Information of spin space group $P\ce{^{1}{\text{-}4}}\ce{^{\text{-}1}{2}}\ce{^{\text{-}1}{m}}\ce{^{\infty m}{1}}$ ( nSSG:81.111.1.1 )

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 ‖ 2001 | 0 }
b, -a, -c | x, y, z	{ 1 ‖ -4+001 | 0 }
-b, a, -c | x, y, z	{ 1 ‖ -4-001 | 0 }
a, -b, -c | -x, -y, -z	{ -1 ‖ 2100 | 0 }
-a, b, -c | -x, -y, -z	{ -1 ‖ 2010 | 0 }
-b, -a, c | -x, -y, -z	{ -1 ‖ m110 | 0 }
b, a, c | -x, -y, -z	{ -1 ‖ m1-10 | 0 }
a, b, c | -x, y, z	{ m ‖ 1 | 0 }
-a, -b, c | -x, y, z	{ m ‖ 2001 | 0 }
b, -a, -c | -x, y, z	{ m ‖ -4+001 | 0 }
-b, a, -c | -x, y, z	{ m ‖ -4-001 | 0 }
a, -b, -c | x, -y, -z	{ 2 ‖ 2100 | 0 }
-a, b, -c | x, -y, -z	{ 2 ‖ 2010 | 0 }
-b, -a, c | x, -y, -z	{ 2 ‖ m110 | 0 }
b, a, c | x, -y, -z	{ 2 ‖ m1-10 | 0 }

Wyckoff Positions

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

Spin Brillouin Zone

4/mmmP

Band Representations (Magnon)

4m

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

8o

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