EXAFS and cumulant expansion studies of an amorphous Se90P10 alloy produced by mechanical alloying

We investigated the local atomic order of an amorphous Se₉₀P₁₀ alloy produced by Mechanical Alloying through EXAFS measurements on Se K edge at five temperatures followed by a cumulant expansion analysis. We obtained a lot of structural information such as average coordination numbers and interatomic distances, structural and thermal disorder, asymmetry of the pair distribution functions g_ij(r), anharmonicity of the interatomic potential, thermal expansion and Einstein and Debye temperatures for the Se-Se and Se-P pairs. We also reconstructed the g_ij(r) functions for the Se-Se and Se-P pairs at the temperatures investigated.     

Clustering of continuous distributions of charge

An investigation is carried out of the association and clustering of equimolar mixtures of oppositely charged Gaussian charge distributions (CDs) of the form ～ exp ( ? r²?/2α²), where r is the separation between the centres of charge and α governs the extent of charge spreading (α→0 is the point charge limit). The results of molecular dynamics (MD) and Ornstein–Zernike integral equation with the mean spherical approximation (MSA) and hypernetted-chain (HNC) closures are compared for these systems. The MD and HNC radial distribution functions, g(r), agree very well for not too small α. The MD and MSA, g(r), also agree well for α ≈ 1 and greater. The potential energy per particle for the three methods also agrees well over a wider range of α values, better than might be expected from inspection of the radial distribution functions, because the dominant contributions to U come predominantly from intermediate and long distance ranges where there is good agreement between the g(r) from the MSD and HNC closures. The nature of the association and clustering of the charges as a function of α is explored through the mean nearest neighbour distance for unlike and like species and the mean and root-mean-square force. The velocity and force autocorrelation functions are also calculated; they show increasingly oscillatory behaviour in the small α limit, originating in vibrations of a pair of CDs of opposite sign.     

The 2‐matrix of the spin‐polarized electron gas: contraction sum rules and spectral resolutions

The spin‐polarized homogeneous electron gas with densities ρ_↑ and ρ_↓ for electrons with spin ‘up’ (↑) and spin ‘down’ (↓), respectively, is systematically analyzed with respect to its lowest‐order reduced densities and density matrices and their mutual relations. The three 2‐body reduced density matrices γ_↑↑, γ_↓↓, γ_a are 4‐point functions for electron pairs with spins ↑↑, ↓↓, and antiparallel, respectively. From them, three functions G_↑↑(x,y), G_↓↓(x,y), G_a(x,y), depending on only two variables, are derived. These functions contain not only the pair densities according to g_↑↑(r) = G_↑uarr;(0,r), g_↓↓(r) = G_↓↓(0,r), g_a(r) = G_a(0,r) with r = | r ₁ ‐ r ₂|, but also the 1‐body reduced density matrices γ_↑ and γ_↓ being 2‐point functions according to γ_s = ρ_sf_s and f_s(r) = G_ss(r, ∞) with s = ↑,↓ and r = | r ₁ ‐ r ^′₁|. The contraction properties of the 2‐body reduced density matrices lead to three sum rules to be obeyed by the three key functions G_ss, G_a. These contraction sum rules contain corresponding normalization sum rules as special cases. The momentum distributions n_↑(k) and n_↓(k), following from f_↑(r) and f_↓(r) by Fourier transform, are correctly normalized through f_s(0) = 1. In addition to the non‐negativity conditions n_s(k),g_ss(r),g_a(r) ≥ 0 [these quantities are probabilities], it holds n_s(k) ≤ 1 and g_ss(0) = 0 due to the Pauli principle and g_a(0) ≤ 1 due to the Coulomb repulsion. Recent parametrizations of the pair densities of the spin‐unpolarized homogeneous electron gas in terms of 2‐body wave functions (geminals) and corresponding occupancies are generalized (i) to the spin‐polarized case and (ii) to the 2‐body reduced density matrix giving thus its spectral resolutions.     

Perturbation theory for the radial distribution function

Perturbation theory is used to consider expansions for the radial distribution function, g ₂(r), of a fluid with a soft core. We consider the Lennard-Jones (12, 6) potential and divide it into repulsive and attractive regions. In the repulsive region we expand the function exp (β u(r))g ₂(r) about a hard sphere value. For the first-order contribution of the attractive region we consider a simple approximation to the exact analytical expression. The resulting g ₂(r) is accurate at densities below about ρσ ³=0·5.     

Limiting behaviour of fluid distribution functions for small distances

An expression for the distribution function y(r) = exp {βu(r)}g(r), where g(r) is the radial distribution function, is obtained in the limit r →0 for a general fluid. The logarithm of y(0) is found to be given by a rapidly convergent series in βε, where β = 1/kT, T is the temperature, and ε is the depth of the potential. Extensions of this result to mixtures and higher-order distribution functions are also given.     

Virial expansion for the radial distribution function of a fluid using the 6 : 12 potential

The radial distribution function can be expressed in a virial expansion. Using the 6 : 12 potential the second-order density coefficient, g ₂(r), is numerically calculated for a wide range of temperatures and intermolecular separations. These results are used to calculate the second-order density coefficient, c ₂(r), in the expansion of the direct correlation function and to calculate the fourth virial coefficient, B ₄. In addition, approximate results for g ₂(r), c ₂(r), and B ₄ are calculated on the basis of the Percus-Yevick, hypernetted chain, and the self-consistent approximations of Hurst and Rowlinson. These approximate results are compared with the exact results. The Percus-Yevick theory is in good agreement with the exact results at high temperatures but is unsatisfactory at low temperatures. The hyper-netted-chain approximation is in fair agreement with the exact results at high temperatures, is in poor agreement at intermediate temperatures, but is in good agreement at low temperatures. The self-consistent approximations are in reasonably good agreement with the exact calculations at all temperatures.     

Exact short time dynamics for steeply repulsive potentials

The autocorrelation functions for the force on a particle, the velocity of a particle and the transverse momentum flux are studied for the power law potential v(r)=ε(σr)^ν (soft spheres). The latter two correlation functions characterize the Green–Kubo expressions for the self-diffusion coefficient and shear viscosity. The short-time dynamics is calculated exactly as a function of ν. The dynamics is characterized by a universal scaling function S(τ), where τ=t/τ_ν and τ_ν is the mean time to traverse the core of the potential divided by ν. In the limit of asymptotically large ν this scaling function leads to delta function in time contributions in the correlation functions for the force and momentum flux. It is shown that this singular limit agrees with the special Green–Kubo representation for hard-sphere transport coefficients. The domain of the scaling law is investigated by comparison with recent results from molecular dynamics simulation for this potential.     

Calculation of the Eigenstates and Eigenvalues for the Power and Inverse-Power Hypercentral Potentials

The bound-state solutions to the hyperradial Schr?dinger equation is constructed for any general case comprising any hypercentral power and inverse-power potentials. The hypercentral potential depends only on the hyperradius which itself is a function of Jacobi relative coordinates that are functions of particle positions (r ₁,r ₂, … , and r _N ). This paper is mainly devoted to the demonstration of the fact that any ψ of the form ψ = power series × exp(polynomial) = [f(x) exp (g(x))] is potentially a solution of the Schr?dinger equation, where the polynomial g(x) is an ansatz depending on the interaction potential.     

Correlated motion of two particles in a fluid

Conditional velocity cross correlation functions of the form <v_i (0)v_j (t); r_ij (0)> in the Lennard-Jones fluid are investigated by molecular dynamics simulation. As shown in previous work, these cross correlation functions may be related to memory functions in a similar manner as the usual velocity auto-correlation function. To compute the memory functions, a modified version of Detyna and Singer's algorithm has been used.     

Non-polar solute-water pair correlation functions

The microscopic theory of the solvation of non-polar solutes in water proposed by Pratt and Chandler has been generalized in order to obtain separate solute-oxygen and solute-hydrogen radial distribution functions, g(r). The g(r)s predicted by this method for a spherical solute have been tested by comparison with the corresponding functions from two computer simulation studies. The water-water interactions were described by the configuration intereaction potential (CI) in both cases. The agreement between theoretical and simulation results is good for the solute-oxygen g(r), less so for the solute-hydrogen function. Moreover, the influence of the model of water on the calculated solute-solvent g(r)s has been examined by comparing results obtained with partial structure functions derived from the CI model and from recent neutron diffraction measurements. It is found that CI model and real water yield remarkably different solute-water radial distribution functions. Finally, the solvation of a model two-site solute has been studied for various bond lengths and the results confirm that when the space between the sites is sufficient to host a water molecule, the solvation of each site is the same as that of an isolated site, with respect to oxygen as well as hydrogen.     

E.S.R. evidence for the planarity of the t-butyl radical

Monte Carlo calculations are reported for the radial distribution function g ₂(r; λ) of a fluid in which the intermolecular pair potential is [u ^ref(r) + λu ^p(r)], u ^ref(r) being the Weeks-Chandler-Andersen (WCA) reference fluid, and [u ^ref(r) + u ^p(r)] being the Lennard-Jones (6, 12) fluid. The calculations are performed for λ values in the range 0 to 1, at the state condition ρσ³ = 0·80, kT/ε = 0·719. It is shown that at high densities the perturbation expansion of g ₂(r; λ = 1) about g ₂(r; λ = 0) is rapidly convergent, but that the corresponding expansion for y ₂(r; λ) = exp [βu(r; λ)] × g ₂(r; λ) is not. In addition Monte Carlo estimates of the individual terms that contribute to the first-order perturbation term, (?g ₂/?λ)_λ=0, are presented. It is shown that these terms are individually large, but that (?g ₂/?λ)_λ=0 is small because there is strong cancellation between the various terms. Consequently, the calculation of (?g ₂/?λ)_λ=0 is highly sensitive to the approximation used to evaluate the individual terms.     

Self-consistent Overhauser model for the pair distribution function of an electron gas at finite temperature

We present calculations of the spin-averaged pair distribution function g(r) in a homogeneous gas of electrons moving in dimensionality D=3 or D=2 at finite temperature. The model involves the solution of a two-electron scattering problem via an effective potential, which embodies many-body effects through a self-consistent Hartree approximation, leading to two-body wave functions to be averaged over a temperature-dependent distribution of relative momentum for electron pairs. We report illustrative numerical results for g(r) in an intermediate-coupling regime and interpret them in terms of changes of short-range order with increasing temperature.     

A simplified approach to kinetic theory of dense fluids

Time correlation functions may be related to the fluctuating force correlation function in the sub-ensemble with fixed initial particle velocity. The latter may be evaluated by following the trajectories of particle pairs in the fluid.     

New explicit nonperiodic solutions of the homogeneous wave equation

We exhibit a newansatz for the solution of the homogeneous three-dimensional time-dependent wave equation in spherical coordinates of the form Φ(r,t)=Y(θ, φ)(I(r)+G(g)), whereg ≡ct/r. FunctionG(g) has explicit solution in terms of three independent nonperiodic functionss _ℓ,t _ℓ,u _ℓ (s _ℓ andt _ℓ are related to the associated Legendre functions of the first and second kinds).G(g) is nonperiodic and may be cast as a superposition of incoming and outgoing waves. To obtainG(g), we solved a nonhomogeneous associated Legendre equation (this solution, to our knowledge, is also new).G(g) may prove useful in many microscopic and macroscopic problems, representable by homogeneous wave equations.     

Molecular dynamics investigation of the size effect upon the β → α transformation in Zr nanocrystals

The stability of the β phase in cubic zirconium nanoparticles has been calculated as a function of the size r (r varies in the range from 2.5 to 11.5 nm) by the molecular dynamics method with the many-body interatomic interaction potential obtained within the embedded-atom model. It has been demonstrated that the temperature T _k at which the cubic cluster of body-centered cubic zirconium becomes structurally unstable depends nonlinearly on the particle size. The curve T _k (r) exhibits a pronounced maximum in the range r ≈ 4.3−4.7 nm. It has been established that the mechanism of the structural transition from the body-centered cubic phase to the hexagonal close-packed phase depends substantially on the particle size. For particles with sizes in the range from 2.5 to 5.0 nm, there exists a temperature range in which the transition from the body-centered cubic phase to the hexagonal close-packed phase remains incomplete for a long time. In this case, two phases coexist and the initial particle undergoes a strong deformation along the habit plane.     

Static structure factor of electrons around a heavy positively charged impurity in a one-component quantum rare plasma

An expression for the static structure factor,g ⁺⁻ (r), of electrons at a distancer from an infinitely heavy positively charged particle in a one component quantum rare plasma has been obtained in linear response theory using an appropriate quantum dielectric function of the rare plasma. The expression is a complicated function of the electron plasma frequency, Debye screening length andr, but reduces to that of classical plasma when quantum corrections are neglected. Forr<r _s (2r _s being the mean distance between two electrons), the temperature dependentg ⁺⁻ (r) has larger values in quantum case in comparison to that in classical situation and keeps increasing with decrease inr, more so at low temperatures when de-Broglie wavelength becomes larger and a considerable fraction ofr _s.     

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.     

Effect of short and long range forces on the structure of water. II. Orientational ordering and the dielectric constant

The effects of short and long range interactions on the structure of water, both spatial and orientational, has been studied in detail by computing the full pair correlation function, site-site correlation functions, 2-dimensional site-site correlation functions in the (r _OO, r _OH) and (r _OO, r _HH) planes, dipole-dipole correlation function, radial Kirkwood g factor, and the dielectric constant. Two model potentials, the T1P4P and ST2, and their short range versions have been considered at ambient and elevated temperatures and under supercritical conditions. The Ewald summation under different conditions has been used to investigate also their effect on results. An analysis of the results shows that although all site-site correlation functions for the short and long range systems are similar, the orientational ordering in systems of different range may be considerably different, this evidence being provided mainly by the dipole-dipole correlation function and the radial Kirkwood factor. The orientational ordering is only short range in long range systems, whereas in short range systems the hydrogen bonding gives rise to a damped long range regular pattern of alignment. Nonetheless, the resulting dielectric constants for the short and long range systems coincide within the combined error bars. All findings are more pronounced at low temperatures but otherwise they are only marginally temperature and density dependent.     

Calculation of bridge function coefficients via modified Mayer-samplings

Two modified Mayer-sampling methods are described based on Transition Matrix Monte Carlo (TMMC) and overlap sampling for calculating the integrated diagrams appearing in the coefficients of the bridge function; b_n (r) constructed in terms of the total correlation function h(r) for a hard-sphere system. Calculations are performed using prescribed h(r) for reduced densities at 0.2, 0.5 and 0.8 up to the third-order expansion in density. The results from these methods compared with the generic Monte Carlo Mayer-sampling are analysed in detail. It was found that the TMMC Mayer-sampling approach shows better precision over core and tail regions of b ₂(r) and b ₃(r), and the overlap sampling method shows overall improvement in the precision of the bridge coefficients. Both methods can be straightforwardly applied to calculate higher order bridge function coefficients and to any model systems with relatively simple modifications.