A Statistical Analysis of Fourier Coefficients of X-Ray Diffraction Line and Microstrain Calculated from the Fourier Coefficients.

The Fourier analysis of the broadened X-ray diffraction profiles from polycrystalline materials allows the determination of the microstrain in materials. To obtain an accurate microstrain, it is important to measure the Fourier coefficients of the diffraction profiles precisely. The equations for calculating the standard deviation of the Fourier coefficients are derived analytically. From these equations, the size of the variation in the Fourier coefficients due to counting statistics can be obtained. The variances in the measured Fourier coefficients result in an error in the microstrain calculated from them. An equation for calculating the standard deviation of the microstrain was derived by taking account of the variances of the Fourier coefficients. The Fourier coefficients and the microstrain calculated from the Fourier coefficients were determined for quenched and tempered steel, and their standard deviations were calculated by using the present equations. As a result, the relative error of the microstrain was remarkably larger than that of the Fourier coefficient.