Information of spin space group $P_{a}\ce{^{1}{2/}}\ce{^{1}{c}}\ce{^{\infty m}{1}}$ ( nSSG:13.13.2.4 )

General Positions

The continuously infinite group ^∞1 is omitted.

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

Wyckoff Positions

WP	Site symmetry	Representative
4a	$\ce{^{1}{-1}}\ce{^{\infty m}{1}} $	(0,0,0 | 0,0,z)
4b	$\ce{^{-1}{-1}}\ce{^{\infty m}{1}} $	(1/4,1/2,0 | 0,0,0)
4c	$\ce{^{1}{-1}}\ce{^{\infty m}{1}} $	(0,1/2,0 | 0,0,z)
4d	$\ce{^{-1}{-1}}\ce{^{\infty m}{1}} $	(1/4,0,0 | 0,0,0)
4e	$\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,b,1/4 | 0,0,z)
4f	$\ce{^{-1}{2}}\ce{^{\infty m}{1}} $	(1/4,b,1/4 | 0,0,0)
8g	$\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:(1/2,0,1/2)	$\ce{^{1}{2/}}\ce{^{1}{m}}\ce{^{\infty /mm}{1}} $
B:(0,0,1/2)	$\ce{^{1}{2/}}\ce{^{1}{m}}\ce{^{\infty /mm}{1}} $
C:(1/2,1/2,0)	$\ce{^{1}{2/}}\ce{^{1}{m}}\ce{^{\infty /mm}{1}} $
D:(0,1/2,1/2)	$\ce{^{1}{2/}}\ce{^{1}{m}}\ce{^{\infty /mm}{1}} $
E:(1/2,1/2,1/2)	$\ce{^{1}{2/}}\ce{^{1}{m}}\ce{^{\infty /mm}{1}} $
Γ:(0,0,0)	$\ce{^{1}{2/}}\ce{^{1}{m}}\ce{^{\infty /mm}{1}} $
Y:(1/2,0,0)	$\ce{^{1}{2/}}\ce{^{1}{m}}\ce{^{\infty /mm}{1}} $
Z:(0,1/2,0)	$\ce{^{1}{2/}}\ce{^{1}{m}}\ce{^{\infty /mm}{1}} $
F:(u,0,w)	$\ce{^{m}{2/}}\ce{^{1}{m}}\ce{^{\infty 2}{1}} $
G:(u,1/2,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:(1/2,v,1/2)	$\ce{^{1}{2/}}\ce{^{m}{m}}\ce{^{\infty 2}{1}} $
V:(0,v,1/2)	$\ce{^{1}{2/}}\ce{^{m}{m}}\ce{^{\infty 2}{1}} $
W:(1/2,v,0)	$\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(4)
A:(1/2,0,1/2)	$A_{1}^{S}(4) $
B:(0,0,1/2)	$B_{1}^{S}(4) $
C:(1/2,1/2,0)	$C_{1}^{S,+}(2)⊕C_{2}^{S,+}(2) $
D:(0,1/2,1/2)	$D_{1}^{S}(4) $
E:(1/2,1/2,1/2)	$E_{1}^{S}(4) $
Γ:(0,0,0)	$Γ_{1}^{S,+}(2)⊕Γ_{2}^{S,+}(2) $
Y:(1/2,0,0)	$Y_{1}^{S,+}(2)⊕Y_{2}^{S,+}(2) $
Z:(0,1/2,0)	$Z_{1}^{S,+}(2)⊕Z_{2}^{S,+}(2) $
F:(u,0,w)	$F_{1}^{S}(2)⊕F_{2}^{S}(2) $
G:(u,1/2,w)	$G_{1}^{S}(2)⊕G_{2}^{S}(2) $
Λ:(0,v,0)	$Λ_{1}^{S}(2)⊕Λ_{2}^{S}(2) $
U:(1/2,v,1/2)	$U_{1}^{S}U_{2}^{S}(4) $
V:(0,v,1/2)	$V_{1}^{S}V_{2}^{S}(4) $
W:(1/2,v,0)	$W_{1}^{S}(2)⊕W_{2}^{S}(2) $
GP:(u,v,w)	$2GP_{1}^{S}(2) $

4c

k-vector	Ag↑G(4)
A:(1/2,0,1/2)	$A_{1}^{S}(4) $
B:(0,0,1/2)	$B_{1}^{S}(4) $
C:(1/2,1/2,0)	$C_{1}^{S,-}(2)⊕C_{2}^{S,-}(2) $
D:(0,1/2,1/2)	$D_{1}^{S}(4) $
E:(1/2,1/2,1/2)	$E_{1}^{S}(4) $
Γ:(0,0,0)	$Γ_{1}^{S,+}(2)⊕Γ_{2}^{S,+}(2) $
Y:(1/2,0,0)	$Y_{1}^{S,+}(2)⊕Y_{2}^{S,+}(2) $
Z:(0,1/2,0)	$Z_{1}^{S,-}(2)⊕Z_{2}^{S,-}(2) $
F:(u,0,w)	$F_{1}^{S}(2)⊕F_{2}^{S}(2) $
G:(u,1/2,w)	$G_{1}^{S}(2)⊕G_{2}^{S}(2) $
Λ:(0,v,0)	$Λ_{1}^{S}(2)⊕Λ_{2}^{S}(2) $
U:(1/2,v,1/2)	$U_{1}^{S}U_{2}^{S}(4) $
V:(0,v,1/2)	$V_{1}^{S}V_{2}^{S}(4) $
W:(1/2,v,0)	$W_{1}^{S}(2)⊕W_{2}^{S}(2) $
GP:(u,v,w)	$2GP_{1}^{S}(2) $

4e

k-vector	A↑G(4)
A:(1/2,0,1/2)	$A_{1}^{S}(4) $
B:(0,0,1/2)	$B_{1}^{S}(4) $
C:(1/2,1/2,0)	$C_{1}^{S,+}(2)⊕C_{1}^{S,-}(2) $
D:(0,1/2,1/2)	$D_{1}^{S}(4) $
E:(1/2,1/2,1/2)	$E_{1}^{S}(4) $
Γ:(0,0,0)	$Γ_{1}^{S,+}(2)⊕Γ_{1}^{S,-}(2) $
Y:(1/2,0,0)	$Y_{1}^{S,+}(2)⊕Y_{1}^{S,-}(2) $
Z:(0,1/2,0)	$Z_{1}^{S,+}(2)⊕Z_{1}^{S,-}(2) $
F:(u,0,w)	$F_{1}^{S}(2)⊕F_{2}^{S}(2) $
G:(u,1/2,w)	$G_{1}^{S}(2)⊕G_{2}^{S}(2) $
Λ:(0,v,0)	$2Λ_{1}^{S}(2) $
U:(1/2,v,1/2)	$U_{1}^{S}U_{2}^{S}(4) $
V:(0,v,1/2)	$V_{1}^{S}V_{2}^{S}(4) $
W:(1/2,v,0)	$2W_{1}^{S}(2) $
GP:(u,v,w)	$2GP_{1}^{S}(2) $

8g

k-vector	A↑G(8)
A:(1/2,0,1/2)	$2A_{1}^{S}(4) $
B:(0,0,1/2)	$2B_{1}^{S}(4) $
C:(1/2,1/2,0)	$C_{1}^{S,+}(2)⊕C_{1}^{S,-}(2)⊕C_{2}^{S,+}(2)⊕C_{2}^{S,-}(2) $
D:(0,1/2,1/2)	$2D_{1}^{S}(4) $
E:(1/2,1/2,1/2)	$2E_{1}^{S}(4) $
Γ:(0,0,0)	$Γ_{1}^{S,+}(2)⊕Γ_{1}^{S,-}(2)⊕Γ_{2}^{S,+}(2)⊕Γ_{2}^{S,-}(2) $
Y:(1/2,0,0)	$Y_{1}^{S,+}(2)⊕Y_{1}^{S,-}(2)⊕Y_{2}^{S,+}(2)⊕Y_{2}^{S,-}(2) $
Z:(0,1/2,0)	$Z_{1}^{S,+}(2)⊕Z_{1}^{S,-}(2)⊕Z_{2}^{S,+}(2)⊕Z_{2}^{S,-}(2) $
F:(u,0,w)	$2F_{1}^{S}(2)⊕2F_{2}^{S}(2) $
G:(u,1/2,w)	$2G_{1}^{S}(2)⊕2G_{2}^{S}(2) $
Λ:(0,v,0)	$2Λ_{1}^{S}(2)⊕2Λ_{2}^{S}(2) $
U:(1/2,v,1/2)	$2U_{1}^{S}U_{2}^{S}(4) $
V:(0,v,1/2)	$2V_{1}^{S}V_{2}^{S}(4) $
W:(1/2,v,0)	$2W_{1}^{S}(2)⊕2W_{2}^{S}(2) $
GP:(u,v,w)	$4GP_{1}^{S}(2) $