Jubilite: A 4-,8-connected cubic structural pattern in space Group Pm3

Abstract

In the course of investigating structural modifications of the 3-,4-connected net known as the Pt<SUB>3</SUB>O<SUB>4</SUB> structure-type (waserite), a novel 4-,8-connected structure-type was discovered. This lattice is generated by replacing the 3-connected trigonal planar vertices of the Pt<SUB>3</SUB>O<SUB>4</SUB> structure-type with 4-connected tetrahedral vertices, to achieve a structure which possesses a generic empirical formula of JK<SUB>6</SUB>L<SUB>8</SUB>. In such a topological modification, the four 3-fold axes of the parent cubic, Pm3n, Pt<SUB>3</SUB>O<SUB>4</SUB> structure-type are retained. Thus the 4-connected tetrahedral vertices are oriented so as to preserve cubic symmetry in the resulting Pm3, JK<SUB>6</SUB>L<SUB>8</SUB> (jubilite) lattice. The unit cell contains a single 8-connected cubecentered vertex, six 4-connected distorted square planar vertices and eight 4-connected distorted tetrahedral vertices. It is a Wellsean structure with a Wells point symbol given by (4<SUP>16</SUP>6<SUP>4</SUP>8<SUP>4</SUP>)(4<SUP>2</SUP>8<SUP>2</SUP>)<SUB>6</SUB>(4<SUP>3</SUP>8<SUP>3</SUP>)<SUB>8</SUB> and a Schläfli symbol of (5<SUP>3/4</SUP>, 4.2667). This latter index reveals a decrease in the lattice's polygonality and concomitant increase in the connectivity through the transformation from waserite to jubilite. The topology of the parent waserite lattice (Pt<SUB>3</SUB>O<SUB>4</SUB>) corresponds to that of the Catalan structures with the Wells point symbol (8<SUP>4</SUP>)<SUB>3</SUB>(8<SUP>3</SUP>)<SUB>4</SUB>, which has the Schläfli symbol (8, 3.4285). Finally, it can be seen that a sequence of structure-types starting with waserite (Pt<SUB>3</SUB>O<SUB>4</SUB>) and moving to jubilite (JK<SUB>6</SUB>L<SUB>8</SUB>) and finally to fluorite (CaF<SUB>2</SUB>) represents a continuous crystallographic structural transformation in which the symmetry and topology undergo concomitant changes from one structure-type (waserite) to the other structure-types. The topology of the fluorite lattice, represented by the Wells point symbol (4<SUP>24</SUP>)(4<SUP>6</SUP>)<SUB>2</SUB>, and the Schläfli symbol (4, 5<SUP>1/3</SUP>), indicates a discontinuous topological transformation from the intermediate jubilite lattice; like the discontinuous topological transformation from Pt<SUB>3</SUB>O<SUB>4</SUB> to JK<SUB>6</SUB>L<SUB>8</SUB>; in which the polygonality is again reduced, in this step from 5<SUP>3/4</SUP> to 4, and the connectivity is concomitantly increased, from 4.2667 to 5<SUP>1/3</SUP>, during the continuous structural transformation. The l index, a topological measure of the form of a structure-type in terms of the ratio of the weighted average polygonality to the weighted average connectivity in the unit cell, decreases in the sequence from Pt<SUB>3</SUB>O<SUB>4</SUB> (2.3333) to jubilite (1.3476) to fluorite (0.7500). This indicates the discontinuous, though monotonic, appearance of more closed networks upon increasing the connectivity and concomitantly decreasing the polygonality in the structural sequence. Interestingly, the ratios of the form indexes of the adjacent members in this series: l<SUB>waserite</SUB>/l<SUB>jubilite</SUB> and l<SUB>jubilite</SUB>/l<SUB>fluorite</SUB>, are approximately equal to each other.