Interpretation of the initial slope of the Fourier-coefficient-against-order curve for X-ray diffraction line profiles

The Fourier coefficient for a given order of a pure diffraction line profile is the product of the Fourier coefficients for the same order of its constituent profiles, such as particle size, strain, stacking and twinning faults, variable interlayer spacings, crystallite-size distribution etc. The Fourier coefficient for the crystallite-size effect is known to be A^P(n) = (〈N〉−|n|), where is the average particle size and n is the order of the Fourier coefficient. According to Warren, in the presence of stacking and twinning faults, the Fourier coefficient derivative at zero order is a function of particle size and a linear combination of stacking and twinning fault probabilities. It has been shown here that Warren's calculation involves some approximations which are not needed. The Fourier coefficient for stacking and twinning faults as well as for variable interlayer spacing is of the type φⁿ, where φ is a function of these fault parameters. The strain-dependent Fourier coefficient is known to be of the type exp ( − n²e²) where e is a function of r.m.s. strain and its derivatives. Combining all these, it has been shown that for all cases

where A(n) is the Fourier coefficient for the original physical line profile uncorrected for strain and faults.