A simple general proof of the Krazer–Prym theorem and related famous formulae resolving convergence properties of Coulomb series in crystals

Kholopov, E. V.

doi:10.1088/1751-8113/40/23/007

Cited by 8 publications

(10 citation statements)

References 38 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In order to discuss this approach, we consider a Bravais lattice composed of unit point charges and immersed in a neutralizing uniform background. It is easy to show that the interaction of a background with the bulk potential field vanishes and the same is right for the background contribution to the sum over reciprocal lattice in expression (10) [5,22,27]. As a result, the effect of lattice summation in (10) is associated with the contribution of point charges alone.…”

Section: Optimization Of Spreading Parametersmentioning

confidence: 93%

“…First of all, here we develop the treatment appropriate to this case. As mentioned earlier [5], it is based on the Coulomb characteristic of a Bravais lattice, the parameter put forward by Harris and Monkhorst [27]. In order to discuss this approach, we consider a Bravais lattice composed of unit point charges and immersed in a neutralizing uniform background.…”

Section: Optimization Of Spreading Parametersmentioning

confidence: 99%

“…where the terms on the left-hand side stand for the contributions of series over reciprocal and direct lattices, respectively, with singling out their principal dependence on λ. Note that C is the value of the modulated lattice potential at the site of a unit point charge and so C/2 = E, where E is the total energy per point charge in question [5]. It is significant that except for C, all the terms in relation (58) are positive.…”

Section: Optimization Of Spreading Parametersmentioning

confidence: 99%

“…It is known that the Ewald approach [1] is the most widespread implementation of the Poisson summation formula in crystals [2,3,4,5]. Bearing in mind that such a treatment proposes that the overall summation is divided into sums over direct and reciprocal space, the enhancement of the rate of convergence is traditional in this problem and assumes the investigation of ranges of summation in either of those sums.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Convergence peculiarities of lattice summation upon multiple charge spreading generalizing the Bertaut approach

Kholopov

2010

Physica B: Condensed Matter

View full text Add to dashboard Cite

Within investigating the multiple charge spreading generalizing the Bertaut approach, a set of confined spreading functions with a polynomial behaviour, but defined so as to enhance the rate of convergence of Coulomb series even upon a single spreading, is proposed. It is shown that multiple spreading is ultimately effective especially in the case when the spreading functions of neighbouring point charges overlap. In the cases of a simple exponential and a Gaussian spreading functions the effect of multiplicity of spreading on the rate of convergence is discussed along with an additional optimization of the spreading parameter in dependence on the cut-off parameters of lattice summation. All the effects are demonstrated on a simple model NaCl structure.

show abstract

Section: Optimization Of Spreading Parametersmentioning

confidence: 93%

Section: Optimization Of Spreading Parametersmentioning

confidence: 99%

Section: Optimization Of Spreading Parametersmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Convergence peculiarities of lattice summation upon multiple charge spreading generalizing the Bertaut approach

Kholopov

2010

Physica B: Condensed Matter

View full text Add to dashboard Cite

show abstract

“…The second sum still converges slowly, but an application of the Poisson summation formula will result in a sum over the reciprocal lattice involving the Fourier transform of Φ(·)(1 − F (·)), which (provided F is sufficiently smooth) will again converge rapidly. Further discussion of the underlying structure of the Ewald method can be found in [104,89] and the fact that it resolves appropriately the issues of conditional convergence is demonstrated in [46] for a particular class of sums. As well as its mathematical efficacy, this method can easily be interpreted physically; see, for example, [108].…”

Section: Ewald Representationsmentioning

confidence: 99%

Lattice Sums for the Helmholtz Equation

Linton¹

2010

SIAM Rev.

133

138

View full text Add to dashboard Cite

A survey of different representations for lattice sums for the Helmholtz equation is made. These sums arise naturally when dealing with wave scattering by periodic structures. One of the main objectives is to show how the various forms depend on the dimension d of the underlying space and the lattice dimension d Λ . Lattice sums are related to, and can be calculated from, the quasi-periodic Green's function and this object serves as the starting point of the analysis. LATTICE SUMS FOR THE HELMHOLTZ EQUATION 631to mathematically challenging problems. Attempts to evaluate the sum in (1.1) in closed form have so far been unsuccessful, though some remarkable identities have been derived in the process [23,113].The lattice sum in (1.1) arises from a sum of singular solutions of Laplace's equation. Here we will be concerned with related sums in which the underlying physics is governed by the Helmholtz equationwhich arises naturally in many contexts, particularly when investigating linear wave phenomena at a given frequency. For example, in acoustics u represents fluctuations in pressure, whereas in elasticity it would be some component of the displacement vector, and in electromagnetism a component of the electric or magnetic field. The quantity k (assumed real and positive) is the wavenumber, related to the frequency ω via some appropriate dispersion relation. In diffraction theory we are often faced with trying to solve (1.2) in a region exterior to some scatterer(s) (and maybe inside as well) subject to boundary conditions on the surface of the scatterer(s). The case where the scatterer is periodic represents an important class of such problems; examples include the study of diffraction gratings, scattering by periodic surfaces, and wave propagation through composite materials. Equation (1.2) is also equivalent to the time-independent Schrödinger equation in the presence of a constant potential, in which case k is related to the total energy of the particle under consideration, and as a result the Helmholtz equation finds application in the study of electron scattering in solids. A common simplification used is the so-called muffin-tin approximation (due to Slater [99]), in which the potential is supposed to be spherically symmetric within spheres surrounding each atom and constant in the region inbetween these spheres. This then leads naturally to the need to solve (1.2) on a periodic domain.The study of wave propagation in periodic structures has a long history and leads to a whole range of interesting mathematical problems. The classic text by Brillouin [11] arguably still serves as the best introduction to the subject, though many significant advances have been made since it was written. For example, the mathematical theory of diffraction gratings (based on functional analysis) is developed in books such as [120] and there is an ever-growing literature describing the rapidly developing field of photonic and phononic crystals; an excellent text is [39]. One of the consequences of periodicity is that a problem on...

show abstract

Multiple charge spreading as a generalization of the Bertaut approach to lattice summation of Coulomb series in crystals

Kholopov¹

2010

Physica B: Condensed Matter

View full text Add to dashboard Cite

The Bertaut approach associated with charge spreading so as to enhance the rate of convergence of Coulomb series in crystals is extended to the case of an arbitrary multiple spreading with a given initial spreading function. It is shown that the effect of spreading may in general be treated as a uniform transformation of space, providing that zero mean potential as a universal spatial property is sustained. As a result, electrostatic potentials driven by different orders of multiple spreading can be obtained from the same energy functional in a consistent manner. It is found that the effect of multiple spreading gives rise to more advanced forms described, for example, by simple exponential decrease, but the functional description based on a Gaussian spreading turns out to be invariant. In addition, the effects of a multiple charge spreading based on simple exponential and Gaussian spreading functions are compared as typical of molecular calculations.

show abstract

A simple general proof of the Krazer–Prym theorem and related famous formulae resolving convergence properties of Coulomb series in crystals

Cited by 8 publications

References 38 publications

Convergence peculiarities of lattice summation upon multiple charge spreading generalizing the Bertaut approach

Convergence peculiarities of lattice summation upon multiple charge spreading generalizing the Bertaut approach

Lattice Sums for the Helmholtz Equation

Multiple charge spreading as a generalization of the Bertaut approach to lattice summation of Coulomb series in crystals

Contact Info

Product

Resources

About