Lattice Of Charges Research Articles

A simple approximation technique for the solution of the Poisson-Boltzmann equation for fixed charge systems

A simple two step approximation technique is described which yields expressions for the mobile ion concentration, electrostatic potential and space charge density profiles inside a lattice of fixed (immobilized) charges when the latter is immersed in an ionic solution. Examples are presented of the results obtained with the approximation technique for a single fixed charge lattice and a membrane consisting of two fixed charge lattices (of opposite sign) in contact. It is shown that the results obtained are in close agreement with iterative computer solutions for the profiles.

Electrical forces between particles with discrete periodic surface charge distributions in ionic solution

The interaction, across aqueous electrolyte, of two charged plates is studied. The charges on each plate take up a discrete periodic lattice structure. Our treatment is based on a Debye–Huckel procedure and use of Fourier–Bloch functions similar to those used in solid state physics. The interaction energy depends on the relative configuration of the charge lattices on the particles. For like charges, the energy is always positive giving a repulsive force. The minimum energy is obtained when the charge lattices interlace each other and leads to a primary maximum in the total interaction energy which is depressed considerably below the smeared out result. The maximum energy is sufficiently large that one would expect particles never to approach with superposed charge lattices.For oppositely charged plates, the minimum energy is for superposed charged lattices and is considerably more negative than the smeared out result. The former tends to –∞ as l→ 0; the latter tends to a finite value. For interlaced equal and opposite charged lattices, we obtain a new result namely that the interaction energy may becomes positive at small separations giving a repulsive force.

The Double Fixed Charge Membrane: Low Frequency Dielectric Dispersion

An analysis is made of the AC characteristics of a membrane consisting of two fixed charge regions of opposite sign, in contact. It is shown that the equivalent parallel capacitance and conductance of such a membrane undergo a strong dispersion at low frequencies. The dielectric dispersion is a result of polarization effects in the diffusion of coions in each of the two fixed charge lattices. This, at low frequencies, gives rise to a very large diffusion capacitance. The form of the dispersion characteristics is very similar to those observed for synthetic-fused anion-cation membranes and various cellular membranes.

Potential of charge distribution in point charge lattices. I. The method

A new method is proposed for the calculation of the potential of an extended charge distribution due to a lattice of point charges. The charge distribution is assumed to arise from the product of two exponential functions (Slate type orbitals). This method, which is exact for a one-centre charge distribution within the point charge lattice model, differs from the conventional 'crystal field potential' treatment in that the integration is first carried out analytically before the summation. This results in two types of lattice sum, called the 'exponential' and the 'harmonic' lattice sums. The convergence behaviours of these lattice sums are discussed. The exponential lattice sum is absolutely convergent and may be evaluated directly. Lower members of the harmonic lattice sums are conditionally convergent and transformation techniques are desirable. Explicit formulae valid for the general triclinic lattice, are derived for their evaluation, using the planewise summation scheme of de Wette.

Potential of charge distributions in point charge lattices. II. Applications

For pt. I, see ibid., vol. 4, 263 (1971). The expressions derived for the calculations of the lattice potential are applied to crystals of different symmetry. These formulae are found to be simple to use and seem to give results superior to other approximate methods. In particular a comparison is made with values obtained by using a point potential (PP) approximation, that is, regarding the extended charge distribution as a single point. In general it is found that the lower the crystal symmetry, the poorer the PP approximation as compared to the more accurate method proposed here. The relative importance of the two lattice sums is also examined. For large orbital exponents, the exponential lattice sum is negligibly small, and the contribution to the lattice potential comes predominantly from the harmonic lattice sum. A practical way of evaluating the auxiliary Incomplete Gamma function is also described.

Comments on the approximate calculation of lattice potential

The potential of a charge distribution due to a lattice of point charges may be evaluated by the classical multipole expansion method. The leading terms in the resultant expressions are just those used in some of our previous calculations [1–3]. In addition, for cases where the leading terms vanish because of the effect of orthogonality of the basis functions upon the Mulliken expansion (this being especially serious in the case of a one-centre charge distribution), we have derived the first nonvanishing term, involving 〈χμ|r|χν〉. In other cases it may be necessary to proceed to still higher multipole terms before a non-zero contribution is obtained. The entire procedure is formulated in such a way that it can be easily applied to LCAO-MO calculations for polyatomic ions in ionic lattices.

Symmetry based potential expansion technique for molecules and point charge lattices

A systematic method is presented for evaluating the electrostatic potential of point charge systems with inversion symmetry. By considering two simple ‘dumbbell’ structures, concise general potential expressions are deduced for point charge symmetries of the point groups containing the inversion operation. The explicit formulae presented for the most common basic symmetries are used as a basis to write down the potential for more complicated geometries formed by the combination of any number of basic symmetries. The equations are also valid for geometries obtained by certain distortions of the basic symmetry. The method presented here gives the potential at any general space point. The use of expressions in terms of the real spherical harmonics facilitates matrix element calculations, and with the potentials in such a form, symmetry arguments can be directly applied to eliminate many terms which vanish.

Linearized Superposition Theory of a Classical One-Component Plasma

The equilibrium theory of a one-component nondegenerate charged gas, in a uniform continuum of a neutralizing charge, is studied on the basis of a linear-closure procedure for the BBGKY hierarchy which, in the present instance, corresponds to the use of the linearized Kirkwood superposition approximation, and leads to a second-order linear differential equation for the radial distribution function. Its solution is studied, numerically and analytically, for a wide range of the plasma parameter γ = e2/kTΛD, where ΛD is the Debye length. Results for γ ≪ 1 are compared with corrections to the Debye-Hückel theory derived by Abe and others. Solutions for γ ≫ 1, which approach an ordered state of the system as γ increases, are compared with two simple ``model'' calculations of the equilibrium theory of a body-centered cubic lattice of charges in a uniform background of compensating charge. The thermodynamic properties of these ``Coulomb lattice'' models are in qualitative agreement with those of the plasma as computed via our closure approximation for γ ≫ 1. The possibility is therefore suggested that, even for large γ, the linear closure approximation has a larger degree of validity than might be expected a priori.

Spin Paramagnetism of an Electron Gas in a Lattice of Positive Charges

The spin paramagnetism of an electron gas in a lattice of positive charges is studied along the line of Bellemans and De Leener. The results are compared with Pines' and also with experimental data on Li and Na.

Ground-State Energy of an Electron Gas in a Lattice of Positive Point Charges

The ground state energy epsilon of an electron gas in a rigid lattice of positive point charges is calculated as a function of the Bohr radius between electrons, by quantum statistical methods. The couplings between pairs of electrons, and between electrons and lattice points, are taken into account. The value of epsilon in a positive continuum is derived as a special case. As an example, epsilon is calculated for an H lattice. (T.F.H.)

DBands in Cubic Lattices. III

The results of a previous, perturbation theoretic treatment of $d$ bands in the body-centered cubic lattice are extended in several respects: The methods of the previous calculation are applied to determine energy levels at the points $\ensuremath{\Gamma}$ and $X$ in the Brillouin zone of the face-centered cubic lattice. As before, the crystal potential is that of a lattice of point charges, screened by a uniform distribution of electrons. The perturbation expansion of the wave function of a $d$ electron is developed for the body-centered cubic lattice. Calculations are reported for two states near the top and bottom of the $d$ band, including terms of first order in the potential. These functions have the characteristic property that the wave function of a state near the top of the $d$ band is more compact than that belonging to a state near the bottom. The energies of four states for the body-centered lattice are computed as a function of the binding parameter $\mathrm{Za}$ by a more accurate method than that employed in the previous work, making possible an estimation of the accuracy of perturbation theory and the dependence of bandwidth on binding parameter. The role of crystal field effects in the tight-binding limit is discussed, and the circumstances are determined under which the $d$ band may split into sub-bands based on functions of different cubic symmetries. Estimation of the value of the binding parameter for which such separation occurs strongly suggests that this split does not occur for the actual transition metals. Finally, the effects of spin-orbit coupling on the band structure are studied in the tight-binding approximation. A formulation of k\ifmmode\cdot\else\textperiodcentered\fi{}p perturbation theory for $d$ bands is given.

DBands in Cubic Lattices. II

The results of a previous, perturbation theoretic treatment of $d$ bands in the body-centered cubic lattice are extended in several respects: The methods of the previous calculation are applied to determine energy levels at the points $\ensuremath{\Gamma}$ and $X$ in the Brillouin zone of the face-centered cubic lattice. As before, the crystal potential is that of a lattice of point charges, screened by a uniform distribution of electrons. The perturbation expansion of the wave function of a $d$ electron is developed for the body-centered cubic lattice. Calculations are reported for two states near the top and bottom of the $d$ band, including terms of first order in the potential. These functions have the characteristic property that the wave function of a state near the top of the $d$ band is more compact than that belonging to a state near the bottom. The energies of four states for the body-centered lattice are computed as a function of the binding parameter $\mathrm{Za}$ by a more accurate method than that employed in the previous work, making possible an estimation of the accuracy of perturbation theory and the dependence of bandwidth on binding parameter. The role of crystal field effects in the tight-binding limit is discussed, and the circumstances are determined under which the $d$ band may split into sub-bands based on functions of different cubic symmetries. Estimation of the value of the binding parameter for which such separation occurs strongly suggests that this split does not occur for the actual transition metals. Finally, the effects of spin-orbit coupling on the band structure are studied in the tight-binding approximation. A formulation of k\ifmmode\cdot\else\textperiodcentered\fi{}p perturbation theory for $d$ bands is given.

Cubic Field Splitting ofDLevels in Metals

The splitting of the fivefold degeneracy of free atom $d$ electron states by nonspherical components of a crystalline field is calculated. The crystal potential employed is that of a lattice of positive point charges screened by a uniform distribution of electrons. The calculation is done to first order in the cubic field, using hydrogenic electron wave functions. The triply degenerate $d$ state is lowered with respect to the doubly degenerate one in both body-centered and face-centered cubic lattices. Numerical results are given for both lattices. Finally, analytic atomic wave functions are used to estimate the splitting in iron and copper at the observed lattice spacing. The crystal field splitting of these levels is found to be much smaller than the overlap splitting as obtained in previous calculations for both materials.

Coulomb Interactions in the Uniform-Background Lattice Model

The uniform-background lattice model consists of a body-centered-cubic lattice of point charges of like sign embedded in a uniform background charge of the opposite sign. The method used by Fuchs to apply the Ewald sum formula to the case of static shear deformations is extended to apply to spatially periodic deformations. The dispersion relations (frequency as a function of wave number) are presented for propagation vectors in the [100], [110], [111], and [210] directions. Comparison is made with the dispersion relations for the same directions in a Born-von K\'arm\'an model of a body-centered cubic lattice in which only central interactions between nearest and next-nearest neighbors are considered. The values of the "Coulomb part" of the macroscopic elastic constants as calculated by Fuchs are used for this model. The problem of treating a model in which the background responds to the displacement of point charges is discussed.