Scherrer formula: estimation of error in determining small nanoparticle size

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 K hkl 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.

NiCo alloy nanoparticles electrodeposited on an electrochemically reduced nitrogen-doped graphene oxide/carbon-ceramic electrode: a low cost electrocatalyst towards methanol and ethanol oxidation

“…where s is the mean size of the crystal, K is a dimensionless shape factor, with a value close to 1, l is the X-ray wavelength (0.15406 nm), b is the line broadening at half the maximum intensity (FWHM) and q is the Bragg angle. 52 The calculated average size of electrodeposited NiCo nanoparticles on the ErN-GO/CCE according to the diffraction peak of Ni (111) is collected as 23.25 nm which is in good agreement with the result of FESEM. NiCo alloy nanoparticles can remarkably decrease the formation of the inefficient g-NiOOH species and on the other hand stabilize the b-NiOOH form in the alkaline media.…”

sp amorphous carbons in view of multianalytical consideration: Normal, expeсted and new

“…However, such a change is big for CB624 only while the dispersion in maximum positions for the other three ACs can easily be attributed to standard statistical errors as well as to errors caused by the difference in the resolution functions of the neutron and X-ray spectrometers. The c CSR L values listed in Table 1 show much more pronounced XRPD/NDP difference, which indicates that the peak broadening is caused by not only size of BSU stacks, but other factors as well, such as inhomogeneous distribution of the stacks over their lateral size and thickness [18][19][20], the latter due to the difference of both BSU layers dimension and interlayer distance caused by changing chemical framing of the BSUs circumference (chemical gradient [19]). The obtained data evidently support a common opinion that the application of Scherrer's formula for obtaining size of nanostructure elements from experimental diffraction data has mainly an evaluative character [16,[18][19][20].…”

Hidrólisis de sacarosa por invertasa de Saccharomyces cerevisiae inmovilizada sobre nanopartículas magnéticas de ferrita de cobalto

“…Los espectros se obtuvieron en el rango de 5 -90° en 2θ. Se realizó refinamiento de Rietveld con la base de datos COD Cristalographic data y el cálculo del tamaño de partícula mediante la Ecuación 2 de Debye-Scherrer (Vorokh, 2018).…”