This July, I spent a week in Brussels - a capital well-known for its striking silver Atomium structure that stands out as a futuristic symbol on the city skyline. Beyond the Atomium, Brussels is home to a richer science scene: including its Institute of Natural Sciences. This museum is most visited for its impressive collection of dinosaur remains and early hominid ancestors, yet hidden in the basement is a dazzling exhibition of mineral specimens that revealed the complex molecular geometry behind the facets and vertices of crystal topology.
One thing that I noticed as I walked around the exhibition was the fact that so many of the samples - disguised in a full array of colours, shapes and sizes - were all ultimately the same mineral: quartz. Quartz is the second most abundant mineral in the Earth's crust after feldspar, but this does little to account for the fact that every variety of quartz is so strikingly unique - even the majority of sand grains on the beach are, in reality, miniature quartz crystals. So what makes each quartz deposit so different?
To understand this, we must first distinguish between the theoretical crystal form of quartz and the true crystal habits it exhibits in real life as a result of variations in its formation. The form of quartz is a model of how the 'perfect' crystal should look in its pure form, unaffected by any chemical or physical factors.
Quartz also goes by the name of silica, or silicon dioxide, with a molecular formula of SiO2. Since silicon is tetravalent like carbon, it also takes on a similar molecular shape when it is in a lattice. Each oxygen atom in the crystalline lattice is shared between two silicon atoms, so that each silicon atom has a coordination number of four, meaning that there are four oxygen atoms around each silicon despite the overall empirical formula being SiO2. The compound therefore takes on a tetrahedral arrangement with a bond angle of 109.5 degrees between each Si-O covalent bond. This shape produces a non-planar hexagon which can be visualised in this simplified representation of a section of the structure that I generated using MolView 3D. .png)
.png)
The agglomeration of the tetrahedral units produces a hexagonal prism shape. This has 4 distinct axes in its geometry: 3 are of equal length and on the same plane and 120 degrees apart from each other, with a fourth perpendicular to this from the point of intersection and at a different length. As crystallisation occurs (in ideal conditions), the molecules are added on the same plane as existing molecules, causing the formation of these neat edges and faces instead of a curved or irregular structure.
Another element to consider is the actual process of crystallisation, or nucleation. At a high temperature and pressure, the silica is able to dissolve. Then, as the solution cools, it enters a state known as metastable supersaturation in which the concentration of the solution is higher than the solubility yet the mineral still remains in solution. This is considered metastable because the solution is currently in a local minimum energy - not the lowest possible energy state (achieved by overcoming the nucleation barrier) but still a stable intermediate. Upon further cooling, the boundary of this metastable state is reached and as it is overcome, crystallisation occurs. A true theoretical crystal achieves this homogenously without a scaffold of foreign particles. In reality, it is much faster and easier to form a nucleus on a heterogenous base of an existing impurity (link how a snowflake forms its crystal structure on the nucleus of a dust particle in the atmosphere). Nucleation - in spite of the initial energy barrier that must be overcome in crystallisation - is energetically favourable and facilitates further crystal growth. Because supersaturation makes it easier to form nuclei, the crystals formed in a highly supersaturated condition are smaller than those formed under less supersaturation because nucleation is favoured over growth of existing crystals. Likewise, for low supersaturation, the crystals will be larger because growth is more favourable than forming an entirely new crystal.
The habit of a crystal is how it actually forms if the conditions and specific differences of formation are taken into account. It is the variation in habit of quartz which results in the formation of several different kinds of appearance and structure that are visible in nature.
Structural deformations to crystals often include interstitial defects (when extra atoms are present in a space that they are not typically found) and the inverse, vacancy defects. These often occur at the same time within the structure. For example, the Frenkel defect is marked by the movement of an atom from one place to another, causing an interstitial defect in the destination and a vacancy defect in the position it used to occupy.
Impurities are the result of unwanted chemicals that are present in the structure during formation. However, in the commercial world these are rarely unwanted changes - in fact, the wide spectrum of quartz colours is desired and manufacturers often treat certain kinds of quartz with heat to cause them to change colour. An example of this is the production of citrine from amethyst: amethyst is a variety of quartz given its striking purple colour by the presence of iron (IV) ions within its lattice. Transition metals such as iron are able to absorb and reflect certain wavelengths of light, giving them bright colours. The particular wavelength of light that is emitted is relative to the oxidation state the ion is in, and changes as the ion is either oxidised or reduced. In the formation of golden citrine from amethyst, high temperatures are used to reduce the iron (IV) ions to iron (III) ions, which causes a shift in the wavelength of light reflected, thus changing the mineral colour. This is a widely-used manufacturing process because citrine is so rare in nature, since it requires the natural amethyst to undergo similar naturally-occurring high levels of heat. 
Finally, another kind of deformation is twinning. This is when multiple crystals join together as they form. The Brussels Institute has handily colour coded the faces in the diagram below of a quartz that shows how multiple quartz crystals can form together and still retain the consistency of their planes. Even though this diagram shows quartz crystals all facing the same direction attached by the red-coded faces, twinning can also happen as another crystal joins to one of the green or yellow-coded faces - during which the crystal will grow at a sharp angle to create an interesting (and much spikier) structure.
Above is a quartz crystal in my collection - you can easily see how its faces at the perfectly clear tip match up with the shaded diagram in the exhibition and the hexagonal model presented earlier. This proves just how predictable a natural phenomenon can be with the use of geometric and thermodynamic rules. The other end of the crystal, however, has become crushed and deformed over many years of weathering and wear into a unique and disorderly shape. Using mathematics can only do so much to predict how they will form: as I observed each specimen in the Institute's collection, I realised that we must instead learn to translate the intricate stories about the past written directly on the facets of each individual crystal.
References:
All accessed 22nd September 2024
https://www.unearthedstore.com/blogs/geology-unearthed/how-do-crystals-form-into-different-shapes?srsltid=AfmBOoru01fMjNVAAtup8PlD9HHJOs6kmzet4s5pqo4xen0BOzXW9Bpi
https://www.mt.com/gb/en/home/applications/L1_AutoChem_Applications/L2_Crystallization/Supersaturation_Application.html
https://en.wikipedia.org/wiki/Crystal#Defects,_impurities,_and_twinning
https://en.wikipedia.org/wiki/Crystal_growth