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Small-angle X-ray and neutron scattering


Small-angle scattering (SAS) is a technique that is able to probe the structural organization of matter and quantify its response to changes in external conditions. X-ray and neutron scattering profiles measured from bulk materials or materials deposited at surfaces arise from nanostructural inhomogeneities of electron or nuclear density. The analysis of SAS data from coherent scattering events provides information about the length scale distributions of material components. Samples for SAS studies may be prepared in situ or under near-native conditions and the measurements performed at various temperatures, pressures, flows, shears or stresses, and in a time-resolved fashion. In this Primer, we provide an overview of SAS, summarizing the types of instrument used, approaches for data collection and calibration, available data analysis methods, structural information that can be obtained using the method, and data depositories, standards and formats. Recent applications of SAS in structural biology and the soft-matter and hard-matter sciences are also discussed.

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Fig. 1: Transmission and GISAS.
Fig. 2: Synchrotron SAXS instruments.
Fig. 3: SANS instruments.
Fig. 4: q-Axis calibration and I(q) absolute scaling.
Fig. 5: Basic scheme for data reduction.
Fig. 6: Example of beam geometry correction — slit smearing.
Fig. 7: Data merging.
Fig. 8: Representative results.
Fig. 9: Schematic display of scattering patterns from rod-like cylindrical particles in different concentration regimes and orientations.
Fig. 10: Example applications.


  1. 1.

    Glatter, O. A new method for the evaluation of small-angle scattering data. J. Appl. Cryst. 10, 415–421 (1977).

    Google Scholar 

  2. 2.

    Feigin, L. A. & Svergun, D. I. Structure Analysis by Small-Angle X-ray and Neutron Scattering (Plenum, 1987).

  3. 3.

    Fratzl, P. Small-angle scattering in materials science — a short review of applications in alloys, ceramics and composite materials. J. Appl. Cryst. 36, 397–404 (2003).

    Google Scholar 

  4. 4.

    Glatter, O. & Kratky, O. Small Angle X-ray Scattering (Academic, 1982).

  5. 5.

    Mortensen, K. in Advanced Functional Molecules and Polymers Vol. 2 (ed Nalwa, H. S.) 223–269 (Overseas Publishers Association, 2001).

  6. 6.

    Mühlbauer, S. et al. Magnetic small-angle neutron scattering. Rev. Mod. Phys. 91, 015004 (2019).

    ADS  MathSciNet  Google Scholar 

  7. 7.

    Radlinski, A. P. & Hinde, A. L. Small angle neutron scattering and petroleum geology. Neutron. N. 13, 10–14 (2002).

    Google Scholar 

  8. 8.

    Svergun, D. I., Koch, M. H. J., Timmins, P. A., & May, R. P. Small Angle X-Ray and Neutron Scattering from Solutions of Biological Macromolecules 1st edn (Oxford Univ. Press, 2013).

  9. 9.

    Stieger, M., Richtering, W., Pedersen, J. S. & Lindner, P. Small-angle neutron scattering study of structural changes in temperature sensitive microgel colloids. J. Chem. Phys. 120, 6197–6206 (2004).

    ADS  Google Scholar 

  10. 10.

    Higgins, J. S. & Benoit, H. C. Polymers and Neutron Scattering (Clarendon, 1994).

  11. 11.

    Renaud, G. et al. Real-time monitoring of growing nanoparticles. Science 300, 1416–1419 (2003).

    ADS  Google Scholar 

  12. 12.

    Sinha, S. K., Sirota, E. B., Garoff, S. & Stanley, H. B. X-ray and neutron scattering from rough surfaces. Phys. Rev. B Condens. Matter 38, 2297–2311 (1988).

    ADS  Google Scholar 

  13. 13.

    Müller-Buschbaum, P. in Applications of Synchrotron Light to Scattering and Diffraction in Materials and Life Sciences (eds Gomez, M., Nogales, A., Garcia-Gutierrez, M. C. & Ezquerra, T. A.) 61–89 (Springer, 2009).

  14. 14.

    Hexemer, A. & Muller-Buschbaum, P. Advanced grazing-incidence techniques for modern soft-matter materials analysis. IUCrJ 2, 106–125 (2015).

    Google Scholar 

  15. 15.

    Müller-Buschbaum, P. Grazing incidence small-angle neutron scattering: challenges and possibilities. Polym. J. 45, 34–42 (2013).

    Google Scholar 

  16. 16.

    Meisburger, S. P., Thomas, W. C., Watkins, M. B. & Ando, N. X-ray scattering studies of protein structural dynamics. Chem. Rev. 117, 7615–7672 (2017).

    Google Scholar 

  17. 17.

    Krueger, J. K. & Wignall, G. D. in Neutron Scattering in Biology Biological and Medical Physics, Biomedical Engineering Ch. 8 (eds Fitter, J., Gutberlet, T. & Katsaras, J.) 127–160 (Springer, 2006).

  18. 18.

    Lakey, J. H. Neutrons for biologists: a beginner’s guide, or why you should consider using neutrons. J. R. Soc. Interface R. Society 6 (Suppl. 5), S567–S573 (2009).

    Google Scholar 

  19. 19.

    Koch, M. H., Vachette, P. & Svergun, D. I. Small-angle scattering: a view on the properties, structures and structural changes of biological macromolecules in solution. Q. Rev. Biophys. 36, 147–227 (2003).

    Google Scholar 

  20. 20.

    Krueger, S. SANS provides unique information on the structure and function of biological macromolecules in solution. Phys. B: Condens. Matter 241–243, 1131–1137 (1997).

    ADS  Google Scholar 

  21. 21.

    Putnam, C. D., Hammel, M., Hura, G. L. & Tainer, J. A. X-ray solution scattering (SAXS) combined with crystallography and computation: defining accurate macromolecular structures, conformations and assemblies in solution. Q. Rev. Biophys. 40, 191–285 (2007).

    Google Scholar 

  22. 22.

    Lipfert, J. & Doniach, S. Small-angle X-ray scattering from RNA, proteins, and protein complexes. Annu. Rev. Biophys. Biomol. Struct. 36, 307–327 (2007).

    Google Scholar 

  23. 23.

    Bizien, T. et al. A brief survey of state-of-the-art BioSAXS. Protein Peptide Lett. 23, 217–231 (2016).

    Google Scholar 

  24. 24.

    Da Vela, S. et al. Kinetics of liquid–liquid phase separation in protein solutions exhibiting LCST phase behavior studied by time-resolved USAXS and VSANS. Soft Matter 12, 9334–9341 (2016).

    ADS  Google Scholar 

  25. 25.

    Martin, E. W. et al. A multi-step nucleation process determines the kinetics of prion-like domain phase separation. Nat. Commun. 12, 4513 (2021).

    ADS  Google Scholar 

  26. 26.

    Renaud, G., Lazzari, R. & Leroy, F. Probing surface and interface morphology with grazing incidence small angle X-ray scattering. Surf. Sci. Rep. 64, 255–380 (2009).

    ADS  Google Scholar 

  27. 27.

    Smilgies, D. M., Busch, P., Papadakis, C. M. & Posselt, D. Characterization of polymer thin films with small-angle X-ray scattering under grazing incidence (GISAXS). Synchrotron Radiat. N. 15, 35–42 (2002).

    Google Scholar 

  28. 28.

    Allen, A. J. & Thomas, J. J. Analysis of C–S–H gel and cement paste by small-angle neutron scattering. Cem. Concr. Res. 37, 319–324 (2007).

    Google Scholar 

  29. 29.

    Adams, C. P. et al. Small and ultra-small angle neutron scattering studies of commercial milk. Food Struct. 21, 100120 (2019).

    Google Scholar 

  30. 30.

    Peyronel, F., Marangoni, A. G. & Pink, D. A. Using the USAXS technique to reveal the fat globule and casein micelle structures of bovine dairy products. Food Res. Int. 129, 108846 (2020).

    Google Scholar 

  31. 31.

    Fister, T. T. in XAFS Techniques for Catalysts, Nanomaterials, and Surfaces (eds Iwasawa, Y., Asakura, Y., & Tada, M.) 237–250 (Springer International, 2017).

  32. 32.

    Sahle, C. J. et al. Planning, performing and analyzing X-ray Raman scattering experiments. J. Synchrotron Radiat. 22, 400–409 (2015).

    Google Scholar 

  33. 33.

    Lundberg, M. & Wernet, P. in Synchrotron Light Sources and Free-Electron Lasers: Accelerator Physics, Instrumentation and Science Applications (eds Jaeschke, E., Khan, S., Schneider, J. R. & Hastings, J. B.) 1–52 (Springer International, 2019).

  34. 34.

    Ament, L. J. P., van Veenendaal, M., Devereaux, T. P., Hill, J. P. & van den Brink, J. Resonant inelastic X-ray scattering studies of elementary excitations. Rev. Mod. Phys. 83, 705–767 (2011).

    ADS  Google Scholar 

  35. 35.

    Dziarzhytski, S. et al. The TRIXS end-station for femtosecond time-resolved resonant inelastic X-ray scattering experiments at the soft X-ray free-electron laser FLASH. Struct. Dyn. 7, 054301 (2020).

    Google Scholar 

  36. 36.

    Slowik, J. M., Son, S. K., Dixit, G., Jurek, Z. & Santra, R. Incoherent X-ray scattering in single molecule imaging. N. J. Phys. 16, 073042 (2014).

    Google Scholar 

  37. 37.

    Sears, V. F. Neutron scattering lengths and cross sections. Neutron. N. 3, 26–37 (1992).

    Google Scholar 

  38. 38.

    Squires, G. L. Introduction to the Theory of Thermal Neutron Scattering 3rd edn (Cambridge Univ. Press, 2012).

  39. 39.

    Zaccai, G. & Jacrot, B. Small angle neutron scattering. Annu. Rev. Biophys. Bioeng. 12, 139–157 (1983).

    Google Scholar 

  40. 40.

    Andreani, C., Colognesi, D., Mayers, J., Reiter, G. F. & Senesi, R. Measurement of momentum distribution of lightatoms and molecules in condensed matter systems using inelastic neutron scattering. Adv. Phys. 54, 377–469 (2005).

    ADS  Google Scholar 

  41. 41.

    Andreani, C., Krzystyniak, M., Romanelli, G., Senesi, R. & Fernandez-Alonso, F. Electron-volt neutron spectroscopy: beyond fundamental systems. Adv. Phys. 66, 1–73 (2017).

    ADS  Google Scholar 

  42. 42.

    Middendorf, H. D. Biophysical applications of quasi-elastic and inelastic neutron scattering. Annu. Rev. Biophys. Bioeng. 13, 425–451 (1984).

    Google Scholar 

  43. 43.

    Ibel, K. & Stuhrmann, H. B. Comparison of neutron and X-ray scattering of dilute myoglobin solutions. J. Mol. Biol. 93, 255–265 (1975).

    Google Scholar 

  44. 44.

    Heller, W. Small-angle neutron scattering and contrast variation: a powerful combination for studying biological structures. Acta Crystallogr. Sect. D 66, 1213–1217 (2010).

    Google Scholar 

  45. 45.

    Whitten, A. E. & Trewhella, J. in Micro and Nano Technologies in Bioanalysis: Methods and Protocols (eds Foote, R. S. & Lee, J. W.) 307–323 (Humana, 2009).

  46. 46.

    Grishaev, A., Anthis, N. J. & Clore, G. M. Contrast-matched small-angle X-ray scattering from a heavy-atom-labeled protein in structure determination: application to a lead-substituted calmodulin–peptide complex. J. Am. Chem. Soc. 134, 14686–14689 (2012).

    Google Scholar 

  47. 47.

    Garcia-Diez, R., Gollwitzer, C. & Krumrey, M. Nanoparticle characterization by continuous contrast variation in small-angle X-ray scattering with a solvent density gradient. J. Appl. Cryst. 48, 20–28 (2015).

    Google Scholar 

  48. 48.

    Hua, D. W., D’Souza, J. V., Schmidt, P. W. & Smith, D. M. in Studies in Surface Science and Catalysis Vol. 87 (eds Rouquerol, J., Rodríguez-Reinoso, F., Sing, K. S. W. & Unger, K. K.) 255–261 (Elsevier, 1994).

  49. 49.

    Larsen, A. H., Pedersen, J. S. & Arleth, L. Assessment of structure factors for analysis of small-angle scattering data from desired or undesired aggregates. J. Appl. Cryst. 53, 991–1005 (2020).

    Google Scholar 

  50. 50.

    Santoro, G. & Yu, S. in X-ray Scattering (ed. Ares, A. E.) 29–60 (IntechOpen, 2017).

  51. 51.

    Guinier, A. La diffraction des rayons X aux tres petits angles; application a l’etude de phenomenes ultramicroscopiques [French]. Ann. Phys. 12, 161–237 (1939).

    MATH  Google Scholar 

  52. 52.

    Guinier, A. & Fournet, G. Small Angle Scattering of X-Rays (Wiley, 1955).

  53. 53.

    Stuhrmann, H. B. & Kirste, R. G. Elimination der intrapartikulären Untergrundstreuung bei der Röntgenkleinwinkelstreuung an kompakten Teilchen (Proteinen) [German]. Z. für Physikalische Chem. 46, 247 (1965).

    Google Scholar 

  54. 54.

    Stuhrmann, H. Interpretation of small-angle scattering functions of dilute solutions and gases. A representation of the structures related to a one-particle scattering function. Acta Cryst. Sect. A 26, 297–306 (1970).

    ADS  Google Scholar 

  55. 55.

    Sivia, D. S. Elementary Scattering Theory: For X-ray and Neutron Users (Oxford Univ. Press, 2011).

  56. 56.

    Fernandez-Alonso, F. & Price, D. L. in Experimental Methods in the Physical Sciences Vol. 44 (Academic, 2013).

  57. 57.

    Lovesey, S. W. Theory of Neutron Scattering from Condensed Matter Vols 1 and 2 (Clarendon, 1984).

  58. 58.

    Pauw, B. R. Everything SAXS: small-angle scattering pattern collection and correction. J. Phys. Condens. Matter 25, 383201 (2013).

    ADS  Google Scholar 

  59. 59.

    Jeffries, C. M. et al. Preparing monodisperse macromolecular samples for successful biological small-angle X-ray and neutron-scattering experiments. Nat. Protoc. 11, 2122–2153 (2016).

    Google Scholar 

  60. 60.

    Grishaev, A. Sample preparation, data collection, and preliminary data analysis in biomolecular solution X-ray scattering. Curr. Protoc. ProteSci. 70, 17.14.11–17.14.18 (2012).

    Google Scholar 

  61. 61.

    Krueger, S. in Biological Small Angle Scattering: Techniques, Strategies and Tips (eds Chaudhuri, B., Muñoz, I. G., Qian, S. & Urban, V. S.) 65–85 (Springer, 2017).

  62. 62.

    Sarachan, K. L., Curtis, J. E. & Krueger, S. Small-angle scattering contrast calculator for protein and nucleic acid complexes in solution. J. Appl. Cryst. 46, 1889–1893 (2013).

    Google Scholar 

  63. 63.

    Whitten, A. E., Cai, S. & Trewhella, J. MULCh: modules for the analysis of small-angle neutron contrast variation data from biomolecular assemblies. J. Appl. Cryst. 41, 222–226 (2008).

    Google Scholar 

  64. 64.

    Allen, A. J. in International Tables for Crystallography Ch. 5.8, 673–696 (Wiley, 2019).

  65. 65.

    Fritz, G. & Bergmann, A. SAXS instruments with slit collimation: investigation of resolution and flux. J. Appl. Cryst. 39, 64–71 (2006).

    Google Scholar 

  66. 66.

    Bruetzel, L. K. et al. A Mo-anode-based in-house source for small-angle X-ray scattering measurements of biological macromolecules. Rev. Sci. Instrum. 87, 025103 (2016).

    ADS  Google Scholar 

  67. 67.

    Skarzynski, T. Collecting data in the home laboratory: evolution of X-ray sources, detectors and working practices. Acta Cryst. Sect. D. 69, 1283–1288 (2013).

    Google Scholar 

  68. 68.

    Tache, O. et al. MOMAC: a SAXS/WAXS laboratory instrument dedicated to nanomaterials. J. Appl. Cryst. 49, 1624–1631 (2016).

    Google Scholar 

  69. 69.

    Larsson, D. H., Takman, P. A. C., Lundström, U., Burvall, A. & Hertz, H. M. A 24 keV liquid-metal-jet X-ray source for biomedical applications. Rev. Sci. Instrum. 82, 123701 (2011).

    ADS  Google Scholar 

  70. 70.

    Hemberg, O., Otendal, M. & Hertz, H. M. Liquid-metal-jet anode electron-impact X-ray source. Appl. Phys. Lett. 83, 1483–1485 (2003).

    ADS  Google Scholar 

  71. 71.

    Lyngso, J. & Pedersen, J. S. A high-flux automated laboratory small-angle X-ray scattering instrument optimized for solution scattering. J. Appl. Cryst. 54, 295–305 (2021).

    Google Scholar 

  72. 72.

    Bergmann, A., Orthaber, D., Scherf, G. & Glatter, O. Improvement of SAXS measurements on Kratky slit systems by Gobel mirrors and imaging-plate detectors. J. Appl. Cryst. 33, 869–875 (2000).

    Google Scholar 

  73. 73.

    Soliman, M., Jungnickel, B.-J. & Meister, E. Stable desmearing of slit-collimated SAXS patterns by adequate numerical conditioning. Acta Cryst. Sect. A 54, 675–681 (1998).

    Google Scholar 

  74. 74.

    Lake, J. An iterative method of slit-correcting small angle X-ray data. Acta Cryst. 23, 191–194 (1967).

    Google Scholar 

  75. 75.

    Glatter, O. A new iterative method for collimation correction in small-angle scattering. J. Appl. Cryst. 7, 147–153 (1974).

    Google Scholar 

  76. 76.

    Singh, M. A., Ghosh, S. S. & Shannon Jnr, R. F. A direct method of beam-height correction in small-angle X-ray scattering. J. Appl. Cryst. 26, 787–794 (1993).

    Google Scholar 

  77. 77.

    Strobl, G. A new method of evaluating slit-smeared small-angle X-ray scattering data. Acta Crystallogr. Sect. A 26, 367–375 (1970).

    ADS  Google Scholar 

  78. 78.

    Vonk, C. A procedure for desmearing X-ray small-angle scattering curves. J. Appl. Cryst. 4, 340–342 (1971).

    Google Scholar 

  79. 79.

    Bonse, U. & Hart, M. Tailless X-ray single-crystal reflection curves obtained by multiple reflection. Appl. Phys. Lett. 7, 238–240 (1965).

    ADS  Google Scholar 

  80. 80.

    Pedersen, J. S. in Modern Aspects of Small-Angle Scattering (ed Brumberger, H.) 57–91 (Springer, 1995).

  81. 81.

    Gravatt, C. C. & Brady, G. W. Slit smearing effects in the Bonse–Hart small-angle X-ray diffractometer. J. Appl. Cryst. 2, 289–295 (1969).

    Google Scholar 

  82. 82.

    Bolze, J. & Gateshki, M. Highly versatile laboratory X-ray scattering instrument enabling (nano-)material structure analysis on multiple length scales by covering a scattering vector range of almost five decades. Rev. Sci. Instrum. 90, 123103 (2019).

    ADS  Google Scholar 

  83. 83.

    Round, A. et al. BioSAXS Sample Changer: a robotic sample changer for rapid and reliable high-throughput X-ray solution scattering experiments. Acta Cryst. Sect. D. 71, 67–75 (2015).

    Google Scholar 

  84. 84.

    Martel, A., Liu, P., Weiss, T. M., Niebuhr, M. & Tsuruta, H. An integrated high-throughput data acquisition system for biological solution X-ray scattering studies. J. Synchrotron Radiat. 19, 431–434 (2012).

    Google Scholar 

  85. 85.

    Hura, G. L. et al. Robust, high-throughput solution structural analyses by small angle X-ray scattering (SAXS). Nat. Methods 6, 606–612 (2009).

    Google Scholar 

  86. 86.

    David, G. & Perez, J. Combined sampler robot and high-performance liquid chromatography: a fully automated system for biological small-angle X-ray scattering experiments at the Synchrotron SOLEIL SWING beamline. J. Appl. Cryst. 42, 892–900 (2009).

    Google Scholar 

  87. 87.

    Osaka, K. et al. High-throughput and automated SAXS/USAXS experiment for industrial use at BL19B2 in SPring-8. AIP Conf. Proc. 1741, 030003 (2016).

    Google Scholar 

  88. 88.

    Lutz-Bueno, V. et al. Scanning-SAXS of microfluidic flows: nanostructural mapping of soft matter. Lab.Chip 16, 4028–4035 (2016).

    Google Scholar 

  89. 89.

    Pham, N. et al. Coupling high throughput microfluidics and small-angle X-ray scattering to study protein crystallization from solution. Anal. Chem. 89, 2282–2287 (2017).

    Google Scholar 

  90. 90.

    Buffet, A. et al. P03, the microfocus and nanofocus X-ray scattering (MiNaXS) beamline of the PETRA III storage ring: the microfocus endstation. J. Synchrotron Radiat. 19, 647–653 (2012).

    Google Scholar 

  91. 91.

    Hubbell, J. H. & Seltzer, S. M. X-Ray Mass Attenuation Coefficients. NIST Standard Reference Database 126 (2004).

    Article  Google Scholar 

  92. 92.

    Ballauff, M. & Jusufi, A. Anomalous small-angle X-ray scattering: analyzing correlations and fluctuations in polyelectrolytes. Colloid Polym. Sci. 284, 1303–1311 (2006).

    Google Scholar 

  93. 93.

    Seifert, S., Winans, R. E., Tiede, D. M. & Thiyagarajan, P. Design and performance of a ASAXS instrument at the Advanced Photon Source. J. Appl. Cryst. 33, 782–784 (2000).

    Google Scholar 

  94. 94.

    Berger, M. J. et al. XCOM: Photon Cross Sections Database, NIST Standard Reference Database 8 (XGAM) (NIST, 2010).

  95. 95.

    Stuhrmann, H. B. Resonant scattering in macromolecular structure research. Applications in molecular biology and material science. Inorganica Chim. Acta 79, 87–88 (1983).

    Google Scholar 

  96. 96.

    Stuhrmann, H. Contrast variation in X-ray and neutron scattering. J. Appl. Cryst. 40, s23–s27 (2007).

    Google Scholar 

  97. 97.

    Sztucki, M., Di Cola, E. & Narayanan, T. Anomalous small-angle X-ray scattering from charged soft matter. Eur. Phys. J. Spec. Top. 208, 319–331 (2012).

    Google Scholar 

  98. 98.

    Das, R. et al. Counterion distribution around DNA probed by solution X-ray scattering. Phys. Rev. Lett. 90, 188103 (2003).

    ADS  Google Scholar 

  99. 99.

    Zettl, T. et al. Absolute intramolecular distance measurements with angstrom-resolution using anomalous small-angle X-ray scattering. Nano Lett. 16, 5353–5357 (2016).

    ADS  Google Scholar 

  100. 100.

    Gruzinov, A. Y. et al. Anomalous SAXS at P12 beamline EMBL Hamburg: instrumentation and applications. J. Synchrotron Radiat. 28, 812–823 (2021).

    Google Scholar 

  101. 101.

    Wieland, D. C. F. et al. ASAXS measurements on ferritin and apoferritin at the bioSAXS beamline P12 (PETRA III, DESY). J. Appl. Cryst. 54, 830–838 (2021).

    Google Scholar 

  102. 102.

    Goerigk, G., Schweins, R., Huber, K. & Ballauff, M. The distribution of Sr2+ counterions around polyacrylate chains analyzed by anomalous small-angle X-ray scattering. Europhys. Lett. 66, 331–337 (2004).

    ADS  Google Scholar 

  103. 103.

    Yu, C., Koh, S., Leisch, J. E., Toney, M. F. & Strasser, P. Size and composition distribution dynamics of alloy nanoparticle electrocatalysts probed by anomalous small angle X-ray scattering (ASAXS). Faraday Discuss. 140, 283–296 (2009).

    ADS  Google Scholar 

  104. 104.

    Narayanan, T. et al. A multipurpose instrument for time-resolved ultra-small-angle and coherent X-ray scattering. J. Appl. Cryst. 51, 1511–1524 (2018).

    Google Scholar 

  105. 105.

    Graceffa, R. et al. Sub-millisecond time-resolved SAXS using a continuous-flow mixer and X-ray microbeam. J. Synchrotron Radiat. 20, 820–825 (2013).

    Google Scholar 

  106. 106.

    Graber, T. et al. BioCARS: a synchrotron resource for time-resolved X-ray science. J. Synchrotron Radiat. 18, 658–670 (2011).

    Google Scholar 

  107. 107.

    Cho, H. S., Schotte, F., Dashdorj, N., Kyndt, J. & Anfinrud, P. A. Probing anisotropic structure changes in proteins with picosecond time-resolved small-angle X-ray scattering. J. Phys. Chem. B 117, 15825–15832 (2013).

    Google Scholar 

  108. 108.

    Burian, M. et al. Picosecond pump-probe X-ray scattering at the Elettra SAXS beamline. J. Synchrotron Radiat. 27, 51–59 (2020).

    Google Scholar 

  109. 109.

    Weigand, S. J. & Keane, D. T. DND-CAT’s new triple area detector system for simultaneous data collection at multiple length scales. Nucl. Instrum. Methods Phys. Res. A Accel. Spectrom. Detect. Assoc. Equip. 649, 61–63 (2011).

    ADS  Google Scholar 

  110. 110.

    Takata, S.-i. et al. The design and q resolution of the small and wide angle neutron scattering instrument (TAIKAN) in J-PARC. JPS Conf. Proc. 8, 036020 (2015).

    Google Scholar 

  111. 111.

    Kirby, N. M. et al. A low-background-intensity focusing small-angle X-ray scattering undulator beamline. J. Appl. Cryst. 46, 1670–1680 (2013).

    Google Scholar 

  112. 112.

    Pernot, P. et al. Upgraded ESRF BM29 beamline for SAXS on macromolecules in solution. J. Synchrotron Radiat. 20, 660–664 (2013).

    Google Scholar 

  113. 113.

    Blanchet, C. E. et al. Versatile sample environments and automation for biological solution X-ray scattering experiments at the P12 beamline (PETRA III, DESY). J. Appl. Cryst. 48, 431–443 (2015).

    Google Scholar 

  114. 114.

    Cowieson, N. P. et al. Beamline B21: high-throughput small-angle X-ray scattering at Diamond Light Source. J. Synchrotron Radiat. 27, 1438–1446 (2020).

    Google Scholar 

  115. 115.

    Jeng, U.-S. et al. A small/wide-angle X-ray scattering instrument for structural characterization of air–liquid interfaces, thin films and bulk specimens. J. Appl. Cryst. 43, 110–121 (2010).

    Google Scholar 

  116. 116.

    Liu, G. et al. Upgraded SSRF BL19U2 beamline for small-angle X-ray scattering of biological macromolecules in solution. J. Appl. Cryst. 51, 1633–1640 (2018).

    Google Scholar 

  117. 117.

    Bras, W. et al. Recent experiments on a small-angle/wide-angle X-ray scattering beam line at the ESRF. J. Appl. Cryst. 36, 791–794 (2003).

    Google Scholar 

  118. 118.

    Ilavsky, J. et al. Ultra-small-angle X-ray scattering instrument at the advanced photon source: history, recent development, and current status. Metall. Mater. Trans. A 44A, 68–76 (2013).

    ADS  Google Scholar 

  119. 119.

    Sztucki, M. & Narayanan, T. Development of an ultra-small-angle X-ray scattering instrument for probing the microstructure and the dynamics of soft matter. J. Appl. Cryst. 40, s459–s462 (2007).

    Google Scholar 

  120. 120.

    Andersen, K. H. et al. The instrument suite of the European Spallation Source. Nucl. Instrum. Methods Phys. Res. A Accel. Spectrom. Detect. Assoc. Equip. 957, 163402 (2020).

    Google Scholar 

  121. 121.

    Arai, M. & Maekawa, F. Japan Spallation Neutron Source (JSNS) of J-PARC. Nucl. Phys. N. 19, 34–39 (2009).

    Google Scholar 

  122. 122.

    Mason, T. E. et al. The Spallation Neutron Source in Oak Ridge: a powerful tool for materials research. Phys. B: Condens. Matter 385–386, 955–960 (2006).

    ADS  Google Scholar 

  123. 123.

    Heller, W. T. et al. The suite of small-angle neutron scattering instruments at Oak Ridge National Laboratory. J. Appl. Cryst. 51, 242–248 (2018).

    Google Scholar 

  124. 124.

    Richards, J. J., Gagnon, C. V. L., Krzywon, J. R., Wagner, N. J. & Butler, P. D. Dielectric RheoSANS — simultaneous interrogation of impedance, rheology and small angle neutron scattering of complex fluids. JoVE 122, e55318 (2017).

    Google Scholar 

  125. 125.

    Gurnon, A. K. et al. Measuring material microstructure under flow using 1–2 plane flow-small angle neutron scattering. JoVE 84, e51068 (2014).

    Google Scholar 

  126. 126.

    Navarro-López, A. et al. Furnace for in situ and simultaneous studies of nano-precipitates and phase transformations in steels by SANS and neutron diffraction. Rev. Sci. Instrum. 91, 123903 (2020).

    ADS  Google Scholar 

  127. 127.

    Widmann, T. et al. Flexible sample environment for the investigation of soft matter at the european spallation source: Part II — the GISANS setup. Appl. Sci. 11, 4036 (2021).

    Google Scholar 

  128. 128.

    Carl, N., Prévost, S., Schweins, R. & Huber, K. Contrast variation of micelles composed of Ca2+ and block copolymers of two negatively charged polyelectrolytes. Colloid Polym. Sci. 298, 663–679 (2020).

    Google Scholar 

  129. 129.

    Shibayama, M., Matsunaga, T. & Nagao, M. Evaluation of incoherent scattering intensity by transmission and sample thickness. J. Appl. Cryst. 42, 621–628 (2009).

    Google Scholar 

  130. 130.

    Avers, K. E. et al. Broken time-reversal symmetry in the topological superconductor UPt3. Nat. Phys. 16, 531–535 (2020).

    Google Scholar 

  131. 131.

    Wood, K. et al. QUOKKA, the pinhole small-angle neutron scattering instrument at the OPAL Research Reactor, Australia: design, performance, operation and scientific highlights. J. Appl. Cryst. 51, 294–314 (2018).

    Google Scholar 

  132. 132.

    Babcock, E. & Ioffe, A. Polarized 3He neutron spin filter program at the JCNS. Phys. B: Condens. Matter 406, 2448–2452 (2011).

    ADS  Google Scholar 

  133. 133.

    Koizumi, S. et al. Focusing and polarized neutron small-angle scattering spectrometer (SANS-J-II). The challenge of observation over length scales from an angstrom to a micrometre. J. Appl. Cryst. 40, s474–s479 (2007).

    Google Scholar 

  134. 134.

    Gabel, F. et al. Protein dynamics studied by neutron scattering. Q. Rev. Biophys. 35, 327–367 (2002).

    Google Scholar 

  135. 135.

    Svergun, D. I. & Nierhaus, K. H. A map of protein–rRNA distribution in the 70S Escherichia coli ribosome. J. Biol. Chem. 275, 14432–14439 (2000).

    Google Scholar 

  136. 136.

    Svergun, D. I. et al. Solution scattering structural analysis of the 70S Escherichia coli ribosome by contrast variation. I. Invariants and validation of electron microscopy models. J. Mol. Biol. 271, 588–601 (1997).

    Google Scholar 

  137. 137.

    Svergun, D. I. et al. Solution scattering structural analysis of the 70S Escherichia coli ribosome by contrast variation. II. A model of the ribosome and its RNA at 3.5 nm resolution. J. Mol. Biol. 271, 602–618 (1997).

    Google Scholar 

  138. 138.

    Dewhurst, C. Novel multiple-beam very small angle neutron scattering (VSANS) using a conventional SANS instrument. J. Appl. Cryst. 47, 1180–1189 (2014).

    Google Scholar 

  139. 139.

    Van Every, E., Deyhim, A. & Kulesza, J. New very small angle neutron scattering (VSANS) instrument. J. Phys. Conf. Ser. 746, 012025 (2016).

    Google Scholar 

  140. 140.

    Vogtt, K. et al. A new time-of-flight small-angle scattering instrument at the Helmholtz-Zentrum Berlin: V16/VSANS. J. Appl. Cryst. 47, 237–244 (2014).

    Google Scholar 

  141. 141.

    Abbas, S. et al. On the design and experimental realization of a multislit-based very small angle neutron scattering instrument at the European Spallation Source. J. Appl. Cryst. 48, 1242–1253 (2015).

    Google Scholar 

  142. 142.

    Brûlet, A. et al. Toward a new lower limit for the minimum scattering vector on the very small angle neutron scattering spectrometer at Laboratoire Leon Brillouin. J. Appl. Cryst. 41, 161–166 (2008).

    Google Scholar 

  143. 143.

    Barker, J. G. et al. Design and performance of a thermal-neutron double-crystal diffractometer for USANS at NIST. J. Appl. Cryst. 38, 1004–1011 (2005).

    ADS  Google Scholar 

  144. 144.

    Rekveldt, M. T. et al. Spin-echo small angle neutron scattering in Delft. Rev. Sci. Instrum. 76, 033901 (2005).

    ADS  Google Scholar 

  145. 145.

    Andersson, R., Van Heijkamp, L. F., De Schepper, I. M. & Bouwman, W. G. Analysis of spin-echo small-angle neutron scattering measurements. J. Appl. Cryst. 41, 868–885 (2008).

    Google Scholar 

  146. 146.

    Rehm, C., Barker, J., Bouwman, W. G. & Pynn, R. DCD USANS and SESANS: a comparison of two neutron scattering techniques applicable for the study of large-scale structures. J. Appl. Cryst. 46, 354–364 (2013).

    Google Scholar 

  147. 147.

    Krouglov, T. et al. Structural transitions of hard-sphere colloids studied by spin-echo small-angle neutron scattering. J. Appl. Cryst. 36, 1417–1423 (2003).

    Google Scholar 

  148. 148.

    Tromp, R. H. & Bouwman, W. G. A novel application of neutron scattering on dairy products. Food Hydrocoll. 21, 154–158 (2007).

    Google Scholar 

  149. 149.

    Parnell, S. R. et al. Porosity of silica Stöber particles determined by spin-echo small angle neutron scattering. Soft Matter 12, 4709–4714 (2016).

    ADS  Google Scholar 

  150. 150.

    Smith, G. N. et al. Spin-echo small-angle neutron scattering (SESANS) studies of diblock copolymer nanoparticles. Soft Matter 15, 17–21 (2019).

    ADS  Google Scholar 

  151. 151.

    Schmitt, J. et al. Mesoporous silica formation mechanisms probed using combined spin-echo modulated small-angle neutron scattering (SEMSANS) and small-angle neutron scattering (SANS). ACS Appl. Mater. Interfaces 12, 28461–28473 (2020).

    Google Scholar 

  152. 152.

    Biswas, P., Sen, D. & Bouwman, W. Structural characterization of spray-dried microgranules by spin-echo small-angle neutron scattering. Powder Technol. 378, 680–684 (2021).

    Google Scholar 

  153. 153.

    Glinka, C. et al. Sub-millisecond time-resolved small-angle neutron scattering measurements at NIST. J. Appl. Cryst. 53, 598–604 (2020).

    Google Scholar 

  154. 154.

    Glinka, C. J. et al. The 30 m small-angle neutron scattering instruments at the National Institute of Standards and Technology. J. Appl. Cryst. 31, 430–445 (1998).

    Google Scholar 

  155. 155.

    Lindner, P. & Schweins, R. The D11 small-angle scattering instrument: a new benchmark for SANS. Neutron. N. 21, 15–18 (2010).

    Google Scholar 

  156. 156.

    Feoktystov, A. V. et al. KWS-1 high-resolution small-angle neutron scattering instrument at JCNS: current state. J. Appl. Cryst. 48, 61–70 (2015).

    Google Scholar 

  157. 157.

    Dewhurst, C. D. et al. The small-angle neutron scattering instrument D33 at the Institut Laue-Langevin. J. Appl. Cryst. 49, 1–14 (2016).

    Google Scholar 

  158. 158.

    Nelson, A. R. J. & Dewhurst, C. D. Towards a detailed resolution smearing kernel for time-of-flight neutron reflectometers. Corrigendum. J. Appl. Cryst. 47, 1162 (2014).

    Google Scholar 

  159. 159.

    Hammouda, B. & Mildner, D. F. R. Small-angle neutron scattering resolution with refractive optics. J. Appl. Cryst. 40, 250–259 (2007).

    Google Scholar 

  160. 160.

    Sokolova, A. et al. Performance and characteristics of the BILBY time-of-flight small-angle neutron scattering instrument. J. Appl. Cryst. 52, 1–12 (2019).

    Google Scholar 

  161. 161.

    Kampmann, R. et al. Horizontal ToF-neutron reflectometer REFSANS at FRM-II Munich/Germany: first tests and status. Phys. B Condens. Matter 385-386, 1161–1163 (2006).

    ADS  Google Scholar 

  162. 162.

    Heenan, R. K. et al. Small angle neutron scattering using SANS2D. Neutron. N. 22, 19–21 (2011).

    ADS  Google Scholar 

  163. 163.

    Koizumi, S. et al. Advanced small-angle scattering instrument available in the Tokyo area. Time-of-flight, small-angle neutron scattering developed on the iMATERIA diffractometer at the high intensity pulsed neutron source J-PARC. Quantum Beam Sci. 4, 42 (2020).

    ADS  Google Scholar 

  164. 164.

    Rauscher, M. et al. Grazing incidence small angle X-ray scattering from free-standing nanostructures. J. Appl. Phys. 86, 6763–6769 (1999).

    ADS  Google Scholar 

  165. 165.

    Huang, T. C., Toraya, H., Blanton, T. N. & Wu, Y. X-ray powder diffraction analysis of silver behenate, a possible low-angle diffraction standard. J. Appl. Cryst. 26, 180–184 (1993).

    Google Scholar 

  166. 166.

    Keiderling, U., Gilles, R. & Wiedenmann, A. Application of silver behenate powder for wavelength calibration of a SANS instrument — a comprehensive study of experimental setup variations and data processing techniques. J. Appl. Cryst. 32, 456–463 (1999).

    Google Scholar 

  167. 167.

    Nygård, K., Bunk, O., Perret, E., David, C. & van der Veen, J. F. Diffraction gratings as small-angle X-ray scattering calibration standards. J. Appl. Cryst. 43, 350–351 (2010).

    Google Scholar 

  168. 168.

    Pauw, B. R., Smith, A. J., Snow, T., Terrill, N. J. & Thunemann, A. F. The modular small-angle X-ray scattering data correction sequence. J. Appl. Cryst. 50, 1800–1811 (2017).

    Google Scholar 

  169. 169.

    Pedersen, J. S. Resolution effects and analysis of small-angle neutron-scattering data. J. Phys. Iv 3, 491–498 (1993).

    Google Scholar 

  170. 170.

    Mildner, D. F. R. & Carpenter, J. M. Optimization of the experimental resolution for small-angle scattering. J. Appl. Cryst. 17, 249–256 (1984).

    Google Scholar 

  171. 171.

    Rennie, A. R. et al. Learning about SANS instruments and data reduction from round robin measurements on samples of polystyrene latex. J. Appl. Cryst. 46, 1289–1297 (2013).

    Google Scholar 

  172. 172.

    Hajizadeh, N. R., Franke, D. & Svergun, D. I. Integrated beamline control and data acquisition for small-angle X-ray scattering at the P12 BioSAXS beamline at PETRAIII storage ring DESY. J. Synchrotron Radiat. 25, 906–914 (2018).

    Google Scholar 

  173. 173.

    Franke, D., Kikhney, A. G. & Svergun, D. I. Automated acquisition and analysis of small angle X-ray scattering. Data. Nuc. Instr. Meth Phys. Res. Sect. A 689, 52–59 (2012).

    ADS  Google Scholar 

  174. 174.

    Mühlbauer, S. et al. The new small-angle neutron scattering instrument SANS-1 at MLZ — characterization and first results. Nucl. Instrum. Methods Phys. Res. A Accel. Spectrom. Detect. Assoc. Equip. 832, 297–305 (2016).

    ADS  Google Scholar 

  175. 175.

    Karge, L., Gilles, R. & Busch, S. Calibrating SANS data for instrument geometry and pixel sensitivity effects: access to an extended Q range. J. Appl. Cryst. 50, 1382–1394 (2017).

    Google Scholar 

  176. 176.

    Lindner, P., Leclercq, F. & Damay, P. Analysis of water scattering used for calibration of small-angle neutron scattering (SANS) measurements. Phys. B Condens. Matter 291, 152–158 (2000).

    ADS  Google Scholar 

  177. 177.

    Brûlet, A., Lairez, D., Lapp, A. & Cotton, J.-P. Improvement of data treatment in small-angle neutron scattering. J. Appl. Cryst. 40, 165–177 (2007).

    Google Scholar 

  178. 178.

    Strzalka, J. A corrective prescription for GISAXS. IUCrJ 5, 661–662 (2018).

    Google Scholar 

  179. 179.

    Liu, J. & Yager, K. G. Unwarping GISAXS data. IUCrJ 5, 737–752 (2018).

    Google Scholar 

  180. 180.

    Brönnimann, C. & Trüb, P. in Synchrotron Light Sources and Free-Electron Lasers: Accelerator Physics, Instrumentation and Science Applications (eds Jaeschke, E. J., Khan, S., Schneider, J. R., & Hastings,J. B.) 995–1027 (Springer International, 2016).

  181. 181.

    Casanas, A. et al. EIGER detector: application in macromolecular crystallography. Acta Crystallogr. Sect. D 72, 1036–1048 (2016).

    Google Scholar 

  182. 182.

    Maj, P. et al. HyPix-3000 - a large area single-photon counting detector with two discriminator thresholds. 2014 IEEE Nuclear Science Symposium and Medical Imaging Conference (NSS/MIC) (2014).

    Article  Google Scholar 

  183. 183.

    Kanaki, K. et al. A novel small-angle neutron scattering detector geometry. J. Appl. Cryst. 46, 1031–1037 (2013).

    Google Scholar 

  184. 184.

    Seibert, J., Boone, J. & Lindfors, K. Flat-Field Correction Technique for Digital Detectors Vol. 3336 (SPIE, 1998).

  185. 185.

    Zhang, F. et al. Glassy carbon as an absolute intensity calibration standard for small-angle scattering. Metall. Mater. Trans. A 41, 1151–1158 (2010).

    Google Scholar 

  186. 186.

    Fan, L., Degen, M., Bendle, S., Grupido, N. & Ilavsky, J. The absolute calibration of a small-angle scattering instrument with a laboratory X-ray source. J. Phys. Conf. Ser. 247, 012005 (2010).

    Google Scholar 

  187. 187.

    Orthaber, D., Bergmann, A. & Glatter, O. SAXS experiments on absolute scale with Kratky systems using water as a secondary standard. J. Appl. Cryst. 33, 218–225 (2000).

    Google Scholar 

  188. 188.

    Wignall, G. D. & Bates, F. S. Absolute calibration of small-angle neutron scattering data. J. Appl. Cryst. 20, 28–40 (1987).

    Google Scholar 

  189. 189.

    Dreiss, C. A., Jack, K. S. & Parker, A. P. On the absolute calibration of bench-top small-angle X-ray scattering instruments: a comparison of different standard methods. J. Appl. Cryst. 39, 32–38 (2006).

    Google Scholar 

  190. 190.

    Allen, A. J., Zhang, F., Kline, R. J., Guthrie, W. F. & Ilavsky, J. NIST Standard Reference Material 3600: absolute intensity calibration standard for small-angle X-ray scattering. J. Appl. Cryst. 50, 462–474 (2017).

    Google Scholar 

  191. 191.

    Bras, W., Koizumi, S. & Terrill, N. J. Beyond simple small-angle X-ray scattering: developments in online complementary techniques and sample environments. IUCrJ 1, 478–491 (2014).

    Google Scholar 

  192. 192.

    Georgiadis, M. et al. 3D scanning SAXS: a novel method for the assessment of bone ultrastructure orientation. Bone 71, 42–52 (2015).

    Google Scholar 

  193. 193.

    Salditt, T. & Köster, S. in Nanoscale Photonic Imaging (eds Salditt, T., Egner, A., & Luke, D. R.) 405–433 (Springer International, 2020).

  194. 194.

    Rai, D. K. et al. High-pressure small-angle X-ray scattering cell for biological solutions and soft materials. J. Appl. Cryst. 54, 111–122 (2021).

    Google Scholar 

  195. 195.

    Narayanan, T., Dattani, R., Möller, J. & Kwaśniewski, P. A microvolume shear cell for combined rheology and X-ray scattering experiments. Rev. Sci. Instrum. 91, 085102 (2020).

    ADS  Google Scholar 

  196. 196.

    Nogales, A., Thornley, S. A. & Mitchell, G. R. Shear cell for in situ WAXS, SAXS, and SANS experiments on polymer melts under flow fields. J. Macromol. Sci. B 43, 1161–1170 (2004).

    Google Scholar 

  197. 197.

    Zákutná, D. et al. In situ magnetorheological SANS setup at Institut Laue-Langevin. Colloid Polym. Sci. 299, 281–288 (2021).

    Google Scholar 

  198. 198.

    Schroer, M. A. et al. Smaller capillaries improve the small-angle X-ray scattering signal and sample consumption for biomacromolecular solutions. J. Synchrotron Radiat. 25, 1113–1122 (2018).

    Google Scholar 

  199. 199.

    Bucciarelli, S. et al. Size-exclusion chromatography small-angle X-ray scattering of water soluble proteins on a laboratory instrument. J. Appl. Cryst. 51, 1623–1632 (2018).

    Google Scholar 

  200. 200.

    Johansen, N. T., Pedersen, M. C., Porcar, L., Martel, A. & Arleth, L. Introducing SEC–SANS for studies of complex self-organized biological systems. Acta Crystallogr. Sect. D. 74, 1178–1191 (2018).

    Google Scholar 

  201. 201.

    Mathew, E., Mirza, A. & Menhart, N. Liquid-chromatography-coupled SAXS for accurate sizing of aggregating proteins. J. Synchrotron Radiat. 11, 314–318 (2004).

    Google Scholar 

  202. 202.

    Pérez, J. & Vachette, P. in Biological Small Angle Scattering: Techniques, Strategies and Tips (eds Chaudhuri, B., Muñoz, I. G., Qian, S. & Urban, V. S.) 183–199 (Springer, 2017).

  203. 203.

    Hutin, S., Brennich, M., Maillot, B. & Round, A. Online ion-exchange chromatography for small-angle X-ray scattering. Acta Crystallogr. Sect. D. 72, 1090–1099 (2016).

    Google Scholar 

  204. 204.

    Brennich, M. E., Round, A. R. & Hutin, S. Online size-exclusion and ion-exchange chromatography on a SAXS beamline. JoVE 119, e54861 (2017).

    Google Scholar 

  205. 205.

    Graewert, M. A. et al. Adding size exclusion chromatography (SEC) and light scattering (LS) devices to obtain high-quality small angle X-ray scattering (SAXS) data. Crystals 10, 975 (2020).

    Google Scholar 

  206. 206.

    Cammarata, M. et al. Tracking the structural dynamics of proteins in solution using time-resolved wide-angle X-ray scattering. Nat. Methods 5, 881–886 (2008).

    Google Scholar 

  207. 207.

    Möller, J., Léonardon, J., Gorini, J., Dattani, R. & Narayanan, T. A sub-ms pressure jump setup for time-resolved X-ray scattering. Rev. Sci. Instrum. 87, 125116 (2016).

    ADS  Google Scholar 

  208. 208.

    Rössle, M. et al. Time-resolved small-angle neutron scattering of proteins in solution. Phys. B Condens. Matter 276-278, 532–533 (2000).

    ADS  Google Scholar 

  209. 209.

    Terashima, T. et al. In situ and time-resolved small-angle neutron scattering observation of star polymer formation via arm-linking reaction in ruthenium-catalyzed living radical polymerization. Macromolecules 43, 8218–8232 (2010).

    ADS  Google Scholar 

  210. 210.

    Mason, T. G. & Lin, M. Y. Time-resolved small angle neutron scattering measurements of asphaltene nanoparticle aggregation kinetics in incompatible crude oil mixtures. J. Chem. Phys. 119, 565–571 (2003).

    ADS  Google Scholar 

  211. 211.

    Grillo, I. Applications of stopped-flow in SAXS and SANS. Curr. Opin. Colloid Interface Sci. 14, 402–408 (2009).

    Google Scholar 

  212. 212.

    Helfer, E., Panine, P., Carlier, M.-F. & Davidson, P. The interplay between viscoelastic and thermodynamic properties determines the birefringence of F-actin gels. Biophys. J. 89, 543–553 (2005).

    Google Scholar 

  213. 213.

    Babonneau, D. FitGISAXS: software package for modelling and analysis of GISAXS data using IGOR Pro. J. Appl. Cryst. 43, 929–936 (2010).

    Google Scholar 

  214. 214.

    Chourou, S. T., Sarje, A., Li, X. S., Chan, E. R. & Hexemer, A. HipGISAXS: a high-performance computing code for simulating grazing-incidence X-ray scattering data. J. Appl. Cryst. 46, 1781–1795 (2013).

    Google Scholar 

  215. 215.

    Lazzari, R. IsGISAXS: a program for grazing-incidence small-angle X-ray scattering analysis of supported islands. J. Appl. Cryst. 35, 406–421 (2002).

    Google Scholar 

  216. 216.

    Pospelov, G. et al. BornAgain: software for simulating and fitting grazing-incidence small-angle scattering. J. Appl. Cryst. 53, 262–276 (2020).

    Google Scholar 

  217. 217.

    Tian, B. et al. Small angle neutron scattering quantifies the hierarchical structure in fibrous calcium caseinate. Food Hydrocoll. 106, 105912 (2020).

    Google Scholar 

  218. 218.

    Stribeck, N., Almendarez Camarillo, A., Nöchel, U., Bösecke, P. & Bayer, R. K. Early oriented isothermal crystallization of polyethylene studied by high-time-resolution SAXS/WAXS. Anal. Bioanal. Chem. 387, 649 (2006).

    Google Scholar 

  219. 219.

    Krishnamurthy, A. et al. Multiscale polymer dynamics in hierarchical carbon nanotube grafted glass fiber reinforced composites. ACS Appl. Polym. Mater. 1, 1905–1917 (2019).

    Google Scholar 

  220. 220.

    Huang, X. et al. In situ constructing the kinetic roadmap of octahedral nanocrystal assembly toward controlled superlattice fabrication. J. Am. Chem. Soc. 143, 4234–4243 (2021).

    Google Scholar 

  221. 221.

    Akcora, P. et al. Anisotropic self-assembly of spherical polymer-grafted nanoparticles. Nat. Mater. 8, 354–359 (2009).

    ADS  Google Scholar 

  222. 222.

    Beaucage, G. Approximations leading to a unified exponential/power-law approach to small-angle scattering. J. Appl. Cryst. 28, 717–728 (1995).

    Google Scholar 

  223. 223.

    Beaucage, G. & Schaefer, D. W. Structural studies of complex systems using small-angle scattering: a unified Guinier/power-law approach. J. Noncryst. Solids 172–174, 797–805 (1994).

    ADS  Google Scholar 

  224. 224.

    Mylonas, E. & Svergun, D. I. Accuracy of molecular mass determination of proteins in solution by small-angle X-ray scattering. J. Appl. Cryst. 40, S245–S249 (2007).

    Google Scholar 

  225. 225.

    Stribeck, N. & Nochel, U. A method for merging of ultra-small-angle X-ray scattering and smeared small-angle X-ray scattering patterns of fibers. J. Appl. Cryst. 41, 715–722 (2008).

    Google Scholar 

  226. 226.

    Skou, S., Gillilan, R. E. & Ando, N. Synchrotron-based small-angle X-ray scattering of proteins in solution. Nat. Protoc. 9, 1727–1739 (2014).

    Google Scholar 

  227. 227.

    Porod, G. Die Rontgenkleinwinkelstreuung Von Dichtgepackten Kolloiden Systemen. 1 [German]. Kolloid Z. Z Polym. 124, 83–114 (1951).

    Google Scholar 

  228. 228.

    Debye, P., Anderson, H. R. Jr & Brumberger, H. Scattering by an inhomogeneous solid. II. The correlation function and its application. J. Appl. Phys. 28, 679–683 (1957).

    ADS  Google Scholar 

  229. 229.

    Rambo, R. P. & Tainer, J. A. Characterizing flexible and intrinsically unstructured biological macromolecules by SAS using the Porod–Debye law. Biopolymers 95, 559–571 (2011).

    Google Scholar 

  230. 230.

    Franke, D., Jeffries, C. M. & Svergun, D. I. Machine learning methods for X-ray scattering data analysis from biomacromolecular solutions. Biophys. J. 114, 2485–2492 (2018).

    ADS  Google Scholar 

  231. 231.

    Hammouda, B. A new Guinier–Porod model. J. Appl. Cryst. 43, 716–719 (2010).

    Google Scholar 

  232. 232.

    Receveur-Brechot, V. & Durand, D. How random are intrinsically disordered proteins? A small angle scattering perspective. Curr. Protein Peptide Sci. 13, 55–75 (2012).

    Google Scholar 

  233. 233.

    Svergun, D. I., Semenyuk, A. V. & Feigin, L. A. Small-angle-scattering-data treatment by the regularization method. Acta Crystallogr. Sect. A 44, 244–250 (1988).

    Google Scholar 

  234. 234.

    Hansen, S. & Pedersen, J. S. A comparison of 3 different methods for analyzing small-angle scattering data. J. Appl. Cryst. 24, 541–548 (1991).

    Google Scholar 

  235. 235.

    Glatter, O. Convolution square root of band-limited symmetrical functions and its application to small-angle scattering data. J. Appl. Cryst. 14, 101–108 (1981).

    Google Scholar 

  236. 236.

    Glatter, O. & Hainisch, B. Improvements in real-space deconvolution of small-angle scattering data. J. Appl. Cryst. 17, 435–441 (1984).

    Google Scholar 

  237. 237.

    Oliveira, C. L. P. et al. Gaussian deconvolution: a useful method for a form-free modeling of scattering data from mono- and multilayered planar systems. J. Appl. Cryst. 45, 1278–1286 (2012).

    Google Scholar 

  238. 238.

    Rayleigh, L. The incidence of light upon a transparent sphere of dimensions comparable with the wave-length. P R. Soc. Lond. Conta 84, 25–46 (1910).

    ADS  MATH  Google Scholar 

  239. 239.

    Pedersen, J. S. Analysis of small-angle scattering data from colloids and polymer solutions: modeling and least-squares fitting. Adv. Colloid Interfac. 70, 171–210 (1997).

    Google Scholar 

  240. 240.

    Pedersen, J. S. in Neutrons, X-Rays and Light: Scattering Methods Applied to Soft Condensed Matter (eds Lindner, P. & Zemb, T.) 391–420 (Elsevier, 2002).

  241. 241.

    Konarev, P. V. & Svergun, D. I. A posteriori determination of the useful data range for small-angle scattering experiments on dilute monodisperse systems. IUCrJ 2, 352–360 (2015).

    Google Scholar 

  242. 242.

    Franke, D., Jeffries, C. M. & Svergun, D. I. Correlation map, a goodness-of-fit test for one-dimensional X-ray scattering spectra. Nat. Methods 12, 419–422 (2015).

    Google Scholar 

  243. 243.

    Svergun, D. I. & Pedersen, J. S. Propagating errors in small-angle scattering data treatment. J. Appl. Cryst. 27, 241–248 (1994).

    Google Scholar 

  244. 244.

    Pedersen, J. S., Posselt, D. & Mortensen, K. Analytical treatment of the resolution function for small-angle scattering. J. Appl. Cryst. 23, 321–333 (1990).

    Google Scholar 

  245. 245.

    Caponetti, E., Floriano, M. A., Didio, E. & Triolo, R. On the shape of the radial-distribution function of an assembly of monodisperse ellipsoidal scatterers. J. Appl. Cryst. 26, 612–615 (1993).

    Google Scholar 

  246. 246.

    Glatter, O. Determination of particle-size distribution-functions from small-angle scattering data by means of the indirect transformation method. J. Appl. Cryst. 13, 7–11 (1980).

    Google Scholar 

  247. 247.

    Pedersen, J. S. Determination of size distributions from small-angle scattering data for systems with effective hard-sphere interactions. J. Appl. Cryst. 27, 595–608 (1994).

    Google Scholar 

  248. 248.

    Pauw, B. R., Kastner, C. & Thunemann, A. F. Nanoparticle size distribution quantification: results of a small-angle X-ray scattering inter-laboratory comparison. J. Appl. Cryst. 50, 1280–1288 (2017).

    Google Scholar 

  249. 249.

    Pedersen, J. S. Analysis of small-angle scattering data from micelles and microemulsions: free-form approaches and model fitting. Curr. Opin. Colloid Interface Sci. 4, 190–196 (1999).

    Google Scholar 

  250. 250.

    Shipovskov, S. et al. Water-in-oil micro-emulsion enhances the secondary structure of a protein by confinement. Chemphyschem 13, 3179–3184 (2012).

    Google Scholar 

  251. 251.

    Kirste, R. G., Oberthur, R. C. in Small Angle X-ray Scattering (eds Glatter, O. & Kratky, O.) 387–431 (Academic, 1982).

  252. 252.

    Menon, S. V. G., Manohar, C. & Srinivasa Rao, K. A new interpretation of the sticky hard sphere model. J. Chem. Phys. 95, 9186–9190 (1991).

    ADS  Google Scholar 

  253. 253.

    Percus, J. K. & Yevick, G. J. Analysis of classical statistical mechanics by means of collective coordinates. Phys. Rev. 110, 1–13 (1958).

    ADS  MathSciNet  MATH  Google Scholar 

  254. 254.

    Sharma, R. V. & Sharma, K. C. The structure factor and the transport properties of dense fluids having molecules with square well potential, a possible generalization. Phys. A Stat. Mech. Appl. 89, 213–218 (1977).

    Google Scholar 

  255. 255.

    Kinning, D. J. & Thomas, E. L. Hard-sphere interactions between spherical domains in diblock copolymers. Macromolecules 17, 1712–1718 (1984).

    ADS  Google Scholar 

  256. 256.

    Debye, P. Molecular-weight determination by light scattering. J. Phys. Colloid Chem. 51, 18–32 (1947).

    Google Scholar 

  257. 257.

    Benoit, H. On the effect of branching and polydispersity on the angular distribution of the light scattered by Gaussian coils. J. Polym. Sci. 11, 507–510 (1953).

    ADS  Google Scholar 

  258. 258.

    Svaneborg, C. & Pedersen, J. S. A formalism for scattering of complex composite structures. I. Applications to branched structures of asymmetric sub-units. J. Chem. Phys. 136, 104105 (2012).

    ADS  Google Scholar 

  259. 259.

    Pedersen, J. S. & Gerstenberg, M. C. Scattering form factor of block copolymer micelles. Macromolecules 29, 1363–1365 (1996).

    ADS  Google Scholar 

  260. 260.

    Burchard, W. & Kajiwara, K. Statistics of stiff chain molecules.1. Particle scattering factor. Proc. R. Soc. Lon. A 316, 185 (1970).

    ADS  Google Scholar 

  261. 261.

    Debye, P. X-ray dispersal. Ann. Phys. Berlin 46, 809–823 (1915).

    ADS  Google Scholar 

  262. 262.

    Bernado, P., Mylonas, E., Petoukhov, M. V., Blackledge, M. & Svergun, D. I. Structural characterization of flexible proteins using small-angle X-ray scattering. J. Am. Chem. Soc. 129, 5656–5664 (2007).

    Google Scholar 

  263. 263.

    Franke, D. & Svergun, D. I. DAMMIF, a program for rapid ab-initio shape determination in small-angle scattering. J. Appl. Cryst. 42, 342–346 (2009).

    Google Scholar 

  264. 264.

    Svergun, D., Barberato, C. & Koch, M. H. J. CRYSOL — a program to evaluate X-ray solution scattering of biological macromolecules from atomic coordinates. J. Appl. Cryst. 28, 768–773 (1995).

    Google Scholar 

  265. 265.

    Svergun, D. I. Restoring low resolution structure of biological macromolecules from solution scattering using simulated annealing. Biophys. J. 77, 2896–2896 (1999).

    Google Scholar 

  266. 266.

    Panjkovich, A. & Svergun, D. I. Deciphering conformational transitions of proteins by small angle X-ray scattering and normal mode analysis. Phys. Chem. Chem. Phys. 18, 5707–5719 (2016).

    Google Scholar 

  267. 267.

    Petoukhov, M. V. et al. Reconstruction of quaternary structure from X-ray scattering by equilibrium mixtures of biological macromolecules. Biochemistry 52, 6844–6855 (2013).

    Google Scholar 

  268. 268.

    Petoukhov, M. V. & Svergun, D. I. Global rigid body modeling of macromolecular complexes against small-angle scattering data. Biophys. J. 89, 1237–1250 (2005).

    Google Scholar 

  269. 269.

    Schneidman-Duhovny, D., Hammel, M. & Sali, A. FoXS: a web server for rapid computation and fitting of SAXS profiles. Nucleic Acids Res. 38, W540–W544 (2010).

    Google Scholar 

  270. 270.

    Grudinin, S., Garkavenko, M. & Kazennov, A. Pepsi-SAXS: an adaptive method for rapid and accurate computation of small-angle X-ray scattering profiles. Acta Cryst. Sect. D. 73, 449–464 (2017).

    Google Scholar 

  271. 271.

    Svergun, D. I. et al. Protein hydration in solution: experimental observation by X-ray and neutron scattering. Proc. Natl Acad. Sci. USA 95, 2267–2272 (1998).

    ADS  Google Scholar 

  272. 272.

    Schneidman-Duhovny, D., Hammel, M., Tainer, John, A. & Sali, A. Accurate SAXS profile computation and its assessment by contrast variation experiments. Biophys. J. 105, 962–974 (2013).

    ADS  Google Scholar 

  273. 273.

    Schneidman-Duhovny, D., Hammel, M., Tainer, J. A. & Sali, A. FoXS, FoXSDock and MultiFoXS: single-state and multi-state structural modeling of proteins and their complexes based on SAXS profiles. Nucleic Acids Res. 44, W424–W429 (2016).

    Google Scholar 

  274. 274.

    Petoukhov, M. V. et al. New developments in the ATSAS program package for small-angle scattering data analysis. J. Appl. Cryst. 45, 342–350 (2012).

    Google Scholar 

  275. 275.

    Svergun, D. I., Petoukhov, M. V. & Koch, M. H. J. Determination of domain structure of proteins from X-ray solution scattering. Biophys. J. 80, 2946–2953 (2001).

    Google Scholar 

  276. 276.

    Kozin, M. B. & Svergun, D. I. Automated matching of high- and low-resolution structural models. J. Appl. Cryst. 34, 33–41 (2001).

    Google Scholar 

  277. 277.

    Tria, G., Mertens, H. D. T., Kachala, M. & Svergun, D. I. Advanced ensemble modelling of flexible macromolecules using X-ray solution scattering. IUCrJ 2, 207–217 (2015).

    Google Scholar 

  278. 278.

    Arnold, T. et al. Implementation of a beam deflection system for studies of liquid interfaces on beamline I07 at Diamond. J. Synchrotron Radiat. 19, 408–416 (2012).

    Google Scholar 

  279. 279.

    Seeck, O. H. et al. The high-resolution diffraction beamline P08 at PETRA III. J. Synchrotron Radiat. 19, 30–38 (2012).

    Google Scholar 

  280. 280.

    Annighöfer, B. et al. A high pressure cell using metallic windows to investigate the structure of molecular solutions up to 600 MPa by small-angle neutron scattering. Rev. Sci. Instrum. 90, 025106 (2019).

    ADS  Google Scholar 

  281. 281.

    Schroer, M. A. et al. High-pressure SAXS study of folded and unfolded ensembles of proteins. Biophys. J. 99, 3430–3437 (2010).

    ADS  Google Scholar 

  282. 282.

    Singh, I., Themistou, E., Porcar, L. & Neelamegham, S. Fluid shear induces conformation change in human blood protein von Willebrand factor in solution. Biophys. J. 96, 2313–2320 (2009).

    ADS  Google Scholar 

  283. 283.

    Josts, I. et al. Structural kinetics of MsbA investigated by stopped-flow time-resolved small-angle X-ray scattering. Structure 28, 348–354.e3 (2020).

    Google Scholar 

  284. 284.

    Ibrahim, Z. et al. Time-resolved neutron scattering provides new insight into protein substrate processing by a AAA+ unfoldase. Sci. Rep. 7, 40948 (2017).

    ADS  Google Scholar 

  285. 285.

    Weiss, T. M. et al. Dynamics of the self-assembly of unilamellar vesicles. Phys. Rev. Lett. 94, 038303 (2005).

    ADS  Google Scholar 

  286. 286.

    Mertens, H. D. T. & Svergun, D. I. Structural characterization of proteins and complexes using small-angle X-ray solution scattering. J. Struct. Biol. 172, 128–141 (2010).

    Google Scholar 

  287. 287.

    Schneidman-Duhovny, D. & Hammel, M. Modeling structure and dynamics of protein complexes with SAXS profiles. Methods Mol. Biol. 1764, 449–473 (2018).

    Google Scholar 

  288. 288.

    Kikhney, A. G. & Svergun, D. I. A practical guide to small angle X-ray scattering (SAXS) of flexible and intrinsically disordered proteins. FEBS Lett. 589, 2570–2577 (2015).

    Google Scholar 

  289. 289.

    Sagar, A., Svergun, D. & Bernadó, P. in Intrinsically Disordered Proteins: Methods and Protocols (eds Kragelund, B. B. & Skriver, K.) 249–269 (Springer, 2020).

  290. 290.

    Konarev, P. V., Volkov, V. V., Sokolova, A. V., Koch, M. H. J. & Svergun, D. I. PRIMUS: a Windows PC-based system for small-angle scattering data analysis. J. Appl. Cryst. 36, 1277–1282 (2003).

    Google Scholar 

  291. 291.

    Konarev, P. V. & Svergun, D. I. Direct shape determination of intermediates in evolving macromolecular solutions from small-angle scattering data. IUCrJ 5, 402–409 (2018).

    Google Scholar 

  292. 292.

    Burian, M. & Amenitsch, H. Dummy-atom modelling of stacked and helical nanostructures from solution scattering data. IUCrJ 5, 390–401 (2018).

    Google Scholar 

  293. 293.

    Mertens, H. D. T. & Svergun, D. I. Combining NMR and small angle X-ray scattering for the study of biomolecular structure and dynamics. Arch. Biochem. Biophys. 628, 33–41 (2017).

    Google Scholar 

  294. 294.

    Evrard, G., Mareuil, F., Bontems, F., Sizun, C. & Perez, J. DADIMODO: a program for refining the structure of multidomain proteins and complexes against small-angle scattering data and NMR-derived restraints. J. Appl. Cryst. 44, 1264–1271 (2011).

    Google Scholar 

  295. 295.

    Zhao, C. & Shukla, D. SAXS-guided enhanced unbiased sampling for structure determination of proteins and complexes. Sci. Rep. 8, 17748 (2018).

    ADS  Google Scholar 

  296. 296.

    Grant, T. D. Ab initio electron density determination directly from solution scattering data. Nat. Methods 15, 191–193 (2018).

    Google Scholar 

  297. 297.

    Konarev, P. V. & Svergun, D. I. Limitations of the iterative electron density reconstruction algorithm from solution scattering data. Nat. Methods 18, 244–245 (2021).

    Google Scholar 

  298. 298.

    Grant, T. D. Reply to: Limitations of the iterative electron density reconstruction algorithm from solution scattering data. Nat. Methods 18, 246–248 (2021).

    Google Scholar 

  299. 299.

    Perez, J. & Koutsioubas, A. Memprot: a program to model the detergent corona around a membrane protein based on SEC-SAXS data. Acta Crystallogr.D. 71, 86–93 (2015).

    Google Scholar 

  300. 300.

    Pedersen, M. C., Arleth, L. & Mortensen, K. WillItFit: a framework for fitting of constrained models to small-angle scattering data. J. Appl. Cryst. 46, 1894–1898 (2013).

    Google Scholar 

  301. 301.

    Trewhella, J., Gallagher, S. C., Krueger, J. K. & Zhao, J. Neutron and X-ray solution scattering provide insights into biomolecular structure and function. Sci. Prog. 81, 101–122 (1998).

    Google Scholar 

  302. 302.

    Olah, G. A., Rokop, S. E., Wang, C. L. A., Blechner, S. L. & Trewhella, J. Troponin I encompasses an extended Troponin C in the Ca2+-bound complex: a small-angle X-ray and neutron scattering study. Biochemistry 33, 8233–8239 (1994).

    Google Scholar 

  303. 303.

    Olah, G. A. & Trewhella, J. A model structure of the muscle protein complex 4Ca2+·Troponin C·Troponin I derived from small-angle scattering data: implications for regulation. Biochemistry 33, 12800–12806 (1994).

    Google Scholar 

  304. 304.

    Jeffries, C. M., Pietras, Z. & Svergun, D. I. The basics of small-angle neutron scattering (SANS for new users of structural biology). EPJ Web Conf. 236, 03001 (2020).

    Google Scholar 

  305. 305.

    Cornilescu, G. et al. Structural analysis of multi-helical RNAs by NMR–SAXS/WAXS: application to the U4/U6 di-snRNA. J. Mol. Biol. 428, 777–789 (2016).

    Google Scholar 

  306. 306.

    Schwieters, C. D. et al. Solution structure of the 128 kDa enzyme I dimer from Escherichia coli and Its 146 kDa complex with HPr using residual dipolar couplings and small- and wide-angle X-ray scattering. J. Am. Chem. Soc. 132, 13026–13045 (2010).

    Google Scholar 

  307. 307.

    Mahieu, E. & Gabel, F. Biological small-angle neutron scattering: recent results and development. Acta Cryst. D. 74, 715–726 (2018).

    Google Scholar 

  308. 308.

    Zhao, J., Hoye, E., Boylan, S., Walsh, D. A. & Trewhella, J. Quaternary structures of a catalytic subunit-regulatory subunit dimeric complex and the holoenzyme of the cAMP-dependent protein kinase by neutron contrast variation. J. Biol. Chem. 273, 30448–30459 (1998).

    Google Scholar 

  309. 309.

    Krueger, J. K., Gallagher, S. C., Wang, C. L. A. & Trewhella, J. Calmodulin remains extended upon binding to smooth muscle caldesmon: a combined small-angle scattering and Fourier transform infrared spectroscopy study. Biochemistry 39, 3979–3987 (2000).

    Google Scholar 

  310. 310.

    Whitten, A. E. et al. The structure of the KinA–Sda complex suggests an allosteric mechanism of histidine kinase inhibition. J. Mol. Biol. 368, 407–420 (2007).

    Google Scholar 

  311. 311.

    Maric, S. et al. Time-resolved small-angle neutron scattering as a probe for the dynamics of lipid exchange between human lipoproteins and naturally derived membranes. Sci. Rep. 9, 7591 (2019).

    ADS  Google Scholar 

  312. 312.

    Maric, S. et al. Stealth carriers for low-resolution structure determination of membrane proteins in solution. Acta Cryst. D. Biol. Cryst. 70, 317–328 (2014).

    Google Scholar 

  313. 313.

    Koutsioubas, A., Berthaud, A., Mangenot, S. & Pérez, J. Ab initio and all-atom modeling of detergent organization around Aquaporin-0 based on SAXS data. J. Phys. Chem. B 117, 13588–13594 (2013).

    Google Scholar 

  314. 314.

    Bengtsen, T. et al. Structure and dynamics of a nanodisc by integrating NMR, SAXS and SANS experiments with molecular dynamics simulations. eLife 9, e56518 (2020).

    Google Scholar 

  315. 315.

    Whitten, A. E., Jeffries, C. M., Harris, S. P. & Trewhella, J. Cardiac myosin-binding protein C decorates F-actin: implications for cardiac function. Proc. Natl Acad. Sci. USA 105, 18360–18365 (2008).

    ADS  Google Scholar 

  316. 316.

    Majorošová, J. et al. Effect of the concentration of protein and nanoparticles on the structure of biohybrid nanocomposites. Biopolymers 111, e23342 (2020).

    Google Scholar 

  317. 317.

    Johansen, D., Jeffries, C. M. J., Hammouda, B., Trewhella, J. & Goldenberg, D. P. Effects of macromolecular crowding on an intrinsically disordered protein characterized by small-angle neutron scattering with contrast matching. Biophys. J. 100, 1120–1128 (2011).

    ADS  Google Scholar 

  318. 318.

    Hirai, M. et al. Observation of protein and lipid membrane structures in a model mimicking the molecular-crowding environment of cells using neutron scattering and cell debris. J. Phys. Chem. B 123, 3189–3198 (2019).

    Google Scholar 

  319. 319.

    Giehm, L., Svergun, D. I., Otzen, D. E. & Vestergaard, B. Low-resolution structure of a vesicle disrupting α-synuclein oligomer that accumulates during fibrillation. Proc. Natl Acad. Sci. USA 108, 3246–3251 (2011).

    ADS  Google Scholar 

  320. 320.

    Langkilde, A. E., Morris, K. L., Serpell, L. C., Svergun, D. I. & Vestergaard, B. The architecture of amyloid-like peptide fibrils revealed by X-ray scattering, diffraction and electron microscopy. Acta Cryst. Sect. D 71, 882–895 (2015).

    Google Scholar 

  321. 321.

    Vestergaard, B. et al. A helical structural nucleus is the primary elongating unit of insulin amyloid fibrils. PLoS Biol. 5, e134 (2007).

    Google Scholar 

  322. 322.

    Zheng, W. et al. Probing the action of chemical denaturant on an intrinsically disordered protein by simulation and experiment. J. Am. Chem. Soc. 138, 11702–11713 (2016).

    Google Scholar 

  323. 323.

    Fuertes, G. et al. Decoupling of size and shape fluctuations in heteropolymeric sequences reconciles discrepancies in SAXS vs. FRET measurements. Proc. Natl Acad. Sci. USA 114, E6342–E6351 (2017).

    Google Scholar 

  324. 324.

    Vestergaard, B. Analysis of biostructural changes, dynamics, and interactions — small-angle X-ray scattering to the rescue. Arch. Biochem. Biophys. 602, 69–79 (2016).

    Google Scholar 

  325. 325.

    Kirby, N. M. & Cowieson, N. P. Time-resolved studies of dynamic biomolecules using small angle X-ray scattering. Curr. Opin. Struct. Biol. 28, 41–46 (2014).

    Google Scholar 

  326. 326.

    Czajka, A. & Armes, S. P. Time-resolved small-angle X-ray scattering studies during aqueous emulsion polymerization. J. Am. Chem. Soc. 143, 1474–1484 (2021).

    Google Scholar 

  327. 327.

    Lund, R. et al. Structural observation and kinetic pathway in the formation of polymeric micelles. Phys. Rev. Lett. 102, 188301 (2009).

    ADS  Google Scholar 

  328. 328.

    Jensen, G. V. et al. Direct observation of the formation of surfactant micelles under nonisothermal conditions by synchrotron SAXS. J. Am. Chem. Soc. 135, 7214–7222 (2013).

    Google Scholar 

  329. 329.

    Kalkowski, J. et al. In situ measurements of polymer micellization kinetics with millisecond temporal resolution. Macromolecules 52, 3151–3157 (2019).

    ADS  Google Scholar 

  330. 330.

    Hayward, D. W., Chiappisi, L., Prévost, S., Schweins, R. & Gradzielski, M. A small-angle neutron scattering environment for in-situ observation of chemical processes. Sci. Rep. 8, 7299 (2018).

    ADS  Google Scholar 

  331. 331.

    Keidel, R. et al. Time-resolved structural evolution during the collapse of responsive hydrogels: the microgel-to-particle transition. Sci. Adv. 4, eaao7086 (2018).

    ADS  Google Scholar 

  332. 332.

    Sankhala, K. et al. Self-assembly of block copolymers during hollow fiber spinning: an in situ small-angle X-ray scattering study. Nanoscale 11, 7634–7647 (2019).

    Google Scholar 

  333. 333.

    Räntzsch, V. et al. Polymer crystallization studied by hyphenated rheology techniques: rheo-NMR, rheo-SAXS, and rheo-microscopy. Macromol. Mater. Eng. 304, 1800586 (2018).

    Google Scholar 

  334. 334.

    Zhu, P. et al. The segmental responses to orientation and relaxation of thermoplastic poly(ether-ester) elastomer during cyclic deformation: an in-situ WAXD/SAXS study. Polymer 188, 122120 (2020).

    Google Scholar 

  335. 335.

    McCulloch, B. et al. Dynamics of magnetic alignment in rod–coil block copolymers. Macromolecules 46, 4462–4471 (2013).

    ADS  Google Scholar 

  336. 336.

    Schaff, F. et al. Six-dimensional real and reciprocal space small-angle X-ray scattering tomography. Nature 527, 353–356 (2015).

    ADS  Google Scholar 

  337. 337.

    Liebi, M. et al. Nanostructure surveys of macroscopic specimens by small-angle scattering tensor tomography. Nature 527, 349–352 (2015).

    ADS  Google Scholar 

  338. 338.

    Conceição, A. L. C., Perlich, J., Haas, S. & Funari, S. S. SAXS-CT: a nanostructure resolving microscopy for macroscopic biologic specimens. Biomed. Phys. Eng. Express 6, 035012 (2020).

    Google Scholar 

  339. 339.

    Lee, S. et al. In situ study of ABC triblock terpolymer self-assembly under solvent vapor annealing. Macromolecules 52, 1853–1863 (2019).

    ADS  Google Scholar 

  340. 340.

    Meyer, A. et al. In situ grazing-incidence small-angle X-ray scattering observation of block-copolymer templated formation of magnetic nanodot arrays and their magnetic properties. Nano Res. 10, 456–471 (2017).

    Google Scholar 

  341. 341.

    Li, Z. et al. Linking experiment and theory for three-dimensional networked binary metal nanoparticle–triblock terpolymer superstructures. Nat. Commun. 5, 3247 (2014).

    ADS  Google Scholar 

  342. 342.

    Li, T., Senesi, A. J. & Lee, B. Small angle X-ray scattering for nanoparticle research. Chem. Rev. 116, 11128–11180 (2016).

    Google Scholar 

  343. 343.

    Liu, X. et al. Effects of geometric confinement on caging and dynamics of polymer-tethered nanoparticle suspensions. Macromolecules 54, 426–439 (2021).

    ADS  Google Scholar 

  344. 344.

    Kusano, T. et al. Interplay between interparticle potential and adsorption structure in nanoparticle dispersions with polymer addition as displayed by small-angle scattering. Langmuir 37, 7503–7512 (2021).

    Google Scholar 

  345. 345.

    Toso, S. et al. Multilayer diffraction reveals that colloidal superlattices approach the structural perfection of single crystals. ACS Nano 15, 6243–6256 (2021).

    Google Scholar 

  346. 346.

    Dey, A. et al. State of the art and prospects for halide perovskite nanocrystals. ACS Nano 15, 10775–10981 (2021).

    Google Scholar 

  347. 347.

    De Geuser, F. & Deschamps, A. Precipitate characterisation in metallic systems by small-angle X-ray or neutron scattering. Comptes Rendus Phys. 13, 246–256 (2012).

    ADS  Google Scholar 

  348. 348.

    Preston, G. The diffraction of X-rays by age-hardening aluminium copper alloys. Proc. R. Soc. A Math. Phys. Sci. 167, 526–538 (1938).

    ADS  Google Scholar 

  349. 349.

    Zhang, F. et al. In situ structural characterization of ageing kinetics in aluminum alloy 2024 across angstrom-to-micrometer length scales. Acta Mater. 111, 385–398 (2016).

    ADS  Google Scholar 

  350. 350.

    Mathon, M. H. et al. A SANS investigation of the irradiation-enhanced α–α′ phases separation in 7–12 Cr martensitic steels. J. Nucl. Mater. 312, 236–248 (2003).

    ADS  Google Scholar 

  351. 351.

    Kelly, T. F. in Springer Handbook of Microscopy (eds Hawkes, P. W. & Spence, J. C. H.) 715–763 (Springer International, 2019).

  352. 352.

    Briggs, S. A. et al. A combined APT and SANS investigation of α′ phase precipitation in neutron-irradiated model FeCrAl alloys. Acta Mater. 129, 217–228 (2017).

    ADS  Google Scholar 

  353. 353.

    Ohnuma, M. et al. A new method for the quantitative analysis of the scale and composition of nanosized oxide in 9Cr-ODS steel. Acta Mater. 57, 5571–5581 (2009).

    ADS  Google Scholar 

  354. 354.

    Anovitz, L. M. & Cole, D. R. Characterization and analysis of porosity and pore structures. Rev. Mineral. Geochem. 80, 61–164 (2015).

    Google Scholar 

  355. 355.

    Hall, P. L., Mildner, D. F. R. & Borst, R. L. Small-angle scattering studies of the pore spaces of shaly rocks. J. Geophys. Res. Solid. Earth 91, 2183–2192 (1986).

    Google Scholar 

  356. 356.

    Radlinski, A. P., Boreham, C. J., Wignall, G. D. & Lin, J. S. Microstructural evolution of source rocks during hydrocarbon generation: a small-angle-scattering study. Phys. Rev. B 53, 14152–14160 (1996).

    ADS  Google Scholar 

  357. 357.

    Anovitz, L. M. et al. A new approach to quantification of metamorphism using ultra-small and small angle neutron scattering. Geochim. Cosmochim. Acta 73, 7303–7324 (2009).

    ADS  Google Scholar 

  358. 358.

    Clarkson, C. R. et al. Pore structure characterization of North American shale gas reservoirs using USANS/SANS, gas adsorption, and mercury intrusion. Fuel 103, 606–616 (2013).

    Google Scholar 

  359. 359.

    Allen, A. J., Thomas, J. J. & Jennings, H. M. Composition and density of nanoscale calcium–silicate–hydrate in cement. Nat. Mater. 6, 311–316 (2007).

    ADS  Google Scholar 

  360. 360.

    Allen, A. Time-resolved phenomena in cements, clays and porous rocks. J. Appl. Cryst. 24, 624–634 (1991).

    Google Scholar 

  361. 361.

    Thomas, J. J., Jennings, H. M. & Allen, A. J. The surface area of hardened cement paste as measured by various techniques. Concr. Sci. Eng. 1, 45–64 (1999).

    Google Scholar 

  362. 362.

    Paul, T. et al. Quantification of thermal oxidation in metallic glass powder using ultra-small angle X-ray scattering. Sci. Rep. 9, 6836 (2019).

    ADS  Google Scholar 

  363. 363.

    Lee, S. et al. Oxidative decomposition of methanol on subnanometer palladium clusters: the effect of catalyst size and support composition. J. Phys. Chem. C. 114, 10342–10348 (2010).

    Google Scholar 

  364. 364.

    Lee, S., Lee, B., Seifert, S., Vajda, S. & Winans, R. E. Simultaneous measurement of X-ray small angle scattering, absorption and reactivity: a continuous flow catalysis reactor. Nucl. Instrum. Methods Phys. Res. A Accel. Spectrom. Detect. Assoc. Equip. 649, 200–203 (2011).

    ADS  Google Scholar 

  365. 365.

    Jiang, Z., Lee, D. R., Narayanan, S., Wang, J. & Sinha, S. K. Waveguide-enhanced grazing-incidence small-angle X-ray scattering of buried nanostructures in thin films. Phys. Rev. B 84, 075440 (2011).

    ADS  Google Scholar 

  366. 366.

    Trewhella, J. et al. Report of the wwPDB Small-Angle Scattering Task Force: data requirements for biomolecular modeling and the PDB. Structure 21, 875–881 (2013).

    Google Scholar 

  367. 367.

    Trewhella, J. et al. 2017 publication guidelines for structural modelling of small-angle scattering data from biomolecules in solution: an update. Acta Crystallogr. D Struct. Biol. 73, 710–728 (2017).

    Google Scholar 

  368. 368.

    Pearson, K. X. On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling. London, Edinburgh Dublin Philos. Mag. J. Sci. 50, 157–175 (1900).

    MATH  Google Scholar 

  369. 369.

    Kikhney, A. G., Borges, C. R., Molodenskiy, D. S., Jeffries, C. M. & Svergun, D. I. SASBDB: towards an automatically curated and validated repository for biological scattering data. Protein Sci. 29, 66–75 (2020).

    Google Scholar 

  370. 370.

    Valentini, E., Kikhney, A. G., Previtali, G., Jeffries, C. M. & Svergun, D. I. SASBDB, a repository for biological small-angle scattering data. Nucleic Acids Res. 43, D357–D363 (2015).

    Google Scholar 

  371. 371.

    Varadi, M. et al. pE-DB: a database of structural ensembles of intrinsically disordered and of unfolded proteins. Nucleic Acids Res. 42, D326–D335 (2014).

    Google Scholar 

  372. 372.

    Vallat, B., Webb, B., Westbrook, J. D., Sali, A. & Berman, H. M. Development of a prototype system for archiving integrative/hybrid structure models of biological macromolecules. Structure 26, 894–904.e2 (2018).

    Google Scholar 

  373. 373.

    Ward, A. B., Sali, A. & Wilson, I. A. Biochemistry. Integrative structural biology. Science 339, 913–915 (2013).

    ADS  Google Scholar 

  374. 374.

    Sali, A. et al. Outcome of the First wwPDB Hybrid/Integrative Methods Task Force Workshop. Structure 23, 1156–1167 (2015).

    Google Scholar 

  375. 375.

    Hall, S. R., Allen, F. H. & Brown, I. D. The Crystallographic Information File (CIF): a new standard archive file for crystallography. Acta Crystallogr. Sect. A 47, 655–685 (1991).

    Google Scholar 

  376. 376.

    Adams, P. D. et al. Announcing mandatory submission of PDBx/mmCIF format files for crystallographic depositions to the Protein Data Bank (PDB). Acta Crystallogr. Sect. D. 75, 451–454 (2019).

    Google Scholar 

  377. 377.

    Kachala, M., Westbrook, J. & Svergun, D. Extension of the sasCIF format and its applications for data processing and deposition. J. Appl. Cryst. 49, 302–310 (2016).

    Google Scholar 

  378. 378.

    Konnecke, M. et al. The NeXus data format. J. Appl. Cryst. 48, 301–305 (2015).

    Google Scholar 

  379. 379.

    Jemian, P. R. et al. NXcanSAS: standard to store reduced SAS data of any dimension. Acta Crystallogr. Sect. A 73, C1442 (2017).

    Google Scholar 

  380. 380.

    Petoukhov, M. V. & Svergun, D. I. Ambiguity assessment of small-angle scattering curves from monodisperse systems. Acta Crystallogr. Sect. D 71, 1051–1058 (2015).

    Google Scholar 

  381. 381.

    Stachowski, T. R., Snell, M. E. & Snell, E. H. SAXS studies of X-ray induced disulfide bond damage: engineering high-resolution insight from a low-resolution technique. PLoS ONE 15, e0239702 (2020).

    Google Scholar 

  382. 382.

    Brooks-Bartlett, J. C. et al. Development of tools to automate quantitative analysis of radiation damage in SAXS experiments. J. Synchrotron Radiat. 24, 63–72 (2017).

    Google Scholar 

  383. 383.

    Hopkins, J. B. & Thorne, R. E. Quantifying radiation damage in biomolecular small-angle X-ray scattering. J. Appl. Cryst. 49, 880–890 (2016).

    Google Scholar 

  384. 384.

    Jeffries, C. M., Graewert, M. A., Svergun, D. I. & Blanchet, C. E. Limiting radiation damage for high-brilliance biological solution scattering: practical experience at the EMBL P12 beamline PETRAIII. J. Synchrotron Radiat. 22, 273–279 (2015).

    Google Scholar 

  385. 385.

    Castellví, A., Pascual-Izarra, C., Crosas, E., Malfois, M. & Juanhuix, J. Improving data quality and expanding BioSAXS experiments to low-molecular-weight and low-concentration protein samples. Acta Crystallogr. D 76, 971–981 (2020).

    Google Scholar 

  386. 386.

    Kirby, N. et al. Improved radiation dose efficiency in solution SAXS using a sheath flow sample environment. Acta Crystallogr. D 72, 1254–1266 (2016).

    Google Scholar 

  387. 387.

    Ryan, T. M. et al. An optimized SEC-SAXS system enabling high X-ray dose for rapid SAXS assessment with correlated UV measurements for biomolecular structure analysis. J. Appl. Cryst. 51, 97–111 (2018).

    Google Scholar 

  388. 388.

    Kuwamoto, S., Akiyama, S. & Fujisawa, T. Radiation damage to a protein solution, detected by synchrotron X-ray small-angle scattering: dose-related considerations and suppression by cryoprotectants. J. Synchrotron Radiat. 11, 462–468 (2004).

    Google Scholar 

  389. 389.

    Lopez, C. G. et al. Microfluidic devices for small-angle neutron scattering. J. Appl. Cryst. 51, 570–583 (2018).

    Google Scholar 

  390. 390.

    Ghosh, R. E. & Rennie, A. R. Assessment of detector calibration materials for SANS experiments. J. Appl. Cryst. 32, 1157–1163 (1999).

    Google Scholar 

  391. 391.

    Do, C. et al. Understanding inelastically scattered neutrons from water on a time-of-flight small-angle neutron scattering (SANS) instrument. Nucl. Instrum. Methods Phys. Res. A: Accel. Spectrom. Detect. Assoc. Equip. 737, 42–46 (2014).

    ADS  Google Scholar 

  392. 392.

    Oba, Y. et al. Magnetic scattering in the simultaneous measurement of small-angle neutron scattering and Bragg edge transmission from steel. J. Appl. Cryst. 49, 1659–1664 (2016).

    Google Scholar 

  393. 393.

    Tremsin, A. S. et al. Energy-resolved neutron imaging options at a small angle neutron scattering instrument at the Australian Center for Neutron Scattering. Rev. Sci. Instrum. 90, 035114 (2019).

    ADS  Google Scholar 

  394. 394.

    Rubinson, K. A., Stanley, C. & Krueger, S. Small-angle neutron scattering and the errors in protein structures that arise from uncorrected background and intermolecular interactions. J. Appl. Cryst. 41, 456–465 (2008).

    Google Scholar 

  395. 395.

    Jacrot, B. & Zaccai, G. Determination of molecular weight by neutron scattering. Biopolymers 20, 2413–2426 (1981).

    Google Scholar 

  396. 396.

    Chen, P.-c, Masiewicz, P., Perez, K. & Hennig, J. Structure-based screening of binding affinities via small-angle X-ray scattering. IUCrJ 7, 644–655 (2020).

    Google Scholar 

  397. 397.

    Inoue, R. et al. Newly developed laboratory-based size exclusion chromatography small-angle X-ray scattering system (La-SSS). Sci. Rep. 9, 12610 (2019).

    ADS  Google Scholar 

  398. 398.

    Brookes, E., Vachette, P., Rocco, M. & Perez, J. US-SOMO HPLC-SAXS module: dealing with capillary fouling and extraction of pure component patterns from poorly resolved SEC–SAXS data. J. Appl. Cryst. 49, 1827–1841 (2016).

    Google Scholar 

  399. 399.

    Meisburger, S. P., Xu, D. & Ando, N. REGALS: a general method to deconvolve X-ray scattering data from evolving mixtures. IUCrJ 8, 225–237 (2021).

    Google Scholar 

  400. 400.

    Baxter, R. J. Percus–Yevick equation for hard spheres with surface adhesion. J. Chem. Phys. 49, 2770–2774 (1968).

    ADS  Google Scholar 

  401. 401.

    De Kruif, C. G. et al. Adhesive hard-sphere colloidal dispersions. A small-angle neutron-scattering study of stickiness and the structure factor. Langmuir 5, 422–428 (1989).

    Google Scholar 

  402. 402.

    Derjaguin, B. V., Churaev, N. V. & Muller, V. M. in Surface Forces 293–310 (Springer, 1987).

  403. 403.

    Kotlarchyk, M., Stephens, R. B. & Huang, J. S. Study of Schultz distribution to model polydispersity of microemulsion droplets. J. Phys. Chem. 92, 1533–1538 (1988).

    Google Scholar 

  404. 404.

    Copley, J. The significance of multiple scattering in the interpretation of small-angle neutron scattering experiments. J. Appl. Cryst. 21, 639–644 (1988).

    Google Scholar 

  405. 405.

    Jensen, G. V. & Barker, J. G. Effects of multiple scattering encountered for various small-angle scattering model functions. J. Appl. Cryst. 51, 1455–1466 (2018).

    Google Scholar 

  406. 406.

    Frielinghaus, H. Strategies for removing multiple scattering effects revisited. Nucl. Instrum. Methods Phys. Res. A: Accel. Spectrom. Detect. Assoc. Equip. 904, 9–14 (2018).

    ADS  Google Scholar 

  407. 407.

    Allen, A. J. & Berk, N. F. Analysis of small-angle scattering data dominated by multiple scattering for systems containing eccentrically shaped particles or pores. J. Appl. Cryst. 27, 878–891 (1994).

    Google Scholar 

  408. 408.

    Folkertsma, L. et al. Synchrotron SAXS and impedance spectroscopy unveil nanostructure variations in redox-responsive porous membranes from poly(ferrocenylsilane) poly(ionic liquid)s. Macromolecules 50, 296–302 (2017).

    ADS  Google Scholar 

  409. 409.

    Metwalli, E. et al. A novel experimental approach for nanostructure analysis: simultaneous small-angle X-ray and neutron scattering. J. Appl. Cryst. 53, 722–733 (2020).

    Google Scholar 

  410. 410.

    Koczwara, C. et al. Towards real-time ion-specific structural sensitivity in nanoporous carbon electrodes using in situ anomalous small-angle X-ray scattering. ACS Appl. Mater. Interfaces 11, 42214–42220 (2019).

    Google Scholar 

  411. 411.

    Martinez, N. et al. Real time monitoring of water distribution in an operando fuel cell during transient states. J. Power Sour. 365, 230–234 (2017).

    ADS  Google Scholar 

  412. 412.

    Morin, A. et al. Quantitative multi-scale operando diagnosis of water localization inside a fuel cell. J. Electrochem. Soc. 164, F9–F21 (2016).

    Google Scholar 

  413. 413.

    Martinez, N. et al. Combined operando high resolution SANS and neutron imaging reveals in-situ local water distribution in an operating fuel cell. ACS Appl. Energy Mater. 2, 8425–8433 (2019).

    Google Scholar 

  414. 414.

    Bauer, P. S. et al. In-situ aerosol nanoparticle characterization by small angle X-ray scattering at ultra-low volume fraction. Nat. Commun. 10, 1122 (2019).

    ADS  Google Scholar 

  415. 415.

    Guizar-Sicairos, M., Georgiadis, M. & Liebi, M. Validation study of small-angle X-ray scattering tensor tomography. J. Synchrotron Radiat. 27, 779–787 (2020).

    Google Scholar 

  416. 416.

    Lee, J. et al. In situ measurement of ionomer water content and liquid water saturation in fuel cell catalyst layers by high-resolution small-angle neutron scattering. ACS Appl. Energy Mater. 3, 8393–8401 (2020).

    ADS  Google Scholar 

  417. 417.

    Lahey-Rudolph, J. M. et al. Rapid screening of in cellulo grown protein crystals via a small-angle X-ray scattering/X-ray powder diffraction synergistic approach. J. Appl. Cryst. 53, 1169–1180 (2020).

    Google Scholar 

  418. 418.

    Cai, J., Townsend, J. P., Dodson, T. C., Heiney, P. A. & Sweeney, A. M. Eye patches: protein assembly of index-gradient squid lenses. Science 357, 564–569 (2017).

    ADS  Google Scholar 

  419. 419.

    Yip, K. M., Fischer, N., Paknia, E., Chari, A. & Stark, H. Atomic-resolution protein structure determination by cryo-EM. Nature 587, 157–161 (2020).

    ADS  Google Scholar 

  420. 420.

    Nakane, T. et al. Single-particle cryo-EM at atomic resolution. Nature 587, 152–156 (2020).

    ADS  Google Scholar 

  421. 421.

    Amann-Winkel, K. et al. X-ray and neutron scattering of water. Chem. Rev. 116, 7570–7589 (2016).

    Google Scholar 

  422. 422.

    Kim, HenryS. et al. SAXS/SANS on supercharged proteins reveals residue-specific modifications of the hydration shell. Biophys. J. 110, 2185–2194 (2016).

    ADS  Google Scholar 

  423. 423.

    Gabel, F. & Bellissent-Funel, M.-C. C-Phycocyanin hydration water dynamics in the presence of trehalose: an incoherent elastic neutron scattering study at different energy resolutions. Biophys. J. 92, 4054–4063 (2007).

    ADS  Google Scholar 

  424. 424.

    Schirò, G. et al. Translational diffusion of hydration water correlates with functional motions in folded and intrinsically disordered proteins. Nat. Commun. 6, 6490 (2015).

    ADS  Google Scholar 

  425. 425.

    Niskanen, J. et al. Compatibility of quantitative X-ray spectroscopy with continuous distribution models of water at ambient conditions. Proc. Natl Acad. Sci. USA 116, 4058–4063 (2019).

    Google Scholar 

  426. 426.

    Bu, W. & Schlossman, M. L. in Synchrotron Light Sources and Free-Electron Lasers: Accelerator Physics, Instrumentation and Science Applications (eds Jaeschke, E. J., Khan, S., Schneider, J. R. & Hastings, J. B.) 1897–1933 (Springer International, 2020).

  427. 427.

    Franke, D. et al. ATSAS 2.8: a comprehensive data analysis suite for small-angle scattering from macromolecular solutions. J. Appl. Cryst. 50, 1212–1225 (2017).

    Google Scholar 

  428. 428.

    Grishaev, A., Guo, L., Irving, T. & Bax, A. Improved fitting of solution X-ray scattering data to macromolecular structures and structural ensembles by explicit water modeling. J. Am. Chem. Soc. 132, 15484–15486 (2010).

    Google Scholar 

  429. 429.

    Hopkins, J. B., Gillilan, R. E. & Skou, S. BioXTAS RAW: improvements to a free open-source program for small-angle X-ray scattering data reduction and analysis. J. Appl. Cryst. 50, 1545–1553 (2017).

    Google Scholar 

  430. 430.

    Perkins, S. J. et al. Atomistic modelling of scattering data in the Collaborative Computational Project for Small Angle Scattering (CCP-SAS). J. Appl. Cryst. 49, 1861–1875 (2016).

    Google Scholar 

  431. 431.

    Ginsburg, A. et al. D+: software for high-resolution hierarchical modeling of solution X-ray scattering from complex structures. J. Appl. Cryst. 52, 219–242 (2019).

    Google Scholar 

  432. 432.

    Filik, J. et al. Processing two-dimensional X-ray diffraction and small-angle scattering data in DAWN 2. J. Appl. Cryst. 50, 959–966 (2017).

    Google Scholar 

  433. 433.

    Benecke, G. et al. A customizable software for fast reduction and analysis of large X-ray scattering data sets: applications of the new DPDAK package to small-angle X-ray scattering and grazing-incidence small-angle X-ray scattering. J. Appl. Cryst. 47, 1797–1803 (2014).

    Google Scholar 

  434. 434.

    Hammersley, A. FIT2D: a multi-purpose data reduction, analysis and visualization program. J. Appl. Cryst. 49, 646–652 (2016).

    Google Scholar 

  435. 435.

    Dewhurst, C. GRASP: Graphical Reduction and Analysis SANS Program for Matlab. ILL Neutrons for Society (2003).

  436. 436.

    Bergmann, A., Fritz, G. & Glatter, O. Solving the generalized indirect Fourier transformation (GIFT) by Boltzmann simplex simulated annealing (BSSA). J. Appl. Cryst. 33, 1212–1216 (2000).

    Google Scholar 

  437. 437.

    Jiang, Z. GIXSGUI: a MATLAB toolbox for grazing-incidence X-ray scattering data visualization and reduction, and indexing of buried three-dimensional periodic nanostructured films. J. Appl. Cryst. 48, 917–926 (2015).

    Google Scholar 

  438. 438.

    Ilavsky, J. & Jemian, P. R. Irena: tool suite for modeling and analysis of small-angle scattering. J. Appl. Cryst. 42, 347–353 (2009).

    Google Scholar 

  439. 439.

    Arnold, O. et al. Mantid — data analysis and visualization package for neutron scattering and μ SR experiments. Nucl. Instrum. Methods Phys. Res. A Accel. Spectrom. Detect. Assoc. Equip. 764, 156–166 (2014).

    ADS  Google Scholar 

  440. 440.

    Ilavsky, J. Nika: software for two-dimensional data reduction. J. Appl. Cryst. 45, 324–328 (2012).

    Google Scholar 

  441. 441.

    Kline, S. Reduction and analysis of SANS and USANS data using IGOR Pro. J. Appl. Cryst. 39, 895–900 (2006).

    Google Scholar 

  442. 442.

    Förster, S., Apostol, L. & Bras, W. Scatter: software for the analysis of nano- and mesoscale small-angle scattering. J. Appl. Cryst. 43, 639–646 (2010).

    Google Scholar 

  443. 443.

    Breßler, I., Kohlbrecher, J. & Thunemann, A. F. SASfit: a tool for small-angle scattering data analysis using a library of analytical expressions. J. Appl. Cryst. 48, 1587–1598 (2015).

    Google Scholar 

  444. 444.

    Liu, C.-H. et al. sasPDF: pair distribution function analysis of nanoparticle assemblies from small-angle scattering data. J. Appl. Cryst. 53, 699–709 (2020).

    Google Scholar 

  445. 445.

    Brookes, E. & Rocco, M. Recent advances in the UltraScan SOlution MOdeller (US-SOMO) hydrodynamic and small-angle scattering data analysis and simulation suite. Eur. Biophys. J. 47, 855–864 (2018).

    Google Scholar 

  446. 446.

    Knight, C. J. & Hub, J. S. WAXSiS: a web server for the calculation of SAXS/WAXS curves based on explicit-solvent molecular dynamics. Nucleic Acids Res. 43, W225–W230 (2015).

    Google Scholar 

  447. 447.

    With, S. et al. Fast diffusion-limited lyotropic phase transitions studied in situ using continuous flow microfluidics/microfocus-SAXS. Langmuir 30, 12494–12502 (2014).

    Google Scholar 

  448. 448.

    Konishi, T. et al. Origin of SAXS intensity in the low-q region during the early stage of polymer crystallization from both the melt and glassy state. Phys. Rev. Mater. 2, 105602 (2018).

    Google Scholar 

  449. 449.

    Fleury, G. et al. Micellar-mediated block copolymer ordering dynamics revealed by in situ grazing incidence small-angle X-ray scattering during spin coating. Adv. Funct. Mater. 29, 1806741 (2019).

    Google Scholar 

  450. 450.

    Mable, C. J. et al. Time-resolved SAXS studies of the kinetics of thermally triggered release of encapsulated silica nanoparticles from block copolymer vesicles. Macromolecules 50, 4465–4473 (2017).

    ADS  Google Scholar 

  451. 451.

    Shim, J., Bates, F. S. & Lodge, T. P. Bicontinuous microemulsions in partially charged ternary polymer blends. ACS Macro Lett. 8, 1166–1171 (2019).

    Google Scholar 

  452. 452.

    Tashiro, K. & Yamamoto, H. Structural evolution mechanism of crystalline polymers in the isothermal melt-crystallization process: a proposition based on simultaneous WAXD/SAXS/FTIR measurements. Polymers 11, 1316 (2019).

    Google Scholar 

  453. 453.

    Samant, S. et al. Ordering pathway of block copolymers under dynamic thermal gradients studied by in situ GISAXS. Macromolecules 49, 8633–8642 (2016).

    ADS  Google Scholar 

  454. 454.

    Yu, F. et al. Block copolymer self-assembly-directed and transient laser heating-enabled nanostructures toward phononic and photonic quantum materials. ACS Nano 14, 11273–11282 (2020).

    Google Scholar 

  455. 455.

    Lopez, C. G., Watanabe, T., Martel, A., Porcar, L. & Cabral, J. T. Microfluidic-SANS: flow processing of complex fluids. Sci. Rep. 5, 7727 (2015).

    ADS  Google Scholar 

  456. 456.

    Philipp, M. et al. Sorption of water and initial stages of swelling of thin PNIPAM films using in situ GISAXS microfluidics. Langmuir 31, 9619–9627 (2015).

    Google Scholar 

  457. 457.

    Lin, Y.-J., Chuang, W.-T. & Hsu, S.-H. Gelation mechanism and structural dynamics of chitosan self-healing hydrogels by in situ SAXS and coherent X-ray scattering. ACS Macro Lett. 8, 1449–1455 (2019).

    Google Scholar 

  458. 458.

    Dunderdale, G. J., Davidson, S. J., Ryan, A. J. & Mykhaylyk, O. O. Flow-induced crystallisation of polymers from aqueous solution. Nat. Commun. 11, 3372 (2020).

    ADS  Google Scholar 

  459. 459.

    Schlenk, M. et al. Parallel and perpendicular alignment of anisotropic particles in free liquid microjets and emerging microdroplets. Langmuir 34, 4843–4851 (2018).

    Google Scholar 

  460. 460.

    Defebvin, J., Barrau, S., Stoclet, G., Rochas, C. & Lefebvre, J.-M. In situ SAXS/WAXS investigation of the structural evolution of poly(vinylidene fluoride) upon uniaxial stretching. Polymer 84, 148–157 (2016).

    Google Scholar 

  461. 461.

    Pepin, J., Gaucher, V., Rochas, C. & Lefebvre, J.-M. In-situ SAXS/WAXS investigations of the mechanically-induced phase transitions in semi-crystalline polyamides. Polymer 175, 87–98 (2019).

    Google Scholar 

  462. 462.

    Zhao, H. et al. A real-time WAXS and SAXS study of the structural evolution of LLDPE bubble. J. Polym. Sci. Part. B Polym. Phys. 56, 1404–1412 (2018).

    ADS  Google Scholar 

  463. 463.

    Cheshire, M. C. et al. Wellbore cement porosity evolution in response to mineral alteration during CO2 flooding. Environ. Sci. Technol. 51, 692–698 (2017).

    ADS  Google Scholar 

  464. 464.

    Coakley, J. et al. Precipitation processes in the β-titanium alloy Ti–5Al–5Mo–5V–3Cr. J. Alloy. Compd. 646, 946–953 (2015).

    Google Scholar 

  465. 465.

    Alauhdin, M. et al. Monitoring morphology evolution within block copolymer microparticles during dispersion polymerisation in supercritical carbon dioxide: a high pressure SAXS study. Polym. Chem. 10, 860–871 (2019).

    Google Scholar 

  466. 466.

    Bahadur, J. et al. Determination of closed porosity in rocks by small-angle neutron scattering. J. Appl. Cryst. 49, 2021–2030 (2016).

    Google Scholar 

  467. 467.

    Hu, T. et al. Nondestructive and quantitative characterization of bulk injection-molded polylactide using SAXS microtomography. Macromolecules 53, 6498–6509 (2020).

    ADS  Google Scholar 

  468. 468.

    Leu, L. et al. Multiscale description of shale pore systems by scanning SAXS and WAXS microscopy. Energy Fuels 30, 10282–10297 (2016).

    ADS  Google Scholar 

  469. 469.

    Reichardt, M. et al. X-ray structural analysis of single adult cardiomyocytes: tomographic imaging and microdiffraction. Biophys. J. 119, 1309–1323 (2020).

    ADS  Google Scholar 

  470. 470.

    Gopinadhan, M. et al. Controlling orientational order in block copolymers using low-intensity magnetic fields. Proc. Natl Acad. Sci. USA 114, E9437–E9444 (2017).

    Google Scholar 

  471. 471.

    Gu, Y. et al. Photoswitching topology in polymer networks with metal–organic cages as crosslinks. Nature 560, 65–69 (2018).

    ADS  Google Scholar 

  472. 472.

    Harada, M. & Katagiri, E. Mechanism of silver particle formation during photoreduction using in situ time-resolved SAXS analysis. Langmuir 26, 17896–17905 (2010).

    Google Scholar 

  473. 473.

    Bikondoa, O., Carbone, D., Chamard, V. & Metzger, T. H. Ageing dynamics of ion bombardment induced self-organization processes. Sci. Rep. 3, 1850 (2013).

    ADS  Google Scholar 

  474. 474.

    Olichwer, N., Meyer, A., Yesilmen, M. & Vossmeyer, T. Gold nanoparticle superlattices: correlating chemiresistive responses with analyte sorption and swelling. J. Mater. Chem. C 4, 8214–8225 (2016).

    Google Scholar 

  475. 475.

    Hejral, U., Müller, P., Balmes, O., Pontoni, D. & Stierle, A. Tracking the shape-dependent sintering of platinum–rhodium model catalysts under operando conditions. Nat. Commun. 7, 10964 (2016).

    ADS  Google Scholar 

  476. 476.

    Mochizuki, T. et al. Temperature- and humidity-controlled SAXS analysis of proton-conductive ionomer membranes for fuel cells. ChemSusChem 7, 729–733 (2014).

    Google Scholar 

  477. 477.

    Schwartzkopf, M. et al. Role of sputter deposition rate in tailoring nanogranular gold structures on polymer surfaces. ACS Appl. Mater. Interfaces 9, 5629–5637 (2017).

    Google Scholar 

  478. 478.

    Zhang, P. et al. Manipulating the assembly of spray-deposited nanocolloids: in situ study and monolayer film preparation. Langmuir 32, 4251–4258 (2016).

    Google Scholar 

  479. 479.

    Dudenas, P. J. & Kusoglu, A. Evolution of ionomer morphology from dispersion to film: an in situ X-ray study. Macromolecules 52, 7779–7785 (2019).

    ADS  Google Scholar 

  480. 480.

    Dendooven, J. et al. Mobile setup for synchrotron based in situ characterization during thermal and plasma-enhanced atomic layer deposition. Rev. Sci. Instrum. 87, 113905 (2016).

    ADS  Google Scholar 

  481. 481.

    Ruge, M., Golks, F., Zegenhagen, J., Magnussen, O. M. & Stettner, J. In operando GISAXS studies of mound coarsening in electrochemical homoepitaxy. Phys. Rev. Lett. 112, 055503 (2014).

    ADS  Google Scholar 

  482. 482.

    Schaffer, C. J. et al. Morphological degradation in low bandgap polymer solar cells — an in operando study. Adv. Energy Mater. 6, 1600712 (2016).

    Google Scholar 

  483. 483.

    Kabir, S. et al. Elucidating the dynamic nature of fuel cell electrodes as a function of conditioning: an ex situ material characterization and in situ electrochemical diagnostic study. ACS Appl. Mater. Interfaces 11, 45016–45030 (2019).

    Google Scholar 

  484. 484.

    Hattendorff, J., Seidlmayer, S., Gasteiger, H. A. & Gilles, R. Li-ion half-cells studied operando during cycling by small-angle neutron scattering. J. Appl. Cryst. 53, 210–221 (2020).

    Google Scholar 

  485. 485.

    Allen, A. J., Ilavsky, J., Jemian, P. R. & Braun, A. Evolution of electrochemical interfaces in solid oxide fuel cells (SOFC): a Ni and Zr resonant anomalous ultra-small-angle X-ray scattering study with elemental and spatial resolution across the cell assembly. RSC Adv. 4, 4676–4690 (2014).

    ADS  Google Scholar 

  486. 486.

    Milsom, A. et al. The persistence of a proxy for cooking emissions in megacities: a kinetic study of the ozonolysis of self-assembled films by simultaneous small & wide angle X-ray scattering (SAXS/WAXS) and Raman microscopy. Faraday Discuss. 226, 364–381 (2021).

    ADS  Google Scholar 

  487. 487.

    Geuchies, J. J. et al. In situ study of the formation mechanism of two-dimensional superlattices from PbSe nanocrystals. Nat. Mater. 15, 1248–1254 (2016).

    ADS  Google Scholar 

  488. 488.

    Yang, F. et al. Investigation of the interaction between nafion ionomer and surface functionalized carbon black using both ultrasmall angle X-ray scattering and cryo-TEM. ACS Appl. Mater. Interfaces 9, 6530–6538 (2017).

    Google Scholar 

  489. 489.

    Rebollar, E. et al. In situ monitoring of laser-induced periodic surface structures formation on polymer films by grazing incidence small-angle X-ray scattering. Langmuir 31, 3973–3981 (2015).

    Google Scholar 

  490. 490.

    Ilavsky, J. et al. Ultra-small-angle X-ray scattering at the advanced photon source. J. Appl. Cryst. 42, 469–479 (2009).

    Google Scholar 

  491. 491.

    Ilavsky, J. et al. Development of combined microstructure and structure characterization facility for in situ and operando studies at the Advanced Photon Source. J. Appl. Cryst. 51, 867–882 (2018).

    Google Scholar 

  492. 492.

    Blanton, T. N. et al. JCPDS — International Centre for Diffraction Data round robin study of silver behenate. A possible low-angle X-ray diffraction calibration standard. Powder Diffr. 10, 91–95 (1995).

    ADS  Google Scholar 

  493. 493.

    Paula, F. Ld. O. SAXS analysis of magnetic field influence on magnetic nanoparticle clusters. Condens. Matter 4, 55 (2019).

    Google Scholar 

  494. 494.

    Moore, P. Small-angle scattering. Information content and error analysis. J. Appl. Cryst. 13, 168–175 (1980).

    Google Scholar 

  495. 495.

    Svergun, D. Determination of the regularization parameter in indirect-transform methods using perceptual criteria. J. Appl. Cryst. 25, 495–503 (1992).

    Google Scholar 

  496. 496.

    Provencher, S. W. A constrained regularization method for inverting data represented by linear algebraic or integral equations. Computer Phys. Commun. 27, 213–227 (1982).

    ADS  Google Scholar 

  497. 497.

    Liao, F. et al. Self-assembly of a silicon-containing side-chain liquid crystalline block copolymer in bulk and in thin films: kinetic pathway of a cylinder to sphere transition. Nanoscale 11, 285–293 (2019).

    Google Scholar 

  498. 498.

    Andrews, R. N., Serio, J., Muralidharan, G. & Ilavsky, J. An in situ USAXS–SAXS–WAXS study of precipitate size distribution evolution in a model Ni-based alloy. J. Appl. Cryst. 50, 734–740 (2017).

    Google Scholar 

  499. 499.

    Piiadov, V., Ares de Araújo, E., Oliveira Neto, M., Craievich, A. F. & Polikarpov, I. SAXSMoW 2.0: online calculator of the molecular weight of proteins in dilute solution from experimental SAXS data measured on a relative scale. Protein Sci. 28, 454–463 (2019).

    Google Scholar 

  500. 500.

    Hajizadeh, N. R., Franke, D., Jeffries, C. M. & Svergun, D. I. Consensus Bayesian assessment of protein molecular mass from solution X-ray scattering data. Sci. Rep. 8, 7204 (2018).

    ADS  Google Scholar 

  501. 501.

    Rambo, R. P. & Tainer, J. A. Accurate assessment of mass, models and resolution by small-angle scattering. Nature 496, 477–481 (2013).

    ADS  Google Scholar 

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The authors acknowledge the support of iNEXT-Discovery, project number 871037, funded by the Horizon 2020 programme of the European Commission and the Bundesministerium für Bildung und Forschung (BMBF) grant 16QK10A ‘SAS-BSOFT’ (to D.I.S.).

Author information




Introduction (C.M.J., D.I.S., J.S.P. and A.V.S.); Experimentation (C.M.J., D.I.S., J.I., A.Ma., S.H., A.Me., J.S.P. and A.V.S.); Results (C.M.J., D.I.S., J.I., J.S.P. and A.V.S.); Applications (C.M.J., D.I.S., J.I., S.H. and A.Me.); Reproducibility and data deposition (C.M.J., D.I.S. and J.I.); Limitations and optimizations (C.M.J., D.I.S., J.I., A.Ma., S.H., A.Me. and A.V.S.); Outlook (C.M.J., D.I.S., A.Ma. and A.V.S.); Overview of the Primer (D.I.S. and C.M.J.).

Corresponding authors

Correspondence to Cy M. Jeffries or Dmitri I. Svergun.

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Nature Reviews Methods Primers thanks B. Abecassis, J. Lipfert, D. Zákutná and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Argonne National Laboratory SAXS and USAXS packages (Irena, Nika Indra):

As low as reasonably achievable (ALARA):,time%2C%20distance%2C%20and%20shielding


canSAS initiative:

CCP-SAS web project:

ESRF FIT2D home page:

ESRF SAXS program package:

ESRF SAXS software tools:

ESRF Scatter program for data analysis:

European Molecular Biology Laboratory ATSAS data analysis software:

European Molecular Biology Laboratory ATSAS online and web services:


GISAXS Community Website:

GISAXS Community Website from the University of Hamburg:

GISAXS Community Website table of contents:




Institut Laue-Langevin (ILL) - GRASP SANS analysis and data reduction:



MULCh: ModULes for the analysis of Contrast variation data:

NIST Neutron activation and scattering calculator:

NIST SANS and USANS data reduction and analysis software:

NIST Scattering Length Density Calculator:

NIST XCOM: Photon Cross Sections Database:

NIST X-Ray Mass Attenuation Coefficients:

OneDep System:

Paul Scherrer Institute, PSI, SASfit package:

PDB-Dev databank:


Protein Ensemble Database:

Resources from the National Institute of Standards and Technology:

SAS Portal:



Saxier SAS forum:

SAXSMoW (SAXS Molecular Weight):

ScÅtter, Lawrence Berkeley National Laboratory, Advanced Light Source:


Small Angle Scattering Biological Data Bank (SASBDB):

UniProt nomenclature:




Summed convolution

The scattering intensities, I(q), may be conceptualized as the squared sum of the scattering wave amplitudes emanating from each scattering point. For coherent-elastic scattering events, if the distances between the scattering centres are spatially correlated, then the magnitude of the final scattered wave amplitudes (Fig. 1b) is dependent on the number and distribution of individual scattering centres and their respective scattering ‘power’ (the scattering length density).

Transmission geometry

A small-angle scattering (SAS) instrument configuration in which an incident X-ray or neutron beam travels through a sample that is placed in the beam path (typically perpendicular to the incident beam direction). The level at which the incident beam transmits through the sample is determined by numerous factors including the X-ray or neutron energy, the absorption and scattering properties of the sample and its thickness.

Grazing incidence

A small-angle scattering (SAS) instrument configuration in which an incident X-ray or neutron beam is directed at a very low incoming incident angle (the grazing incidence angle) towards a sample that is deposited on a surface (Fig. 2). The incident beam and the reflected beam generate scattering events that are dependent on numerous factors including the incident X-ray or neutron wavelength, the absorption and scattering properties of the sample, and the tilt angle of the sample surface relative to the incident beam.

Form factor

A term describing the squared magnitude of the q-dependent coherent scattering amplitudes arising from regions of excess scattering length density after background scattering contributions have been subtracted. The form factor represents scattering intensities from the distribution of distances between spatially correlated scattering centres within the particle and does not account for the distribution/interactions between the particles, which are described by the structure factor.

Radius of gyration

(Rg). The root mean squared distance calculated from the centre of contrast (typically the centre of mass).

Probable frequency of real-space distances p(r)

(Otherwise known as pair-distance distribution function). The inverse Fourier transform of the form factor that converts the reciprocal space scattering I(q) versus q into a frequency distribution, p, of real-space distances, r; that is, p(r) versus r.


The process of applying proton bombardment to eject fragments from heavy metal target materials. Used to produce high-flux neutron beams without nuclear fission chain reaction.

Bragg reflections

Reflections that occur for periodic structures with a spacing d (such as crystal matrices) at a scattering angle θ that is described by the Bragg relation, 2dsinθ = , where n is a positive integer and λ the radiation wavelength. This can be observed in bulk or for ordered materials deposited on surfaces when a grazing incidence beam illuminates a 2D lattice with well-defined symmetry-related periodicity.

Core–shell and multiple-shell particles

Particles consisting of contiguous but spatially distinct layered regions of different average scattering length density. For example, a detergent micelle that in water forms an external layer of higher electron density (the hydrophilic heads) surrounded by a less electron-dense core (the hydrophobic tails).


Systems containing a distribution of sizes, or displaying a level of non-uniformity of structural states. For example, a monomer–dimer particle equilibrium (a mixture of different molecular weights) or disordered polymers (that may have the same molecular weight but, when viewed as a population, sample different conformations in solution).

Centrosymmetrical potentials

Energy potentials that are distributed symmetrically with respect to a central point.

Mean-free path

The average distance travelled between successive collisions of an X-ray or neutron with the atoms of a material, which modifies the direction or energy of the X-rays or neutrons (for example, between multiple scattering events).

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Jeffries, C.M., Ilavsky, J., Martel, A. et al. Small-angle X-ray and neutron scattering. Nat Rev Methods Primers 1, 70 (2021).

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