Definitions and conventions

• Definitions and conventions
• • Basis vectors
  • Atomic point coordinates
  • Space group operation and change of basis
  • • Symmetry operation $(\mathbf{W}, \mathbf{w})$
    • Transformation matrix $\mathbf{P}$ and origin shift $\mathbf{p}$
    • Difference between rotation and transformation matrices
  • Spglib conventions of standardized unit cell
  • • Transformation to the primitive cell
    • Idealization of unit cell structure
    • Rotation introduced by idealization

Basis vectors

In this package, basis vectors are represented by three-column vectors:

\[\mathbf{a} = \begin{bmatrix} a_x \\ a_y \\ a_z \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} b_x \\ b_y \\ b_z \end{bmatrix}, \quad \mathbf{c} = \begin{bmatrix} c_x \\ c_y \\ c_z \end{bmatrix},\]

in Cartesian coordinates.

Therefore, a lattice is represented as

\[\mathbf{A} = \begin{bmatrix} \mathbf{a} & \mathbf{b} & \mathbf{c} \end{bmatrix} = \begin{bmatrix} a_x & b_x & c_x \\ a_y & b_y & c_y \\ a_z & b_z & c_z \end{bmatrix}.\]

Depending on the situation, $\begin{bmatrix} \mathbf{a}_1 & \mathbf{a}_2 & \mathbf{a}_3 \end{bmatrix}$ is used instead of $\begin{bmatrix} \mathbf{a} & \mathbf{b} & \mathbf{c} \end{bmatrix}$.

A reciprocal lattice is its inverse, represented as three columns vectors, too:

\[\mathbf{B} = \bigl(\mathbf{A}^{-1}\bigr)^\intercal = \begin{bmatrix} \mathbf{b}_1 & \mathbf{b}_2 & \mathbf{b}_3 \end{bmatrix}\]

so that

\[\mathbf{A} \mathbf{B}^\intercal = \mathbf{B}^\intercal \mathbf{A} = \mathbf{I},\]

where $\mathbf{I}$ is the $3 \times 3$ identity matrix.

We choose this convention because it is convenient for converting reduced reciprocal coordinates $\mathbf{x}^\ast$ to Cartesian coordinates using the expression $\mathbf{B} \mathbf{x}^\ast$.

This is analogous to the convention employed in Atomic point coordinates for the definition of reduced coordinates.

Atomic point coordinates

Coordinates of an atomic point $\mathbf{x}$ are represented as three fractional values relative to basis vectors as follows,

\[\mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix},\]

where $0 \le x_i < 1$. A position vector $\mathbf{r}$ in Cartesian coordinates is obtained by

\[\mathbf{r} = \mathbf{A} \mathbf{x} = \begin{bmatrix} \mathbf{a}_1 & \mathbf{a}_2 & \mathbf{a}_3 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}\]

or

\[\mathbf{r} = \sum_i x_i \mathbf{a}_i.\]

[ [ a_x, b_x, c_x ],
  [ a_y, b_y, c_y ],
  [ a_z, b_z, c_z ] ]

Space group operation and change of basis

Symmetry operation $(\mathbf{W}, \mathbf{w})$

A symmetry operation consists of a pair of the rotation part $\mathbf{W}$ and translation part $\mathbf{w}$, and is represented as $(\mathbf{W}, \mathbf{w})$. The symmetry operation transfers $\mathbf{x}$ to $\tilde{\mathbf{x}}$ as follows:

\[\tilde{\mathbf{x}} = \mathbf{W} \mathbf{x} + \mathbf{w}.\]

Transformation matrix $\mathbf{P}$ and origin shift $\mathbf{p}$

The transformation matrix $\mathbf{P}$ changes choice of basis vectors as follows

\[\begin{bmatrix} \mathbf{a} & \mathbf{b} & \mathbf{c} \end{bmatrix} = \begin{bmatrix} \mathbf{a}_\text{s} & \mathbf{b}_\text{s} & \mathbf{c}_\text{s} \end{bmatrix} \mathbf{P},\]

where $\begin{bmatrix} \mathbf{a} & \mathbf{b} & \mathbf{c} \end{bmatrix}$ and $\begin{bmatrix} \mathbf{a}_\text{s} & \mathbf{b}_\text{s} & \mathbf{c}_\text{s} \end{bmatrix}$ are the basis vectors of an arbitrary system and of a standardized system, respectively. In general, the transformation matrix is not limited for the transformation from the standardized system, but can be used in between any systems possibly transformed. It has to be emphasized that the transformation matrix does not rotate a crystal in Cartesian coordinates, but just changes the choices of basis vectors.

Difference between rotation and transformation matrices

A rotation matrix rotates (or mirrors, inverts) the crystal body with respect to origin. A transformation matrix changes the choice of the basis vectors, but does not rotate the crystal body.

Active/reverse/alibi transformation

A space group operation having no translation part sends an atom to another point by

\[\tilde{\mathbf{x}} = \mathbf{W} \mathbf{x},\]

where $\tilde{\mathbf{x}}$ and $\mathbf{x}$ are represented with respect to the same basis vectors $\begin{bmatrix} \mathbf{a} & \mathbf{b} & \mathbf{c} \end{bmatrix}$. Equivalently the rotation is achieved by rotating the basis vectors:

\[\begin{bmatrix} \tilde{\mathbf{a}} & \tilde{\mathbf{b}} & \tilde{\mathbf{c}} \end{bmatrix} = \begin{bmatrix} \mathbf{a} & \mathbf{b} & \mathbf{c} \end{bmatrix} \mathbf{W}\]

with keeping the representation of the atomic point coordinates $\mathbf{x}$ because

\[\tilde{\mathbf{x}} = \begin{bmatrix} \mathbf{a} & \mathbf{b} & \mathbf{c} \end{bmatrix} \tilde{\mathbf{x}} = \begin{bmatrix} \mathbf{a} & \mathbf{b} & \mathbf{c} \end{bmatrix} \mathbf{W} \mathbf{x} = \begin{bmatrix} \tilde{\mathbf{a}} & \tilde{\mathbf{b}} & \tilde{\mathbf{c}} \end{bmatrix} \mathbf{x}.\]

Passive/forward/alias transformation

The transformation matrix changes the choice of the basis vectors as:

\[\begin{bmatrix} \mathbf{a}' & \mathbf{b}' & \mathbf{c}' \end{bmatrix} = \begin{bmatrix} \mathbf{a} & \mathbf{b} & \mathbf{c} \end{bmatrix} \mathbf{P}.\]

The atomic position vector is not altered by this transformation, i.e.,

\[\begin{bmatrix} \mathbf{a}' & \mathbf{b}' & \mathbf{c}' \end{bmatrix} \mathbf{x}' = \begin{bmatrix} \mathbf{a} & \mathbf{b} & \mathbf{c} \end{bmatrix} \mathbf{x}.\]

However the representation of the atomic point coordinates changes as follows:

\[\mathbf{P} \mathbf{x}' = \mathbf{x}.\]

because

\[\begin{bmatrix} \mathbf{a} & \mathbf{b} & \mathbf{c} \end{bmatrix} \mathbf{P} \mathbf{x}' = \begin{bmatrix} \mathbf{a}' & \mathbf{b}' & \mathbf{c}' \end{bmatrix} \mathbf{x}' = \begin{bmatrix} \mathbf{a} & \mathbf{b} & \mathbf{c} \end{bmatrix} \mathbf{x}.\]

Spglib conventions of standardized unit cell

Transformation to the primitive cell

In the standardized unit cells, there are five different centring types available, base centerings of A and C, rhombohedral (R), body-centred (I), and face-centred (F). The transformation is applied to the standardized unit cell by

\[\begin{bmatrix} \mathbf{a}_\text{p} & \mathbf{b}_\text{p} & \mathbf{c}_\text{p} \end{bmatrix} = \begin{bmatrix} \mathbf{a}_\text{s} & \mathbf{b}_\text{s} & \mathbf{c}_\text{s} \end{bmatrix} \mathbf{P}\]

where $\mathbf{a}_\text{p}$, $\mathbf{b}_\text{p}$, and $\mathbf{c}_\text{p}$ are the basis vectors of the primitive cell and $\mathbf{P}$ is the transformation matrix from the standardized unit cell to the primitive cell. Matrices $\mathbf{P}$ for different centring types are given as follows:

\[\mathbf{P}_\text{A} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \dfrac{1}{2} & \dfrac{-1}{2} \\ 0 & \dfrac{1}{2} & \dfrac{1}{2} \end{bmatrix}, \quad \mathbf{P}_\text{C} = \begin{bmatrix} \dfrac{1}{2} & \dfrac{1}{2} & 0 \\ \dfrac{-1}{2} & \dfrac{1}{2} & 0 \\ 0 & 0 & 1 \end{bmatrix}, \quad \mathbf{P}_\text{R} = \begin{bmatrix} \dfrac{2}{3} & \dfrac{-1}{3} & \dfrac{-1}{3} \\ \dfrac{1}{3} & \dfrac{1}{3} & \dfrac{\bar{2}}{3} \\ \dfrac{1}{3} & \dfrac{1}{3} & \dfrac{1}{3} \end{bmatrix}, \quad \mathbf{P}_\text{I} = \begin{bmatrix} \dfrac{-1}{2} & \dfrac{1}{2} & \dfrac{1}{2} \\ \dfrac{1}{2} & \dfrac{-1}{2} & \dfrac{1}{2} \\ \dfrac{1}{2} & \dfrac{1}{2} & \dfrac{-1}{2} \end{bmatrix}, \quad \mathbf{P}_\text{F} = \begin{bmatrix} 0 & \dfrac{1}{2} & \dfrac{1}{2} \\ \dfrac{1}{2} & 0 & \dfrac{1}{2} \\ \dfrac{1}{2} & \dfrac{1}{2} & 0 \end{bmatrix}.\]

The choice of transformation matrix depends on the purpose.

For rhombohedral lattice systems with the H setting (hexagonal lattice), $\mathbf{P}_\text{R}$ is applied to obtain primitive basis vectors. However, with the R setting (rhombohedral lattice), no transformation matrix is used because it is already a primitive cell.

Idealization of unit cell structure

Spglib allows tolerance parameters to match a slightly distorted unit cell structure to a space group type with some higher symmetry. Using obtained symmetry operations, the distortion is removed to idealize the unit cell structure. The coordinates of atomic points are idealized using respective site-symmetries^[1]. The basis vectors are idealized by forcing them into respective lattice shapes as follows. In this treatment, except for triclinic crystals, crystals can be rotated in Cartesian coordinates, which is the different type of transformation from that of the change-of-basis transformation explained above.

Triclinic lattice

• Niggli-reduced cell is used for choosing $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$.
• $\mathbf{a}$ is set along $+x$ direction of Cartesian coordinates.
• $\mathbf{b}$ is set in $x$-$y$ plane of Cartesian coordinates so that $\mathbf{a}\times\mathbf{b}$ is along $+z$ direction of Cartesian coordinates.

Monoclinic lattice

• The $b$-axis is taken as the unique axis.
• $\alpha = 90^\circ$ and $\gamma = 90^\circ$, while $90^\circ < \beta < 120^\circ$.
• $\mathbf{a}$ is set along $+x$ direction of Cartesian coordinates.
• $\mathbf{b}$ is set along $+y$ direction of Cartesian coordinates.
• $\mathbf{c}$ is set in $x$-$z$ plane of Cartesian coordinates.

Orthorhombic lattice

• $\alpha = \beta = \gamma = 90^\circ$.
• $\mathbf{a}$ is set along $+x$ direction of Cartesian coordinates.
• $\mathbf{b}$ is set along $+y$ direction of Cartesian coordinates.
• $\mathbf{c}$ is set along $+z$ direction of Cartesian coordinates.

Tetragonal lattice

• $\alpha = \beta = \gamma = 90^\circ$.
• $a=b$.
• $\mathbf{a}$ is set along $+x$ direction of Cartesian coordinates.
• $\mathbf{b}$ is set along $+y$ direction of Cartesian coordinates.
• $\mathbf{c}$ is set along $+z$ direction of Cartesian coordinates.

Rhombohedral lattice

• $\alpha = \beta = \gamma$.
• $a=b=c$.
• Let $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ projected on $x$-$y$ plane in Cartesian coordinates be $\mathbf{a}_{xy}$, $\mathbf{b}_{xy}$, and $\mathbf{c}_{xy}$, respectively, and their angles be $\alpha_{xy}$, $\beta_{xy}$, $\gamma_{xy}$, respectively.
• Let $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ projected along $z$-axis in Cartesian coordinates be $\mathbf{a}_{z}$, $\mathbf{b}_{z}$, and $\mathbf{c}_{z}$, respectively.
• $\mathbf{a}_{xy}$ is set along the ray $30^\circ$ rotated counter-clockwise from the $+x$ direction of Cartesian coordinates, and $\mathbf{b}_{xy}$ and $\mathbf{c}_{xy}$ are placed by angles $120^\circ$ and $240^\circ$ from $\mathbf{a}_{xy}$ counter-clockwise, respectively.
• $\alpha_{xy} = \beta_{xy} = \gamma_{xy} = 120^\circ$.
• $a_{xy} = b_{xy} = c_{xy}$.
• $a_{z} = b_{z} = c_{z}$.

Hexagonal lattice

• $\alpha = \beta = 90^\circ$, $\gamma = 120^\circ$.
• $a=b$.
• $\mathbf{a}$ is set along $+x$ direction of Cartesian coordinates.
• $\mathbf{b}$ is set in $x$-$y$ plane of Cartesian coordinates.
• $\mathbf{c}$ is set along $+z$ direction of Cartesian coordinates.

Cubic lattice

• $\alpha = \beta = \gamma = 90^\circ$.
• $a=b=c$.
• $\mathbf{a}$ is set along $+x$ direction of Cartesian coordinates.
• $\mathbf{b}$ is set along $+y$ direction of Cartesian coordinates.
• $\mathbf{c}$ is set along $+z$ direction of Cartesian coordinates.

Rotation introduced by idealization

In the idealization step presented above, the input unit cell crystal structure can be rotated in the Cartesian coordinates. The rotation matrix $\mathbf{R}$ of this rotation is defined by

\[\begin{bmatrix} \bar{\mathbf{a}}_\text{s} & \bar{\mathbf{b}}_\text{s} & \bar{\mathbf{c}}_\text{s} \end{bmatrix} = \mathbf{R} \begin{bmatrix} \mathbf{a}_\text{s} & \mathbf{b}_\text{s} & \mathbf{c}_\text{s} \end{bmatrix}.\]

This rotation matrix rotates the standardized crystal structure before idealization $\begin{bmatrix} \mathbf{a}_\text{s} & \mathbf{b}_\text{s} & \mathbf{c}_\text{s} \end{bmatrix}$ to that after idealization $\begin{bmatrix} \bar{\mathbf{a}}_\text{s} & \bar{\mathbf{b}}_\text{s} & \bar{\mathbf{c}}_\text{s} \end{bmatrix}$ in Cartesian coordinates of the given input unit cell.