Information of spin space group $P\ce{^{1}{2/}}\ce{^{\text{-}1}{m}}\ce{^{\infty m}{1}}$ ( nSSG:3.10.1.1 )

General Positions

The continuously infinite group ^∞1 is omitted.

Coordinates	Seitz symbol
a, b, c | x, y, z	{ 1 ‖ 1 | 0 }
-a, b, -c | x, y, z	{ 1 ‖ 2010 | 0 }
-a, -b, -c | -x, -y, -z	{ -1 ‖ -1 | 0 }
a, -b, c | -x, -y, -z	{ -1 ‖ m010 | 0 }
a, b, c | -x, y, z	{ m ‖ 1 | 0 }
-a, b, -c | -x, y, z	{ m ‖ 2010 | 0 }
-a, -b, -c | x, -y, -z	{ 2 ‖ -1 | 0 }
a, -b, c | x, -y, -z	{ 2 ‖ m010 | 0 }

Wyckoff Positions

WP	Site symmetry	Representative
1a	$\ce{^{1}{2}}/\ce{^{-1}{m}}\ce{^{\infty m}{1}} $	(0,0,0 | 0,0,0)
1b	$\ce{^{1}{2}}/\ce{^{-1}{m}}\ce{^{\infty m}{1}} $	(0,1/2,0 | 0,0,0)
1c	$\ce{^{1}{2}}/\ce{^{-1}{m}}\ce{^{\infty m}{1}} $	(0,0,1/2 | 0,0,0)
1d	$\ce{^{1}{2}}/\ce{^{-1}{m}}\ce{^{\infty m}{1}} $	(1/2,0,0 | 0,0,0)
1e	$\ce{^{1}{2}}/\ce{^{-1}{m}}\ce{^{\infty m}{1}} $	(1/2,1/2,0 | 0,0,0)
1f	$\ce{^{1}{2}}/\ce{^{-1}{m}}\ce{^{\infty m}{1}} $	(0,1/2,1/2 | 0,0,0)
1g	$\ce{^{1}{2}}/\ce{^{-1}{m}}\ce{^{\infty m}{1}} $	(1/2,0,1/2 | 0,0,0)
1h	$\ce{^{1}{2}}/\ce{^{-1}{m}}\ce{^{\infty m}{1}} $	(1/2,1/2,1/2 | 0,0,0)
2i	$\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,b,0 | 0,0,z)
2j	$\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(1/2,b,0 | 0,0,z)
2k	$\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,b,1/2 | 0,0,z)
2l	$\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(1/2,b,1/2 | 0,0,z)
2m	$\ce{^{-1}{m}}\ce{^{\infty m}{1}} $	(a,0,c | 0,0,0)
2n	$\ce{^{-1}{m}}\ce{^{\infty m}{1}} $	(a,1/2,c | 0,0,0)
4o	$\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:(1/2,0,1/2)	$\ce{^{1}{2/}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
B:(0,0,1/2)	$\ce{^{1}{2/}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
C:(1/2,1/2,0)	$\ce{^{1}{2/}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
D:(0,1/2,1/2)	$\ce{^{1}{2/}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
E:(1/2,1/2,1/2)	$\ce{^{1}{2/}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
Γ:(0,0,0)	$\ce{^{1}{2/}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
Y:(1/2,0,0)	$\ce{^{1}{2/}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
Z:(0,1/2,0)	$\ce{^{1}{2/}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
F:(u,0,w)	$\ce{^{m}{2/}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
G:(u,1/2,w)	$\ce{^{m}{2/}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
Λ:(0,v,0)	$\ce{^{1}{2/}}\ce{^{-1}{m}}\ce{^{\infty}{1}} $
U:(1/2,v,1/2)	$\ce{^{1}{2/}}\ce{^{-1}{m}}\ce{^{\infty}{1}} $
V:(0,v,1/2)	$\ce{^{1}{2/}}\ce{^{-1}{m}}\ce{^{\infty}{1}} $
W:(1/2,v,0)	$\ce{^{1}{2/}}\ce{^{-1}{m}}\ce{^{\infty}{1}} $
GP:(u,v,w)	$\ce{^{-1}{-1}}\ce{^{\infty}{1}} $

Spin Brillouin Zone

Band Representations (Magnon)

2i

k-vector	k-vector-G↑	A↑G(2)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2) $
A:(1/2,0,1/2)	A:(1/2,0,1/2)	$A_{1}^{S}(2) $
B:(0,0,1/2)	B:(0,0,1/2)	$B_{1}^{S}(2) $
C:(1/2,1/2,0)	C:(1/2,1/2,0)	$C_{1}^{S}(2) $
D:(0,1/2,1/2)	D:(0,1/2,1/2)	$D_{1}^{S}(2) $
E:(1/2,1/2,1/2)	E:(1/2,1/2,1/2)	$E_{1}^{S}(2) $
Y:(1/2,0,0)	Y:(1/2,0,0)	$Y_{1}^{S}(2) $
Z:(0,1/2,0)	Z:(0,1/2,0)	$Z_{1}^{S}(2) $
Λ:(0,v,0)	Λ:(0,v,0)	$Λ_{1}^{S}(2) $
U:(1/2,v,1/2)	U:(1/2,v,1/2)	$U_{1}^{S}(2) $
V:(0,v,1/2)	V:(0,v,1/2)	$V_{1}^{S}(2) $
W:(1/2,v,0)	W:(1/2,v,0)	$W_{1}^{S}(2) $
F:(u,0,w)	F:(u,0,w)	$F_{1}^{S}(2) $
G:(u,1/2,w)	G:(u,1/2,w)	$G_{1}^{S}(2) $
GP:(u,v,w)	GP:(u,v,w)	$GP_{1}^{S}(2) $

2k

k-vector	k-vector-G↑	A↑G(2)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2) $
A:(1/2,0,1/2)	A:(1/2,0,1/2)	$A_{2}^{S}(2) $
B:(0,0,1/2)	B:(0,0,1/2)	$B_{2}^{S}(2) $
C:(1/2,1/2,0)	C:(1/2,1/2,0)	$C_{1}^{S}(2) $
D:(0,1/2,1/2)	D:(0,1/2,1/2)	$D_{2}^{S}(2) $
E:(1/2,1/2,1/2)	E:(1/2,1/2,1/2)	$E_{2}^{S}(2) $
Y:(1/2,0,0)	Y:(1/2,0,0)	$Y_{1}^{S}(2) $
Z:(0,1/2,0)	Z:(0,1/2,0)	$Z_{1}^{S}(2) $
Λ:(0,v,0)	Λ:(0,v,0)	$Λ_{1}^{S}(2) $
U:(1/2,v,1/2)	U:(1/2,v,1/2)	$U_{2}^{S}(2) $
V:(0,v,1/2)	V:(0,v,1/2)	$V_{2}^{S}(2) $
W:(1/2,v,0)	W:(1/2,v,0)	$W_{1}^{S}(2) $
F:(u,0,w)	F:(u,0,w)	$F_{1}^{S}(2) $
G:(u,1/2,w)	G:(u,1/2,w)	$G_{1}^{S}(2) $
GP:(u,v,w)	GP:(u,v,w)	$GP_{1}^{S}(2) $

2j

k-vector	k-vector-G↑	A↑G(2)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2) $
A:(1/2,0,1/2)	A:(1/2,0,1/2)	$A_{2}^{S}(2) $
B:(0,0,1/2)	B:(0,0,1/2)	$B_{1}^{S}(2) $
C:(1/2,1/2,0)	C:(1/2,1/2,0)	$C_{2}^{S}(2) $
D:(0,1/2,1/2)	D:(0,1/2,1/2)	$D_{1}^{S}(2) $
E:(1/2,1/2,1/2)	E:(1/2,1/2,1/2)	$E_{2}^{S}(2) $
Y:(1/2,0,0)	Y:(1/2,0,0)	$Y_{2}^{S}(2) $
Z:(0,1/2,0)	Z:(0,1/2,0)	$Z_{1}^{S}(2) $
Λ:(0,v,0)	Λ:(0,v,0)	$Λ_{1}^{S}(2) $
U:(1/2,v,1/2)	U:(1/2,v,1/2)	$U_{2}^{S}(2) $
V:(0,v,1/2)	V:(0,v,1/2)	$V_{1}^{S}(2) $
W:(1/2,v,0)	W:(1/2,v,0)	$W_{2}^{S}(2) $
F:(u,0,w)	F:(u,0,w)	$F_{1}^{S}(2) $
G:(u,1/2,w)	G:(u,1/2,w)	$G_{1}^{S}(2) $
GP:(u,v,w)	GP:(u,v,w)	$GP_{1}^{S}(2) $

2l

k-vector	k-vector-G↑	A↑G(2)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2) $
A:(1/2,0,1/2)	A:(1/2,0,1/2)	$A_{1}^{S}(2) $
B:(0,0,1/2)	B:(0,0,1/2)	$B_{2}^{S}(2) $
C:(1/2,1/2,0)	C:(1/2,1/2,0)	$C_{2}^{S}(2) $
D:(0,1/2,1/2)	D:(0,1/2,1/2)	$D_{2}^{S}(2) $
E:(1/2,1/2,1/2)	E:(1/2,1/2,1/2)	$E_{1}^{S}(2) $
Y:(1/2,0,0)	Y:(1/2,0,0)	$Y_{2}^{S}(2) $
Z:(0,1/2,0)	Z:(0,1/2,0)	$Z_{1}^{S}(2) $
Λ:(0,v,0)	Λ:(0,v,0)	$Λ_{1}^{S}(2) $
U:(1/2,v,1/2)	U:(1/2,v,1/2)	$U_{1}^{S}(2) $
V:(0,v,1/2)	V:(0,v,1/2)	$V_{2}^{S}(2) $
W:(1/2,v,0)	W:(1/2,v,0)	$W_{2}^{S}(2) $
F:(u,0,w)	F:(u,0,w)	$F_{1}^{S}(2) $
G:(u,1/2,w)	G:(u,1/2,w)	$G_{1}^{S}(2) $
GP:(u,v,w)	GP:(u,v,w)	$GP_{1}^{S}(2) $

4o

k-vector	k-vector-G↑	A↑G(4)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{2}^{S}(2) $
A:(1/2,0,1/2)	A:(1/2,0,1/2)	$A_{1}^{S}(2)⊕A_{2}^{S}(2) $
B:(0,0,1/2)	B:(0,0,1/2)	$B_{1}^{S}(2)⊕B_{2}^{S}(2) $
C:(1/2,1/2,0)	C:(1/2,1/2,0)	$C_{1}^{S}(2)⊕C_{2}^{S}(2) $
D:(0,1/2,1/2)	D:(0,1/2,1/2)	$D_{1}^{S}(2)⊕D_{2}^{S}(2) $
E:(1/2,1/2,1/2)	E:(1/2,1/2,1/2)	$E_{1}^{S}(2)⊕E_{2}^{S}(2) $
Y:(1/2,0,0)	Y:(1/2,0,0)	$Y_{1}^{S}(2)⊕Y_{2}^{S}(2) $
Z:(0,1/2,0)	Z:(0,1/2,0)	$Z_{1}^{S}(2)⊕Z_{2}^{S}(2) $
Λ:(0,v,0)	Λ:(0,v,0)	$Λ_{1}^{S}(2)⊕Λ_{2}^{S}(2) $
U:(1/2,v,1/2)	U:(1/2,v,1/2)	$U_{1}^{S}(2)⊕U_{2}^{S}(2) $
V:(0,v,1/2)	V:(0,v,1/2)	$V_{1}^{S}(2)⊕V_{2}^{S}(2) $
W:(1/2,v,0)	W:(1/2,v,0)	$W_{1}^{S}(2)⊕W_{2}^{S}(2) $
F:(u,0,w)	F:(u,0,w)	$2F_{1}^{S}(2) $
G:(u,1/2,w)	G:(u,1/2,w)	$2G_{1}^{S}(2) $
GP:(u,v,w)	GP:(u,v,w)	$2GP_{1}^{S}(2) $