular potential.

Optimized cluster theory, the Lennard-Jones fluid, and the liquid-gas phase transition

The optimized cluster theory is applied to the Lennard-Jones fluid. The coexistence curve obtained from the theory is presented. It represents the most accurate microscopic calculation of the liquid-gas phase diagram for a fluid with a realistic Hamiltonian. The theory is shown to be quantitatively accurate for all fluid states except those very close to the critical point. Phenomenological procedures are devised which extrapolate towards the critical point from regions in which the optimized cluster theory is accurate. The extrapolation calculations predict nonclassical values for the critical exponents which are in fairly good agreement with experiment. In addition to investigating the liquid-gas phase transition, the equilibrium pair correlation functions are calculated at several states. Further, the internal consistency of the optimized cluster theory is discussed.

Generalization of the Theory of Ichimaru and the Physical Properties of an Interacting-Electron Gas

An expression is derived for the dielectric response function of a strongly correlated degenerate electron system from the solution of the second equation of the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy. A modification over Ichimaru's procedure is suggested by the adjustment of the pair-correlation function to the external perturbation brought about via the density derivative of the equilibrium pair-correlation function. The present theory has the merit of satisfying the compressibility sum rule almost exactly and at the same time leading to satisfactory results of the pair-correlation function. Results of self-consistent numerical calculations for different physical quantities are presented for the electron gas in the metallic density range (${r}_{s}$ ranging from 1 to 6).

Cluster expansions for hydrogen-bonded fluids. I. Molecular association in dilute gases

The cluster expansion method of Mayer is formulated in a way that is convenient for calculating the equilibrium pair correlation function of hydrogen-bonded fluids. The topological properties of cluster diagrams for these fluids is investigated, and a cancellation theorem is presented which permits the neglect of certain types of diagrams in these cluster expansions. A method is presented for renormalizing the hydrogen bond in such expansions, and a technique for summing the graphs which define the renormalized hydrogen bond is discussed. The present paper deals only with low density gases, and future work will extend the results to dense liquids.

Derivation of an integral equation for pair correlation functions in molecular fluids

A new integral equation for equilibrium pair correlation functions in molecular fluids is derived from a systematic functional Taylor expansion. The derivation employs a new Ornstein-Zernike-like equation and a generalization of the functional Taylor expansion techniques frequently employed to derive integral equations for radial distribution functions in simple atomic liquids.

Spin-lattice relaxation and molecular reorientations near Tc with application to NH4Cl

Reorientations of the ammonium tetrahedra around C2 axes in the neighborhood of the structural phase transition in a NH4Cl crystal are described by a master equation. Within each group of two sterically different tetrahedral configurations characterized by a pseudospin variable σ=± 1 distinction is made of a=12 sterically equivalent positions. Cooperative dynamical behavior is introduced by extending Glauber's theory of the dynamics of an Ising system to the tetrahedral motion in molecular crystals. The (approximate) solution of the master equation yields a critical slowing down behavior for the pseudospin correlation function. This explains quadrupole spin-lattice relaxation T1Q experiments. In addition the decay time τ for the probability P(σa;t) of finding a tetrahedron in a given position σ, α∈ 12 is found to depend on the equilibrium pair correlation function g=<σiσi+1>. By relating this quantity to the thermal expansion, the proton spin-lattice relaxation time T1P can be expressed in terms of the anomalous thermal expansion. The theory explains the anomalous behaviour found experimentally for T1P around the phase transition at low pressure and high pressure.

Electron Correlations at Metallic Densities. V

In this paper we present a modification of an earlier theory of Singwi et al. of electron correlations at metallic densities. The modification consists in allowing for the change of the pair correlation function in an external weak field via the density derivative of the equilibrium pair correlation function. This results in a new expression for the local-field correction. The present theory has the merit of satisfying almost exactly the compressibility sum rule and of giving a satisfactory pair correlation function. Results of self-consistent numerical calculations for the static pair correlation function, correlation energy, compressibility, and plasmon dispersion relation for the electron liquid in the metallic-density range are presented. For those interested in the application of the results of the present paper, numerical values of the local-field correction as a function of wave number have been tabulated in the density range ${r}_{s}=1\ensuremath{-}6$.

Numerical Study of Nonequilibrium Diagram Theory

In this paper we numerically analyze the first few diagrams in a Boltzmann-like collision operator that occurs in Severne's exact kinetic equation for the singlet distribution function. A similar analysis was used by Allen and Cole in deriving their singlet and doublet kinetic equations. Our analysis shows that the diagrams neglected by Allen and Cole in their kinetic equations are not negligible and these should be incorporated into dense fluid theories. The Allen–Cole kinetic transport coefficients and equilibrium pair correlation function are presented and calculated for dense argon. These results are not promising.

Lattice dynamics and transport equations for quantum crystals including singular interactions

The technique of non-equilibrium Green's functions is used to describe equilibrium properties and to derive transport equations for quantum crystals. The resulting equations are of the same structure as in anharmonic perturbation theory involving, however, an effective pair potential. This is formed by means of the equilibrium pair correlation function and is also defined for singular interactions.

Theory of Phenomenological Coefficients in Solid-State Diffusion. III. Limiting Law Conductance for Pure Ionic Crystals

The cycle diagrams of the previously derived cluster expansions are studied in detail for pure ionic crystals containing Schottky defects. The chain diagram approximation for the equilibrium pair correlation function is employed throughout. Some diagrams are then zero and the remainder yield, in the continuum limit, the Onsager limiting law for the relaxation effect for the conductance at zero frequency. The coefficient of the correction of lowest order in κa to this limiting law which arises when the cycle diagrams are evaluated allowing for the discreteness of the lattice is determined for the NaCl structure. The predictions of this correction are compared with exact numerical evaluations of the cycle diagrams. The corrections from the time-dependent correlation terms are well approximated by the analytic result and are very close to those predicted by the Pitts ion-size correction for electrolyte solutions. However, the remaining corrections from the time-independent correlation terms are but poorly represented by the analytic result and tend to cancel the other corrections. It is concluded that the major approximation in the use of the Onsager limiting law is the use of the chain diagram approximation for the pair correlation function (equivalent to the Debye–Hückel linearized result in the continuum limit) and this is probably relatively more important than effects arising from the discreteness of the lattice or corrections arising from the dynamics of vacancy pairs, triplets, etc. Further numerical study of this point is required. The generalization of the results to Frenkel defects is considered briefly but without detailed results.

Statistical-Mechanical Derivation of the Partial Molecular Stress Tensors in Isothermal Multicomponent Systems

Commencing with the general equations of molecular dynamics as used by Bearman and Kirkwood, the hydrodynamic equations of motion are derived for each component in an isothermal multicomponent system consisting of sensibly spherical nonreacting molecules. It was the primary purpose of the investigation of which this is a report to attempt to obtain a microscopic statistical basis for the macroscopic phenomenologic development of Snell and Spangler, especially in regard to their new formulation for the partial molecular stress tensors. An expansion of the number densities in pair space of perturbed nonequilibrium states alternate to that used by Bearman and Kirkwood is employed. Our expansion esplicitly recognizes the position vector arguments of both the number densities in singlet space and the local mean velocities of the individual components, as well as for the equilibrium pair correlation function. The quantities are then expanded about the central position vector in a Taylor's series. The results are not only corroborative of the previous phenomenologic development when the flows of the individual components are solenoidal and attention is confined to systems in which the kinetic contribution to the stress tensor is negligible in comparison to the contribution arising from intermolecular interaction, but they also provide a new microscopic theory of hydrodynamical flow in multicomponent systems.

Onsager Relations and the Spectrum of Critical Opalescence

The necessity of extending the hydrodynamic equations near the critical point is investigated on the basis of Onsager's assumption concerning the regression of fluctuations. It is shown that it follows from the Onsager relations that in general the new equations are nonlocal. The most important coefficient corresponding to the inverse compressibility in the original equations is obtained from the Onsager relations and expressed in terms of the equilibrium pair correlation function. Fixman's extension of hydrodynamics follows when the pair correlation is given by the Ornstein—Zernike formula. The consequences for the spectrum of light scattering near the critical point are studied and the width and relative intensity of the central scattering peak are expressed in terms of the equilibrium pair correlation function.