# Encyclopedia of Crystallographic Prototypes

• M. J. Mehl, D. Hicks, C. Toher, O. Levy, R. M. Hanson, G. L. W. Hart, and S. Curtarolo, The AFLOW Library of Crystallographic Prototypes: Part 1, Comp. Mat. Sci. 136, S1-S828 (2017). (doi=10.1016/j.commatsci.2017.01.017)
• D. Hicks, M. J. Mehl, E. Gossett, C. Toher, O. Levy, R. M. Hanson, G. L. W. Hart, and S. Curtarolo, The AFLOW Library of Crystallographic Prototypes: Part 2, Comp. Mat. Sci. 161, S1-S1011 (2019). (doi=10.1016/j.commatsci.2018.10.043)
• D. Hicks, M.J. Mehl, M. Esters, C. Oses, O. Levy, G.L.W. Hart, C. Toher, and S. Curtarolo, The AFLOW Library of Crystallographic Prototypes: Part 3, Comp. Mat. Sci. 199, 110450 (2021). (doi=10.1016/j.commatsci.2021.110450)

## The Cubic Crystal System

The cubic crystal system is defined as having the symmetry of a cube: the conventional unit cell can be rotated by 90$^\circ$ about any axis, or by 180$^\circ$ around an axis running through the center of two opposing cube edges, or by 120$^\circ$ around a body diagonal, and retain the same shape. The conventional cell then takes the form $\begin{array}{ccc} \mathbf{A}_1 & = & a \, \mathbf{\hat{x}} \nonumber \\ \mathbf{A}_2 & = & a \, \mathbf{\hat{y}} \nonumber \\ \mathbf{A}_3 & = & a \, \mathbf{\hat{z}}, \end{array}$ with unit cell volume $V = a^3.$
This is the limiting case of both the orthorhombic and tetragonal systems when all primitive vectors are equal in length. There are three Bravais lattices in the cubic system.

### Lattice 12: Simple Cubic

The simple cubic system is identical to the conventional cubic unit cell $\begin{array}{ccc} \mathbf{a}_1 & = & a \, \mathbf{\hat{x}} \nonumber \\ \mathbf{a}_2 & = & a \, \mathbf{\hat{y}} \nonumber \\ \mathbf{a}_3 & = & a \, \mathbf{\hat{z}}, \end{array}$ with volume $V = a^3.$
This can also be considered as a rhombohedral lattice with $\alpha = \pi/2$. The space groups associated with this lattice are $\begin{array}{lll} 195. ~ \text{P23} & 198. ~ \text{P2_{1}3} & 200. ~ \text{Pm\overline{3}} \\ 201. ~ \text{Pn\overline{3}} & 205. ~ \text{Pa\overline{3}} & 207. ~ \text{P432} \\ 208. ~ \text{P4_{2}32} & 212. ~ \text{P4_{3}32} & 213. ~ \text{P4_{1}32} \\ 215. ~ \text{P\overline{4}3m} & 218. ~ \text{P\overline{4}3n} & 221. ~ \text{Pm\overline{3}m} \\ 222. ~ \text{Pn\overline{3}n} & 223. ~ \text{Pm\overline{3}n} & 224. ~ \text{Pn\overline{3}m} \\ \end{array}$

### Lattice 13: Face-Centered Cubic

The face-centered cubic lattice has the same periodicity as its simple cubic parent with the addition of a translation from one corner of the cube to the center of any face. Our standard face-centered cubic primitive vectors have the form $\begin{array}{ccc} \mathbf{a}_1 & = & \frac{a}{2} \, \mathbf{\hat{y}} + \frac{a}{2} \, \mathbf{\hat{z}} \\ \mathbf{a}_2 & = & \frac{a}{2} \, \mathbf{\hat{x}} + \frac{a}{2} \, \mathbf{\hat{z}} \\ \mathbf{a}_3 & = & \frac{a}{2} \, \mathbf{\hat{x}} + \frac{a}{2} \, \mathbf{\hat{y}}, \end{array}$ and the primitive cell volume is $V = \frac{a^3}{4}.$
There are four face-centered cubic primitive cells in the conventional cubic cell. The face-centered cubic lattice can be considered as a rhombohedral lattice where $\alpha = 60^\circ$. The space groups associated with this lattice are $\begin{array}{lll} 196. ~ \text{F23} & 202. ~ \text{Fm\overline{3}} & 203. ~ \text{Fd\overline{3}} \\ 209. ~ \text{F432} & 210. ~ \text{F4_{1}32} & 216. ~ \text{F\overline{4}3m} \\ 219. ~ \text{F4\overline{3}c} & 225. ~ \text{Fm\overline{3}m} & 226. ~ \text{Fm\overline{3}c} \\ 227. ~ \text{Fd\overline{3}m} & 228. ~ \text{Fd\overline{3}c} & \\ \end{array}$

### Lattice 14: Body-Centered Cubic

Like its predecessors in the orthorhombic and tetragonal systems, the body-centered cubic crystal has the same periodicity as its parent with the addition of a translation from one corner of the cube to its center. Our standard body-centered cubic primitive vectors have the form $\begin{array}{ccc} \mathbf{a}_1 & = & - \frac{a}{2} \, \mathbf{\hat{x}} + \frac{a}{2} \, \mathbf{\hat{y}} + \frac{a}{2} \, \mathbf{\hat{z}} \\ \mathbf{a}_2 & = & ~ \frac{a}{2} \, \mathbf{\hat{x}} - \frac{a}{2} \, \mathbf{\hat{y}} + \frac{a}{2} \, \mathbf{\hat{z}} \\ \mathbf{a}_3 & = & ~ \frac{a}{2} \, \mathbf{\hat{x}} + \frac{a}{2} \, \mathbf{\hat{y}} - \frac{a}{2} \, \mathbf{\hat{z}}, \end{array}$ and the primitive cell volume is $V = \frac{a^3}{2}.$
There are two body-centered cubic primitive cells in the conventional cubic cell. The body-centered cubic lattice can be considered as a rhombohedral lattice where $\alpha = \cos^{-1} (-1/3) \approx 109.47^\circ$. The space groups associated with this lattice are $\begin{array}{lll} 197. ~ \text{I23} & 199. ~ \text{I2_{1}3} & 204. ~ \text{Im\overline{3}} \\ 206. ~ \text{Ia\overline{3}} & 211. ~ \text{I432} & 214. ~ \text{I4_{1}32} \\ 217. ~ \text{I\overline{4}3m} & 220. ~ \text{I\overline{4}3d} & 229. ~ \text{Im\overline{3}m} \\ 230. ~ \text{Ia\overline{3}d} & ~ & ~ \\ \end{array}$