A crystal system classifies crystals based on the specific geometric arrangement of their constituent atoms, ions, or molecules. This ordered, repeating structure is known as the crystal lattice, and its fundamental building block is the unit cell. The unit cell’s shape and symmetry determine which crystal system a material belongs to. The scientific community recognizes seven fundamental crystal systems that categorize all known crystalline solids.
The Basis of Crystal Classification
The classification of crystals into one of the seven systems is based on the geometry of the unit cell, which is the smallest repeating volume that shows the complete symmetry of the crystal structure. This geometry is defined by six parameters: the lengths of the three crystallographic axes (\(a\), \(b\), and \(c\)) and the three angles between those axes (\(\alpha\), \(\beta\), and \(\gamma\)). The axes \(a\), \(b\), and \(c\) represent the edges of the unit cell.
The angle \(\alpha\) is between the \(b\) and \(c\) axes, \(\beta\) is between \(a\) and \(c\), and \(\gamma\) is between \(a\) and \(b\). The relationships between these six parameters—whether the axial lengths are equal or unequal, and whether the angles are \(90^{\circ}\)—dictate the overall symmetry of the unit cell. Higher symmetry systems have more constraints on these lengths and angles, while lower symmetry systems have fewer constraints.
The Seven Crystal Systems
The seven crystal systems are categorized in descending order of symmetry, moving from the cubic system to the triclinic system. The most symmetrical is the Cubic system, where all three axes are of equal length (\(a=b=c\)) and all three angles are \(90^{\circ}\) (\(\alpha=\beta=\gamma=90^{\circ}\)). This structure provides a high degree of symmetry, exemplified by common materials like table salt (sodium chloride) and the mineral iron.
The Tetragonal system retains three axes at right angles but features two axes of equal length and one that is different (\(a=b \neq c\), \(\alpha=\beta=\gamma=90^{\circ}\)). This means the unit cell is essentially a square prism, which is the structure found in the mineral zircon.
A reduction in symmetry leads to the Orthorhombic system, where all three axes are of different lengths, but they all remain perpendicular to one another (\(a \neq b \neq c\), \(\alpha=\beta=\gamma=90^{\circ}\)). Minerals such as topaz and barite crystallize with this geometry.
The Hexagonal system is described by four axes instead of three. Three equal-length axes lie in a plane at \(120^{\circ}\) to each other, and the fourth axis is perpendicular to that plane and of a different length (\(a_1=a_2 \neq c\), \(\alpha=\beta=90^{\circ}\), \(\gamma=120^{\circ}\)). This distinctive six-sided symmetry is characteristic of ice and the mineral beryl.
Closely related is the Trigonal system, often also called the rhombohedral system, which is defined by three equal axes that are equally inclined but not at \(90^{\circ}\) (\(a=b=c\), \(\alpha=\beta=\gamma \neq 90^{\circ}\)). Calcite is a common example of a material that forms in the trigonal system.
The Monoclinic system is characterized by three unequal axes, with two pairs of axes perpendicular and the third pair not perpendicular (\(a \neq b \neq c\), \(\alpha=\gamma=90^{\circ}\), \(\beta \neq 90^{\circ}\)). This means that one axis is tilted relative to the others, which is the crystal structure of the common mineral gypsum.
The least symmetrical is the Triclinic system, where all three axes are of different lengths and all three angles are unequal and not \(90^{\circ}\) (\(a \neq b \neq c\), \(\alpha \neq \beta \neq \gamma \neq 90^{\circ}\)). This irregular geometry is found in minerals like turquoise and plagioclase feldspar.
How Structure Influences Material Properties
The geometric constraints of the seven crystal systems directly determine how a material will respond to external forces or energy, such as light, heat, or electricity. Highly symmetrical systems, like the cubic system, exhibit properties that are the same regardless of the direction in which they are measured, a characteristic called isotropy. For example, the thermal conductivity of a cubic crystal like iron is uniform whether heat is applied along the \(a\), \(b\), or \(c\) axis.
Conversely, crystals belonging to lower-symmetry systems, such as tetragonal, monoclinic, or triclinic, are anisotropic, meaning their physical properties vary depending on the crystallographic direction. The difference in atomic packing density along the different axes causes properties like electrical conductivity or the speed of light passing through the material to change based on the direction of measurement. An important example of anisotropy is the phenomenon of double refraction in hexagonal crystals like quartz, where light entering the material is split into two rays traveling at different velocities. This inherent link between the atomic-level symmetry and the bulk properties explains why understanding the crystal system is fundamental to materials science.