Symmetry operations

A SymOperation{D} is a representation of a spatial symmetry operation $g=\{\mathbf{W}|\mathbf{w}\}$, composed of a rotational $\mathbf{W}$ and a translation part $\mathbf{w}$. The rotational and translation parts are assumed to share the same basis setting; by default, operations returned by Crystalline.jl are in the conventional setting of the International Tables of Crystallography, Volume A (ITA).

SymOperations can be constructed in two ways, either by explicitly specifying the $\mathbf{W}$ and $\mathbf{w}$:

using Crystalline, StaticArrays
W, w = (@SMatrix [1 0 0; 0 0 1; 0 1 0]), (@SVector [0, 0.5, 0])
op = SymOperation(W, w)

{m₀₋₁₁|0,½,0} ──────────────── (x,z+1/2,y)
 ┌ 1  0  0 ╷   0 ┐
 │ 0  0  1 ┆ 1/2 │
 └ 0  1  0 ╵   0 ┘

or by its equivalent triplet form

op = SymOperation{3}("x,z+1/2,y")

There is also a string macro accessor @S_str that allows triplet input via S"x,z+1/2,y".

In the above output, three equivalent notations for the symmetry operation are given: first, the Seitz notation {m₀₋₁₁|0,½,0}, then the triplet notation (x,z+1/2,y), and finally the explicit matrix notation.

Components

The rotation and translation parts $\mathbf{W}$ and $\mathbf{w}$ of a SymOperation{D} $\{\mathbf{W}|\mathbf{w}\}$ can be accessed via rotation and translation, returning an SMatrix{D, D, Float64} and an SVector{D, Float64}, respectively. The "augmented" matrix $[\mathbf{W}|\mathbf{w}]$ can similarly be obtained via matrix.

Operator composition

Composition of two operators $g_1$ and $g_2$ is defined by

\[g_1 \circ g_2 = \{\mathbf{W}_1|\mathbf{w}_1\} \circ \{\mathbf{W}_2|\mathbf{w}_2\} = \{\mathbf{W}_1\mathbf{W}_2|\mathbf{w}_1 + \mathbf{W}_1\mathbf{w}_2\}\]

We can compose two SymOperations in Crystalline via:

op1 = S"z,x,y" # 3₁₁₁⁺
op2 = S"z,y,x" # m₋₁₀₁
op1 * op2

m₀₋₁₁ ──────────────────────────── (x,z,y)
 ┌ 1  0  0 ╷ 0 ┐
 │ 0  0  1 ┆ 0 │
 └ 0  1  0 ╵ 0 ┘

which is accessed by an overloaded call to Base.*, i.e. the multiplication operator (this enables us to also call derived methods of *, such as integer powers (e.g., S"-y,x-y,z"^3 == S"x,y,z"). Note that composition is taken modulo integer lattice translations by default, such that

op2′ = S"z,y,x+1" # {m₋₁₀₁|001}
op1 * op2′ # equivalent to compose(op1, op2′, true)

m₀₋₁₁ ──────────────────────────── (x,z,y)
 ┌ 1  0  0 ╷ 0 ┐
 │ 0  0  1 ┆ 0 │
 └ 0  1  0 ╵ 0 ┘

rather than S"x+1,z,y", which is the result of direct application of the above composition rule. To compute "unreduced" composition, the more precise compose variant of * can be used with an optional third argument false:

compose(op1, op2′, false)

{m₀₋₁₁|1,0,0} ────────────────── (x+1,z,y)
 ┌ 1  0  0 ╷ 1 ┐
 │ 0  0  1 ┆ 0 │
 └ 0  1  0 ╵ 0 ┘

Operator inverses

The operator inverse is defined as $\{\mathbf{W}|\mathbf{w}\} = \{\mathbf{W}^{-1}|-\mathbf{W}^{-1}\mathbf{w}\}$ and can be computed via

inv(op1) # inv(3₁₁₁⁺)

3₁₁₁⁻ ──────────────────────────── (y,z,x)
 ┌ 0  1  0 ╷ -0 ┐
 │ 0  0  1 ┆ -0 │
 └ 1  0  0 ╵ -0 ┘

Action of symmetry operators

A SymOperation can act on vectors in direct (RVec) or reciprocal (KVec) space. When acting in reciprocal space, translation parts of a SymOperation have no effect.

Magnetic symmetry operations

Magnetic symmetry operations that may incorporate composition with an anti-unitary time-reversal operation can be created via MSymOperation (see also mspacegroup).