The corundum crystal structure, found for \(\mathrm{Al}_{2} \mathrm{O}_{3},\) consists of an \(\mathrm{HCP}\) arrangement of \(\mathrm{O}^{2-}\) ions; the \(\mathrm{Al}^{3+}\) ions occupy octahedral positions. (a) What fraction of the available octahedral positions are filled with \(\mathrm{Al}^{3+}\) ions? (b) Sketch two close-packed \(\mathrm{O}^{2-}\) planes stacked in an \(A B\) sequence, and note octahedral positions that will be filled with the \(\mathrm{Al}^{3+}\) ions.

When we delve into the intriguing world of crystal structures, it's essential to understand the arrangement of atoms or ions within the lattice. One common arrangement is the Hexagonal Close-Packed (HCP) structure. In this layout, ions are packed in a way that maximizes their number within a given volume, which results in an efficient use of space.

In the HCP arrangement, each ion lies at the corners of a hexagon and also at the center of the hexagon. Imagine a layer of marbles lying in a tray where each marble touches three others to its left and three to its right, forming a triangle both above and below it. Now, stack another layer exactly on top, and you will have portions of two opposite triangles of the first layer in between four marbles of the second layer. This is a small glimpse into how HCP is structured.

Furthermore, for materials like corundum (Al2O3), the oxygen ions (O2-) form the HCP structure. This specific arrangement influences various physical properties of the material like density, hardness, and slippage planes, making it fundamental in understanding how such materials will behave in different applications.

Where Atoms Find a Home in Crystals

Within the HCP structure, there exist little 'pockets' or interstices called octahedral interstitial sites. Imagine a void surrounded by six ions forming an octahedron — that's an octahedral interstitial site. In corundum's case, these sites serve as a home for aluminum ions (Al3+).

Now, imagine you're stacking oranges in a box: In between the layers, you can nestle smaller cherries in the gaps. These cherries are like interstitial atoms fitting into the gaps between the larger close-packed ions. Not every gap is filled; in Al2O3, only two-thirds of these octahedral sites contain an aluminum ion. This partial filling is crucial—it's what maintains the crystal's overall charge balance and stability, and consequently, determines its material properties.

Getting the Proportions Right

Stoichiometry, in the context of crystal chemistry, refers to the definitive ratio of atoms in a compound. For aluminum oxide or corundum (Al2O3), the stoichiometry is reflected in its formula: two aluminum ions (Al3+) for every three oxygen ions (O2-).

This isn't just a matter of counting atoms; it's about the entire structure's balance. Since oxygen ions form the close-packed layers and make up the majority of the crystal, you can think of them as the 'backbone' or framework. Aluminum ions, the 'guests,' are nestled within this framework in the octahedral sites. Thanks to the 2-to-3 ratio, the structure achieves electrical neutrality and the geometry that gives corundum its characteristic properties, such as hardness and resistance to corrosion. Understanding the stoichiometry of Al2O3 is not just about numbers; it's about the harmony of construction that allows the material to function in various applications, from abrasives to laser components.