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FINDSPINGROUP: Identify Spin Space Group and Related Properties

File NameOriented Spin Space GroupG0 SymbolL0 Symbolitik
\(0.607.RuO2\)
\(P ^{\bar{1}}4_{2}/ ^{1}m ^{\bar{1}}n ^{1}m ^{\infty_{001}m}1\) \((136.65.1.1.L)\)
\(P4_2/mnm\ (136)\)\(Cmmm\ (65)\)\(2\)\(1\)
Space Group Magnetic Space Group Spin-space PGConfigurationMagnetic PhaseAltermagnet here means the core identified a collinear AFM with k-dependent spin splitting in the non-SOC route.
\(P4_2/mnm\ (136)\)\[\begin{aligned}&P4_2'/mnm'(136.499)\\ &Type\ III\end{aligned}\]\[\begin{aligned}∞/mm\\ D∞h\end{aligned}\]\(Collinear\)\(AFM(Altermagnet)\)
Little Cogroup at General PositionSpin SplittingSpin TextureExpressions are written in the special Cartesian frame used by the SSG convention: x is parallel to a, y lies in the ab plane perpendicular to x and chosen toward b, and z = x × y. The listed spin textures are symmetry-allowed leading terms. They describe the lowest-order terms not forbidden by symmetry; the actual low-energy or Fermi-surface spin texture may start at a higher order. For example, a symmetry-allowed p-wave term may vanish for microscopic reasons, and the leading observable texture near the Fermi surface may instead be f-wave or higher-order.
without SOCwith SOCwithout SOCwith SOCwithout SOCwith SOC
$$^{m}-1^{\infty }1$$$$1$$\(k-dependent\)\(Yes\)
\(\mathrm{d-wave}\)
\(C_{1}\left(k_{x}k_{y}\,\sigma_{z}\right)\)
\(\mathrm{d-wave}\)
\(C_{1}\left(k_{y}k_{z}\,\sigma_{x} + k_{x}k_{z}\,\sigma_{y}\right) + C_{2}\left(k_{x}k_{y}\,\sigma_{z}\right)\)
Effective Magnetic Point GroupEffective Magnetic Point Group is obtained from the spin space group by mapping spin rotations with det = -1 to time reversal and then removing the translation part. Anomalous Hall Effect
$$4/mmm1'$$ without SOC with SOC
xyz
x$$0$$$$0$$$$0$$
y$$0$$$$0$$$$0$$
z$$0$$$$0$$$$0$$
xyz
x$$0$$$$0$$$$0$$
y$$0$$$$0$$$$0$$
z$$0$$$$0$$$$0$$
Effective MPG Constraint on rank-3 Tensor
without SOC with SOC
T-odd Quantum metric dipole $$0$$ $$0$$
T-odd Inversed mass dipole $$0$$ $$0$$
T-even Berry curvature dipole $$0$$ $$0$$

G0-std Unit Magnetic Cell:

a b c alpha beta gamma
4.492 4.492 3.1061 90.0 90.0 90.0
Transformation from input to G0-standard setting Transformation from G0-standard to magnetic primitive setting
Transformation Matrix (T) Origin Shift (τ) Transformation Matrix (Tp) Origin Shift (τp)
\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\] \[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\] \[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]
T * r_input + τ = r_G0std Tp * r_G0std + τp = r_prim

Fractional Coordinates & Moments (in G0-std Unit Magnetic Cell)

No. Atom Site Moment (Oriented)
1

O

[0.1944, 0.8056, 0.5]

[0.0, 0.0, 0.0]
2

O

[0.3056, 0.3056, 0.0]

[0.0, 0.0, 0.0]
3

O

[0.6944, 0.6944, 0.0]

[0.0, 0.0, 0.0]
4

O

[0.8056, 0.1944, 0.5]

[0.0, 0.0, 0.0]
5

Ru

[0.0, 0.0, 0.0]

[0.0, -0.0, 0.0161]
6

Ru

[0.5, 0.5, 0.5]

[-0.0, 0.0, -0.0161]
ElementSG WyckoffSSG WyckoffFor magnetic atoms, the number in parentheses after an SSG or MSG Wyckoff symbol is the constrained magnetic freedom degree of that magnetic site.MSG WyckoffFor magnetic atoms, the number in parentheses after an SSG or MSG Wyckoff symbol is the constrained magnetic freedom degree of that magnetic site.
O4f4f4f
Ru2a2a(1)2a(1)

Elements of the convention SSG:

Real Space Elements are under the basis of the lattice given above.

Spin components are expressed in the lattice basis (a, b, c).

Collinear presentation
This operation list shows the convention nSSG for the collinear case. nSSG point part: -1 (Ci). Spin-only component: ∞m (C∞v). Collinear direction (oriented spin frame): 0,0,1.
No.Spin RotationSpace RotationSpace TranslationSeitz Symbol
1\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, 1 \,\middle|\, 0,0,0 \right\}\)
2\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, m_{1-10} \,\middle|\, 0,0,0 \right\}\)
3\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, m_{110} \,\middle|\, 0,0,0 \right\}\)
4\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, m_{001} \,\middle|\, 0,0,0 \right\}\)
5\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ -1 \,\middle\|\, 4^{1}_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{2} \right\}\)
6\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ -1 \,\middle\|\, 4^{3}_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{2} \right\}\)
7\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ -1 \,\middle\|\, m_{100} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{2} \right\}\)
8\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ -1 \,\middle\|\, m_{010} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{2} \right\}\)
9\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, 2_{001} \,\middle|\, 0,0,0 \right\}\)
10\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, 2_{1-10} \,\middle|\, 0,0,0 \right\}\)
11\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, 2_{110} \,\middle|\, 0,0,0 \right\}\)
12\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ -1 \,\middle\|\, -4^{1}_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{2} \right\}\)
13\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ -1 \,\middle\|\, -4^{3}_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{2} \right\}\)
14\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ -1 \,\middle\|\, 2_{100} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{2} \right\}\)
15\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ -1 \,\middle\|\, 2_{010} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{2} \right\}\)
16\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, -1 \,\middle|\, 0,0,0 \right\}\)
Collinear presentation
This operation list shows the convention nSSG for the collinear case. nSSG point part: -1 (Ci). Spin-only component: ∞m (C∞v). Collinear direction (oriented spin frame): 0,0,1.
No.Spin RotationSpace RotationSpace TranslationSeitz Symbol
5\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ -1 \,\middle\|\, 4^{1}_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{2} \right\}\)
4\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, m_{001} \,\middle|\, 0,0,0 \right\}\)
7\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ -1 \,\middle\|\, m_{100} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{2} \right\}\)
3\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, m_{110} \,\middle|\, 0,0,0 \right\}\)
No.Spin RotationSpace RotationSpace TranslationSeitz Symbol
1\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, 1 \,\middle|\, 0,0,0 \right\}\)
Collinear presentation
This operation list shows the convention nSSG for the collinear case. nSSG point part: -1 (Ci). Spin-only component: ∞m (C∞v). Collinear direction (oriented spin frame): 0,0,1.
No.Spin RotationSpace RotationSpace TranslationSeitz Symbol
1\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, 1 \,\middle|\, 0,0,0 \right\}\)
No.Spin RotationSpace RotationSpace TranslationSeitz Symbol
1\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, 1 \,\middle|\, 0,0,0 \right\}\)
2\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, m_{1-10} \,\middle|\, 0,0,0 \right\}\)
3\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, m_{110} \,\middle|\, 0,0,0 \right\}\)
4\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, m_{001} \,\middle|\, 0,0,0 \right\}\)
9\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, 2_{001} \,\middle|\, 0,0,0 \right\}\)
10\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, 2_{1-10} \,\middle|\, 0,0,0 \right\}\)
11\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, 2_{110} \,\middle|\, 0,0,0 \right\}\)
16\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, -1 \,\middle|\, 0,0,0 \right\}\)
No.Spin RotationSpace RotationSpace TranslationSeitz Symbol
1\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, 1 \,\middle|\, 0,0,0 \right\}\)
2\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, m_{1-10} \,\middle|\, 0,0,0 \right\}\)
3\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, m_{110} \,\middle|\, 0,0,0 \right\}\)
4\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, m_{001} \,\middle|\, 0,0,0 \right\}\)
5\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ -1 \,\middle\|\, 4^{1}_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{2} \right\}\)
6\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ -1 \,\middle\|\, 4^{3}_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{2} \right\}\)
7\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ -1 \,\middle\|\, m_{100} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{2} \right\}\)
8\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ -1 \,\middle\|\, m_{010} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{2} \right\}\)
9\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, 2_{001} \,\middle|\, 0,0,0 \right\}\)
10\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, 2_{1-10} \,\middle|\, 0,0,0 \right\}\)
11\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, 2_{110} \,\middle|\, 0,0,0 \right\}\)
12\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ -1 \,\middle\|\, -4^{1}_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{2} \right\}\)
13\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ -1 \,\middle\|\, -4^{3}_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{2} \right\}\)
14\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ -1 \,\middle\|\, 2_{100} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{2} \right\}\)
15\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ -1 \,\middle\|\, 2_{010} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{2} \right\}\)
16\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, -1 \,\middle|\, 0,0,0 \right\}\)
No.Spin RotationSpace RotationSpace TranslationSeitz Symbol
1\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, 1 \,\middle|\, 0,0,0 \right\}\)
2\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ m_{110} \,\middle\|\, m_{110} \,\middle|\, 0,0,0 \right\}\)
3\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ m_{1-10} \,\middle\|\, m_{1-10} \,\middle|\, 0,0,0 \right\}\)
4\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 2_{001} \,\middle\|\, m_{001} \,\middle|\, 0,0,0 \right\}\)
5\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ 2_{010} \,\middle\|\, m_{010} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{2} \right\}\)
6\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ 2_{100} \,\middle\|\, m_{100} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{2} \right\}\)
7\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ -4^{3}_{001} \,\middle\|\, 4^{3}_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{2} \right\}\)
8\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ -4^{1}_{001} \,\middle\|\, 4^{1}_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{2} \right\}\)
9\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ m_{110} \,\middle\|\, 2_{110} \,\middle|\, 0,0,0 \right\}\)
10\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ m_{1-10} \,\middle\|\, 2_{1-10} \,\middle|\, 0,0,0 \right\}\)
11\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 2_{001} \,\middle\|\, 2_{001} \,\middle|\, 0,0,0 \right\}\)
12\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ 2_{010} \,\middle\|\, 2_{010} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{2} \right\}\)
13\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ 2_{100} \,\middle\|\, 2_{100} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{2} \right\}\)
14\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & -1.0 & 0.0 \\ 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ -4^{3}_{001} \,\middle\|\, -4^{3}_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{2} \right\}\)
15\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 & 1.0 & 0.0 \\ -1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.5 \\ 0.5 \\ 0.5 \end{bmatrix}\]\(\left\{ -4^{1}_{001} \,\middle\|\, -4^{1}_{001} \,\middle|\, \frac{1}{2},\frac{1}{2},\frac{1}{2} \right\}\)
16\[\begin{bmatrix} 1.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 \\ 0.0 & 0.0 & 1.0\end{bmatrix}\]\[\begin{bmatrix} -1.0 & 0.0 & 0.0 \\ 0.0 & -1.0 & 0.0 \\ 0.0 & 0.0 & -1.0\end{bmatrix}\]\[\begin{bmatrix} 0.0 \\ 0.0 \\ 0.0 \end{bmatrix}\]\(\left\{ 1 \,\middle\|\, -1 \,\middle|\, 0,0,0 \right\}\)