{ "id": "epeyh5emr0", "slug": "total-scattering-diffraction", "title": "Total scattering (diffraction)", "updated": "2026-10-15", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.physics", "label": "Physics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.physics" }, { "id": "physical.physics.matter-and-radiation", "label": "Matter and radiation physics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.physics.matter-and-radiation" }, { "id": "physical.physics.matter-and-radiation.condensed-matter", "label": "Condensed matter physics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.physics.matter-and-radiation.condensed-matter" }, { "id": "physical.physics.matter-and-radiation.condensed-matter.cm-structures", "label": "Crystal and structural condensed matter", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.physics.matter-and-radiation.condensed-matter.cm-structures" } ], "geo": [ { "id": "geo.nongeo.t1946.physical.physics.matter-and-radiation.condensed-matter.cm-structures", "label": "Non-geographic · 1946 to 2000: Crystal and structural condensed matter", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.physics.matter-and-radiation.condensed-matter.cm-structures", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1946", "label": "Non-geographic · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946" }, { "id": "geo.nongeo.t1946.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical" }, { "id": "geo.nongeo.t1946.physical.physics", "label": "Physics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.physics" }, { "id": "geo.nongeo.t1946.physical.physics.matter-and-radiation", "label": "Matter and radiation physics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.physics.matter-and-radiation" }, { "id": "geo.nongeo.t1946.physical.physics.matter-and-radiation.condensed-matter", "label": "Condensed matter physics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.physics.matter-and-radiation.condensed-matter" }, { "id": "geo.nongeo.t1946.physical.physics.matter-and-radiation.condensed-matter.cm-structures", "label": "Crystal and structural condensed matter", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.physics.matter-and-radiation.condensed-matter.cm-structures" } ] } ], "excerpt": "Total scattering is a diffraction method analyzing the complete scattering pattern, Bragg peaks and diffuse scattering alike, to obtain the atomic pair distribution function, which reveals local structure.", "snippet": "Total scattering is a diffraction method analyzing the complete scattering pattern, Bragg peaks and diffuse scattering alike, to obtain the atomic pair distribution function, which reveals local structure.", "node": "physical.physics.matter-and-radiation.condensed-matter.cm-structures", "markdown": "# Total scattering (diffraction)\n\nTotal scattering is a diffraction method that analyzes the complete scattering pattern, Bragg peaks and diffuse scattering together, to extract the atomic pair distribution function (PDF), a real-space histogram of interatomic distances that reveals local structure in crystalline, nanocrystalline, and amorphous materials.[1](https://journals.iucr.org/j/issues/2021/01/00/gj5253/) Conventional crystallography uses only the sharp Bragg peaks and assumes periodic order; total scattering instead uses all the information in the pattern, including elastic diffuse scattering from static local distortions and inelastic diffuse scattering from dynamics.[2](https://par.nsf.gov/servlets/purl/10587291) Because the PDF does not require periodic order in the sample, it applies equally to glasses, liquids, nanoparticles, and disordered crystals.[3](https://www.diffpy.org/doc/pdfgui/Farrow-jpcm-2007.pdf)\n\n| Key fact | Detail |\n|---|---|\n| Output | The reduced pair distribution function \\( G(r) \\), a bond-length histogram averaged over the sample[2](https://par.nsf.gov/servlets/purl/10587291) |\n| Real-space resolution | \\( \\Delta r \\approx 2\\pi/Q_{\\mathrm{max}} \\); \\( Q_{\\mathrm{max}} \\) of at least 15–20 Å⁻¹ is needed for atomic-scale information[4](https://neutrons.ornl.gov/sites/default/files/KPage_Neutron_PDF_2025.pdf) • [5](https://www.chimia.ch/chimia/article/download/2021_368/172/10810) |\n| Synchrotron conditions | Energies typically >60 keV; \\( Q_{\\mathrm{max}} \\) around 25 Å⁻¹; seconds to minutes per pattern[6](https://pmc.ncbi.nlm.nih.gov/articles/PMC10000805/) • [7](https://journals.iucr.org/m/issues/2025/05/00/it5039/) |\n| Neutron conditions | NOMAD: 0.2–40 Å⁻¹ from 30–100 mg in ~1 h; POWGEN: 3–10 g in ~3 h[4](https://neutrons.ornl.gov/sites/default/files/KPage_Neutron_PDF_2025.pdf) |\n| Main modeling routes | Small-box least-squares refinement (PDFgui, TOPAS) and large-box reverse Monte Carlo (RMCProfile)[4](https://neutrons.ornl.gov/sites/default/files/KPage_Neutron_PDF_2025.pdf) |\n| Key artifact | Termination ripples from the finite \\( Q_{\\mathrm{max}} \\), growing as \\( Q_{\\mathrm{max}} \\) decreases[8](https://mdpi-res.com/d_attachment/materials/materials-12-01347/article_deploy/materials-12-01347.pdf?version=1556172641) |\n\n## How it works\n\nThe coherent scattering intensity from a collection of atoms was derived by Pieter Debye in 1915:\n\n\\[ I_{\\mathrm{coh}}(Q) = \\sum_{i} \\sum_{j} f_{i}(Q) \\cdot f_{j}(Q) \\frac{\\sin(Q \\cdot r_{ij})}{Q \\cdot r_{ij}} \\]\n\nwhere \\( f_{i} \\) and \\( f_{j} \\) are form factors for atoms \\( i \\) and \\( j \\) separated by \\( r_{ij} \\).[2](https://par.nsf.gov/servlets/purl/10587291) The measured intensity is normalized to the structure function \\( S(Q) \\), and the reduced PDF is obtained as the (truncated) [Fourier transform](https://www.edgepedia.org/fourier-transform) of \\( F(Q) = Q \\cdot [S(Q) - 1] \\):[9](https://royalsocietypublishing.org/rsta/article-pdf/doi/10.1098/rsta.2018.0413/246013/rsta.2018.0413.pdf)\n\n\\[ G(r) = 4\\pi r [\\rho(r) - \\rho_{0}] = \\frac{2}{\\pi} \\int Q \\cdot [S(Q) - 1] \\sin(Qr)\\, dQ \\]\n\nwhere \\( \\rho(r) \\) is the atomic pair density and \\( \\rho_{0} \\) the average number density.[10](https://www.ias.ac.in/article/fulltext/pram/071/04/0713-0719) The PDF is essentially a histogram of interatomic distances, spatially and temporally averaged over the whole sample.[2](https://par.nsf.gov/servlets/purl/10587291) Real-space resolution is dominated by the maximum measured momentum transfer, \\( \\Delta r \\approx 2\\pi/Q_{\\mathrm{max}} \\).[4](https://neutrons.ornl.gov/sites/default/files/KPage_Neutron_PDF_2025.pdf)\n\n## How it is done\n\nData are collected at synchrotron beamlines or spallation-neutron diffractometers. Reduction is more demanding than for Bragg diffraction: background subtraction, normalization, proper scaling, and corrections for absorption, multiple scattering, and inelastic scattering are all required.[11](https://powder.ornl.gov/total_scattering/intro.html) The measured intensity is typically modeled as \\( I_{m}(Q) = \\alpha(Q) \\cdot I_{c}(Q) + \\beta(Q) \\), where additive corrections cover Compton and background scattering and multiplicative corrections cover self-absorption and polarization; for neutrons \\( S(Q) = [I_{c}(Q) - \\langle b^{2} \\rangle + \\langle b \\rangle^{2}]/\\langle b \\rangle^{2} \\).[12](https://www.osti.gov/servlets/purl/1439450) X-ray form-factor corrections are handled either explicitly during normalization, ad hoc, or by forward calculation in the refined model.[1](https://journals.iucr.org/j/issues/2021/01/00/gj5253/) Shorter neutron wavelengths are preferred because they integrate over a greater range of energy transfers at closer to constant Q.[13](https://www.tandfonline.com/doi/abs/10.1080/0889311X.2020.1797708) Common reduction software includes PDFgetX3, PDFgetN3, GudrunX/GudrunN, GSAS-II, TOPAS, and the mantidtotalscattering framework at Oak Ridge.[2](https://par.nsf.gov/servlets/purl/10587291) • [11](https://powder.ornl.gov/total_scattering/intro.html) [Instrumental](https://www.edgepedia.org/instrumental) resolution and damping parameters (for example \\( Q_{\\mathrm{damp}} = 0.0035 \\) and \\( Q_{\\mathrm{broad}} = 0.0017 \\) at one synchrotron setup) are determined by refining a NIST Si 640d standard.[6](https://pmc.ncbi.nlm.nih.gov/articles/PMC10000805/)\n\nDedicated X-ray PDF beamlines typically operate above 60 keV, for example 76.6 keV at I15-1 ([Diamond Light Source](https://www.edgepedia.org/diamond-light-source)) and 86.7 keV at 11-ID-B ([Advanced Photon Source](https://www.edgepedia.org/advanced-photon-source)).[6](https://pmc.ncbi.nlm.nih.gov/articles/PMC10000805/) Neutron total scattering is done at spallation-source diffractometers such as NOMAD, POWGEN, and SNAP at the [Spallation Neutron Source](https://www.edgepedia.org/spallation-neutron-source), and at ISIS, J-PARC, and reactor instruments such as D4 at the ILL.[11](https://powder.ornl.gov/total_scattering/intro.html) • [6](https://pmc.ncbi.nlm.nih.gov/articles/PMC10000805/) Typical samples are 30–100 mg at NOMAD (~1 h) and 3–10 g at POWGEN (~3 h); mail-in X-ray PDF at 11-ID-B takes under a second to seconds on about 10 mg.[4](https://neutrons.ornl.gov/sites/default/files/KPage_Neutron_PDF_2025.pdf) X-ray free-electron laser total scattering has reached normalized data over 0.35 Å⁻¹ < Q < 16.6 Å⁻¹ from a single ~30 fs pulse at the HED instrument of the European XFEL.[7](https://journals.iucr.org/m/issues/2025/05/00/it5039/)\n\n## Origin\n\nThe method's roots lie in the Debye scattering equation, published by P. Debye in 1915 in [Annalen der Physik](https://www.edgepedia.org/annalen-der-physik).[14](https://doi.org/10.1002/andp.19153510606) In 1927, F. Zernike and J. A. Prins derived the Fourier relationship between the real-space pair density and that equation in Zeitschrift für Physik, at which point, as one historical account puts it, the PDF was born.[15](https://doi.org/10.1007/bf01391926) • [1](https://journals.iucr.org/j/issues/2021/01/00/gj5253/) B. E. Warren developed the approach for glasses in 1934 in [Physical Review](https://www.edgepedia.org/physical-review).[16](https://doi.org/10.1103/physrev.45.657) For roughly fifty years the method was largely the domain of liquid and amorphous structure studies, often a last resort for materials with limited order, before being rediscovered in the late 1980s as a way to uncover disorder within crystals.[13](https://www.tandfonline.com/doi/abs/10.1080/0889311X.2020.1797708) • [5](https://www.chimia.ch/chimia/article/download/2021_368/172/10810) The first synchrotron X-ray PDF experiments were carried out at [Cornell High Energy Synchrotron Source](https://www.edgepedia.org/cornell-high-energy-synchrotron-source) (CHESS) by Takeshi Egami in the mid-1980s, and the seminal work marking the PDF's potential for local structure in crystals was by S. J. L. Billinge and T. Egami in the early 1990s on Nd₂₋ₓCeₓCuO₄₋ᵧ, published in Physical Review B in 1993.[9](https://royalsocietypublishing.org/rsta/article-pdf/doi/10.1098/rsta.2018.0413/246013/rsta.2018.0413.pdf) • [17](https://doi.org/10.1103/physrevb.47.14386) Publications mentioning X-ray PDF have grown enormously since about 2000, supported by dedicated beamlines.[9](https://royalsocietypublishing.org/rsta/article-pdf/doi/10.1098/rsta.2018.0413/246013/rsta.2018.0413.pdf)\n\n## Variants\n\nThe main variants are X-ray PDF, neutron PDF, and electron-diffraction PDF (ePDF), which dates from the 1960s and enables nanometer-scale spatially resolved PDF mapping in 4D-STEM mode.[2](https://par.nsf.gov/servlets/purl/10587291) Modeling splits into small-box least-squares refinement, for structures up to several hundred atoms, and large-box reverse [Monte Carlo](https://www.edgepedia.org/monte-carlo), for 20,000 or more atoms.[4](https://neutrons.ornl.gov/sites/default/files/KPage_Neutron_PDF_2025.pdf) Full-profile real-space refinement was introduced with PDFFIT by Th. Proffen and S. J. L. Billinge in 1999, and its successor PDFfit2/PDFgui fits lattice constants, atomic positions, anisotropic displacement parameters, correlated motion, and experimental factors with symmetry constraints.[18](https://doi.org/10.1107/s0021889899003532) • [19](https://doi.org/10.1088/0953-8984/19/33/335219) The reverse [Monte Carlo method](https://www.edgepedia.org/monte-carlo-method) was introduced in Molecular Simulation, and RMCProfile extended it to polycrystalline materials in Journal of Physics Condensed Matter, with later big-box extensions to magnetic total scattering.[20](https://doi.org/10.1080/08927028808080958) • [21](https://doi.org/10.1088/0953-8984/19/33/335218) • [13](https://www.tandfonline.com/doi/abs/10.1080/0889311X.2020.1797708) Refining as a function of r-range (box-car fitting) can estimate domain size.[10](https://www.ias.ac.in/article/fulltext/pram/071/04/0713-0719)\n\n## Applications\n\nFor gold nanoparticles of average size about 3.6 nm measured on NPDF, PDF peaks diminish at distances corresponding to the particle diameter, providing a complete structural fingerprint without requiring periodicity.[10](https://www.ias.ac.in/article/fulltext/pram/071/04/0713-0719) In disordered crystalline materials, PDFgui has been demonstrated on a temperature series of neutron PDF data from LaMnO₃, following an orthorhombic to pseudo-cubic phase transition near 750 K without a change in space group.[3](https://www.diffpy.org/doc/pdfgui/Farrow-jpcm-2007.pdf) Glasses, liquids, and amorphous carbons remain traditional strongholds of the method.[5](https://www.chimia.ch/chimia/article/download/2021_368/172/10810) Machine learning has entered analysis: MLstructureMining, an XGBoost classifier trained on PDFs simulated from 10,833 Crystallography Open Database structures, identifies candidate structure models from PDF data and has been installed on the DanMAX beamline at MAX IV.[22](https://pmc.ncbi.nlm.nih.gov/articles/PMC11094694/) IsoDAT2D uses unsupervised machine learning to separate thin-film total scattering from single-crystal substrate Bragg spots in 2D detector images.[23](https://www.osti.gov/biblio/2569456) Browser-based WebPDF runs \\( S(Q) \\rightarrow G(r) \\) transformation without installation, and modern laboratory diffractometers with Ag Kα sources can reach \\( Q_{\\mathrm{max}} \\) of approximately 20–22 Å⁻¹, while Mo Kα sources are limited to roughly 17 Å⁻¹, broadening access beyond synchrotrons.[24](https://link.springer.com/article/10.1007/s44211-026-00917-x)\n\n## Limitations and alternatives\n\nThe finite Q-range introduces termination ripples whose amplitudes grow as \\( Q_{\\mathrm{max}} \\) decreases, and the \\( Q_{\\mathrm{damp}} \\) parameter is directly related to the finite Q-resolution.[8](https://mdpi-res.com/d_attachment/materials/materials-12-01347/article_deploy/materials-12-01347.pdf?version=1556172641) Highly absorbing samples with \\( \\mu \\cdot R > 1 \\) always require dilution, because absorption reduces low-angle Bragg intensities and degrades background subtraction; only correlation-length differences above 0.35 Å may be resolved.[6](https://pmc.ncbi.nlm.nih.gov/articles/PMC10000805/) High-resolution instruments such as NPDF allow atom–atom correlations beyond 200 Å.[10](https://www.ias.ac.in/article/fulltext/pram/071/04/0713-0719) Compared with EXAFS, which provides element-specific local information only up to the first three atomic shells or about 6 Å, a PDF covers all atom pairs and can provide information beyond 1000 Å depending on reciprocal-space resolution.[6](https://pmc.ncbi.nlm.nih.gov/articles/PMC10000805/) In one published comparison, instrumental parameters refined by PDF and by Rietveld agreed closely, with a Si lattice parameter of 5.4302 Å by PDF versus 5.4311 Å by Rietveld.[6](https://pmc.ncbi.nlm.nih.gov/articles/PMC10000805/) Symmetric peak fitting should only be carried out using PDF functions such as \\( G_{\\mathrm{PDF}}(r) \\) or \\( D(r) \\), a notation and low-Q pitfall documented in a published corrigendum.[1](https://journals.iucr.org/j/issues/2021/01/00/gj5253/)\n\n## References\n\n1. [Illustrated formalisms for total scattering data: a guide for new practitioners (Peterson, Olds, McDonnell & Page, J. Appl. Cryst. 54, 317–332, 2021), with corrigendum (gj5273)](https://journals.iucr.org/j/issues/2021/01/00/gj5253/)\n2. [Local structure determination using total scattering data (book chapter, NSF PAR)](https://par.nsf.gov/servlets/purl/10587291)\n3. [PDFfit2 and PDFgui: computer programs for studying nanostructure in crystals (Farrow et al., J. Phys.: Condens. Matter 19, 335219, 2007)](https://www.diffpy.org/doc/pdfgui/Farrow-jpcm-2007.pdf)\n4. [Neutron PDF lecture slides (K. Page, ORNL, 2025)](https://neutrons.ornl.gov/sites/default/files/KPage_Neutron_PDF_2025.pdf)\n5. [Characterization of Nanomaterials with Total Scattering and Pair Distribution Function Analysis (Jensen, Chimia 2021)](https://www.chimia.ch/chimia/article/download/2021_368/172/10810)\n6. [Total scattering measurements at the Australian Synchrotron Powder Diffraction beamline: capabilities and limitations (J. Synchrotron Radiat., 2023)](https://pmc.ncbi.nlm.nih.gov/articles/PMC10000805/)\n7. [High-quality ultra-fast total scattering and pair distribution function data using an X-ray free-electron laser (IUCr, 2025)](https://journals.iucr.org/m/issues/2025/05/00/it5039/)\n8. [A Comparative Study of Experimental Configurations in Synchrotron Pair Distribution Function (Materials 12, 2019)](https://mdpi-res.com/d_attachment/materials/materials-12-01347/article_deploy/materials-12-01347.pdf?version=1556172641)\n9. [The rise of the X-ray atomic pair distribution function method: a series of fortunate events (Billinge, Phil. Trans. R. Soc. A, 2019)](https://royalsocietypublishing.org/rsta/article-pdf/doi/10.1098/rsta.2018.0413/246013/rsta.2018.0413.pdf)\n10. [Total neutron scattering: The key to the local and medium range structure of complex materials (Proffen et al., Pramana 71, 2008)](https://www.ias.ac.in/article/fulltext/pram/071/04/0713-0719)\n11. [Introduction, Total scattering data reduction at ORNL (mantidtotalscattering)](https://powder.ornl.gov/total_scattering/intro.html)\n12. [PDFgetN3: Atomic Pair Distribution Functions From Neutron Powder Diffraction Data Using ad hoc Corrections (OSTI full text)](https://www.osti.gov/servlets/purl/1439450)\n13. [Total scattering and the pair distribution function in crystallography (Keen, Crystallography Reviews 26, 2020)](https://www.tandfonline.com/doi/abs/10.1080/0889311X.2020.1797708)\n14. [P. Debye (1915). Zerstreuung von Röntgenstrahlen. Annalen der Physik.](https://doi.org/10.1002/andp.19153510606)\n15. [F. Zernike, J. A. Prins (1927). Die Beugung von Röntgenstrahlen in Flüssigkeiten als Effekt der Molekülanordnung. Zeitschrift für Physik A Hadrons and Nuclei.](https://doi.org/10.1007/bf01391926)\n16. [B. E. Warren (1934). The Diffraction of X-Rays in Glass. Physical Review.](https://doi.org/10.1103/physrev.45.657)\n17. [S. J. L. Billinge, T. Egami (1993). Short-range atomic structure of Nd 2 − x Ce x CuO 4 − y determined by real-space refinement of neutron-powder-diffraction data. Physical review. B, Condensed matter.](https://doi.org/10.1103/physrevb.47.14386)\n18. [Th. Proffen, S. J. L. Billinge (1999). PDFFIT , a program for full profile structural refinement of the atomic pair distribution function. Journal of Applied Crystallography.](https://doi.org/10.1107/s0021889899003532)\n19. [C L Farrow and colleagues (2007). PDFfit2 and PDFgui: computer programs for studying nanostructure in crystals. Journal of Physics Condensed Matter.](https://doi.org/10.1088/0953-8984/19/33/335219)\n20. [R. L. McGreevy, L. Pusztai (1988). Reverse Monte Carlo Simulation: A New Technique for the Determination of Disordered Structures. Molecular Simulation.](https://doi.org/10.1080/08927028808080958)\n21. [Matthew G Tucker and colleagues (2007). RMCProfile: reverse Monte Carlo for polycrystalline materials. Journal of Physics Condensed Matter.](https://doi.org/10.1088/0953-8984/19/33/335218)\n22. [MLstructureMining: a machine learning tool for structure identification from X-ray pair distribution functions](https://pmc.ncbi.nlm.nih.gov/articles/PMC11094694/)\n23. [Distinguishing isotropic and anisotropic signals for X-ray total scattering using machine learning (IsoDAT2D, Acta Crystallographica Section A)](https://www.osti.gov/biblio/2569456)\n24. [WebPDF: a browser-based software application for calculating X-ray pair distribution function (Analytical Sciences, Springer)](https://link.springer.com/article/10.1007/s44211-026-00917-x)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Physics › Matter and radiation physics › Condensed matter physics › Crystal and structural condensed matter*\n\n*Initially written Sep 29, 2026 · Edited: Oct 15, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [], "url": "https://www.edgepedia.org/total-scattering-diffraction", "markdown_url": "https://www.edgepedia.org/total-scattering-diffraction.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"Total scattering (diffraction)\", Edgepedia (EdgeChat), https://www.edgepedia.org/total-scattering-diffraction. Edgepedia Community License 1.0.", "credit_md": "\"[Total scattering (diffraction)](https://www.edgepedia.org/total-scattering-diffraction)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/total-scattering-diffraction](https://www.edgepedia.org/total-scattering-diffraction). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"Total scattering (diffraction)\", Edgepedia (EdgeChat), https://www.edgepedia.org/total-scattering-diffraction. Edgepedia Community License 1.0.", "speakable": "Total scattering is a diffraction method analyzing the complete scattering pattern, Bragg peaks and diffuse scattering alike, to obtain the atomic pair distribution function, which reveals local structure." }