On the electrostatic potential due to a slab shaped and a semi-infinite NaCl-type lattice

Consider N layers of a NaCl-type ionic lattice such that in every layer one has an infinite square lattice of positive and negative unit point charges. We present formulae in which the electrostatic potential in an arbitrary field point is expressed as a sum of two rapidly converging lattice sums. For N→∞ we obtain formulae applicable for a semi-infinite lattice.     

On the force of electrostatic interaction between two conducting spheres

A compact solution is obtained to the problem on the force of interaction between two conducting spheres with preset charges on their surfaces in zero external field. The derivation is based on exact solution of the problem of the potential distribution in the bispherical coordinate system. The expression for the force was derived by differentiating the potential energy of interaction between the spheres with respect to the distance between their centers. It is shown using numerical calculations that with decreasing distance between the spheres, the ratio of their charges for which the forces of interaction between the charges are zero tends to the ratio of the charges of contacting spheres. It follows hence that for any ratio of charges of the same polarity, which differs from the ratio of charges of the contacting spheres, there always exists a small distance between the spheres, at which they attract each other.     

Multiple Image Method for the Two Conductor Spheres in a Uniform Electrostatic Field

A method using multiple mirror images of point charges is put forward to analyze the polarization of two identical conductor spheres in a uniform electrostatic field. By use of the method, the electric field distribution and the interaction force between two spheres can be calculated accurately even for very small gap between two spheres. Our results show that the magnitude of the product of the gap between two spheres and the local electric field in the center of the gap is approximately in the same order and the interaction between two spheres increases very fast as the two spheres are close to each other. We also show that the interaction force between two conductor spheres is almost same with that between two dielectric spheres with high permittivity.     

The one particle theory of periodic point interactions

We solve explicitly and without approximation the problem of a quantum-mechanical particle inR ³ subjected to point interactions that are periodic inR ³ with periodicity of the typeZ, Z ², andZ ³. In the first case we get a model of an infinite straight polymer, in the second case we get a model of a monomolecular layer and in the third case we get a model of a crystal. In all three cases the unit cell of the Bravais lattice is allowed to contain any finite number of interaction sites (atomes), placed arbitrarily and with arbitrary interaction strength. In the case: one interaction site per unit cell we find explicit formulas for the resonance bands and energy bands and their corresponding wavefunctions.     

A dipole approximation for a dielectric mixture based on the equal field energy method

In this paper, a dipole-energy approximation for calculating the electric field distribution and saturation charge of spheres in an infinitely large dielectric mixture has been carried out. The approximation rests on the assumption that the field energy of mixture calculated using two different methods have the same value. One way is considering that the mixture in a uniform field E₀ as a uniform object of effective permittivity ?_eff from a macroscopic point of view, where ?_eff is seen as the average characteristic parameter of the object. The other way is assuming that the spheres in mixture are in the equivalent external field E₀′, and E₀′ related to the dielectric mismatch and the ratio of the sphere radius to the average distance between neighboring spheres has been obtained. Based on dipole-energy model, the approximate formulas for calculating the maximum field strength and saturation charge of spheres are derived separately.     

Potential-energy surfaces for asymmetric heavy-ion reactions

We calculate the macroscopic potential energy of deformation as a function of mass asymmetry and distance between mass centers for shape configurations of interest in heavy-ion reactions. For the system³⁰⁰120 we also study the effect of adding microscopic shell and pairing corrections to the macroscopic potential energy. The shape configurations are generated by bringing together two separated spheres of unequal size. After the spheres touch the shapes are constructed by filling in the neck while keeping constant the radii of the end spheres, the nuclear density and the total nuclear volume. The macroscopic energy is calculated as the sum of a Coulomb energy and a nuclear macroscopic energy that takes into account the finite range of the nuclear force. For systems throughout the periodic table we display the calculated energy as a function of distance between mass centers and mass asymmetry in the form of contour maps. Some important features of the contour maps are the stationary points of the potential energy and how they change in character and location as functions of the nuclear system considered. For example, for light systems there is a maximum in the potential energy for symmetric shapes. As we move to heavier systems this peak in the potential-energy surface splits into two asymmetric peaks that are separated by a symmetric saddle point. This occurs when Z²/A ≈ 30 for the total system. As the systems become still heavier the peaks become more and more asymmetric. In heavy-ion reactions for which the asymmetry of the system is smaller than that corresponding to the peak, the smaller nucleus tends to suck up the larger one. For larger asymmetries the larger nucleus tends to suck up the smaller one. For heavy systems the binary fission saddle point is lower than the maximum in the one-dimensional interaction barrier. The penetrability calculated for the multidimensional potential-energy surface is therefore increased relative to that for the one-dimensional barrier. The microscopic shell and pairing corrections lower the potential energy for configurations in which the target and/or projectile are magic or nearly magic. This effect persists to somewhat inside the point of touching. These corrections also lower the energy near the ground state.     

A quantitative mean-field theory of the hydrophobic effect of neutral and charged molecules of arbitrary geometry

A self-consistent two-length scale theory of the interaction between a hydrophobic molecule and a water environment is considered. This theory allows the width of the hydrophobic layer to be calculated for molecules of arbitrary geometry by explicitly taking into account the water structure through the correlation function of a pure liquid. This approach is used to calculate the density profile ρ(r) around a molecule of arbitrary geometry and the solvation free energy ΔG(R) related to the transport of the molecule from a vacuum to a liquid. The model parameters are adjusted by comparing the results of numerical Monte Carlo simulations taken from the literature with predictions of the model for molecules of spherical geometry. The free energy of the interaction Δ G(D) between two spheres of radius R separated by distance D is also determined using the developed approach. The model is generalized to electrostatic interactions within the framework of a self-consistent scheme in which water is modeled by a gas of point dipoles. Analysis of the derived equations shows that this theory coincides with the electrostatic theory of a continuous medium with an effective permittivity in the limit of weak electric fields.     

Classical models of confinement II

We generalize the class of Abelian models of paper I which lead to a linear potential between opposite charges. The electric fieldE is here taken as a power series in the electric displacementD raised to the σ^th power. We solve to first order the case of two opposite static point charges asymptotically for large separations.     

Semiempirical study of perturbations of the Landé g factors of the electronic-vibrational-rotational levels of hydrogen: I. Theory

Theoretical analysis of perturbations of the Landé g factors of the electronic-vibrational-rotational levels of a diatomic molecule is performed for the case of interactions between electronic states whose number is arbitrary finite and that are not limited by the smallness of the parameter describing these interactions, with regard for the interaction of rovibrational states with an arbitrary finite number of vibrational-rotational levels of individual perturbing electronic states. The spin-multiplet interaction between rovibrational states was disregarded. As a result of general consideration, formulas are obtained for the g factors of rovibrational levels for the following cases: (i) mutual perturbation of a pair of levels; (ii) an nl complex of terms; and (iii) the interaction between an arbitrary number of vibrational-rotational levels of electronic states (whose number is also not limited) considered in the first order of the perturbation theory. The formulas obtained are given in the form of dependences on differences in observed (perturbed) values of rovibrational terms and matrix elements of vibrational wave functions dependent on the internuclear distance, which, in turn, are matrix elements of the electron wave functions of different operators that take into account the interaction between the electrons and nuclei of a molecule. The possibilities of using the obtained expressions in semiempirical study of perturbations and of determining the absolute dependences of the g factors of rovibrational levels of the electronic states of diatomic molecules (in particular, the hydrogen molecule) on the vibrational and rotational quantum numbers are analyzed.     

Stieltjes electrostatic model interpretation for bound state problems

In this paper, it is shown that Stieltjes electrostatic model and quantum Hamilton Jacobi formalism are analogous to each other. This analogy allows the bound state problem to mimic as n unit moving imaginary charges \(i\hbar \) , which are placed in between the two fixed imaginary charges arising due to the classical turning points of the potential. The interaction potential between n unit moving imaginary charges \(i\hbar \) is given by the logarithm of the wave function. For an exactly solvable potential, this system attains stable equilibrium position at the zeros of the orthogonal polynomials depending upon the interval of the classical turning points.     

Non-relativistic limit of Randall-Sundrum model: solutions, applications and constraints

In the Randall-Sundrum model with one brane, we found the approximate and exact solutions for gravitational potentials and accelerations of test bodies in these potentials for different geometrical configurations. We applied these formulas for calculation of the gravitational interaction between two spheres and found the approximate and exact expressions for the relative force corrections to the Newton’s gravitational force. We demonstrated that the difference between relative force corrections for the approximate and exact cases increases with the parameter l (for the fixed distance r between centers of the spheres). On the other hand, this difference increases with decreasing of the distance between the centers of the spheres (for the fixed curvature scale parameter l). We got the upper limit for the curvature scale parameter l ≲ 10 μm. For these values of l, the difference between the approximate and exact solutions is negligible.     

Collective plasma corrections to thermonuclear reaction rates in dense plasmas

General kinetic equations are derived for nuclear reactions in dense plasmas by taking into account first-order collective plasma effects. We show that, apart from the corrections proportional to the product of the charges Z _i and Z _j of two reacting nuclei i and j, new corrections comparable in magnitude and proportional to the squares of the nuclear charges Z _i ² and Z _j ² arise. The Salpeter corrections [1] to the nuclear reaction probabilities due to the plasma screening of the interaction potential are shown to be at least a factor of r/d (r is the nuclear size and d is the Debye screening length) smaller than those assumed previously. These are zero in the approximation where the terms of order r/d are disregarded. The correlation corrections proportional to Z _iZ_j have a different physical meaning than those in [1], can have a different sign, and arise for reactions with zero Salpeter corrections. For the correlation corrections that substitute for the previously used Salpeter corrections, strong correlations are difficult to describe analytically. The interpolation formulas between weak and strong Salpeter screenings previously used in many astrophysical applications are inapplicable, because the interpolation formulas between weak and strong correlations cannot yet be obtained. We found a new type of corrections that are proportional to the squares of the charges of reacting nuclei. These are attributable to a change in the collective electrostatic self-energy of the plasma system during nuclear reactions. Plasma corrections for the hydrogen-cycle nuclear reactions are numerically calculated for the temperature, density, and abundances in the solar interior.     

Effect of the size of macroparticles on their electrostatic interaction in a plasma

The electrostatic interaction of two spherical macroparticles in a plasma has been considered. Primary attention has been focused on investigating the electrostatic interaction at short distances where polarization effects of the surface charge of finite-size macroparticles begin to play a dominant role. The first part of this study is devoted to the interaction of a point charge with a charged conducting sphere in an equilibrium plasma. It has been shown that the presence of a plasma in the system leads to a decrease in the potential barrier when two like-charged macroparticles approach each other and that this decrease proves to be the most significant in the case where the macroparticle radius is comparable to the Debye screening length. The second part of this study is concerned with the interaction of two conducting spheres in the bispherical system of the coordinates under the assumption that the charges of the conducting spheres are constant and under the assumption that the surface potentials of the spheres are constant. The latter case is closer to the physics of electrostatic interaction of two macroparticles in a plasma medium where the electrostatic potential of their surface is determined by the floating potential of the plasma. It has been demonstrated that the interaction potentials in these two cases are substantially different from each other and that, at constant macroparticle charges, the energy of the electrostatic field is an interaction potential, but, in the case of macroparticles with constant surface potentials, which are independent of the interparticle distance, the energy of the electrostatic field is not an interaction potential. In the latter case, account must be taken of the work done by external sources on the macroparticle potentials to maintain them constant. The form of the interaction potential has been established in this case from the analysis of the interaction force in terms of the Maxwell tension tensor. In the third part of this study, the interaction of two macroparticles has been considered in the spherical system of coordinates and analytical expressions for the interaction potentials have been derived for both the case of constant macroparticle charges and the case of constant surface potentials of the macroparticles.     

Light propagation in linear arrays of spherical particles

A propagation of dipolar radiation in a finite length linear chain of identical dielectric spheres is investigated using the multisphere Mie scattering formalism (MSMS). A frequency pass band is shown to be formed near every Mie resonances inherent in the spheres. The manifestation of the pass band depends on the polarization of the travelling radiation. To prove this effect, a point dipole placed by the end of the chain is used as an external source of radiation. It is found that, if this dipole is directed parallel to the chain axis, the frequency pass bands exist if the refractive index of dielectric spheres is sufficiently large nr>1.9. For the dipole normal to the chain axis, the pass band can always be formed if the chain is sufficiently long. Such a distinction is due to different behavior of the far-field dipolar interaction between the spheres induced by the external source. The edges of the pass bands are defined by the guiding wave criterion based on the light-cone constraint. The criterion of creation of the pass bands correlate with condition of formation of high quality factor modes in these systems found in our previous papers. A comparison with the results available for infinite chains is made. In particular, we clarify the nature of braking down the band structure for small enough wavevectors.     

Two magnetic impurities with arbitrary spins in open boundary t-J model

From the open boundary t-J model an impurity model is constructed in which magnetic impurities of arbitrary spins are coupled to the edges of the strongly correlated electron system. The boundary R matrices are given explicitly. The interaction parameters between magnetic impurities and electrons are related to the potentials of the impurities to preserve the integrability of the system. The Hamiltonian of the impurity model is diagonalized exactly. The integral equations of the ground state are derived and the ground state properties are discussed in detail. We discuss also the string solutions of the Bethe ansatz equations, which describe the bound states of the charges and spins. By minimizing the thermodynamic potential we get the thermodynamic Bethe ansatz equations. The finite size correction of the free energy contributed by the magnetic impurities is obtained explicitly. The properties of the system at some special limits are discussed and the boundary bound states are obtained.     

Anisotropy of electrooptical interaction in LiNbO3 crystals

Results of studying the conditions for optimum electrooptical (EO) interaction in lithium niobate crystals are presented. Analytical formulas are obtained for calculating refractive indices of natural waves and their polarizations for basis X-, Y-, and Z-cuts of crystals for arbitrary orientations of the wave vector of an optical wave. Conditions are found for the appearance of intermode EO interaction causing the orientation of polarization plane of natural waves to become a function of the external electric field.     

Three-sphere low-Reynolds-number swimmer with a cargo container

A recently introduced model for an autonomous swimmer at low Reynolds number that is comprised of three spheres connected by two arms is considered when one of the spheres has a large radius. The Stokes hydrodynamic flow associated with the swimming strokes and net motion of this system can be studied analytically using the Stokes Green's function of a point force in front of a sphere of arbitrary radius R provided by Oseen. The swimming velocity is calculated, and shown to scale as 1/R ³ with the radius of the sphere.     

Symmetry of electrostatic interaction between pyrophosphate DNA molecules

We study chiral electrostatic interaction between artificial ideal homopolymer DNA-like molecules in which a number of phosphate groups of the sugar-phosphate backbone are exchanged for the pyrophosphate ones. We employ a model in which the DNA is considered as a one-dimensional lattice of dipoles and charges corresponding to base pairs and (pyro)phosphate groups, respectively. The interaction between molecules of the DNA is described by a pair potential U of electrostatic forces between the two sets of dipoles and charges belonging to respective lattices describing the molecules. Minima of the potential U indicate orientational ordering of the molecules and thus liquid crystalline phases of the DNA. We use numerical methods for finding the set of minima in conjunction with symmetries verified by the potential U . The symmetries form a non-commutative group of 8th order, S . Using the group S we suggest a classification of liquid crystalline phases of the DNA, which allows several cholesteric phases, that is polymorphism. Pyrophosphate forms of the DNA could clarify the role played by charges in their liquid crystalline phases, and open experimental research, important for nano-technological and bio-medical applications.     

Thermodynamics of Two-Component Log-Gases with Alternating Charges

We consider a one-dimensional gas of positive and negative unit charges interacting via a logarithmic potential, which is in thermal equilibrium at the (dimensionless) inverse temperature β. In a previous paper [?amaj, L. in J. Stat. Phys. 105:173–191, 2001], the exact thermodynamics of the unrestricted log-gas of pointlike charges was obtained using an equivalence with a (1+1)-dimensional boundary sine-Gordon model. The present aim is to extend the exact study of the thermodynamics to the log-gas on a line with alternating ± charges. The formula for the ordered grand partition function is obtained by using the exact results of the Thermodynamic Bethe ansatz. The complete thermodynamics of the ordered log-gas with pointlike charges is checked by a small-β expansion and at the collapse point β _c =1. The inclusion of a small hard core around particles permits us to go beyond the collapse point. The differences between the unconstrained and ordered versions of the log-gas are pointed out.