Information of spin space group $P\ce{^{1}{6_{4}}}\ce{^{\text{-}1}{2}}\ce{^{\text{-}1}{2}}\ce{^{\infty m}{1}}$ ( nSSG:172.181.1.1 )

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

Wyckoff Positions

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

Spin Brillouin Zone

Band Representations (Magnon)

6e

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

6f

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

12k

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