The Tetragonal System: Zircon, Rutile and Square-Based Crystals
In the tetragonal system, two axes are equal and the third differs, all perpendicular. Zircon, rutile, vesuvianite, wulfenite and apophyllite illustrate it.
Key points
| Axes | a = b ≠ c |
|---|---|
| Angles | α = β = γ = 90° |
| Other name | Quadratic (the traditional French term) |
| Characteristic symmetry | One four-fold rotation axis |
| Symmetry classes | 7 of 32 |
| Bravais lattices | 2 (primitive and body-centred) |
| Typical minerals | Zircon, rutile, cassiterite, vesuvianite, apophyllite, scheelite, wulfenite, chalcopyrite |
| Optics | Uniaxial: one direction without double refraction |
A cube stretched or squashed
Imagine a cube stretched in a single direction: you get a brick with a square base. That is the picture of the tetragonal system (also called quadratic). Two horizontal axes, a and b, have the same length; the third, c, which is vertical, has a different length. The three axes remain perpendicular.
The true signature of the system is a four-fold symmetry axis: turning the crystal a quarter turn around the c axis leaves it identical to itself. It is this that gives the crystals their square cross-section.
Values to remember
| Quantity | Value |
|---|---|
| Axis lengths | a = b ≠ c |
| Angles between axes | 90°, 90°, 90° |
| Minimum symmetry | 1 four-fold axis (order 4) |
| Symmetry classes | 7 |
| Bravais lattices | 2: primitive (P) and body-centred (I) |
The c/a ratio varies from one mineral to another. Calculated from published cell parameters, it is about 0.64 for rutile (a = 4.594 Å, c = 2.959 Å) and about 0.91 for zircon (a = 6.607 Å, c = 5.982 Å). The ångström (Å) is one ten-billionth of a metre. These ratios describe the unit cell, not the shape of the visible crystal: rutile, although its c axis is short, most often forms crystals elongated along c, because growth does not simply follow the proportions of the cell.
Forms and habits
Tetragonal crystals are often recognised by their square cross-section. The main forms are:
- The square prism: four side faces perpendicular to one another in pairs, like a square bar (zircon, vesuvianite, rutile).
- The square bipyramid: two square-based pyramids joined base to base, resembling an octahedron that has been “squashed” or “stretched” (zircon, wulfenite in pointed crystals, scheelite).
- The pinacoid: a flat face at the top, perpendicular to c; it gives crystals in the form of square tablets (wulfenite, apophyllite).
- The sphenoid: a wedge-shaped form that gives chalcopyrite the look of small, distorted tetrahedra.
This resemblance to the cubic system is why apophyllite, with its almost cubic crystals, long fooled the eye. An examination in polarised light or a measurement of angles settles the doubt.
Minerals to know
| Mineral | Simplified formula | Mohs hardness | Density | Remark |
|---|---|---|---|---|
| Zircon | ZrSiO4 | 6.5 to 7.5 | 4.6 to 4.7 | Adamantine lustre; the oldest known grains are about 4.4 billion years old |
| Rutile | TiO2 | 6 to 6.5 | 4.2 to 4.3 | Golden needles in quartz |
| Cassiterite | SnO2 | 6 to 7 | 6.8 to 7.1 | Main tin ore, very dense |
| Vesuvianite (idocrase) | silicate of calcium, magnesium and aluminium | 6.5 | 3.3 to 3.4 | Green to brown, striated prisms |
| Chalcopyrite | CuFeS2 | 3.5 to 4 | 4.1 to 4.3 | Iridescent brass-yellow, copper ore |
| Apophyllite | hydrated silicate of potassium and calcium | 4.5 to 5 | 2.3 to 2.4 | Transparent crystals, pearly lustre on the upper face |
| Wulfenite | PbMoO4 | 2.5 to 3 | 6.5 to 7 | Orange tablets, contains lead |
Zircon is a special case worth remembering: it can contain a little uranium and thorium, whose radioactivity gradually damages its structure. A very old zircon can become “metamict”, that is, almost amorphous, with a lower density and hardness. It is also thanks to zircon that rocks can be dated: a zircon grain from the Jack Hills, in Western Australia, has been dated to about 4.404 billion years (study published in 2001), making it the oldest dated terrestrial material. See our pages zircon, idocrase and apophyllite.
Optics: a uniaxial crystal
In a tetragonal crystal, light has one particular speed along the c axis and another in the perpendicular directions. The crystal is uniaxial: it has a single direction (the optic axis) in which double refraction disappears. In all other directions, a ray splits in two. This produces dichroism (two possible colours depending on the viewing angle) in coloured crystals, and strong double refraction in zircon, which gives the impression that the back edge of a facet is doubled when viewed through a loupe. See colour and pleochroism.
Knee twins and pitfalls
Rutile and cassiterite form “knee” twins: two crystals joined together and bent like a knee. Rutile also forms six- or eight-pointed stars through repeated twinning. These figures are natural and do not signal an imitation.
Common pitfalls: a “stone” sold as zircon in jewellery is sometimes in fact cubic zirconia (synthetic ZrO2, cubic and not tetragonal, density about 5.6 to 6). And the golden needles in a quartz are rutile inclusions, not quartz crystals. When in doubt about a valuable piece, a laboratory certificate is the only reliable answer; see labels and certificates.
Precaution for wulfenite: it contains lead and molybdenum. Wash your hands after handling it, do not put it in your mouth, and do not sand it.
Frequently asked questions
What is the difference between quadratic and tetragonal?
None: they are two names for the same system. “Tetragonal” is the standard term in English, while “quadratic” is the traditional term in French.
Which minerals are tetragonal?
Zircon, rutile, cassiterite, vesuvianite, apophyllite, scheelite, wulfenite and chalcopyrite, among others.
How can you recognise a tetragonal crystal by eye?
By its square cross-section and its faces meeting at right angles, with a four-fold axis. But only a measurement of angles or an optical analysis can confirm it.
Is zircon a radioactive stone?
It can contain very small amounts of uranium and thorium, which has no practical effect on a piece of jewellery. This natural radioactivity is mainly used to date rocks.
Why does rutile form needles in quartz?
These are rutile crystals that grew before or during the growth of the quartz, often along a precise direction. They are elongated along their c axis.
Also read
The meanings mentioned belong to tradition and symbolism. They are neither promises of results nor medical advice. Back to Mineralogy for beginners.