Information of spin space group $P\ce{^{1}{\text{-}3}}\ce{^{\infty m}{1}}$ ( nSSG:147.147.1.1 )

General Positions

The continuously infinite group ^∞1 is omitted.

Coordinates	Seitz symbol
a, b, c | x, y, z	{ 1 ‖ 1 | 0 }
-b, a-b, c | x, y, z	{ 1 ‖ 3+001 | 0 }
-a+b, -a, c | x, y, z	{ 1 ‖ 3-001 | 0 }
-a, -b, -c | x, y, z	{ 1 ‖ -1 | 0 }
b, -a+b, -c | x, y, z	{ 1 ‖ -3+001 | 0 }
a-b, a, -c | x, y, z	{ 1 ‖ -3-001 | 0 }
a, b, c | -x, y, z	{ m ‖ 1 | 0 }
-b, a-b, c | -x, y, z	{ m ‖ 3+001 | 0 }
-a+b, -a, c | -x, y, z	{ m ‖ 3-001 | 0 }
-a, -b, -c | -x, y, z	{ m ‖ -1 | 0 }
b, -a+b, -c | -x, y, z	{ m ‖ -3+001 | 0 }
a-b, a, -c | -x, y, z	{ m ‖ -3-001 | 0 }

Wyckoff Positions

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

Spin Brillouin Zone

Band Representations (Magnon)

1a

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

1b

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

2c

k-vector	A1↑G(2)
A:(0,0,1/2)	$A_{1}^{S,+}(1)⊕A_{1}^{S,-}(1) $
Γ:(0,0,0)	$Γ_{1}^{S,+}(1)⊕Γ_{1}^{S,-}(1) $
H:(1/3,1/3,1/2)	$2H_{1}^{S}(1) $
K:(1/3,1/3,0)	$2K_{1}^{S}(1) $
L:(1/2,0,1/2)	$L_{1}^{S,+}(1)⊕L_{1}^{S,-}(1) $
M:(1/2,0,0)	$M_{1}^{S,+}(1)⊕M_{1}^{S,-}(1) $
Δ:(0,0,w)	$2Δ_{1}^{S}(1) $
P:(1/3,1/3,w)	$2P_{1}^{S}(1) $
B:(u,v,0)	$2B_{1}^{S}(1) $
C:(u,u,w)	$2C_{1}^{S}(1) $
D:(u,0,w)	$2D_{1}^{S}(1) $
E:(u,v,1/2)	$2E_{1}^{S}(1) $
Λ:(u,u,0)	$2Λ_{1}^{S}(1) $
Q:(u,u,1/2)	$2Q_{1}^{S}(1) $
R:(u,0,1/2)	$2R_{1}^{S}(1) $
Σ:(u,0,0)	$2Σ_{1}^{S}(1) $
U:(1/2,0,w)	$2U_{1}^{S}(1) $
GP:(u,v,w)	$2GP_{1}^{S}(1) $

2d

k-vector	A1↑G(2)
A:(0,0,1/2)	$A_{1}^{S,+}(1)⊕A_{1}^{S,-}(1) $
Γ:(0,0,0)	$Γ_{1}^{S,+}(1)⊕Γ_{1}^{S,-}(1) $
H:(1/3,1/3,1/2)	$H_{2}^{S}H_{3}^{S}(2) $
K:(1/3,1/3,0)	$K_{2}^{S}K_{3}^{S}(2) $
L:(1/2,0,1/2)	$L_{1}^{S,+}(1)⊕L_{1}^{S,-}(1) $
M:(1/2,0,0)	$M_{1}^{S,+}(1)⊕M_{1}^{S,-}(1) $
Δ:(0,0,w)	$2Δ_{1}^{S}(1) $
P:(1/3,1/3,w)	$P_{2}^{S}P_{3}^{S}(2) $
B:(u,v,0)	$2B_{1}^{S}(1) $
C:(u,u,w)	$2C_{1}^{S}(1) $
D:(u,0,w)	$2D_{1}^{S}(1) $
E:(u,v,1/2)	$2E_{1}^{S}(1) $
Λ:(u,u,0)	$2Λ_{1}^{S}(1) $
Q:(u,u,1/2)	$2Q_{1}^{S}(1) $
R:(u,0,1/2)	$2R_{1}^{S}(1) $
Σ:(u,0,0)	$2Σ_{1}^{S}(1) $
U:(1/2,0,w)	$2U_{1}^{S}(1) $
GP:(u,v,w)	$2GP_{1}^{S}(1) $

3e

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

3f

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

6g

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