Bravais Lattice

Crystal systems

cubic, tetragonal, and orthorhombic

lattice constants and angles:

$$\alpha = \beta = \gamma = 90^\circ.$$ (1)

$$\mathrm{cubic:} \quad a = b = c$$ (2)

$$\mathrm{tetragonal:} \quad a = b \neq c$$ (3)

$$\mathrm{orthorhombic:} \quad a \neq b \neq c$$ (4)

"axis" vectors:

$$\bm{a}_1 = \begin{pmatrix}a \\ 0 \\ 0 \end{pmatrix} \;\; \bm{a}... ... \\ 0 \end{pmatrix} \;\; \bm{a}_3 = \begin{pmatrix}0 \\ 0 \\ c \end{pmatrix}$$ (5)

"axis" reciprocal vectors:

$$\bm{q}_1 = \frac{2\pi}{a} \begin{pmatrix}1 & 0 & 0 \end{pmatrix} ... ...{pmatrix} \;\; \bm{q}_3 = \frac{2\pi}{c} \begin{pmatrix}0 & 0 & 1 \end{pmatrix}$$ (6)

Monoclinic

lattice constants and angles:

$$a \neq b \neq c, \;\; \alpha = \beta = 90^\circ, \;\; \gamma \neq 90^\circ.$$ (7)

"axis" vectors:

$$\bm{a}_1 = a \begin{pmatrix}\sin \gamma \\ \cos \gamma \\ 0 \en... ... \\ 0 \end{pmatrix} \;\; \bm{a}_3 = \begin{pmatrix}0 \\ 0 \\ c \end{pmatrix}$$ (8)

"axis" reciprocal vectors:

$$\bm{q}_1 = \frac{2\pi}{a \sin \gamma} \begin{pmatrix}1 & 0 & 0 \e... ...atrix} , \quad \bm{q}_3 = \frac{2\pi}{c} \begin{pmatrix}0 & 0 & 1 \end{pmatrix}$$ (9)

Triclinic

lattice constants and angles:

$$a \neq b \neq c, \;\; \alpha \neq 90^\circ, \; \beta \neq 90^\circ, \; \gamma \neq 90^\circ.$$ (10)

"axis" vectors:

$$\bm{a}_1 = a \begin{pmatrix}1 \\ 0 \\ 0 \end{pmatrix} \;\; \bm{... ...os \beta \\ \cos \alpha' \sin \beta \\ \sin \alpha' \sin \beta \end{pmatrix},$$ (11)

where

$$\cos \alpha' = \frac{ \cos \alpha - \cos \beta \cos \gamma } {\sin \beta \sin \gamma}$$ (12)

$$\sin \alpha' = \frac{\sqrt{\sin^2 \gamma - \cos^2 \beta - \cos^2 \alpha + 2 \cos \alpha \cos \beta \cos \gamma} } { \sin \beta \sin \gamma}$$ (13)

Hexagonal

lattice constants and angles:

$$a = b \neq c, \;\; \alpha = \beta = 90^\circ, \; \gamma = 120^\circ.$$ (14)

"axis" vectors:

$$\bm{a}_1 = a \begin{pmatrix}\frac{\sqrt{3}}{2} \\ -\frac{1}{2} \... ... 0 \end{pmatrix} \;\; \bm{a}_3 = c \begin{pmatrix}0 \\ 0 \\ 1 \end{pmatrix},$$ (15)

"axis" reciprocal vectors:

$$\bm{q}_1 = \frac{2\pi}{a} \begin{pmatrix}\frac{2}{\sqrt{3}} & 0 &... ...atrix} , \quad \bm{q}_3 = \frac{2\pi}{c} \begin{pmatrix}0 & 0 & 1 \end{pmatrix}$$ (16)

Trigonal (Rhombohedral)

We treat trigonal (rhombohedral) lattice as a hexagonal crystal system with sublattices at $(\frac{2}{3}, \frac{1}{3}, \frac{1}{3})$ and $(\frac{1}{3}, \frac{2}{3}, \frac{2}{3})$.

Bravais Lattices

simple(for all crystal systems)

The primitive translation vectors:

$$\bm{t}_1 = \bm{a}_1, \; \bm{t}_2 = \bm{a}_2, \; \bm{t}_3 = \bm{a}_3.$$ (17)

The primitive reciprocal vectors:

$$\bm{g}_1 = \bm{q}_1, \; \bm{g}_2 = \bm{q}_2, \; \bm{g}_3 = \bm{q}_3.$$ (18)

body centered ( for cubic, tetragonal and orthorhombic)

The primitive translation vectors:

$$\bm{t}_1 = -\frac{1}{2}\bm{a}_1 + \frac{1}{2}\bm{a}_2 + \frac{1}{2}\bm{a}_3$$ (19)

$$\bm{t}_2 = \frac{1}{2}\bm{a}_1 - \frac{1}{2}\bm{a}_2 + \frac{1}{2}\bm{a}_3$$ (20)

$$\bm{t}_3 = \frac{1}{2}\bm{a}_1 + \frac{1}{2}\bm{a}_2 - \frac{1}{2}\bm{a}_3$$ (21)

The primitive reciprocal vectors:

$$\bm{g}_1 = \bm{q}_2 + \bm{q}_3$$ (22)

$$\bm{g}_2 = \bm{q}_1 + \bm{q}_3$$ (23)

$$\bm{g}_3 = \bm{q}_l + \bm{q}_2$$ (24)

face centered ( for cubic and orthorhombic)

The primitive translation vectors:

$$\bm{t}_1 = \frac{1}{2}\bm{a}_2 + \frac{1}{2}\bm{a}_3$$ (25)

$$\bm{t}_2 = \frac{1}{2}\bm{a}_1 + \frac{1}{2}\bm{a}_3$$ (26)

$$\bm{t}_3 = \frac{1}{2}\bm{a}_1 + \frac{1}{2}\bm{a}_2$$ (27)

The primitive reciprocal vectors:

$$\bm{g}_1 = -\bm{q}_1 + \bm{q}_2 + \bm{q}_3$$ (28)

$$\bm{g}_2 = \bm{q}_1 - \bm{q}_2 + \bm{q}_3$$ (29)

$$\bm{g}_3 = \bm{q}_l + \bm{q}_2 - \bm{q}_3$$ (30)

base centered ( for orthorhombic)

(CXY)

The primitive translation vectors:

$$\bm{t}_1 = \frac{1}{2}\bm{a}_1 - \frac{1}{2}\bm{a}_2$$ (31)

$$\bm{t}_2 = \frac{1}{2}\bm{a}_1 + \frac{1}{2}\bm{a}_2$$ (32)

$$\bm{t}_3 = \frac{1}{2}\bm{a}_3$$ (33)

The primitive reciprocal vectors:

$$\bm{g}_1 = \bm{q}_1 - \bm{q}_2$$ (34)

$$\bm{g}_2 = \bm{q}_1 + \bm{q}_2$$ (35)

$$\bm{g}_3 = \bm{q}_3$$ (36)

rhombohedral

The primitive translation vectors:

$$\bm{t}_1 = \frac{2}{2}\bm{a}_1 + \frac{1}{3}\bm{a}_2 + \frac{1}{3}\bm{a}_3$$ (37)

$$\bm{t}_2 = -\frac{1}{3}\bm{a}_1 + \frac{1}{3}\bm{a}_2 + \frac{1}{3}\bm{a}_3$$ (38)

$$\bm{t}_3 = -\frac{1}{3}\bm{a}_1 - \frac{2}{2}\bm{a}_2 + \frac{1}{3}\bm{a}_3$$ (39)

The primitive reciprocal vectors:

$$\bm{g}_1 = \bm{q}_1 + \bm{q}_3$$ (40)

$$\bm{g}_2 = -\bm{q}_1 + \bm{q}_2 + \bm{q}_3$$ (41)

$$\bm{g}_3 = -\bm{q}_2 + \bm{q}_3$$ (42)

Transformation between Hexagonal and Rhombohedral

Transformation between hexagonal coordinates $(x_H, y_H, z_H)$ and rhombohedral coordinates $(x_R, y_R, z_R)$ are as follows.

$$\begin{pmatrix}x_H \\ y_H \\ z_H \end{pmatrix} = \begin{pmatrix... ... 1/3 & 1/3 & 1/3 \end{pmatrix} \begin{pmatrix}x_R \\ y_R \\ z_R \end{pmatrix}$$ (43)

$$\begin{pmatrix}x_R \\ y_R \\ z_R \end{pmatrix} = \begin{pmatrix... ...1 \\ 0 & -1 & 1 \end{pmatrix} \begin{pmatrix}x_H \\ y_H \\ z_H \end{pmatrix}$$ (44)

Coordinate Systems

In the following, we use following three coordinate systems.

• xyz : cartesian coordinates.
• prm : relative coordinates for the primitive translation vectors.
• abc : relative coordinates for the primitive axis vectors.

$$\bm{r} = x_{prm} \bm{e}_x + y_{prm} \bm{e}_y + z_{prm} \bm{e}_z =... ...}_2 + z_{abc} \bm{a}_3 = x_{prm} \bm{t}_1 + y_{prm} \bm{t}_2 + z_{prm} \bm{t}_3$$ (45)

$$\bm{k} = \xi_{prm} \bm{e}_x + \zeta_{prm} \bm{e}_y + \eta_{prm} \... ...abc} \bm{q}_3 = \xi_{prm} \bm{g}_1 + \zeta_{prm} \bm{g}_2 + \eta_{prm} \bm{g}_3$$ (46)

The prm and abc coordinate systems are identical for simple and hexagonal lattices. We must distinguish them for body centered, face centered, base centered and trigonal lattices.