Intensities, structure factors and their variances

Abstract

Hypothetical variances are checked and improved and observations are tested for normal and independent distribution by means of χ2 distributions. For the conversion of σ2I into σ2F and of I into F the quantity X(I) = sign (I) [I]1/2 instead of F = pI1/2 (p = Lorentz correction etc.) is considered. With the probability density function f(I) for I the variance of F is: σ2F = p2 \Biggl \{ {\int_{-\infty}^{+\infty}} X^{2}(I) f(I) {\rm d}I - \Biggl [ {\int_{-\infty}^{+\infty}}X(I)f(I) {\rm d}I \Biggr ]^{2}\Biggr \}. This formula is also valid if the observed intensity is zero or negative. Unexpectedly, the approximate σ'F = pσI/(2I1/2) are in most cases smaller than the correct ones. The simple formula F = pI1/2 yields F that in the mean are too low. The correction formula valid for all ratios I/σ1 is Fcorr = p(2I1/2 - A), where I is the observed intensity and A the second integral in the above formula for σF2. Since the expressions σF2 and Fcorr are known, there is no other reason to discard weak intensities than the fact that this saves computing time. Examples are given and the normal probability plot is applied.