The Triclinic Crystal System

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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

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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 \[ \begin{equation} V = a \, b \, c_z \, \sin\gamma. \end{equation} \] The space groups associated with the triclinic lattice are \begin{array}{ll} 1. ~ \mbox{P1} & 2. ~ \mbox{P}\overline{1} \\ \end{array}