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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "https://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> <html xmlns="http://www.w3.org/1999/xhtml"> <head> <meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> <title>(IUCr) Parameterized absorptive electron scattering factors</title> <link rel="schema.DC" href="http://purl.org/dc/elements/1.1/" /> <link rel="schema.DCTERMS" href="http://purl.org/dc/terms/" /> <link rel="schema.prism" href="http://prismstandard.org/namespaces/1.2/basic/" /> <meta http-equiv="Content-Type" content="text/html; charset=utf-8"/> <meta name="twitter:image" content="https://journals.iucr.org/a/issues/2024/02/00/pl5034/pl5034largethumb.jpg" /> <meta name="twitter:title" content="Parameterized absorptive electron scattering factors" /> <meta name="twitter:site" content="@ActaCrystA" /> <meta name="twitter:description" content="This paper provides a rapid parameterized calculation of absorptive scattering factors for 103 elements as neutral, spherical atoms, which reduces calculation time considerably." /> <meta name="twitter:card" content="summary" /> <meta property="og:updated_time" content="2024-03-04T13:23:45Z" /> <meta property="og:url" content="https://journals.iucr.org/a/issues/2024/02/00/pl5034/" /> <meta property="og:description" content="This paper provides a rapid parameterized calculation of absorptive scattering factors for 103 elements as neutral, spherical atoms, which reduces calculation time considerably." /> <meta property="og:image" content="https://journals.iucr.org/a/issues/2024/02/00/pl5034/pl5034largethumb.jpg" /> <meta property="og:locale" content="en_US" /> <meta property="og:title" content="Parameterized absorptive electron scattering factors" /> <meta property="og:site_name" content="Acta Crystallographica Section A" /> <meta property="og:image:alt" content="Image from Parameterized absorptive electron scattering factors" /> <meta property="og:type" content="article" /> <meta property="og:image:type" content="image/jpeg" /> <meta property="og:image:height" content="300" /> <meta property="og:image:secureurl" content="https://journals.iucr.org/a/issues/2024/02/00/pl5034/pl5034largethumb.jpg" /> <meta property="og:image:width" content="300" /> <meta name="DC.date" content="2024-01-25" /> <meta name="DC.relation" content="" /> <meta name="DC.identifier" content="doi:10.1107/S2053273323010963" /> <meta name="DC.format" content="text/html" /> <meta name="DC.language" content="en" /> <meta name="description" content="Parameterized absorptive electron scattering factors" /> <meta name="title" content="Parameterized absorptive electron scattering factors" /> <meta name="DC.title" content="Parameterized absorptive electron scattering factors" /> <meta name="DC.type" content="research papers" /> <meta name="DC.coverage" content="" /> <meta name="DC.teaser" content="This paper provides a rapid parameterized calculation of absorptive scattering factors for 103 elements as neutral, spherical atoms, which reduces calculation time considerably." /> <meta name="DC.description" content="In electron diffraction, thermal atomic motion produces incoherent scattering over a relatively wide angular range, which appears as a diffuse background that is usually subtracted from measurements of Bragg spot intensities in structure solution methods. The transfer of electron flux from Bragg spots to diffuse scatter is modelled using complex scattering factors f + if′ in the Bloch wave methodology. In a two-beam Einstein model the imaginary `absorptive' scattering factor f′ can be obtained by the evaluation of an integral containing f over all possible scattering angles. While more sophisticated models of diffuse scatter are widely used in the electron microscopy community, it is argued in this paper that this simple model is appropriate for current structure solution and refinement methods. The two-beam model is a straightforward numerical calculation, but even this simplistic approach can become time consuming for simulations of materials with large numbers of atoms in the unit cell and/or many incident beam orientations. Here, a parameterized form of f′ is provided for 103 elements as neutral, spherical atoms that reduces calculation time considerably." /> <meta name="DC.publisher" content="International Union of Crystallography" /> <meta name="DC.subject" content="ELECTRON DIFFRACTION" /> <meta name="DC.subject" content="ABSORPTION" /> <meta name="DC.subject" content="3D-ED" /> <meta name="DC.subject" content="THREE-DIMENSIONAL ELECTRON DIFFRACTION" /> <meta name="DC.subject" content="THERMAL DIFFUSE SCATTERING" /> <meta name="DC.copyright" content="https://creativecommons.org/licenses/by/4.0/" /> <meta name="DC.source" content="urn:issn:2053-2733" /> <meta name="DC.rights" content="https://creativecommons.org/licenses/by/4.0/" /> <meta name="DC.link" content="https://journals.iucr.org/paper?pl5034" /> <meta name="DC.creator" content="Thomas, M." /> <meta name="DC.creator" content="Cleverley, A." /> <meta name="DC.creator" content="Beanland, R." /> <meta name="authors" content="Thomas, M.|Cleverley, A.|Beanland, R." /> <meta name="DCTERMS.abstract" content="In electron diffraction, thermal atomic motion produces incoherent scattering over a relatively wide angular range, which appears as a diffuse background that is usually subtracted from measurements of Bragg spot intensities in structure solution methods. The transfer of electron flux from Bragg spots to diffuse scatter is modelled using complex scattering factors f + if′ in the Bloch wave methodology. In a two-beam Einstein model the imaginary `absorptive' scattering factor f′ can be obtained by the evaluation of an integral containing f over all possible scattering angles. While more sophisticated models of diffuse scatter are widely used in the electron microscopy community, it is argued in this paper that this simple model is appropriate for current structure solution and refinement methods. The two-beam model is a straightforward numerical calculation, but even this simplistic approach can become time consuming for simulations of materials with large numbers of atoms in the unit cell and/or many incident beam orientations. Here, a parameterized form of f′ is provided for 103 elements as neutral, spherical atoms that reduces calculation time considerably." /> <meta name="prism.number" content="2" /> <meta name="prism.volume" content="80" /> <meta name="prism.startingPage" content="146" /> <meta name="prism.publicationName" content="Acta Crystallographica Section A: Foundations and Advances" /> <meta name="prism.endingPage" content="150" /> <meta name="prism.doi" content="https://doi.org/10.1107/S2053273323010963" /> <meta name="prism.issn" content="2053-2733" /> <meta name="prism.eissn" content="2053-2733" /> <meta name="prism.rightsAgent" content="med@iucr.org" /> <meta name="prism.publicationDate" content="2024-01-25" /> <meta name="prism.section" content="research papers" /> <meta name="prism.copyright" content="https://creativecommons.org/licenses/by/4.0/" /> <meta name="keywords" content="ELECTRON DIFFRACTION; ABSORPTION; 3D-ED; THREE-DIMENSIONAL ELECTRON DIFFRACTION; THERMAL DIFFUSE SCATTERING" /> <meta name="copyright" content="https://creativecommons.org/licenses/by/4.0/" /> <meta name="ROBOTS" content="NOARCHIVE" /> <meta name="citation_fulltext_url" content="https://journals.iucr.org/a/issues/2024/02/00/pl5034/" /> <meta name="citation_fulltext_world_readable" content=""/> <meta name="citation_year" content="2024" /> <meta name="citation_journal_abbrev" content="Acta Cryst A" /> <meta name="citation_journal_abbrev" content="Acta Cryst Sect A" /> <meta name="citation_journal_abbrev" content="Acta Crystallogr A" /> <meta name="citation_journal_abbrev" content="Acta Crystallogr Sect A" /> <meta name="citation_journal_abbrev" content="Acta Crystallogr A Cryst Phys Diffr Theor Gen Crystallogr" /> <meta name="citation_journal_abbrev" content="Acta Crystallogr Sect A Cryst Phys Diffr Theor Gen Crystallogr" /> <meta name="citation_title" content="Parameterized absorptive electron scattering factors" /> <meta name="citation_abstract" content="In electron diffraction, thermal atomic motion produces incoherent scattering over a relatively wide angular range, which appears as a diffuse background that is usually subtracted from measurements of Bragg spot intensities in structure solution methods. The transfer of electron flux from Bragg spots to diffuse scatter is modelled using complex scattering factors f + if′ in the Bloch wave methodology. In a two-beam Einstein model the imaginary `absorptive' scattering factor f′ can be obtained by the evaluation of an integral containing f over all possible scattering angles. While more sophisticated models of diffuse scatter are widely used in the electron microscopy community, it is argued in this paper that this simple model is appropriate for current structure solution and refinement methods. The two-beam model is a straightforward numerical calculation, but even this simplistic approach can become time consuming for simulations of materials with large numbers of atoms in the unit cell and/or many incident beam orientations. Here, a parameterized form of f′ is provided for 103 elements as neutral, spherical atoms that reduces calculation time considerably." /> <meta name="citation_publisher" content="International Union of Crystallography" /> <meta name="citation_doi" content="10.1107/S2053273323010963" /> <meta name="citation_firstpage" content="146" /> <meta name="citation_abstract_html_url" content="https://journals.iucr.org/paper?pl5034" /> <meta name="citation_lastpage" content="150" /> <meta name="citation_issn" content="2053-2733" /> <meta name="citation_date" content="2024-03-01" /> <meta name="citation_issue" content="2" /> <meta name="citation_pdf_url" content="https://journals.iucr.org/a/issues/2024/02/00/pl5034/pl5034.pdf" /> <meta name="citation_language" content="en" /> <meta name="citation_journal_title" content="Acta Crystallographica Section A: Foundations and Advances" /> <meta name="citation_keywords" content="ELECTRON DIFFRACTION; ABSORPTION; 3D-ED; THREE-DIMENSIONAL ELECTRON DIFFRACTION; THERMAL DIFFUSE SCATTERING" /> <meta name="citation_author" content="Thomas, M." /> <meta name="citation_author_institution" content="Department of Physics, University of Warwick, Coventry CV4 7AL, UK" /> <meta name="citation_author_email" content="milo.thomas@warwick.ac.uk" /> <meta name="citation_author" content="Cleverley, A." /> <meta name="citation_author_institution" content="Department of Chemistry, University of Warwick, Coventry CV4 7AL, UK" /> <meta name="citation_author_email" content="anton.cleverley@warwick.ac.uk" /> <meta name="citation_author" content="Beanland, R." /> <meta name="citation_author_institution" content="Department of Physics, University of Warwick, Coventry CV4 7AL, UK" /> <meta name="citation_author_email" content="r.beanland@warwick.ac.uk" /> <meta name="citation_volume" content="80" /> <meta name="citation_reference" content="citation_author=Allen L. 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text-align: right; width: 100px;"> <script src="https://crossmark-cdn.crossref.org/widget/v2.0/widget.js"></script> <a data-target="crossmark"><img alt="crossmark logo" src="https://crossmark-cdn.crossref.org/widget/v2.0/logos/CROSSMARK_Color_square_no_text.svg" width="60" /></a> </div>  <div id="aug"> <div class="au"> <b> <a href="https://scripts.iucr.org/cgi-bin/citedin?search_on=name&author_name=Thomas%2C%20M%2E"><span class="au">M. Thomas</span></a>,<a href="#oida"><sup>a</sup></a> <a href="https://scripts.iucr.org/cgi-bin/citedin?search_on=name&author_name=Cleverley%2C%20A%2E"><span class="au">A. Cleverley</span></a><a href="#oidb"><sup>b</sup></a> and <a href="https://scripts.iucr.org/cgi-bin/citedin?search_on=name&author_name=Beanland%2C%20R%2E"><span class="au">R. Beanland</span></a><a href="#oida"><sup>a</sup></a><a href="#cor"><sup>*</sup></a></b> </div> <div id="aff"> <p><span class="font_size_2"><a id="oida"><sup><b>a</b></sup></a>Department of Physics, University of Warwick, Coventry CV4 7AL, UK, and <a id="oidb"><sup><b>b</b></sup></a>Department of Chemistry, University of Warwick, Coventry CV4 7AL, UK<br /><a id="cor"></a><sup>*</sup>Correspondence e-mail: <a href="mailto:r.beanland%40warwick.ac.uk">r.beanland@warwick.ac.uk</a></span></p> </div> </div> <div id="editdetails"><span class="editor">Edited by P. M. Dominiak, University of Warsaw, Poland</span> (<span class="ed_rec">Received 23 September 2023;</span> <span class="ed_acc">accepted 21 December 2023;</span> <span class="ed_web">online 25 January 2024</span>)</div> <div id="abs"> <p>In electron diffraction, thermal atomic motion produces <a target='Navigator' class="ref_lookup_orange hideorange" href='https://goldbook.iupac.org/I03000.html' onclick="return makeSubWindow("https://goldbook.iupac.org/I03000.html", 'Navigator')">incoherent scattering</a> over a relatively wide angular range, which appears as a diffuse background that is usually subtracted from measurements of Bragg spot intensities in structure solution methods. The transfer of electron <a target='Navigator' class="ref_lookup_yellow hideyellow" href='https://dictionary.iucr.org/Flux' onclick="return makeSubWindow("https://dictionary.iucr.org/Flux", 'Navigator')">flux</a> from Bragg spots to diffuse scatter is modelled using complex scattering factors <span class="it"><i>f</i></span> + <span class="it"><i>if</i></span>′ in the Bloch wave methodology. In a two-beam Einstein model the imaginary `absorptive' scattering factor <span class="it"><i>f</i></span>′ can be obtained by the evaluation of an integral containing <span class="it"><i>f</i></span> over all possible scattering angles. While more sophisticated models of diffuse scatter are widely used in the <a target='Navigator' class="ref_lookup_orange hideorange" href='https://goldbook.iupac.org/E02002.html' onclick="return makeSubWindow("https://goldbook.iupac.org/E02002.html", 'Navigator')">electron microscopy</a> community, it is argued in this paper that this simple model is appropriate for current structure solution and <a target='Navigator' class="ref_lookup_yellow hideyellow" href='https://dictionary.iucr.org/Refinement' onclick="return makeSubWindow("https://dictionary.iucr.org/Refinement", 'Navigator')">refinement</a> methods. The two-beam model is a straightforward numerical calculation, but even this simplistic approach can become time consuming for simulations of materials with large numbers of atoms in the <a target='Navigator' class="ref_lookup_yellow hideyellow" href='https://dictionary.iucr.org/Unit_cell' onclick="return makeSubWindow("https://dictionary.iucr.org/Unit_cell", 'Navigator')">unit cell</a> and/or many incident beam orientations. Here, a parameterized form of <span class="it"><i>f</i></span>′ is provided for 103 elements as neutral, spherical atoms that reduces calculation time considerably.</p> </div> <div id="kwdg"> <p><span class="kwdg_head">Keywords: </span> <a href="https://scripts.iucr.org/cgi-bin/full_search?words=electron%20diffraction&Action=Search">electron diffraction</a>; <a href="https://scripts.iucr.org/cgi-bin/full_search?words=absorption&Action=Search">absorption</a>; <a href="https://scripts.iucr.org/cgi-bin/full_search?words=3D%2DED&Action=Search">3D-ED</a>; <a href="https://scripts.iucr.org/cgi-bin/full_search?words=three%2Ddimensional%20electron%20diffraction&Action=Search">three-dimensional electron diffraction</a>; <a href="https://scripts.iucr.org/cgi-bin/full_search?words=thermal%20diffuse%20scattering&Action=Search">thermal diffuse scattering</a>.</p></div> <div class="art_codelinks"> </div> <div class="ica_readmore"> <a href="https://scripts.iucr.org/cgi-bin/similar?wordList=ELECTRON%20DIFFRACTION%20or%20ABSORPTION%20or%203D-ED%20or%20THREE-DIMENSIONAL%20ELECTRON%20DIFFRACTION%20or%20THERMAL%20DIFFUSE%20SCATTERING&from=pl5034">Similar articles</a></div> <script type="text/javascript" src="//api.growkudos.com/widgets/article/10.1107/S2053273323010963"></script> </div> </div> <div id="body"> <div id="DIVSEC1" class="sec1"> <h3><a id="SEC1"></a>1. Introduction</h3> <p><a target='Navigator' class="ref_lookup_yellow hideyellow" href='https://dictionary.iucr.org/Crystal_structure' onclick="return makeSubWindow("https://dictionary.iucr.org/Crystal_structure", 'Navigator')">Crystal structure</a> solution and <a target='Navigator' class="ref_lookup_yellow hideyellow" href='https://dictionary.iucr.org/Refinement' onclick="return makeSubWindow("https://dictionary.iucr.org/Refinement", 'Navigator')">refinement</a> using electron diffraction has been practised for some time (Vainshtein <span class="it"><i>et al.</i></span>, 1964<a id="sourceBB35"></a><a href="#BB35"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Vainshtein, B. K., Feigl, E. & Spink, J. A. (1964). Structure Analysis by Electron Diffraction. Oxford: Pergamon.]" title="Vainshtein, B. K., Feigl, E. & Spink, J. A. (1964). Structure Analysis by Electron Diffraction. Oxford: Pergamon." /></a>) and is currently undergoing a revival. This renewed interest is based on advances in the technique over the past two decades (collectively known as three-dimensional electron diffraction, 3D-ED) (Gemmi <span class="it"><i>et al.</i></span>, 2019<a id="sourceBB9"></a><a href="#BB9"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Gemmi, M., Mugnaioli, E., Gorelik, T., Kolb, U., Palatinus, L., Boullay, P. & Abrahams, J. (2019). ACS Cent. Sci. 5, 1315-1329.]" title="Gemmi, M., Mugnaioli, E., Gorelik, T., Kolb, U., Palatinus, L., Boullay, P. & Abrahams, J. (2019). ACS Cent. Sci. 5, 1315-1329." /></a>; Gruene <span class="it"><i>et al.</i></span>, 2021<a id="sourceBB10"></a><a href="#BB10"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Gruene, T., Holstein, J. J., Clever, G. H. & Keppler, B. (2021). Nat. Rev. Chem. 5, 660-668.]" title="Gruene, T., Holstein, J. J., Clever, G. H. & Keppler, B. (2021). Nat. Rev. Chem. 5, 660-668." /></a>) and the arrival of new dedicated electron diffractometers (Ito <span class="it"><i>et al.</i></span>, 2021<a id="sourceBB15"></a><a href="#BB15"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Ito, S., White, F. J., Okunishi, E., Aoyama, Y., Yamano, A., Sato, H., Ferrara, J. D., Jasnowski, M. & Meyer, M. (2021). CrystEngComm, 23, 8622-8630.]" title="Ito, S., White, F. J., Okunishi, E., Aoyama, Y., Yamano, A., Sato, H., Ferrara, J. D., Jasnowski, M. & Meyer, M. (2021). CrystEngComm, 23, 8622-8630." /></a>). However, <a target='Navigator' class="ref_lookup_yellow hideyellow" href='https://dictionary.iucr.org/Dynamical_diffraction' onclick="return makeSubWindow("https://dictionary.iucr.org/Dynamical_diffraction", 'Navigator')">dynamical diffraction</a> effects often limit the accuracy of structures obtained by applying a kinematic scattering model to 3D-ED data, requiring the use of models that take multiple scattering effects into account (Klar <span class="it"><i>et al.</i></span>, 2023<a id="sourceBB17"></a><a href="#BB17"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Klar, P. B., Krysiak, Y., Xu, H., Steciuk, G., Cho, J., Zou, X. & Palatinus, L. (2023). Nat. Chem. 15, 848-855.]" title="Klar, P. B., Krysiak, Y., Xu, H., Steciuk, G., Cho, J., Zou, X. & Palatinus, L. (2023). Nat. Chem. 15, 848-855." /></a>). While most simulations of dynamical electron scattering in <a target='Navigator' class="ref_lookup_orange hideorange" href='https://goldbook.iupac.org/E02002.html' onclick="return makeSubWindow("https://goldbook.iupac.org/E02002.html", 'Navigator')">electron microscopy</a> are currently performed using the multislice method, the Bloch wave methodology retains some advantages, in particular for continuous-rotation electron diffraction (cRED) data in which low-index zone axes are rare. Bloch wave calculations impose the periodicity of the crystal on the allowed solutions to Schrodinger's equation but, unlike multislice methods, are not atomistic in their application of boundary conditions, allowing an arbitrary crystal orientation to be simulated without artefacts. In comparison with kinematic intensities calculated from the structure-factor equation, dynamical calculations of intensity are more complex and time consuming, which is a serious concern for iterative <a target='Navigator' class="ref_lookup_yellow hideyellow" href='https://dictionary.iucr.org/Refinement' onclick="return makeSubWindow("https://dictionary.iucr.org/Refinement", 'Navigator')">refinement</a> of <a target='Navigator' class="ref_lookup_yellow hideyellow" href='https://dictionary.iucr.org/Crystal_structure' onclick="return makeSubWindow("https://dictionary.iucr.org/Crystal_structure", 'Navigator')">crystal structure.</a> Furthermore, a consequence of only including Bloch waves in a simulation is that only Bragg scattering is accounted for. The inclusion of non-Bragg scattering adds another time penalty to the calculation, which can become appreciable for unit cells containing many atoms.</p><p>Here, we provide a rapid parameterized calculation of absorptive scattering factors that eliminates this additional penalty. Current methods do not generally include absorptive scattering factors; their implementation may allow better <span class="it"><i>R</i></span> factors to be obtained and more accurate determinations of <a target='Navigator' class="ref_lookup_yellow hideyellow" href='https://dictionary.iucr.org/Crystal_structure' onclick="return makeSubWindow("https://dictionary.iucr.org/Crystal_structure", 'Navigator')">crystal structure</a> in future 3D-ED methods.</p><p>Strong non-Bragg scattering, in the form of diffuse scatter in electron diffraction patterns, was noticed from the earliest days of electron diffraction (Kikuchi, 1928<a id="sourceBB16"></a><a href="#BB16"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Kikuchi, S. (1928). Jpn. J. Phys. 5, 83.]" title="Kikuchi, S. (1928). Jpn. J. Phys. 5, 83." /></a>; Beeching, 1936<a id="sourceBB3"></a><a href="#BB3"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Beeching, R. (1936). Electron Diffraction. London: Methuen.]" title="Beeching, R. (1936). Electron Diffraction. London: Methuen." /></a>) and was soon identified as being due to <a target='Navigator' class="ref_lookup_orange hideorange" href='https://goldbook.iupac.org/I03025.html' onclick="return makeSubWindow("https://goldbook.iupac.org/I03025.html", 'Navigator')">inelastic scattering</a> from two principal sources: (<span class="it"><i>a</i></span>) displacement of atoms from their nominal sites due to thermal vibrations (thermal diffuse scattering, TDS), <span class="it"><i>i.e.</i></span> electron–phonon scattering, in which the energy loss is low; and (<span class="it"><i>b</i></span>) other <a target='Navigator' class="ref_lookup_orange hideorange" href='https://goldbook.iupac.org/I03025.html' onclick="return makeSubWindow("https://goldbook.iupac.org/I03025.html", 'Navigator')">inelastic scattering,</a> principally due to the excitation of plasmons, <span class="it"><i>Bremsstrahlung</i></span>, or ionization of the material's constituent atoms (a much greater energy loss than for TDS, typically larger than 1 eV). By the 1960s its influence both on diffraction and <a target='Navigator' class="ref_lookup_orange hideorange" href='https://goldbook.iupac.org/T06481.html' onclick="return makeSubWindow("https://goldbook.iupac.org/T06481.html", 'Navigator')">transmission electron microscopy</a> (TEM) images began to be quantified (Hirsch <span class="it"><i>et al.</i></span>, 1966<a id="sourceBB13"></a><a href="#BB13"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Hirsch, P., Howie, A., Nicholson, R., Pashley, D. & M.J, W. (1966). Electron Microscopy of Thin Crystals. London: Butterworths.]" title="Hirsch, P., Howie, A., Nicholson, R., Pashley, D. & M.J, W. (1966). Electron Microscopy of Thin Crystals. London: Butterworths." /></a>), under the term `anomalous absorption' by analogy with the <a target='Navigator' class="ref_lookup_yellow hideyellow" href='https://dictionary.iucr.org/Borrmann_effect' onclick="return makeSubWindow("https://dictionary.iucr.org/Borrmann_effect", 'Navigator')">Borrmann effect</a> seen in X-ray scattering (Borrmann, 1941<a id="sourceBB5"></a><a href="#BB5"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Borrmann, G. (1941). Z. Phys. 42, 157-162.]" title="Borrmann, G. (1941). Z. Phys. 42, 157-162." /></a>; Authier & Klapper, 2007<a id="sourceBB2"></a><a href="#BB2"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Authier, A. & Klapper, H. (2007). Phys. Status Solidi A, 204, 2515-2527.]" title="Authier, A. & Klapper, H. (2007). Phys. Status Solidi A, 204, 2515-2527." /></a>). This term now seems dated but follows from two characteristics – first, while high-energy electrons are not actually absorbed by a thin crystal, the `absorption' of electrons into a diffuse background and consequent attenuation of Bragg scattering follows a similar law to true absorption, and second, the effect is dependent on many parameters, including crystal orientation, scattering vector and incident beam energy.</p><p>The first good description of TDS (Hall & Hirsch, 1965<a id="sourceBB12"></a><a href="#BB12"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Hall, C. R. & Hirsch, P. B. (1965). Proc. R. Soc. London Ser. A, 286, 158-177.]" title="Hall, C. R. & Hirsch, P. B. (1965). Proc. R. Soc. London Ser. A, 286, 158-177." /></a>) used a simple model, <span class="it"><i>i.e.</i></span> a crystal with spherical atoms of only one type and one atom per lattice point, which vibrate harmonically and independently (the Einstein model). Using the fact that electrons propagate in a crystal as Bloch waves rather than plane waves, for a given scattering vector <span class="b"><b>s</b></span> they calculated the difference between total <a target='Navigator' class="ref_lookup_orange hideorange" href='https://goldbook.iupac.org/E01916.html' onclick="return makeSubWindow("https://goldbook.iupac.org/E01916.html", 'Navigator')">elastic scattering</a> <img src="teximages/pl5034fi1.svg" alt="[I_{\rm tot}({\bf s})]" class="img_align_bottom mathimage" style="position: relative; display: inline; padding-top: 0px; padding-bottom: 0px; vertical-align: -2.10816pt;" height="13" width="32" /> (<span class="it"><i>i.e.</i></span> including the diffuse component) and Bragg scattering <img src="teximages/pl5034fi2.svg" alt="[I_{\rm Bragg}({\bf s})]" class="img_align_bottom mathimage" style="position: relative; display: inline; padding-top: 0px; padding-bottom: 0px; vertical-align: -3.524257pt;" height="15" width="45" /> for a two-beam condition. The total non-Bragg scattering is then given by the difference, integrated over the <a target='Navigator' class="ref_lookup_yellow hideyellow" href='https://dictionary.iucr.org/Ewald_sphere' onclick="return makeSubWindow("https://dictionary.iucr.org/Ewald_sphere", 'Navigator')">Ewald sphere</a> defining all possible scattering vectors <span class="b"><b>s</b></span>. This framing allows, in the Bloch wave formalism, the transfer of intensity to be dealt with using a complex electron scattering factor of the form <span class="it"><i>f</i></span><span class="inf"><sub><span class="it"><i>g</i></span></sub></span> + <span class="it"><i>if</i></span><span class="inf"><sub><span class="it"><i>g</i></span></sub></span>′, where <span class="it"><i>f</i></span><span class="inf"><sub><span class="it"><i>g</i></span></sub></span> is the usual Born electron scattering factor for a diffraction vector with magnitude <span class="it"><i>g</i></span> and <span class="it"><i>f</i></span><span class="inf"><sub><span class="it"><i>g</i></span></sub></span>′ is an imaginary component that depends upon the isotropic Debye–Waller factor <span class="it"><i>B</i></span><span class="inf"><sub>iso</sub></span> (Humphreys & Hirsch, 1968<a id="sourceBB14"></a><a href="#BB14"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Humphreys, C. & Hirsch, P. (1968). Philos. Mag.: A J. Theor. Exp. Appl. Phys. 18, 115-122.]" title="Humphreys, C. & Hirsch, P. (1968). Philos. Mag.: A J. Theor. Exp. Appl. Phys. 18, 115-122." /></a>; Hirsch <span class="it"><i>et al.</i></span>, 1966<a href="#BB13"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Hirsch, P., Howie, A., Nicholson, R., Pashley, D. & M.J, W. (1966). Electron Microscopy of Thin Crystals. London: Butterworths.]" title="Hirsch, P., Howie, A., Nicholson, R., Pashley, D. & M.J, W. (1966). Electron Microscopy of Thin Crystals. London: Butterworths." /></a>; Peng, 1997<a id="sourceBB21"></a><a href="#BB21"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Peng, L.-M. (1997). Acta Cryst. A53, 663-672.]" title="Peng, L.-M. (1997). Acta Cryst. A53, 663-672." /></a>, 1999<a id="sourceBB22"></a><a href="#BB22"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Peng, L.-M. (1999). Micron, 30, 625-648.]" title="Peng, L.-M. (1999). Micron, 30, 625-648." /></a>). This complex scattering factor is then used in combination with the temperature factor, <img src="teximages/pl5034fi3.svg" alt="[\exp(-B_{\rm iso}s^{2})]" class="img_align_bottom mathimage" style="position: relative; display: inline; padding-top: 0px; padding-bottom: 0px; vertical-align: -2.58458pt;" height="16" width="75" />; for example the <a target='Navigator' class="ref_lookup_yellow hideyellow" href='https://dictionary.iucr.org/Structure_factor' onclick="return makeSubWindow("https://dictionary.iucr.org/Structure_factor", 'Navigator')">structure factor</a> for a reflection <span class="b"><b>g</b></span> is </p><div class="FD"><p class="mathmode display_math"><a id="FD1"></a><img src="teximages/pl5034fd1.svg" alt="[F_{g} = \textstyle\sum\limits_{j = 1}^{n}[f_{g}^{(j)}+i{f_{g}^{(j)}}^{\prime}]\exp{\left[2\pi i {\bf g}\cdot{\bf r}^{(j)}\right]}\exp{\left[-B_{\rm iso}^{(j)}s^{2}\right]},\eqno(1)]" class="mathimage" style="max-width: 100%; height: auto; width: auto;" height="31" width="355" /></p></div><p>where <span class="it"><i>s</i></span> = sin(<span class="symbol">θ</span><span class="inf"><sub>B</sub></span>)/<span class="symbol">λ</span> = <span class="it"><i>g</i></span>/2 and the summation is performed over all <span class="it"><i>n</i></span> atoms in the <a target='Navigator' class="ref_lookup_yellow hideyellow" href='https://dictionary.iucr.org/Unit_cell' onclick="return makeSubWindow("https://dictionary.iucr.org/Unit_cell", 'Navigator')">unit cell.</a> It is found that <span class="it"><i>f</i></span>′ decays more rapidly with angle, and is typically an order of magnitude smaller than <span class="it"><i>f</i></span>. Due to the computational cost of the Hall and Hirsch approach, most working calculations at the time instead used a proportional model in which <span class="it"><i>f</i></span><span class="inf"><sub><span class="it"><i>g</i></span></sub></span>′ = <span class="symbol">α</span><span class="it"><i>f</i></span><span class="inf"><sub><span class="it"><i>g</i></span></sub></span>, typically with <span class="symbol">α</span> ∼ 0.1 (Humphreys & Hirsch, 1968<a href="#BB14"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Humphreys, C. & Hirsch, P. (1968). Philos. Mag.: A J. Theor. Exp. Appl. Phys. 18, 115-122.]" title="Humphreys, C. & Hirsch, P. (1968). Philos. Mag.: A J. Theor. Exp. Appl. Phys. 18, 115-122." /></a>). Subsequent work expanded the approach to include core-loss scattering (Radi, 1970<a id="sourceBB26"></a><a href="#BB26"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Radi, G. (1970). Acta Cryst. A26, 41-56.]" title="Radi, G. (1970). Acta Cryst. A26, 41-56." /></a>; Rossouw & Bursill, 1985<span class="it"><i>b</i></span><a id="sourceBB31"></a><a href="#BB31"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Rossouw, C. J. & Bursill, L. A. (1985b). Acta Cryst. A41, 320-327.]" title="Rossouw, C. J. & Bursill, L. A. (1985b). Acta Cryst. A41, 320-327." /></a>, 1986<a id="sourceBB32"></a><a href="#BB32"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Rossouw, C. J. & Bursill, L. A. (1986). Proc. R. Soc. London Ser. A, 408, 149-164.]" title="Rossouw, C. J. & Bursill, L. A. (1986). Proc. R. Soc. London Ser. A, 408, 149-164." /></a>; Rossouw, 1985<a id="sourceBB28"></a><a href="#BB28"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Rossouw, C. (1985). Ultramicroscopy, 16, 241-254.]" title="Rossouw, C. (1985). Ultramicroscopy, 16, 241-254." /></a>; Allen & Rossouw, 1990<a id="sourceBB1"></a><a href="#BB1"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Allen, L. J. & Rossouw, C. J. (1990). Phys. Rev. B, 42, 11644-11654.]" title="Allen, L. J. & Rossouw, C. J. (1990). Phys. Rev. B, 42, 11644-11654." /></a>). Rossouw also extended the TDS model to a full dynamical <span class="it"><i>n</i></span>-beam case for both the incident and scattered waves (Rossouw & Bursill, 1985<span class="it"><i>a</i></span><a id="sourceBB30"></a><a href="#BB30"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Rossouw, C. J. & Bursill, L. A. (1985a). Acta Cryst. B41, 248-254.]" title="Rossouw, C. J. & Bursill, L. A. (1985a). Acta Cryst. B41, 248-254." /></a>,<span class="it"><i>b</i></span><a href="#BB31"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Rossouw, C. J. & Bursill, L. A. (1985b). Acta Cryst. A41, 320-327.]" title="Rossouw, C. J. & Bursill, L. A. (1985b). Acta Cryst. A41, 320-327." /></a>). In the context of microscopy, TDS has received a great deal of attention as the primary signal in atomic resolution <a target='Navigator' class="ref_lookup_orange hideorange" href='https://goldbook.iupac.org/S05486.html' onclick="return makeSubWindow("https://goldbook.iupac.org/S05486.html", 'Navigator')">scanning transmission electron microscopy</a> (STEM) (<span class="it"><i>e.g.</i></span> Pennycook & Jesson, 1991<a id="sourceBB25"></a><a href="#BB25"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Pennycook, S. & Jesson, D. (1991). Ultramicroscopy, 37, 14-38.]" title="Pennycook, S. & Jesson, D. (1991). Ultramicroscopy, 37, 14-38." /></a>; Rossouw <span class="it"><i>et al.</i></span>, 2003<a id="sourceBB29"></a><a href="#BB29"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Rossouw, C., Allen, L., Findlay, S. & Oxley, M. (2003). Ultramicroscopy, 96, 299-312.]" title="Rossouw, C., Allen, L., Findlay, S. & Oxley, M. (2003). Ultramicroscopy, 96, 299-312." /></a>; Croitoru <span class="it"><i>et al.</i></span>, 2006<a id="sourceBB8"></a><a href="#BB8"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Croitoru, M., Van Dyck, D., Van Aert, S., Bals, S. & Verbeeck, J. (2006). Ultramicroscopy, 106, 933-940.]" title="Croitoru, M., Van Dyck, D., Van Aert, S., Bals, S. & Verbeeck, J. (2006). Ultramicroscopy, 106, 933-940." /></a>; Klenov & Stemmer, 2006<a id="sourceBB18"></a><a href="#BB18"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Klenov, D. O. & Stemmer, S. (2006). Ultramicroscopy, 106, 889-901.]" title="Klenov, D. O. & Stemmer, S. (2006). Ultramicroscopy, 106, 889-901." /></a>; Rosenauer <span class="it"><i>et al.</i></span>, 2008<a id="sourceBB27"></a><a href="#BB27"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Rosenauer, A., Schowalter, M., Titantah, J. T. & Lamoen, D. (2008). Ultramicroscopy, 108, 1504-1513.]" title="Rosenauer, A., Schowalter, M., Titantah, J. T. & Lamoen, D. (2008). Ultramicroscopy, 108, 1504-1513." /></a>).</p><p>The sophistication of the latter models of diffuse electron scattering is essential for a complete description of a diffraction pattern, particularly when the crystal is aligned to a low-index <a target='Navigator' class="ref_lookup_yellow hideyellow" href='https://dictionary.iucr.org/Zone_axis' onclick="return makeSubWindow("https://dictionary.iucr.org/Zone_axis", 'Navigator')">zone axis</a> with many beams excited simultaneously. This comes at a computational cost – Rossouw's <span class="it"><i>n</i></span>-beam dynamical calculations of TDS scale as <span class="it"><i>n</i></span><span class="sup"><sup>8</sup></span> – which is impractical to include in iterations of models when refining a <a target='Navigator' class="ref_lookup_yellow hideyellow" href='https://dictionary.iucr.org/Crystal_structure' onclick="return makeSubWindow("https://dictionary.iucr.org/Crystal_structure", 'Navigator')">crystal structure</a> obtained from cRED data. It is therefore important to find the best compromise, <span class="it"><i>i.e.</i></span> use a model of sufficient accuracy for the technique of interest whilst minimizing the computational overhead. In the measurement of diffracted intensities, describing diffuse scatter as `absorption' equates to an assumption that these electrons do not return into Bragg scattered spots (or do so in a way that can readily be subtracted). For such measurements, an important distinction should be made between TDS and higher energy-loss <a target='Navigator' class="ref_lookup_orange hideorange" href='https://goldbook.iupac.org/I03025.html' onclick="return makeSubWindow("https://goldbook.iupac.org/I03025.html", 'Navigator')">inelastic scattering,</a> since the latter is strong only at very small scattering angles, while TDS produces a broad diffuse background across the whole pattern. Thus, <a target='Navigator' class="ref_lookup_orange hideorange" href='https://goldbook.iupac.org/I03025.html' onclick="return makeSubWindow("https://goldbook.iupac.org/I03025.html", 'Navigator')">inelastic scattering</a> effectively acts to blur a diffracted spot, or dynamical features in convergent-beam patterns (Tanaka <span class="it"><i>et al.</i></span>, 2002<a id="sourceBB33"></a><a href="#BB33"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Tanaka, M., Terauchi, M., Tsuda, K. & Saitoh, K. (2002). Convergent Beam Electron Diffraction IV. Tokyo: JEOL Ltd.]" title="Tanaka, M., Terauchi, M., Tsuda, K. & Saitoh, K. (2002). Convergent Beam Electron Diffraction IV. Tokyo: JEOL Ltd." /></a>) and will still be included in a measurement of diffracted intensity. Conversely, the broad TDS background is usually subtracted from a measurement (Palatinus <span class="it"><i>et al.</i></span>, 2019<a id="sourceBB20"></a><a href="#BB20"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Palatinus, L., Brázda, P., Jelínek, M., Hrdá, J., Steciuk, G. & Klementová, M. (2019). Acta Cryst. B75, 512-522.]" title="Palatinus, L., Brázda, P., Jelínek, M., Hrdá, J., Steciuk, G. & Klementová, M. (2019). Acta Cryst. B75, 512-522." /></a>) and therefore considering it to be `absorption' is an appropriate model. For structure solution using cRED, where low-index zone axes are encountered infrequently and absorption is already a second-order effect, a simple model will suffice. While Rossouw did not quantify the difference between a two-beam model and an <span class="it"><i>n</i></span>-beam model, the effect is principally to change the distribution of diffuse intensity at zone axes where multiple channelling pathways exist (Rossouw & Bursill, 1985<span class="it"><i>b</i></span><a href="#BB31"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Rossouw, C. J. & Bursill, L. A. (1985b). Acta Cryst. A41, 320-327.]" title="Rossouw, C. J. & Bursill, L. A. (1985b). Acta Cryst. A41, 320-327." /></a>). Furthermore, as shown by Peng (1997<a href="#BB21"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Peng, L.-M. (1997). Acta Cryst. A53, 663-672.]" title="Peng, L.-M. (1997). Acta Cryst. A53, 663-672." /></a>), anisotropic thermal vibrations produce similar changes in both real and imaginary parts of the complex crystal potential, indicating that it should generally be acceptable to simply replace <span class="it"><i>f</i></span><span class="inf"><sub><span class="it"><i>g</i></span></sub></span> with <span class="it"><i>f</i></span><span class="inf"><sub><span class="it"><i>g</i></span></sub></span> + <span class="it"><i>if</i></span><span class="inf"><sub><span class="it"><i>g</i></span></sub></span>′ in an electron diffraction refinement.</p></div> <div id="DIVSEC2" class="sec1"> <h3><a id="SEC2"></a>2. Calculation</h3> <p>Calculations of absorptive TDS scattering factors <span class="it"><i>f</i></span><span class="inf"><sub><span class="it"><i>g</i></span></sub></span>′ based on the two-beam model were given by Bird & King (1990<a id="sourceBB4"></a><a href="#BB4"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Bird, D. M. & King, Q. A. (1990). Acta Cryst. A46, 202-208.]" title="Bird, D. M. & King, Q. A. (1990). Acta Cryst. A46, 202-208." /></a>) and Weickenmeier & Kohl (1991<a id="sourceBB36"></a><a href="#BB36"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Weickenmeier, A. & Kohl, H. (1991). Acta Cryst. A47, 590-597.]" title="Weickenmeier, A. & Kohl, H. (1991). Acta Cryst. A47, 590-597." /></a>), who provided some tabulated values and computer code allowing their calculation. While some tabulated values were given for a limited set of elements and compounds for 100 kV electrons by Peng <span class="it"><i>et al.</i></span> (1996<span class="it"><i>a</i></span><a id="sourceBB23"></a><a href="#BB23"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Peng, L.-M., Ren, G., Dudarev, S. L. & Whelan, M. J. (1996a). Acta Cryst. A52, 456-470.]" title="Peng, L.-M., Ren, G., Dudarev, S. L. & Whelan, M. J. (1996a). Acta Cryst. A52, 456-470." /></a>,<span class="it"><i>b</i></span><a id="sourceBB24"></a><a href="#BB24"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Peng, L.-M., Ren, G., Dudarev, S. L. & Whelan, M. J. (1996b). Acta Cryst. A52, 257-276.]" title="Peng, L.-M., Ren, G., Dudarev, S. L. & Whelan, M. J. (1996b). Acta Cryst. A52, 257-276." /></a>), no general parameterized version exists that would enable a rapid calculation for all elements, which is our purpose here. We use the calculation of Bird & King (1990<a href="#BB4"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Bird, D. M. & King, Q. A. (1990). Acta Cryst. A46, 202-208.]" title="Bird, D. M. & King, Q. A. (1990). Acta Cryst. A46, 202-208." /></a>), who employed an elegant change of variable to give </p><div class="FD"><p class="mathmode display_math"><a id="FD2"></a><img src="teximages/pl5034fd2.svg" alt="[\eqalignno{f^{\prime}(s,B_{\rm iso})& = {{2h} \over {\beta m_{0}c}}\int\, {\rm d}^{2}{\bf s}^{\prime}f_{g }\left(\left|{{{\bf s}} \over {2}}+{\bf s}^{\prime}\right|\right)f_{g}\left (\left|{{{\bf s}} \over {2}}-{\bf s}^{\prime}\right|\right)&\cr &\quad\times\left\{1-\exp\left[-2 B_{\rm iso}\left({s^{\prime}}^{2}-{{s^{2}} \over {4}}\right)\right]\right\},&(2)}]" class="mathimage" style="max-width: 100%; height: auto; width: auto;" height="76" width="363" /></p></div><p>integrated over the <a target='Navigator' class="ref_lookup_yellow hideyellow" href='https://dictionary.iucr.org/Ewald_sphere' onclick="return makeSubWindow("https://dictionary.iucr.org/Ewald_sphere", 'Navigator')">Ewald sphere</a> for <span class="it"><i>s</i></span>′, where <span class="it"><i>h</i></span> is Planck's constant, <span class="it"><i>m</i></span><span class="inf"><sub>0</sub></span> the rest mass of the electron, <span class="it"><i>c</i></span> the speed of light and <span class="symbol">β</span> the velocity ratio <span class="it"><i>v</i></span>/<span class="it"><i>c</i></span>. Usefully, this framing of the problem gives a fourfold symmetry of the integrand, allowing numerical integration to be performed over 0 ≤ <span class="it"><i>s</i></span>′ < ∞ in two dimensions which avoids most issues with poor convergence.</p><p>We evaluate the integral of equation (2<a href="#FD2"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[link]" /></a>), moving the factor <span class="symbol">β</span> to give a result that is independent of accelerating voltage, using the Born electron scattering factors of Lobato & Van Dyck (2014<a id="sourceBB19"></a><a href="#BB19"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Lobato, I. & Van Dyck, D. (2014). Acta Cryst. A70, 636-649.]" title="Lobato, I. & Van Dyck, D. (2014). Acta Cryst. A70, 636-649." /></a>) for neutral atoms. For each element we give a parameterized <span class="symbol">β</span><span class="it"><i>f</i></span>′(<span class="it"><i>s</i></span>, <span class="it"><i>B</i></span><span class="inf"><sub>iso</sub></span>) that is a sum of four Gaussian functions, each of which is determined by two parameters <span class="it"><i>a</i></span><span class="inf"><sub><span class="it"><i>i</i></span></sub></span>, <span class="it"><i>b</i></span><span class="inf"><sub><span class="it"><i>i</i></span></sub></span>. This is in addition to a single constant term <span class="it"><i>c</i></span>, giving a total of nine parameters tabulated for each value of <span class="it"><i>B</i></span><span class="inf"><sub>iso</sub></span>, <span class="it"><i>i.e.</i></span> </p><div class="FD"><p class="mathmode display_math"><a id="FD3"></a><img src="teximages/pl5034fd3.svg" alt="[\beta f^{\prime}(B_{\rm iso},s) = \textstyle\sum\limits_{i = 1}^{4}a_{i}\exp{(-b_{i}s^{2})}+c. \eqno(3)]" class="mathimage" style="max-width: 100%; height: auto; width: auto;" height="31" width="311" /></p></div><p>A non-linear least-squares fit of equation (3<a href="#FD3"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[link]" /></a>) to equation (2<a href="#FD2"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[link]" /></a>), evaluated over 100 values of <span class="it"><i>s</i></span> (0 ≤ <span class="it"><i>s</i></span> ≤ 6 Å<span class="sup"><sup>−1</sup></span>), for 103 elements was performed using a Trust Region reflective algorithm (Branch <span class="it"><i>et al.</i></span>, 1999<a id="sourceBB6"></a><a href="#BB6"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Branch, M. A., Coleman, T. F. & Li, Y. (1999). SIAM J. Sci. Comput. 21, 1-23.]" title="Branch, M. A., Coleman, T. F. & Li, Y. (1999). SIAM J. Sci. Comput. 21, 1-23." /></a>), providing a parameterized approximation of <span class="it"><i>f</i></span>′ for a given value of <span class="it"><i>B</i></span><span class="inf"><sub>iso</sub></span> (0.1 ≤ <span class="it"><i>B</i></span><span class="inf"><sub>iso</sub></span> ≤ 4 Å<span class="sup"><sup>2</sup></span>). This approach is more straightforward in use than the <span class="it"><i>f</i></span>′/<span class="it"><i>f</i></span> over a grid of <span class="it"><i>B</i></span><span class="inf"><sub>iso</sub></span><span class="it"><i>s</i></span><span class="sup"><sup>2</sup></span> and <span class="it"><i>B</i></span><span class="inf"><sub>iso</sub></span> given by Bird & King (1990<a href="#BB4"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Bird, D. M. & King, Q. A. (1990). Acta Cryst. A46, 202-208.]" title="Bird, D. M. & King, Q. A. (1990). Acta Cryst. A46, 202-208." /></a>). In equation (1<a href="#FD1"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[link]" /></a>), multiplying by the temperature factor, <img src="teximages/pl5034fi3.svg" alt="[\exp(-B_{\rm iso}s^{2})]" class="img_align_bottom mathimage" style="position: relative; display: inline; padding-top: 0px; padding-bottom: 0px; vertical-align: -2.58458pt;" height="16" width="75" />, ensures that the function smoothly asymptotes to zero. Without it, the absorptive form factor instead behaves as <img src="teximages/pl5034fi5.svg" alt="[-\exp(s)]" class="img_align_bottom mathimage" style="position: relative; display: inline; padding-top: 0px; padding-bottom: 0px; vertical-align: -2.58458pt;" height="14" width="49" /> for large <span class="it"><i>s</i></span> paired with any <span class="it"><i>B</i></span><span class="inf"><sub>iso</sub></span> > 0.05 Å<span class="sup"><sup>2</sup></span> (Peng <span class="it"><i>et al.</i></span>, 1996<span class="it"><i>b</i></span><a href="#BB24"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Peng, L.-M., Ren, G., Dudarev, S. L. & Whelan, M. J. (1996b). Acta Cryst. A52, 257-276.]" title="Peng, L.-M., Ren, G., Dudarev, S. L. & Whelan, M. J. (1996b). Acta Cryst. A52, 257-276." /></a>). This results in large negative <span class="symbol">β</span><span class="it"><i>f</i></span>′ at large <span class="it"><i>s</i></span>, which implies amplification, rather than absorption, of the electron beam and we consider this to be unphysical. We therefore set <span class="symbol">β</span><span class="it"><i>f</i></span>′ to zero where equation (2<a href="#FD2"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[link]" /></a>) returns a negative value. Note that equation (1<a href="#FD1"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[link]" /></a>) does not present this behaviour to such an extreme; only when the temperature factor is removed do the unphysical results strongly present themselves. Intermediate values of <span class="symbol">β</span><span class="it"><i>f</i></span>′ may be obtained by linear interpolation of the four nearest values, while negative values may be disregarded. In any implementation the accelerating voltage must be taken into account both by including <span class="symbol">β</span> and multiplying by the relativistic correction <span class="symbol">γ</span> = 1/(1 − <span class="it"><i>v</i></span><span class="sup"><sup>2</sup></span>/<span class="it"><i>c</i></span><span class="sup"><sup>2</sup></span>)<span class="sup"><sup>1/2</sup></span>, to give the complex scattering factor <span class="symbol">γ</span><span class="it"><i>f</i></span> + <span class="it"><i>i</i></span><span class="symbol">γ</span><span class="it"><i>f</i></span>′.</p></div> <div id="DIVSEC3" class="sec1"> <h3><a id="SEC3"></a>3. Results</h3> <p>Fig. 1<a href="#FIG1"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[link]" /></a> shows the absorptive electron scattering factor over a range of <span class="it"><i>B</i></span><span class="inf"><sub>iso</sub></span> and <span class="it"><i>s</i></span> for selected light, intermediate and heavy atoms, <span class="it"><i>i.e.</i></span> C, Ga and Pb, respectively (note different scales; like normal scattering factors, absorptive scattering factors increase with atomic number). The maximum value of <span class="it"><i>f</i></span>′ at <span class="it"><i>s</i></span> = 0 increases rapidly with <span class="it"><i>B</i></span><span class="inf"><sub>iso</sub></span>, but rapidly drops to zero around <span class="it"><i>s</i></span> ∼ 1 Å<span class="sup"><sup>−1</sup></span> at high temperatures (large <span class="it"><i>B</i></span><span class="inf"><sub>iso</sub></span>). Conversely, below <span class="it"><i>B</i></span><span class="inf"><sub>iso</sub></span> ∼ 0.5 Å<span class="sup"><sup>2</sup></span> it extends to much larger <span class="it"><i>s</i></span>. Thirteen values of <span class="it"><i>B</i></span><span class="inf"><sub>iso</sub></span> were chosen to be parameterized, as a compromise between the best accuracy and a compact calculation that does not require a large number of parameters (here, 9 × 13 = 117) for each element. In general, the parameterized value of <span class="it"><i>f</i></span>′ is well within 0.1% of that resulting from equation (2<a href="#FD2"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[link]" /></a>) and the curves are indistinguishable in Fig. 1<a href="#FIG1"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[link]" /></a>. However, the curvature of the surfaces in Fig. 1<a href="#FIG1"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[link]" /></a> causes the parameterized <span class="it"><i>f</i></span>′, which is obtained using a simple linear interpolation, to be slightly less reliable between the chosen values of <span class="it"><i>B</i></span><span class="inf"><sub>iso</sub></span>. The extent of these errors is shown in Fig. 2<a href="#FIG2"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[link]" /></a>, which gives a map of the difference between parameterized and directly calculated absorptive scattering factors. To avoid artefacts these maps include the temperature factor <img src="teximages/pl5034fi6.svg" alt="[\exp({-Bs^{2})}]" class="img_align_bottom mathimage" style="position: relative; display: inline; padding-top: 0px; padding-bottom: 0px; vertical-align: -2.58458pt;" height="16" width="63" />; they are otherwise dominated by errors in very small values of <span class="it"><i>f</i></span>′ at the boundary where the curve passes through zero, which occur due to evaluation over a finite grid. These errors are negligible in practice since they are suppressed by the factor <img src="teximages/pl5034fi6.svg" alt="[\exp({-Bs^{2})}]" class="img_align_bottom mathimage" style="position: relative; display: inline; padding-top: 0px; padding-bottom: 0px; vertical-align: -2.58458pt;" height="16" width="63" />. In order to allow comparison between different elements, the error is normalized by the maximum value of <img src="teximages/pl5034fi8.svg" alt="[\beta f^{\prime}\exp({-Bs^{2})}]" class="img_align_bottom mathimage" style="position: relative; display: inline; padding-top: 0px; padding-bottom: 0px; vertical-align: -2.58458pt;" height="16" width="85" /> in the grid. While this means that the absolute values are somewhat arbitrary, it shows that they remain very small fractions of the calculated <span class="symbol">β</span><span class="it"><i>f</i></span>′.</p><div class="fig"> <table summary="[Figure 1]" cellpadding="5" class="fig" width="100%"> <tbody> <tr> <td class="td_align_center width_20"> <a href="pl5034fig1.html"><img width="100" height="27" src="pl5034fig1thm.gif" class="figlnkthm img_align_middle" alt="[Figure 1]" /> <br /></a> </td> <td> <span class="font_size_3"><b><a href="pl5034fig1.html" id="FIG1">Figure 1</a></b></span> <br /><span class="font_size_2 caption">Plot of <span class="symbol">β</span><span class="it"><i>f</i></span>′ for carbon (<span class="it"><i>Z</i></span> = 6), gallium (<span class="it"><i>Z</i></span> = 31) and lead (<span class="it"><i>Z</i></span> = 82) over the range 0 < <span class="it"><i>s</i></span> < 4 and 0 < <span class="it"><i>B</i></span><span class="inf"><sub>iso</sub></span> < 4 calculated from equation (2<a href="#FD2"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[link]" /></a>) (blue) and parameterized at 13 values of <span class="it"><i>B</i></span><span class="inf"><sub>iso</sub></span> (red).</span></td> </tr> </tbody> </table> </div> <div class="fig"> <table summary="[Figure 2]" cellpadding="5" class="fig" width="100%"> <tbody> <tr> <td class="td_align_center width_20"> <a href="pl5034fig2.html"><img width="100" height="31" src="pl5034fig2thm.gif" class="figlnkthm img_align_middle" alt="[Figure 2]" /> <br /></a> </td> <td> <span class="font_size_3"><b><a href="pl5034fig2.html" id="FIG2">Figure 2</a></b></span> <br /><span class="font_size_2 caption">Errors resulting from linear interpolation between the parameterized values of <img src="teximages/pl5034fi8.svg" alt="[\beta f^{\prime}\exp({-Bs^{2})}]" class="img_align_bottom mathimage" style="position: relative; display: inline; padding-top: 0px; padding-bottom: 0px; vertical-align: -2.58458pt;" height="16" width="85" />, compared with those calculated from the integral of equation (2)<a href="#FD2"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[link]" /></a> for carbon (<span class="it"><i>Z</i></span> = 6), gallium (<span class="it"><i>Z</i></span> = 31) and lead (<span class="it"><i>Z</i></span> = 82). The range of scattering vector [<span class="it"><i>s</i></span> = sin(<span class="symbol">θ</span>)/<span class="symbol">λ</span> = <span class="it"><i>g</i></span>/2] is 0 < <span class="it"><i>s</i></span> < 3 and isotropic Debye–Waller factor is 0 < <span class="it"><i>B</i></span><span class="inf"><sub>iso</sub></span> < 4. Values are given as fractions of the maximum value of <img src="teximages/pl5034fi8.svg" alt="[\beta f^{\prime}\exp({-Bs^{2})}]" class="img_align_bottom mathimage" style="position: relative; display: inline; padding-top: 0px; padding-bottom: 0px; vertical-align: -2.58458pt;" height="16" width="85" /> over the mapped range. In the blank region, <span class="it"><i>f</i></span>′ = 0.</span></td> </tr> </tbody> </table> </div> <p>Measurements of the speed improvement derived from parameterized <span class="it"><i>f</i></span>′ in comparison with the direct calculation are given in Fig. 3<a href="#FIG3"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[link]" /></a>(<span class="it"><i>a</i></span>), evaluated for 1000 function calls to a Python script on a Windows 11 machine. Parameterized <span class="it"><i>f</i></span>′ were typically returned in 30 µs, while the direct calculation required between 300 and 600 ms. While significant variability is present, a small improvement with increasing <a target='Navigator' class="ref_lookup_orange hideorange" href='https://goldbook.iupac.org/A00499.html' onclick="return makeSubWindow("https://goldbook.iupac.org/A00499.html", 'Navigator')">atomic number</a> is apparent. Fig. 3<a href="#FIG3"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[link]" /></a>(<span class="it"><i>b</i></span>) gives an evaluation of errors using the same method as in Fig. 2<a href="#FIG2"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[link]" /></a>. The lack of dependence on <a target='Navigator' class="ref_lookup_orange hideorange" href='https://goldbook.iupac.org/A00499.html' onclick="return makeSubWindow("https://goldbook.iupac.org/A00499.html", 'Navigator')">atomic number</a> indicates that the method gives good results, which should be generally applicable to all elements and compounds. A Python subroutine that returns the complex scattering factor <span class="symbol">γ</span><span class="it"><i>f</i></span> + <span class="it"><i>i</i></span><span class="symbol">γ</span><span class="it"><i>f</i></span>′ for input <span class="it"><i>B</i></span><span class="inf"><sub>iso</sub></span>, <span class="it"><i>s</i></span> and accelerating voltage <span class="it"><i>V</i></span> is provided in the supporting information and is also available online (Thomas, 2023<a id="sourceBB34"></a><a href="#BB34"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Thomas, M. (2023). A Python Subroutine Returning βf′. https://github.com/WarwickMicroscopy/fprime.]" title="Thomas, M. (2023). A Python Subroutine Returning βf′. https://github.com/WarwickMicroscopy/fprime." /></a>).</p><div class="fig"> <table summary="[Figure 3]" cellpadding="5" class="fig" width="100%"> <tbody> <tr> <td class="td_align_center width_20"> <a href="pl5034fig3.html"><img width="100" height="36" src="pl5034fig3thm.gif" class="figlnkthm img_align_middle" alt="[Figure 3]" /> <br /></a> </td> <td> <span class="font_size_3"><b><a href="pl5034fig3.html" id="FIG3">Figure 3</a></b></span> <br /><span class="font_size_2 caption">(<span class="it"><i>a</i></span>) Increase in speed of calculation as measured on a Windows 11 64 bit desktop. Variations are mainly due to competing windows processes but generally exceed 10 000× with a slight improvement at higher atomic numbers. (<span class="it"><i>b</i></span>) The maximum error in <img src="teximages/pl5034fi8.svg" alt="[\beta f^{\prime}\exp({-Bs^{2})}]" class="img_align_bottom mathimage" style="position: relative; display: inline; padding-top: 0px; padding-bottom: 0px; vertical-align: -2.58458pt;" height="16" width="85" /> resulting from linear interpolation between the parameterized values at fixed <span class="it"><i>B</i></span><span class="inf"><sub>iso</sub></span>, evaluated in the same way as in Fig. 2<a href="#FIG2"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[link]" /></a>.</span></td> </tr> </tbody> </table> </div> </div> <div id="DIVSEC4" class="sec1"> <h3><a id="SEC4"></a>4. Discussion and conclusions</h3> <p>In summary, we have parameterized absorptive scattering factors using the method described by Bird & King (1990<a href="#BB4"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Bird, D. M. & King, Q. A. (1990). Acta Cryst. A46, 202-208.]" title="Bird, D. M. & King, Q. A. (1990). Acta Cryst. A46, 202-208." /></a>). The impact of `absorption' into a diffuse background on the measurement of diffracted intensities, used for structure solution with electron diffraction, is currently not quantified. It is, however, clear that <a target='Navigator' class="ref_lookup_yellow hideyellow" href='https://dictionary.iucr.org/Crystal_structure' onclick="return makeSubWindow("https://dictionary.iucr.org/Crystal_structure", 'Navigator')">crystal structure</a> refinements based on dynamical simulations show significant improvements over kinematical ones (Klar <span class="it"><i>et al.</i></span>, 2023<a href="#BB17"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Klar, P. B., Krysiak, Y., Xu, H., Steciuk, G., Cho, J., Zou, X. & Palatinus, L. (2023). Nat. Chem. 15, 848-855.]" title="Klar, P. B., Krysiak, Y., Xu, H., Steciuk, G., Cho, J., Zou, X. & Palatinus, L. (2023). Nat. Chem. 15, 848-855." /></a>; Cleverley & Beanland, 2023<a id="sourceBB7"></a><a href="#BB7"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Cleverley, A. & Beanland, R. (2023). IUCrJ, 10, 118-130.]" title="Cleverley, A. & Beanland, R. (2023). IUCrJ, 10, 118-130." /></a>) and it is probable that, with sufficiently accurate dynamical simulations and high-quality data, these absorption effects will become evident to a similar extent to that seen in the more established methods of CBED (convergent-beam electron diffraction), TEM and STEM. The method applied here neglects diffuse scattering due to higher-energy (plasmon and core-loss) <a target='Navigator' class="ref_lookup_orange hideorange" href='https://goldbook.iupac.org/I03025.html' onclick="return makeSubWindow("https://goldbook.iupac.org/I03025.html", 'Navigator')">inelastic scattering</a> on the grounds that it is generally limited to relatively small angles in comparison with TDS. This assumption will inevitably hold less well for thicker crystals, for which the TDS also becomes inelastically scattered through these mechanisms. As shown by Yang <span class="it"><i>et al.</i></span> (2022<a id="sourceBB37"></a><a href="#BB37"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Yang, T., Xu, H. & Zou, X. (2022). J. Appl. Cryst. 55, 1583-1591.]" title="Yang, T., Xu, H. & Zou, X. (2022). J. Appl. Cryst. 55, 1583-1591." /></a>), the removal of <a target='Navigator' class="ref_lookup_orange hideorange" href='https://goldbook.iupac.org/I03025.html' onclick="return makeSubWindow("https://goldbook.iupac.org/I03025.html", 'Navigator')">inelastic scattering</a> by energy filtering can give significant improvements in quality-of-fit indices such as <span class="it"><i>R</i></span><span class="inf"><sub>1</sub></span>, although the details of diffuse background subtraction in their data processing (using code developed for X-ray diffraction) and the thickness of the crystals they used were not given. Further work is still required to determine whether the effort needed to quantify, or remove, this additional effect by energy filtering is worthwhile for structure solution and <a target='Navigator' class="ref_lookup_yellow hideyellow" href='https://dictionary.iucr.org/Refinement' onclick="return makeSubWindow("https://dictionary.iucr.org/Refinement", 'Navigator')">refinement.</a> In addition, it is important to note that multiple scattering of electrons in the diffuse background also takes place, adding structure such as Kikuchi lines and complex variations of intensity at low-index zone axes, which may affect measurements of the intensity of a Bragg peak. It is thus certain that, as precision and accuracy improve in electron diffraction methods, these effects will become more visible.</p><p>Anisotropic thermal vibrations are routinely determined in <a target='Navigator' class="ref_lookup_yellow hideyellow" href='https://dictionary.iucr.org/Crystal_structure' onclick="return makeSubWindow("https://dictionary.iucr.org/Crystal_structure", 'Navigator')">crystal structure</a> refinements. As a first approach to an anisotropic model in electron diffraction, it may be sufficient to simply use the complex <a target='Navigator' class="ref_lookup_yellow hideyellow" href='https://dictionary.iucr.org/Structure_factor' onclick="return makeSubWindow("https://dictionary.iucr.org/Structure_factor", 'Navigator')">structure factor</a> <span class="it"><i>f</i></span> + <span class="it"><i>if</i></span>′ in place of <span class="it"><i>f</i></span>. However, a more appropriate method to obtain an anisotropic form of <span class="it"><i>f</i></span>′ would be to reframe the integral of equation (2<a href="#FD2"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[link]" /></a>) using the anisotropic tensor form of <span class="it"><i>f</i></span>, which will give a slightly different result. A further important aspect is the effect of charge transfer, ionicity and multipolar atomic models. The change in <a target='Navigator' class="ref_lookup_yellow hideyellow" href='https://dictionary.iucr.org/Structure_factor' onclick="return makeSubWindow("https://dictionary.iucr.org/Structure_factor", 'Navigator')">structure factor</a> can be significant at low <span class="it"><i>s</i></span> and there is already clear evidence that it is readily detectable in cRED electron diffraction data (Gruza <span class="it"><i>et al.</i></span>, 2020<a id="sourceBB11"></a><a href="#BB11"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Gruza, B., Chodkiewicz, M. L., Krzeszczakowska, J. & Dominiak, P. M. (2020). Acta Cryst. A76, 92-109.]" title="Gruza, B., Chodkiewicz, M. L., Krzeszczakowska, J. & Dominiak, P. M. (2020). Acta Cryst. A76, 92-109." /></a>). Inclusion of these effects in the simple absorption model used here can simply be achieved by use of the appropriate Born scattering factor <span class="it"><i>f</i></span> in equation (2<a href="#FD2"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[link]" /></a>). However, like the other more sophisticated models mentioned above, it remains to be determined whether the additional computational cost is worthwhile in structure solution and <a target='Navigator' class="ref_lookup_yellow hideyellow" href='https://dictionary.iucr.org/Refinement' onclick="return makeSubWindow("https://dictionary.iucr.org/Refinement", 'Navigator')">refinement</a> using electron diffraction.</p><p>While evaluation of the integral of equation (2<a href="#FD2"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[link]" /></a>) is not difficult using modern numerical methods, it remains relatively slow and could limit simulations of materials with large unit cells and many atoms. It is hoped that the parameterized versions provided here will allow the inclusion of absorption in the rapid calculations that will be necessary in routine refinements of crystal structure.</p></div> <div id="DIVSEC5" class="sec1"> <h3><a id="SEC5"></a>5. Software and data availability</h3> <p>All code used in this work is available online (Thomas, 2023<a href="#BB34"><img class="linkarrow img_align_bottom img_border_0" src="../../../../../../logos/arrows/a_arr.gif" width="17" height="16" alt="[Thomas, M. (2023). A Python Subroutine Returning βf′. https://github.com/WarwickMicroscopy/fprime.]" title="Thomas, M. (2023). A Python Subroutine Returning βf′. https://github.com/WarwickMicroscopy/fprime." /></a>) and on the Warwick Research Archive Portal <a href="https://wrap.warwick.ac.uk/181354/">https://wrap.warwick.ac.uk/181354/</a>.</p></div> <div class="supmat"> <a id="suppinfoanchor"></a> <h3 id="suppinfo">Supporting information</h3>  <div class="art_codelinks"> </div> <div class="file_links_other"> <p>Python code for returning the complex scattering factor. DOI: <a href="https://doi.org/10.1107/S2053273323010963/pl5034sup1.zip">https://doi.org/10.1107/S2053273323010963/pl5034sup1.zip</a></p> </div> <br />   </div> </div> <div id="bm"> <div id="funding"><h3>Funding information</h3><p>MT acknowledges the support of a Warwick University URSS bursary for this work. 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