Euler Transformation Research Articles

Redistribution of dopant impurities in oxidizing ambients

The redistribution of dopant impurities in an oxidizing ambient is considered using a new series method. The results obtained are valid for a wide class of oxide growth functions including as special cases the parabolic and linear laws. More importantly, the method copes readily with the practically useful mixed linear-parabolic growth law. The solution retains the flexibility of the related Chen and Chen series and is applicable for a wide range of initial dopant profiles and for different boundary conditions at the oxide-semiconductor interface. As an example of the power of this new technique results are presented for the previously analytically unsolved problem of redistribution from an initial erfc profile, with a segregation type boundary condition, assuming a linear-parabolic growth law. The convergence difficulties of our general series solution have been surmounted by the use of the Euler transformation, providing solutions which are valid for realistic diffusion times. Our analysis permits a comparison with the earlier work of Guckel and Hall on the redistribution from a step profile.

On reducing to zero the remainder of an asymptotic series

AbstractIf a weighted Euler transformation is applied to the asymptotic series for ezE1(z) the remainder can be expressed as an integral. Examination of this integral shows that for a transformation of given order the smallest term of the resulting series remains at approximately a constant distance from the start of the series. If, however, there is no restriction on the order of transformation the remainder may be decreased to zero by increasing the number of terms used, but if z is close to the negative real axis the rate of decrease is small. A more general theorem for alternating real series and Taylor's series is also given.

Redistribution of dopant impurities in a semiconductor using the series method of Chen and Chen

The series method of Chen and Chen is re-examined and applied to a variety of redistribution problems. It is seen that this analysis can play an important role in deriving closed-form solutions of the diffusion equation useful in semiconductor technology. In particular, series solutions are formulated for two alternative Si/SiO 2 interface boundary conditions. Moreover, taken in conjunction with the Euler transformation it is shown to provide exact solutions to several redistribution problems for a wide class of initial dopant profiles and for realistic diffusion times. All solutions have been compared where ever possible with existing analytical techniques, and in all other cases reference has been made to results obtained from the numerical Crank-Nicolson technique.

Extended Stokes series: laminar flow through a loosely coiled pipe

Dean's series for steady fully developed laminar flow through a toroidal pipe of small curvature ratio has been extended by computer to 24 terms. Analysis suggests that convergence is limited by a square-root singularity on the negative axis of the square of the Dean number. An Euler transformation and extraction of the leading and secondary singularities at infinity render the series accurate for all Dean numbers. For curvature ratios no greater than$\frac{1}{250} $, experimental measurements of the laminar friction factor agree with the theory over a wide range of Dean numbers. In particular, they confirm our conclusion that the friction in a loosely coiled pipe grows asymptotically as the one-quarter power of the Dean number based on mean flow speed. This contradicts a number of incomplete boundary-layer analyses in the literature, which predict a square-root variation.

Numerical evaluation of the distribution of the product of two normal variables

This paper describes the numerical evaluation of the mathematical forms of the distribution of the product , , , where x 1 and x 2 follow a bivariate normal distribution . The methods used are general for the evaluation of distribution functions found by the inversion of the characteristic function of the random variable. Special cases of the distribution are the central and noncentral γ 2 with one degree of freedom. The distribution function of the product is expressed as an integral involving sine, cosine, and exponential functions. A computer code has been written to generate the distribution function table for fixed δ 1 δ 2, and ρ. The computer code uses Romberg integration and Euler's transformation to evaluate the infinite integrals. Several tables are provided to support the accuracy of the computer code.

Bone marrow cell scene segmentation by computer-aided color cytophotometry.

Computer scene segmentation of touching cell images in bone marrow, on the basis of color information, is achieved using digitized scans at three different wavelengths of light. With trivariate histograms and Euler's coordinate transformation, it is possible cytophotometrically to isolate, on the basis of chromatic differences, individual heterogeneous cells located in cell groups. The ability of the described computer methods to isolate correctly the touching cell images is determined by visual comparison of the cells as seen in the microscope and the computer-generated displays of the scanned and segmented scenes.

Temperature of a cylindrical cavity wall heated by a periodic flux

Response of the surface temperature of an initially uniform temperature material space bounded internally at radius r = a and heated with a flux Q = Q ̄ sin(ωt + ε) is studied. This is done with the use of Duhamel's integral together with a previously obtained solution to the constant step heating problem. Application of the Euler transformation to short time expansions and the use of various asymptotic expansions result in analytic solution estimates useful for different ranges of ω and t. Together these estimates can be used to predict the entire history of the surface temperature for arbitrary ω and ε. They can easily be used to compute the r = a surface temperature history under conditions of arbitrary periodic heating.

Three theorems on relative motion

The three theorems treat the analytical aspects of the relative motion of non-interacting particles influenced by arbitrary force fields. As special cases motion in gravitational and central force fields are discussed. The theorems generalize Encke's method, Euler's transformation and Sundman's regularization.

Second-order boundary-layer solutions for flow past a blunted wedge

The second-order effects of the longitudinal curvature and displacement for flow past a blunted wedge have been studied employing the Gortler power series in streamwise coordinate, σ. The first five terms for each of the effects have been computed. The results, in general, have a restricted region of validity in the downstream direction due to the limited radius of convergence. The results are improved by using Euler transformation technique and it is found that the results are good even as σ→∞. For a parabola, the present results are compared with the exact numerical solution of the Navier-Stokes equations and it is shown that the second-order boundary-layer equations can be usefully employed down to Reynolds number 103.

The Gibbs Phenomenon for Generalized Taylor and Euler Transforms

Let f be a real-valued function satisfying the Dirichlet conditions in a neighborhood of x = x0, at which point f has a jump discontinuity. If {Sn(x)} is the sequence of partial sums of the Fourier series of f at x, then ﹛Sn(x)﹜ cannot converge uniformly at x — x0. Moreover, for any number , there exists a sequence ﹛tn﹜, where tn → x0 and

A Note on the van Wijngaarden Transformation

Theory and applications are given of the van Wijngaarden transformation by means of which slowly convergent or even divergent series may be converted into rapidly convergent series. Using the technique of generating functions it is shown that by the van Wijngaarden transformation the Borel sum is kept invariant. The van Wijngaarden transformation is shown to be the Laplace transform of the Euler transformation.

Nonequilibrium phenomena in dusty supersonic flow past blunt bodies of revolution

A theory is developed for inviscid supersonic flow past blunt bodies of revolution containing small spherical cloud particles of uniform size. It is assumed that there are no collisions between the particles. The only interactions between the gas and the particles are drag force and heat exchange. The problem is solved by using the direct method of a given body and the technique of series truncation. The convergence of the series is improved by recasting it in a modified Euler transformation. It is found that the relaxation zone is much larger than the shock layer thickness, hence the flow is nonequilibrium in the entire subsonic region. The solutions for the case without the particles agree with the known blunt‐body solutions, and those with the particles resemble the known dusty air solutions behind a normal shock wave.

Perturbation solution of the spin hamiltonian of low symmetry using the projector technique

Second-order perturbation solutions of the spin hamiltonian are presented for the case of most general symmetry conditions, when the principal axes of the different tensors may be non-collinear and the tensors g and A may be asymmetric. In order to obtain a concise formalism we introduce projectors on the quantization directions of the electron and nuclear spins, which make the determination of complete Euler transformation matrices unnecessary. The angular dependence of the intensities are discussed both for the allowed and different first-order for bidden transitions for randomly oriented samples.

Analytic Continuation of the Euler Transform

The Euler transform is defined by an integral over the unit interval $[0,1]$ if the transform variables have positive real parts. If the function to be transformed is holomorphic on a neighborhood of $[0,1]$, its transform can be represented for all complex values of the transform variables by an integral around a contour which encircles $[0,1]$. The integrand contains a ${}_2 F_1 $-function. This type of contour integral represents the principal branch of Appell’s double hypergeometric function $F_1$ for all values of the parameters and variables.

Two-dimensional radiative equilibrium: A semi-infinite medium subjected to a finite strip of radiation

Exact numerical results are presented for the emissive power and radiative flux at the boundary of a two-dimensional, absorbing-emitting, semi-infinite medium bounded by (1) a strip of collimated radiation and (2) a constant temperature black strip. The method of super-position is used to obtain the finite strip solutions in terms of cosine varying solutions. The infinite integrals arising in the solutions are converted to an alternating series of finite integrals. The Euler transformation is then applied to speed up convergence. Error bounds are determined whereby the two-dimensional finite strip analysis can be approximated by the simpler one-dimensional solution.

The end-point distribution of self-avoiding walks on a crystal lattice. II. Loose-packed lattices

A study is made of the mean-square lengths of self-avoiding walks on a number of loose-packed lattices. It is found that the even-odd oscillations characteristic of this type of lattice may be largely removed by the use of an Euler transformation. From an analysis of the transformed series and of other moments from 1 to 10 of the distribution the mean-square length indices are estimated as gamma =1.500+or-0.005 in two dimensions and gamma =1.20+or-0.02 in three dimensions. These estimates are in good agreement with the exact simple fractions 3/2 and 6/5 favoured by several workers.

Perturbation Solutions for Spherical Solidification of Saturated Liquids

A perturbation solution is obtained for outward and partial inward spherical solidification of a liquid initially at the freezing temperature. The constant-wall-temperature boundary condition is considered with the properties of the solidified material assumed as constants. A nonlinear transformation is applied to the sequence of partial sums in the perturbation solution to increase its range of applicability. For inward solidification it is found that the regular perturbation solution diverges for front positions close to the center. An Euler transformation and an overall energy balance are then used to obtain a modified series solution which is compared with numerical results.

Numerical calculation of Fourier-transform integrals

A Fourier-transform integral can be transformed into an integral of a periodic function over a finite interval. This integral can be calculated efficiently by the trapezium rule, while the periodic function, which is expressed as an infinite series, can be evaluated using Euler's transformation. A FORTRAN program implementing this method is presented.

Errata

Due to delays caused by a strike at the Université de Montréal, the author's proof corrections of the paper ‘A note on the generalised Euler transformation’ by P. Wynn (this Journal, Vol. 14, No. 4, pp. 437-441) were received after the printer's deadline. The following corrections should be made:

Paramagnetic resonance of chromium-doped yttrium orthoaluminate

The paramagnetic resonance spectrum of Cr 3+ in yttrium orthoaluminate has been examined at X band and has been compared with the resonance spectrum of Cr 3+ in europium orthoaluminate. Following van der Ziel's approach an attempt was made to explain the contribution of the Cr 3+-Eu 3+ exchange interaction to the Hamiltonian of the europium orthoaluminate. A Hamiltonian of the form D(α 2 2 − 1 3 ) + E(α 1 2−α 2 2) is found to apply, where the direction cosines α i refer to a set of axes obtained from the crystallographic axes by an Euler transformation. Using g = 1.98, the spin Hamiltonian parameters are found to be D = −0.041 cm −1 and E = −0.010 cm − for the diamagnetic substance, as compared to D = −0.235 cm −1 and E = 0.0522 cm −1 for the europium orthoaluminate. The value of the europium orthoaluminate effective exchange interaction constant is found here to be 7.5 cm −1, as compared to the europium gallium garnet effective exchange interaction constant value of 9 cm −1. Measurements of line widths and spin-lattice relaxation times are also presented.