Щодо поширення похибки визначення міжатомних відстаней в моделі зв`язкової валентності

Abstract

In the last four decades, the bond valence model (BVM) [derived from the Pauling rules for ionic crystals and developed to its modern form by I.D. Brown and by several other researchers] has found wide use in solid state physics, structural inorganic chemistry, crystal chemistry and mineralogy – as a convenient and reliable tool (i) for independently validating the structural models of interest and (ii) for predicting the bond lengths in the structures of known chemical composition and presupposed bond-network topology. Within the BVM framework, the bond valence (BV) sAX is defined as the part of the "classical" atomic valence shared with each A–X cation-anion bond in a given [AXn] coordination sphere. The valence of a bond sAX (measured in valence units, v.u.) is considered to be a unique nonlinear function of the bond length dAX. The most commonly adopted empirical expression for the dAX – sAX relationship is the equation sAX = exp[(r0 – dAX)/b], where r0 and b (BV parameters) are the empirically determined constants for a specific A/X ion (atom) pair. However, analysis of the literature on the BVM has revealed the fact that the propagation of the |DdAX| error (uncertainty) on the sAX value(s) has not been properly and rigorously studied; so the main goal of the present work was to develop a calculation scheme for estimating the |DdAX| error (uncertainty) propagation in the BVM. Using the well established calculus-based approach for estimating the error propagation, the following formula has been derived for the relative error |DsAX|/|sAX| introduced by the small |DdAX| error: |DsAX|/|sAX| » |DdAX|/|b|. This formula indicates the critical dependence of |DsAX|/|sAX| on the particular b value relevant to a specific A/X ion (atom) pair. Given the range of 0.2¸0.7 Å for the observed b values, the relative error |DsAX|/|sAX| corresponding to the typical experimental |DdAX| error of 0.01 Å is expected to vary from 0.05 to 0.014. Hence, any conclusion about the significance of the observed discrepancy between the calculated and expected sAX values can be made only after evaluating the |DsAX|/|sAX| value corresponding to a given b parameter and to the error |DdAX| = 0.01 Å. Additionally, dealing with the ion pairs characterized by relatively small b parameters, a higher precision of the dAX values is required in order to avoid introducing of serious round-off errors. Keywords: crystal structures; bond valence model; error analysis.