Studies on the Radial Distribution Analysis in Diffraction Methods

I. Errors due to the Approximation of Integration by Summation

抄録

In the calculation of a radial distribution function
rD(r)=(1⁄2π²)∫₀^∞sI(s)sinsr ds
we usually approximate the integral by such a sum as
rD_Σ(r)=(Δs⁄2π²) ∑\limits_n=1^∞nΔs·I(nΔs)·sin (nΔsr).
In this paper the mathematical property of rD_Σ(r) is derived and the error due to the approximation is proved to be expressed as
rD_Σ(r)−rD(r)=∑_{\substackm=−1
≠0}\limits^∞′(r+2mπ⁄Δs)D(r+2mπ⁄Δs).
Furthermore, the formula for the estimation of the error is derived. It turned out that in order to obtain a high accuracy from the approximation we must make the spacing Δs less than π⁄R₀, where R₀ is the distance beyond which rD(r) is regarded practically zero.