Number Of Wave Vectors Research Articles

The suppression of finite size effect within a few lattice sites

Boundary modes localized on the boundaries of a finite-size lattice experience a finite size effect (FSE) that could result in unwanted couplings, crosstalks and formation of gaps even in topological boundary modes. It is commonly believed that the FSE decays exponentially with the size of the system and thus requires many lattice sites before eventually becoming negligibly small. Here we consider a two-dimensional strip geometry that is periodic along one direction and truncated along the other direction, in which we identify a special type of FSE of some boundary modes that apparently vanishes at some particular wave vectors along the periodic direction. Meanwhile, the number of wave vectors where the FSE vanishes equals the number of lattice sites across the strip. We analytically prove this type of FSE in a simple model and prove this peculiar feature. We also provide a physical system consisting of a plasmonic sphere array where this FSE is present. Our work points to the possibility of almost arbitrarily tunning of the FSE, which facilitates unprecedented manipulation of the coupling strength between modes or channels such as the integration of multiple waveguides and photonic non-abelian braiding.

Persistent Currents and Bethe Ansatz

Persistent current appears in a small metallic ring penetrated by perpendicular magnetic field. This quantum effect is remarkable due to the non-dissipative character of this current. It was predicted by Buttiker [1] in 1983 and later the experimental evidence was obtained by Levy [2]. Due to the progress in nanometer technology there is still interest in persistent currents in nanorings that contain a small numbers of electrons [3]. These currents are rather sensitive to the quantum coherence, and thus to electron interaction and their interaction with impurities, serving as a convenient instrument for studying the subtle properties of strongly interacting electron system. The important role belongs to one-dimensional (1D) models that enable to obtain the analytical solutions and understand the main features of the above phenomena. The most popular method of analyzing the above systems is the so-called Bethe Ansatz [4] (see also Ref. [5]) based on the assumption that the electron system wave function can be constructed as a superposition of plane waves with the restricted number of wave vectors coordinated with the rules of elastic scattering. For instance, using this Ansatz Li and Ma [6] considered the system of many electrons interacting with an impurity. However, despite numerous applications the motivation of the assumption is still lacking. The goal of the present work is to illustrate the applicability of the Bethe Ansatz using the most simple model system of two electrons moving in 1D ring and interacting with a single impurity via Dirac function type interaction, and comparing the results with those obtained by means of the exact numerical diagonalization of the Hamiltonian.

Efficient calculation of the exact exchange energy in periodic systems using a truncated Coulomb potential

We present a scheme for calculating the exact exchange energy in periodic solids within a Kohn--Sham or Hartree--Fock approach, which removes the need to treat the integrable singularities via an auxiliary function. In the exchange integrals, we use a modified Coulomb potential, which tends to the exact potential as the number of $\mathbf{k}$ points increases yet has no singularities, and which is also very simple to implement. We compare this approach to the auxiliary function scheme for diamond, graphite, and two allotropes of silicon carbide and show that it converges more rapidly with the number of wave vectors.

Negative refraction and slab imaging of photonic crystals

Imaging using lenses with curved surfaces is a common experience in our daily life, as for example when we use digital or cellular-phone cameras or overhead projectors. However, the imaging properties of such conventional lens systems are limited by aberration and diffraction. Careful optical design can significantly reduce the former, but not the latter, because it is due to the fundamental nature of light, which blurs the image of a geometrical point object into a finite-sized spot. In 2000, John Pendry1 performed insightful simulations showing that a perfect lens could be realized in the form of a thin planar slab with refractive index n = −1. When light is refracted in a direction opposite to that of conventional refraction, the process is termed negative refraction as illustrated in Figure 1. Materials that can display a negative index include metals, silicon carbide, some polymers and dyes, as well as more versatilely engineered metamaterials and photonic crystals (PhCs).2 PhCs are nanostructured materials in which a periodic variation of the dielectric constant results in a photonic band gap. Those made from dielectric materials are particularly well-suited for optical imaging since there is little or no material loss. However, the negative refraction and imaging mechanisms of PhCs differ from those of metals and metamaterials. Since PhCs are periodic structures, electromagnetic Bloch modes that consist of an infinite number of wavevectors can form in the photonic lattice. Therefore, the light-lattice interaction is governed by diffraction, scattering, reflection, and resonance effects, which makes analytical methods generally impractical to characterize their imaging properties. Bloch wave propagation studies have shown that the effective index of refraction, neff, can be extracted from band structures. It is negative at the second band because the group velocity (i.e. the speed at which a pulse propagates in a PhC and equal to the energy velocity in periodic media) moves towards the scattering center Γ . Figure 1. Positive refraction (left) and negative refraction (right).

Metals at finite temperature: a modified smearing scheme

The specification of electronic eigenstate occupation numbers, in density-functional theory, has two purposes: simulating the canonical ensemble for the electrons at non-zero temperature (Fermi–Dirac occupation numbers), and improving the convergence with respect to the number of electronic wavevectors sampling the Brillouin zone. We describe a scheme which combines both, providing finite-temperature eigenstate occupations with an additional smearing to improve sampling convergence. We present the correction formula obtained for the free energy, and test the method on a small system of metallic aluminium for temperatures under 3000 K. In this regime, the Fermi–Dirac smearing alone gives only a modest reduction in the number of wavevectors needed for convergence.

On the wavevector dependent shear viscosity of a simple fluid

Non-equilibrium molecular dynamics (NEMD) simulations employing the sinusoidal transverse force (STF) method are used to calculate the wavevector and strain rate dependent shear viscosity of a WCA fluid at a zero wavevector number density of 0.8442 and zero wavevector temperature of 0.722. The functional form of the dependence of shear viscosity on the strain rate and wavevector are examined, in the zero field limit. The ramifications of these results for the possible existence of the Burnett coefficients are discussed. The shear rate dependent viscosity for a range of wavevectors can be fitted to both a Cross analytical form and a formula derived from Quentrec's local order theory. Also it can be fitted to two square-root regimes. The extrapolated zero strain rate viscosities are consistent with a Lorentzian functional form, a Taylor series expansion in q 2 and also to a q 3/2 variation at low wavevector. This study reveals that with current computational resources it is still not possible to establish the validity of the Burnett coefficients.

Average square atomic displacement: A comparison of the Monte‐Carlo and Green's function results

AbstractHighly accurate results for the mean‐square displacement (MSD) of an atom for a nearest‐neighbor Lennard‐Jones model of a f.c.c. solid are presented. These results are obtained from the Monte‐Carlo (MC) method and the Green's function method. The dependence of MSD on the sample size (N) in the MC method and similarly the dependence of MSD on the number of wave vectors used in the calculation by the Green's function method is discussed in detail. The results are presented by both methods for the infinite sample size limit as well as for the finite sizes, i.e., 32‐, 108‐, and 256‐atom sample sizes and the same number of corresponding wave vectors. It is shown that an analytical method like the Green's function method reproduces almost exactly the results of the MC method (a completely numerical procedure) for all temperatures except at the melting point (Tm). The computing time required in the former method is only a fraction of the latter one.

Chebyshev expansions for the scattering matrices in full-potential Korringa-Kohn-Rostoker band-structure calculations.

The search for electronic energy bands in full-potential Korringa-Kohn-Rostoker (KKR) band theory requires the evaluation of the scattering matrices at many different energies for a certain number of wave vectors. We develop Chebyshev-function approximations for the scattering matrices, which can save a factor of 2 to 3 in computing time for the case of silicon. Moreover, the Chebyshev coefficients of the scattering matrices can be used in a polynomial expansion of the KKR secular matrix, which transforms the search for the energy bands to an eigenvalue problem. In a test calculation for silicon it turns out that if one wants to reproduce energy bands to within 1 mRy one needs to include cubic terms in the polynomial expansions. Calculations with polynomial coefficients from a Chebyshev series expansion are found to give more precise energy eigenvalues than calculations where the polynomial coefficients were obtained from a Taylor series expansion without using any additional computing time.

Phonon frequencies of cesium

Phonon frequencies of cesium are obtained at a number of wave vectors along the principal symmetry directions (i) by exploiting the homology of experimental phonon frequencies of lithium, sodium, potassium and rubidium, as well as (ii) by using a six force constant model of the authors. These frequencies are compared with the theoretical frequencies calculated by Animalu et al., Ho, and Price et al. using different pseudopotentials. It is shown that cesium is homologous with other alkali metals and its homologous frequencies which correspond to their experimental values are in complete agreement with the model frequencies, but differ from others at some wave vectors.

An inhomogeneous Landau-like equation

The problem of the mixing of two plasmas is very difficult to solve if the methods of statistical mechanics are used. When both plasmas are homogeneous in space, and the limit of weak coupling is appropriate, the Landau equation describes the motion. An equation, similar to the Landau equation, is developed for the case when one of the plasmas is initially inhomogeneous in space. The equation is a power series in time and involves wave vectors in space. The short-time-after-mixing limit is considered, which restricts the number of wave vectors. A sample problem is presented of the short time development of an infinite-temperature spike interacting with a homogeneous plasma. The spike is found to move very rapidly from the origin owing to the long range of the forces. Extension of the time development past the initial short time requires consideration of destructive wave vectors in addition to the wave vector generation considered here.