Information of spin space group $C_{a}\ce{^{1}{2/}}\ce{^{1}{m}}\ce{^{\infty m}{1}}$ ( nSSG:12.10.2.1 )

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 ‖ 2010 | 0 }
-a, -b, -c | x, y, z	{ 1 ‖ -1 | 0 }
a, -b, c | x, y, z	{ 1 ‖ m010 | 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 ‖ 2010 | 1/2 1/2 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 ‖ m010 | 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 ‖ 2010 | 1/2 0 0 }
-a+1/2, -b, -c | -x, -y, -z	{ -1 ‖ -1 | 1/2 0 0 }
a+1/2, -b, c | -x, -y, -z	{ -1 ‖ m010 | 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 ‖ 2010 | 0 1/2 0 }
-a, -b+1/2, -c | -x, -y, -z	{ -1 ‖ -1 | 0 1/2 0 }
a, -b+1/2, c | -x, -y, -z	{ -1 ‖ m010 | 0 1/2 0 }
a, b, c | -x, y, z	{ m ‖ 1 | 0 }
-a, b, -c | -x, y, z	{ m ‖ 2010 | 0 }
-a, -b, -c | -x, y, z	{ m ‖ -1 | 0 }
a, -b, c | -x, y, z	{ m ‖ m010 | 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 ‖ 2010 | 1/2 1/2 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 ‖ m010 | 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 ‖ 2010 | 1/2 0 0 }
-a+1/2, -b, -c | x, -y, -z	{ 2 ‖ -1 | 1/2 0 0 }
a+1/2, -b, c | x, -y, -z	{ 2 ‖ m010 | 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 ‖ 2010 | 0 1/2 0 }
-a, -b+1/2, -c | x, -y, -z	{ 2 ‖ -1 | 0 1/2 0 }
a, -b+1/2, c | x, -y, -z	{ 2 ‖ m010 | 0 1/2 0 }

Wyckoff Positions

WP	Site symmetry	Representative
4a	$\ce{^{1}{2}}/\ce{^{1}{m}}\ce{^{\infty m}{1}} $	(0,0,0 | 0,0,z)
4b	$\ce{^{1}{2}}/\ce{^{-1}{m}}\ce{^{\infty m}{1}} $	(0,1/4,0 | 0,0,0)
4c	$\ce{^{1}{2}}/\ce{^{1}{m}}\ce{^{\infty m}{1}} $	(0,0,1/2 | 0,0,z)
4d	$\ce{^{-1}{2}}/\ce{^{1}{m}}\ce{^{\infty m}{1}} $	(1/4,0,0 | 0,0,0)
4e	$\ce{^{-1}{2}}/\ce{^{-1}{m}}\ce{^{\infty m}{1}} $	(1/4,1/4,0 | 0,0,0)
4f	$\ce{^{1}{2}}/\ce{^{-1}{m}}\ce{^{\infty m}{1}} $	(0,1/4,1/2 | 0,0,0)
4g	$\ce{^{-1}{2}}/\ce{^{1}{m}}\ce{^{\infty m}{1}} $	(1/4,0,1/2 | 0,0,0)
4h	$\ce{^{-1}{2}}/\ce{^{-1}{m}}\ce{^{\infty m}{1}} $	(1/4,1/4,1/2 | 0,0,0)
8i	$\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,b,0 | 0,0,z)
8j	$\ce{^{-1}{2}}\ce{^{\infty m}{1}} $	(1/4,b,0 | 0,0,0)
8k	$\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,b,1/2 | 0,0,z)
8l	$\ce{^{-1}{2}}\ce{^{\infty m}{1}} $	(1/4,b,1/2 | 0,0,0)
8m	$\ce{^{1}{m}}\ce{^{\infty m}{1}} $	(a,0,c | 0,0,z)
8n	$\ce{^{-1}{m}}\ce{^{\infty m}{1}} $	(a,1/4,c | 0,0,0)
16o	$\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
A:(0,0,1/2)	$\ce{^{1}{2/}}\ce{^{1}{m}}\ce{^{\infty /mm}{1}} $
Γ:(0,0,0)	$\ce{^{1}{2/}}\ce{^{1}{m}}\ce{^{\infty /mm}{1}} $
M:(0,1,1/2)	$\ce{^{1}{2/}}\ce{^{1}{m}}\ce{^{\infty /mm}{1}} $
Y:(0,1,0)	$\ce{^{1}{2/}}\ce{^{1}{m}}\ce{^{\infty /mm}{1}} $
L:(1/2,1/2,1/2)	$\ce{^{1}{-1}}\ce{^{\infty /mm}{1}} $
V:(1/2,1/2,0)	$\ce{^{1}{-1}}\ce{^{\infty /mm}{1}} $
B:(u,0,w)	$\ce{^{m}{2/}}\ce{^{1}{m}}\ce{^{\infty 2}{1}} $
Λ:(0,v,0)	$\ce{^{1}{2/}}\ce{^{m}{m}}\ce{^{\infty 2}{1}} $
U:(0,v,1/2)	$\ce{^{1}{2/}}\ce{^{m}{m}}\ce{^{\infty 2}{1}} $
GP:(u,v,w)	$\ce{^{m}{-1}}\ce{^{\infty 2}{1}} $

Spin Brillouin Zone

Band Representations (Magnon)

4a

k-vector	Ag↑G(2)
A:(0,0,1/2)	$A_{1}^{S,+}(2) $
Γ:(0,0,0)	$Γ_{1}^{S,+}(2) $
M:(0,1,1/2)	$M_{1}^{S,+}(2) $
Y:(0,1,0)	$Y_{1}^{S,+}(2) $
L:(1/2,1/2,1/2)	$L_{1}^{S,+}(2) $
V:(1/2,1/2,0)	$V_{1}^{S,+}(2) $
B:(u,0,w)	$B_{1}^{S}(2) $
Λ:(0,v,0)	$Λ_{1}^{S}(2) $
U:(0,v,1/2)	$U_{1}^{S}(2) $
GP:(u,v,w)	$GP_{1}^{S}(2) $

4c

k-vector	Ag↑G(2)
A:(0,0,1/2)	$A_{2}^{S,-}(2) $
Γ:(0,0,0)	$Γ_{1}^{S,+}(2) $
M:(0,1,1/2)	$M_{2}^{S,-}(2) $
Y:(0,1,0)	$Y_{1}^{S,+}(2) $
L:(1/2,1/2,1/2)	$L_{1}^{S,-}(2) $
V:(1/2,1/2,0)	$V_{1}^{S,+}(2) $
B:(u,0,w)	$B_{1}^{S}(2) $
Λ:(0,v,0)	$Λ_{1}^{S}(2) $
U:(0,v,1/2)	$U_{2}^{S}(2) $
GP:(u,v,w)	$GP_{1}^{S}(2) $

8i

k-vector	A↑G(4)
A:(0,0,1/2)	$A_{1}^{S,+}(2)⊕A_{1}^{S,-}(2) $
Γ:(0,0,0)	$Γ_{1}^{S,+}(2)⊕Γ_{1}^{S,-}(2) $
M:(0,1,1/2)	$M_{1}^{S,+}(2)⊕M_{1}^{S,-}(2) $
Y:(0,1,0)	$Y_{1}^{S,+}(2)⊕Y_{1}^{S,-}(2) $
L:(1/2,1/2,1/2)	$L_{1}^{S,+}(2)⊕L_{1}^{S,-}(2) $
V:(1/2,1/2,0)	$V_{1}^{S,+}(2)⊕V_{1}^{S,-}(2) $
B:(u,0,w)	$B_{1}^{S}(2)⊕B_{2}^{S}(2) $
Λ:(0,v,0)	$2Λ_{1}^{S}(2) $
U:(0,v,1/2)	$2U_{1}^{S}(2) $
GP:(u,v,w)	$2GP_{1}^{S}(2) $

8k

k-vector	A↑G(4)
A:(0,0,1/2)	$A_{2}^{S,+}(2)⊕A_{2}^{S,-}(2) $
Γ:(0,0,0)	$Γ_{1}^{S,+}(2)⊕Γ_{1}^{S,-}(2) $
M:(0,1,1/2)	$M_{2}^{S,+}(2)⊕M_{2}^{S,-}(2) $
Y:(0,1,0)	$Y_{1}^{S,+}(2)⊕Y_{1}^{S,-}(2) $
L:(1/2,1/2,1/2)	$L_{1}^{S,+}(2)⊕L_{1}^{S,-}(2) $
V:(1/2,1/2,0)	$V_{1}^{S,+}(2)⊕V_{1}^{S,-}(2) $
B:(u,0,w)	$B_{1}^{S}(2)⊕B_{2}^{S}(2) $
Λ:(0,v,0)	$2Λ_{1}^{S}(2) $
U:(0,v,1/2)	$2U_{2}^{S}(2) $
GP:(u,v,w)	$2GP_{1}^{S}(2) $

8m

k-vector	A'↑G(4)
A:(0,0,1/2)	$A_{1}^{S,+}(2)⊕A_{2}^{S,-}(2) $
Γ:(0,0,0)	$Γ_{1}^{S,+}(2)⊕Γ_{2}^{S,-}(2) $
M:(0,1,1/2)	$M_{1}^{S,+}(2)⊕M_{2}^{S,-}(2) $
Y:(0,1,0)	$Y_{1}^{S,+}(2)⊕Y_{2}^{S,-}(2) $
L:(1/2,1/2,1/2)	$L_{1}^{S,+}(2)⊕L_{1}^{S,-}(2) $
V:(1/2,1/2,0)	$V_{1}^{S,+}(2)⊕V_{1}^{S,-}(2) $
B:(u,0,w)	$2B_{1}^{S}(2) $
Λ:(0,v,0)	$Λ_{1}^{S}(2)⊕Λ_{2}^{S}(2) $
U:(0,v,1/2)	$U_{1}^{S}(2)⊕U_{2}^{S}(2) $
GP:(u,v,w)	$2GP_{1}^{S}(2) $

16o

k-vector	A↑G(8)
A:(0,0,1/2)	$A_{1}^{S,+}(2)⊕A_{1}^{S,-}(2)⊕A_{2}^{S,+}(2)⊕A_{2}^{S,-}(2) $
Γ:(0,0,0)	$Γ_{1}^{S,+}(2)⊕Γ_{1}^{S,-}(2)⊕Γ_{2}^{S,+}(2)⊕Γ_{2}^{S,-}(2) $
M:(0,1,1/2)	$M_{1}^{S,+}(2)⊕M_{1}^{S,-}(2)⊕M_{2}^{S,+}(2)⊕M_{2}^{S,-}(2) $
Y:(0,1,0)	$Y_{1}^{S,+}(2)⊕Y_{1}^{S,-}(2)⊕Y_{2}^{S,+}(2)⊕Y_{2}^{S,-}(2) $
L:(1/2,1/2,1/2)	$2L_{1}^{S,+}(2)⊕2L_{1}^{S,-}(2) $
V:(1/2,1/2,0)	$2V_{1}^{S,+}(2)⊕2V_{1}^{S,-}(2) $
B:(u,0,w)	$2B_{1}^{S}(2)⊕2B_{2}^{S}(2) $
Λ:(0,v,0)	$2Λ_{1}^{S}(2)⊕2Λ_{2}^{S}(2) $
U:(0,v,1/2)	$2U_{1}^{S}(2)⊕2U_{2}^{S}(2) $
GP:(u,v,w)	$4GP_{1}^{S}(2) $