A Priori Bond-Valence and Bond-Length Calculations in Rock-Forming Minerals

Abstract

Synopsis A priori bond-valences and bond-lengths are calculated for a series of rock-forming minerals. Comparison of a priori and observed bond-lengths allows structural strain to be assessed for those minerals.Abstract Within the framework of the bond-valence model, one may write equations describing the valence-sum rule and the loop rule in terms of the constituent bond-valences. These are collectively called the network equations, and can be solved for a specific bond topology to calculate its a priori bond-valences. A priori bond-valences are the ideal values of bond strengths intrinsic to a given bond topology that depend strictly on the formal valences of the ion at each site in the structure, and the bond-topological characteristics of the structure (i.e., the ion connectivity). The a priori bondvalences are calculated for selected rock-forming minerals, beginning with a simple example (magnesiochromite, = 1.379 bits/atom) and progressing through a series of gradually more complex minerals (grossular, diopside, forsterite, fluoro-phlogopite, phlogopite, fluoro-tremolite, tremolite, albite) to finish with epidote ( = 4.187 bits/atom). The effects of weak bonds (hydrogen bonds, long Na + -O 2bonds) on the calculation of a priori bond-valences and bond-lengths are examined. For the selected set of minerals, a priori and observed bond-valences and bond-lengths scatter closely about the 1:1 line with an average deviation of 0.04 v.u. and 0.048 Å and maximum deviations of 0.16 v.u. and 0.620 Å. The scatter of the corresponding a priori and observed bond-lengths is strongly a function of the Lewis acidity of the constituent cation. For cations of high Lewis acidity, the range of differences between the a priori and observed bond-lengths is small, whereas for cations of low Lewis acidity, the range of differences between the a priori and observed bond-lengths is large. These calculations allow assessment of the strain in a crystal structure and provide a way to examine the effect of bond topology on variation in observed bond-lengths for the same ion-pair in different bond topologies. Keywords: bond valence, bond length, network equations, Lewis acidity, structural strain IntroductionThe Bond-Valence Model (Brown, 2002(Brown, , 2016 is used extensively in Crystallography and Mineralogy to validate structural arrangements in crystals, and to interpret many aspects of crystal structure in terms of constituent chemical composition and bond topology. There are two distinct aspects of the bond-valence model: (1) relating observed bond-lengths to bond valences through experimentally determined bond-valence curves (e.g., Brown & Shannon, 1973;Brown & Altermatt, 1985;Brese & O'Keeffe, 1991;Gagné & Hawthorne, 2015), and (2) using bond-valence theory to understand chemical and topological aspects of atomic arrangements.There are two important theorems in the bond-valence model (Brown, 2002(Brown, , 2016: [1] the valencesum rule, which states that the sum of the bond valences at each atom is equal to the magnitude of the ato...