Wentzel Kramers Brillouin Research Articles

The waveguide invariant for a Pekeris waveguide.

A closed-form waveguide invariant β for a Pekeris waveguide is derived. It is based on the modal Wentzel-Kramers-Brillouin (WKB) dispersion equation and implicit differentiation, in conjunction with the concept of the "effective boundary depth," ΔH(θ), where θ is the propagation angle. First, an explicit formula for β(m,n) between mode pairs is obtained assuming an ideal waveguide of the effective waveguide depth, H+ΔH(θ), and provides an excellent agreement with the reference value for the Pekeris waveguide of depth H obtained using the normal mode program kraken. Then, a closed-form expression for a group of adjacent modes is derived: β=(H+ΔH(θ))/(H/ cos2 θ-ΔH(θ)), which can be approximated by β=cos2 θ as ΔH(θ)/H≪1, the analytical expression for an ideal waveguide.

Quasi-Compact Model of Direct Source-to-Drain Tunneling Current in Ultrashort-Channel Nanosheet MOSFETs by Wavelet Transform

We present an analytical approach for the calculation of direct source-to-drain tunneling (DSDT) probability of electrons in gate-all-around (GAA) silicon nanosheet (SiNS) MOSFETs. The used method is based on the wavelet transform and leads to a quasi-compact model (QCM) for the DSDT current of ultrashort-channel devices. Among them, we introduce a four-piece parabolic approximation method for the conduction band edge and present analytical expressions for the tunneling distances of electrons with different energy levels. The development of a QCM is achieved by limiting the number of interpolation points for the tunneling current density to seven specific electron energies, distributed around the energy level that makes the largest contribution to the tunneling current. A further simplification is achieved by the Gaussian approximation of the tunneling current density in transverse direction so that only the center and surface potentials ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Phi _ {\text {C}}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Phi _ {\text {S}}$ </tex-math></inline-formula> ) at the barrier are of interest for the modeling. For comparison, all those approximations are also implemented in the Wentzel–Kramer–Brillouin (WKB) approximation. Furthermore, the approach is verified by nonequilibrium Green’s function (NEGF) simulation.

Light scattering by horizontally oriented square pyramid in the Wentzel–Kramers–Brillouin approximation

In this research, we studied the sensitivity of the form factor of a pyramidal ice crystal in the Wentzel–Kramers–Brillouin (WKB) approximation to its geometrical characteristics: the shape, spatial orientation, aspect ratio and size parameter. Using the WKB method, we derive an analytical formula of the form factor of horizontally oriented square pyramid. We will begin this work by applying the WKB approximation to the case of a particle which rotates about its main axis. Then, we will move to deal with the case of a particle which rotates about an axis perpendicular to its main axis in a future work. In addition, the coefficient of extinction is also given. To illustrate our analytical results, some numerical examples are analyzed.

Enhancement effect of frequency chirp on vacuum electron-positron pair production in strong field

In this review article, we show an important aspect of electron-positron pair production from vacuum under strong background field where the frequency chirping plays a key role in enhancing the pair production. A series of researches on the enhancement effect of frequency chirp on electron-positron pair production in strong field is summarized. Three approaches are introduced, i.e. the Dirac-Heisenberg-Wigner formalism used to treat the spatial inhomogeneous field or/and multidimensional homogeneous time-dependent field, quantum Vlasov equation to cope with the one-dimensional homogeneous time-dependent field, and the computation quantum field theory employed to study the problem with external potential. Some interesting results about the momentum spectrum structure of created particle and the yielding of pair numbers are demonstrated for various different field parameters such as field strength and central frequency, in particular their significant influence on results when the frequency chirping form or/and strength are changed. In general, the number density can be improved by 2-3 orders of magnitude with the strengthening of frequency chirping in comparison with that without chirping for low frequency field, which is attributed to the effect that the dynamically assisted mechanism plays a significant role since the chirping expands the frequency spectrum of field. For high frequency field, however, this effect is suppressed so that the number density is enhanced by about a few times. For spatially inhomogeneous field, field changing on a small scale does not make the number density so high and the frequency chirping can enhance the yield in the order of magnitude, while the field changing on a large scale makes the number density to approach to that of homogeneous field and the chirping increases the yield by a few times. These numerical results can be understood by the Wentzel-Kramer-Brillouin (WKB) approximation and the structure of turning points. Finally the possible applicable prospects of the electron-positron pair production by the frequency chirping are presented briefly.

Analytical Study of Skew-cosh-Gaussian Laser Beam Propagation through Collisionless Plasma

In the present theoretical investigation the domains of the order of skew-cosh-Gaussian (skew-chG) laser beams for agiven set of skewness parameters and its significant effect on the self-focusing/defocusing in collisionless plasma has beenstudied. The nonlinearity in the dielectric function of collisionless plasma is mainly due to the ponderomotive force. Byfollowing parabolic wave equation approach under Wentzel-Kramers-Brillouin (WKB) and paraxial approximations,nonlinear coupled differential equations are established for the beam width parameters. These equations have been solvednumerically by using fourth-order Runge-Kutta method. The results obtained are presented graphically and discussed.

Some explicit solutions of the three-dimensional Euler equations with a free surface

We present a family of radial solutions (given in Eulerian coordinates) to the three-dimensional Euler equations in a fluid domain with a free surface and having finite depth. The solutions that we find exhibit vertical structure and a non-constant vorticity vector. Moreover, the flows described by these solutions display a density that depends on the depth. While the velocity field and the pressure function corresponding to these solutions are given explicitly through (relatively) simple formulas, the free surface defining function is specified (in general) implicitly by a functional equation which is analysed by functional analytic methods. The elaborate nature of the latter functional equation becomes simpler when the density function has a particular form leading to an explicit formula of the free surface. We subject these solutions to a stability analysis by means of a Wentzel–Kramers–Brillouin (WKB) ansatz.

Propagation of two relativistic hollow Gaussian laser beams in magnetized plasma: extended paraxial theory

This work presents an analytical and numerical study for the mutual interaction and propagation of two intense hollow Gaussian laser beams (HGLBs) at difference frequency in a magnetized plasma. This work has been done in the presence of relativistic nonlinearity, which depends on the intensity of both laser beams. In this process, the propagation of both laser beams in the plasma is described by cross-focusing, as the behavior of one laser beam affects that of another laser beam. The nonlinear differential equations for the beamwidth of HGLBs have been obtained within the Wentzel–Kramers–Brillouin (WKB) method and higher-order paraxial theory. The expression for the nonlinear dielectric constant of the plasma has been obtained from the maximum irradiance of the HGLBs. The obtained equations have been solved numerically to explore the effect of the well-established laser and plasma parameters on the focusing of both laser beams in a magnetized plasma. The results have been compared with paraxial ray theory and Gaussian profile of the laser beams. The focusing/intensity of both laser beams increases in the extended paraxial region compared to the paraxial region in the magnetized plasma. The focusing of both laser beams increases even at high values of beam order and electron cyclotron frequency in the extended paraxial region.

Efficient Modeling of Charge Trapping at Cryogenic Temperatures—Part I: Theory

Charge trapping is arguably the most important detrimental mechanism distorting the ideal characteristics of MOS transistors, and nonradiative multiphonon (NMP) models have been demonstrated to provide a very accurate description. For the calculation of the NMP rates at room temperature or above, simple semiclassical approximations have been successfully used to describe this intricate mechanism. However, for the computation of charge transition rates at cryogenic temperatures, it is necessary to use the full quantum mechanical description based on Fermi’s golden rule. Since this is computationally expensive and often not feasible, we discuss an efficient method based on the Wentzel–Kramers–Brillouin (WKB) approximation in combination with the saddle point method and benchmark this approximation against the full model. We show that the approximation delivers excellent results and can, hence, be used to model charge trapping behavior at cryogenic temperatures.

Fluid focusing and viscosity allow high gain and stability of the cochlear response.

This paper discusses the role of two-dimensional (2-D)/three-dimensional (3-D) cochlear fluid hydrodynamics in the generation of the large nonlinear dynamical range of the basilar membrane (BM) and pressure response, in the decoupling between cochlear gain and tuning, and in the dynamic stabilization of the high-gain BM response in the peak region. The large and closely correlated dependence on stimulus level of the BM velocity and fluid pressure gain [Dong, W., and Olson, E. S. (2013). Biophys. J. 105(4), 1067-1078] is consistent with a physiologically oriented schematization of the outer hair cell (OHC) mechanism if two hydrodynamic effects are accounted for: amplification of the differential pressure associated with a focusing phenomenon, and viscous damping at the BM-fluid interface. The predictions of the analytical 2-D Wentzel-Kramers-Brillouin (WKB) approach are compared to solutions of a 3-D finite element model, showing that these hydrodynamic phenomena yield stable high-gain response in the peak region and a smooth transition among models with different effectiveness of the active mechanism, mimicking the cochlear nonlinear response over a wide stimulus level range. This study explains how an effectively anti-damping nonlinear outer hair cells (OHC) force may yield large BM and pressure dynamical ranges along with an almost level-independent admittance.

Thermodynamics of Bardeen regular black hole with generalized uncertainty principle

This study explores the emission of massive charged spin-1 particles from the background of Bardeen regular spacetime by the semi-classical method used to study the Hawking radiation spectrum. We employed the Hamilton–Jacobi method and Wentzel–Kramers–Brillouin approximation technique with the suitable form of the wave function to solve the Proca field equation. We calculated the tunneling probability of outgoing spin-1 particles and the corresponding thermodynamic temperature. Furthermore, we obtained the modified thermodynamic quantities like temperature, entropy as well as heat capacity by utilizing the quadratic form of generalized uncertainty principle (GUP) and minimal length. In the end, we investigated the local stability as well as phase transitions of the Bardeen black hole in the context of GUP-modified heat capacity.

The MHD Newtonian hybrid nanofluid flow and mass transfer analysis due to super-linear stretching sheet embedded in porous medium

The steady magnetohydrodynamics (MHD) incompressible hybrid nanofluid flow and mass transfer due to porous stretching surface with quadratic velocity is investigated in the presence of mass transpiration and chemical reaction. The basic laminar boundary layer equations for momentum and mass transfer, which are non-linear partial differential equations, are converted into non-linear ordinary differential equations by means of similarity transformation. The mass equation in the presence of chemical reaction is a differential equation with variable coefficients, which is transformed to a confluent hypergeometric differential equation. The mass transfer is analyzed for two different boundary conditions of concentration field that are prescribed surface concentration (PSC) and prescribed mass flux (PMF). The asymptotic solution of concentration filed for large Schmidt number is analyzed using Wentzel-Kramer-Brillouin (WKB) method. The parameters influence the flow are suction/injection, superlinear stretching parameter, porosity, magnetic parameter, hybrid nanofluid terms, Brinkman ratio and the effect of these are analysed using graphs.

Retrieving the refractive index of a sphere from the phase spectrum of its light-scattering profile

We studied the Fourier spectrum of the light-scattering profiles of single particles in the Rayleigh-Gans-Debye (RGD) and Wentzel–Kramers–Brillouin (WKB) approximations. In the case of a homogeneous sphere, we found the relationship between the key parameters of the spectrum (including its phase) and the sphere characteristics – both analytically and numerically in the framework of the approximations and the rigorous Lorentz–Mie theory, respectively. Based on these results, we have improved the existing spectral characterization method for spheres extending the applicability range to particles with a higher refractive index.

Modeling of the Gate Tunneling Current in MFIS NCFETs

In this article, we present a model for the gate tunneling current (GTC) in metal-ferroelectric-insulator-semiconductor (MFIS) negative capacitance FETs (NCFETs), which, to the best of our knowledge, is the first such report. The model is numerical in nature, and is developed using the Tsu–Esaki formulation, employing the Wentzel–Kramer–Brillouin (WKB) approximation, in order to estimate the transmission coefficients of the carriers through the barriers. The ferroelectric (FE) material considered is HfO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> , and is modeled using the Landau phase transition theory. Simulation results reveal a remarkable nonmonotonic dependence of the GTC on the FE layer thickness, an effect that we explain through the Landau model. Furthermore, it is shown how this GTC can be reduced by orders of magnitude without changing the overall dielectric capacitance—a feature that may prove to be beneficial in low-power circuit designs. Additionally, it is seen that the GTC is a weak function of the remanent polarization and coercive field of the FE. All the model predictions are validated through a comparison with the results obtained from Sentaurus 2-D TCAD simulations. The novel results presented in this work should serve as a guide for detailed experimental studies on the gate current characteristics of MFIS NCFETs.

Quasinormal modes and shadow of a Schwarzschild black hole with GUP

We consider quantum corrections for the Schwarzschild black hole metric by using the generalized uncertainty principle (GUP) to investigate quasinormal modes, shadow and their relationship in the eikonal limit. We calculate the quasinormal frequencies of the quantum-corrected Schwarzschild black hole by using the sixth-order Wentzel–Kramers–Brillouin (WKB) approximation, and also perform a numerical analysis that confirms the results obtained from this approach. We also find that the shadow radius is nonzero even at very small mass limit for finite GUP parameter.

Analytical WKB theory for high-harmonic generation and its application to massive Dirac electrons

We propose an analytical approach to high-harmonic generation (HHG) for nonperturbative low-frequency and high-intensity fields based on the (Jeffreys-)Wentzel-Kramers-Brillouin (WKB) approximation. By properly taking into account Stokes phenomena of WKB solutions, we obtain wave functions that systematically include the repetitive dynamics of production and acceleration of electron-hole pairs and quantum interference due to phase accumulation between different pair production times (St\"uckelberg phase). Using the obtained wave functions without relying on any phenomenological assumptions, we explicitly compute electric current (including intra- and interband contributions) as the source of HHG for a massive Dirac system in $(1+1)$ dimensions under an ac electric field. We demonstrate that the WKB approximation agrees well with numerical results obtained by solving the time-dependent Schr\"odinger equation and point out that the quantum interference is important in HHG. We also predict in the deep nonperturbative regime that (1) harmonic intensities oscillate with respect to electric-field amplitude ${E}_{0}$ and frequency $\mathrm{\ensuremath{\Omega}}$, with a period determined by the St\"uckelberg phase, (2) the cutoff order of HHG is determined by $2e{E}_{0}/\ensuremath{\hbar}{\mathrm{\ensuremath{\Omega}}}^{2}$, with $e$ being the electron charge, and that (3) noninteger harmonics, controlled by the St\"uckelberg phase, appear as a transient effect. Our WKB theory is particularly suited for a parameter regime, where the Keldysh parameter $\ensuremath{\gamma}=(\mathrm{\ensuremath{\Delta}}/2)\mathrm{\ensuremath{\Omega}}/e{E}_{0}$, with $\mathrm{\ensuremath{\Delta}}$ being the gap size, is small. This parameter regime corresponds to intense lasers in the terahertz regime for realistic massive Dirac materials. Our analysis implies that the so-called HHG plateau can be observed at the terahertz frequency within the current technology.

Uniform-over-size approximation of the internal fields for scatterers with low refractive-index contrast

We improved the Wentzel-Kramers-Brillouin (WKB) approximation for calculating the electric field inside a scatterer. The new method, named WKBr, additionally takes into account refraction (rotation) of the incident rays at the particle-medium interface. For large particles the WKBr is equivalent to the geometric optics in the limit of relative refractive index m approaching 1. Thus, the resulting accuracy is determined by (m − 1) uniformly for all particle sizes, in contrast to the original WKB, which accuracy deteriorates with increasing size. We extensively studied several versions WKBr for the case of a homogeneous sphere. These versions additionally account for Fresnel transmission and wavefront focusing for rays entering the scatterer, combination of intersecting rays, and the vanishing of the electric field in the shadow region. While keeping the same order of errors with varying (m − 1), they significantly reduce the errors for moderate values of m (from 1.1 to 1.3). Further, we tested the WKB and WKBr as an initial guess in the iterative solution of the linear system in the framework of the discrete dipole approximation (DDA). It allowed us to accelerate the DDA solution without compromising the final accuracy. The acceleration becomes more pronounced with increasing size, while the WKBr becomes clearly superior to the WKB. Moreover, the new initialization of the iterative solver extends the practical applicability of the DDA to size parameter equal to 250, 160, and 140 for m = 1.1, 1.2, and 1.3, respectively. They could not be reached previously due to the stagnation of the iterative solution. We expect similar DDA acceleration for particles of other shapes. One can easily test it for the case of the WKB, since it is already implemented in the open-source ADDA code.

The waveguide invariant for a Pekeris waveguide

A closed-form waveguide invariant β for a Pekeris waveguide is derived. It is based on the modal Wentzel–Kramers–Brillouin (WKB) dispersion equation and implicit differentiation, in conjunction with the concept of the angle-dependent “effective boundary depth,'' ΔH(θ). First, an explicit expression for β(m,n) between mode pairs is derived assuming an ideal waveguide of the effective waveguide depth H+ΔH(θ), which provides an excellent agreement with the exact value calculated for the Pekeris waveguide of depth H using KRAKEN. Then, a closed-form expression for a group of adjacent modes is derived: β = (H + ΔH)/(H/cos2θ − ΔH), which reduces to β = cos2θ as ΔH/H ≪ 1, the analytical expression for an ideal waveguide. For applications to source-range estimation, the waveguide invariant β is used to derive the theoretical beam-time migration (i.e., dispersion curve) in a Pekeris waveguide.

A single-mode approximation for gravity waves in the thermosphere

In previous work (Knight et al., 2019) a multilayer Fourier method was introduced for obtaining solutions for a partial differential equation (PDE) associated with a dispersion relation for linear gravity waves in a vertically varying, anelastic, nonhydrostatic, viscous, and thermally diffusive atmosphere over some altitude range, given boundary conditions that allow waves to enter at the lower boundary and exit at the top. The previous work described an exact solution method for the dispersion-relation PDE using all wave modes. The current work describes a type of approximate solution, based on a single upgoing gravity-wave mode for each Fourier component, which is computationally cheaper and simpler to implement. Two different approaches to defining the single-mode solution are presented, one based on the Wentzel–Kramers–Brillouin (WKB) method and the other based on a simplified version of the multilayer approximation. The two approaches are shown to be mathematically equivalent. The accuracy of the single-mode approximation is investigated through comparison with the all-mode solution for a realistic example in which gravity waves generated by a convective storm in the troposphere propagate into the upper thermosphere and for which the single-mode approximation gives reasonably accurate results. It is necessary to include imaginary frequency shifts in some Fourier components in order for the different types of modes (e.g., upgoing vs. downgoing and gravity-wave vs. dissipative) to be well-defined in the presence of molecular viscosity. A systematic approach to determining the smallest possible imaginary frequency shift for any given Fourier component is introduced. The new method efficiently suppresses a specific type of numerical artifact associated with large imaginary frequency shifts.

Systematic study of two-proton radioactivity with a screened electrostatic barrier * *Supported in part by the National Natural Science Foundation of China (11205083, 11505100, 11705055), the Construct Program of the Key Discipline in Hunan Province, the Research Foundation of Education Bureau of Hunan Province, China (18A237), the Natural Science Foundation of Hunan Province, China (2018JJ3324), the Innovation Group of Nuclear

In this study, a phenomenological model is proposed based on Wentzel-Kramers-Brillouin (WKB) theory and applied to investigate the two-proton ( ) radioactive half-lives of nuclei near or beyond the proton drip line. The total diproton-daughter nucleus interaction potential is composed of the Hulthen-type electrostatic term and the centrifugal term. The calculated radioactive half-lives can accurately reproduce the existing 10 experimental datasets of five true radioactive nuclei with σ = 0.736. In addition, we extend this model to predict the half-lives of possible radioactive nuclei whose radioactivity is energetically allowed or observed but not yet quantified in NUBASE2016. The predicted results are in agreement with those obtained using the Gamow-like model, generalized liquid drop model, Sreeja formula, and Liu formula.

Effect of initial 3α cluster configurations in C12 on the direct decay of its Hoyle state

We investigate the implications of initial $3\ensuremath{\alpha}$ configurations in $^{12}\mathrm{C}$ corresponding to different decay modes of its Hoyle state on the penetrability ratios. Considering the second ${2}^{+}$ (10.03 MeV) state to be a collective excitation of the Hoyle state, the direct $3\ensuremath{\alpha}$ decay width for the Hoyle state has been calculated using the ratio of the barrier penetration probability of the Hoyle state to the ${2}^{+}$ state. Semiclassical Wentzel--Kramers--Brillouin (WKB) approximation has been employed to determine the penetrability ratio, resulting in an upper limit on the branching ratio of the direct decay of the Hoyle state in ``equal phase-space'' ($\mathrm{DD}\ensuremath{\phi}$) mode as $\frac{{\mathrm{\ensuremath{\Gamma}}}_{3\ensuremath{\alpha}}}{\mathrm{\ensuremath{\Gamma}}}&lt;3.1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$. However, this limit for ``linear chain'' (DDL) decay is $\frac{{\mathrm{\ensuremath{\Gamma}}}_{3\ensuremath{\alpha}}}{\mathrm{\ensuremath{\Gamma}}}&lt;2.6\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}7}$, which is one order of magnitude smaller than the $\mathrm{DD}\ensuremath{\phi}$ decay and the limit for ``equal energy'' (DDE) decay is $\frac{{\mathrm{\ensuremath{\Gamma}}}_{3\ensuremath{\alpha}}}{\mathrm{\ensuremath{\Gamma}}}&lt;1.5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$, which is greater than both $\mathrm{DD}\ensuremath{\phi}$ and DDL decays. It implies that the limit on direct decay probability is strongly dependent on the initial configuration of the $3\ensuremath{\alpha}$ cluster. A further probe using a bent-arm-like $3\ensuremath{\alpha}$ initial configuration shows that the direct decay probability is maximum when the angle of the bent arm is $\ensuremath{\approx}{120}^{\ensuremath{\circ}}$, an important ingredient for understanding the Hoyle-state structure.