Information of spin space group $P_{I}\ce{^{1}{4_{1}}}\ce{^{1}{2_{1}}}\ce{^{1}{2}}\ce{^{\infty m}{1}}$ ( nSSG:92.98.2.1 )

General Positions

The continuously infinite group ^∞1 is omitted.

Coordinates	Seitz symbol
a, b, c | x, y, z	{ 1 ‖ 1 | 0 }
-b+1/2, a+1/2, c+1/4 | x, y, z	{ 1 ‖ 4+001 | 1/2 1/2 1/4 }
b+1/2, -a+1/2, c+3/4 | x, y, z	{ 1 ‖ 4-001 | 1/2 1/2 3/4 }
a+1/2, -b+1/2, -c+3/4 | x, y, z	{ 1 ‖ 2100 | 1/2 1/2 3/4 }
-a+1/2, b+1/2, -c+1/4 | x, y, z	{ 1 ‖ 2010 | 1/2 1/2 1/4 }
-a, -b, c+1/2 | x, y, z	{ 1 ‖ 2001 | 0 0 1/2 }
b, a, -c | x, y, z	{ 1 ‖ 2110 | 0 }
-b, -a, -c+1/2 | x, y, z	{ 1 ‖ 21-10 | 0 0 1/2 }
a+1/2, b+1/2, c+1/2 | -x, -y, -z	{ -1 ‖ 1 | 1/2 1/2 1/2 }
-b, a, c+3/4 | -x, -y, -z	{ -1 ‖ 4+001 | 0 0 3/4 }
b, -a, c+1/4 | -x, -y, -z	{ -1 ‖ 4-001 | 0 0 1/4 }
a, -b, -c+1/4 | -x, -y, -z	{ -1 ‖ 2100 | 0 0 1/4 }
-a, b, -c+3/4 | -x, -y, -z	{ -1 ‖ 2010 | 0 0 3/4 }
-a+1/2, -b+1/2, c | -x, -y, -z	{ -1 ‖ 2001 | 1/2 1/2 0 }
b+1/2, a+1/2, -c+1/2 | -x, -y, -z	{ -1 ‖ 2110 | 1/2 1/2 1/2 }
-b+1/2, -a+1/2, -c | -x, -y, -z	{ -1 ‖ 21-10 | 1/2 1/2 0 }
a, b, c | -x, y, z	{ m ‖ 1 | 0 }
-b+1/2, a+1/2, c+1/4 | -x, y, z	{ m ‖ 4+001 | 1/2 1/2 1/4 }
b+1/2, -a+1/2, c+3/4 | -x, y, z	{ m ‖ 4-001 | 1/2 1/2 3/4 }
a+1/2, -b+1/2, -c+3/4 | -x, y, z	{ m ‖ 2100 | 1/2 1/2 3/4 }
-a+1/2, b+1/2, -c+1/4 | -x, y, z	{ m ‖ 2010 | 1/2 1/2 1/4 }
-a, -b, c+1/2 | -x, y, z	{ m ‖ 2001 | 0 0 1/2 }
b, a, -c | -x, y, z	{ m ‖ 2110 | 0 }
-b, -a, -c+1/2 | -x, y, z	{ m ‖ 21-10 | 0 0 1/2 }
a+1/2, b+1/2, c+1/2 | x, -y, -z	{ 2 ‖ 1 | 1/2 1/2 1/2 }
-b, a, c+3/4 | x, -y, -z	{ 2 ‖ 4+001 | 0 0 3/4 }
b, -a, c+1/4 | x, -y, -z	{ 2 ‖ 4-001 | 0 0 1/4 }
a, -b, -c+1/4 | x, -y, -z	{ 2 ‖ 2100 | 0 0 1/4 }
-a, b, -c+3/4 | x, -y, -z	{ 2 ‖ 2010 | 0 0 3/4 }
-a+1/2, -b+1/2, c | x, -y, -z	{ 2 ‖ 2001 | 1/2 1/2 0 }
b+1/2, a+1/2, -c+1/2 | x, -y, -z	{ 2 ‖ 2110 | 1/2 1/2 1/2 }
-b+1/2, -a+1/2, -c | x, -y, -z	{ 2 ‖ 21-10 | 1/2 1/2 0 }

Wyckoff Positions

WP	Site symmetry	Representative
4a	$\ce{^{-1}{2}}.\ce{^{1}{2}}\ce{^{-1}{2}}\ce{^{\infty m}{1}} $	(3/4,3/4,0 | 0,0,0)
4b	$\ce{^{-1}{2}}.\ce{^{1}{2}}\ce{^{-1}{2}}\ce{^{\infty m}{1}} $	(3/4,3/4,1/2 | 0,0,0)
8c	$\ce{^{-1}{2}}..\ce{^{\infty m}{1}} $	(3/4,3/4,c | 0,0,0)
8d	$..\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(a,a,0 | 0,0,z)
8e	$..\ce{^{-1}{2}}\ce{^{\infty m}{1}} $	(a,-a+1/2,0 | 0,0,0)
8f	$.\ce{^{-1}{2}}.\ce{^{\infty m}{1}} $	(a,0,1/8 | 0,0,0)
16g	$\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:(1/2,1/2,1/2)	$\ce{^{1}{4}}\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{\infty /mm}{1}} $
Γ:(0,0,0)	$\ce{^{1}{4}}\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{\infty /mm}{1}} $
M:(1/2,1/2,0)	$\ce{^{1}{4}}\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{\infty /mm}{1}} $
Z:(0,0,1/2)	$\ce{^{1}{4}}\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{\infty /mm}{1}} $
R:(0,1/2,1/2)	$\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{\infty /mm}{1}} $
X:(0,1/2,0)	$\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{\infty /mm}{1}} $
Λ:(0,0,w)	$\ce{^{1}{4}}\ce{^{m}{2}}\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
V:(1/2,1/2,w)	$\ce{^{1}{4}}\ce{^{m}{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}} $
S:(u,u,1/2)	$\ce{^{m}{2}}\ce{^{1}{2}}\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
Σ:(u,u,0)	$\ce{^{m}{2}}\ce{^{1}{2}}\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
T:(u,1/2,1/2)	$\ce{^{1}{2}}\ce{^{m}{2}}\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
U:(0,v,1/2)	$\ce{^{m}{2}}\ce{^{1}{2}}\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
W:(0,1/2,w)	$\ce{^{m}{2}}\ce{^{m}{2}}\ce{^{1}{2}}\ce{^{\infty 2}{1}} $
Y:(u,1/2,0)	$\ce{^{1}{2}}\ce{^{m}{2}}\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
B:(0,v,w)	$\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
C:(u,u,w)	$\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
D:(u,v,0)	$\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
E:(u,v,1/2)	$\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
F:(u,1/2,w)	$\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
GP:(u,v,w)	$\ce{^{1}{1}}\ce{^{\infty 2}{1}} $

Spin Brillouin Zone

Band Representations (Magnon)

8d

k-vector	A↑G(8)
A:(1/2,1/2,1/2)	$A_{1}^{S}A_{2}^{S}(8) $
Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{4}^{S}(2)⊕Γ_{5}^{S}(4) $
M:(1/2,1/2,0)	$M_{1}^{S}M_{4}^{S}(4)⊕M_{5}^{S}(4) $
Z:(0,0,1/2)	$Z_{1}^{S}(4)⊕Z_{2}^{S}(4) $
R:(0,1/2,1/2)	$R_{1}^{S}R_{3}^{S}(4)⊕R_{2}^{S}R_{4}^{S}(4) $
X:(0,1/2,0)	$2X_{1}^{S}(4) $
Λ:(0,0,w)	$Λ_{1}^{S}(2)⊕Λ_{2}^{S}(2)⊕Λ_{3}^{S}(2)⊕Λ_{4}^{S}(2) $
V:(1/2,1/2,w)	$V_{1}^{S}V_{2}^{S}(4)⊕V_{3}^{S}V_{4}^{S}(4) $
Δ:(0,v,0)	$2Δ_{1}^{S}(2)⊕2Δ_{2}^{S}(2) $
S:(u,u,1/2)	$2S_{1}^{S}S_{2}^{S}(4) $
Σ:(u,u,0)	$3Σ_{1}^{S}(2)⊕Σ_{2}^{S}(2) $
T:(u,1/2,1/2)	$T_{1}^{S}T_{1}^{S}(4)⊕T_{2}^{S}T_{2}^{S}(4) $
U:(0,v,1/2)	$2U_{1}^{S}U_{2}^{S}(4) $
W:(0,1/2,w)	$2W_{1}^{S}W_{2}^{S}(4) $
Y:(u,1/2,0)	$2Y_{1}^{S}Y_{2}^{S}(4) $
B:(0,v,w)	$4B_{1}^{S}(2) $
C:(u,u,w)	$4C_{1}^{S}(2) $
D:(u,v,0)	$4D_{1}^{S}(2) $
E:(u,v,1/2)	$2E_{1}^{S}E_{1}^{S}(4) $
F:(u,1/2,w)	$2F_{1}^{S}F_{1}^{S}(4) $
GP:(u,v,w)	$4GP_{1}^{S}(2) $

16g

k-vector	A↑G(16)
A:(1/2,1/2,1/2)	$2A_{1}^{S}A_{2}^{S}(8) $
Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{2}^{S}(2)⊕Γ_{3}^{S}(2)⊕Γ_{4}^{S}(2)⊕2Γ_{5}^{S}(4) $
M:(1/2,1/2,0)	$M_{1}^{S}M_{4}^{S}(4)⊕M_{2}^{S}M_{3}^{S}(4)⊕2M_{5}^{S}(4) $
Z:(0,0,1/2)	$2Z_{1}^{S}(4)⊕2Z_{2}^{S}(4) $
R:(0,1/2,1/2)	$2R_{1}^{S}R_{3}^{S}(4)⊕2R_{2}^{S}R_{4}^{S}(4) $
X:(0,1/2,0)	$4X_{1}^{S}(4) $
Λ:(0,0,w)	$2Λ_{1}^{S}(2)⊕2Λ_{2}^{S}(2)⊕2Λ_{3}^{S}(2)⊕2Λ_{4}^{S}(2) $
V:(1/2,1/2,w)	$2V_{1}^{S}V_{2}^{S}(4)⊕2V_{3}^{S}V_{4}^{S}(4) $
Δ:(0,v,0)	$4Δ_{1}^{S}(2)⊕4Δ_{2}^{S}(2) $
S:(u,u,1/2)	$4S_{1}^{S}S_{2}^{S}(4) $
Σ:(u,u,0)	$4Σ_{1}^{S}(2)⊕4Σ_{2}^{S}(2) $
T:(u,1/2,1/2)	$2T_{1}^{S}T_{1}^{S}(4)⊕2T_{2}^{S}T_{2}^{S}(4) $
U:(0,v,1/2)	$4U_{1}^{S}U_{2}^{S}(4) $
W:(0,1/2,w)	$4W_{1}^{S}W_{2}^{S}(4) $
Y:(u,1/2,0)	$4Y_{1}^{S}Y_{2}^{S}(4) $
B:(0,v,w)	$8B_{1}^{S}(2) $
C:(u,u,w)	$8C_{1}^{S}(2) $
D:(u,v,0)	$8D_{1}^{S}(2) $
E:(u,v,1/2)	$4E_{1}^{S}E_{1}^{S}(4) $
F:(u,1/2,w)	$4F_{1}^{S}F_{1}^{S}(4) $
GP:(u,v,w)	$8GP_{1}^{S}(2) $