title: The Scherrer equation and the dynamical theory of X-ray diffraction.
abstract: The Scherrer equation is a widely used tool to determine the crystallite size of polycrystalline samples. However, it is not clear if one can apply it to large crystallite sizes because its derivation is based on the kinematical theory of X-ray diffraction. For large and perfect crystals, it is more appropriate to use the dynamical theory of X-ray diffraction. Because of the appearance of polycrystalline materials with a high degree of crystalline perfection and large sizes, it is the authors' belief that it is important to establish the crystallite size limit for which the Scherrer equation can be applied. In this work, the diffraction peak profiles are calculated using the dynamical theory of X-ray diffraction for several Bragg reflections and crystallite sizes for Si, LaB6 and CeO2. The full width at half-maximum is then extracted and the crystallite size is computed using the Scherrer equation. It is shown that for crystals with linear absorption coefficients below 2117.3 cm(-1) the Scherrer equation is valid for crystallites with sizes up to 600 nm. It is also shown that as the size increases only the peaks at higher 2θ angles give good results, and if one uses peaks with 2θ > 60° the limit for use of the Scherrer equation would go up to 1 µm.
authors.affiliation: Departamento de Física, Universidade Federal do Ceará, Caixa Postal 6030, Fortaleza, Ceará 60440-970, Brazil.
authors.forename: Francisco Tiago Leitão
authors.initials: FT
authors.lastname: Muniz
authors.affiliation: Departamento de Física, Universidade Federal do Ceará, Caixa Postal 6030, Fortaleza, Ceará 60440-970, Brazil.
authors.forename: Marcus Aurélio Ribeiro
authors.initials: MA
authors.lastname: Miranda
authors.affiliation: Departamento de Física, Universidade Federal do Ceará, Caixa Postal 6030, Fortaleza, Ceará 60440-970, Brazil.
authors.forename: Cássio
authors.initials: C
authors.lastname: Morilla Dos Santos
authors.affiliation: Departamento de Física, Universidade Federal do Ceará, Caixa Postal 6030, Fortaleza, Ceará 60440-970, Brazil.
authors.forename: José Marcos
authors.initials: JM
authors.lastname: Sasaki
keywords: Scherrer equation; X-ray diffraction; dynamical theory; powder diffraction