Crystal Systems

The hexagonal lattice has one axis perpendicular to two other axes that are equal in length and intersect at a $120^\circ$ angle. It is another close-packed structure (HCP). Lattice parameters: $a = b \neq c$, and angles $\alpha = \beta = 90^\circ$, $\gamma = 120^\circ$. Examples: Magnesium, Zinc, Titanium, Graphite.

The monoclinic system has three unequal axes. Two axes intersect at an oblique angle, while the third is perpendicular to the plane of the other two. Lattice parameters: $a \neq b \neq c$, and angles $\alpha = \gamma = 90^\circ$, $\beta \neq 90^\circ$. Examples: Gypsum, Orthoclase.

The orthorhombic system consists of three mutually orthogonal axes, all of which are of different lengths. It can be thought of as a rectangular prism. Lattice parameters: $a \neq b \neq c$, and angles $\alpha = \beta = \gamma = 90^\circ$. Examples: Sulfur (rhombic), Olivine.

Also known as trigonal, the rhombohedral system is defined by three equal axes that are equally inclined to each other at an angle other than $90^\circ$. It can be thought of as a cube stretched along its body diagonal. Lattice parameters: $a = b = c$, and angles $\alpha = \beta = \gamma \neq 90^\circ$. Examples: Quartz, Calcite, Bismuth.

OverviewThe simple cubic (sc) structure is the simplest crystal lattice system. In this arrangement, atoms are located exactly at the eight corners of a cube. While it is fundamental to understanding crystallography, very few elements (only Polonium, $\text{Po}$) crystallize in this structure under normal conditions because of its low packing efficiency.Geometrical PropertiesThe simple cubic system has equal edge lengths and all angles are right angles. The lattice parameters are defined as: $$ a = b = c $$ $$ \alpha = \beta = \gamma = 90^\circ $$Atomic Packing Factor (APF)The Atomic Packing Factor represents the fraction of volume in a crystal structure that is occupied by constituent particles.For a simple cubic unit cell, the atoms strictly touch along the cube edges. Therefore, the relationship between the atomic radius ($r$) and the lattice constant ($a$) is: $$ a = 2r $$Since there is exactly $1$ atom per unit cell (8 corners $\times \frac{1}{8}$ atom per corner), the total volume of atoms in the cell ($V_{atoms}$) is: $$ V_{atoms} = 1 \times \left( \frac{4}{3} \pi r^3 \right) $$The total volume of the cubic unit cell ($V_{cell}$) is simply: $$ V_{cell} = a^3 = (2r)^3 = 8r^3 $$Thus, the Atomic Packing Factor is calculated as: $$ \text{APF} = \frac{V_{atoms}}{V_{cell}} = \frac{\frac{4}{3} \pi r^3}{8r^3} = \frac{\pi}{6} \approx 0.524 $$ This means that only about $52.4%$ of the unit cell volume is filled with atoms, leaving $47.6%$ as empty space.

The tetragonal crystal system results from stretching a cubic lattice along one of its vectors, so that the cube becomes a rectangular prism with a square base. Lattice parameters: $a = b \neq c$, and angles $\alpha = \beta = \gamma = 90^\circ$. Examples: White Tin, Rutile (TiO$_2$).

The triclinic system is the least symmetric of all crystal systems. All three axes are of unequal length, and none of them intersect at right angles. Lattice parameters: $a \neq b \neq c$, and angles $\alpha \neq \beta \neq \gamma \neq 90^\circ$. Examples: Microcline, Turquoise.