Information of spin space group $I\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{\text{-}1}{(1/2 \ 1/2 \ 1/2)}}\ce{^{\infty m}{1}}$ ( nSSG:23.21.2.1 )

General Positions

The continuously infinite group ^∞1 is omitted.

Coordinates	Seitz symbol
a, b, c | x, y, z	{ 1 ‖ 1 | 0 }
a, -b, -c | x, y, z	{ 1 ‖ 2100 | 0 }
-a, b, -c | x, y, z	{ 1 ‖ 2010 | 0 }
-a, -b, c | x, y, z	{ 1 ‖ 2001 | 0 }
a+1/2, b+1/2, c+1/2 | x, y, z	{ 1 ‖ 1 | 1/2 1/2 1/2 }
a+1/2, -b+1/2, -c+1/2 | x, y, z	{ 1 ‖ 2100 | 1/2 1/2 1/2 }
-a+1/2, b+1/2, -c+1/2 | x, y, z	{ 1 ‖ 2010 | 1/2 1/2 1/2 }
-a+1/2, -b+1/2, c+1/2 | x, y, z	{ 1 ‖ 2001 | 1/2 1/2 1/2 }
a, b, c+1/2 | -x, -y, -z	{ -1 ‖ 1 | 0 0 1/2 }
a, -b, -c+1/2 | -x, -y, -z	{ -1 ‖ 2100 | 0 0 1/2 }
-a, b, -c+1/2 | -x, -y, -z	{ -1 ‖ 2010 | 0 0 1/2 }
-a, -b, c+1/2 | -x, -y, -z	{ -1 ‖ 2001 | 0 0 1/2 }
a+1/2, b+1/2, c | -x, -y, -z	{ -1 ‖ 1 | 1/2 1/2 0 }
a+1/2, -b+1/2, -c | -x, -y, -z	{ -1 ‖ 2100 | 1/2 1/2 0 }
-a+1/2, b+1/2, -c | -x, -y, -z	{ -1 ‖ 2010 | 1/2 1/2 0 }
-a+1/2, -b+1/2, c | -x, -y, -z	{ -1 ‖ 2001 | 1/2 1/2 0 }
a, b, c | -x, y, z	{ m ‖ 1 | 0 }
a, -b, -c | -x, y, z	{ m ‖ 2100 | 0 }
-a, b, -c | -x, y, z	{ m ‖ 2010 | 0 }
-a, -b, c | -x, y, z	{ m ‖ 2001 | 0 }
a+1/2, b+1/2, c+1/2 | -x, y, z	{ m ‖ 1 | 1/2 1/2 1/2 }
a+1/2, -b+1/2, -c+1/2 | -x, y, z	{ m ‖ 2100 | 1/2 1/2 1/2 }
-a+1/2, b+1/2, -c+1/2 | -x, y, z	{ m ‖ 2010 | 1/2 1/2 1/2 }
-a+1/2, -b+1/2, c+1/2 | -x, y, z	{ m ‖ 2001 | 1/2 1/2 1/2 }
a, b, c+1/2 | x, -y, -z	{ 2 ‖ 1 | 0 0 1/2 }
a, -b, -c+1/2 | x, -y, -z	{ 2 ‖ 2100 | 0 0 1/2 }
-a, b, -c+1/2 | x, -y, -z	{ 2 ‖ 2010 | 0 0 1/2 }
-a, -b, c+1/2 | x, -y, -z	{ 2 ‖ 2001 | 0 0 1/2 }
a+1/2, b+1/2, c | x, -y, -z	{ 2 ‖ 1 | 1/2 1/2 0 }
a+1/2, -b+1/2, -c | x, -y, -z	{ 2 ‖ 2100 | 1/2 1/2 0 }
-a+1/2, b+1/2, -c | x, -y, -z	{ 2 ‖ 2010 | 1/2 1/2 0 }
-a+1/2, -b+1/2, c | x, -y, -z	{ 2 ‖ 2001 | 1/2 1/2 0 }

Wyckoff Positions

WP	Site symmetry	Representative
4a	$\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,0,0 | 0,0,z)
4b	$\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,1/2,0 | 0,0,z)
4c	$\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(1/2,0,1/4 | 0,0,0)
4d	$\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,0,1/4 | 0,0,0)
8e	$\ce{^{1}{2}}..\ce{^{\infty m}{1}} $	(a,0,0 | 0,0,z)
8f	$\ce{^{-1}{2}}..\ce{^{\infty m}{1}} $	(a,0,1/4 | 0,0,0)
8g	$.\ce{^{1}{2}}.\ce{^{\infty m}{1}} $	(0,b,0 | 0,0,z)
8h	$.\ce{^{-1}{2}}.\ce{^{\infty m}{1}} $	(0,b,1/4 | 0,0,0)
8i	$..\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,0,c | 0,0,z)
8j	$..\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,1/2,c | 0,0,z)
8k	$..\ce{^{-1}{2}}\ce{^{\infty m}{1}} $	(1/4,1/4,c | 0,0,0)
16l	$\ce{^{1}{1}}\ce{^{\infty m}{1}} $	(a,b,c | 0,0,z)

Spin Brillouin Zone & k-little co-groups

Wavevector-k	Little co-group
Γ:(0,0,0)	$\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{\infty /mm}{1}} $
X:(1,1,1)	$\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{\infty /mm}{1}} $
R:(1/2,0,1/2)	$\ce{^{1}{2}}\ce{^{\infty /mm}{1}} $
S:(0,1/2,1/2)	$\ce{^{1}{2}}\ce{^{\infty /mm}{1}} $
T:(1/2,1/2,0)	$\ce{^{1}{2}}\ce{^{\infty /mm}{1}} $
W:(1/2,1/2,1/2)	$\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{\infty 2}{1}} $
Δ:(0,v,0)	$ \ce{^{m}{2}}\ce{^{1}{2}}\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
Λ:(0,0,w)	$\ce{^{m}{2}}\ce{^{m}{2}}\ce{^{1}{2}}\ce{^{\infty 2}{1}} $
Σ:(u,0,0)	$\ce{^{1}{2}}\ce{^{m}{2}}\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
A:(0,v,w)	$\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
B:(u,0,w)	$\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
C:(u,v,0)	$\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
D:(u,1/2,1/2)	$\ce{^{1}{2}}\ce{^{\infty 2}{1}} $
P:(1/2,1/2,w)	$\ce{^{1}{2}}\ce{^{\infty 2}{1}} $
Q:(1/2,v,1/2)	$\ce{^{1}{2}}\ce{^{\infty 2}{1}} $
GP:(u,v,w)	$\ce{^{1}{1}}\ce{^{\infty 2}{1}} $

Spin Brillouin Zone

Band Representations (Magnon)

4a

k-vector	A↑G(2)
Γ:(0,0,0)	$Γ_{1}^{S}(2) $
X:(1,1,1)	$X_{1}^{S}(2) $
R:(1/2,0,1/2)	$R_{1}^{S}(2) $
S:(0,1/2,1/2)	$S_{1}^{S}(2) $
T:(1/2,1/2,0)	$T_{1}^{S}(2) $
W:(1/2,1/2,1/2)	$W_{1}^{S}(2) $
Δ:(0,v,0)	$Δ_{1}^{S}(2) $
Λ:(0,0,w)	$Λ_{1}^{S}(2) $
Σ:(u,0,0)	$Σ_{1}^{S}(2) $
A:(0,v,w)	$A_{1}^{S}(2) $
B:(u,0,w)	$B_{1}^{S}(2) $
C:(u,v,0)	$C_{1}^{S}(2) $
D:(u,1/2,1/2)	$D_{1}^{S}(2) $
P:(1/2,1/2,w)	$P_{1}^{S}(2) $
Q:(1/2,v,1/2)	$Q_{1}^{S}(2) $
GP:(u,v,w)	$GP_{1}^{S}(2) $

4b

k-vector	A↑G(2)
Γ:(0,0,0)	$Γ_{1}^{S}(2) $
X:(1,1,1)	$X_{1}^{S}(2) $
R:(1/2,0,1/2)	$R_{1}^{S}(2) $
S:(0,1/2,1/2)	$S_{2}^{S}(2) $
T:(1/2,1/2,0)	$T_{2}^{S}(2) $
W:(1/2,1/2,1/2)	$W_{4}^{S}(2) $
Δ:(0,v,0)	$Δ_{1}^{S}(2) $
Λ:(0,0,w)	$Λ_{1}^{S}(2) $
Σ:(u,0,0)	$Σ_{1}^{S}(2) $
A:(0,v,w)	$A_{1}^{S}(2) $
B:(u,0,w)	$B_{1}^{S}(2) $
C:(u,v,0)	$C_{1}^{S}(2) $
D:(u,1/2,1/2)	$D_{2}^{S}(2) $
P:(1/2,1/2,w)	$P_{2}^{S}(2) $
Q:(1/2,v,1/2)	$Q_{1}^{S}(2) $
GP:(u,v,w)	$GP_{1}^{S}(2) $

8e

k-vector	A↑G(4)
Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{3}^{S}(2) $
X:(1,1,1)	$X_{1}^{S}(2)⊕X_{3}^{S}(2) $
R:(1/2,0,1/2)	$R_{1}^{S}(2)⊕R_{2}^{S}(2) $
S:(0,1/2,1/2)	$2S_{1}^{S}(2) $
T:(1/2,1/2,0)	$T_{1}^{S}(2)⊕T_{2}^{S}(2) $
W:(1/2,1/2,1/2)	$W_{1}^{S}(2)⊕W_{3}^{S}(2) $
Δ:(0,v,0)	$Δ_{1}^{S}(2)⊕Δ_{2}^{S}(2) $
Λ:(0,0,w)	$Λ_{1}^{S}(2)⊕Λ_{2}^{S}(2) $
Σ:(u,0,0)	$2Σ_{1}^{S}(2) $
A:(0,v,w)	$2A_{1}^{S}(2) $
B:(u,0,w)	$2B_{1}^{S}(2) $
C:(u,v,0)	$2C_{1}^{S}(2) $
D:(u,1/2,1/2)	$2D_{1}^{S}(2) $
P:(1/2,1/2,w)	$P_{1}^{S}(2)⊕P_{2}^{S}(2) $
Q:(1/2,v,1/2)	$Q_{1}^{S}(2)⊕Q_{2}^{S}(2) $
GP:(u,v,w)	$2GP_{1}^{S}(2) $

8g

k-vector	A↑G(4)
Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{4}^{S}(2) $
X:(1,1,1)	$X_{1}^{S}(2)⊕X_{4}^{S}(2) $
R:(1/2,0,1/2)	$2R_{1}^{S}(2) $
S:(0,1/2,1/2)	$S_{1}^{S}(2)⊕S_{2}^{S}(2) $
T:(1/2,1/2,0)	$T_{1}^{S}(2)⊕T_{2}^{S}(2) $
W:(1/2,1/2,1/2)	$W_{1}^{S}(2)⊕W_{4}^{S}(2) $
Δ:(0,v,0)	$2Δ_{1}^{S}(2) $
Λ:(0,0,w)	$Λ_{1}^{S}(2)⊕Λ_{2}^{S}(2) $
Σ:(u,0,0)	$Σ_{1}^{S}(2)⊕Σ_{2}^{S}(2) $
A:(0,v,w)	$2A_{1}^{S}(2) $
B:(u,0,w)	$2B_{1}^{S}(2) $
C:(u,v,0)	$2C_{1}^{S}(2) $
D:(u,1/2,1/2)	$D_{1}^{S}(2)⊕D_{2}^{S}(2) $
P:(1/2,1/2,w)	$P_{1}^{S}(2)⊕P_{2}^{S}(2) $
Q:(1/2,v,1/2)	$2Q_{1}^{S}(2) $
GP:(u,v,w)	$2GP_{1}^{S}(2) $

8i

k-vector	A↑G(4)
Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{2}^{S}(2) $
X:(1,1,1)	$X_{1}^{S}(2)⊕X_{2}^{S}(2) $
R:(1/2,0,1/2)	$R_{1}^{S}(2)⊕R_{2}^{S}(2) $
S:(0,1/2,1/2)	$S_{1}^{S}(2)⊕S_{2}^{S}(2) $
T:(1/2,1/2,0)	$2T_{1}^{S}(2) $
W:(1/2,1/2,1/2)	$W_{1}^{S}(2)⊕W_{2}^{S}(2) $
Δ:(0,v,0)	$Δ_{1}^{S}(2)⊕Δ_{2}^{S}(2) $
Λ:(0,0,w)	$2Λ_{1}^{S}(2) $
Σ:(u,0,0)	$Σ_{1}^{S}(2)⊕Σ_{2}^{S}(2) $
A:(0,v,w)	$2A_{1}^{S}(2) $
B:(u,0,w)	$2B_{1}^{S}(2) $
C:(u,v,0)	$2C_{1}^{S}(2) $
D:(u,1/2,1/2)	$D_{1}^{S}(2)⊕D_{2}^{S}(2) $
P:(1/2,1/2,w)	$2P_{1}^{S}(2) $
Q:(1/2,v,1/2)	$Q_{1}^{S}(2)⊕Q_{2}^{S}(2) $
GP:(u,v,w)	$2GP_{1}^{S}(2) $

8j

k-vector	A↑G(4)
Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{2}^{S}(2) $
X:(1,1,1)	$X_{1}^{S}(2)⊕X_{2}^{S}(2) $
R:(1/2,0,1/2)	$R_{1}^{S}(2)⊕R_{2}^{S}(2) $
S:(0,1/2,1/2)	$S_{1}^{S}(2)⊕S_{2}^{S}(2) $
T:(1/2,1/2,0)	$2T_{2}^{S}(2) $
W:(1/2,1/2,1/2)	$W_{3}^{S}(2)⊕W_{4}^{S}(2) $
Δ:(0,v,0)	$Δ_{1}^{S}(2)⊕Δ_{2}^{S}(2) $
Λ:(0,0,w)	$2Λ_{1}^{S}(2) $
Σ:(u,0,0)	$Σ_{1}^{S}(2)⊕Σ_{2}^{S}(2) $
A:(0,v,w)	$2A_{1}^{S}(2) $
B:(u,0,w)	$2B_{1}^{S}(2) $
C:(u,v,0)	$2C_{1}^{S}(2) $
D:(u,1/2,1/2)	$D_{1}^{S}(2)⊕D_{2}^{S}(2) $
P:(1/2,1/2,w)	$2P_{2}^{S}(2) $
Q:(1/2,v,1/2)	$Q_{1}^{S}(2)⊕Q_{2}^{S}(2) $
GP:(u,v,w)	$2GP_{1}^{S}(2) $

16k

k-vector	A↑G(8)
Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{2}^{S}(2)⊕Γ_{3}^{S}(2)⊕Γ_{4}^{S}(2) $
X:(1,1,1)	$X_{1}^{S}(2)⊕X_{2}^{S}(2)⊕X_{3}^{S}(2)⊕X_{4}^{S}(2) $
R:(1/2,0,1/2)	$2R_{1}^{S}(2)⊕2R_{2}^{S}(2) $
S:(0,1/2,1/2)	$2S_{1}^{S}(2)⊕2S_{2}^{S}(2) $
T:(1/2,1/2,0)	$2T_{1}^{S}(2)⊕2T_{2}^{S}(2) $
W:(1/2,1/2,1/2)	$W_{1}^{S}(2)⊕W_{2}^{S}(2)⊕W_{3}^{S}(2)⊕W_{4}^{S}(2) $
Δ:(0,v,0)	$2Δ_{1}^{S}(2)⊕2Δ_{2}^{S}(2) $
Λ:(0,0,w)	$2Λ_{1}^{S}(2)⊕2Λ_{2}^{S}(2) $
Σ:(u,0,0)	$2Σ_{1}^{S}(2)⊕2Σ_{2}^{S}(2) $
A:(0,v,w)	$4A_{1}^{S}(2) $
B:(u,0,w)	$4B_{1}^{S}(2) $
C:(u,v,0)	$4C_{1}^{S}(2) $
D:(u,1/2,1/2)	$2D_{1}^{S}(2)⊕2D_{2}^{S}(2) $
P:(1/2,1/2,w)	$2P_{1}^{S}(2)⊕2P_{2}^{S}(2) $
Q:(1/2,v,1/2)	$2Q_{1}^{S}(2)⊕2Q_{2}^{S}(2) $
GP:(u,v,w)	$4GP_{1}^{S}(2) $