US 2007/0168130 A1Published application

Method for computing crystal shapes from X-ray diffraction data (XRD) of a substance

AssigneeCouncil of Scientific and Industrial Research

InventorsDaniel Sherwood, Bosco Emmanuel

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Abstract

The present invention relates to a method for computing external crystal shapes from X-Ray Diffraction Data (XRD) of a substance. Each diffraction peak arises from a set of crystal planes and the peak width is related to the thickness of the crystal in a direction perpendicular to these set of planes. The crystal shape is actually given by the mathematical envelope of the pairs of planes corresponding to each diffraction peak.

Claims10

Dependent claims

A method of computing crystal shapes from X-Ray Diffraction Data (XRD) of a substance, the method comprising steps of:

a) obtaining input parameters associated with said substance, said parameters comprising:
(i) number of XRD peaks (n),
(ii) (2·θ), wherein θ represents the angle of incidence on said substance of X-rays, and
(iii) the peak-width at half maximum, represented as B, for each of the XRD peaks;
b) indexing by assigning the Miller indices (h, k, l) to each of the XRD peaks;
c) determining at least one unit cell parameter associated with said substance;
d) determining crystal thickness associated with said substance in different directions based at least in part on the data gathered from step (a) and step (b) in Scherrer formula (thkl=0.9λ/B Cos θhkl), wherein λ represents the wavelength of said X-rays, θhkl represents angle of incidence of X-rays on the planes having Miller indices (h, k, l), and thkl represents a thickness of crystal perpendicular to said (h, k, l) planes;
e) based at least partly on the data gathered from steps (a) to (d), determining a set of crystal planes which determine the mathematical envelope for the crystal shape associated with said substance, wherein said crystal planes are associated with crystal faces;
f) determining co-ordinates of each point of intersection of said crystal planes;
g) determining a set of all real points of intersection;
h) for each of said crystal planes, identifying a set of real points of intersection which fall on said plane;
i) transforming coordinates of said real points of intersection from Global to Local frames, wherein said real points of intersection are associated with vertices of said crystal faces;
j) determining a polar co-ordinate[θi] of each of the vertices ordered in an ascending series in each of the crystal faces;
k) determining a transformation, represented by T, which reshuffles the indices;
l) based at least partly on the transformation T obtained from step (k), ordering the vertices on each crystal plane and computing the areas and Miller indices of exposed crystal faces;
and
m) determining a 3D crystal shape associated with said substance.

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Dependencies are inferred from the claim text.

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Abstract

A method of computing crystal shapes from X-Ray Diffraction Data (XRD) of a substance, the method comprising steps of:

a) obtaining input parameters associated with said substance, said parameters comprising:
(i) number of XRD peaks (n),
(ii) (2·θ), wherein θ represents the angle of incidence on said substance of X-rays, and
(iii) the peak-width at half maximum, represented as B, for each of the XRD peaks;
b) indexing by assigning the Miller indices (h, k, l) to each of the XRD peaks;
c) determining at least one unit cell parameter associated with said substance;
d) determining crystal thickness associated with said substance in different directions based at least in part on the data gathered from step (a) and step (b) in Scherrer formula (thkl=0.9λ/B Cos θhkl), wherein λ represents the wavelength of said X-rays, θhkl represents angle of incidence of X-rays on the planes having Miller indices (h, k, l), and thkl represents a thickness of crystal perpendicular to said (h, k, l) planes;
e) based at least partly on the data gathered from steps (a) to (d), determining a set of crystal planes which determine the mathematical envelope for the crystal shape associated with said substance, wherein said crystal planes are associated with crystal faces;
f) determining co-ordinates of each point of intersection of said crystal planes;
g) determining a set of all real points of intersection;
h) for each of said crystal planes, identifying a set of real points of intersection which fall on said plane;
i) transforming coordinates of said real points of intersection from Global to Local frames, wherein said real points of intersection are associated with vertices of said crystal faces;
j) determining a polar co-ordinate[θi] of each of the vertices ordered in an ascending series in each of the crystal faces;
k) determining a transformation, represented by T, which reshuffles the indices;
l) based at least partly on the transformation T obtained from step (k), ordering the vertices on each crystal plane and computing the areas and Miller indices of exposed crystal faces;
and
m) determining a 3D crystal shape associated with said substance.

14 more elements