The determination of crystal size and disorder from X-ray diffraction photographs of polymer fibres. 1. The accuracy of determination of Fourier coefficients of the intensity profile of a reflection

1989 ◽  
Vol 22 (4) ◽  
pp. 363-371 ◽  
Author(s):  
R. Somashekar ◽  
I. H. Hall ◽  
P. D. Carr
Keyword(s):  
Crystal Size ◽  
Fourier Coefficients ◽  
Intensity Profile ◽  
X Ray Diffraction ◽  
Scattering Angle ◽  
X Ray ◽  
Fourier Cosine ◽  
High Harmonics ◽  
Lattice Planes

Methods which determine the number and disorder of lattice planes in a crystal from the Fourier cosine coefficients of the intensity profile of an X-ray reflection use only the low harmonics and require that the coefficients be normalized so that the zero harmonic is unity. Experimentally, the profiles can only be recorded over a smaller range of scattering angle than required by the theory, and it is necessary to subtract background, which is likely to be estimated with considerable error, before determining the coefficients. It is shown that with polymer fibres this causes serious errors in the normalization, and in the values of those low harmonics used in the size and disorder determination, and prevents reliable values being obtained. Methods which avoid normalization and use only high harmonics are needed. It is shown that disorder may be obtained in such a way, but not size, for which low-order normalized coefficients are essential. A method of extrapolation is described and tested which enables the accurate high harmonics to be used to improve the estimates of the low ones. Whilst this will yield more reliable values of crystal size than are obtainable from existing methods, the accuracy depends entirely on the validity of the extrapolation, which cannot be tested in many cases of interest.

X-Ray Diffraction Determination of Texture and Stress in Damascene Fabricated Copper Interconnects

1999 ◽  
Vol 563 ◽  
Author(s):  
Delrose Winter ◽  
Paul R. Besser
Keyword(s):  
Incident Beam ◽  
Preferred Orientation ◽  
Copper Interconnects ◽  
X Rays ◽  
X Ray Diffraction ◽  
X Ray ◽  
Diffraction Determination ◽  
Lattice Planes ◽  
Excellent Tool

AbstractX-Ray diffraction (XRD) provides an excellent tool for the measurement of both stress and texture (preferred orientation) on fabricated damascene interconnect structures. Since x-ray diffraction provides a direct measurement of lattice spacings, film strain can be measured directly. Also, since the intensity of diffracted x-rays is proportional to the density of lattice planes oriented in diffracting condition with respect to the incident beam, both the direction and extent of preferred orientation can be accurately measured. Special techniques and considerations are necessary when examining damascene interconnect structures with XRD which are not necessary with blanket films. These techniques are discussed and described in order to aid in obtaining meaningful XRD data and a correct interpretation of the results.

The Determination of Crystallite Size and Size Distribution from Broadened X-Ray Diffraction Lines

1961 ◽  
Vol 5 ◽  
pp. 94-103 ◽  
Author(s):  
H. F. Quinn ◽  
P. Cherin
Keyword(s):  
Harmonic Analysis ◽  
Size Distribution ◽  
Fourier Coefficients ◽  
Distribution Functions ◽  
Composite Sample ◽  
X Ray Diffraction ◽  
X Ray ◽  
Mean Size ◽  
The Mean

AbstractMagnesium oxide crystallites having mean dimensions in the range of 25–1000 A can be prepared by controlled thermal decomposition of the carbonate.Following some earlier investigations of Birks and Friedman, we have determined the mean size and size distribution of several such MgO samples from the broadened X-ray diffraction lines which they exhibit. Contrary to the procedure of the above investigators, the harmonic analysis due to Stokes has been used to correct for instrumental broadening and values of mean-size and size-distribution functions obtained from the Fourier coefficients by the methods of Warren and Averbach.The results obtained are compared with average sizes and distributions obtained by direct examination of the samples in an electron microscope.A composite sample has been prepared by mixing known quantities of the sample previously studied. The distribution function obtained by harmonic analysis of one diffraction line of the composite sample is compared with the function calculated from the distributions of its components.Conclusions are drawn concerning the significance of the results obtained by the Warren technique: in particular, the average sizes obtained by this method are compared with those given by the approximate method used by Birks and Friedman.

scholarly journals FXD-CSD-GUI: a graphical user interface for the X-ray-diffraction-based determination of crystallite size distributions

2019 ◽  
Vol 52 (6) ◽  
pp. 1437-1439
Author(s):  
Sigmund H. Neher ◽  
Helmut Klein ◽  
Werner F. Kuhs
Keyword(s):  
User Interface ◽  
Graphical User Interface ◽  
Crystal Size ◽  
Data Handling ◽  
Size Distributions ◽  
Two Dimensional ◽  
X Ray Diffraction ◽  
X Ray ◽  
User Friendly

Bragg intensities can be used to analyse crystal size distributions in a method called FXD-CSD, which is based on the fast measurement of many Bragg spots using two-dimensional detectors. This work presents the Python-based software and its graphical user interface FXD-CSD-GUI. The GUI enables user-friendly data handling and processing and provides both graphical and numerical crystal size distribution results.

The determination of crystal size and disorder from the X-ray diffraction photograph of polymer fibres. 2. Modelling intensity profiles

1991 ◽  
Vol 24 (6) ◽  
pp. 1051-1059 ◽  
Author(s):  
I. H. Hall ◽  
R. Somashekar
Keyword(s):  
Structural Parameters ◽  
Intensity Profile ◽  
Model Parameters ◽  
X Ray Diffraction ◽  
X Ray ◽  
Fourier Cosine ◽  
Lattice Disorder ◽  
Diffraction Photograph ◽  
Diffraction Patterns ◽  
Experimental Values

The intensity profile of the X-ray reflection from a crystalline material is related to the lattice disorder and the distribution of crystal sizes through its Fourier cosine coefficients. However, existing methods of obtaining these structural parameters from the coefficients require more than one order of reflection and this is seldom available with polymer fibres. They also rely heavily on the low-order harmonics which are those determined with least accuracy. The development and testing of a method which overcomes this weakness and which is suitable for use with a single order is described. The coefficients are calculated for a model with paracrystalline disorder and an assumed distribution of crystal sizes and the parameters describing this model are refined to minimize the discrepancy between the calculated and experimental values of the coefficients. Provided the distribution of lengths is asymmetric this discrepancy is no greater than would be expected from experimental error and so the assumed model cannot be rejected on the evidence available. Since a range of model parameters all gave equally good agreement with experiment, it was not possible with a single order to obtain a well defined set of values. Diffraction patterns displaying two orders had been chosen and results from the second order were consistent with the first, only a narrow range satisfying both simultaneously. The method was further developed by calculating the intensity profile from the harmonics and using this in the refinement. There was no advantage over using harmonics; indeed, on occasions the refinement algorithm was unstable producing unreliable results.

Direct Determination of the Reciprocal Lattice Spacing and the Radial Interference Distribution by the Fourier Method

1968 ◽  
Vol 12 ◽  
pp. 354-371
Keyword(s):  
Fourier Coefficients ◽  
Reciprocal Lattice ◽  
Fourier Method ◽  
Direct Determination ◽  
Annealed Specimen ◽  
X Ray Diffraction ◽  
X Ray ◽  
Free Material ◽  
Unfolding Method

AbstractA procedure for extracting the interference distribution from the X-ray diffraction line using measured instrumental and wavelength distributions has been applied to the study of residual stress in copper following extension.In the direct unfolding method the principal instrumental distributions due to the source and to specimen absorption are unfolded by the Fourier method. The remaining instrumental aberrations are extracted using the centroid displacements. The resultant scattering distribution is transformed by the relations of Mitchell and de Wolff and the spectral distribution unfolded. Conditions limiting the Fourier coefficients are applied to preserve stability. Measurements have been carried out with a diffractometer equipped with a diffracted beam monochromator.The 420 and 331 interference distributions in copper following 10% extension have been compared with those of stress - free material. The distributions are symmetric both in the annealed specimen and after extension. In extension the distributions are displaced by an amount equivalent to a surface compressive stress of 6 kg/mm2, The symmetry of the interference distributions indicates that the displacement is due to a macrostress distribution.

Sem and X-Ray Analysis of Renal and Biliary Calculi

1976 ◽  
Vol 34 ◽  
pp. 306-307
Author(s):  
R. J. Narconis ◽  
G. L. Johnson
Keyword(s):  
Chemical Constituents ◽  
Natural State ◽  
X Ray Diffraction ◽  
Chemical Methods ◽  
X Ray ◽  
Additional Dimension ◽  
Biliary Calculi ◽  
Calculus Formation ◽  
Scanning Electron

Analysis of the constituents of renal and biliary calculi may be of help in the management of patients with calculous disease. Several methods of analysis are available for identifying these constituents. Most common are chemical methods, optical crystallography, x-ray diffraction, and infrared spectroscopy. The application of a SEM with x-ray analysis capabilities should be considered as an additional alternative.A scanning electron microscope equipped with an x-ray “mapping” attachment offers an additional dimension in its ability to locate elemental constituents geographically, and thus, provide a clue in determination of possible metabolic etiology in calculus formation. The ability of this method to give an undisturbed view of adjacent layers of elements in their natural state is of advantage in determining the sequence of formation of subsequent layers of chemical constituents.

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