Moments and Cumulants of Diffraction Profiles Broadened by Stacking Faults


Line broadening in a diffraction intensity profile of powdered crystalline materials due to stacking fault has been characterized in terms of the zeroth, first, second, third, and fourth moments and the fourth cumulant. Calculations have been derived showing that the first moment causes a shift in the peak position of the profile while the third moment affects its shape. The intensity expression has been derived on the basis of usual Cartesian coordinates and also of polar coordinates indicated by the probability of the fault and the reciprocal lattice parameter as the two axes. The expressions for the fourth cumulant have also been so derived. Here we have used three different approaches to determine methods for calculating the fourth cumulant due to stacking faults. The three forms of the equations derived here are for different coordinate systems, but will arrive at the same answers.

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G. Mitra, "Moments and Cumulants of Diffraction Profiles Broadened by Stacking Faults," Journal of Crystallization Process and Technology, Vol. 3 No. 3, 2013, pp. 103-107. doi: 10.4236/jcpt.2013.33017.

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The authors declare no conflicts of interest.


[1] A. J. C. Wilson, “Refraction Broadening in Powder Diffractometry,” Proceedings of the Physical Society, Vol. 80, No. 1, 1962, pp. 303-305. doi:10.1088/0370-1328/80/1/134
[2] G. B. Mitra, “The Fourth Moment of Diffraction Profiles,” British Journal of Applied Physics, Vol. 15, No. 8, 1964, pp. 917-921. doi:10.1088/0508-3443/15/8/305
[3] M. Cernansky, “Cumulants and Moments in the Line Profile Analysis,” Zeitschrift für Kristallographie Supplements, Vol. 2008, No. 27, 2008, pp. 127-133. doi:10.1524/zksu.2008.0017
[4] A. J. C. Wilson, “On Variance as a Measure of Line Broadening in Diffractometry II: Mistakes and Strain,” Proceedings of the Physical Society, Vol. 81, No. 1, 1963, p. 41. doi:10.1088/0370-1328/81/1/309
[5] B. O. Pierce and R. M. Foster, “A Short Table of Integrals,” 4th Edition, Blaisdell Publishing Co., New York, 1966, p. 96.
[6] B. E. Warren and E. P. Warekois, “Stacking Faults in Cold Worked Alpha-Brass,” Acta Metallurgica, Vol. 3, No. 5, 1955, pp. 473-479. doi:10.1016/0001-6160(55)90138-3
[7] T. Ida, “New Measures of Sharpness for Symmetric Powder Diffraction Peak Profiles,” Journal of Applied Crystallography, Vol. 41, No. 2, 2008, pp. 393-401. doi:10.1107/S0021889807067659
[8] I. S. Gradshteyn and I. M. Ryzhik, “Table of Integrals, Series and Products,” 4th Edition, Academic Press, Inc. Orlando, 1980, p. 1160.
[9] A. J. C. Wilson, “The Moments of a Powder Diffraction Profile in the Kinematic Tangent Plane Approximation,” Acta Crystallographica Section A, Vol. 27, No. 6, 1971, pp. 599-604. doi:10.1107/S0567739471001323
[10] K. Stokbro and K. W. Jacobsen, “Simple Model of Stacking-Fault Energies,” Physical Review B, Vol. 47, No. 9, 1993, pp. 4916-4921. doi:10.1103/PhysRevB.47.4916
[11] C.-C. Pei, “Theory of Interface Energies,” Physical Review B, Vol. 18, No. 6, 1978, pp. 2583-2590. doi:10.1103/PhysRevB.18.2583
[12] B. E. Warren, “X-Ray Diffraction,” Addison-Wesley Publishing Co., Reading, 1969, p. 381.
[13] H. Jagodzinski, “Eindimensionale Fehlordnung in Kristallen und ihr Einfluss auf die Rontgeninterferenzen. III. Vergleich der Berechnungen mit Experimentellen Ergebnissen,” Acta Crystallographica, Vol. 2, No. 8, 1949, pp. 298-304. doi:10.1107/S0365110X49000771
[14] R. Gevers, “X-Ray Diffraction by Close-Packed Crystals with ‘Growth’ and ‘Deformation or Transformation Stacking Faults’ Assuming an ‘n-Layer Influence’,” Acta Crystallographica, Vol. 7, No. 11, 1954, pp. 740-744. doi:10.1107/S0365110X54002241
[15] J. Gjonnes, “On the Fourier Treatment of Distortion Broadening in X-Ray Diffraction,” Acta Crystallographica, Vol. 12, No. 6, 1959, pp. 439-442. doi:10.1107/S0365110X59001335
[16] E. Estevez-Rams, A. Penton Madrigal, P. Scardi, et al., “Powder Diffraction Characterization of Stacking Disorder,” Zeitschrift für Kristallographie Supplements, Vol. 2007, Suppl. 26, 2007, pp. 99-104.
[17] D. Pandey, S. Lele and P. Krishna, “X-Ray Diffraction from One-Dimensionally Disordered 2H Crystals Under-going Solid State Transformation to the 6H Structure. III. Comparison with Experimental Observations on SiC,” Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, Vol. 369, No. 1739, 1980, pp. 463-477.
[18] C. Thompson, M. E. Misenheimer and S. C. Moss, “Hendricks-Teller Analysis of Stacking-Fault-Broadened Profiles in C24Rb,” Acta Crystallographica Section B, Vol. 44, No. 1, 1988, pp. 6-11. doi:10.1107/S010876818700898X

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