Search for a command to run...
The empirical similarity in the behavior of scattering-factor curves for different atoms is used to replace the structure-factor formulae by an approximating system. This system is shown to lead to $m'\mathrm{th}$ degree algebraic equations, designated fundamental equations, whose coefficients are calculable in terms of crystal structure and atomic scattering factors. The roots of these equations give the coordinates of the unknown atomic positions. Thus if the values of a sufficient number ($2m$ in general; $m+1$ where there is a center of symmetry) of sequential structure-factors are known, the problem of direct structure determination is solved. If only the absolute magnitudes are known, one may derive reciprocal equations of the $m(m\ensuremath{-}1)\mathrm{th}$ degree whose coefficients are calculable in terms of the squared moduli of the structure factors. We can then write fundamental equations of degree $\frac{m(m\ensuremath{-}1)}{2}$ whose roots give the coordinates of the interatomic distances from which the unknown parameters themselves may readily be determined. If the number of unknown parameters in any projection is $m$, the minimum number of structure factor moduli required in this case is $m(m\ensuremath{-}1)+1$. The method developed in the paper is applied to the K${\mathrm{H}}_{2}$P${\mathrm{O}}_{4}$ crystal, and the results are found to compare very well with the values found by West on the basis of a trial-error and Fourier synthesis investigation.