Alain J. Phares

The occupation statistics for indistinguishable dumbbells on a rectangular lattice space. I

The method of Hock and McQuistan used recently to solve the occupation statistics for indistinguishable dumbbells (or dimers) on a 2×2×N lattice is extended further to obtain, for the L×M×N lattice, general expressions for the normalization, expectation, and dispersion of the statistics, and their limit as N becomes very large. In particular, an explicit expression of the partition function in the thermodynamic limit Ξ(x) is obtained for any value of the absolute activity x of dimers. The developed mathematical formalism is then applied to planar lattices, 1×M×N, with M=1, 2, 3, and 4. The known results for M=1 and 2 are recovered, and some new ones are obtained. The recurrence relation for the number A(q,N) of arrangements of q dumbbells on a 1×M×N lattice which has 3 and 5 terms when M=1 and 2, respectively, is found to have 15 and 65 terms for M=3 and 4. Analysis and extrapolation of the results enable one to predict the expectation 〈θ〉1MN on a planar 1×M×N lattice to be 63.4%, in the limit as both M a...

Thermodynamics of dimers on a rectangular L×M×N lattice

The exact closed‐form analytic solution of the problem of dimers on infinite two‐dimensional and three‐dimensional lattices is obtained. Entropy, isothermal compressibility, and constant pressure heat capacity of the system are given in terms of the normalized number density of dimers. The absolute activity of dimers is also given in terms of the normalized number density; it exhibits a behavior near close packing with a critical exponent exactly equal to 2, and with an amplitude 1/(4φ), where φ is the molecular freedom per dimer at close packing.

Decoupling of a system of partial difference equations with constant coefficients and application

Consider D multi‐variable functions, Aj(n), j=1 to D, where n stands for the evaluation point in the associated multi‐dimensional space of coordinates (n1,n2,...). Let us assume that the Aj’s satisfy a system of D linearly coupled finite difference equations: the value of each function Ai at the evaluation point n is given as a linear combination of the values of this function and others at shifted evaluation points. By introducing D suitable generating functions, Gj, j=1 to D, one is able to replace the D coupled difference equations by a system of D linear equations where the Gj’s play the role of the D unknowns. After solving this new system of equations, it is then possible to construct a difference equation for each of the Aj’s relating the value of Ai at the evaluation point n to the values of Ai itself at shifted arguments. The solution of such a decoupled equation can then be handled using the multi‐dimensional combinatorics function technique.

An approximate solution of the monomer–dimer problem on a square lattice. II

The mathematical method developed in paper I is applied to obtain the partition function and thermodynamical properties of the monomer–dimer problem for a square lattice in terms of the absolute activity x. We also obtain by extrapolation an approximate expression of the partition function which is accurate to better than 0.1% in the range 0≤x≤10. The expectation of the statistics 〈θ(x)〉 is calculated in two different ways and numerical results agree to better than 3%, thus showing the consistency of the underlying mathematical method. Consistent with earlier studies, there is no phase transition. Approximation methods used in earlier work are also found to be in good agreement with our analytic study.

Thermodynamics and molecular freedom of dimers on plane triangular lattices

This is an extension of three previous papers dealing with dimers on rectangular lattices (one, two, and three dimensions). The technique presented in the first paper in this series continues to be fruitful for dimers on plane triangular lattices. Entropy, isothermal compressibility, constant pressure heat capacity, and molecular freedom per dimer at close packing are obtained exactly for lattices infinite in one direction and finite in the other. Observations made in the third paper of the series concerning molecular freedom per dimer at close packing on rectangular lattices are used to extrapolate our results to infinite plane triangular lattices. At close packing, the molecular freedom per dimer on an infinite plane triangular lattice is calculated to be 2.356 527... in agreement with the value obtained by Nagle. Based on our earlier findings, the value of 2.356 527... was used to obtain the analytic fit for the thermodynamic quantities in terms of the normalized number density.

The entropy curves for interacting dimers on a square lattice

Abstract We report the presence of cusps in the curves of the entropy versus the coverage for nearest neighbour interacting dimers on a semi-infinite square lattice.

Monomer adsorption on bcc (110) surfaces with first- and second-neighbor interactions

The low-temperature phases of monomer adsorption on body-centered-cubic (110) surfaces, infinite in one direction and of finite width, M, with non-periodic boundaries, are obtained with first- and second-neighbor interactions. These ordered phases, far below the critical temperature, are observed at values of the external gas pressure within specific ranges, and are characterized by M and the values of the interaction energies. One finds two distinct sets of phases, depending on whether first-neighbor interactions are repulsive or attractive. As in the case of the square lattice, most of the numerical results fit exact closed-form expressions in M, allowing analytic extrapolations to the infinite twodimensional surface (M=2). The conditions under which a transition from one ordered phase to another ordered phase are also determined, depending on M and the first- and second-neighbor interaction energies. To the extent to which these interactions are predominant, the model shows that it is possible to obtain them from an experimental observation of the phases and the external gas pressure at which transitions between phases occur. The finite width results are of particular interest when considering adsorption on terraces.

COADSORPTION OF MONOMERS AND DIMERS WITH FIRST NEIGHBOUR INTERACTIONS ON SQUARE LATTICES

Abstract We model surface coadsorption on a square lattice infinite in length and of finite width M . The adsorbed species occupy either one site (monomers) or two nearest neighbour sites (dimers) and have first neighbour interactions. The quantities computer per site at thermal equilibrium are the entropy, the number of each species, and the numbers of first neighbours (monomer-monomer, dimer-dimer, monomer-dimer). Numerical results are obtained for M ≤ 5. Structural orderings are observed and the detailed analysis of the corresponding phase diagrams is presented. Following our analysis, the boundary lines between phases are found to be linear for all M , and their exact analytic expressions are obtained for arbitrary values of the interaction energies. A number of extrapolations for any value of M is possible leading to predictions on the infinite two-dimensional lattice.

Adsorption on bcc (110) surfaces with first-, second-, and third-neighbor interactions

Abstract Third-neighbor adsorbate–adsorbate interactions are included to extend the low-temperature adsorption study recently published on body-centered-cubic (110) surfaces, infinite in one direction and of finite width, M, with non-periodic boundaries. In addition to the six phases reported earlier on the infinite two-dimensional surface (M→8), 11 new phases have been found for an exhaustive total of two at (1/4) coverage, two at (1/3), four at (1/2), five at (2/3), and four at (3/4) coverage. Comparison with the experimental data focuses mainly on the adsorption of bromine on Cr (110), the chemisorbed system of atomic hydrogen on Fe (110), and the adsorption of oxygen atoms on W (110).

The linear potential wavefunctions

We present an exact analytic solution to the Schrodinger equation for two particles interacting via a central linear potential of the from V (r) =V0+kr. The solution is given in terms of the generalized hypergeometric functions, and is especially useful when discussing problems where the radial parameter is small.