Information of spin space group $P\ce{^{1}{6_{5}}}\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{\text{-}1}{(0 \ 0 \ 1/2)}}\ce{^{\infty m}{1}}$ ( nSSG:179.181.2.1 )

General Positions

The continuously infinite group ^∞1 is omitted.

Coordinates	Seitz symbol
a, b, c | x, y, z	{ 1 ‖ 1 | 0 }
a-b, a, c+5/6 | x, y, z	{ 1 ‖ 6+001 | 0 0 5/6 }
-b, a-b, c+2/3 | x, y, z	{ 1 ‖ 3+001 | 0 0 2/3 }
-a, -b, c+1/2 | x, y, z	{ 1 ‖ 2001 | 0 0 1/2 }
-a+b, -a, c+1/3 | x, y, z	{ 1 ‖ 3-001 | 0 0 1/3 }
b, -a+b, c+1/6 | x, y, z	{ 1 ‖ 6-001 | 0 0 1/6 }
a-b, -b, -c | x, y, z	{ 1 ‖ 2100 | 0 }
b, a, -c+2/3 | x, y, z	{ 1 ‖ 2110 | 0 0 2/3 }
-a, -a+b, -c+1/3 | x, y, z	{ 1 ‖ 2010 | 0 0 1/3 }
a, a-b, -c+5/6 | x, y, z	{ 1 ‖ 2210 | 0 0 5/6 }
-a+b, b, -c+1/2 | x, y, z	{ 1 ‖ 2120 | 0 0 1/2 }
-b, -a, -c+1/6 | x, y, z	{ 1 ‖ 21-10 | 0 0 1/6 }
a, b, c+1/2 | -x, -y, -z	{ -1 ‖ 1 | 0 0 1/2 }
a-b, a, c+1/3 | -x, -y, -z	{ -1 ‖ 6+001 | 0 0 1/3 }
-b, a-b, c+1/6 | -x, -y, -z	{ -1 ‖ 3+001 | 0 0 1/6 }
-a, -b, c | -x, -y, -z	{ -1 ‖ 2001 | 0 }
-a+b, -a, c+5/6 | -x, -y, -z	{ -1 ‖ 3-001 | 0 0 5/6 }
b, -a+b, c+2/3 | -x, -y, -z	{ -1 ‖ 6-001 | 0 0 2/3 }
a-b, -b, -c+1/2 | -x, -y, -z	{ -1 ‖ 2100 | 0 0 1/2 }
b, a, -c+1/6 | -x, -y, -z	{ -1 ‖ 2110 | 0 0 1/6 }
-a, -a+b, -c+5/6 | -x, -y, -z	{ -1 ‖ 2010 | 0 0 5/6 }
a, a-b, -c+1/3 | -x, -y, -z	{ -1 ‖ 2210 | 0 0 1/3 }
-a+b, b, -c | -x, -y, -z	{ -1 ‖ 2120 | 0 }
-b, -a, -c+2/3 | -x, -y, -z	{ -1 ‖ 21-10 | 0 0 2/3 }
a, b, c | -x, y, z	{ m ‖ 1 | 0 }
a-b, a, c+5/6 | -x, y, z	{ m ‖ 6+001 | 0 0 5/6 }
-b, a-b, c+2/3 | -x, y, z	{ m ‖ 3+001 | 0 0 2/3 }
-a, -b, c+1/2 | -x, y, z	{ m ‖ 2001 | 0 0 1/2 }
-a+b, -a, c+1/3 | -x, y, z	{ m ‖ 3-001 | 0 0 1/3 }
b, -a+b, c+1/6 | -x, y, z	{ m ‖ 6-001 | 0 0 1/6 }
a-b, -b, -c | -x, y, z	{ m ‖ 2100 | 0 }
b, a, -c+2/3 | -x, y, z	{ m ‖ 2110 | 0 0 2/3 }
-a, -a+b, -c+1/3 | -x, y, z	{ m ‖ 2010 | 0 0 1/3 }
a, a-b, -c+5/6 | -x, y, z	{ m ‖ 2210 | 0 0 5/6 }
-a+b, b, -c+1/2 | -x, y, z	{ m ‖ 2120 | 0 0 1/2 }
-b, -a, -c+1/6 | -x, y, z	{ m ‖ 21-10 | 0 0 1/6 }
a, b, c+1/2 | x, -y, -z	{ 2 ‖ 1 | 0 0 1/2 }
a-b, a, c+1/3 | x, -y, -z	{ 2 ‖ 6+001 | 0 0 1/3 }
-b, a-b, c+1/6 | x, -y, -z	{ 2 ‖ 3+001 | 0 0 1/6 }
-a, -b, c | x, -y, -z	{ 2 ‖ 2001 | 0 }
-a+b, -a, c+5/6 | x, -y, -z	{ 2 ‖ 3-001 | 0 0 5/6 }
b, -a+b, c+2/3 | x, -y, -z	{ 2 ‖ 6-001 | 0 0 2/3 }
a-b, -b, -c+1/2 | x, -y, -z	{ 2 ‖ 2100 | 0 0 1/2 }
b, a, -c+1/6 | x, -y, -z	{ 2 ‖ 2110 | 0 0 1/6 }
-a, -a+b, -c+5/6 | x, -y, -z	{ 2 ‖ 2010 | 0 0 5/6 }
a, a-b, -c+1/3 | x, -y, -z	{ 2 ‖ 2210 | 0 0 1/3 }
-a+b, b, -c | x, -y, -z	{ 2 ‖ 2120 | 0 }
-b, -a, -c+2/3 | x, -y, -z	{ 2 ‖ 21-10 | 0 0 2/3 }

Wyckoff Positions

WP	Site symmetry	Representative
6a	$\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,0,3/4 | 0,0,0)
6b	$\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{-1}{2}}\ce{^{\infty m}{1}} $	(0,0,0 | 0,0,0)
6c	$\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(1/2,0,3/4 | 0,0,0)
6d	$\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{-1}{2}}\ce{^{\infty m}{1}} $	(1/2,0,0 | 0,0,0)
12e	$\ce{^{-1}{2}}..\ce{^{\infty m}{1}} $	(0,0,c | 0,0,0)
12f	$\ce{^{-1}{2}}..\ce{^{\infty m}{1}} $	(1/2,0,c | 0,0,0)
12g	$.\ce{^{-1}{2}}.\ce{^{\infty m}{1}} $	(a,0,3/4 | 0,0,0)
12h	$.\ce{^{1}{2}}.\ce{^{\infty m}{1}} $	(a,0,0 | 0,0,z)
12i	$..\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(a,2a,3/4 | 0,0,z)
12j	$..\ce{^{-1}{2}}\ce{^{\infty m}{1}} $	(a,2a,0 | 0,0,0)
24k	$\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}{6}}\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{\infty /mm}{1}} $
Γ:(0,0,0)	$\ce{^{1}{6}}\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{\infty /mm}{1}} $
H:(1/3,1/3,1/2)	$\ce{^{m}{6}}\ce{^{1}{2}}\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
K:(1/3,1/3,0)	$\ce{^{m}{6}}\ce{^{1}{2}}\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
L:(1/2,0,1/2)	$\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{\infty /mm}{1}} $
M:(1/2,0,0)	$\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{\infty /mm}{1}} $
Δ:(0,0,w)	$\ce{^{1}{6}}\ce{^{m}{2}}\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
P:(1/3,1/3,w)	$\ce{^{1}{3}}\ce{^{1}{1}}\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
Λ:(u,u,0)	$\ce{^{m}{2}}\ce{^{1}{2}}\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
Σ:(u,0,0)	$\ce{^{1}{2}}\ce{^{m}{2}}\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
Q:(u,u,1/2)	$\ce{^{m}{2}}\ce{^{1}{2}}\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
R:(u,0,1/2)	$\ce{^{1}{2}}\ce{^{m}{2}}\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
U:(1/2,0,w)	$\ce{^{m}{2}}\ce{^{m}{2}}\ce{^{1}{2}}\ce{^{\infty 2}{1}} $
B:(u,v,0)	$\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
C:(u,u,w)	$\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
D:(u,0,w)	$\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
E:(u,v,1/2)	$\ce{^{m}{2}}\ce{^{\infty 2}{1}} $
GP:(u,v,w)	$\ce{^{1}{1}}\ce{^{\infty 2}{1}}$

Spin Brillouin Zone

Band Representations (Magnon)

12h

k-vector	A↑G(12)
A:(0,0,1/2)	$A_{1}^{S}(4)⊕A_{2}^{S}(4)⊕A_{3}^{S}(4) $
Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{3}^{S}(2)⊕Γ_{5}^{S}(4)⊕Γ_{6}^{S}(4) $
H:(1/3,1/3,1/2)	$H_{1}^{S}H_{2}^{S}(4)⊕2H_{3}^{S}(4) $
K:(1/3,1/3,0)	$2K_{1}^{S}(2)⊕2K_{3}^{S}(4) $
L:(1/2,0,1/2)	$3L_{1}^{S}(4) $
M:(1/2,0,0)	$2M_{1}^{S}(2)⊕M_{2}^{S}(2)⊕2M_{3}^{S}(2)⊕M_{4}^{S}(2) $
Δ:(0,0,w)	$Δ_{1}^{S}(2)⊕Δ_{2}^{S}(2)⊕Δ_{3}^{S}(2)⊕Δ_{4}^{S}(2)⊕Δ_{5}^{S}(2)⊕Δ_{6}^{S}(2) $
P:(1/3,1/3,w)	$2P_{1}^{S}(2)⊕2P_{2}^{S}(2)⊕2P_{3}^{S}(2) $
Λ:(u,u,0)	$4Λ_{1}^{S}(2)⊕2Λ_{2}^{S}(2) $
Σ:(u,0,0)	$3Σ_{1}^{S}(2)⊕3Σ_{2}^{S}(2) $
Q:(u,u,1/2)	$3Q_{1}^{S}Q_{2}^{S}(4) $
R:(u,0,1/2)	$3R_{1}^{S}R_{2}^{S}(4) $
U:(1/2,0,w)	$3U_{1}^{S}(2)⊕3U_{2}^{S}(2) $
B:(u,v,0)	$6B_{1}^{S}(2) $
C:(u,u,w)	$6C_{1}^{S}(2) $
D:(u,0,w)	$6D_{1}^{S}(2) $
E:(u,v,1/2)	$3E_{1}^{S}E_{1}^{S}(4) $
GP:(u,v,w)	$6GP_{1}^{S}(2) $

12i

k-vector	A↑G(12)
A:(0,0,1/2)	$A_{1}^{S}(4)⊕A_{2}^{S}(4)⊕A_{3}^{S}(4) $
Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{4}^{S}(2)⊕Γ_{5}^{S}(4)⊕Γ_{6}^{S}(4) $
H:(1/3,1/3,1/2)	$H_{1}^{S}H_{2}^{S}(4)⊕2H_{3}^{S}(4) $
K:(1/3,1/3,0)	$K_{1}^{S}(2)⊕K_{2}^{S}(2)⊕2K_{3}^{S}(4) $
L:(1/2,0,1/2)	$3L_{1}^{S}(4) $
M:(1/2,0,0)	$2M_{1}^{S}(2)⊕M_{2}^{S}(2)⊕M_{3}^{S}(2)⊕2M_{4}^{S}(2) $
Δ:(0,0,w)	$Δ_{1}^{S}(2)⊕Δ_{2}^{S}(2)⊕Δ_{3}^{S}(2)⊕Δ_{4}^{S}(2)⊕Δ_{5}^{S}(2)⊕Δ_{6}^{S}(2) $
P:(1/3,1/3,w)	$2P_{1}^{S}(2)⊕2P_{2}^{S}(2)⊕2P_{3}^{S}(2) $
Λ:(u,u,0)	$3Λ_{1}^{S}(2)⊕3Λ_{2}^{S}(2) $
Σ:(u,0,0)	$4Σ_{1}^{S}(2)⊕2Σ_{2}^{S}(2) $
Q:(u,u,1/2)	$3Q_{1}^{S}Q_{2}^{S}(4) $
R:(u,0,1/2)	$3R_{1}^{S}R_{2}^{S}(4) $
U:(1/2,0,w)	$3U_{1}^{S}(2)⊕3U_{2}^{S}(2) $
B:(u,v,0)	$6B_{1}^{S}(2) $
C:(u,u,w)	$6C_{1}^{S}(2) $
D:(u,0,w)	$6D_{1}^{S}(2) $
E:(u,v,1/2)	$3E_{1}^{S}E_{1}^{S}(4) $
GP:(u,v,w)	$6GP_{1}^{S}(2) $

24k

k-vector	A↑G(24)
A:(0,0,1/2)	$2A_{1}^{S}(4)⊕2A_{2}^{S}(4)⊕2A_{3}^{S}(4) $
Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{2}^{S}(2)⊕Γ_{3}^{S}(2)⊕Γ_{4}^{S}(2)⊕2Γ_{5}^{S}(4)⊕2Γ_{6}^{S}(4) $
H:(1/3,1/3,1/2)	$2H_{1}^{S}H_{2}^{S}(4)⊕4H_{3}^{S}(4) $
K:(1/3,1/3,0)	$2K_{1}^{S}(2)⊕2K_{2}^{S}(2)⊕4K_{3}^{S}(4) $
L:(1/2,0,1/2)	$6L_{1}^{S}(4) $
M:(1/2,0,0)	$3M_{1}^{S}(2)⊕3M_{2}^{S}(2)⊕3M_{3}^{S}(2)⊕3M_{4}^{S}(2) $
Δ:(0,0,w)	$2Δ_{1}^{S}(2)⊕2Δ_{2}^{S}(2)⊕2Δ_{3}^{S}(2)⊕2Δ_{4}^{S}(2)⊕2Δ_{5}^{S}(2)⊕2Δ_{6}^{S}(2) $
P:(1/3,1/3,w)	$4P_{1}^{S}(2)⊕4P_{2}^{S}(2)⊕4P_{3}^{S}(2) $
Λ:(u,u,0)	$6Λ_{1}^{S}(2)⊕6Λ_{2}^{S}(2) $
Σ:(u,0,0)	$6Σ_{1}^{S}(2)⊕6Σ_{2}^{S}(2) $
Q:(u,u,1/2)	$6Q_{1}^{S}Q_{2}^{S}(4) $
R:(u,0,1/2)	$6R_{1}^{S}R_{2}^{S}(4) $
U:(1/2,0,w)	$6U_{1}^{S}(2)⊕6U_{2}^{S}(2) $
B:(u,v,0)	$12B_{1}^{S}(2) $
C:(u,u,w)	$12C_{1}^{S}(2) $
D:(u,0,w)	$12D_{1}^{S}(2) $
E:(u,v,1/2)	$6E_{1}^{S}E_{1}^{S}(4) $
GP:(u,v,w)	$12GP_{1}^{S}(2) $