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An elementary method to generate asymmetric profiles and some applications in X-ray diffraction analysis

Published online by Cambridge University Press:  06 December 2024

Abraham Gutierrez  and
Manfred Wiessner Open the ORCID record for Manfred Wiessner [Opens in a new window]
Abraham Gutierrez
Affiliation:
Institute of Analysis and Number Theory, TUGraz , 8010 Graz, Austria
Manfred Wiessner*
Affiliation:
Anton Paar GmbH , Anton-Paar-Str. 20, 8054 Graz, Austria
*
a)Author to whom correspondence should be addressed. Electronic mail: [email protected]
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Abstract

We present a versatile method to generate asymmetric profiles and use it to create Gaussian-like, Cauchy-like, and Pseudo-Voigt-like profiles in terms of elementary functions. Furthermore, this method guarantees that the position and magnitude of the global maximum are independent of the asymmetry parameter, which substantially facilitates the convergence of an optimizer when fitting the peaks to real data. This investigation shows that the method developed here exhibits favorable practical properties and is particularly well suited for various applications where asymmetric peak profiles are observed. For example, in X-ray diffraction (XRD) measurements, the use of asymmetric profiles is essential for obtaining accurate outcomes. This is because diffractometers can introduce asymmetry into the diffraction peaks due to factors such as axial divergence in the beam path. By taking this asymmetry into account during the modeling process, the resulting data obtained can be corrected for instrumental effects. The results of the study show that the evaluation of XRD using nearly defect-free LaB6 allows a precise characterization of the peak broadening caused by the diffractometer itself. Additional size-strain effects of ZnO are determined by considering the asymmetric peak profile of the diffractometer.

Keywords

asymmetric profilesasymmetric profile constructionelementary functionsXRD measurementsinstrumental effects correction
Type
Technical Article
Information
Powder Diffraction , Volume 40 , Issue 1 , March 2025 , pp. 21 - 35
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of International Centre for Diffraction Data

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References

REFERENCES

Alexander, L. 1954. “The Synthesis of X-Ray Spectrometer Line Profiles with Application to Crystallite Size Measurements.” Journal of Applied Physics 25: 155–61.CrossRefGoogle Scholar
Anton Paar. 2021. https://www.anton-paar.com/us-en/products/details/xrdynamic-500/ (download 20/10/2024).Google Scholar
Balzar, D., and Ledbetter, H. . 1994. “Accurate Modelling of Size and Strain Broadening in the Rietveld Refinement: The ‘Double-Voigt’ Approach.” Advances in X-ray Analysis 38: 397–404.Google Scholar
Black, D. R., Mendenhall, M. H., Brown, C. M., Henins, A., Filliben, J., and Cline, J. P. . 2020. “Certification of Standard Reference Material 660c for Powder Diffraction.” Powder Diffraction 35: 17–22.CrossRefGoogle ScholarPubMed
Bragg, W. H., and Bragg, W. L. . 1913. “The Reflection of X-Rays by Crystals.” Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 88: 428–38.Google Scholar
Cheary, R. W., and Coelho, A. . 1992. “A Fundamental Parameters Approach to X-Ray Line-Profile Fitting.” Journal of Applied Crystallography 25: 109–21.CrossRefGoogle Scholar
Cheary, R. W., and Coelho, A. . 1994. “Synthesizing and Fitting Linear Position-Sensitive Detector Step-Scanned Line Profiles.” Journal of Applied Crystallography 27: 673–81.CrossRefGoogle Scholar
Cheary, R. W., and Coelho, A. A. . 1998. “Axial Divergence in a Conventional X-Ray Powder Diffractometer. I. Theoretical Foundations.” Journal of Applied Crystallography 31: 851–61.CrossRefGoogle Scholar
Cheary, R. W., Coelho, A. A., and Cline, J. P. . 2004. “Fundamental Parameters Line Profile Fitting in Laboratory Diffractometers.” Journal of Research of the National Institute of Standards and Technology 109: 1–25.CrossRefGoogle ScholarPubMed
Cline, J. P., Mendenhall, M. H., Ritter, J. J., Black, D., Henins, A., Bonevich, J. E., Whitfield, P. S., and Filliben, J. J. . 2020. “The Certification of Standard Reference Material 1979: Powder Diffraction Line Profile Standard for Crystallite Size Analysis.” Journal of Research of the National Institute of Standards & Technology 125: 1–39. doi:10.6028/jres.125.020.CrossRefGoogle ScholarPubMed
Coelho, A. A. 2018. “TOPAS and TOPAS-Academic: An Optimization Program Integrating Computer Algebra and Crystallographic Objects Written in C++.” Journal of Applied Crystallography 51: 210–18.CrossRefGoogle Scholar
Davis, B. L. 1989. Quantitative Reference Intensity Analysis: Methodology and Means for Verification of Results in Hašek, J. (Ed.), X-Ray and Neutron Structure Analysis in Materials Science (pp. 15–21). Boston, MA, Springer.CrossRefGoogle Scholar
Davis, B. L. 1992. Quantitative Phase Analysis With Reference Intensity Ratio in E. Prince & J. K. Stalick (Eds.), Accuracy in Powder Diffraction II: Proceedings of the International Conference May 26-29, 1992(pp. 7–16).Google Scholar
Degen, T., Sadki, M., Bron, E., König, U., and Nénert, G. . 2014. “The HighScore Suite.” Powder Diffraction 29: 13–18.CrossRefGoogle Scholar
Denney, J. J., Mattei, G. S., Mendenhall, M. H., Cline, J. P., Khalifah, P. G., and Toby, B. H. . 2022. “Determination of Physically Based Pseudo-Voigt Powder Diffraction Profile Terms From the Fundamental Parameters Approach.” Journal of Applied Crystallography 55: 289–95.CrossRefGoogle Scholar
Faddeeva, V. N., and Terent'Ev, N. M. . 1961. Tables of Values of the Function W (z). Oxford/London/New York/Paris, Pergamon Press.Google Scholar
Gates-Rector, S., and Blanton, T. . 2019. “The Powder Diffraction File: A Quality Materials Characterization Database.” Powder Diffraction 34: 352–60.CrossRefGoogle Scholar
Gupta, A. K., Nguyen, T. T., and Sanqui, J. A. T. . 2004. “Characterization of the Skew-Normal Distribution.” Annals of the institute of statistical Mathematics 56: 351–60.CrossRefGoogle Scholar
Huang, W.-J., and Chen, Y.-H. . 2007. “Generalized Skew-Cauchy Distribution.” Statistics & Probability Letters 77: 1137–47.CrossRefGoogle Scholar
Hubbard, C. R., Evans, E. H. V. I., and Sedlovets, D. M.. 1976. “The Reference Intensity Ratio, I/Ic, for computer Simulated Powder Patterns.” Journal of Applied Crystallography 9: 169–74.CrossRefGoogle Scholar
Kern, A. 1998. Hochtemperatur-Rietveldanalysen: Möglichkeiten und Grenzen. Heidelberg, Ruprecht-Karls-Univ.Google Scholar
Korepanov, V. I., and Sedlovets, D. M. . 2018. “An Asymmetric Fitting Function for Condensed-Phase Raman Spectroscopy.” Analyst 143: 2674–9.CrossRefGoogle ScholarPubMed
Mittemeijer, E. J., and Scardi, P. . (Eds.). 2013. Diffraction Analysis of the Microstructure of Materials. Berlin, Heidelberg, Springer Science & Business Media.Google Scholar
Rietveld, H. M. 1967. “Line Profiles of Neutron Powder-Diffraction Peaks for Structure Refinement.” Acta Crystallographica 22: 151–2.CrossRefGoogle Scholar
Rietveld, H. M. 1969. “A Profile Refinement Method for Nuclear and Magnetic Structures.” Journal of Applied Crystallography 2: 65–71.Google Scholar
Sampoorna, M., Nagendra, K. N., and Frisch, H. . 2007. “Generalized Voigt Functions and Their Derivatives.” Journal of Quantitative Spectroscopy and Radiative Transfer 104: 71–85.CrossRefGoogle Scholar
The MathWorks Inc . 2023. MATLAB Version: 23.2.0.2380103 (R2023b). Natick, MA: The MathWorks Inc. https://www.mathworks.com.Google Scholar
Thomson, P., Cox, D. E., and Hastings, J. B. . 1987. “Rietveld Refinement of Debye-Scherrer Synchrotron X-Ray Data From Al2O3.” Journal of Applied Crystallography 20: 79–83.CrossRefGoogle Scholar
Toby, B. H. 2006. “R Factors in Rietveld Analysis: How Good is Good Enough?” Powder Diffraction 21: 67–70.CrossRefGoogle Scholar
Toby, B. H., and Von Dreele, R. B. . 2013. “GSAS-II: The Genesis of a Modern Open-Source All Purpose Crystallography Software Package.” Journal of Applied Crystallography 46: 544–9.CrossRefGoogle Scholar
Wießner, M., Kleber, S., and Kulmburg, A. . 2005. “In-Situ Investigation During Tempering of a High Speed Steel With X-Ray Diffraction.” Particle & Particle Systems Characterization 22: 407–17.CrossRefGoogle Scholar
Wiessner, M., Hipp, M., Gutierrez, A., and Bachofner, B. . 2023. XRDanalysis 1.2 (Computer Software). Graz, Anton Paar GmbH.Google Scholar
Wolberg, J. 2006. Data Analysis Using the Method of Least Squares. Berlin, Heidelberg, New York, Springer.Google Scholar
Zak, A. K., Majid, W. A., Abrishami, M. E., and Yousefi, R. . 2011. “X-Ray Analysis of ZnO Nanoparticles by Williamson–Hall and Size–Strain Plot Methods.” Solid State Sciences 13: 251–6.Google Scholar