## The Triclinic Crystal System

The triclinic is the most general crystal system. All of the other crystal systems can be considered special cases of the triclinic. The primitive vectors are also completely general: their lengths ($a$, $b$, $c$) and angles ($\alpha$, $\beta$, $\gamma$) may have arbitrary values. The triclinic system has one Bravais lattice, which is also the conventional lattice for this system.

### Lattice 1: Triclinic

There are many choices for the primitive vectors in the triclinic system. We make the choice $\begin{eqnarray} \mathbf{a}_1 & = & a \, \mathbf{\hat{x}} \nonumber \\ \mathbf{a}_2 & = & b \, \cos\gamma \, \mathbf{\hat{x}} + b \, \sin\gamma \,\mathbf{\hat{y}} \nonumber \\ \mathbf{a}_3 & = & c_x \, \mathbf{\hat{x}} + c_y \, \mathbf{\hat{y}} + c_z \, \mathbf{\hat{z}}, \end{eqnarray}$ where $\begin{eqnarray*} c_x & = & c \, \cos\beta \\ c_y & = & \frac{c \, (\cos\alpha - \cos\beta \, \cos\gamma)}{\sin\gamma} \end{eqnarray*}$ and $\begin{eqnarray*} c_z & = & \sqrt{c^2 - c_x^2 - c_y^2}. \end{eqnarray*}$
The volume of the triclinic unit cell is $$$V = a \, b \, c_z \, \sin\gamma.$$$ The space groups associated with the triclinic lattice are \begin{array}{ll} 1. ~ \mbox{P1} & 2. ~ \mbox{P}\overline{1} \\ \end{array}