Key Crystal Classes to Know for Mathematical Crystallography

Crystal classes are essential in understanding the symmetry and structure of crystals in Mathematical Crystallography. Each class, from triclinic to cubic, showcases unique arrangements of axes and symmetry elements, influencing the properties of various minerals.

1. Triclinic

   • Characterized by three unequal axes that are not perpendicular to each other.
   • Lacks any symmetry elements, making it the least symmetric crystal system.
   • Commonly found in minerals like feldspar and turquoise.
   • The angles between the axes are all different, leading to a complex internal structure.

2. Monoclinic

   • Features two axes of unequal length and one axis that is perpendicular to the other two.
   • Contains a single two-fold rotational axis or a mirror plane, providing some symmetry.
   • Common examples include gypsum and orthoclase.
   • The angles between the axes are not all 90 degrees, creating a distinct slant.

3. Orthorhombic

   • Composed of three mutually perpendicular axes of unequal lengths.
   • Exhibits higher symmetry with three two-fold rotational axes.
   • Commonly found in minerals such as olivine and barite.
   • The rectangular shape of the unit cell simplifies calculations in crystallography.

4. Tetragonal

   • Contains two axes of equal length and one axis that is different, all at right angles.
   • Features a four-fold rotational axis, enhancing its symmetry.
   • Common examples include zircon and rutile.
   • The square base of the unit cell allows for straightforward geometric analysis.

5. Trigonal

   • Characterized by three equal axes that intersect at 120-degree angles.
   • Contains a three-fold rotational axis, contributing to its symmetry.
   • Commonly found in minerals like quartz and calcite.
   • Often considered a subset of the hexagonal system due to its geometric properties.

6. Hexagonal

   • Composed of four axes: three equal axes in a plane at 120 degrees and one perpendicular axis.
   • Exhibits a six-fold rotational axis, providing significant symmetry.
   • Common examples include graphite and beryl.
   • The hexagonal unit cell facilitates the study of layered structures in crystallography.

7. Cubic

   • Features three equal axes that are all perpendicular to each other.
   • Exhibits the highest symmetry with four three-fold rotational axes and multiple mirror planes.
   • Commonly found in minerals like salt and diamond.
   • The simplicity of the cubic structure makes it a fundamental model in crystallography.