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Scattering Theory: Born Series Stefan Blügel This document has been published in Manuel Angst, Thomas Brückel, Dieter Richter, Reiner Zorn (Eds.): Scattering Methods for Condensed Matter Research: Towards Novel Applications at Future Sources Lecture Notes of the 43rd IFF Spring School 2012 Schriften des Forschungszentrums Jülich / Reihe Schlüsseltechnologien / Key Tech- nologies, Vol. 33 JCNS, PGI, ICS, IAS Forschungszentrum Jülich GmbH, JCNS, PGI, ICS, IAS, 2012 ISBN: 978-3-89336-759-7 All rights reserved. A2 Scattering Theory: Born Series 1 Stefan Blügel Peter Grünberg Institut and Institute for Advanced Simulation Forschungszentrum Jülich GmbH Contents 1 Introduction 2 2 The Scattering Problem 2 2.1 The Experimental Situation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Description of Scattering Experiment . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 The Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Lippmann Schwinger Equation 9 4 Born Approximation 11 4.1 Example of Born Approximation: Central Potential . . . . . . . . . . . . . . . 12 4.2 Example of Born Approximation: Square Well Potential . . . . . . . . . . . . 13 4.3 Validity of first Born Approximation . . . . . . . . . . . . . . . . . . . . . . . 14 4.4 Distorted-Wave Born Approximation (DWBA) . . . . . . . . . . . . . . . . . 15 5 Method of Partial Wave Expansion 17 5.1 The Born Approximation for Partial Waves . . . . . . . . . . . . . . . . . . . 20 5.2 Low Energy Scattering: Scattering Phases and Scattering Length . . . . . . . . 21 5.3 S-Wave Scattering at Square Well Potential . . . . . . . . . . . . . . . . . . . 22 5.4 Nuclear Scattering Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 6 Scattering from a Collection of Scatterers 25 Lecture Notes of the 43rd IFF Spring School “Scattering Methods for Condensed Matter Research: Towards 1 Novel Applications at Future Sources” (Forschungszentrum Jülich, 2012). All rights reserved. A2.2 Stefan Blügel 1 Introduction Since Rutherford’s surprise at finding that atoms have their mass and positive charge concen- trated in almost point-like nuclei, scattering methods are of extreme importance for studying the properties of condensed matter at the atomic scale. Electromagnetic waves and particle radiation are used as microscopic probes to study a rich variety of structural and dynamical properties of solids and liquids. Atomistic processes in condensed matter take place at length scales on the order of an Ångström (1Å= 10−10 m) and an energy scale between a meV and a few eV. Obviously, detailed information concerning atomic systems require measurements re- lated to their behavior at very small separations. Such measurements are in general not possible unless the de Broglie wavelength (λ = hp = mv h ) of the relative motion of the probing particle is comparable to these distances. This makes x-ray scattering and neutron scattering, in addition to electron scattering and to a certain extent also Helium scattering, to the outstanding micro- scopic “measurement instruments” for studying condensed matter. To push electromagnetic waves in this area one uses either x-rays with wavelengths of a few Ångströms, but in the keV energy range, or light with energies in the eV range, but wavelengths of some 1000 Å. Neutrons (an

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