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A New Method for Fourier Analysis of Shapes of X-Ray Peaks and its Application to Line Broadening and Integrated Intensity Measurements

  • R. L. Rothman (a1) and J. B. Cohen (a1)

Abstract

A method of Fourier analysis of x-ray line broadening is presented whereby microstrain, incoherent particle size, and fault probability can be calculated using only first-order peaks. This method can thus be used in studies of catalysts, vapor- and electrodeposits or heavily textured specimens for which second-order peaks are too broad or weak, in cases where the effect of particle size is not the same for all orders of a peak, or for multiphase specimens where overlap of peaks sometimes occurs. Examples of deformed FCG and BCC metals and alloys are presented, with comparisons to the method of multiple orders.

One part of the procedure, first demonstrated by Pines and Sirenko, provides for reducing the effects of truncation on the Fourier coefficients. As a result, larger particle sizes can be analyzed and the integrated intensity can be determined more precisely. Application to determinations of volume fraction are given.

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5. Stokes, A. R., “A Numerical Fourier Analysis Method for the the Correction of Widths and Shapes of Lines on X-ray Powder Photographs,” Proc. Phys. Soc. London 61: 382, 1948.
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7. Harrison, J. W., “The Use of Strain Moments in Determining Scrain Distributions in Deformed Crystals,” Acta Cryst. 20:390, 1966.
8. Cohen, J. B. and Rothman, R. L., “The Effects of Faulting on [110] and (321) Planes of a BCC Diffraction Pattern,” Technical Report No. 11, Office of Naval Research, Contract NONR 1228(20), October, 1967.
9. Smith, R. S., “Measurement of Crystallite Size and Strain of Electroplated Films,” IBM J. of Research and Development 4:205, 1960.
10. Pines, B. Ya. and Sirenko, A. F., “Deviation of the Lattice Distortion and Spread in Block Size by Means of Harmonic Analysis Applied to X-ray Reflections,” Soviet Physics-Cryst. 7: 15, 1962.
11. McKeehan, M. and Warren, B. E., “X-ray Study of Cold-Work in Thoriatsd Tungsten,” J. Appl. Phys, 24:52, 1953.
12. Evans, W. P., Ricklefs, R. E. and Millan, J. F., “X-ray and Fatigue Studies of Hardened and Cold-Worked Steels,” in J. B. Cohen and J. E. Hilliard, Editors, Local Atomic Arrangements Studied by X-ray Diffraction, Gordon and Breach Science Publishers, New York, 1966, p. 351.
13. De Angelis, R. J., “Evaluations from X-ray Diffraction Profiles of Fourier Coefficients and the Microstrain Distribution Function,” in J. B. Cohen and J. E. Hilliard, Editors, Local Atomic Arrangements Studied by X-ray Diffraction, Gordon and Breach Science Publishers, New York, 1966, p. 271.
14. Mikkola, D. E. and Cohen, J. B., “The Substructure of Cu3Au After Tensile Deformation and Shock Loading,” Acta Met. 14:105, 1966; D. E. Mikkola, Ph.D. Thesis, Northwestern University, 1964.
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18. Cullity, B. D., “Elements of X-ray Diffraction,” Addison-Wesley Publishers, Reading, Massachusetts, 1956, p. 446.
19. Cohen, J. B. and Wagner, C. N. J., “Determination of Twin Fault Probabilities from the Diffraction Patterns of FCC Metals and Alloys,” J. Appl. Phys, 33 :2073, 1962.
20. Aqua, E. N. and Wagner, C. N. J., “X-ray Diffraction Study of Deformation by Filing in BCC Refractory Metals,” Phil. Mag. 9 :565, 1964.
21. Mikkola, D. E. and Cohen, J. B., “Examples of Applications of Line Broadening,” in J. B. Cohen and J. E. Hilliard, Editors, Local Atomic Arrangements Studied by X-ray Diffraction, Gordon and Breach Science Publishers, New York, 1966, p. 319.
22. Siegel, S., “The Variation of thePrinciple Elastic Moduli of Cu3Au with Temperature.” Phys. Rev. 57: 537, 1940.
23. Warren, B. E. and Warekois, E. P., “Stacking Faults in Cold- Worked Alpha-Brass,” Acta Met. 3:473, 1955.
24. Wagner, C. N. J., (a) “Analysis of the Broadening and Changes in Position of Peaks in an X-ray Powder Pattern,” in J. B. Cohen and J. E. Hilliard, Editors, Local Atomic Arrangements Studied by X-ray Diffraction, Gordon and Breach Science Publishers, New York, 1966, p. 256; (b) Ibid., p. 248.
25. Koistinen, D. P. and Marburger, R. E., “A Simplified Procedure . for Calculating Peak Position in X-ray Residual Stress Measurements on Hardened Steel,” Trans. ASM 51:537, 1959.
26.For some sharp peaks linear dependence of lnAL with L2 may be found; this depends on the actual peak, slit function, etc. The main point here is that any simple functional dependence could be used to obtain A0 and hence a correction to the integrated intensity.

A New Method for Fourier Analysis of Shapes of X-Ray Peaks and its Application to Line Broadening and Integrated Intensity Measurements

  • R. L. Rothman (a1) and J. B. Cohen (a1)

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