On Calculating the Current-Voltage Characteristic of Multi-Diode Models for Organic Solar Cells

We provide an alternative formulation of the exact calculation of the current-voltage characteristic of solar cells which have been modeled with a lumped parameters equivalent circuit with one or two diodes. Such models, for instance, are suitable for describing organic solar cells whose current-voltage characteristic curve has an inflection point, also known as an S-shaped anomaly. Our formulation avoids the risk of numerical overflow in the calculation. It is suitable for implementation in Fortran, C or on micro-controllers

Tutorial: The quantum finite square well and the Lambert W function

We present a solution of the quantum mechanics problem of the allowable energy levels of a bound particle in a one-dimensional finite square well. The method is a geometric-analytic technique utilizing the conformal mapping w -\u3e z = we(w) between two complex domains. The solution of the finite square well problem can be seen to be described by the images of simple geometric shapes, lines, and circles, under this map and its inverse image. The technique can also be described using the Lambert W function. One can work in either of the complex domains, thereby obtaining additional insight into the finite square well problem and its bound energy states. This suggests interesting possibilities for the design of materials that are sensitive to minute changes in their environment such as nanostructures and the quantum well infrared photodetector

Cosmic Background Radiation

It is shown that a collection of photons with nearly the same frequency exhibits a “condensation” type of phenomenon corresponding to a peak intensity. The observed cosmic background radiation can be explained from this standpoint.We have obtained analogous results by extremization of the occupation number for photons with the use of the Lambert W function. Some of the interesting applications of this function are briefly discussed in the context of graphene which exhibits an interesting two dimensional structure with several characteristic properties and diverse practical applications

The Polylogarithm and the Lambert W Functions in Thermoelectrics

In this work, we determine the conditions for the extremum of the figure of merit, theta2, in a degenerate semiconductor for thermoelectric (TE) applications. We study the variation of the function theta2 with respect to the reduced chemical potential mu* using relations involving polylogarithms of both integral and nonintegral orders. We present the relevant equations for the thermopower, thermal, and electrical conductivities that result in optimizing theta2 and obtaining the extremum equations. We discuss the different cases that arise for various values of r, which depends on the type of carrier scattering mechanism present in the semiconductor. We also present the important extremum conditions for theta2 obtained by extremizing the TE power factor and the thermal conductivity separately. In this case, simple functional equations, which lead to solutions in terms of the Lambert W function, result. We also present some solutions for the zeros of the polylogarithms. Our analysis allows for the possibility of considering the reduced chemical potential and the index r of the polylogarithm as complex variables

An Analytical and Numerical Treatment of the Carter Constant for Inclined Elliptical Orbits about a Massive Kerr Black Hole

In an extreme binary black hole system, an orbit will increase its angle of inclination (i) as it evolves in Kerr spacetime. We focus our attention on the behaviour of the Carter constant (Q) for near-polar orbits. The value of Q for bound orbits is non-negative; and an increase in Q corresponds to an increase in i. For a Schwarzschild black hole, the polar orbit represents the boundary between the prograde and retrograde orbits at which Q is at its maximum value. The introduction of spin (S = |J|/M2) to the massive black hole causes this boundary, or Abutment, to be moved towards the retrograde orbits. We consider this characteristic to be important for understanding the evolution of Q for near-polar orbits. We have developed analytical formulae for Q in a polar orbit and at the last stable orbit (LSO) for given values of latus rectum (l) and eccentricity (e). The Abutment is an important analytical and numerical laboratory that allows us to make a detailed investigation of the evolution of Q for a test particle near its LSO

Analytic Models of Brown Dwarfs and the Substellar Mass Limit

We present the analytic theory of brown dwarf evolution and the lower mass limit of the hydrogen burning main-sequence stars and introduce some modifications to the existing models. We give an exact expression for the pressure of an ideal nonrelativistic Fermi gas at a finite temperature, therefore allowing for nonzero values of the degeneracy parameter. We review the derivation of surface luminosity using an entropy matching condition and the first-order phase transition between the molecular hydrogen in the outer envelope and the partially ionized hydrogen in the inner region.We also discuss the results of modern simulations of the plasma phase transition, which illustrate the uncertainties in determining its critical temperature. Based on the existing models and with some simple modification, we find the maximum mass for a brown dwarf to be in the range 0.064��⊙–0.087��⊙. An analytic formula for the luminosity evolution allows us to estimate the time period of the nonsteady state (i.e., non-main-sequence) nuclear burning for substellar objects. We also calculate the evolution of very low mass stars. We estimate that ≃11% of stars take longer than 107 yr to reach the main sequence, and ≃5% of stars take longer than 108 yr

Fourier transform of the continuous gravitational wave signal

The direct detection of continuous gravitational waves from pulsars is a much anticipated discovery in the emerging field of multimessenger gravitational wave (GW) astronomy. Because putative pulsar signals are exceedingly weak large amounts of data need to be integrated to achieve desired sensitivity. Contemporary searches use ingenious ad hoc methods to reduce computational complexity. In this paper we provide analytical expressions for the Fourier transform of realistic pulsar signals. This provides description of the manifold of pulsar signals in the Fourier domain, used by many search methods. We analyze the shape of the Fourier transform and provide explicit formulas for location and size of peaks resulting from stationary frequencies. We apply our formulas to analysis of recently identified outlier at 1891.76 Hz

The Lambert W Function and Quantum Statistics

We present some applications of the Lambert W function (W function) to the formalism of quantum statistics (QS). We consider the problem of finding extrema in terms of energy for a general QS distribution, which involves the solution of a transcendental equation in terms of the W function. We then present some applications of this formula including Bose–Einstein systems in d dimensions, Maxwell–Boltzmann systems, and black body radiation. We also show that for the appropriate parameter values, this formula reduces to an analytic expression in connection with Wien’s displacement law that was found in a previous study. In addition, we show that for Maxwell–Boltzmann and Bose–Einstein systems, the W function allows us to express the temperature of the system as a function of the thermodynamically relevant chemical potential, the particle density, and other parameters. Finally, we explore an indirect relationship of the W function to the polylogarithm function and to the Lambert transform

An Analytic Study of the Wiedemann-Franz Law and the Thermoelectric Figure of Merit

Advances in optimizing thermoelectric material efficiency have seen a parallel activity in theoretical and computational advances. In the current work, it is shown that the calculation of exact Fermi-Dirac integrals enables the generalization of the Wiedemann-Franz law (WF) to optimize the dimensionless thermoelectric figure of merit ZT. This is done by optimizing the Seebeck coefficient, the electrical conductivity and the thermal conductivity. In the calculation of the thermal conductivity, both electronic and phononic contributions are included. The solutions provide insight into the relevant parameter space including the physical significance of complex solutions and their dependence on the scattering parameter r and the reduced chemical potential

Vacuum birefringence, the photon anomalous magnetic moment and the neutron star RX J1856.5−3754

We analyse the spectrum of the Hamiltonian of a photon propagating in a strong magnetic field B ∼ Bcr, where Bcr=m2e≃4.4×1013 role= presentation \u3eBcr=m2e≃4.4×1013 G is the Schwinger critical field. We show that the anomalous magnetic moment of a photon in the one-loop approximation is a non-decreasing function of the magnetic field B in the range 0 ≤ B ≤ 30 Bcr. We provide a numerical representation of the expression for the anomalous magnetic moment in terms of special functions. We find that the anomalous magnetic moment μγ of a photon for B = 30 Bcr is 8/3 of the anomalous magnetic moment of a photon for B = 1/2Bcr. Based on the recent observational evidence for vacuum birefringence from the neutron star RX J1856.5−3764 by Mignani et al., we suggest vacuum birefringence, the anomalous magnetic moment of the photon and the Faraday rotation angle as key observables for future experiments and measurements