Diffraction line profiles and Scherrer constants for materials with cylindrical crystallites

Abstract

Calculations of diffraction line profiles and Scherrer constants for crystallites whose external shape has cubic symmetry are extended to crystallites of cylindrical shape. The analysis includes the limiting cases of acicular crystals and disks and, to a reasonable degree of approximation in many cases, to hexagonal prisms. These shapes have applications in size determination for materials which form prismatic crystallites, particularly those which belong to the hexagonal system or have been derived from substances with this symmetry.

Crystalline domain size and faulting in the new NIST SRM 1979 zinc oxide

“…The slope of a linear fit to the peaks in group 1 of the sample annealed at 400 °C is about 5x10 -4 , indicating negligible strain. The data of figure 2 were analyzed in the context of the cylindrical shape model as proposed by Langford and Louër (1982) to yield the volume weighted crystallite dimensions. This analysis indicated, for the 400 °C data, a diameter of 26.7 nm and a height of 24.7 nm, for the 550 °C data, the dimensions were a diameter of 108.6 nm and a height of 87.5 nm.…”

“…The specimen broadening function used was specific to modeling of cylinders of constant aspect ratio with a distribution of diameters presumed to be log-normal. A numerical approach, based on the model for cylinders proposed by Langford & Louër (1982) and the distribution treatment of Scardi & Leoni (2001) and Leoni & Scardi (2004), was employed to calculate the profile for the distributed domains. The Warren (1969) model for hexagonal stacking faults was used.…”

“…We would like to thank Dr. Katherine Mullen for her assistance in developing the software used to implement the Langford and Louër (1982) model for integral breadth analysis to yield the crystallite diameter and height.…”

Microstructure et profil des raies de diffraction des rayons X

“…En pratique, il est clair que les dimensions réelles (et la forme) des cristallites présentent un intérêt tout particulier. En principe, elles peuvent être obtenues à partir de Ep et Ep pour des cristallites possédant une morphologie bien définie, comme par exemple une sphère [9], un parallélépipède [16], un cylindre [18] ou un prisme droit à base hexagonale [19]. En pratique, la situation est souvent compliquée par la superposition exacte de réflexions, due à la symétrie cristalline du matériau (multiplicité).…”

Line profiles for a collection of identical crystals: effect of a linear transformation of shape

“…This line profile can be evaluated, in principle, for crystallites of arbitrary shape; the problem reduces to the purely geometrical one of determining V(t) and then finding its cosine Fourier transform. In practice most calculations so far have been confined to regular shapes with cubic or higher symmetry (see Langford & Wilson, 1978, for a survey), though some attention has been given to parallelepipeds (Allegra & Ronca, 1978, 1979 and hexagonal and circular cylinders (Langford, Lou~r & Wilson, 1980;Langford & Lou~r, 1982; * Present address: Crystallographic Data Centre, University Chemical Laboratory, Lensfield Road, Cambridge CB2 1EW.0567-7394/83/030280-03501.50Lou~r, Vargas & Langford, 1981). However, as was pointed out by Patterson (1939), results for a regular symmetrical shape may be altered to apply to any non-regular unsymmetrical shapes that can be derived from the regular one by linear transformations of coordinates.…”