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 (oxidation state) 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 bond-valences are calculated for selected rock-forming minerals, beginning with a simple example (spinel, = 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 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.

Bond-Length Distributions for Ions Bonded to Oxygen: Results for the Transition Metals and Quantification of the Factors Underlying Bond-Length Variation in Inorganic Solids

“…41 The main axioms of the bond-valence model, analogous to Kirchhoff's rules for electrical circuits, are: [1] the valence-sum rule, which states that the sum of the directed bond valences around an ion is equal to its oxidation state (essentially, a modernization of Pauling's 2 nd rule), and [2] the path rule which states that the sum of the directed bond valences along any path of bonds in a structure is zero where the path begins and ends on symmetrically equivalent ions. 42 Although the model finds many applications both in solution and in the solid state (summarized in refs [ 41,43 ]), its most common use is to serve as a check on newly refined crystal structures via verification of the valence-sum rule. Key to this model is the relation between the length of a bond and its strength (called its bond valence):…”

“…46 In addition, the valence-sum rule allows inferring the oxidation state of redox-active ions; [47][48][49][50] this is particularly relevant for confirming the oxidation state of transition metals, and to resolve mixed-valence site occupancy. Of greatest relevance to this work, the bond-topological underpinnings of the bond-valence model allow prediction of the a priori bond valences (thus bond lengths) of crystal structures; 42 as we show below, crystal structures often have intrinsic requirements for uneven distribution of bond valences (and thus bond lengths). Next, we investigate the inner-workings of this phenomenon, and the extent to which it results in bondlength variation for transition metals bonded to O 2-.…”

“…Where such constraints are absent or have negligible effect, a crystal structure is observed with the lowestpossible number of crystallographically distinct sites, i.e., equal to the number of distinct elements in the compound (e.g., for spinel: MgAl2O4). In this configuration, cations and anions distribute their a priori (ideal) bond valences evenly (see ref [ 42 ] for their calculation), resulting in coordination polyhedra with identical or similar bond lengths. With considerable constraints acting on the system at the time of crystallization, crystallographically distinct sites may rapidly outgrow the number of distinct elements in the compound.…”

On the Crystal Chemistry of Inorganic Nitrides: Crystal-Chemical Parameters and Opportunities in the Exploration of Their Compositional Space

“…This is likely a consequence of sampling a wide range of meta-stable nitride structures, 3 as opposed to thermodynamically-stable minerals which make up a greater fraction of the oxide and oxysalt data of Gagné & Hawthorne. 59 Meta-stable structures generally entail less-than-perfect mapping of bond-length constraints in three-dimensional space (which can be calculated a priori 72 ) under the constraints of symmetry and periodicity, leading to higher variations in mean bond lengths across structure types in comparison to thermodynamically-stable structures. 73 We compared cation and anion bond-valence sums (BVS) obtained using the parameters of this work and the set of soft bond-valence parameters of Chen & Adams 74 which they adapted to the first coordination shell (but did not evaluate for anion BVS).…”