Information of spin space group $C_{a}\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}}$ ( nSSG:21.16.2.1 )

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 | 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+1/2, b, c | -x, -y, -z	{ -1 ‖ 1 | 1/2 0 0 }
a+1/2, -b, -c | -x, -y, -z	{ -1 ‖ 2100 | 1/2 0 0 }
-a+1/2, b, -c | -x, -y, -z	{ -1 ‖ 2010 | 1/2 0 0 }
-a+1/2, -b, c | -x, -y, -z	{ -1 ‖ 2001 | 1/2 0 0 }
a, b+1/2, c | -x, -y, -z	{ -1 ‖ 1 | 0 1/2 0 }
a, -b+1/2, -c | -x, -y, -z	{ -1 ‖ 2100 | 0 1/2 0 }
-a, b+1/2, -c | -x, -y, -z	{ -1 ‖ 2010 | 0 1/2 0 }
-a, -b+1/2, c | -x, -y, -z	{ -1 ‖ 2001 | 0 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 | -x, y, z	{ m ‖ 1 | 1/2 1/2 0 }
a+1/2, -b+1/2, -c | -x, y, z	{ m ‖ 2100 | 1/2 1/2 0 }
-a+1/2, b+1/2, -c | -x, y, z	{ m ‖ 2010 | 1/2 1/2 0 }
-a+1/2, -b+1/2, c | -x, y, z	{ m ‖ 2001 | 1/2 1/2 0 }
a+1/2, b, c | x, -y, -z	{ 2 ‖ 1 | 1/2 0 0 }
a+1/2, -b, -c | x, -y, -z	{ 2 ‖ 2100 | 1/2 0 0 }
-a+1/2, b, -c | x, -y, -z	{ 2 ‖ 2010 | 1/2 0 0 }
-a+1/2, -b, c | x, -y, -z	{ 2 ‖ 2001 | 1/2 0 0 }
a, b+1/2, c | x, -y, -z	{ 2 ‖ 1 | 0 1/2 0 }
a, -b+1/2, -c | x, -y, -z	{ 2 ‖ 2100 | 0 1/2 0 }
-a, b+1/2, -c | x, -y, -z	{ 2 ‖ 2010 | 0 1/2 0 }
-a, -b+1/2, c | x, -y, -z	{ 2 ‖ 2001 | 0 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}} $	(1/4,0,0 | 0,0,0)
4c	$\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{-1}{2}}\ce{^{\infty m}{1}} $	(0,1/4,0 | 0,0,0)
4d	$\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,0,1/2 | 0,0,z)
4e	$\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(1/4,1/4,0 | 0,0,0)
4f	$\ce{^{1}{2}}\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{\infty m}{1}} $	(1/4,0,1/2 | 0,0,0)
4g	$\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{-1}{2}}\ce{^{\infty m}{1}} $	(0,1/4,1/2 | 0,0,0)
4h	$\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(1/4,1/4,1/2 | 0,0,0)
8i	$\ce{^{1}{2}}..\ce{^{\infty m}{1}} $	(a,0,0 | 0,0,z)
8j	$\ce{^{1}{2}}..\ce{^{\infty m}{1}} $	(a,0,1/2 | 0,0,z)
8k	$\ce{^{-1}{2}}..\ce{^{\infty m}{1}} $	(a,1/4,0 | 0,0,0)
8l	$\ce{^{-1}{2}}..\ce{^{\infty m}{1}} $	(a,1/4,1/2 | 0,0,0)
8m	$.\ce{^{1}{2}}.\ce{^{\infty m}{1}} $	(0,b,0 | 0,0,z)
8n	$.\ce{^{1}{2}}.\ce{^{\infty m}{1}} $	(0,b,1/2 | 0,0,z)
8o	$.\ce{^{-1}{2}}.\ce{^{\infty m}{1}} $	(1/4,b,0 | 0,0,0)
8p	$.\ce{^{-1}{2}}.\ce{^{\infty m}{1}} $	(1/4,b,1/2 | 0,0,0)
8q	$..\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,0,c | 0,0,z)
8r	$..\ce{^{-1}{2}}\ce{^{\infty m}{1}} $	(1/4,0,c | 0,0,0)
8s	$..\ce{^{-1}{2}}\ce{^{\infty m}{1}} $	(0,1/4,c | 0,0,0)
8t	$..\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(1/4,1/4,c | 0,0,z)
16u	$\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}} $
T:(1,0,1/2)	$\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{\infty /mm}{1}} $
Y:(1,0,0)	$\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{\infty /mm}{1}} $
Z:(0,0,1/2)	$\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{\infty /mm}{1}} $
R:(1/2,1/2,1/2)	$\ce{^{1}{2}}\ce{^{\infty /mm}{1}} $
S:(1/2,1/2,0)	$\ce{^{1}{2}}\ce{^{\infty /mm}{1}} $
A:(u,0,1/2)	$\ce{^{1}{2}}\ce{^{m}{2}}\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
B:(0,v,1/2)	$\ce{^{m}{2}}\ce{^{1}{2}}\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
Δ:(0,v,0)	$\ce{^{m}{2}}\ce{^{1}{2}}\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
H:(1,0,w)	$\ce{^{m}{2}}\ce{^{m}{2}}\ce{^{1}{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}} $
D:(1/2,1/2,w)	$\ce{^{1}{2}}\ce{^{\infty 2}{1}} $
K:(0,v,w)	$\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
M:(u,0,w)	$\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
P:(u,v,0)	$\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
Q:(u,v,1/2)	$\ce{^{m}{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) $
T:(1,0,1/2)	$T_{1}^{S}(2) $
Y:(1,0,0)	$Y_{1}^{S}(2) $
Z:(0,0,1/2)	$Z_{1}^{S}(2) $
R:(1/2,1/2,1/2)	$R_{1}^{S}(2) $
S:(1/2,1/2,0)	$S_{1}^{S}(2) $
A:(u,0,1/2)	$A_{1}^{S}(2) $
B:(0,v,1/2)	$B_{1}^{S}(2) $
Δ:(0,v,0)	$Δ_{1}^{S}(2) $
H:(1,0,w)	$H_{1}^{S}(2) $
Λ:(0,0,w)	$Λ_{1}^{S}(2) $
Σ:(u,0,0)	$Σ_{1}^{S}(2) $
D:(1/2,1/2,w)	$D_{1}^{S}(2) $
K:(0,v,w)	$K_{1}^{S}(2) $
M:(u,0,w)	$M_{1}^{S}(2) $
P:(u,v,0)	$P_{1}^{S}(2) $
Q:(u,v,1/2)	$Q_{1}^{S}(2) $
GP:(u,v,w)	$GP_{1}^{S}(2) $

4d

k-vector	A↑G(2)
Γ:(0,0,0)	$Γ_{1}^{S}(2) $
T:(1,0,1/2)	$T_{2}^{S}(2) $
Y:(1,0,0)	$Y_{1}^{S}(2) $
Z:(0,0,1/2)	$Z_{2}^{S}(2) $
R:(1/2,1/2,1/2)	$R_{1}^{S}(2) $
S:(1/2,1/2,0)	$S_{1}^{S}(2) $
A:(u,0,1/2)	$A_{2}^{S}(2) $
B:(0,v,1/2)	$B_{2}^{S}(2) $
Δ:(0,v,0)	$Δ_{1}^{S}(2) $
H:(1,0,w)	$H_{1}^{S}(2) $
Λ:(0,0,w)	$Λ_{1}^{S}(2) $
Σ:(u,0,0)	$Σ_{1}^{S}(2) $
D:(1/2,1/2,w)	$D_{1}^{S}(2) $
K:(0,v,w)	$K_{1}^{S}(2) $
M:(u,0,w)	$M_{1}^{S}(2) $
P:(u,v,0)	$P_{1}^{S}(2) $
Q:(u,v,1/2)	$Q_{1}^{S}(2) $
GP:(u,v,w)	$GP_{1}^{S}(2) $

8i

k-vector	A↑G(4)
Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{3}^{S}(2) $
T:(1,0,1/2)	$T_{1}^{S}(2)⊕T_{3}^{S}(2) $
Y:(1,0,0)	$Y_{1}^{S}(2)⊕Y_{3}^{S}(2) $
Z:(0,0,1/2)	$Z_{1}^{S}(2)⊕Z_{3}^{S}(2) $
R:(1/2,1/2,1/2)	$R_{1}^{S}(2)⊕R_{2}^{S}(2) $
S:(1/2,1/2,0)	$S_{1}^{S}(2)⊕S_{2}^{S}(2) $
A:(u,0,1/2)	$2A_{1}^{S}(2) $
B:(0,v,1/2)	$B_{1}^{S}(2)⊕B_{2}^{S}(2) $
Δ:(0,v,0)	$Δ_{1}^{S}(2)⊕Δ_{2}^{S}(2) $
H:(1,0,w)	$H_{1}^{S}(2)⊕H_{2}^{S}(2) $
Λ:(0,0,w)	$Λ_{1}^{S}(2)⊕Λ_{2}^{S}(2) $
Σ:(u,0,0)	$2Σ_{1}^{S}(2) $
D:(1/2,1/2,w)	$D_{1}^{S}(2)⊕D_{2}^{S}(2) $
K:(0,v,w)	$2K_{1}^{S}(2) $
M:(u,0,w)	$2M_{1}^{S}(2) $
P:(u,v,0)	$2P_{1}^{S}(2) $
Q:(u,v,1/2)	$2Q_{1}^{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_{3}^{S}(2) $
T:(1,0,1/2)	$T_{2}^{S}(2)⊕T_{4}^{S}(2) $
Y:(1,0,0)	$Y_{1}^{S}(2)⊕Y_{3}^{S}(2) $
Z:(0,0,1/2)	$Z_{2}^{S}(2)⊕Z_{4}^{S}(2) $
R:(1/2,1/2,1/2)	$R_{1}^{S}(2)⊕R_{2}^{S}(2) $
S:(1/2,1/2,0)	$S_{1}^{S}(2)⊕S_{2}^{S}(2) $
A:(u,0,1/2)	$2A_{2}^{S}(2) $
B:(0,v,1/2)	$B_{1}^{S}(2)⊕B_{2}^{S}(2) $
Δ:(0,v,0)	$Δ_{1}^{S}(2)⊕Δ_{2}^{S}(2) $
H:(1,0,w)	$H_{1}^{S}(2)⊕H_{2}^{S}(2) $
Λ:(0,0,w)	$Λ_{1}^{S}(2)⊕Λ_{2}^{S}(2) $
Σ:(u,0,0)	$2Σ_{1}^{S}(2) $
D:(1/2,1/2,w)	$D_{1}^{S}(2)⊕D_{2}^{S}(2) $
K:(0,v,w)	$2K_{1}^{S}(2) $
M:(u,0,w)	$2M_{1}^{S}(2) $
P:(u,v,0)	$2P_{1}^{S}(2) $
Q:(u,v,1/2)	$2Q_{1}^{S}(2) $
GP:(u,v,w)	$2GP_{1}^{S}(2) $

8m

k-vector	A↑G(4)
Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{4}^{S}(2) $
T:(1,0,1/2)	$T_{1}^{S}(2)⊕T_{4}^{S}(2) $
Y:(1,0,0)	$Y_{1}^{S}(2)⊕Y_{4}^{S}(2) $
Z:(0,0,1/2)	$Z_{1}^{S}(2)⊕Z_{4}^{S}(2) $
R:(1/2,1/2,1/2)	$R_{1}^{S}(2)⊕R_{2}^{S}(2) $
S:(1/2,1/2,0)	$S_{1}^{S}(2)⊕S_{2}^{S}(2) $
A:(u,0,1/2)	$A_{1}^{S}(2)⊕A_{2}^{S}(2) $
B:(0,v,1/2)	$2B_{1}^{S}(2) $
Δ:(0,v,0)	$2Δ_{1}^{S}(2) $
H:(1,0,w)	$H_{1}^{S}(2)⊕H_{2}^{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) $
D:(1/2,1/2,w)	$D_{1}^{S}(2)⊕D_{2}^{S}(2) $
K:(0,v,w)	$2K_{1}^{S}(2) $
M:(u,0,w)	$2M_{1}^{S}(2) $
P:(u,v,0)	$2P_{1}^{S}(2) $
Q:(u,v,1/2)	$2Q_{1}^{S}(2) $
GP:(u,v,w)	$2GP_{1}^{S}(2) $

8n

k-vector	A↑G(4)
Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{4}^{S}(2) $
T:(1,0,1/2)	$T_{2}^{S}(2)⊕T_{3}^{S}(2) $
Y:(1,0,0)	$Y_{1}^{S}(2)⊕Y_{4}^{S}(2) $
Z:(0,0,1/2)	$Z_{2}^{S}(2)⊕Z_{3}^{S}(2) $
R:(1/2,1/2,1/2)	$R_{1}^{S}(2)⊕R_{2}^{S}(2) $
S:(1/2,1/2,0)	$S_{1}^{S}(2)⊕S_{2}^{S}(2) $
A:(u,0,1/2)	$A_{1}^{S}(2)⊕A_{2}^{S}(2) $
B:(0,v,1/2)	$2B_{2}^{S}(2) $
Δ:(0,v,0)	$2Δ_{1}^{S}(2) $
H:(1,0,w)	$H_{1}^{S}(2)⊕H_{2}^{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) $
D:(1/2,1/2,w)	$D_{1}^{S}(2)⊕D_{2}^{S}(2) $
K:(0,v,w)	$2K_{1}^{S}(2) $
M:(u,0,w)	$2M_{1}^{S}(2) $
P:(u,v,0)	$2P_{1}^{S}(2) $
Q:(u,v,1/2)	$2Q_{1}^{S}(2) $
GP:(u,v,w)	$2GP_{1}^{S}(2) $

8q

k-vector	A↑G(4)
Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{2}^{S}(2) $
T:(1,0,1/2)	$T_{1}^{S}(2)⊕T_{2}^{S}(2) $
Y:(1,0,0)	$Y_{1}^{S}(2)⊕Y_{2}^{S}(2) $
Z:(0,0,1/2)	$Z_{1}^{S}(2)⊕Z_{2}^{S}(2) $
R:(1/2,1/2,1/2)	$2R_{1}^{S}(2) $
S:(1/2,1/2,0)	$2S_{1}^{S}(2) $
A:(u,0,1/2)	$A_{1}^{S}(2)⊕A_{2}^{S}(2) $
B:(0,v,1/2)	$B_{1}^{S}(2)⊕B_{2}^{S}(2) $
Δ:(0,v,0)	$Δ_{1}^{S}(2)⊕Δ_{2}^{S}(2) $
H:(1,0,w)	$2H_{1}^{S}(2) $
Λ:(0,0,w)	$2Λ_{1}^{S}(2) $
Σ:(u,0,0)	$Σ_{1}^{S}(2)⊕Σ_{2}^{S}(2) $
D:(1/2,1/2,w)	$2D_{1}^{S}(2) $
K:(0,v,w)	$2K_{1}^{S}(2) $
M:(u,0,w)	$2M_{1}^{S}(2) $
P:(u,v,0)	$2P_{1}^{S}(2) $
Q:(u,v,1/2)	$2Q_{1}^{S}(2) $
GP:(u,v,w)	$2GP_{1}^{S}(2) $

8t

k-vector	A↑G(4)
Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{2}^{S}(2) $
T:(1,0,1/2)	$T_{1}^{S}(2)⊕T_{2}^{S}(2) $
Y:(1,0,0)	$Y_{1}^{S}(2)⊕Y_{2}^{S}(2) $
Z:(0,0,1/2)	$Z_{1}^{S}(2)⊕Z_{2}^{S}(2) $
R:(1/2,1/2,1/2)	$2R_{2}^{S}(2) $
S:(1/2,1/2,0)	$2S_{2}^{S}(2) $
A:(u,0,1/2)	$A_{1}^{S}(2)⊕A_{2}^{S}(2) $
B:(0,v,1/2)	$B_{1}^{S}(2)⊕B_{2}^{S}(2) $
Δ:(0,v,0)	$Δ_{1}^{S}(2)⊕Δ_{2}^{S}(2) $
H:(1,0,w)	$2H_{1}^{S}(2) $
Λ:(0,0,w)	$2Λ_{1}^{S}(2) $
Σ:(u,0,0)	$Σ_{1}^{S}(2)⊕Σ_{2}^{S}(2) $
D:(1/2,1/2,w)	$2D_{2}^{S}(2) $
K:(0,v,w)	$2K_{1}^{S}(2) $
M:(u,0,w)	$2M_{1}^{S}(2) $
P:(u,v,0)	$2P_{1}^{S}(2) $
Q:(u,v,1/2)	$2Q_{1}^{S}(2) $
GP:(u,v,w)	$2GP_{1}^{S}(2) $

16u

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) $
T:(1,0,1/2)	$T_{1}^{S}(2)⊕T_{2}^{S}(2)⊕T_{3}^{S}(2)⊕T_{4}^{S}(2) $
Y:(1,0,0)	$Y_{1}^{S}(2)⊕Y_{2}^{S}(2)⊕Y_{3}^{S}(2)⊕Y_{4}^{S}(2) $
Z:(0,0,1/2)	$Z_{1}^{S}(2)⊕Z_{2}^{S}(2)⊕Z_{3}^{S}(2)⊕Z_{4}^{S}(2) $
R:(1/2,1/2,1/2)	$2R_{1}^{S}(2)⊕2R_{2}^{S}(2) $
S:(1/2,1/2,0)	$2S_{1}^{S}(2)⊕2S_{2}^{S}(2) $
A:(u,0,1/2)	$2A_{1}^{S}(2)⊕2A_{2}^{S}(2) $
B:(0,v,1/2)	$2B_{1}^{S}(2)⊕2B_{2}^{S}(2) $
Δ:(0,v,0)	$2Δ_{1}^{S}(2)⊕2Δ_{2}^{S}(2) $
H:(1,0,w)	$2H_{1}^{S}(2)⊕2H_{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) $
D:(1/2,1/2,w)	$2D_{1}^{S}(2)⊕2D_{2}^{S}(2) $
K:(0,v,w)	$4K_{1}^{S}(2) $
M:(u,0,w)	$4M_{1}^{S}(2) $
P:(u,v,0)	$4P_{1}^{S}(2) $
Q:(u,v,1/2)	$4Q_{1}^{S}(2) $
GP:(u,v,w)	$4GP_{1}^{S}(2) $