Magnesium silicate, \(\mathrm{Mg}_{2} \mathrm{SiO}_{4}\), forms in the olivine crystal structure that consists of an HCP arrangement of \(\mathrm{O}^{2-}\) ions. (a) Which type of interstitial site will the \(\mathrm{Mg}^{2+}\) ions occupy? Why? (b) Which type of interstitial site will the \(\mathrm{Si}^{4+}\) ions occupy? Why? (c) What fraction of the total tetrahedral sites will be occupied? (d) What fraction of the total octahedral sites will be occupied?

In the study of minerals and crystallography, the hexagonal close-packed (HCP) structure is a highly organized arrangement of atoms or ions within a crystal. Imagine a tight grouping where each layer of atoms is closely packed, with the next layer fitting into the gaps of the previous one, kind of like stacking oranges at a grocery store. This configuration ensures the most efficient use of space.

Within the HCP structure, atoms align in a repeating pattern of two alternating layers, which we often label as A and B layers. The atoms in layer A are positioned directly above and below the gaps in layer B, creating a three-dimensional framework. This pattern is repeated throughout the crystal, resulting in a structure that is both stable and dense.

The HCP structure is fundamental to understanding various properties of crystals, such as their density, strength, and how they interact with other elements and compounds. When it comes to minerals like magnesium silicate (olivine), recognizing how the HCP arrangement allows oxygen ions to form a foundation for other ions to fit in between is critical for understanding the material's overall structure.

Interstitial sites in a crystal are like the tiny nooks and crannies within a sponge, except in this case, the sponge is the regular lattice of ions in a crystal structure. These sites are the spaces between the ions that make up the crystal lattice, where smaller ions can fit. There are majorly two types of interstitial sites in HCP structures: tetrahedral and octahedral.

Tetrahedral Sites

Envision a pyramid with a triangular base; this is the shape of a tetrahedral site. Each site is surrounded by four ions, forming a tetrahedron. The compact nature of these sites makes them suitable for smaller ions.

Octahedral Sites

Now picture a figure with eight faces, each shaped like a diamond — that's an octahedral site. These are surrounded by six ions and can usually accommodate ions that are larger than those in tetrahedral sites but still not too large for the crystal lattice.

The ion's size and charge, along with the crystal's stoichiometry, influence which interstitial site an ion will occupy. For example, in olivine, magnesium ions favor the larger octahedral sites due to their size relative to the available space and the crystal's stoichiometric balance.

Ionic radii refer to the measured sizes of ions within a crystal lattice, which are typically smaller or larger than the neutral atoms from which they’re formed, depending on whether they've lost or gained electrons.

In the context of crystal structures, think of ionic radii as the fitting size for a shoe. If an ion is too large (a big foot), it will look for a roomier space (a bigger shoe), such as an octahedral site in an HCP structure. Conversely, if an ion is smaller (a small foot), it might prefer the snug fit of a tetrahedral site (a smaller shoe).

For example, in the olivine structure, the larger silicon ions, with their greater ionic radius compared to magnesium ions, settle into the smaller tetrahedral sites which offer a better fit. This precise ‘fitting’ is vital in determining the stability and formula of the mineral. Each ion's radius helps us predict where it will reside in the crystal, informing us about the material's properties, reactivity, and potential applications in various industries.