0000000000030839
AUTHOR
Claus Schneider
Element-Selective Magnetic Imaging in Exchange-Coupled Systems by Magnetic Photoemission Microscopy
We have used a photoemission microscope to obtain element-resolved magnetic contrast in stacked magnetic thin film systems. Magnetic information is thereby provided by X-ray magnetic circular dichroism. Elemental sensitivity, which is crucial for studying magnetic coupling phenomena in systems with several different layers, is achieved by tuning the energy of the illuminating photons to atomic absorption edges. We present measurements of a Ni-coated Co micropattern on Cu(001), and a wedged Co/Cr/Fe(001) sample. In the former sample the Ni magnetization is seen to follow the magnetization of the Co pattern, thereby changing from an out-of-plane easy axis in areas without underlying Co to in…
Regularity of the solution to a class of weakly singular fredholm integral equations of the second kind
Continuity and differentiability properties of the solution to a class of Fredholm integral equations of the second kind with weakly singular kernel are derived. The equations studied in this paper arise from e.g. potential problems or problems of radiative equilibrium. Under reasonable assumptions it is proved that the solution possesses continuous derivatives in the interior of the interval of integration but may have mild singularities at the end-points.
Error Bounds for the Numerical Evaluation of Integrals with Weights
This paper is concerned with a procedure of obtaining error bounds for numerically evaluated integrals with weights. If \( - \infty \mathop < \limits_ = a < b\mathop < \limits_ = \infty \), w integrable over [a,b] and positive almost everywhere, then an approximation of \({I_W}f: = \int\limits_a^b {w\left( t \right)f\left( t \right)dt} \) by a quadrature rule \({Q_n}f: = \sum\limits_{i = 0}^n {{\alpha _i}f\left( {{t_i}} \right)} \) is leading to the error Enf ≔ Iwf ‒ Qnf. An algorithm is derived for the computation of bounds for |Enf| depending on the smoothness of the integrand f and on the degree of exactness of Q. As initial values this algorithm needs moments of the weighting function w…
Rational Hermite Interpolation and Quadrature
Rational Hermite interpolation is used in two different ways in order to derive and analyze quadrature rules. One approach yields quadratures of Gaussian-type whereas the other one generalizes Engels’ dual quadratures exhibiting the close connection between rational Hermite interpolation and quadrature in general.
Generalized singly-implicit Runge-Kutta methods with arbitrary knots
The aim of this paper is to derive Butcher's generalization of singly-implicit methods without restrictions on the knots. Our analysis yields explicit computable expressions for the similarity transformations involved which allow the efficient implementation of the first phase of the method, i.e. the solution of the nonlinear equations. Furthermore, simple formulas for the second phase of the method, i.e. computation of the approximations at the next nodal point, are established. Finally, the matrix which governs the stability of the method is studied.
An analysis of Ralston's quadrature
Ralston's quadrature achieves higher accuracy in composite rules than analogous Newton-Cotes or Gaussian formulas. His rules are analyzed, computable expressions for the weights and knots are given, and a more suitable form of the remainder is derived.
Preparation of thin films of the ternary heavy fermion system CeNi 2 Ge 2
Ge2 layers on W(110). In order to produce well-ordered and atomically clean surfaces of the Ce-based intermetallic system the growth was performed under UHV conditions (p<2×10-11 mbar). Both the polycrystalline CeNi2Ge2 compound and the individual elements Ce, Ni, and Ge were used as evaporants. The characterisation of the layers was made with LEED, SEM, and XPS. We find a significant influence of the substrate temperature and the evaporation power on the growth characteristics. The compound material CeNi2Ge2 exhibits complicated behaviour when evaporated. Under carefully selected growth conditions we obtain well-ordered films with a stoichiometry of Ce:Ni:Ge=1:2:2 and a (001) oriented surf…
Quadrature rules for qualocation
Qualocation is a method for the numerical treatment of boundary integral equations on smooth curves which was developed by Chandler, Sloan and Wendland (1988-2000) [1,2]. They showed that the method needs symmetric J–point–quadrature rules on [0, 1] that are exact for a maximum number of 1–periodic functions The existence of 2–point–rules of that type was proven by Chandler and Sloan. For J ∈ {3, 4} such formulas have been calculated numerically in [2]. We show that the functions Gα form a Chebyshev–system on [0, 1/2] for arbitrary indices a and thus prove the existence of such quadrature rules for any J.
Vereinfachte Rekursionen zur Richardson-Extrapolation in Spezialf�llen
Recursions are given for Richardson-extrapolation based on generalized asymptotic expansions for the solution of a finite algorithm depending upon a parameterh>0. In particular, these expansions may contain terms likeh ?·log(h), (?>0). Simplified formulae are established in special cases. They are applicable to numerical integration of functions with algebraic or logarithmic endpoint singularities and provide a Romberg-type quadrature.
Fast elemental mapping and magnetic imaging with high lateral resolution using a novel photoemission microscope
Abstract Using tunable soft X-ray synchrotron radiation and a new-generation photoemission electron microscope with integral sample stage and microarea selector, elemental images and local XANES spectra have been measured. Given the present conditions (PM3 at BESSY), the lateral resolution was in the range of 130 nm with the potential of considerable improvement with high-brilliance sources (a base resolution of 25 nm was obtained in threshold photoemission). Measurements at the oxygen K-edge demonstrate that differences in the local chemical environment of the emitter atom are clearly revealed and can thus be used as a fingerprint technique for its chemical state and geometrical surroundin…
Stroboscopic XMCD–PEEM imaging of standing and propagating spinwave modes in permalloy thin-film structures
Abstract Using synchrotron-based stroboscopic photoemission electron microscopy with X-ray circular dichroism as contrast method, we have investigated the high-frequency response of permalloy thin-film structures. Standing precessional modes have been studied in rectangular elements (16 × 32 μm 2 , 10 nm thick) with a high time resolution of about 15 ps in the low- α mode of BESSY. With increasing amplitude of the applied magnetic AC field the particle is driven from an initial symmetric Landau flux-closure state into an asymmetric state and finally into a single-domain state magnetized perpendicular to the applied field H AC . The electromagnetic microwave field thus can induces a net magn…
Produktintegration mit nicht-�quidistanten St�tzstellen
For the numerical evaluation of $$\int\limits_a^b {(t - a)^{\alpha - 1} x(t)dt}$$ , 0<?<1 andx `smooth', product integration rules are applied. It is known that high-order rules, e.g. Gauss-Legendre quadrature, become `normal'-order rules in this case. In this paper it is shown that the high order is preserved by a nonequidistant spacing. Furthermore, the leading error terms of this product integration method and numerical examples are given.
Product Integration for Weakly Singular Integral Equations In ℝm
In this note we discuss the numerical solution of the second kind Fredholm integral equation: $$ y(t) = f(t) + \lambda \int\limits_{\Omega } {{{\psi }_{\alpha }}(|t - s|)g(t,s)y(s)ds,\;t \in \bar{\Omega },} $$ (1) Where \( \lambda \in ;\not{ \subset }\backslash \{ 0\} \) , the functions f,g are given and continuous, |.| denotes the Euclidean norm, and φα, 0 \alpha > 0} \\ {\left\{ {\begin{array}{*{20}{c}} {\ln (r),} & {j = 0} \\ {{{r}^{{ - j}}}} & {j > 0} \\ \end{array} } \right\},\alpha = m} \\ \end{array} ,} \right. $$ with Cj not depending on r. Here Ω _ is the closure of a bounded domain Ω⊂ℝm.