An analytical model for total absorption of a band

A theoretical model for total absorption of a band is developed based on the Elsasser band model and considering the effects of spectral band contour and low-intensity lines in the spectral band. This model yields a simple and analytical expression and has the correct asymptotic forms. Comparisons with other models are presented.     

Universal functions for estimating total vibration-rotation band absorptance—II. Symmetric tops

The method of universal functions for estimating total band absorptances and their derivatives for nonoverlapping lines of vibration-rotation bands for linear molecules and spherical tops is extended to symmetric top bands. Power and asymptotic series expansions of universal functions have been obtained for the Lorentz line shape.     

Slab band absorptance for molecular gas radiation

Nongray molecular gas radiation can be formulated in terms of slab band absorptance, which can be calculated from correlations of laboratory-measured values of band absorption. Slab-band absorptance is not only useful for obtaining the emissivity or mean beam length but also for the formulation of the radiant flux in a nonisothermal gas. Effects of nongray self-absorption by cold boundary layers can then be readily estimated. Illustrative results are presented for gas in a channel with simple straight-line boundary layer profiles at the walls and constant temperature in the center.     

A band absorptance formulation for Doppler broadening

An expression is presented for the total band absorptance of vibration—rotation bands having Doppler-broadened lines. The result is similar in form to corresponding expressions for pressure broadening, in that the total band absorptance is expressed in terms of a dimensionless pressure path length, a bandwidth parameter, and a line-structure parameter.The band absorptance formulation is employed to estimate pressure levels at which Dopple broadening dominates over pressure broadening for CO₂ bands within the atmospheres of Venus and Mars, and for CH₄ bands within the atmospheres of Jupiter and Saturn.     

Universal functions for estimating total vibration-rotation band absorptance—I. Linear molecules and spherical tops

We estimate total band absorptances and their derivatives for nonoverlapping lines of vibration-rotation bands for linear molecules and spherical tops. We use universal functions obtained by replacing the sums of line contributions by integrals over the rotational quantum numbers. An optical path is introduced for the total band. Only general information is utilized on vibrational transitions and line shapes. Power and asymptotic series have been obtained for Doppler and Lorentz line shapes. For a linear molecule and the Lorentz shape, approximate formulae have been derived for the universal functions.     

Total band absorptance,emissivity, and absorptivity of the pure rotational band of water vapor

Based upon the exponential wide band model, an analytical expression is derived for the total band absorptance of the pure rotational band of water vapor. It is shown that the usual band absorptance expressions, which are based on this model, are inaccurate for this band. The use of the total band absorptance in computing the emissivity and absorptivity is also considered. Representations of these functions is achieved within a simple analytical framework.     

Comparison of an analytical model with approximate models for total band absorptions and their derivatives

The introduction of the total band absorption concept has greatly reduced the mathematical complexity of integrating the spectral function in determining the i.r. radiation by nongrey molecular gases due to vibration-rotation bands.A number of simple but approximate equations for total band absorption have been developed for the investigations of i.r. radiation by nongrey gases. Most of these equations are, however, limited to completely overlapped lines and are not applicable to the strong-line cases. Others, which account for the line effects, have been developed empirically and are not directly related to the spectral narrow-band models. Analytical equations for total band absorption and its derivative were derived directly from a spectral narrow-band model and they are universally applicable to all line strengths.Presented in this paper is the comparison of the analytical model with the approximate wide-band models. Also presented is the comparison of the foregoing models with a spectral function integrated to give total band absorption.     

An analytical expression for the electric field in MSM structures

In this work the conformal mapping is used to determine an analytical expression for the electric field in MSM structures. The expression simplifies the analysis of optically generated carriers transport, i.e. the determination of the response of the MSN photodetectors. It can also be useful in the analysis of MSM electrooptical modulators operation.     

An analytical expression for the specific capacity of lithium ion cells

The concentration of lithium ions in the cathode of lithium ion cells has been obtained by solving the materials balance equation $$\frac{{\partial c}}{{\partial t}} = \varepsilon ^{1/2} D\frac{{\partial ^2 c}}{{\partial x^2 }} + \frac{{aj_n (1--t_ + )}}{\varepsilon }$$ by Laplace transform. On the assumption that the cell is fully discharged when there are zero lithium ions at the current collector of the cathode, the discharge timet _d is obtained as $$\tau = \frac{{r^2 }}{{\pi ^2 \varepsilon ^{1/2} }}\ln \left[ {\frac{{\pi ^2 }}{{r^2 }}\left( {\frac{{\varepsilon ^{1/2} }}{J} + \frac{{r^2 }}{6}} \right)} \right]$$ which, when substituted into the equationC=It _d /M, whereI is the discharge current andM is the mass of the separator and positive electrode, an analytical expression for the specific capacity of the lithium cell is given as $$C = \frac{{IL_c ^2 }}{{\pi {\rm M}D\varepsilon ^{1/2} }}\ln \left[ {\frac{{\pi ^2 }}{2}\left( {\frac{{FDc_0 \varepsilon ^{3/2} }}{{I(1 - t_ + )L_c }} + \frac{1}{6}} \right)} \right]$$      

An analytical expression for the calculation of electron mean free paths in solids

A simple analytical expression for the calculation of the inelastic-scattering mean free path for electrons in free-electron-like metals has been derived on the basis of a dielectric formalism. Good agreement with experimental data for energies greater than 200 eV has been demonstrated. By extending the concept of a one-mode excitation spectrum, an analogous formula applicable to semiconductors, insulators and non-free-electron-like metals has been derived which also compares well with experimental data.     

An analytical expression of the amplitude of stripping reaction in the coulomb approximation

A method is derived for calculating the matrix element of stripping reaction (d,p) in the socalled distorted-wave Born approximation; only Coulomb scattering of incident and outgoing particles is taken into consideration. Coulomb wave functions are expressed by means of integral representation permitting integration over angular variables and, when the radial wave function of the trapped neutron is specially chosen, also calculation of the radial integral for non-zero orbital momentum of the trapped neutron. Relations are given for calculating the differential cross-section of reaction and the polarization of products. The special case, when the orbital momentum of the captured neutron is equal to one or two, is discussed in detail.     

An approximate analytical expression for the energies of the tight-bound states of the Zeeman effect in hydrogen

A simple approximate analytical expression for the energies of the tight-bound states of the Zeeman effect in hydrogen is developed that incorporates both the perturbation series in powers of the field strength and the asymptotic logarithmic expansion for the superstrong-field case. Results for the lowest tight-bound states are compared with very accurate non-perturbative calculations.     

An explicit analytical expression for bed-load layer thickness based on maximum entropy principle

The present study aims to derive an analytical model on bed-load layer thickness in an open channel turbulent flow carrying sediments. Determination of the thickness of the bed-load layer is of utmost importance in the study of bed-load transport as it is required to determine the bed-load transport rate, as well as in the study of suspended load transport as it acts as reference level for the particles in suspension. Apart from the several deterministic approaches available in the literature, the work adopts probabilistic approach based on entropy theory to determine the bed-load layer thickness. The concept of entropy theory developed by Shannon is used and the method of Lagrange multipliers is employed for the maximization of entropy function to find the least biased probability distribution. To calculate the Lagrange multipliers, present in the probabilistic model of dimensionless bed-load layer thickness, two different methodologies are presented. The model of bed-load layer thickness is a function of dimensionless shear stress and also depends on three other parameters which are found to be functions of specific gravity of sediment particle and dimensionless particle diameter from a non-linear regression analysis. The proposed model is validated with wide sets of experimental data available in literature and a good agreement is achieved. Apart from comparison with data, the model is also compared with existing deterministic model and computation of relative percentage error proves the better efficiency of the present model.     

An analytical expression for Eg and Ag modes of pyrochlores

A general analytical expression has been derived for E_g and A_1g modes in terms of stretching and bending force constant for cubic pyrochlore structure compounds. The expression has been applied to different pyrochlore oxides accounting six stretching and four bending force constants. The calculation of E_g and A_1g mode has been carried out by using the earlier reported values of force constants for titanates, hafnates and stannates. The analytical expression can be applied to any pyrochlore structure compounds for evaluating and assigning E_g and A_1g mode if the interatomic force constants are already known.     

Pancharatnam and total phases for one-dimensional photonic band gaps

We study the time evolution of the total and Pancharatnam phase of four-level atoms in the presence of a one-dimensional photonic-band-gap material. We obtain analytical and numerical results under certain parametric conditions and analyze the effect of a Kerr-like medium and the detuning parameter on the dynamics of the Pancharatnam phase for the initial atomic position. We observe an interesting feature of the Pancharatnam phase for the one-dimensional photonic-band-gap material.     

An optimal analytical expression for the superparamagnetic relaxation rate under conditions of a stochastic resonance for the model of bistable systems

An analytical expression for the superparamagnetic relaxation rate of uniaxial particles modulated by a radio-frequency field is chosen that describes adequately the magnetic moment dynamics under conditions of a stochastic resonance in the approximation of discrete orientations. The complex magnetic susceptibility components are calculated for a weak external variable field. A comparison is made with the results of numerical modeling based on a solution of the Fokker-Planck equation with a periodic drift term. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 54–58, May, 2007.     

An analytical model for evaluating the metal-metal surface energy

Metal-metal surface energies are calculated as a function of the electron density at the interface. Their contribution to the adhesion energy is shown to be small.     

An analytical construction of the SRB measures for Baker-type maps

For a class of dynamical systems, called the axiom-A systems, Sinai, Ruelle and Bowen showed the existence of an invariant measure (SRB measure) weakly attracting the temporal average of any initial distribution that is absolutely continuous with respect to the Lebesgue measure. Recently, the SRB measures were found to be related to the nonequilibrium stationary state distribution functions for thermostated or open systems. Inspite of the importance of these SRB measures, it is difficult to handle them analytically because they are often singular functions. In this article, for three kinds of Baker-type maps, the SRB measures are analytically constructed with the aid of a functional equation, which was proposed by de Rham in order to deal with a class of singular functions. We first briefly review the properties of singular functions including those of de Rham. Then, the Baker-type maps are described, one of which is nonconservative but time reversible, the second has a Cantor-like invariant set, and the third is a model of a simple chemical reaction R<-->I<-->P. For the second example, the cases with and without escape are considered. For the last example, we consider the reaction processes in a closed system and in an open system under a flux boundary condition. In all cases, we show that the evolution equation of the distribution functions partially integrated over the unstable direction is very similar to de Rham's functional equation and, employing this analogy, we explicitly construct the SRB measures. (c) 1998 American Institute of Physics.