Information of spin space group $P\ce{^{\text{-}1}{4_{1}}}\ce{^{1}{3}}\ce{^{\text{-}1}{2}}\ce{^{\infty m}{1}}$ ( nSSG:198.213.1.1 )

General Positions

The continuously infinite group ^∞1 is omitted.

Coordinates	Seitz symbol
a, b, c | x, y, z	{ 1 ‖ 1 | 0 }
a+1/2, -b+1/2, -c | x, y, z	{ 1 ‖ 2100 | 1/2 1/2 0 }
-a, b+1/2, -c+1/2 | x, y, z	{ 1 ‖ 2010 | 0 1/2 1/2 }
-a+1/2, -b, c+1/2 | x, y, z	{ 1 ‖ 2001 | 1/2 0 1/2 }
c, a, b | x, y, z	{ 1 ‖ 3+111 | 0 }
b, c, a | x, y, z	{ 1 ‖ 3-111 | 0 }
-b, c+1/2, -a+1/2 | x, y, z	{ 1 ‖ 3-1-1-1 | 0 1/2 1/2 }
-c+1/2, -a, b+1/2 | x, y, z	{ 1 ‖ 3+1-1-1 | 1/2 0 1/2 }
-b+1/2, -c, a+1/2 | x, y, z	{ 1 ‖ 3--11-1 | 1/2 0 1/2 }
c+1/2, -a+1/2, -b | x, y, z	{ 1 ‖ 3+-11-1 | 1/2 1/2 0 }
b+1/2, -c+1/2, -a | x, y, z	{ 1 ‖ 3--1-11 | 1/2 1/2 0 }
-c, a+1/2, -b+1/2 | x, y, z	{ 1 ‖ 3+-1-11 | 0 1/2 1/2 }
a+1/4, -c+1/4, b+3/4 | -x, -y, -z	{ -1 ‖ 4+100 | 1/4 1/4 3/4 }
a+3/4, c+1/4, -b+1/4 | -x, -y, -z	{ -1 ‖ 4-100 | 3/4 1/4 1/4 }
c+3/4, b+1/4, -a+1/4 | -x, -y, -z	{ -1 ‖ 4+010 | 3/4 1/4 1/4 }
-c+1/4, b+3/4, a+1/4 | -x, -y, -z	{ -1 ‖ 4-010 | 1/4 3/4 1/4 }
-b+1/4, a+3/4, c+1/4 | -x, -y, -z	{ -1 ‖ 4+001 | 1/4 3/4 1/4 }
b+1/4, -a+1/4, c+3/4 | -x, -y, -z	{ -1 ‖ 4-001 | 1/4 1/4 3/4 }
b+3/4, a+1/4, -c+1/4 | -x, -y, -z	{ -1 ‖ 2110 | 3/4 1/4 1/4 }
-b+3/4, -a+3/4, -c+3/4 | -x, -y, -z	{ -1 ‖ 21-10 | 3/4 3/4 3/4 }
-a+1/4, c+3/4, b+1/4 | -x, -y, -z	{ -1 ‖ 2011 | 1/4 3/4 1/4 }
-a+3/4, -c+3/4, -b+3/4 | -x, -y, -z	{ -1 ‖ 201-1 | 3/4 3/4 3/4 }
c+1/4, -b+1/4, a+3/4 | -x, -y, -z	{ -1 ‖ 2101 | 1/4 1/4 3/4 }
-c+3/4, -b+3/4, -a+3/4 | -x, -y, -z	{ -1 ‖ 2-101 | 3/4 3/4 3/4 }
a, b, c | -x, y, z	{ m ‖ 1 | 0 }
a+1/2, -b+1/2, -c | -x, y, z	{ m ‖ 2100 | 1/2 1/2 0 }
-a, b+1/2, -c+1/2 | -x, y, z	{ m ‖ 2010 | 0 1/2 1/2 }
-a+1/2, -b, c+1/2 | -x, y, z	{ m ‖ 2001 | 1/2 0 1/2 }
c, a, b | -x, y, z	{ m ‖ 3+111 | 0 }
b, c, a | -x, y, z	{ m ‖ 3-111 | 0 }
-b, c+1/2, -a+1/2 | -x, y, z	{ m ‖ 3-1-1-1 | 0 1/2 1/2 }
-c+1/2, -a, b+1/2 | -x, y, z	{ m ‖ 3+1-1-1 | 1/2 0 1/2 }
-b+1/2, -c, a+1/2 | -x, y, z	{ m ‖ 3--11-1 | 1/2 0 1/2 }
c+1/2, -a+1/2, -b | -x, y, z	{ m ‖ 3+-11-1 | 1/2 1/2 0 }
b+1/2, -c+1/2, -a | -x, y, z	{ m ‖ 3--1-11 | 1/2 1/2 0 }
-c, a+1/2, -b+1/2 | -x, y, z	{ m ‖ 3+-1-11 | 0 1/2 1/2 }
a+1/4, -c+1/4, b+3/4 | x, -y, -z	{ 2 ‖ 4+100 | 1/4 1/4 3/4 }
a+3/4, c+1/4, -b+1/4 | x, -y, -z	{ 2 ‖ 4-100 | 3/4 1/4 1/4 }
c+3/4, b+1/4, -a+1/4 | x, -y, -z	{ 2 ‖ 4+010 | 3/4 1/4 1/4 }
-c+1/4, b+3/4, a+1/4 | x, -y, -z	{ 2 ‖ 4-010 | 1/4 3/4 1/4 }
-b+1/4, a+3/4, c+1/4 | x, -y, -z	{ 2 ‖ 4+001 | 1/4 3/4 1/4 }
b+1/4, -a+1/4, c+3/4 | x, -y, -z	{ 2 ‖ 4-001 | 1/4 1/4 3/4 }
b+3/4, a+1/4, -c+1/4 | x, -y, -z	{ 2 ‖ 2110 | 3/4 1/4 1/4 }
-b+3/4, -a+3/4, -c+3/4 | x, -y, -z	{ 2 ‖ 21-10 | 3/4 3/4 3/4 }
-a+1/4, c+3/4, b+1/4 | x, -y, -z	{ 2 ‖ 2011 | 1/4 3/4 1/4 }
-a+3/4, -c+3/4, -b+3/4 | x, -y, -z	{ 2 ‖ 201-1 | 3/4 3/4 3/4 }
c+1/4, -b+1/4, a+3/4 | x, -y, -z	{ 2 ‖ 2101 | 1/4 1/4 3/4 }
-c+3/4, -b+3/4, -a+3/4 | x, -y, -z	{ 2 ‖ 2-101 | 3/4 3/4 3/4 }

Wyckoff Positions

WP	Site symmetry	Representative
4a	$.\ce{^{1}{3}}\ce{^{-1}{2}}\ce{^{\infty m}{1}} $	(3/8,3/8,3/8 | 0,0,0)
4b	$.\ce{^{1}{3}}\ce{^{-1}{2}}\ce{^{\infty m}{1}} $	(7/8,7/8,7/8 | 0,0,0)
8c	$.\ce{^{1}{3}}.\ce{^{\infty m}{1}} $	(a,a,a | 0,0,z)
12d	$..\ce{^{-1}{2}}\ce{^{\infty m}{1}} $	(1/8,b,b+1/4 | 0,0,0)
24e	$\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
Γ:(0,0,0)	$\ce{^{-1}{4}}\ce{^{1}{3}}\ce{^{-1}{2}}\ce{^{\infty m}{1}} $
R:(1/2,1/2,1/2)	$\ce{^{-1}{4}}\ce{^{1}{3}}\ce{^{-1}{2}}\ce{^{\infty m}{1}} $
M:(1/2,1/2,0)	$\ce{^{-1}{4}}\ce{^{1}{2}}\ce{^{-1}{2}}\ce{^{\infty m}{1}} $
X:(0,1/2,0)	$\ce{^{-1}{4}}\ce{^{1}{2}}\ce{^{-1}{2}}\ce{^{\infty m}{1}} $
Δ:(0,v,0)	$\ce{^{2}{4}}\ce{^{m}{2}}\ce{^{-1}{2}}\ce{^{\infty}{1}} $
T:(1/2,1/2,w)	$\ce{^{2}{4}}\ce{^{m}{2}}\ce{^{-1}{2}}\ce{^{\infty}{1}} $
Λ:(u,u,u)	$\ce{^{1}{3}}\ce{^{-1}{2}}\ce{^{\infty}{1}} $
S:(u,1/2,u)	$\ce{^{2}{2}}\ce{^{m}{2}}\ce{^{-1}{2}}\ce{^{\infty}{1}} $
Σ:(u,u,0)	$\ce{^{-1}{2}}\ce{^{2}{2}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
Z:(u,1/2,0)	$\ce{^{1}{2}}\ce{^{m}{2}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
A:(u,v,0)	$\ce{^{m}{2}}\ce{^{\infty}{1}} $
B:(u,1/2,w)	$\ce{^{m}{2}}\ce{^{\infty}{1}} $
C:(u,u,w)	$\ce{^{-1}{2}}\ce{^{\infty}{1}} $
GP:(u,v,w)	$\ce{^{1}{1}}\ce{^{\infty}{1}} $

Spin Brillouin Zone

Band Representations (Magnon)

8c

k-vector	k-vector-G↑	A1↑G(8)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{4}^{S}(6) $
M:(1/2,1/2,0)	M:(1/2,1/2,0)	$M_{1}^{S}M_{2}^{S}(4)⊕M_{3}^{S}M_{4}^{S}(4) $
R:(1/2,1/2,1/2)	R:(1/2,1/2,1/2)	$R_{1}^{S}R_{3}^{S}(8) $
X:(0,1/2,0)	X:(0,1/2,0)	$2X_{1}^{S}(4) $
Δ:(0,v,0)	Δ:(0,v,0)	$2Δ_{1}^{S}(2)⊕2Δ_{2}^{S}(2) $
Λ:(u,u,u)	Λ:(u,u,u)	$2Λ_{1}^{S}(2)⊕Λ_{2}^{S}(2)⊕Λ_{3}^{S}(2) $
S:(u,1/2,u)	S:(u,1/2,u)	$2S_{1}^{S}S_{1}^{S}(4) $
Σ:(u,u,0)	Σ:(u,u,0)	$4Σ_{1}^{S}(2) $
T:(1/2,1/2,w)	T:(1/2,1/2,w)	$T_{1}^{S}T_{1}^{S}(4)⊕T_{2}^{S}T_{2}^{S}(4) $
Z:(u,1/2,0)	Z:(u,1/2,0)	$4Z_{1}^{S}Z_{2}^{S}(2)$	Spin Splitting
A:(u,v,0)	A:(u,v,0)	$8A_{1}^{S}(1)$	Spin Splitting
B:(u,1/2,w)	B:(u,1/2,w)	$4B_{1}^{S}B_{1}^{S}(2)$	Spin Splitting
C:(u,u,w)	C:(u,u,w)	$4C_{1}^{S}(2) $
GP:(u,v,w)	GP:(u,v,w)	$8GP_{1}^{S}(1)$	Spin Splitting

24e

k-vector	k-vector-G↑	A↑G(24)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{2}^{S}Γ_{3}^{S}(4)⊕3Γ_{4}^{S}(6) $
M:(1/2,1/2,0)	M:(1/2,1/2,0)	$3M_{1}^{S}M_{2}^{S}(4)⊕3M_{3}^{S}M_{4}^{S}(4) $
R:(1/2,1/2,1/2)	R:(1/2,1/2,1/2)	$2R_{1}^{S}R_{3}^{S}(8)⊕R_{2}^{S}R_{2}^{S}(8) $
X:(0,1/2,0)	X:(0,1/2,0)	$6X_{1}^{S}(4) $
Δ:(0,v,0)	Δ:(0,v,0)	$6Δ_{1}^{S}(2)⊕6Δ_{2}^{S}(2) $
Λ:(u,u,u)	Λ:(u,u,u)	$4Λ_{1}^{S}(2)⊕4Λ_{2}^{S}(2)⊕4Λ_{3}^{S}(2) $
S:(u,1/2,u)	S:(u,1/2,u)	$6S_{1}^{S}S_{1}^{S}(4) $
Σ:(u,u,0)	Σ:(u,u,0)	$12Σ_{1}^{S}(2) $
T:(1/2,1/2,w)	T:(1/2,1/2,w)	$3T_{1}^{S}T_{1}^{S}(4)⊕3T_{2}^{S}T_{2}^{S}(4) $
Z:(u,1/2,0)	Z:(u,1/2,0)	$12Z_{1}^{S}Z_{2}^{S}(2)$	Spin Splitting
A:(u,v,0)	A:(u,v,0)	$24A_{1}^{S}(1)$	Spin Splitting
B:(u,1/2,w)	B:(u,1/2,w)	$12B_{1}^{S}B_{1}^{S}(2)$	Spin Splitting
C:(u,u,w)	C:(u,u,w)	$12C_{1}^{S}(2) $
GP:(u,v,w)	GP:(u,v,w)	$24GP_{1}^{S}(1)$	Spin Splitting