Moments of the Percus-Yevick hard-sphere correlation function

A simple recursive relation is derived for the momentsM _n ,n=1, 2,..., of the Percus-Yevick correlation functionh(r) for identical hard spheres. TheM _n are rational functions of the volume fractionw occupied by the spheres; the first ten are given explicitly, and a single-term asymptotic form is obtained to suffice for the rest. Applications of theM _n(w) include testing different approximations forh by numerical integration ofh(r) r ⁿ . We compare exact moments with shell approximationsM _n [h ^s ] corresponding to integration fromr=0 tos+1 fors=3–8, and with hybrid approximationsM _n [h ^s +h ^a ] which supplement the shell approximations with integrals of an asymptotic tail froms+1 to . For a givens, the hybrid approximation is better forw increasing than the shell approximation, andM _n [h ³+h ^a ] is even better thanM _n [h ⁸]     

Analytical representation of the Percus-Yevick hard-sphere radial distribution function

Explicit analytical expressions, written in terms of complex variables and suitable for rapid computer evaluation, are presented for the Percus-Yevick hard-sphere radial distribution function, g(R), for R ˇ- 5σ. Some effects of truncating g(R) to unity past R = 5 σ are discussed.     

Direct correlation function of the hard-sphere fluid

Two analytic approximations for the direct correlation function of a hard-sphere fluid are considered. The first follows from a generalization of the Percus–Yevick result in d dimensions, whereas the second arises in the Rational Function Approximation (RFA) method. Both approximations require the equation of state of the hard-sphere fluid as input. The results, derived after use of the Carnahan–Starling and the Padé 4 Ripoll, MS and Tejero, CF. 1995. Molec. Phys., 85: 423[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar],3 Wertheim, MS. 1963. Phys. Rev. Lett., 10: 321 J. Math. Phys. 5, 643 (1964); E. Thiele, J. Chem. Phys. 39, 474 (1963).[Crossref], [Web of Science ®] , [Google Scholar] equations of state in both approaches, are compared with simulation data. The comparison shows that the first approximation is rather accurate in the region inside the core, but inherits the limitation of the Percus–Yevick theory for distances beyond the hard-sphere diameter. On the other hand, the results of the RFA method are also accurate inside the core and capture well the initial part of the tail beyond the hard-sphere diameter, but fail to account for the subsequent oscillations observed in the simulations. Other merits and limitations of the two approaches are reported.     

Calculation of the radial distribution function of hard-sphere mixtures in the Percus-Yevick approximation

A formulation is given which permits the rapid mechanical computation of the three radial distribution functions g_ij (r) of a binary hard-sphere mixture to any distance r, in the Percus-Yevick (P-Y) approximation. The consistency of the P-Y equation of state obtained by various methods is discussed.     

Callan—symanzik β(g) function in different topological sectors

Using the formalism developed by Amati and the present author for constructing a perturbation theory around an instanton in gauge theories, it is proved that the Callan—Symanzik β(g) function is the same as in the perturbation theory developed around zero.     

Pole expansion determination of the pair correlation function of a simple fluid

An explicit expression for the pair correlation function is given in terms of an expansion in the poles of the structure factor S(q) in the complex q plane. This is derived when the direct correlation function or structure factor is of known analytical form. An illustrative example is given for a hard-sphere-Yukawa fluid in the mean spherical approximation.     

An efficient method for calculating the pair distribution functions of a ternary hard-sphere system in r space within the Percus-Yevick approximation

We present a direct method which allows an accurate and—at the same time—economical calculation of the pair distribution functions (PDFs) g_ij (r) of an additive ternary hard-sphere system within the Percus—Yevick approximation. The approach is based on the fact that for this approximation the Laplace transforms [?_ij (s)] of the PDFs are known analytically, so that the inversion of the ?_ij (s) into r space can be performed exactly. The expressions presented here allow the determination of the ?_ij (r) for r values up to 8R ₁, R ₁ being the diameter of the smallest species; this range in r space should be sufficient for applications in standard algorithms of liquid state theory, such as thermodynamic perturbation theories or integral-equation approaches.     

New measurement of (g−2) of the Muon

The g-factor anomaly, a≡(g?2)/2, has been measured for μ⁺ in the new Muon Storage Ring at CERN. The result is a = (1 165 895 ± 27) × 10^?9. This is (13 ± 29) × 10^?9 below the theoretical value which includes sixth-order QED terms and a hadronic contribution of (73 ± 10) × 10^?9.     

Quantum perturbations to the radial distribution function of a hard-sphere fluid

Expressions derived in the previous paper for quantum corrections to the radial distribution function of a fluid are applied to the hard-sphere fluid. It is found that the perturbation theory given in the paper is valid only at very high values of temperatures when applied to calculate the correction to the distribution function of the hard-sphere fluid and is not valid at the temperatures at which Gibson and others have obtained numerical results for a given value of ?a³, where ? is the number density and a the hard-sphere diameter.     

Monte Carlo calculation of y(r) for the hard-sphere fluid

The function y(r) = exp {βu(r)}g(r) is calculated for hard spheres in the region r < σ using umbrella-sampling Monte Carlo techniques. The resulting values are found to be well represented over the entire range 0 < r < σ by a simple function proposed by Grundke and Henderson.     

The δ-function expansion of the modified two-particle Ursell function of a hard-sphere fluid

The expansion of the modified two-particle Ursell functionU(r) of a hardsphere quantal fluid is obtained in terms of a series of derivatives of δ-function. This expansion has been used to expand the second virial co-efficientB ₂ of the fluid. The expansion is correct up to the fourth power in thermal wavelength and the terms of the order of λ⁸ and λ⁴ in the first expa nsion are new.     

The pair correlations down to r = 0 in a fully ionized dilute and homogeneous hydrogen plasma in equilibrium

The pair correlations in a fully ionized dilute and homogeneous hydrogen plasma in equilibrium are calculated down to r = 0 on the basis of the equilibrium hierarchy of statistical mechanics and the linear superposition approximation. Thereby the Coulomb potential between electron and proton is modified by a damped potential with finite value at r = 0 which is due to quantum corrections given by Hagenow and Koppe. The solution of the hierarchy equations is quite simple: For $\text{r > γ}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{l}{}_{\mathrm{D}}\text{(γ = 5 × plasma parameter, l}{}_{\mathrm{D}}\text{= Debye-length)}$ one obtains the well-known result by Debye and Hückel, for $\text{r < γ}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{l}{}_{\mathrm{D}}$ the pair distribution functions are simply given by the corresponding Boltzmann factors. The calculation of the energy of the system leads to the result that the nonideal part is coming from the Debye-Hückel part of the pair correlations only.     

Transformation of the pair correlation function of galaxies near a caustic

The effect of large-scale caustics in nondissipative dark matter on the pair correlation function of galaxies is investigated. It is shown that if the initial correlation function of the galaxies is of a power-law form, then the presence of caustics in the observation region does not change the form of the function but only decreases its amplitude. Pis’ma Zh. éksp. Teor. Fiz. 65, No. 8, 565–570 (25 April 1997)     

Crystal field effect on the magnetic moment's direction of rare earth ions with low point symmetry in r4co3 (r = Tb, Ho, Dy, Er, Tm) compounds

Neutron diffraction measurements, made on powder samples, show that Ho₄Co₃ and Er₄Co₃ intermetallic compounds are ferrimagnetic at 4.2 K. The magnetic moments of the 2 holmium sites are 8.7 and 2.1 μ_B and those of the erbium sites are equal to 8.7 and 8.1μ_B. The cobal⁺ magnetic moment is 0.2μ_B for both compounds. The easy magnetization direction lies on the hexagonal plane for Ho₄Co₃ while for Er₄Co₃ there are 2 anisotropy directions. Exchange interactions between rare-earth ions of both sites are very weak compared with the total crystal field splitting of the ground state multiplet J. The crystal field parameters are calculated and the magnitude and direction of the rare-earth magnetic moments in each site is determined.     

On the isothermal density derivative ofG(r) and a new theory of the pair correlation function of hard spheres

A new representation is obtained for the isothermal density derivative ofg(r). It explicitly exhibits the contributions of potential energy terms that are not pairwise additive. Consideration of a previously known result shows that one has to be rather cautious when using it to obtain information on the triplet correlation function from the well-known relation between this function andg/, due to large cancellations which take place at high density. By integrating with respect to density the new representation forg/, after a suitable closure has been introduced, we obtain an augmented Percus-Yevick equation for hard spheres which has full thermodynamic consistency. The equation of state and the cavity functiony(r) are very accurate at low density and considerably improve PY at medium density, so that this appears to be a useful new approach to the theory of fluids, but it is necessary to improve the closure in order to treat a dense fluid.This paper is dedicated to Jerry Percus on the occasion of his 65th birthday.     

Critical behavior of the correlation function of a coulombic fluid

An extension of the Ornstein-Zernike theory of critical scattering by a simple fluid to include a type of coulombic system is suggested. The relation between the oscillations of the charge distribution predicted by the second moment condition of Stillinger and Lovett in the restricted primitive model and the critical behavior of the correlation function is also discussed.This work was supported in part by a grant from the National Institutes of Health, GM 20800-03.Contribution No. 3100 from the Department of Chemistry, Indiana University.Supported by CONACYT (Mexico) and on leave from Universidad Autónoma de Puebla, Puebla, Mexico.     

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 inverse problem for the third order equation uxxx + q(x)ux + r(x)u = −iζ3u

The spectral problem u_xxx + q(x)u_x + r(x)u = ?iξ³u is considered. A set of spectral data which is sufficient for the reconstruction of the potentials q(x) and r(x) is found and the problem of this reconstruction, this inverse problem solved.