Mean spherical model for asymmetric electrolytes

The mean spherical approximation for the primitive model of electrolytic solutions is solved for the most general case of an arbitrary charge and size. It is found that the excess thermodynamic properties are scaled to the charge and the size by means of a rational expression involving a single parameter. This parameter is found by solving an algebraic equation. The explicit form of this equation is obtained for the cases of a binary mixture and in the limit of low concentrations.     

Critical parameters of asymmetric primitive model electrolytes in the mean spherical approximation

The effect of size and charge asymmetries on the critical parameters of primitive model electrolytes is comprehensively surveyed by using the mean spherical approximation.     

The mean spherical model for charged hard spheres

The reduced mean electrostatic potential v(r) and the radial distribution functions gij(r) for a system of charged hard spheres of equal diameter are calculated from the solution of the mean spherical model equation given by Waisman and Lebowitz. An analytical solution is given for v(r) and the gij(r) are shown to be the sum of the Percus-Yevick uncharged hard-sphere distribution function and an electrostatic term. The correct qualitative behaviour of the mean potential is predicted at high concentrations but the radial distribution functions are only accurate for low valency electrolytes at high concentrations.     

Relation between the free energy and the direct correlation function in the mean spherical approximation

It is shown that the direct correlation function of a mixture of hard ions in the mean spherical approximation (MSA) can be expressed in terms of overlap functions of charged spherical shells. In particular, if the system is a mixture of pairs of ions of equal size and opposite charge, then the MSA direct correlation function is given by the electrostatic energy of a pair of charged shells, of radius equal to the radius of the hard ion plus 1/(2). This direct correlation function can be derived from a free energy functional, and a simple extension to nonuniform systems is given.     

Generalised mean spherical approximation and the thermodynamics of a classical one-component plasma

The generalised mean spherical approximation has been used in deriving the thermodynamics of a classical one-component plasma using the variational method. The free energy and the excess specific heat are calculated and compared with recent Monte-Carlo results. The agreement was found to be good.     

The direct pair correlation function of liquid indium

The static structure factor S(k) of liquid indium has been measured accurately down to $\text{k = 0.8}\text{A}\text{?}{}^{\mathrm{?1}}$ using CuKα radiation with reflection geometry. The direct pair correlation function in k space is analyzed to demonstrate the utility of this technique in reducing errors in the resulting direct pair correlation function in configuration space.     

Explicit solution of the mean spherical model for ions and dipoles

It is shown that the solution of the mean spherical approximation for the ion-dipole mixtures obtained by Blum, Adelman, and Deutch has an explicit closed form solution which is one of the roots of a cubic equation.     

Calculation of a pair correlation function for the three-dimensional Ising model with long-range exchange forces

A simple difference equation for one type of correlators has been obtained by the method of direct calculation of the correlation functions (CF). It is correct for the Ising model of an arbitrary dimension with the exactness $\text{O}$($\text{1}\text{z}$) (z-number of particles covered by the interaction potential). The equation obtained is used to calculate the pair CF 〈S^z_fS^z_j〉 in the three-dimensional Ising model with long-range forces which are of importance when dealing with the applied magnetic problems. The results of the calculations lead to corrections in the Ornstein-Zernike (OZ) theory.     

Analytical solution of an equation for the pair correlation function

For the hard sphere model, an approximate analytical solution is obtained of a linear integral equation for the pair correlation function derived from the Bogolyubov equation. An equation of state is given for this system which is nearly identical to the expression obtained from the Percus-Yevick equation.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 28–32, August, 1984.The author thanks G. I. Nazin for proposing the problem and also acknowledges useful discussions with É. A. Arinshtein and A. F. Nyashin.     

The mean spherical model in a random external field and the replica method

We provide a quick elementary solution of the mean spherical model in a random external field. This also allows an immediate proof of the self-averaging property of the free energy. We calculate the free energy by means of the replica method, i.e., for any (not necessarily integer) replica numbern, and show that when a phase transition occurs the limits andn 0 are not interchangeable.     

Logarithmic finite-size corrections in the three-dimensional mean spherical model

The finite-size scaling prediction about logarithmic corrections in the free energy arising from corners in the geometry of the system is tested on the three-dimensional mean spherical model. The general case of boundary conditions which are periodic ind 0 dimensions and free or fixed in the remaining 3 –d dimensions is considered. Logarithmic and double-logarithmic size corrections stemming from corners, edges, and surfaces are obtained.     

The self-consistent mean spherical approximation for the one-component plasma

The consequences of choosing the adjustable hard-core diameter in the mean spherical approximation for the one-component plasma so as to achieve thermodynamic consistency between the energy and compressibility equations are investigated. Such a choice is found to be possible only for >8.5 and, although the resulting correlation functions are discontinuous, the height of the main peak in the static structure factor is remarkably accurate. Two especially noteworthy aspects of the thermodynamic results are that the compressibility equation is much more accurate than in any previous approximation free of input from computer simulations and that the nonstatic part of the internal energy has a ^1/4 dependence in the strong coupling limit in agreement with Monte Carlo data.     

Short distance behaviour of the pair correlation function for classical plasmas

The screening effects on two like charges at short distances in a classical plasma are computed using a cluster expansion for the pair correlation function. We obtain numerically accurate results for the intermediate screening region, Λ 1, where Λ is Coulomb interaction parameter.     

Finite-size scaling for the mean spherical model with inverse power law interaction

A new analytical technique based on integral transformations with Mittag-Leffler-type kernels is used to derive the finite-size scaling function for the free energy per particle of the mean spherical model with inverse power law asymptotics of the interaction potential. The asymptotic formation of the singularities in the specific heat and magnetic susceptibility at the bulk critical point is studied.     

Exact solution of the mean spherical model for fluids of non-spherical molecules

The mean spherical model is solved in closed form for a fluid of molecules represented by hard spheres together with short-range angle-dependent repulsion and long-range multipole attraction, with no restriction placed on the parity of the angular functions. This constitutes an extension of a previous report which presented a solution for the same model but with the restriction that the angular terms had even parity.     

Mean spherical model for hard ions and dipoles: Thermodynamics and correlation functions

The solution of the mean spherical model of a mixture of equal-size hard ions and dipoles is reinvestigated. Simple expressions for the coefficients of the Laplace transform of the pair correlation function and the other thermodynamic properties are given.Supported by DOE through the Center of Environment and Energy Research of the University of Puerto Rico and NSF Grant CH 77-14611.     

Susceptibilities and correlation functions of the anisotropic spherical model

The static transverse and longitudinal correlation functions (CF) of a 3-dimensional ferromagnet are calculated for the exactly solvable anisotropic spherical model (ASM) determined as the limit D → ∞ of the classical D- component vector model. The results are nonequivalent to those for the standard spherical model of Berlin and Kac even in the isotropic case. Whereas the transverse CF has the usual Ornstein-Zernike form for small wave vectors, the longitudinal CF shows a nontrivial behavior in the ordered region caused by spin-wave fluctuations. In particular, in the isotropic case below T _c one has S _zz(k) ∝ 1/k (the result of the spin-wave theory) for k ? κ_m ∝ T _c - T.     

The magnetic zener-relaxation and correlation factors in the pair model of chemical reactions

The magnetic Zener-relaxation of a silicon-iron alloy can be explained within the framework of irreversible thermodynamics by a pair model of chemical reactions. In this phenomenological theory two extreme cases are included: isolated silicon pairs in an iron solution and higher silicon complexes. Higher complexes are possible, if in contrast to the first model the surroundings of the vacancy inducing the chemical reactions are thought to be given by randomly distributed silicon or iron atoms. The different surroundings of the vacancy in either model are responsible for different transition frequencies. Only the second model verifies the experimental finding that the transition frequencies depend on the density of silicon in the sample.     

On the asymptotic behavior of the pair correlation function for liquid4He

A new expression for the asymptotic pair correlation function of He II is proposed. Unlike the standard expressions, this contains explicitly the condensate fraction and leans heavily on the corresponding function for He I. The consequences for the long-wavelength liquid structure factor and the asymptotic effective interaction in the system are explored. In passing, the triplet and higher-order correlations are discussed briefly. Whenever possible, the connection with other treatments is pointed out.