Remarks on separation of particle size and strain by line profile analysis using Fourier coefficients

The exact expression for A(t), the tth-order Fourier coefficient of an X-ray diffraction line profile in terms of column length distribution, mean thickness, the r.m.s. strain and various mean strain gradients, has been derived and the assumptions involved in the Warren-Averbach method of separating particle size and r.m.s. strain have been critically examined. It has been observed that the Warren-Averbach method gives a r.m.s. value of the pseudo strain <e²>^1/2 which is equal to the true r.m.s. strain <e²>^1/2 only at t = 0. It has been further shown that ln J(t), where J(t) is the tth-order Fourier coefficient of the strain profile, is not, as is taken in the Warren-Averbach method, linear in l²₀ when the given reflection hkl has been converted into a 00l₀ reflection by suitable changes of axes. In the Warren-Averbach method, this essentially non-linear curve is taken as linear and a straight line passing through two points on this curve is extended to meet the ln A(t) axis, the intercept on which is supposed to give ln N(t), where N(t) is the tth-order Fourier transform of the particle-size profile. Thus approximate values of the mean particle size <T> and the r.m.s. pseudo strain <e²>^1/2 are obtained. Based on the exact expression for A(t) a new multiple-order technique for determining not only the particle size and r.m.s. strain but also various strain derivatives has been developed and its limitations examined. It is found that an error introduced in the r.m.s. strain is of an order less than that of the experimental errors. Similarly, the single-line technique of Mitra & Misra [Acta Cryst. (1967), 23, 867-868] has been shown to give correct values of <T> and <e²>^1/2 within experimental error. The single-line technique has been proved to be quite a perfect and useful technique.