Model pyroxenes I: Ideal pyroxene topologies

Ideal pyroxenes are hypothetical structures based on ideal closest-packed arrangements of O anions. They are modeled after observed pyroxene structures and have the general formula M2M1T2O6, where M2 and M1 represent octahedrally coordinated cations, and T represents tetrahedrally coordinated cations. An algorithm has been created to construct all possible ideal pyroxenes based on closest-packed stacking sequences of length 12 or less. These structures are reported. The only significant structural parameters that vary between different ideal pyroxenes are the M1-T and M2-T distances. We show that the repulsive forces between these pairs of cations distinguishes the energetics of the ideal pyroxenes and may be important in determining the topologies of observed pyroxenes. ers, denoted A, B, and C in the traditional way (Patterson and Kasper 1959). For example, the anion arrangement in ideal P21cn protopyroxene can be described by the stacking sequence ABAC. Since all of the cations between a given pair of monolayers are exclusively tetrahedral or octahedral, we can modify the traditional ABC stacking sequence symbolism with superscripted Ts or Os to indicate tetrahedral or octahedral cation layers, respectively. The complete ideal P21cn protopyroxene can be described as ABAC, with octahedrally coordinated cations between AB and AC and tetrahedrally coordinated cations between BA and CA. It will be important to our discussion to distinguish between monolayer sandwiches with identical letters reversed, e.g., AB and BA. We define AB to mean that the atoms in the A-layer have smaller x-coordinates than those in the B-layer, and vice versa for BA. It will also be important to distinguish between the stacking sequence ABAC and the stacking sequence label ABAC. The former refers to the physical structure, a unique closest-packed arrangement of O anions; the latter refers to the four letters that represent the structure. The label can be manipulated using certain rules that represent changes of basis to derive equivalent labels representing the same stacking sequence or structure (Thompson and Downs 2001b). In this example, BABC, CBCA, ACAB, etc. all represent the same structure. Similarly, “pyroxene ABAC” refers to the unique physical structure, while “pyroxene label ABAC” refers to the non-unique sequence of letters representing that structure. In an ideal pyroxene, we define the basal faces of the tetrahedra as the faces parallel to (100). The two anions that are shared with other tetrahedra at the corners of these basal faces are referred to as the bridging O3 anions (Fig. 1). The nonbridging basal anions are referred to as O2 and the apical anions as O1. This nomenclature is consistent with the traditional labeling of atoms in observed structures. The chain-forming symmetrically equivalent edge-sharing octahedral sites are called M1 and are related to each other by THOMPSON AND DOWNS: MODEL PYROXENES I: IDEAL PYROXENE TOPOLOGIES 654 the c-glide running up the middle of the chain. Tucked into the kinks of the M1 chains are additional cation sites referred to as M2. They are also related to each other by the c-glide, but do not form continuous chains. In an ideal pyroxene structure, both M1 and M2 are at the centers of perfect octahedra. However, electron density analysis of observed structures has shown that M2 can have four, five, six, or eight coordination (Downs 2003). Adjacent octahedral chains within a given monolayer sandwich are linked by basal faces of tetrahedral chains in the monolayer sandwiches above and below (Fig. 2). This connects the structure in the b direction. The apical anions of the tetrahedral chains are shared with octahedral chains so that the tetrahedra also connect the structure in the a* direction. Thompson (1970) noted that in many observed pyroxenes, the O3-O3-O3 angle is about 180∞. He called these “extended structures” and we refer to this sort of tetrahedral chain as an E-chain (Fig. 3) after Papike et al. (1973). Thompson (1970) made model pyroxene chains with regular M1 and T. He pointed out that a rotation of the tetrahedra in a model E-chain in either direction by 30∞ about an axis passing through the apical O1 anion perpendicular to the (100) plane brings the anions into a closest-packed arrangement. If the basal faces of the rotated tetrahedra point in the same direction as the closest parallel octahedral faces in the octahedral chain at the apices of the tetrahedral chain, then he called the rotation an S-rotation and we call the tetrahedral chain an S-chain. A 30∞ rotation in the opposite sense leaves the tetrahedral basal faces pointing opposite to the octahedral faces and is called an O-rotation, producing an O-chain. It has become commonplace to use the O3-O3-O3 angles in natural pyroxenes as a way to quantify the degree of Sor Orotation (Thompson 1970; Papike et al. 1973; for more recent examples c.f. Arlt and Angel 2000; Tribaudino et al. 2002). If a tetrahedral chain in a pyroxene is O-rotated, then its O3-O3O3 angle is described as less than 180∞, while an S-rotated chain is described as having an O3-O3-O3 angle greater than 180∞. Although each tetrahedron in a fully rotated chain is only rotated 30∞ from the extended chain position, the resulting O3O3-O3 angle is formed by two tetrahedra so a fully rotated Ochain has an O3-O3-O3 angle of 120∞, while a fully rotated S-chain has an O3-O3-O3 angle of 240∞. Papike et al. (1973) pointed out that a fully rotated structure containing only Schains is hexagonal closest-packed and a fully rotated structure containing only O-chains is cubic closest-packed. Thompson (1970) used, but did not define, the term “tilt” to FIGURE 1. A portion of an ideal pyroxene structure viewed along a*. The M1O6 and M2O6 groups are illustrated as octahedra and TO4 groups as tetrahedra. Representative O atoms are illustrated as spheres and are labeled to indicate nomenclature. FIGURE 2. A portion of an ideal pyroxene structure viewed along c. Tetrahedra bridge the adjacent octahedral chains in three dimensions, connecting the pyroxene structure. The M1O6 groups are illustrated as octahedra, and M2 as a sphere. FIGURE 3. Portions of three different model pyroxene structures viewed along a* to illustrate chain configurations. An E-chain has an O3-O3-O3 angle of 180∞, and is not closest-packed. An O-chain has an O3-O3-O3 angle of 120∞ and is cubic closest-packed. An S-chain has an O3-O3-O3 angle of 240∞ and is hexagonal closest-packed. In order to avoid confusion when determining O3-O3-O3 angles, imagine that c points north and b points east. The O3-O3-O3 angle is determined by any two adjacent tetrahedra that are pointing along –a* and have a southeastern and a northwestern orientation relative to each other. THOMPSON AND DOWNS: MODEL PYROXENES I: IDEAL PYROXENE TOPOLOGIES 655 describe the orientation of the octahedra in pyroxenes. Papike et al. (1973) compared Thompson’s (1970) ideal pyroxenes to real pyroxenes and formalized the definition of octahedral “tilt” as follows: if the lower triangular face parallel to (100) of an octahedron points in the –c direction, then it has a negative tilt, denoted “–tilt,” and if it points in the +c direction, then it has a positive tilt, denoted “+tilt.” The tetrahedral chains between a given monolayer sandwich have two orientations (Fig. 2). The tetrahedra in half of the chains point down a*, the rest point up a*. Thompson (1970) introduced the parity rule, which states that if the chains pointing up and the chains pointing down between a given monolayer sandwich in an ideal pyroxene are rotated in the same direction (i.e., both are either Sor O-chains), then the octahedral chains above and below them must have the same tilt and if the tetrahedral chains are rotated in the opposite directions (i.e., one Sand one O-chain), then the octahedra must have opposite tilts. Thompson’s parity rule only applies to ideal pyroxenes, and is not obeyed by observed pyroxenes with space groups Pbca and Pbcn (Papike et al. 1973). Pannhorst (1979, 1981) made model pyroxene structures containing both O and E-chains. He argued that both symmetry and M2-O bonding topologies should be included in pyroxene classification schemes. Law and Whittaker (1980) derived space groups for all possible ideal pyroxenes based on stacking sequences of length four and eight. They pointed out that octahedral tilt is arbitrarily dependent on the direction of the c-axis. Consider the six possible pairings of the three different closest-packed O atom monolayers: AB, BA, AC, CA, BC, and CB. If a basis is chosen so that an octahedral layer between an AB pair has a +tilt, then octahedral layers between BC and CA pairs will also have +tilts, while octahedral layers between BA, CB, and AC pairs will have –tilts. Also, if three consecutive monolayers can be described with only two different letters (i.e., ABA), then the tetrahedral chain is an S-chain, while three different letters indicates an O-chain. For example, the ideal pyroxene portion ABAC has one tetrahedral layer. Half of the tetrahedra in this layer are associated with the Oa octahedral chain, half with the Ob octahedral chain. Those associated with Oa form an S-chain while those associated with Ob form an Ochain. If Oa has +tilt, then Ob has –tilt. This partial pyroxene could be described symbolically in the Law and Whittaker (1980) notation as +SO– (equivalents –SO+, –OS+, +OS– can be obtained by changing basis). Completing this pyroxene by placing a tetrahedral layer between the CA monolayer sandwich results in a pyroxene with traditional representation +SO– SO. Law and Whittaker (1980) thus established a correspondence between SO+– pyroxene representation and closest-packing representation. The only reported crystal structure data for an ideal pyroxene prior to this study is from Hattori et al. (2000). They determined that the ideal representation of FeGeO3 was cubic closest-packed (CCP) and showed that FeGeO3 approached the ideal arrangement with increasing pressure. They derived the ideal structure in order to quantify the distortion of their observed crystal from ideal cubic closest-packed. No studies have presented structural data for other ideal pyroxenes. Thompson and Downs (2001a) created an algorithm to quantify the distortion from hexagonal and cubic closest-packing in crystals provided the crystals are not too distorted. In particular, they showed that C2/c pyroxenes with eight-coordinated M2 sites rapidly move toward CCP with pressure and away from it with temperature. Under the assumption that anion-anion interactions are the principle component of the forces governing compression mechanisms in pyroxenes, we undertook a study of the ideal structures. We intended to determine if all pyroxenes move toward ideal closest-packed with pressure. Furthermore, comparing the energetics of ideal analogs may indicate why we see only a few of the many possible pyroxenes in nature and why they behave the way they do with temperature, pressure, and composition. In particular, we are searching for an understanding of the sequences of structures adopted during pyroxene-pyroxene transitions. As a first step, Thompson and Downs (2001b) created an algorithm that generates all symmetrically nonequivalent closest-packed stacking sequences of given length N using group theory. We have taken all of the closest-packed stacking sequences of length 12 or less and designed an algorithm to create each of the 81 possible pyroxenes based on those sequences. Our study is restricted to stacking sequences of length 12 or less because no observed pyroxenes have been reported with closest-packed analogs having longer stacking sequences.