Supercell (crystal)

The basis vectors of unit cell U ${\textstyle ({\vec {a}},{\vec {b}},{\vec {c}})}$ can be transformed to basis vectors of supercell S ${\textstyle ({\vec {a}}',{\vec {b}}',{\vec {c}}')}$ by linear transformation[1]

$${\displaystyle {\begin{pmatrix}{\vec {a}}'&{\vec {b}}'&{\vec {c}}'\\\end{pmatrix}}={\begin{pmatrix}{\vec {a}}&{\vec {b}}&{\vec {c}}\\\end{pmatrix}}{\hat {P}}={\begin{pmatrix}{\vec {a}}&{\vec {b}}&{\vec {c}}\\\end{pmatrix}}{\begin{pmatrix}P_{11}&P_{12}&P_{13}\\P_{21}&P_{22}&P_{23}\\P_{31}&P_{32}&P_{33}\\\end{pmatrix}}}$$

where ${\textstyle {\hat {P}}}$ is a transformation matrix. All elements ${\textstyle P_{ij}}$ should be integers with ${\textstyle \det({\hat {P}})>1}$ (with ${\textstyle \det({\hat {P}})=1}$ the transformation preserves volume). For example, the matrix

$${\displaystyle P_{P\rightarrow I}={\begin{pmatrix}0&1&1\\1&0&1\\1&1&0\\\end{pmatrix}}}$$

transforms a primitive cell to body-centered. Another particular case of the transformation is a diagonal matrix (i.e., ${\textstyle P_{i\neq j}=0}$). This called diagonal supercell expansion and can be represented as repeating of the initial cell over crystallographic axes of the initial cell.

Supercells are also commonly used in computational models of crystal defects to allow the use of periodic boundary conditions[2].

• Crystal structure
• Bravais lattice
• Primitive cell
• Space group

• IUCR online dictionary of crystallography