On derivation of the analytical expression for damping ratios in a low-viscosity approximation

The damping ratios of waves and oscillations in nonlinear dispersion equations are found for planar, cylindrical, and spherical geometries as applied to finite-volume liquids. For a cylindrical jet and a plane interface between viscous liquids, the damping ratios are determined for the first time. When the radius of curvature of the liquid jet surface decreases, so does the damping ratio of capillary waves. In a system of immiscible liquids, the damping ratio may be both larger and smaller than that for the pure liquid depending on the viscosity of the liquids and the ratio of their densities. This is because the damping ratio depends on the kinematic viscosities of pure liquids. The damping ratio is also estimated for waves arising at the liquidgas interface due to a tangential discontinuity of the velocity field.     

An empirical approximation to the Voigt profile

A closed-form approximation to the Voigt profile is developed from an examination of Voigt profile calculations. This approximation matches the Voigt profile within 5 per cent at worst and is generally within 3 per cent or less. In radiant energy transport calculations this approximation is shown to closely reproduce the curves of growth given by use of the Voigt profile expression.     

An alternative derivation of the Wigner-Weisskopf approximation

The Wigner-Weisskopf approximation for the lineshape function for a two-level atom coupled to an electromagnetic field is derived by application of a mathematical approximation known as the “principle of averaging”. The present approach not only affords a new perspective, but also has some advantages over previous ones.     

An efficient approximation for a wavelength-modulated 2nd harmonic lineshape from a Voigt absorption profile

In this paper we report on an efficient approximation for the reconstruction and fitting of 2nd harmonic, wavelength-modulated, absorption spectra from Voigt profiles. The model consists in a proper combination of results recently obtained by other authors, concerning on one hand, Fourier analysis of signals resulting from wavelength modulation spectroscopy and, on the other hand, an empirical analytical approximation to the Voigt lineshape. The fitting algorithm was tested both on simulated and real spectra, corresponding to ammonia rovibrational transitions around 2-μm wavelength. In the case of real spectra, a standard deviation for the normalized residuals at a level of ∼0.2% was found. The accuracy in the reproduction of peak intensities was comparable with that provided by a more complex model, previously developed by other authors.     

An analytical derivation of the optimum source patterns for the pseudospectral time-domain method

In the computational electromagnetics and acoustics, spatially smoothed sources are often utilized to alleviate the aliasing errors in the pseudospectral time-domain (PSTD) algorithms. In our work, an analytical derivation of the optimum source patterns is presented according to the accurately derived expressions of the dominant source-introduced aliasing errors according to the circular discrete convolution and Tailor series expansion method. We quantitatively demonstrate, for the first time in literature, that the aliasing errors can be optimally suppressed and rapidly reduced to the negligible levels by these optimum patterns and with the increment of source cells. We also provide the different implementation schemes of the optimal patterns both for the soft and hard source cases. The numerical calculation and 1D PSTD transient simulations are conducted to verify the excellent performance of these optimum sources.     

An approximation for rotation-projected expectation values of the energy for deformed nuclei and a derivation of the cranking variational equation

An approximation is given for the expectation value of the Hamiltonian with functions: where ¦?〉 is a HFB wave function of a deformed nucleus. This is achieved by expanding the overlap function of the energy 〈?¦ HD(Ω)¦?〉 in derivatives of the overlap function of the norm 〈? D(Ω)¦?〉. The cranking equations for even nuclei are yielded by varying the approximate energy.     

The Gutzwiller approximation for degenerate bands: a formal derivation

We present a detailed derivation of the Gutzwiller approximation for multi-band Hubbard models with density-density Coulomb interactions. For the one-band Hubbard model we introduce a mathematically well-defined formalism which is easily generalized to the multi-band case. In contrast to earlier attempts, our approach allows us to include inter-orbital hopping terms in the Hamiltonian. Received: 9 December 1997 / Revised and accepted: 6 March 1998     

Parametric generation of light in a cavity: An analytical approximation

A new analytical approximate solution is suggested for the problem of nonlinear parametric generation of light in a cavity. This solution is much more accurate than the known ones. Two- and three-cavity lasing schemes are considered and criteria for their adequacy are ascertained. The accuracy of the results is confirmed by computer simulation.     

An efficient method for evaluation of the complex probability function: The Voigt function and its derivatives

An efficient method is developed to evaluate the function w(z)=e^-z2(1+(2i/√π)∫^z₀e^t2dt) for the complex argument z = x + iy. The real part of w(z) is the Voigt function describing spectral line profiles; the imaginary part can be used to compute derivatives of the spectral line shapes, which are useful, e.g. in least-squares fitting procedures. As an example of the method a simple and fast FORTRAN subroutine is listed in the Appendix from which w(z) in the entire y ? 0 half-plane can be calculated, the maximum relative error being less than 2 × 10^-6 and 5 × 10^-6 for the real and imaginary parts, respectively.     

Calculation of the apparatus function for the case of a generalized Voigt contour

The Rayleigh-Bracewell method of finite differences is applied to cases where the apparatus function is either entirely Lorentzian or results from the superposition of Lorentzian and Gaussian components. A firstorder correction is calculated, and order of magnitude estimates are given for the remaining terms in the series expansion representing the correction. The method is employed to correct self-reversed contours of spectral lines.     

A gauge-covariant derivation of the strong-field approximation

A brief discussion of gauge invariance in nonrelativistic quantum mechanics is presented. The strong-field approximation for ionization of an atom by a laser field is rederived in such a way that the resulting ionization amplitude comes out identical in any gauge. This result agrees with the usual length-gauge form.     

An elementary derivation of the density of states function for continuum states

The density of states function for continuum states in the presence of a nuclear Coulomb field is derived by the Sommerfeld-Weyl eigendifferential method.     

The differential equations of the Voigt function

The three second-order partial differential equations of the Voigt function are presented, leading to a powerful and accurate method of determining the Voigt function in the calculation of a line profile in a stellar atmosphere.     

Simple analytical expression for work function in the “nearest neighbour” approximation

Nonlocal operator of potential is suggested, based on the “nearest neighbour” approximation (NNA) for single electron wave function in metals. It is shown that Schrödinger equation with nonlocal potential leads to quite simple analytical expression for work function, which surprisingly well fits to experimental data.     

Rapidly convergent series for high-accuracy calculation of the Voigt function

A rapidly convergent series, based on Fourier expansion of the exponential multiplier, is presented for highly accurate approximation of the Voigt function (VF). The corresponding algorithm enables the rapid calculation, required for its implementation as a subprogram in an interpolation approach. The numerical analysis of this VF approximation suggests that it may be more accurate than 10⁻⁹ in the Humlí?ek regions 3 and 4.     

An analytical methodology for magnetic field control in unilateral NMR

Traditionally, unilateral NMR systems such as the NMR-MOUSE have used the fringe field between two bar magnets joined with a yoke in a 'U' geometry. This allows NMR signals to be acquired from a sensitive volume displaced from the magnets, permitting large samples to be investigated. The drawback of this approach is that the static field (B0) generated in this configuration is inhomogeneous, and has a large, nonlinear, gradient. As a consequence, the sensitive volume of the instrument is both small and ill defined. Empirical redesign of the permanent magnet array producing the B0 field has yielded instruments with magnetic field topologies acceptable for varying applications. The drawback of current approaches is the lack of formalism in the control of B0. Rather than tailoring the magnet geometry to NMR investigations, measurements must be tailored to the available magnet geometry. In this work, we present a design procedure whereby the size, shape, field strength, homogeneity, and gradients in the sensitive spot of a unilateral NMR sensor can be controlled. Our design uses high permeability pole pieces, shaped according to the contours of an analytical expression, to control B0, allowing unilateral NMR instruments to be designed to generate a controlled static field topology. We discuss the approach in the context of previously published design techniques, and explain the advantages inherent in our strategy as compared to other optimization methods. We detail the design, simulation, and construction of a unilateral magnet array using our approach. It is shown that the fabricated array exhibits a B0 topology consistent with the design. The utility of the design is demonstrated in a sample nondestructive testing application. Our design methodology is general, and defines a class of unilateral permanent magnet arrays in which the strength and shape of B0 within the sensitive volume can be controlled.     

Rapid and accurate calculation of the Voigt function

Two general techniques significantly improve both the accuracy and speed of spectral line shapes that use only the complex Voigt function. Such line shape functions include the Voigt line profile, which is closely related to the real part of this function. The first technique is a new algorithm, which trades a small amount of RAM (at most 1.49 megabytes) for a considerable gain in speed and accuracy. The accuracy is one part in 10⁶ of the function itself. In addition, the derivatives of the function with respect to x and y are returned with an accuracy of 0.5%. The algorithm is up to nine times faster than the Drayson or Humlí?ek algorithms and about two orders of magnitude more accurate. A second independent improvement is a set of simple criteria to decide when an evaluation is required and when a value of zero can be assumed. This section alone can reduce the computation time by three orders of magnitude or more and it can be applied to algorithms using any Voigt approximation.     

On a collective pair approximation for low-energy nuclear spectra

We investigate an approximate procedure to evaluate overlaps and matrix elements in a space of collective pairs. The procedures uses recursion formulas based on the commutation relations of the fermion operators. We test the approximation in the case of a pairing hamiltonian acting in a space of S pairs and compare it with the number operator approximation.