On Variance as a Measure of Line Broadening in Diffractometry II: Mistakes and Strain

If J ( t )- iK ( t ) is the mean value of FF * for cells separated a distance t in the hkl direction, the mean-square broadening due to mistakes is W 2θ = - [λΔ(2θ) / 2π 2 cos θ] × [ J '(0) / J (0)] - [λ 2 / 4π 2 cos 2 θ] [ J ''(0) / J (0) + { K '(0) / J (0)} 2 ] where Δ(2θ) is the range of 2θ used in evaluating W 2θ . An argument of Strijk and MacGillavry is extended to show that J '(0) / J (0) = - ( J s - J a ) / TJ s and K '(0) / J (0) = K a / TJ s where J s is the mean value of FF * for cells in the same domain, J a - iK a is the mean value of FF * for cells in adjacent domains, and T is the mean domain thickness in the hkl direction, and that J ''(0) / J (0) = ( J s - 2 J a + J n ) Q '(0) / TJ s . In the last expression J n is the mean value of the real part of FF * for cells in next-but-one domains, and Q ( x ) dx is the fraction of domain thicknesses in the hkl direction lying in the range x to x + dx . For mistakes at random the relation J a 2 = J s J n must hold. A varying strain adds a constant contribution of 4 tan 2 θ × (the variance of the strain) to the mean-square breadth in 2θ, provided that Δ(2θ) is sufficiently large.