Advances in the Parameter Space Concept towards Picometer Precise Crystal Structure Refinement—A Resolution Study

The Parameter Space Concept (PSC) is an alternative approach to solving and refining (partial) crystal structures from very few pre-chosen X-ray or neutron diffraction amplitudes without the use of Fourier inversion. PSC interprets those amplitudes as piecewise analytic hyper-surfaces, so-called isosurfaces, in the Parameter Space, which is spanned by the spatial coordinates of all atoms of interest. The intersections of all isosurfaces constitute the (possibly degenerate) structure solution. The present feasibility study investigates the La and Sr split position of the potential high-temperature super-conductor (La_0.5Sr_1.5)MnO₄, $I4$/mmm, with a postulated total displacement between La and Sr of a few $\mathrm{p}$$\mathrm{m}$ by theoretical amplitudes of pre-selected $00l$ reflections ($l=2,4,\dots ,20$). The revision of 15-year-old results with state-of-the-art computing equipment enhances the former simplified model by varying the scattering power ratio $f_{\begin{array}{c}\hfill errortypeceLa\end{array}}/f_{\begin{array}{c}\hfill errortypeceSr\end{array}}$, as exploitable by means of resonant scattering contrast at synchrotron facilities, and irrevocably reveals one of the two originally proposed solutions as being a “blurred” pseudo-solution. Finally, studying the resolution limits of PSC as a function of intensity errors by means of Monte-Carlo simulations shows both that the split can only be resolved for sufficiently low errors and, particularly for the resonant scattering contrast, a theoretical precision down to $\pm 0.19$$\mathrm{p}$$\mathrm{m}$ can be achieved for this specific structural problem.