Convolution Based Profile Fitting with TOPAS Arnt Kern Title picture can be exchanged in the masterslide. 2011 TOPAS User's Meeting

TOPAS Convolution Based Profile Fitting TOPAS features a direct convolution approach with all parameters refineable Choice of empirical or physically meaningful profile fitting Virtually any peak shape, angle dependence as well as hkl dependence (anisotropic line broadening) can normally be described using a minimum number of profile parameters Quantitative microstructure analysis "Double-Voigt Approch" (e.g. Balzar, 1999) "WPPM" (Scardi & Leoni, 2001, 2004; David et al., 2010; Scardi et al. 2010) Anisotropic strain (Leineweber, 2010, 2011) ... 2011 TOPAS User's Meeting

TOPAS Convolution Based Profile Fitting Kern, A., Cheary, R.W., Coelho, A.A. (2004): Convolution Based Profile Fitting. Diffraction Analysis of the Microstructure of Materials. Editors: Mittemeijer, E.J. & Scardi, P. Springer Series in Materials Science, Vol. 68, 552 pages, ISBN: 978-3-540-40519-1 Cheary, R.W., Coelho, A.A. & Cline, J.P. (2004): Fundamental Parameters Line Profile Fitting in Laboratory Diffractometers. J. Res. Natl. Inst. Stand. Technol., 109, 1-25. Kern, A. (2008): Convolution Based Profile Fitting. Principles and Applications of Powder Diffraction. Editors: Clearfield, A., Bhuvanesh N. & Reibenspies J. Blackwell Publishers, 400 pages. ISBN: 978-1-405-16222-7 2011 TOPAS User's Meeting

TOPAS Convolution Based Profile Fitting Methodology Physically meaningless parametrization of observed line profile shapes Explicit discrimination of instrument and sample contributions to observed line profile shapes Y(2q) = (W  G)  S, with (W  G) = I (Well known approach latest since Klug & Alexander, 1954) Measured I: Based on a standard reference material Calculated I: Fundamental Parameters Approach Y(2q): Observed line profile shape, I: Instrument function, W: Source emission profile, G: Geometric instrument aberations 2011 TOPAS User's Meeting

TOPAS Convolution Based Profile Fitting Profile shape functions (PSFs) are generated by convoluting functions together to form the observed profile shape: Y(2q) = F1(2q)  F2(2q)  ...  Fi(2q)  ...  Fn(2q) In general any combination of appropriate functions for Fi(2q) may be used The functions Fi(2q) can be interpreted as the aberration functions of the diffractometer: FPA 2011 TOPAS User's Meeting

Convolution Based Profile Fitting "Joy of Convolution" The process of convolution is one in which the product of two functions f(2q) and h(2q) is integrated over all space, y(2q) = f(2q)  h(2q) = f(2q') h(2q - 2q') d(2q') where y(2q) is the convolution product, 2q' is the variable of integration in the same 2q domain, and  denotes the convolution process. In simple terms, convolution can be understood as "blending" one function with another, producing a kind of very general "moving average". The convoluted function is obtained by setting down the origin of the first function in every possible position of the second, multiplying the values of both functions in each position, and taking the sum of all operations. See e.g. http://www.jhu.edu/%7Esignals/discreteconv/ 2011 TOPAS User's Meeting

Convolution Based Profile Fitting "Joy of Convolution" Demo http://www.jhu.edu/%7Esignals/discreteconv/ TOPAS 2011 TOPAS User's Meeting

TOPAS Convolution Synthesis of Line Profiles W Fi(2q) Y(2q) Y(2q) = F1(2q)  F2(2q)  ...  Fi(2q)  ...  Fn(2q) 2011 TOPAS User's Meeting

TOPAS Convolution Based Profile Fitting Applicability Virtually any peak shape, angle dependence as well as hkl dependence (anisotropic line broadening) can normally be described using a minimum number of profile parameters Powder Diffraction Laboratory and synchrotron X-ray data CW and TOF neutron data Other X-ray applications PDF, SAXS Other XY data types XRF, Raman, NMR, OES, DTA / DSC ... (peak positions, intensities) 2011 TOPAS User's Meeting

Source Emission Profiles 2011 TOPAS User's Meeting

Source Emission Profiles Generation of the emission profile is the first step in peak generation. An emission profile comprises EMk lines, each of which is a Voigt function comprising the following parameters: la: Area under the emission profile line (relative intensities for more than one line; the line with the largest la will be used for d-spacing calculations by default) lo: Wavelength in [Å] of the emission profile line lh: Lorentzian HW of the emission profile line in [mili-Å] that is convoluted into the emission profile line lg: Gaussian HW of the emission profile line in [mili-Å] that is convoluted into the emission profile line Source emission profiles for most common sealed tube / rotating anode materials are found in the "/lam" directory Instruments with crystal monochromators may require (re)determination of their source emission profile Synchrotron and neutron sources: User-defined 2011 TOPAS User's Meeting

Source Emission Profiles Cu Ka Cu Ka spectrum fitted by four Lorentzians (dotted lines) CuKa4_Holzer.lam Härtwig et al. (1993) 2011 TOPAS User's Meeting

Source Emission Profiles Cu Ka Satellites ("CuKa3") Cu Ka spectrum is better represented using five Lorentzians CuKa5_Berger.lam From Cheary et al. (2004) 2011 TOPAS User's Meeting

Source Emission Profiles Cu Ka Two examples for differently sophisticated Cu Ka emission profiles: 1) By default, d-values are calculated from the line with the largest la 2) Approximates modeling in traditional software 3) Intensity ratio Ka1/Ka2 is 0.514 4) Assumes same shape and width for Ka1 and Ka2  misfit! Emission profile Emission lines lo [Å] la [%] lh [mili-Å] CuKa5_Berger Satellites 1.534753 1.59 3.6854 Ka1a 1.540596 56.91 1) 0.4370 Ka1b 1.541058 7.62 0.6000 Ka2a 1.544410 25.17 0.5200 Ka2b 1.544721 8.71 0.6200 CuKa2_analyt 2) Ka1 66.05 1, 3) 0.5 4) Ka2 1.544493 33.95 3) 2011 TOPAS User's Meeting

Source Emission Profiles Cu Ka Satellites ("CuKa3") LAM cursor for CuKa5_Berger.lam Size-Strain Round Robin, Balzar (2000); "sharp data", Le Bail. 2011 TOPAS User's Meeting

Source Emission Profiles Finite X-Ray Source Width (Tube Tails) Intensity scan with 50 mm wide slit of an image formed through a 10 mm pinhole in platinum of the 0.4 mm wide long fine focus in a Cu anode. X-ray tube set at 40kV, 40 mA. From Cheary et al. (2004) 2011 TOPAS User's Meeting

Source Emission Profiles Finite X-Ray Source Width (Tube Tails) Cu Ka1,2 Cu Ka3 Tube Tail Tube Tail SQRT scale 2011 TOPAS User's Meeting

Source Emission Profiles Absorption Edge Modelling Cu Ka3 Cu Ka1,2 Remnant Cu Kß Remnant Bremsstrahlung Ni Abs. Edge Tube Tail Tube Tail SQRT scale Absorption Edge Modelling

TOPAS Convolution Based Profile Fitting Methodology Empirical parametrization of observed line profile shapes Profile parameters have NO physical meaning Explicit discrimination of instrument and sample contributions Measured Instrument Function Sample related profile parameters have a physical meaning: Absorption, microstructure (size, strain, ...)... Calculated Instrument Function All profile parameters have a physical meaning (Fundamental Parameters Approach) 2011 TOPAS User's Meeting

1. Empirical parametrization of observed line profile shapes 2011 TOPAS User's Meeting

Convolution Based Profile Fitting 1. Empirical Parametrization Convolute any appropriate functions to achieve a best fit Minimize the number of functions / function parameters to minimize parameter correlation Note: An excellent approach for all profile fit applications using any instrument, if micro-structure information is NOT of interest Refined profile parameters have NO physical meaning 2011 TOPAS User's Meeting

Convolution Based Profile Fitting 1. Empirical Parametrization Knowledge of the most common contributions to line profile shapes and their dependence on angle helps, e.g. for laboratory instruments: Do it yourself: Trial and error Two back-to-back circle or exponential functions convoluted on top of a Voigt function (e.g. David & Jorgensen, 1993), each of which parametrized with appropriate dependence on angle and maybe hkl, will fit virtually anything Contribution Convolution Angular dependence Detector (Slit) Hat Constant Crystallite size Lorentzian 1/cos(Th) Strain Gaussian Tan(Th) Axial divergence Circle -1/Tan(Th) 2011 TOPAS User's Meeting

CPD Size-Strain Round Robin – CeO2 Balzar et al. (2004) Size-Strain Round Robin, Balzar (2004); "sharp data".

CPD Size-Strain Round Robin – CeO2 Balzar et al. (2004) Virtually any peak shape and angle dependence can normally be described using a minimum number of (refineable) profile parameters Size-Strain Round Robin, Balzar (2004); "sharp data".

2a. Explicit discrimination of instrument and sample contributions Measured Instrument Functions 2011 TOPAS User's Meeting

Convolution Based Profile Fitting 2a. Measured Instrument Functions Measured Instrument Function Approach: Step 1: Determine an instrument function Obtain data from a suited standard reference material, e.g.: Reflection: LaB6 (NIST SRM 660a) Transmission: Si (NIST SRM 640c) Convolute any appropriate functions to achieve a best fit (Empirical parameterization!, see above) The number of functions / function parameters is irrelevant, the more the better! Fix all refineable profile parameters and save an instrument file Step 2: Refine the actual data using the instrument function Load the instrument file Refine on micro-structure parameters as required 2011 TOPAS User's Meeting

Convolution Based Profile Fitting 2a. Measured Instrument Functions Measured Instrument Function Approach: Note: An excellent approach for all profile fit applications using any instrument Micro-structure information can be derived Allows to deal with higher degrees of peak overlap (with significant impact specifically on quantitative Rietveld analysis) Uses a minimal number of profile parameters (specimen contributions such as size/strain/...) Sample related profile parameters have a physical meaning (Absorption, microstructure (size, strain, ...)... Measured instrument functions need redetermination after instrument alignment / change of configuration 2011 TOPAS User's Meeting

CPD Size-Strain Round Robin – CeO2 Balzar et al. (2004) Size-Strain Round Robin, Balzar et al. (2004); "broad data".

2b. Explicit discrimination of instrument and sample contributions Calculated Instrument Functions (Fundamental Parameters Approach) 2011 TOPAS User's Meeting

Convolution Based Profile Fitting 2b. Calculated Instrument Functions Calculated Instrument Function Approach: Physically based diffractometer models are used by convoluting instrument aberation functions. This can be done for most powder diffractometer configurations including Conventional divergent beam instruments (most convenient!) Parallel beam instruments Instruments used for asymmetric diffraction, (e.g. instruments equipped with position sensitive detectors)  Fundamental Parameters Approach 2011 TOPAS User's Meeting

Convolution Based Profile Fitting 2b. Calculated Instrument Functions Calculated Instrument Function Approach: Note: The ideal approach for all profile fit applications - not only if micro-structure information is of interest The Bragg Brentano Geometry is the most convenient Micro-structure information can be derived Allows to deal with higher degrees of peak overlap (with significant impact specifically on quantitative Rietveld analysis) Uses a minimal number of profile parameters (normally only specimen contributions such are refined) All profile parameters have a physical meaning 2011 TOPAS User's Meeting

Example: Bragg-Brentano Geometry Beam Path and Optics 2011 TOPAS User's Meeting

CPD Size-Strain Round Robin – CeO2 Balzar et al. (2004) Laboratory X-ray data (D8 ADVANCE): Instrument function (FPA) 1 Lorentzian function: Crystallite size broadening Total number of refineable parameters: 1

CPD Size-Strain Round Robin – CeO2 Balzar et al. (2004) D8 ADVANCE, "sharp data"

CPD Size-Strain Round Robin – CeO2 Balzar et al. (2004) 2011 TOPAS User's Meeting

CPD Size-Strain Round Robin – CeO2 Balzar et al. (2004) TOPAS results submitted (Kern, A.)