John Stephenson

Ising‐Model Spin Correlations on the Triangular Lattice. III. Isotropic Antiferromagnetic Lattice

The asymptotic behavior of the pair correlation ω2(r) = 〈σ0σr〉 between two spins at sites 0 and r on an axis of an isotropic antiferromagnetic triangular lattice is investigated with the aid of the theory of Toeplitz determinants as developed by Wu. The leading terms in the asymptotic expansion are obtained for large spin separation at fixed nonzero temperature. Evidence is presented that the zero‐point behavior of the correlation is of the form ω2(r) ∼ e0r−½ cos ⅔πr, where r = |r| is the spin separation and e0=212(E0T)2=0.632226080…,E0T being the decay amplitude of the pair correlation at the Curie point (critical point) of an isotropic ferromagnetic triangular lattice. A special class of fourth‐order correlations ω4(r) = 〈σ0σδσr σr+δ〉 − 〈σ0σδ〉 〈σrσr+δ〉 between the four spins at sites 0, δ, r, and r + δ on the same lattice axis, where δ is a lattice vector, is reconsidered. The asymptotic form of the correlation for large separation of pairs of spins r = |r| is obtained for all fixed temperatures.

Ising‐Model Spin Correlations on the Triangular Lattice. IV. Anisotropic Ferromagnetic and Antiferromagnetic Lattices

A detailed discussion of pair correlations ω2(r) = 〈σ0σr〉 between spins at lattice sites 0 and r on the axes of anisotropic triangular lattices is given. The asymptotic behavior of ω2(r) for large spin separation is obtained for ferromagnetic and antiferromagnetic lattices. The axial pair correlation for the ferromagnetic triangular lattice has the same qualitative behavior as that for the ferromagnetic rectangular lattice: There is long‐range order below the Curie point TC and short‐range order above. It is shown that correlations on the anisotropic antiferromagnetic triangular lattice must be given separate treatment in three different temperature ranges. Below the Neel point TN (antiferromagnetic critical point), the completely anisotropic lattice exhibits antiferromagnetic long‐range order along the two lattice axes with the strongest interactions. Spins along the third axis with the weakest interaction are ordered ferromagnetically. Between TN and a uniquely located temperature TD, there is antiferro...

Some non-linear diffusion equations and fractal diffusion

Some scaling solutions of a class of radially symmetric non-linear diffusion equations in an arbitrary dimension d are obtained, (A) for an initial point source with a fixed total amount of material, and (B) for a radial flux of material through a hyper-spherical surface. In this macroscopic model the flux density depends on powers of the concentration and its (radial) gradient. The dimensional dependence of these solutions is analyzed and comparison made with scaling solutions of the corresponding linear equations for fractal diffusion. The non-linear equations contain arbitrary exponents which can be related to an effective fractal dimension of the underlying diffusion process.

Partition function zeros for the two-dimensional Ising model III☆

The density of the complex temperature zeros of the partition function of the two-dimensional Ising model on completely anisotropic triangular lattices is investigated near real and complex critical points. The non-uniform behaviour of the density of zeros at interior and boundary critical points is studied analytically, and numerically for a lattice with interactions in the ratios 3:2:1. The limiting behaviour of the density at complex critical points depends on the direction of approach in the complex plane. In the neighbourhood of interior critical points one finds only a single layer of zeros. But generally there are two layers or superposed sets of zeros with different distributions, and different limiting densities at boundary critical points. On anisotropic quadratic and on partially anisotropic triangular lattices the two layers become identical, by symmetry of the partition function. The divergence of the (two- dimensional) density of zeros at real critical points is discussed briefly in relation to scaling theory.

Hard and Soft-Core Equations of State for Simple Fluids: IV. Elementary Theory of Termination Temperatures

Abstract An elementary theory of termination temperatures of characteristic curves is developed with the aid of two simple mathematical models. The first model contains two important constant ratios involving all six termination temperatures. The second model interpolates between the first model and the hard-core limit with the aid of a parametric softening temperature, Ts The properties of the temperature ratios and the underlying models are discussed in detail, especially in the hard-core limit.

Hard and Soft-Core Equations of State for Simple Fluids: II Characteristic Curves for Argon

Abstract A detailed discussion is given of the geometrical properties and the physical significance of the characteristic curves of a simple fluid, with specific reference to experimental results for argon.

Hard and Soft-Core Equations of State for Simple Fluids : III Characteristic Curves and Hard-core Equations of State

Abstract Explicit formulae are presented for those characteristic curves which can be constructed for a hard-core equation of state. Excellent agreement is obtained with experimental curves for argon up to fairly high pressures (20 P c) and temperatures (10 T c). It is shown that only three termination temperatures are obtainable from a hard-core type second virial coefficient. The limitations of the hard-core equation of state are discussed briefly.

Hard and Soft - Core Equations of State for Simple Fluids: V. Termination Temperatures for the Lennard - Jones m, n Potential

Abstract The six termination temperatures associated with the ten characteristic curves of a simple fluid are calculated for the Lennard-Jones m, n potential second virial coefficient with n ≥ m > 3. The extreme values in both the hard-core Sutherland potential limit n → x, and in the opposite limit n → m are obtained. The termination temperature ratios Tc/Tb, TF/TC and TD/TA lie within a narrow finite range, with TD/TA → 2 in the hard-core limit, independent of m. The high temperature form of the second virial coefficient is derived, and used to estimate thesoftening temperature Ts. Also, some results for the square-well potential are presented.

Hard and Soft-Core Equations of State for Simple Fluids. IX. Soft-Core Equations of State and Loci of Cp Extrema

Abstract Loci of Cp extrema along isotherms are constructed for model soft-core equations of state, parameterized by an exponent N(= 3/n, where n is the repulsive potential exponent) and a softening temperature Ts . Generally the loci exhibit two branches whose geometry depends on Ts and N. In soft-core type behaviour, a locus of Cp maxima commences at the critical point and terminates on the temperature axis at a temperature Tn where the second virial coefficient has a point of inflexion, and the second branch is located at higher pressures and temperatures. In hard-core type behaviour, a locus of Cp maxima commences at the critical point and turns into a locus of minima before crossing the fusion curve, whereas the second branch, which terminates at TD , is generally a locus of minima lying at high pressures and temperatures. The values of Ts and N at which the geometry of the loci changes is studied in detail.

On the Critical Region of a Simple Fluid. II. Scaling‐Law Equation of State

The scaling‐law equation of state is used to illustrate some consequences of the elementary index inequalities β + γ− > 1 > α− + β. It is shown that (∂2P / ∂T2)V evaluated on the phase boundary below Tc from the homogeneous‐phase side diverges with index (α + β), where α is the (scaling‐law) index for CV=Vc. Also, it is shown that CP=Pc diverges with index [1 − (1 / δ)] on the critical isobar, where δ is the degree of the critical isotherm.