Information of spin space group $C\ce{^{1}{c}}\ce{^{1}{c}}\ce{^{\text{-}1}{e}}\ce{^{\infty m}{1}}$ ( nSSG:37.68.1.1 )

General Positions

The continuously infinite group ^∞1 is omitted.

Coordinates	Seitz symbol
a, b, c | x, y, z	{ 1 ‖ 1 | 0 }
-a, -b+1/2, c | x, y, z	{ 1 ‖ 2001 | 0 1/2 0 }
-a+1/2, b, c+1/2 | x, y, z	{ 1 ‖ m100 | 1/2 0 1/2 }
a+1/2, -b+1/2, c+1/2 | x, y, z	{ 1 ‖ m010 | 1/2 1/2 1/2 }
a, -b+1/2, -c+1/2 | -x, -y, -z	{ -1 ‖ 2100 | 0 1/2 1/2 }
-a, b, -c+1/2 | -x, -y, -z	{ -1 ‖ 2010 | 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, -c | -x, -y, -z	{ -1 ‖ m001 | 1/2 0 0 }
a+1/2, b+1/2, c | x, y, z	{ 1 ‖ 1 | 1/2 1/2 0 }
-a+1/2, -b, c | x, y, z	{ 1 ‖ 2001 | 1/2 0 0 }
-a, b+1/2, c+1/2 | x, y, z	{ 1 ‖ m100 | 0 1/2 1/2 }
a, -b, c+1/2 | x, y, z	{ 1 ‖ m010 | 0 0 1/2 }
a+1/2, -b, -c+1/2 | -x, -y, -z	{ -1 ‖ 2100 | 1/2 0 1/2 }
-a+1/2, b+1/2, -c+1/2 | -x, -y, -z	{ -1 ‖ 2010 | 1/2 1/2 1/2 }
-a, -b, -c | -x, -y, -z	{ -1 ‖ -1 | 0 }
a, b+1/2, -c | -x, -y, -z	{ -1 ‖ m001 | 0 1/2 0 }
a, b, c | -x, y, z	{ m ‖ 1 | 0 }
-a, -b+1/2, c | -x, y, z	{ m ‖ 2001 | 0 1/2 0 }
-a+1/2, b, c+1/2 | -x, y, z	{ m ‖ m100 | 1/2 0 1/2 }
a+1/2, -b+1/2, c+1/2 | -x, y, z	{ m ‖ m010 | 1/2 1/2 1/2 }
a, -b+1/2, -c+1/2 | x, -y, -z	{ 2 ‖ 2100 | 0 1/2 1/2 }
-a, b, -c+1/2 | x, -y, -z	{ 2 ‖ 2010 | 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, -c | x, -y, -z	{ 2 ‖ m001 | 1/2 0 0 }
a+1/2, b+1/2, c | -x, y, z	{ m ‖ 1 | 1/2 1/2 0 }
-a+1/2, -b, c | -x, y, z	{ m ‖ 2001 | 1/2 0 0 }
-a, b+1/2, c+1/2 | -x, y, z	{ m ‖ m100 | 0 1/2 1/2 }
a, -b, c+1/2 | -x, y, z	{ m ‖ m010 | 0 0 1/2 }
a+1/2, -b, -c+1/2 | x, -y, -z	{ 2 ‖ 2100 | 1/2 0 1/2 }
-a+1/2, b+1/2, -c+1/2 | x, -y, -z	{ 2 ‖ 2010 | 1/2 1/2 1/2 }
-a, -b, -c | x, -y, -z	{ 2 ‖ -1 | 0 }
a, b+1/2, -c | x, -y, -z	{ 2 ‖ m001 | 0 1/2 0 }

Wyckoff Positions

WP	Site symmetry	Representative
4a	$\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,1/4,1/4 | 0,0,0)
4b	$\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,1/4,3/4 | 0,0,0)
8c	$\ce{^{-1}{-1}}\ce{^{\infty m}{1}} $	(1/4,3/4,0 | 0,0,0)
8d	$\ce{^{-1}{-1}}\ce{^{\infty m}{1}} $	(0,0,0 | 0,0,0)
8e	$\ce{^{-1}{2}}..\ce{^{\infty m}{1}} $	(a,1/4,1/4 | 0,0,0)
8f	$.\ce{^{-1}{2}}.\ce{^{\infty m}{1}} $	(0,b,1/4 | 0,0,0)
8g	$..\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,1/4,c | 0,0,z)
8h	$..\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(1/4,0,c | 0,0,z)
16i	$\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}{m}}\ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
T:(1,0,1/2)	$\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
Y:(1,0,0)	$\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
Z:(0,0,1/2)	$\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
R:(1/2,1/2,1/2)	$\ce{^{1}{2/}}\ce{^{2}{m}}\ce{^{\infty m}{1}} $
S:(1/2,1/2,0)	$\ce{^{1}{2/}}\ce{^{2}{m}}\ce{^{\infty m}{1}} $
A:(u,0,1/2)	$\ce{^{m}{m}}\ce{^{1}{m}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
B:(0,v,1/2)	$\ce{^{1}{m}}\ce{^{m}{m}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
Δ:(0,v,0)	$\ce{^{1}{m}}\ce{^{m}{m}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
H:(1,0,w)	$\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{\infty}{1}} $
Λ:(0,0,w)	$ \ce{^{1}{m}}\ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{\infty}{1}} $
Σ:(u,0,0)	$\ce{^{m}{m}}\ce{^{1}{m}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
D:(1/2,1/2,w)	$\ce{^{1}{2/}}\ce{^{-1}{m}}\ce{^{\infty}{1}} $
K:(0,v,w)	$\ce{^{-1}{2/}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
M:(u,0,w)	$\ce{^{-1}{2/}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
P:(u,v,0)	$\ce{^{m}{2/}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
Q:(u,v,1/2)	$\ce{^{m}{2/}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
GP:(u,v,w)	$\ce{^{-1}{-1}}\ce{^{\infty}{1}} $

Spin Brillouin Zone

Band Representations (Magnon)

8h

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

8g

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

16i

k-vector	k-vector-G↑	A↑G(8)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{2}^{S}(2)⊕Γ_{3}^{S}(2)⊕Γ_{4}^{S}(2) $
R:(1/2,1/2,1/2)	R:(1/2,1/2,1/2)	$2R_{1}^{S}(2)⊕2R_{2}^{S}(2) $
S:(1/2,1/2,0)	S:(1/2,1/2,0)	$2S_{1}^{S}(2)⊕2S_{2}^{S}(2) $
T:(1,0,1/2)	T:(1,0,1/2)	$T_{1}^{S}T_{2}^{S}(4)⊕T_{3}^{S}T_{4}^{S}(4) $
Y:(1,0,0)	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:(0,0,1/2)	$Z_{1}^{S}Z_{2}^{S}(4)⊕Z_{3}^{S}Z_{4}^{S}(4) $
A:(u,0,1/2)	A:(u,0,1/2)	$2A_{1}^{S}A_{2}^{S}(4) $
B:(0,v,1/2)	B:(0,v,1/2)	$2B_{1}^{S}B_{2}^{S}(4) $
D:(1/2,1/2,w)	D:(1/2,1/2,w)	$2D_{1}^{S}(2)⊕2D_{2}^{S}(2) $
Δ:(0,v,0)	Δ:(0,v,0)	$2Δ_{1}^{S}(2)⊕2Δ_{2}^{S}(2) $
H:(1,0,w)	H:(1,0,w)	$H_{1}^{S}(2)⊕H_{2}^{S}(2)⊕H_{3}^{S}(2)⊕H_{4}^{S}(2) $
Λ:(0,0,w)	Λ:(0,0,w)	$Λ_{1}^{S}(2)⊕Λ_{2}^{S}(2)⊕Λ_{3}^{S}(2)⊕Λ_{4}^{S}(2) $
Σ:(u,0,0)	Σ:(u,0,0)	$2Σ_{1}^{S}(2)⊕2Σ_{2}^{S}(2) $
K:(0,v,w)	K:(0,v,w)	$2K_{1}^{S}(2)⊕2K_{2}^{S}(2) $
M:(u,0,w)	M:(u,0,w)	$2M_{1}^{S}(2)⊕2M_{2}^{S}(2) $
P:(u,v,0)	P:(u,v,0)	$4P_{1}^{S}(2) $
Q:(u,v,1/2)	Q:(u,v,1/2)	$4Q_{1}^{S}(2) $
GP:(u,v,w)	GP:(u,v,w)	$4GP_{1}^{S}(2) $