Scherrer formula: estimation of error in determining small nanoparticle size
https://doi.org/10.17586/2220-8054-2018-9-3-364-369
Abstract
The lower limit of the applicability of the Scherrer formula has been established by calculating the diffraction patterns from model nanoparticles by the Debye formula. Particle size was calculated using the Scherrer formula for different hkl-peaks. The obtained data of particle sizes were compared with “real” sizes of model particles in the same hkl-directions. The form-factor Khkl was analyzed as main correction of Scherrer formula. It was shown that the Scherrer formula error increases nonlinearly at particle sizes less than 4 nm. For any hkl direction, the absolute error of average particle size determination using formula does not exceed 0.3 nm. Analysis shows that average particle size can be determined by Scherrer formula from single diffraction peak of experimental pattern for center-symmetrical particles.
Supporting agencies: The author thanks E. V. Shalaeva for useful discussion and A. L. Syuzyumova for English translation of the article. This work was supported by the Russian Science Foundation (project No. 17-79-20165) and performed at the Institute of Solid State Chemistry UrB RAS.
About the Author
A. S. Vorokh
Institute of Solid State Chemistry of the Ural Branch of the Russian Academy of Sciences
Russian Federation
Russian Federation
91 Pervomaiskaya st., Ekaterinburg
References
1. Ingham B. X-ray scattering characterisation of nanoparticles. Crystallography Reviews, 2015, 21 (4), P. 229–303.
2. Muniz F.T.L., Miranda M.A.R., dos Santos C.M., Sasaki J.M. The Scherrer equation and the dynamical theory of X-ray diffraction. Acta Crystallographica A: Found. & Adv., 2016, 72 (3), P. 385–390.
3. Miranda M.A.R., Sasaki J.M. The limit of application of the Scherrer equation. Acta Crystallographica A: Found. & Adv., 2018, 74 (1), P. 54–65.
4. Uvarov V., Popov I. Metrological characterization of X-ray diffraction methods at different acquisition geometries for determination of crystallite size in nano-scale materials. Materials Characterization, 2013, 85, P. 111–123.
5. Uvarov V., Popov I. Metrological characterization of X-ray diffraction methods for determination of crystallite size in nano-scale materials. Mater. Charact., 2007, 27, P. 883–891.
6. Scardi P., Billinge S.J.L., Neder R., Cervellino A. Celebrating 100 years of the Debye scattering equation. Acta Crystallographica A: Found. & Adv., 2016, 72 (6), P. 589–590.
7. Hall B.D., Zanchet D., Ugarte D. Estimating nanoparticle size from diffraction measurements. J. Appl. Cryst., 2000, 33, P. 1335–1341.
8. Rempel A.A., Vorokh A.S., Neder R., Magerl A. Disordered structure of cadmium sulphide nanoparticles. J. Surf. Investig. X-ray, Synchr. and Neutron Tech., 2011, 5 (6), P. 1028–1031.
9. Vorokh A.S., Rempel A.A. Direct-space visualization of the short and average long-range orders in the noncrystalline sctructure of a single cadmium sulfide nanoparticle. JETP Letters, 2010, 91 (2), P. 100–104.
10. Perez-Demydenko C., Scardi P. Diffraction peak profiles of surface relaxed spherical nanocrystals. Philosophical Magazine, 2017, 97 (26),P. 2317–2346.
11. Sestu M., Navarra G., et al. Whole-nanoparticle atomistic modeling of the schwertmannite structure from total scattering data. J. Appl. Cryst., 2017, 50, P. 1617–1626.
12. Tsybulya S.V., Yatsenko D.A. X-ray diffraction analysis of ultradisperse systems: The Debye formula. J. Struct. Chem., 2012, 53, S150–S165.
13. Yatsenko D., Tsybulya S. DIANNA (diffraction analysis of nanopowders) – a software for structural analysis of nanosized powders. Zeitschrift Fur Kristallographie – Crystalline Materials, 2018, 233 (1), P. 61–66.
14. Cervellino A., Frison R., Bertolotti F., Guagliardi A. DEBUSSY 2.0: the new release of a Debye user system for nanocrystalline and/or disordered materials. J. Appl. Cryst., 2015, 48 (6), P. 2026–2032.
15. Nikulina O., Yatsenko D., et al. Debye function analysis of nanocrystalline gallium oxide gamma-Ga2O3. Zeitschrift Fur Kristallographie – Crystalline Materials, 2016, 231 (5), P. 261–266.
16. Scherrer P. Bestimmung der Gr¨oße und der inneren Struktur von Kolloidteilchen mittels R¨ontgenstrahlen. Nachrichten Gesellschaft Wissenschaft Gottingen, 1918, 2, P. 98–100.
17. Seljakow N. Eine r¨ontgenographische Methode zur Messung der absoluten Dimensionen einzelner Kristalle in K¨orpern von fein kristallinischem Bau. Zeitschrift fuer Physik, 1925, 516, P. 439–444.
18. Langford J.I., Wilson A.J.C. Scherrer after Sixty Years: A Survey and Some New Results in the Determination of Crystallite Size. J. Appl. Cryst., 1978, 11, P. 102–113.
19. James R.W. Optical principles of the Diffraction of X Rays. G. Bell & Sons, 1948, 624 p.
20. Debye P. Zerstreuung von R¨ontgenstrahlen. Annalen der Physik B, 1915, 46, P. 809–823.
21. International Tables for X Ray crystallography IV, Birmingham, England, 1974, 366 p.
22. Mitra G.B. Moments and Cumulants of Diffraction Profiles Broadened by Stacking Faults. Journal of Crystallization Process and Technology, 2013, 3, P. 103–107.
23. Ectors D., Goetz-Neunhoeffer F., Neubauer J. Routine (an)isotropic crystallite size analysis in the double-Voigt approximation done right? Powder Diffraction, 2017, 32, S27–S34.
24. Ida T. New measures of sharpness for symmetric powder diffraction peak profiles. J. Appl. Cryst., 2008, 41, P. 393–401.
25. Li Z. Characterization of Different Shaped Nanocrystallites using X-ray Diffraction Line Profiles. Particle & Particle Systems Characterization, 2011, 28 (1–2), P. 19–24.
Review
For citations:
Vorokh A.S. Scherrer formula: estimation of error in determining small nanoparticle size. Nanosystems: Physics, Chemistry, Mathematics. 2018;9(3):364-369. https://doi.org/10.17586/2220-8054-2018-9-3-364-369
Views:
7
Email this article 