Long-wavelength Approximation Research Articles

Anharmonic processes of scattering and absorption of slow quasi-transverse modesin cubic crystals with positive and negative anisotropies of second-order elasticmoduli

The quasi-transverse ultrasound absorption during anharmonic processes of the scatteringin cubic crystals with positive (Ge, Si, diamond and InSb) and negative (KCl and NaCl)anisotropies of the second-order elastic moduli is studied. Mechanisms underlying therelaxation of the slow quasi-transverse mode by two slow (the SSS mechanism) or two fast(the SFF) modes are discussed in the long-wavelength approximation. Angular dependencesof the ultrasound absorption for the SSS, SFF and Landau–Rumer relaxation mechanismsare analyzed in terms of the anisotropic continuum model. The full absorption of theslow quasi-transverse mode is determined. It is shown that the SSS and SFFrelaxation mechanisms are due to the cubic anisotropy of the crystals, leading tothe interaction between noncollinear phonons. Two most important cases—thewavevectors of phonons are in the cube face plane or the diagonal planes—areconsidered. In crystals with a considerable anisotropy of the elastic energy (Ge,Si, InSb, KCl and NaCl) the total contribution of the SSS and SFF relaxationmechanisms to the full absorption is either several times or one to two orders ofmagnitude larger than the contribution from the Landau–Rumer mechanism dependingon the direction. Much of the dominance of the former relaxation mechanismsover the Landau–Rumer mechanism is explained by second-order elastic moduli.

Reflection of electromagnetic plane waves in a long-wavelength approximation from a multilayer system of anisotropic transparent films on absorbing medium

The propagation of s- and p-polarised electromagnetic plane waves in a N-layer system of anisotropic films on isotropic and homogeneous absorbing substrate is investigated in the long-wavelength limit. The analytical expressions are obtained for the reflection (transmission) coefficients and ellipsometric angles of an anisotropic multilayer system. All analytical results are correlated with the numerical solution of the reflection problem on the basis of rigorous electromagnetic theory for anisotropic layered systems. The possibilities of using obtained approximate formulae for resolving the inverse problem for ultrathin anisotropic dielectric films upon absorbing substrates are discussed.

Vacuum polarization effects on graphitic nanocones

Theory of vacuum polarization by singular background fields is applied to the study of electronic properties of the graphitic monolayer (graphene) with a topological defect. Graphene in the long-wavelength approximation is characterized by the linear dispersion law for the quasiparticle excitations, representing a unique example of the really two-dimensional ‘ultrarelativistic’ electronic system which in the presence of topological defects can possess rather unusual properties. A disclination that rolls up a graphitic sheet into a nanocone is represented by a pointlike pseudomagnetic vortex with a flux which is related to the deficit angle of the cone. The electronic system on a graphitic nanocone is found to acquire the ground state condensate and current of special type, and we determine the dependence of these quantities on the deficit angle of the nanocone, continuous parameter of the boundary condition at the apex, and the distance from the apex.

Harmonic generation: Quantum-electrodynamical theory of the harmonic photon-number spectrum

Simulations of harmonic generation in the single-atom limit usually describe the harmonic radiation as proportional to the squared magnitude of the Fourier transform (FT) of the expectation value of either the dipole moment or the dipole acceleration. However, no derivation of an explicit single-atom representation of the typical experimental measurement (i.e., the number of harmonic photons produced versus the frequency of the harmonic) has been given. By means of the first principles of quantum electrodynamics a formula for this ``harmonic photon-number spectrum'' is derived, which is proportional to the double FT of the time-autocorrelation function of the dipole velocity of the whole system. Within the long-wavelength approximation the time-correlation function can be recast as the FT of the squared magnitude of the expectation value of the dipole velocity of a single representative atom.

Asymptotic construction of a dynamic shell theory: Finite-element-based approach

In the theory of shells, asymptotically correct dimensional reduction from the dynamic equations of three-dimensional (3-D) elasticity to the approximate two-dimensional (2-D) equations of shell theory is possible only within the long-wavelength approximation, just as in statics. However, as pointed out in the literature, such a procedure is not suitable for analysis of complicated high-frequency effects in the short-wavelength regime because displacements may change rapidly through the thickness. Therefore, in addition to the general derivation of 2-D equations within the long-wavelength regime, construction of a dynamic shell theory must involve a dimensional reduction procedure for the high-frequency behavior plus a separate and logically independent step for the short-wavelength regime. In this paper, with the aid of small parameters that are inherent to the shell problem, a dimensional reduction yielding a shell theory distinct from established shell theories is carried out using the variational-asymptotic method. The dimensional procedure rigorously splits the 3-D problem into a linear one-dimensional (1-D) through-the-thickness analysis and a nonlinear 2-D shell analysis, taking into account both low- and high-frequency vibration within the long-wavelength regime. Clearly, this is not sufficient for the short-wavelength regime. Therefore, one more step is used in the present paper—called hyperbolic short-wavelength extrapolation—which enables the analysis to qualitatively describe a 3-D stress state in the short-wavelength regime with a 2-D theory. The main focus of this paper regards the through-thickness analysis, which is solved by a 1-D finite element approximation and which provides a means to calculate both inertia and stiffness coefficients for 2-D composite shell theory. This paper then represents the first attempt to develop a procedure based on finite elements for the through-thickness analysis associated with shell dynamics, which makes more the problem tractable in practice. Finally, numerical results are compared with published analytical solutions. The excellent agreement demonstrates the capabilities of the present model to analyze dynamic structural responses over a wide range of frequencies and wavelengths.

High-frequency corrections to the detector response and their effect on searches for gravitational waves

Searches for gravitational waves with km-scale laser interferometers often involve the long-wavelength approximation to describe the detector response. The prevailing assumption is that the corrections to the detector response due to its finite size are small and the errors due to the long-wavelength approximation are negligible. Recently, however, Baskaran and Grishchuk (2004 Class. Quantum Grav. 21 4041) found that in a simple Michelson interferometer such errors can be as large as 10%. For more accurate analysis, these authors proposed to use a linear-frequency correction to the long-wavelength approximation. In this paper we revisit these calculations. We show that the linear-frequency correction is inadequate for certain locations in the sky and therefore accurate analysis requires taking into account the exact formula, commonly derived from the photon round-trip propagation time. Also, we extend the calculations to include the effect of Fabry–Perot resonators in the interferometer arms. Here we show that a simple approximation which combines the long-wavelength Michelson response with the single-pole approximation to the Fabry–Perot transfer function produces rather accurate results. In particular, the difference between the exact and the approximate formulae is at most 2–3% for those locations in the sky where the detector response is greater than half of its maximum value. We analyse the impact of such errors on detection sensitivity and parameter estimation in searches for periodic gravitational waves emitted by a known pulsar, and in searches for an isotropic stochastic gravitational-wave background. At frequencies up to 1 kHz, the effect of such errors is at most 1–2%. For higher frequencies, or if more accuracy is required, one should use the exact formula for the detector response.

The theory of surface waves in waveguides filled with a smoothly inhomogeneous plasma

Electrostatic surface waves in a smoothly inhomogeneous plasma contained in a waveguide are considered. For the plasma with a linear-constant density profile, four types of surface waves are demonstrated for various wave-number regions. The existence conditions, the complex dispersion relations, and the structures of such waves are determined. Complex frequencies of weakly damped surface waves are obtained for the plasma with an arbitrary density profile in the long-wavelength approximation.

Diffraction of a plane electromagnetic wave by a cylindrical wedge with a rounded apex

We reduce the rigorously formulated problem of diffraction of a plane electromagnetic wave by a perfectly conducting cylindrical wedge with a rounded apex to solving the system of linear algebraic equations of the second kind for unknown coefficients of the Fourier expansions of the diffracted-field components. The expansion coefficients are determined analytically in the long-wavelength approximation. The results of calculations of the diffracted field in the far zone are presented with a given accuracy in the case of an E-polarized wave. It is shown that the rounding of the apex of a cylindrical wedge leads to an increase in the backscattering coefficient of the structure in the long-wavelength range.

Condensate depletion in two-species Bose gases: A variational quantum Monte Carlo study

We investigate two-species Bose gases in traps with various interactions using variational quantum Monte Carlo (VMC) techniques at zero temperature. The bosons are represented by hard spheres whose diameter is equivalent to the $s$-wave scattering length in the low-energy and long-wavelength approximation. We explore the role of repulsive and attractive interspecies or intraspecies interactions on the condensate properties of the mixtures, particularly the condensate fraction of each species as compared to the case when each species is in a separate trap of its own. We model the repulsive interactions by a hard-core (HC) potential and the attractive interactions by a shallow model potential. The VMC density profiles and energies are evaluated at various interactions and two mass ratios of the species.

On the possible induced charge on a graphitic nanocone at finite temperature

Electronic excitations in a graphitic monolayer (graphene) in the long-wavelength approximation are characterized by the linear dispersion law, representing a unique example of the really two-dimensional ‘ultrarelativistic’ fermionic system which in the presence of topological defects possesses rather unusual properties. A disclination that rolls up a graphitic sheet into a nanocone is described by a pointlike pseudomagnetic vortex at the apex of the cone, and the flux of the vortex is related to the deficit angle of the conical surface. A general theory of planar relativistic fermionic systems in the singular vortex background is employed, and we derive the analytical expression for the charge which is induced at finite temperature on some graphitic nanocones.

Light-cone Yang-Mills mechanics: SU(2) vs. SU(3)

We investigate the light-cone SU(n) Yang-Mills mechanics formulated as the leading order of the long-wavelength approximation to the light-front SU(n) Yang-Mills theory. In the framework of the Dirac formalism for degenerate Hamiltonian systems, for models with the structure groups SU(2) and SU(3), we determine the complete set of constraints and classify them. We show that the light-cone mechanics has an extended invariance: in addition to the local SU(n) gauge rotations, there is a new local two-parameter Abelian transformation, not related to the isotopic group, that leaves the Lagrangian system unchanged. This extended invariance has one profound consequence. It turns out that the light-cone SU(2) Yang-Mills mechanics, in contrast to the well-known instant-time SU(2) Yang-Mills mechanics, represents a classically integrable system. For calculations, we use the technique of Grobner bases in the theory of polynomial ideals.

Drastic Nonlinear Optical Response beyond Long-Wavelength Approximation Regime

Drastic Nonlinear Optical Response beyond Long-Wavelength Approximation Regime

Dynamics of a Fabry–Perot cavity in the field of a plane gravitational wave

The interaction of a weak gravitational wave with a Fabry–Perot cavity is analysed beyond the long-wavelength approximation in the input-mirror locally Lorentzian frame of reference taking the light pressure into account. The generalised expressions are obtained for the coefficient of pondermotive optical rigidity, the motion law of the moving mirror of the cavity and the response function of the cavity. It is shown that the latter is a sum of two phase shifts of a circulating light wave: the phase incursion after reflection from the moving mirror and the phase incursion due to the direct interaction of gravitational and light waves in the cavity. The possibility of the resonance detection of high-frequency gravitational waves by using the optical rigidity effect is considered.

Real-time analysis of nonlinear transient response of weakly confined excitons

By means of the nonlocal transient-response theory, we elucidate the characteristics of the femtosecond transient response of thin films with a thickness beyond the long wavelength approximation (LWA) regime. In this regime, the contribution of higher excitonic states with a nondipole-type spatial structure becomes dominant and the interplay between the spatial structures of excitonic wavefunction and the radiation field plays an important role, causing an anomalous enhancement of nonlinear signal at specific size-energy resonant conditions. In addition, in the femtosecond pulse excitation, the interference of the signals from the multiple excitonic states, which are excited simultaneously by the incident pulse with a wide spectral width, can generate a greater diversity of optical response than expected by the steady-state analysis. This suggests the possibility that we can control the optical function of the nano-materials by the selective excitation of the aimed excitonic states using the laser pulse. This study serves as a theoretical basis for the nonlinear transient response by ultrashort pulse excitation of excitons confined in nano-structures.

Modeling electrocortical activity through improved local approximations of integral neural field equations

Neural field models of firing rate activity typically take the form of integral equations with space-dependent axonal delays. Under natural assumptions on the synaptic connectivity we show how one can derive an equivalent partial differential equation (PDE) model that properly treats the axonal delay terms of the integral formulation. Our analysis avoids the so-called long-wavelength approximation that has previously been used to formulate PDE models for neural activity in two spatial dimensions. Direct numerical simulations of this PDE model show instabilities of the homogeneous steady state that are in full agreement with a Turing instability analysis of the original integral model. We discuss the benefits of such a local model and its usefulness in modeling electrocortical activity. In particular, we are able to treat "patchy" connections, whereby a homogeneous and isotropic system is modulated in a spatially periodic fashion. In this case the emergence of a "lattice-directed" traveling wave predicted by a linear instability analysis is confirmed by the numerical simulation of an appropriate set of coupled PDEs.

Anomalous exciton–radiation coupling in the nano-to-bulk crossover regime

Anomalous exciton–radiation coupling in the sample-size regime beyond the long-wavelengthapproximation (LWA) is discussed. In high-quality samples fabricated using thelatest technologies, the long-range excitonic coherence and non-LWA effect cause apeculiar interplay between the spatial structures of the radiation and excitonicwaves, which emerge as an anomalous level structure of the exciton–radiationcoupled system including an extremely large radiative correction. We theoreticallydemonstrated such a situation for a thin film by a microscopic nonlocal theoryin which a size-resonant enhancement of the radiative damping constant andsuccessive interchanges of the confined polariton states can be observed. In order toverify the predicted level structure of confined polaritons, we present the results ofnon-degenerate two-photon scattering spectroscopy investigations and a correspondinganalysis based on the nonlocal theory for the second-order nonlinear process. Thisscheme has successfully elucidated peculiar level structures of confined polaritonsin the non-LWA regime. In the last half of the paper, a new type of resonanthyper-parametric scattering (RHPS) spectroscopy with a non-degenerate inducedexcitation is theoretically proposed as another method to study confined polaritons in thenon-LWA regime. Some calculations for thin films are demonstrated to reveal thepotentiality of this method that provides rich information on confined biexcitons andpolaritons.

On the evolution of tachyonic perturbations at super-Hubble scales

In the slow-roll inflationary scenario, the amplitude of the curvature perturbationsapproaches a constant value soon after the modes leave the Hubble radius. However,relatively recently, it was shown that the amplitude of the curvature perturbations inducedby the canonical scalar field can grow at super-Hubble scales if there is either a transitionto fast roll inflation or if inflation is interrupted for some period of time. In this work, weextend the earlier analysis to the case of a non-canonical scalar field described by theDirac–Born–Infeld action. With the help of a specific example, we show thatthe amplitude of the tachyonic perturbations can be enhanced or suppressed atsuper-Hubble scales if there is a transition from slow roll to fast roll inflation. We alsoillustrate as to how the growth of the entropy perturbations during the fast rollregime proves to be responsible for the change in the amplitude of the curvatureperturbations at super-Hubble scales. Furthermore, following the earlier analysis for thecanonical scalar field, we show that the power spectrum evaluated in the longwavelength approximation matches the exact power spectrum obtained numericallyvery well. Finally, we briefly comment on an application of this phenomenon.

Quantum Hall effect in carbon nanotubes and curved graphene strips

We develop a long-wavelength approximation in order to describe the low-energy states of carbon nanotubes in a transverse magnetic field. We show that in the limit where the square of the magnetic length $l=\sqrt{\ensuremath{\hbar}c∕eB}$ is much larger than the C-C distance times the nanotube radius $R$, the low-energy theory is given by the linear coupling of a two-component Dirac spinor to the corresponding vector potential. We investigate in this regime the evolution of the band structure of zigzag nanotubes for values of $R∕l>1$, showing that for radius $R\ensuremath{\approx}20\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ a clear pattern of Landau levels starts to develop for magnetic field strength $B\ensuremath{\gtrsim}10\phantom{\rule{0.3em}{0ex}}\mathrm{T}$. The levels tend to be fourfold degenerate, and we clarify the transition to the typical twofold degeneracy of graphene as the nanotube is unrolled to form a curved strip. We show that the dynamics of the Dirac fermions leads to states which are localized at the flanks of the nanotube and that carry chiral currents in the longitudinal direction. We discuss the possibility of observing the quantization of the Hall conductivity in thick carbon nanotubes, which should display steps at even multiples of $2{e}^{2}∕h$, with values doubled with respect to those in the odd-integer quantization of graphene.

THE IMPORTANCE OF THE "MAGNETIC" COMPONENTS OF GRAVITATIONAL WAVES IN THE RESPONSE FUNCTIONS OF INTERFEROMETERS

With an enlighting analysis, Baskaran and Grishchuk have recently shown the presence and importance of the so-called "magnetic" components of gravitational waves (GWs), which have to be taken into account in the context of the total response functions of interferometers for GWs propagating from arbitrary directions. In this paper, more detailed angular and frequency dependences of the response functions for the magnetic components are given in the approximation of wavelength much larger than the linear dimensions of the interferometer, with a specific application to the parameters of the LIGO and Virgo interferometers. The results of this paper agree with the work of Baskaran and Grishchuk, in which it has been shown that the identification of "electric" and "magnetic" contributions is unambiguous in the long-wavelength approximation. At the end of this paper, the angular and frequency dependences of the total response functions of the LIGO and Virgo interferometers are given. In the high-frequency regime, the division on "electric" and "magnetic" components becomes ambiguous, thus the full theory of gravitational waves has to be used. Our results are consistent with the ones of Baskaran and Grishchuk for this case.

Surface photoeffect: photoabsorption of surface and image states

The surface photoeffect of Al(001) is studied for a p continuous laser in the long wavelength (LWL), i.e. λ ≥ 100 Å. Perpendicular to a gas–solid interface the potential, the electron density, and consequently the laser fields, vary at the atomic scale and the LWL approximation, corresponding to constant in space laser fields, does not apply. The vector potential associated with these laser fields, written in the temporal gauge U = 0, is obtained by solving the Maxwell and the material equations. In the present model the dielectric function, entering these material equations, is dependent on the electron density ρ(z) perpendicular to the surface calculated from the Schrödinger equation and on the bulk dielectric constant. Using these calculated spatially varying laser fields, the photoabsorption spectrum of the surface, the Fermi and the first image states of Al(001) are calculated using the Fermi golden rule. For the first time in the case of a realistic potential, the photoabsorption yield is analysed in terms of the standard velocity, the surface and the interference contributions. All these terms contribute significantly to the total yield and this yield, calculated up to 25 eV, compares fairly well with the experiment.