General Analytical Formulas Research Articles

Can tunneling current in molecular junctions be so strongly temperature dependent to challenge a hopping mechanism? Analytical formulas answer this question and provide important insight into large area junctions.

Analytical equations like Richardson-Dushman's or Shockley's provided a general, if simplified conceptual background, which was widely accepted in conventional electronics and made a fundamental contribution to advances in the field. In the attempt to develop a (highly desirable, but so far missing) counterpart for molecular electronics, in this work, we deduce a general analytical formula for the tunneling current through molecular junctions mediated by a single level that is valid for any bias voltage and temperature. Starting from this expression, which is exact and obviates cumbersome numerical integration, in the low and high temperature limits we also provide analytical formulas expressing the current in terms of elementary functions. They are accurate for broad model parameter ranges relevant for real molecular junctions. Within this theoretical framework we show that: (i) by varying the temperature, the tunneling current can vary by several orders of magnitude, thus debunking the myth that a strong temperature dependence of the current is evidence for a hopping mechanism, (ii) real molecular junctions can undergo a gradual (Sommerfeld-Arrhenius) transition from a weakly temperature dependent to a strongly ("exponential") temperature dependent current that can be tuned by the applied bias, and (iii) important insight into large area molecular junctions with eutectic gallium indium alloy (EGaIn) top electrodes can be gained. E.g., merely based on transport data, we estimate that the current carrying molecules represent only a fraction of f ≈ 4 × 10-4 out of the total number of molecules in a large area Au-S-(CH2)13-CH3/EGaIn junction.

Analytical Model of Gear Thermal Stiffness Based on the Potential Energy Method

Stiffness is the main internal excitation of gear system vibration and is the basis for studying the static and dynamic characteristics of the system. Based on the relationship between thermal stress and strain in plane strain theory and combined with the potential energy method, a novel gear thermal stiffness model is proposed in this paper. The model deduces more general analytical formulas for gear thermal mesh stiffness and time-varying thermal mesh stiffness. The results show that the gear thermal stiffness decreases as a whole compared with the elastic stiffness. The decrease in stiffness is nonuniform along the line of action. The effects of temperatures, temperature distributions and thermal stress on gear thermal stiffness are investigated. The accuracy of the model is verified by comparisons with existing methods and references.

The evolutions of spin density of tightly focused vector fields with spatially-variant uniform and non-uniform ellipticities

Recently, the emergence of transverse spin has aroused enormous interest research. Here, we present general analytical formulas of the tightly focused electric fields and spin densities for polarized beams with spatially-variant uniform and non-uniform ellipticities. We explore the evolution of spin densities from the Poincaré sphere view using the tightly focused azimuthal-variant first-order cylindrically polarized beams and azimuthal-variant first-order hybrid polarized beams as examples. This reveals the roles and contributions of the relative intensity fraction and initial phase in the modulation of spin density. These two degrees of freedom have different effects on the polarization distribution of these two types of input fields, and thus on the spin density of tightly focused fields. These findings enrich the spin structure of the light field and contribute to the modulation of the spin density.

Neuroendocrine pancreatic tumors: about 13 cases

This article intends to revisit the electro-diffusional theory for the wall shear stress measurement from mass transfer probes of rectangular shape by considering the existence of two components of the wall shear rate (i.e., axial and transversal). General analytical formulas for the effective transfer length and the dimensionless mass transport coefficient were derived as a function of two parameters: a dimensionless angle of the flow direction, relative to the leading edge of the probe, and the aspect ratio between the width and the length of the strip probe. The correctness of the analytical relations for arbitrary flow direction and the aspect ratio was confirmed by numerical solutions of the transport equation in the convective-diffusive regime. It has also been proved that the differences between the Lévêque solution and the general analytical formula exhibit a significant deviation for a specific range of parameters. In the case of the three-dimensional boundary layers, in addition to the magnitude of the wall shear stress, the direction of the fluid flow in the vicinity of the probe’s surface is of paramount importance. Accordingly, a measurement methodology is proposed using two strip probes with different aspect ratios. The resulting equations required to quantify the magnitude of the wall shear rate vector and the dimensionless angle are also derived.

Analytical buckling load formulas for cylindrical shell structures with variable thickness under non-uniform external pressure

The novel contribution of this paper is to develop a quadratic perturbation method to derive general analytical buckling load formulas for cylindrical shell structures with varying wall thickness under non-uniform lateral pressure loads for the first time. Arbitrary thickness and lateral pressure loads are skillfully described and governing differential equations are derived. A quadratic perturbation method is developed to derive buckling load formulas, which establishes the relation among buckling loads, thickness, and load functions. The presented formulas are adequately validated by comparing with recent results for variable wall thickness or non-uniform external pressure, and exhibit a great advantage in determining buckling loads. For the purpose of engineering application, buckling of a cylinder with linear thickness under linear liquid pressure and wind load is deeply studied by the presented formulas. This study can provide a highly efficient method for buckling load computation, and a guide to design the thickness of the shell under non-uniform lateral loads.

On the impedance of a high-frequency capacitive discharge with different excitation methods

Consideration is given to a low-pressure ( << ) capacitive high-frequency discharge with large-area electrodes when it is excited by an electromagnetic field with a frequency from 13 to 900 MHz. General analytical formulas are obtained for the amplitudes of natural waves and the impedance of the discharge. It is taken into account that the excitation of surface waves and higher nonpropagating modes occurs both due to the axial inhomogeneity of the plasma-metal layer structure and due to edge effects at the electrode cut. The higher ampli-tude of the resonance modes in this case (in comparison with the excitation of the discharge by a TEM wave) leads to a greater irregularity in the dependence of the discharge impedance on the electron density. This conclusion is confirmed by the direct calculation of impedance using the Comsol Multiphysics® software.

Gravitational production of a conformal dark sector

Dark sectors with purely gravitational couplings to the Standard Model are unavoidably populated from the SM plasma by graviton exchange, and naturally provide dark matter candidates. We examine the production in the relativistic regime where the dark sector is approximately scale invariant, providing general analytical formulas that depend solely on the central charge of the dark sector. We then assess the relevance of interactions that can lead to a variety of phenomena including thermalisation, non-perturbative mass gaps, out-of-equilibrium phase transitions and cannibalism in the dark sector. As an illustrative example we consider the dark glueball scenario in this light and show it to be a viable dark matter candidate due to the suppression of gravitational production. We go on to extend these results to strongly coupled CFTs and their holographic duals at large-N with the dark dilaton as the dark matter candidate.

Equilibrium models with heterogeneous agents under rational expectations and its numerical solution

In this work we assume rational expectations to pose general equilibrium models with heterogeneous firms that can enter or exit the industry. More precisely, we assume a general Ito process for the dynamics of the agents productivity, including the main dynamics in the literature. A Hamilton-Jacobi-Bellman (HJB) formulation models the endogenous decision of firms to remain or exit the industry. All firms that exit are immediately replaced by a group of new ones, so that the probability density function of firms satisfies an appropriate Kolmogorov-Fokker-Plank (KFP) equation with source term. Equilibrium models are completed with the household problem formulation and the feasibility conditions. In the evolutive and general stationary settings, analytical or semi-analytical formulas are not available, so that appropriate numerical methods are required. We propose a Crank-Nicolson scheme for the time discretization of the evolutive problems. Moreover, we use an augmented Lagrangian active set (ALAS) method combined with a finite difference discretization for the HJB formulation and a suitable finite differences discretization for the KFP problem. For the global equilibrium problem we propose a Steffensen algorithm. Numerical examples illustrate the performance of the proposed numerical methodologies as well as the expected behaviours of the computed economic variables.

A revisit of the electro-diffusional theory for the wall shear stress measurement

This article intends to revisit the electro-diffusional theory for the wall shear stress measurement from mass transfer probes of rectangular shape by considering the existence of two components of the wall shear rate (i.e., axial and transversal). General analytical formulas for the effective transfer length and the dimensionless mass transport coefficient were derived as a function of two parameters: a dimensionless angle of the flow direction, relative to the leading edge of the probe, and the aspect ratio between the width and the length of the strip probe. The correctness of the analytical relations for arbitrary flow direction and the aspect ratio was confirmed by numerical solutions of the transport equation in the convective-diffusive regime. It has also been proved that the differences between the Lévêque solution and the general analytical formula exhibit a significant deviation for a specific range of parameters. In the case of the three-dimensional boundary layers, in addition to the magnitude of the wall shear stress, the direction of the fluid flow in the vicinity of the probe’s surface is of paramount importance. Accordingly, a measurement methodology is proposed using two strip probes with different aspect ratios. The resulting equations required to quantify the magnitude of the wall shear rate vector and the dimensionless angle are also derived.

Outer Radiation Belt Electron Lifetime Model Based on Combined Van Allen Probes and Cluster VLF Measurements

AbstractThe flux of energetic electrons in the outer radiation belt shows a high variability. The interactions of electrons with very low frequency (VLF) chorus waves play a significant role in controlling the flux variation of these particles. Quantifying the effects of these interactions is crucially important for accurately modeling the global dynamics of the outer radiation belt and to provide a comprehensive description of electron flux variations over a wide energy range (from the source population of 30 keV electrons up to the relativistic core population of the outer radiation belt). Here, we use a synthetic chorus wave model based on a combined database compiled from the Van Allen Probes and Cluster spacecraft VLF measurements to develop a comprehensive parametric model of electron lifetimes as a function of L‐shell, electron energy, and geomagnetic activity. The wave model takes into account the wave amplitude dependence on geomagnetic latitude, wave normal angle distribution, and variations of wave frequency with latitude. We provide general analytical formulas to estimate electron lifetimes as a function of L‐shell (for L = 3.0 to L = 6.5), electron energy (from 30 keV to 2 MeV), and geomagnetic activity parameterized by the AE index. The present model lifetimes are compared to previous studies and analytical results and also show a good agreement with measured lifetimes of 30 to 300 keV electrons at geosynchronous orbit.

On the Prospects of Multiport Devices for Photon-Number-Resolving Detection

Ideal photon-number-resolving detectors form a class of important optical components in quantum optics and quantum information theory. In this article, we theoretically investigate the potential of multiport devices having reconstruction performances approaching that of the Fock-state measurement. By recognizing that all multiport devices are minimally complete, we first provide a general analytical framework to describe the tomographic accuracy (or quality) of these devices. Next, we show that a perfect multiport device with an infinite number of output ports functions as either the Fock-state measurement when photon losses are absent or binomial mixtures of Fock-state measurements when photon losses are present and derive their respective expressions for the tomographic transfer function. This function is the scaled asymptotic mean squared error of the reconstructed photon-number distributions uniformly averaged over all distributions in the probability simplex. We then supply more general analytical formulas for the transfer function for finite numbers of output ports in both the absence and presence of photon losses. The effects of photon losses on the photon-number resolving power of both infinite- and finite-size multiport devices are also investigated.

Series-, Parallel-, and Inter-Connection of Solid-State Arbitrary Fractional-Order Capacitors: Theoretical Study and Experimental Verification

In this paper, general analytical formulas are introduced for the determination of equivalent impedance, magnitude, and phase, i.e., order, for ${n}$ arbitrary fractional-order capacitors (FoCs) connected in series, parallel, and their interconnection. The approach presented helps to evaluate these relevant quantities in the fractional domain, since the order of each element has a significant effect on the impedance of each FoC and their equivalent capacitance cannot be considered. Three types of solid-state fractional-order passive capacitors of different orders, using ferroelectric polymer and reduced graphene oxide-percolated P(VDF-TrFE-CFE) composite structures, are fabricated and characterized. Using an impedance analyzer, the behavior of the devices was found to be stable in the frequency range 0.2 – 20 MHz, with a phase angle deviation of ±4°. Multiple numerical and experimental case studies are given, in particular for two and three connected FoCs. The fundamental issues of the measurement units of the FoCs connected in series and parallel are derived. A MATLAB open-access source code is given in the Appendix for easy calculation of the equivalent FoC magnitude and phase. The experimental results are in good agreement with the theoretical assumptions.

A study on elastic properties of carbon nanocones based on a temperature-related quasi-continuum model

ABSTRACTIn the present paper, general analytical formulas for calculating the Young's modulus and Poisson's ratio of single-walled carbon nanocones (SWCNCs) at finite temperatures are derived based on the proposed temperature-related multiscale quasi-continuum (QC) model. To this end, a temperature-related higher-order Cauchy-Born (THCB) rule is employed to establish the hyper-elastic constitutive model of SWCNCs. With use of the proposed approach, the influences of the temperature, apex angle, rotation angle of cutting lines and top end radius on the Young's moduli and Poisson's ratios of SWCNCs are investigated systematically.

Ray Tracing and Modal Methods for Modeling Radio Propagation in Tunnels With Rough Walls.

At the ultrahigh frequencies common to portable radios, tunnels such as mine entries are often modeled by hollow dielectric waveguides. The roughness condition of the tunnel walls has an influence on radio propagation, and therefore should be taken into account when an accurate power prediction is needed. This paper investigates how wall roughness affects radio propagation in tunnels, and presents a unified ray tracing and modal method for modeling radio propagation in tunnels with rough walls. First, general analytical formulas for modeling the influence of the wall roughness are derived, based on the modal method and the ray tracing method, respectively. Second, the equivalence of the ray tracing and modal methods in the presence of wall roughnesses is mathematically proved, by showing that the ray tracing-based analytical formula can converge to the modal-based formula through the Poisson summation formula. The derivation and findings are verified by simulation results based on ray tracing and modal methods.

Efficiency of Metasurface Antennas

Different types of efficiencies are introduced for metasurface (MTS) antennas based on single point-source excitation: 1) the feed efficiency $\varepsilon _{\mathrm{ feed}} $ is the power delivered by the feed to surface wave (SW) over the input power; 2) the tapering efficiency $\varepsilon _{\mathrm{ tap}} $ quantifies the deviation with respect to a uniform illumination; 3) the conversion efficiency $\varepsilon _{\mathrm{ conv}} $ defines the capability of the MTS to transform the SW power into leaky-wave power; and 4) the ohmic efficiency $\varepsilon _{\Omega } $ accounts for the antenna losses. The product of all of these factors provides the overall efficiency of the antenna. New, quite general analytical formulas are suggested to estimate any individual efficiency. In particular, it is seen that $\varepsilon _{\mathrm{ feed}} $ can reach also 90% by using planar or quasi-planar circularly symmetric sources. It is also proven that the product $\varepsilon _{\mathrm{ conv}} \varepsilon _{\mathrm{ tap}} $ cannot be larger than 58% for a uniform modulation; however, by using a nonuniform modulation, it can be increased to reach $\varepsilon _{\mathrm{ conv}} \varepsilon _{\mathrm{ tap}} \approx a / \lambda /(a / \lambda +2)$ , where $a/\lambda $ is the antenna radius in free-space wavelengths. A new closed-form formula for estimating the losses is given. In accordance with the prescription given here, one can reach 75% of global efficiency, including losses, until a radius of 20 wavelengths with commercially available substrates.

Unified semiclassical approach to electronic transport from diffusive to ballistic regimes**Project supported by the National Basic Research Program of China (Grant Nos. 2015CB921202 and 2014CB921103) and the National Natural Science Foundation of China (Grant No. 11225420).

We show that by integrating out the electric field and incorporating proper boundary conditions, a Boltzmann equation can describe electron transport properties, continuously from the diffusive to ballistic regimes. General analytical formulas of the conductance in D = 1,2,3 dimensions are obtained, which recover the Boltzmann–Drude formula and Landauer–Büttiker formula in the diffusive and ballistic limits, respectively. This intuitive and efficient approach can be applied to investigate the interplay of system size and impurity scattering in various charge and spin transport phenomena, when the quantum interference effect is not important.

Static electric and magnetic multipole susceptibilities for Dirac one-electron atoms in the ground state

We present tabulated data for several families of static electric and magnetic multipole susceptibilities for hydrogenic atoms with nuclear charge numbers from the range 1⩽Z⩽137. Atomic nuclei are assumed to be point-like and spinless. The susceptibilities considered include the multipole electric polarizabilities αEL→EL and magnetizabilities (magnetic susceptibilities) χML→ML with 1⩽L⩽4 (i.e., the dipole, quadrupole, octupole and hexadecapole ones), the electric-to-magnetic cross-susceptibilities αEL→M(L−1) with 2⩽L⩽5 and αEL→M(L+1) with 1⩽L⩽4, the magnetic-to-electric cross-susceptibilities χML→E(L−1) with 2⩽L⩽5 and χML→E(L+1) with 1⩽L⩽4 (it holds that χML→E(L∓1)=αE(L∓1)→ML), and the electric-to-toroidal-magnetic cross-susceptibilities αEL→TL with 1⩽L⩽4. Numerical values are computed from general exact analytical formulas, derived by us elsewhere within the framework of the Dirac relativistic quantum mechanics, and involving generalized hypergeometric functions 3F2 of the unit argument.

Combined effect of oriented strain and external magnetic field on electrical properties of superlattice-graphene nanoribbons

Using the tight-binding model, electronic quantum transport properties of strained superlattice-graphene nanoribbons (SGNRs) attached to two semi-infinite metallic leads are studied in real space and numerically. Then, with the combination of mode space (MS) and renormalization methods, general analytical formulas for the conductance and energy band gap of the strained system are derived. The formulas are useful in studying the impact of slice-like defects and uniaxial strains on the electronic transport properties of the system as well as in reducing the computation time. The calculations are based on the Green’s function method, in which the effects of uniaxial strains and the concentration of boron nitride (BN) slices as well as magnetic fields on the electrical conductance and the band gap of the armchair SGNR are studied. It is shown that the conductance of the system reduces with increasing BN concentration so a gap opens and its value increases. Both uniaxial strains and the BN impurities cause the metal–semiconductor phase transition, while the magnetic fields induce a periodic metal–semiconductor transition. The electronic transport properties of the systems can be controlled by tunable parameters such as the BN concentration, oriented strain and magnetic field.

Breaking classical Lie groups to finite subgroups – an automated approach

The decomposition of representations of compact classical Lie groups into representations of finite subgroups is discussed. A Mathematica package is presented that can be used to compute these branching rules using the Weyl character formula. For some low order finite groups including A4 and Δ(27) general analytical formulas are presented for the branching rules of arbitrary representations of their smallest Lie super-groups.

A Numerical Model of MHD Waves in a 3D Twisted Solar Flux Tube

We generalize our analytical and numerical models of the solar flux tube on twisted magnetic-field lines. The basic equations and numerical methods have much in common with those of Murawski et al. (Astron. Astrophys. 577, A126, 2015); the new and important issue is the twisted magnetic-field component that couples explored torsional Alfven and magnetoacoustic waves. In these models we specify a magnetic-flux function and derive general analytical formulas for the equilibrium mass density and a gas pressure. We use the developed models, which can be adopted for any axisymmetric structure with twisted and untwisted magnetic lines, to simulate the MHD waves. These waves are excited by a localized pulse in the azimuthal velocity component that is launched at the top of the solar photosphere. Their propagation through the solar chromosphere and transition region to the solar corona reveals a complex scenario of twisted magnetic-field lines and flows associated with torsional Alfven and magnetoacoustic waves.