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Oliver Bauer, Moritz Sokolowski Institute for Physical and Theoretical Chemistry University of Bonn Wegelerstrasse 12, Bonn, Germany X-Ray Standing Waves experiments and their evaluation XSWAVES, version 2.x

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1)Introduction to X-Ray Standing Waves 2)Computation of XSW Data - XSWAVES Outline

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Introduction to XSW – the Physics behind… Literature: (1)B.W. Batterman, H. Cole, Reviews of Modern Physics 36 (1964) (2)J. Zegenhagen, Surface Science Reports 18 (1993) (3)D.P. Woodruff, Progress in Surface Science 57 (1998) (4)D.P. Woodruff, Reports on Progress in Physics 68 (2005)

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Introduction to XSW (NI)XSW = (Normal Incidence) X-ray Standing Waves –Absorption spectroscopy based on diffraction / Photoemission spectroscopy at photon energies E Bragg –Determination of adsorption heights and adsorption geometries (molecular distortions upon adsorption?) single-crystalline substrate

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Introduction to XSW Within the finite width of the Bragg reflection there is interference between the incoming and the Bragg-reflected wave standing wave field (phase (E)). Bragg-reflected x-ray plane wave incoming x-ray plane wave wave fronts z I XSW dHdH B max I XSW dHdH crystal surface J. Zegenhagen, Surf. Sci. Rep. 18 (1993) 199. / D.P. Woodruff, Rep. Prog. Phys. 68 (2005) 743. / B.W. Batterman, H. Cole, Rev. Mod. Phys. 36 (1964) 681.

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Typical NIXSW profiles Bragg-reflected wave incoming wave interference of incoming and reflected wave F H : coherent fraction P H : coherent position S R, |S I |, :non-dipolar parameters Introduction to XSW

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Non-dipolar contributions Introduction to XSW Bragg-reflected wave incoming wave interference of incoming and reflected wave F H : coherent fraction P H : coherent position S R, |S I |, :non-dipolar parameters

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The Physics behind XSW… The XSW absorption profile as a function of coherent fraction and coherent position is taken as (3,4): where and are : p and l are the partial phase shifts for the outgoing p- and d-waves, respectively (photoemission from an s-state). Q and are tabulated. = S R = |S I | M.B. Trzhaskovskaya et al., Atomic Data and Nuclear Data Tables 77 (2001) 97 and 82 (2002) 257. NIST Electron Elastic-Scattering Cross-Section Database 3.1 (June 2003)

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The reflectivity curve R is calculated as (1-4): where is (in terms of photon energy): is a complex number since the structure factors are complex. Polarisation factor P is taken as cos(2 * Bragg ) (normal incidence => polarisation, P = 1). The above formula is only valid for centrosymmetric crystals since the pre-factor F H / F -H is omitted = 1 for centrosymmetric crystals The Physics behind XSW…

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The phase shift (or …) between the incoming and the outgoing X-ray plane wave is computed as (1-4): where is : and The Physics behind XSW… conditions inverted in XSWAVES source code: ( ) (E) J. Zegenhagen, Surf. Sci. Rep. 18 (1993) 199. / D.P. Woodruff, Rep. Prog. Phys. 68 (2005) 743. / B.W. Batterman, H. Cole, Rev. Mod. Phys. 36 (1964) 681.

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Computation of XSW data: XSWAVES – an XSW data evaluation routine for ORIGIN ® 8 XSWAVES (open-source): ORIGIN (commercial):

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Requirements: –Open-source routine –Sophisticated, reliable fitting engine –Full access to fit parameters –Batch processing –User-friendly interface Computation of XSW data *.txt file input: parameters reflectivity exp. XSW profile NLSF fitting engine numerical and graphical results output

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XSWAVES Exemplary fit result

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Computation of XSW data Experimental broadening Si(111) double-crystal monochromator

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XSWAVES source code FFT convolution funtion F(E) step-wise convolution: trapezoidal rule

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XSWAVES source code The theoretical curves (formulae given on pages 25 and 26) are convoluted with two further functions, namely a Gaussian and the squared reflectivity of the Monochromator, and then fitted to the experimental data employing the ORIGIN fitting engine NLSF. The Gaussian function (width wG, center xcG) resembles the instrumental broadening due to substrate mosaicity, e.g. The X-ray beam energy spread is explicitly mimiced by convolution with the squared reflectivity of the Si(111) double-crystal monochromator. The convolution of the squared monochromator reflectivity with the respective theoretical curve is done via FFT convolution which is an ORIGIN C global function. This results in an ideal curve named f which is then convoluted with a Gaussian g by explicitly solving the integral over (source code: integral over t from t_initial to t_final): If the Gaussian function is incorporated in FFT convolution, artefacts are observed (i.e. wiggling of the fit curve) which can be overcome by an increased convolution number of points. This is of course very time-consuming. Computational details: The integral is explicitly solved for the exp. photon energy range plus 2.5 eV (or more) in both directions (lower / higher photon energy); this avoids artefacts in the fit result on the boundaries of the exp. energy range. Stepsize d is chosen to be 0.1 eV or smaller (depending on exp. photon energy stepsize). Computation costs about 2 min in total for the template dataset (31 data points, 0.2 eV photon energy stepsize, default settings) on an Intel ® Pentium ® 4 CPU, 3.20 GHz, with 1.00 GB RAM. Allow the fit to a typical experimental data set (about 50 points, photon energy stepsize: 0.1 – 0.2 eV) to take around 5 to 10 min in total …

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fit result

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XSWAVES benchmarking Fit of synthetic data sets for Ag(111): –Exemplary data sets were created with EXCEL simulation sheet by Bruce Cowie. –Neither error weighting for reflectivity fit nor for XSW absorption profile fit. –Non-dipolar parameters : Q = 0, = 0. –Modification of the response function is NOT enabled during XSW profile fit. simulationXSWAVES ver. 2.0 Data setCFCPCFCP Test Test Test Test Test Test

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XSWAVES benchmarking Ag(111), Test 2: –Simulation: CF = 1.0 CP = 0.7 –Fit: CF = 1.000CP = 0.698

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Summary XSWAVES – an XSW data evaluation routine for ORIGIN ® 8: –Open-source routine with user-friendly interface

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