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**X-Ray Standing Waves experiments and their evaluation**

Oliver Bauer, Moritz Sokolowski Institute for Physical and Theoretical Chemistry University of Bonn Wegelerstrasse 12, Bonn, Germany XSWAVES, version 2.x

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**Outline Introduction to X-Ray Standing Waves**

Computation of XSW Data - XSWAVES

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**Introduction to XSW – the Physics behind…**

Literature: B.W. Batterman, H. Cole, Reviews of Modern Physics 36 (1964) J. Zegenhagen, Surface Science Reports 18 (1993) D.P. Woodruff, Progress in Surface Science 57 (1998) 1-60. D.P. Woodruff, Reports on Progress in Physics 68 (2005)

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**single-crystalline substrate**

Introduction to XSW (NI)XSW = (Normal Incidence) X-ray Standing Waves Absorption spectroscopy based on diffraction / Photoemission spectroscopy at photon energies EBragg 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 IXSW dH lB max IXSW crystal surface J. Zegenhagen, Surf. Sci. Rep. 18 (1993) / D.P. Woodruff, Rep. Prog. Phys. 68 (2005) / B.W. Batterman, H. Cole, Rev. Mod. Phys. 36 (1964) 681.

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**interference of incoming and reflected wave**

Introduction to XSW Typical NIXSW profiles Bragg-reflected wave incoming interference of incoming and reflected wave FH: coherent fraction PH: coherent position SR, |SI|, : non-dipolar parameters

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**interference of incoming and reflected wave**

Introduction to XSW Non-dipolar contributions Bragg-reflected wave incoming interference of incoming and reflected wave FH: coherent fraction PH: coherent position SR, |SI|, : 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 D are tabulated. = SR = |SI| 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 Physics behind XSW…**

The reflectivity curve R is calculated as (1-4): where h is (in terms of photon energy): h 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 FH / F-H is omitted = 1 for centrosymmetric crystals

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**The Physics behind XSW…**

The phase shift (or F…) between the incoming and the outgoing X-ray plane wave is computed as (1-4): where is : and conditions inverted in XSWAVES source code: () → (E) J. Zegenhagen, Surf. Sci. Rep. 18 (1993) / D.P. Woodruff, Rep. Prog. Phys. 68 (2005) / 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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**Computation of XSW data**

Requirements: Open-source routine Sophisticated, reliable fitting engine Full access to fit parameters Batch processing User-friendly interface *.txt file input: parameters reflectivity exp. XSW profile NLSF fitting engine numerical and graphical results output NLSF = LabTalk object that can be called from ORIGIN C; robust, widely tested and documented fitting engine LabTalk = scripting language in ORIGIN NLFitSession = ORIGIN C class; faster, but hardlay any documentation, no chance of error weighting (version 8.0) …

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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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**“step-wise“ convolution:**

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 … ORIGIN C global functions : hirachy = global function <= header <= library

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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. simulation XSWAVES ver. 2.0 Data set CF CP Test 1 0.5 0.513 0.496 Test 2 1.0 0.7 1.000 0.698 Test 3 0.3 0.300 0.696 Test 4 0.8 0.1 0.812 0.099 Test 5 0.8 0.809 0.799 Test 6 0.6 0.615 0.299 Agreement within 1 – 2 % between synthetic data and XSWAVES fit result.

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**XSWAVES benchmarking Ag(111), Test 2: Simulation: CF = 1.0 CP = 0.7**

Fit: CF = CP = 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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