Information of spin space group $F\ce{^{1}{d}}\ce{^{1}{d}}\ce{^{1}{2}}\ce{^{\text{-}1}{(1/2 \ 1/2 \ 1/2)}}\ce{^{\infty m}{1}}$ ( nSSG:43.34.2.1 )

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 ‖ 2001 | 0 }
-a+1/4, b+1/4, c+1/4 | x, y, z	{ 1 ‖ m100 | 1/4 1/4 1/4 }
a+1/4, -b+1/4, c+1/4 | x, y, z	{ 1 ‖ m010 | 1/4 1/4 1/4 }
a, b+1/2, c+1/2 | x, y, z	{ 1 ‖ 1 | 0 1/2 1/2 }
-a, -b+1/2, c+1/2 | x, y, z	{ 1 ‖ 2001 | 0 1/2 1/2 }
-a+1/4, b+3/4, c+3/4 | x, y, z	{ 1 ‖ m100 | 1/4 3/4 3/4 }
a+1/4, -b+3/4, c+3/4 | x, y, z	{ 1 ‖ m010 | 1/4 3/4 3/4 }
a+1/2, b, c+1/2 | x, y, z	{ 1 ‖ 1 | 1/2 0 1/2 }
-a+1/2, -b, c+1/2 | x, y, z	{ 1 ‖ 2001 | 1/2 0 1/2 }
-a+3/4, b+1/4, c+3/4 | x, y, z	{ 1 ‖ m100 | 3/4 1/4 3/4 }
a+3/4, -b+1/4, c+3/4 | x, y, z	{ 1 ‖ m010 | 3/4 1/4 3/4 }
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 ‖ 2001 | 1/2 1/2 0 }
-a+3/4, b+3/4, c+1/4 | x, y, z	{ 1 ‖ m100 | 3/4 3/4 1/4 }
a+3/4, -b+3/4, c+1/4 | x, y, z	{ 1 ‖ m010 | 3/4 3/4 1/4 }
a, b, c+1/2 | -x, -y, -z	{ -1 ‖ 1 | 0 0 1/2 }
-a, -b, c+1/2 | -x, -y, -z	{ -1 ‖ 2001 | 0 0 1/2 }
-a+1/4, b+1/4, c+3/4 | -x, -y, -z	{ -1 ‖ m100 | 1/4 1/4 3/4 }
a+1/4, -b+1/4, c+3/4 | -x, -y, -z	{ -1 ‖ m010 | 1/4 1/4 3/4 }
a, b+1/2, c | -x, -y, -z	{ -1 ‖ 1 | 0 1/2 0 }
-a, -b+1/2, c | -x, -y, -z	{ -1 ‖ 2001 | 0 1/2 0 }
-a+1/4, b+3/4, c+1/4 | -x, -y, -z	{ -1 ‖ m100 | 1/4 3/4 1/4 }
a+1/4, -b+3/4, c+1/4 | -x, -y, -z	{ -1 ‖ m010 | 1/4 3/4 1/4 }
a+1/2, b, c | -x, -y, -z	{ -1 ‖ 1 | 1/2 0 0 }
-a+1/2, -b, c | -x, -y, -z	{ -1 ‖ 2001 | 1/2 0 0 }
-a+3/4, b+1/4, c+1/4 | -x, -y, -z	{ -1 ‖ m100 | 3/4 1/4 1/4 }
a+3/4, -b+1/4, c+1/4 | -x, -y, -z	{ -1 ‖ m010 | 3/4 1/4 1/4 }
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 ‖ 2001 | 1/2 1/2 1/2 }
-a+3/4, b+3/4, c+3/4 | -x, -y, -z	{ -1 ‖ m100 | 3/4 3/4 3/4 }
a+3/4, -b+3/4, c+3/4 | -x, -y, -z	{ -1 ‖ m010 | 3/4 3/4 3/4 }
a, b, c | -x, y, z	{ m ‖ 1 | 0 }
-a, -b, c | -x, y, z	{ m ‖ 2001 | 0 }
-a+1/4, b+1/4, c+1/4 | -x, y, z	{ m ‖ m100 | 1/4 1/4 1/4 }
a+1/4, -b+1/4, c+1/4 | -x, y, z	{ m ‖ m010 | 1/4 1/4 1/4 }
a, b+1/2, c+1/2 | -x, y, z	{ m ‖ 1 | 0 1/2 1/2 }
-a, -b+1/2, c+1/2 | -x, y, z	{ m ‖ 2001 | 0 1/2 1/2 }
-a+1/4, b+3/4, c+3/4 | -x, y, z	{ m ‖ m100 | 1/4 3/4 3/4 }
a+1/4, -b+3/4, c+3/4 | -x, y, z	{ m ‖ m010 | 1/4 3/4 3/4 }
a+1/2, b, c+1/2 | -x, y, z	{ m ‖ 1 | 1/2 0 1/2 }
-a+1/2, -b, c+1/2 | -x, y, z	{ m ‖ 2001 | 1/2 0 1/2 }
-a+3/4, b+1/4, c+3/4 | -x, y, z	{ m ‖ m100 | 3/4 1/4 3/4 }
a+3/4, -b+1/4, c+3/4 | -x, y, z	{ m ‖ m010 | 3/4 1/4 3/4 }
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 ‖ 2001 | 1/2 1/2 0 }
-a+3/4, b+3/4, c+1/4 | -x, y, z	{ m ‖ m100 | 3/4 3/4 1/4 }
a+3/4, -b+3/4, c+1/4 | -x, y, z	{ m ‖ m010 | 3/4 3/4 1/4 }
a, b, c+1/2 | x, -y, -z	{ 2 ‖ 1 | 0 0 1/2 }
-a, -b, c+1/2 | x, -y, -z	{ 2 ‖ 2001 | 0 0 1/2 }
-a+1/4, b+1/4, c+3/4 | x, -y, -z	{ 2 ‖ m100 | 1/4 1/4 3/4 }
a+1/4, -b+1/4, c+3/4 | x, -y, -z	{ 2 ‖ m010 | 1/4 1/4 3/4 }
a, b+1/2, c | x, -y, -z	{ 2 ‖ 1 | 0 1/2 0 }
-a, -b+1/2, c | x, -y, -z	{ 2 ‖ 2001 | 0 1/2 0 }
-a+1/4, b+3/4, c+1/4 | x, -y, -z	{ 2 ‖ m100 | 1/4 3/4 1/4 }
a+1/4, -b+3/4, c+1/4 | x, -y, -z	{ 2 ‖ m010 | 1/4 3/4 1/4 }
a+1/2, b, c | x, -y, -z	{ 2 ‖ 1 | 1/2 0 0 }
-a+1/2, -b, c | x, -y, -z	{ 2 ‖ 2001 | 1/2 0 0 }
-a+3/4, b+1/4, c+1/4 | x, -y, -z	{ 2 ‖ m100 | 3/4 1/4 1/4 }
a+3/4, -b+1/4, c+1/4 | x, -y, -z	{ 2 ‖ m010 | 3/4 1/4 1/4 }
a+1/2, b+1/2, c+1/2 | x, -y, -z	{ 2 ‖ 1 | 1/2 1/2 1/2 }
-a+1/2, -b+1/2, c+1/2 | x, -y, -z	{ 2 ‖ 2001 | 1/2 1/2 1/2 }
-a+3/4, b+3/4, c+3/4 | x, -y, -z	{ 2 ‖ m100 | 3/4 3/4 3/4 }
a+3/4, -b+3/4, c+3/4 | x, -y, -z	{ 2 ‖ m010 | 3/4 3/4 3/4 }

Wyckoff Positions

WP	Site symmetry	Representative
16a	$..\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,0,c | 0,0,z)
16b	$..\ce{^{-1}{2}}\ce{^{\infty m}{1}} $	(0,1/4,c | 0,0,0)
32c	$\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}{2}}\ce{^{\infty /mm}{1}} $
T:(1,0,0)	$\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{1}{2}}\ce{^{\infty /mm}{1}} $
Y:(0,1,0)	$\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{1}{2}}\ce{^{\infty /mm}{1}} $
Z:(0,0,1)	$\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{1}{2}}\ce{^{\infty /mm}{1}} $
L:(1/2,1/2,1/2)	$\ce{^{1}{1}}\ce{^{\infty /mm}{1}} $
A:(u,0,1)	$\ce{^{m}{m}}\ce{^{1}{m}}\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
B:(0,v,1)	$\ce{^{1}{m}}\ce{^{m}{m}}\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
Δ:(0,v,0)	$\ce{^{1}{m}}\ce{^{m}{m}}\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
H:(0,1,w)	$\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{1}{2}}\ce{^{\infty 2}{1}} $
Λ:(0,0,w)	$\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{1}{2}}\ce{^{\infty 2}{1}} $
Σ:(u,0,0)	$\ce{^{m}{m}}\ce{^{1}{m}}\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
E:(0,v,w)	$\ce{^{1}{m}}\ce{^{\infty 2}{1}} $
J:(u,0,w)	$\ce{^{1}{m}}\ce{^{\infty 2}{1}} $
M:(u,v,0)	$\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
GP:(u,v,w)	$\ce{^{1}{1}}\ce{^{\infty 2}{1}} $

Spin Brillouin Zone

Band Representations (Magnon)

16a

k-vector	A↑G(4)
Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{2}^{S}(2) $
T:(1,0,0)	$T_{1}^{S}(4) $
Y:(0,1,0)	$Y_{1}^{S}(4) $
Z:(0,0,1)	$Z_{1}^{S}Z_{2}^{S}(4) $
L:(1/2,1/2,1/2)	$2L_{1}^{S}(2) $
A:(u,0,1)	$A_{1}^{S}A_{2}^{S}(4) $
B:(0,v,1)	$B_{1}^{S}B_{2}^{S}(4) $
Δ:(0,v,0)	$Δ_{1}^{S}(2)⊕Δ_{2}^{S}(2) $
H:(0,1,w)	$H_{1}^{S}(4) $
Λ:(0,0,w)	$Λ_{1}^{S}(2)⊕Λ_{2}^{S}(2) $
Σ:(u,0,0)	$Σ_{1}^{S}(2)⊕Σ_{2}^{S}(2) $
E:(0,v,w)	$E_{1}^{S}(2)⊕E_{2}^{S}(2) $
J:(u,0,w)	$J_{1}^{S}(2)⊕J_{2}^{S}(2) $
M:(u,v,0)	$2M_{1}^{S}(2) $
GP:(u,v,w)	$2GP_{1}^{S}(2) $

32c

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:(1,0,0)	$2T_{1}^{S}(4) $
Y:(0,1,0)	$2Y_{1}^{S}(4) $
Z:(0,0,1)	$Z_{1}^{S}Z_{2}^{S}(4)⊕Z_{3}^{S}Z_{4}^{S}(4) $
L:(1/2,1/2,1/2)	$4L_{1}^{S}(2) $
A:(u,0,1)	$2A_{1}^{S}A_{2}^{S}(4) $
B:(0,v,1)	$2B_{1}^{S}B_{2}^{S}(4) $
Δ:(0,v,0)	$2Δ_{1}^{S}(2)⊕2Δ_{2}^{S}(2) $
H:(0,1,w)	$2H_{1}^{S}(4) $
Λ:(0,0,w)	$Λ_{1}^{S}(2)⊕Λ_{2}^{S}(2)⊕Λ_{3}^{S}(2)⊕Λ_{4}^{S}(2) $
Σ:(u,0,0)	$2Σ_{1}^{S}(2)⊕2Σ_{2}^{S}(2) $
E:(0,v,w)	$2E_{1}^{S}(2)⊕2E_{2}^{S}(2) $
J:(u,0,w)	$2J_{1}^{S}(2)⊕2J_{2}^{S}(2) $
M:(u,v,0)	$4M_{1}^{S}(2) $
GP:(u,v,w)	$4GP_{1}^{S}(2) $