Radial distribution function and second virial coefficient for interacting bosons

We have studied the radial distribution function and the second virial coefficient of interacting bosons. The second virial coefficient has been deduced theoretically and is in good agreement with experimental values. The third virial coefficient has been calculated from the experimental values of the pressure.     

Delta-function expansion of Mayer function with application to virial coefficients

The Mayer cluster integrals of a fluid with smooth, repulsive interactions are expanded in orders of a well-defined softness parameter. To first but not second order in softness, all virial coefficients are given by their hard-sphere forms with an effective diameter. A closed asymptotic expression is derived for the third virial coefficient which gives excellent results for the inverse power and exponential potentials.     

Diverging seventh virial coefficient of sticky discs

The seventh virial coefficient of a two-dimensional system of particles interacting with a hard-core square-well pair potential is studied. The Ree-Hoover type cluster integrals were examined and it was found that a graph in the form of a hexagonal wheel with all the bonds of the attractive square-well type does not allow Baxter's ‘sticky sphere’ limit to be achieved. The value of that particular cluster integral was calculated. It was shown that when approaching the sticky limit the cluster integral corresponding to the hexagonal wheel diverges linearly with the height of the peak in the Mayer f function at the location of the potential square-well. As a consequence, the seventh virial coefficient of the sticky disc system does not have a finite value.     

Non-universal thermodynamics of a strongly interacting inhomogeneous Fermi gas using the quantum virial expansion

Within a quantum virial expansion, we investigate theoretically the violation of universal thermodynamics for a strongly interacting unitary Fermi gas trapped in a harmonic potential. The violation is caused by the existence and anisotropy of the trapping potential and a finite-range of the two-body interaction. We calculate the second virial coefficient by solving a two-fermion problem in 3D uniform harmonic traps, as well as in anisotropic traps. In the unitarity limit, the universal value of the trapped second virial coefficient is 1/4. We discuss in detail the non-universal correction to the second virial coefficient and to the equation of state.     

H-H and C-H coupling constants in neutral and protonated azines

New measurements are reported of the density dependent depolarization ratio for argon, krypton, xenon, methane and sulphur hexafluoride, and the results are analysed to provide values for the second and third depolarization virial coefficients. The relationships between the second depolarization virial coefficient, the zeroth moment of the two-body Rayleigh spectrum and the second Kerr virial coefficient are considered, and it is shown that they now provide consistent results for the collision-induced pair polarizability anisotropy. Former inconsistencies are attributed to insufficient allowance for the effects of three-body interactions. Calculations of the second and third depolarization virial coefficients based on the DID model and using the Maitland-Smith potential are in excellent agreement with the experimental results for argon, krypton and xenon.     

Critical properties of non-spherical molecule fluids from the virial expansion

Critical constants of pure fluids (as important reference data in constructing vapour-liquid phase diagrams and basic input of various estimation methods) were determined for systems of non-spherical Kihara molecules; values of the critical temperature, density, compression factor and pressure of fluids composed of prolate and oblate molecules were evaluated from the fourth-order virial expansion. The second and third virial coefficients of the Kihara molecules were determined by applying the recently proposed method in which the effect of molecular core geometry and functional dependence of a pair interaction on the surface-surface distance are factorized and the former contribution determined from a formula for the corresponding hard convex body virial coefficient. The virial expansion for non-spherical Kihara molecules is applied to determine the critical constants of n-alkanes (methane to octane) and cyclic hydrocarbons (cyclopentane, cyclohexane, benzene and naphthalene); a fair agreement with experimental data was found.     

On the formal equivalence of the Frisch-Berne and Watts perturbation theories to evaluate transport coefficients

A system of hard spheres in contact with a structureless wall is considered. The potential between the wall and the particles is of Lennard-Jones type. The second, the third and the fourth three-dimensional virial coefficient of the adsorption isotherm are calculated. The result being that over an extremely large temperature range the two-dimensional virial treatment is a rather good approximation. Conclusions are drawn about the deviations from Henry's law and the coverage, which can be described by virial expansions. It is doubted whether, from experimentally determined virial coefficients, information about the potential between the adatoms can be extracted.     

Geometry of three convex bodies applicable to three-molecule clusters in polyatomic gases

An expression for the configuration integral for three overlapping convex bodies, which is a generalization of Hadwiger-Isihara's formula for two convex bodies, has been found. As an application of this expression, two- and three-molecule cluster integrals (or second and third virial coefficients) for polyatomic molecules in gases are discussed on the basis of a squarewell potential with convex cores.     

Evaluating virial coefficients for multicomponent mixtures: hard sphere mixtures and flexible chains

A new algorithm to compute the virial coefficients of multicomponent mixtures is proposed. The number of graphs that must be evaluated increases dramatically in a multicomponent mixture so that it becomes difficult to enumerate and compute all possible graphs. However, once all of them are known and evaluated, the virial coefficient of the mixture can be evaluated for any composition. If one is interested in the virial coefficient of a mixture of a certain composition, then a simpler approach can be followed. Starting from the graphs of a pure fluid, we assign a random chemical identity to each of the molecules of the graph. The probability of assigning a given chemical identity is taken from the composition of the mixture. In this way composition is treated as a random variable within the Monte Carlo procedure which determines the virial coefficient. The algorithm is checked by comparison with the virial coefficients of binary hard spheres mixtures which are well known. Good agreement is found. The procedure is then extended to multicomponent mixtures of hard spheres. Finally the procedure is applied to the determination of the virial coefficients of a flexible molecule. For flexible molecules the possible configurations of the molecules are treated as different components of the mixture. In this way we present what appears to be the first determination of the third and fourth virial coefficients of polymers in the continuum.     

Universality in a 2-component fermi system at finite temperature

Thermodynamic properties of a Fermi system close to the unitarity limit, where the 2-body scattering length a approaches +/-infinity, are studied in the high temperature Boltzmann regime. For dilute systems the virial expansion coefficients in the Boltzmann regime are expected, from general arguments, to be universal. A model independent finite temperature T calculation of the third virial coefficient b3(T) is presented. At the unitarity limit, b3infinity approximately 1.11 is a universal number. The energy density up to the third virial expansion is derived. These calculations are of interest in dilute neutron matter and could be tested in current atomic experiments on dilute Fermi gases near the Feshbach resonance.     

On the quantum corrections to the virial coefficients of the equation of state of a fluid

The first quantum correction to the virial coefficients of the equation of state of a fluid is derived in the presence of a weak three-body potential ?(i, j, k). Results for the third and fourth virial coefficients are given. Representing the potential energy of interaction of a pair and a triplet, by the Lennard-Jones (12-6) model and the triple dipole dispersion potential model of Axilrod and Teller, the first quantum correction to the third virial coefficient is calculated for many values of T*. The theoretical result is compared with the experimental data of helium.     

The second and third virial coefficients of a two-dimensional gas

Numerical values of reduced second and third virial coefficients and their temperature derivatives are reported for a two-dimensional gas, for use in interpreting adsorption isotherms of monolayers on isoenergetic substrates. Correct values of the third virial coefficient have not been available before. Examples of earlier treatments of adsorption data are given to demonstrate that values of the higher virial coefficients are essential for adequate discussions of adsorption isotherms. When these are used the high-temperature adsorption isotherms of argon on graphitized carbon black in the monolayer region are completely described in terms of the unperturbed bulk-gas parameters.     

Continuous Particles in the Canonical Ensemble as an Abstract Polymer Gas

We revisit the expansion recently proposed by Pulvirenti and Tsagkarogiannis for a system of N continuous particles in the Canonical Ensemble. Under the sole assumption that the particles interact via a tempered and stable pair potential and are subjected to the usual free boundary conditions, we show the analyticity of the Helmholtz free energy at low densities and, using the Penrose tree graph identity, we establish a lower bound for the convergence radius which happens to be identical to the lower bound of the convergence radius of the virial series in the Grand Canonical ensemble established by Lebowitz and Penrose in 1964. We also show that the free energy can be written as a series in powers of the density whose k-th order coefficient coincides, modulo terms o(N)/N, with the k-th order virial coefficient divided by k+1, according to its expression in terms of the m-th order (with m≤k+1) simply connected cluster integrals first given by Mayer in 1942. We finally give an upper bound for the k-th order virial coefficient which slightly improves, at high temperatures, the bound obtained by Lebowitz and Penrose.     

Thermodynamics of the Stockmayer fluid in an applied field

The thermodynamic properties of the Stockmayer fluid in an applied field are studied using theory and computer simulation. Theoretical expressions for the second and third virial coefficients are obtained in terms of the dipolar coupling constant (λ, measuring the strength of dipolar interactions as compared to thermal energy) and dipole–field interaction energy (α, being proportional to the applied field strength). These expressions are tested against numerical results obtained by Mayer sampling calculations. The expression for the second virial coefficient contains terms up to λ⁴, and is found to be accurate over realistic ranges of dipole moment and temperature, and over the entire range of the applied field strength (from zero to infinity). The corresponding expression for the third virial coefficient is truncated at λ³, and is not very accurate: higher order terms are very difficult to calculate. The virial coefficients are incorporated in to a thermodynamic theory based on a logarithmic representation of the Helmholtz free energy. This theory is designed to retain the input virial coefficients, and account for some higher order terms in the sense of a resummation. The compressibility factor is obtained from the theory and compared to results from molecular dynamics simulations with a typical value λ = 1. Despite the mathematical approximations of the virial coefficients, the theory captures the effects of the applied field very well. Finally, the vapour–liquid critical parameters are determined from the theory, and compared to published simulation results; the agreement between the theory and simulations is good.     

Two Hard Spheres in a Spherical Pore: Exact Analytic Results in Two and Three Dimensions

The partition function and the one- and two-body distribution functions are evaluated for two hard spheres with different sizes constrained into a spherical pore. The equivalent problem for hard disks is addressed too. We establish a relation valid for any dimension between these partition functions, second virial coefficient for inhomogeneous systems in a spherical pore, and third virial coefficients for polydisperse hard spheres mixtures. Using the established relation we were able to evaluate the cluster integral b ₂(V) related with the second virial coefficient for the Hard Disc system into a circular pore. Finally, we analyse the behaviour of the obtained expressions near the maximum density.     

Critical temperature of infinitely long chains from Wertheim's perturbation theory

Wertheim's theory is used to determine the critical properties of chains formed by m tangent spheres interacting through the pair potential u(r). It is shown that within Wertheim's theory the critical temperature and compressibility factor reach a finite non-zero value for infinitely long chains, whereas the critical density and pressure vanish as m ^-1.5. Analysing the zero density limit of Wertheim's equation or state for chains it is found that the critical temperature of the infinitely long chain can be obtained by solving a simple equation which involves the second virial coefficient of the reference monomer fluid and the second virial coefficient between a monomer and a dimer. According to Wertheim's theory, the critical temperature of an infinitely long chain (i.e. the Θ temperature) corresponds to the temperature where the second virial coefficient of the monomer is equal to 2/3 of the second virial coefficient between a monomer and dimer. This is a simple and useful result. By computing the second virial coefficient of the monomer and that between a monomer and a dimer, we have determined the Θ temperature that follows from Wertheim's theory for several kinds of chains. In particular, we have evaluated Θ for chains made up of monomer units interacting through the Lennard-Jones potential, the square well potential and the Yukawa potential. For the square well potential, the Θ temperature that follows from Wertheim's theory is given by a simple analytical expression. It is found that the ratio of Θ to the Boyle and critical temperatures of the monomer decreases with the range of the potential.     

Scattering problem with nonlocal separable potential of rank-two: Application to statistical mechanics

In this paper, we present a formulation of statistical mechanics of a thermodynamic system consisting of free particles and independent correlated pairs interacting via nonlocal potential in terms of the scattering properties. Some quantum statistical properties such as energy, heat capacity, second virial coefficient, virial pressure and quantum correction of kinetic energy are described analytically. The difference between the resolvents of the interacting and free Hamiltonians, represented as , that is associated with particle correlations is used for the evaluation of the properties. The statistical properties are related to correlated states, when making a pole expansion of the analytically continued momentum matrix element of . The present work illustrates these relations for a three-dimensional nonlocal separable potential of rank-two.     

Cluster expansion for the dielectric constant of a polarizable suspension

We derive a cluster expansion for the electric susceptibility kernel of a dielectric suspension of spherically symmetric inclusions in a uniform background. This also leads to a cluster expansion for the effective dielectric constant. It is shown that the cluster integrals of any order are absolutely convergent, so that the dielectric constant is well defined and independent of the shape of the sample in the limit of a large system. We compare with virial expansions derived earlier in statistical mechanics for the dielectric constant of a nonpolar gas. In these expansions the virial coefficients are given by integrals which are only conditionally convergent.     

Sedimentation equilibria of ferrofluids: II. Experimental osmotic equations of state of magnetite colloids

The first experimental osmotic equation of state is reported for well-defined magnetic colloids that interact via a dipolar hard-sphere potential. The osmotic pressures are determined from the sedimentation equilibrium concentration profiles in ultrathin capillaries using a low-velocity analytical centrifuge, which is the subject of the accompanying paper I. The pressures of the magnetic colloids, measured accurately to values as low as a few pascals, obey Van 't Hoff's law at low concentrations, whereas at increasing colloid densities non-ideality appears in the form of a negative second virial coefficient. This virial coefficient corresponds to a dipolar coupling constant that agrees with the coupling constant obtained via independent magnetization measurements. The coupling constant manifests an attractive potential of mean force that is significant but yet not quite strong enough to induce dipolar chain formation. Our results disprove van der Waals-like phase behavior of dipolar particles for reasons that are explained.     

An approximate multi-property empirical isotropic interatomic pair potential for the krypton dimer

An approximate empirical isotropic interatomic potential for krypton interaction is developed by simultaneously fitting the Morse-Morse-Morse-Spline-van der Waals potential form to the pressure second virial coefficient, viscosity, thermal conductivity and depolarized interaction-induced light scattering data. Absolute zeroth and second moments of the two-and three-body spectra, the pressure third virial coefficient and isotopic thermal diffusion factor have been measured and compared with theoretical calculations using various models for the interatomic potential. The results show that it is the most accurate potential yet reported for this system. The use of the new potential in lattice sum calculations yields good results for several properties of solid krypton.