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# Scherrer equation

About: Scherrer equation is a research topic. Over the lifetime, 1339 publications have been published within this topic receiving 38094 citations.

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TL;DR: An exact derivation of the Scherrer equation is given for particles of spherical shape, values of the constant for half-value breadth and for integral breadth being obtained in this article, and various approximation methods which have been used are compared with the exact calculation.

Abstract: An exact derivation of the Scherrer equation is given for particles of spherical shape, values of the constant for half-value breadth and for integral breadth being obtained. Various approximation methods which have been used are compared with the exact calculation. The tangent plane approximation of v. Laue is shown to be quite satisfactory, but some doubt is cast on the use of approximation functions. It is suggested that the calculation for the ellipsoidal particle based on the tangent plane approximation will provide a satisfactory basis for future work.

6,907 citations

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TL;DR: In this article, the Scherrer constants of simple regular shapes have been determined for all low-angle reflections (h2 + k2 + l2 ≤ 100) for four measures of breadth.

Abstract: Existing knowledge about Scherrer constants is reviewed and a summary is given of the interpretation of the broadening arising from small crystallites. Early work involving the half-width as a measure of breadth has been completed and Scherrer constants of simple regular shapes have been determined for all low-angle reflections (h2 + k2 + l2 ≤ 100) for four measures of breadth. The systematic variation of Scherrer constant with hkl is discussed and a convenient representation in the form of contour maps is applied to simple shapes. The relation between the `apparent' crystallite size, as determined by X-ray methods, and the `true' size is considered for crystallites having the same shape. If they are of the same size, then the normal Scherrer constant applies, but if there is a distribution of sizes, a modified Scherrer constant must be used.

3,018 citations

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TL;DR: Paul Scherrer and Peter Debye developed powder X-ray diffraction together, but it was Scherrer who figured out how to determine the size of crystallites from the broadening of diffraction peaks.

Abstract: Paul Scherrer and Peter Debye developed powder X-ray diffraction together, but it was Scherrer who figured out how to determine the size of crystallites from the broadening of diffraction peaks.

1,970 citations

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TL;DR: In this article, a modified Scherrer equation (MSE) method was proposed to calculate the nano-scale size of bovine bone using XRD radiation of wavelength λ (nm) from measuring full width at half maximum of peaks (β) in radian located at any 2π in the pattern.

Abstract: Scherrer Equation, L=Kλ/β.cosθ, was developed in 1918, to calculate the nano crystallite size (L) by XRD radiation of wavelength λ (nm) from measuring full width at half maximum of peaks (β) in radian located at any 2θ in the pattern. Shape factor of K can be 0.62 - 2.08 and is usually taken as about 0.89. But, if all of the peaks of a pattern are going to give a similar value of L, then β.cosθ must be identical. This means that for a typical 5nm crystallite size and λ Cukα1 = 0.15405 nm the peak at 2θ = 170° must be more than ten times wide with respect to the peak at 2θ = 10°, which is never observed. The purpose of modified Scherrer equation given in this paper is to provide a new approach to the kind of using Scherrer equation, so that a least squares technique can be applied to minimize the sources of errors. Modified Scherrer equation plots lnβ against ln(1/cosθ) and obtains the intercept of a least squares line regression, ln=Kλ/L, from which a single value of L is obtained through all of the available peaks. This novel technique is used for a natural Hydroxyapatite (HA) of bovine bone fired at 600°C, 700°C, 900°C and 1100°C from which nano crystallite sizes of 22.8, 35.5, 37.3 and 38.1 nm were respectively obtained and 900°C was selected for biomaterials purposes. These results show that modified Scherrer equation method is promising in nano materials applications and can distinguish between 37.3 and 38.1 nm by using the data from all of the available peaks.

1,418 citations

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TL;DR: This study investigates colloidally prepared, highly monodisperse CoPt3 nanoparticles by transmission electron microscopy, small-angleX-ray scattering (SAXS), and powder X-ray diffraction (XRD), and examines to which extent agreement is obtained by the different techniques when applied to small nanocrystals in the size range below 10 nm.

Abstract: One of the most fundamental tasks in nanoscience is the accurate determination of particle sizes. Various methods have been developed to elucidate the mean particle diameter and the standard deviation for an ensemble of nanocrystals. However, good agreement between the results from different methods is not always encountered in the literature. In this study, we investigate colloidally prepared, highly monodisperse CoPt3 nanoparticles by transmission electron microscopy (TEM), small-angle X-ray scattering (SAXS), and powder X-ray diffraction (XRD). The results are compared in order to examine to which extent agreement is obtained by the different techniques when applied to small nanocrystals in the size range below 10 nm. In particular, the applicability of the simple Scherrer formula for size determination from the broadening of XRD reflections is checked. When the different techniques are correctly applied, the results from all methods are in good agreement.

644 citations