## Research

#### Our Research Interests

Figure 1: Cu_{41}Zn_{59}, Cu_{36.5}Al_{63.5}, and Cd_{3}Cu_{4}. |

Metals are important. Three-quarters of the elements on the periodic table are metals, and when metallic elements are combined, they make more metals. These combinations of metals have some amazingly complicated structures, some of which are shown here. Cu_{41}Zn_{59} (Figure 1, left panel) has a crystalline unit cell with 10 million atoms in it. Cu_{36.5}Al_{63.5} (Figure 1, center panel) has a cell axis of 1000 Å. And while Cu has 4 atoms in its cubic unit cell and Cd has 2, Cd_{3}Cu_{4} (Figure 1, right panel) has 1124. And these are just a few examples.

Figure 2: Pd_{59}Cd_{217} and Pd_{105}Cd_{343}. |

Figure 3: Perpendicular pseudo 5-fold axes in Cd_{3}Cu_{4}. |

Our group synthesizes intermetallic compounds with complex crystal structures (including the Pd-Cd compounds shown in Figure 2), and develops theoretical models to understand their structures. One structure of interest to us is that of Cd_{3}Cu_{4}. This compound has pseudo 5-fold symmetry along its [110] and [110] directions (Figure 3, in reciprocal space). This is interesting for two reasons. First, it’s interesting because 5-fold rotational symmetry is incompatible with crystalline translational symmetry, which has already been explored in the context of quasicrystals and their approximants. Of more interest to us is the fact that [110] and [110] are perpendicular to each other, but no 3-D point group has perpendicular 5-fold axes. How is it possible for Cd_{3}Cu_{4} to have perpendicular pseudo 5-fold axes, when this feature is not consistent with any 3-D point group? If you’re interested in the answer, see our recent article. (Berger, R.F.; Lee, S.; Johnson, J.; Nebgen, B.; Sha, F.; Xu, J. “The mystery of perpendicular 5-fold axes and the fourth dimension in intermetallic structures”. *Chem. Eur. J.* **2008**, *14*, 3908-3930.)

Figure 4: Ionic and metallic regions of NaCd_{2}. |

In another related mystery, we trace the complicated structure of NaCd_{2} (with 1192 atoms in its cubic unit cell), as studied using simple electronic structure calculations, to periodic minimal surfaces (Figure 4). You can read the full story in our recent article. (Fredrickson, D.C.; Lee, S.; Hoffmann, R. “Interpenetrating polar and nonpolar sublattices in intermetallics: The NaCd_{2} structure”. *Angew. Chem. Int. Ed.* **2007**, *46*, 1958-1976.)