Information of spin space group $R\ce{^{1}{3}}\ce{^{\text{-}1}{m}}\ce{^{\infty m}{1}}$ ( nSSG:146.160.1.1 )

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, b, c | -x, -y, -z	{ -1 ‖ m100 | 0 }
-b, -a, c | -x, -y, -z	{ -1 ‖ m110 | 0 }
a, a-b, c | -x, -y, -z	{ -1 ‖ m010 | 0 }
a+2/3, b+1/3, c+1/3 | x, y, z	{ 1 ‖ 1 | 2/3 1/3 1/3 }
-b+2/3, a-b+1/3, c+1/3 | x, y, z	{ 1 ‖ 3+001 | 2/3 1/3 1/3 }
-a+b+2/3, -a+1/3, c+1/3 | x, y, z	{ 1 ‖ 3-001 | 2/3 1/3 1/3 }
-a+b+2/3, b+1/3, c+1/3 | -x, -y, -z	{ -1 ‖ m100 | 2/3 1/3 1/3 }
-b+2/3, -a+1/3, c+1/3 | -x, -y, -z	{ -1 ‖ m110 | 2/3 1/3 1/3 }
a+2/3, a-b+1/3, c+1/3 | -x, -y, -z	{ -1 ‖ m010 | 2/3 1/3 1/3 }
a+1/3, b+2/3, c+2/3 | x, y, z	{ 1 ‖ 1 | 1/3 2/3 2/3 }
-b+1/3, a-b+2/3, c+2/3 | x, y, z	{ 1 ‖ 3+001 | 1/3 2/3 2/3 }
-a+b+1/3, -a+2/3, c+2/3 | x, y, z	{ 1 ‖ 3-001 | 1/3 2/3 2/3 }
-a+b+1/3, b+2/3, c+2/3 | -x, -y, -z	{ -1 ‖ m100 | 1/3 2/3 2/3 }
-b+1/3, -a+2/3, c+2/3 | -x, -y, -z	{ -1 ‖ m110 | 1/3 2/3 2/3 }
a+1/3, a-b+2/3, c+2/3 | -x, -y, -z	{ -1 ‖ m010 | 1/3 2/3 2/3 }
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, b, c | x, -y, -z	{ 2 ‖ m100 | 0 }
-b, -a, c | x, -y, -z	{ 2 ‖ m110 | 0 }
a, a-b, c | x, -y, -z	{ 2 ‖ m010 | 0 }
a+2/3, b+1/3, c+1/3 | -x, y, z	{ m ‖ 1 | 2/3 1/3 1/3 }
-b+2/3, a-b+1/3, c+1/3 | -x, y, z	{ m ‖ 3+001 | 2/3 1/3 1/3 }
-a+b+2/3, -a+1/3, c+1/3 | -x, y, z	{ m ‖ 3-001 | 2/3 1/3 1/3 }
-a+b+2/3, b+1/3, c+1/3 | x, -y, -z	{ 2 ‖ m100 | 2/3 1/3 1/3 }
-b+2/3, -a+1/3, c+1/3 | x, -y, -z	{ 2 ‖ m110 | 2/3 1/3 1/3 }
a+2/3, a-b+1/3, c+1/3 | x, -y, -z	{ 2 ‖ m010 | 2/3 1/3 1/3 }
a+1/3, b+2/3, c+2/3 | -x, y, z	{ m ‖ 1 | 1/3 2/3 2/3 }
-b+1/3, a-b+2/3, c+2/3 | -x, y, z	{ m ‖ 3+001 | 1/3 2/3 2/3 }
-a+b+1/3, -a+2/3, c+2/3 | -x, y, z	{ m ‖ 3-001 | 1/3 2/3 2/3 }
-a+b+1/3, b+2/3, c+2/3 | x, -y, -z	{ 2 ‖ m100 | 1/3 2/3 2/3 }
-b+1/3, -a+2/3, c+2/3 | x, -y, -z	{ 2 ‖ m110 | 1/3 2/3 2/3 }
a+1/3, a-b+2/3, c+2/3 | x, -y, -z	{ 2 ‖ m010 | 1/3 2/3 2/3 }

Wyckoff Positions

WP	Site symmetry	Representative
3a	$\ce{^{1}{3}}\ce{^{-1}{m}}.\ce{^{\infty m}{1}} $	(0,0,c | 0,0,0)
9b	$.\ce{^{-1}{m}}.\ce{^{\infty m}{1}} $	(a,-a,c | 0,0,0)
18c	$\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}{3}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
T:(0,0,3/2)	$\ce{^{1}{3}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
F:(0,1/2,1)	$\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
L:(-1/2,1/2,1/2)	$\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
Λ:(0,0,w)	$\ce{^{1}{3}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
Σ:(u,-2*u,0)	$\ce{^{-1}{m}}\ce{^{\infty}{1}} $
Y:(u,u,3/2)	$\ce{^{-1}{m}}\ce{^{\infty}{1}} $
C:(u,-u,w)	$\ce{^{2}{m}}\ce{^{\infty}{1}} $
GP:(u,v,w)	$\ce{^{1}{1}}\ce{^{\infty}{1}} $

Spin Brillouin Zone

Band Representations (Magnon)

18c

k-vector	k-vector-G↑	A↑G(6)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{2}^{S}Γ_{3}^{S}(4) $
F:(0,1/2,1)	F:(0,1/2,1)	$3F_{1}^{S}(2) $
L:(-1/2,1/2,1/2)	L:(-1/2,1/2,1/2)	$3L_{1}^{S}(2) $
T:(0,0,3/2)	T:(0,0,3/2)	$T_{1}^{S}(2)⊕T_{2}^{S}T_{3}^{S}(4) $
Λ:(0,0,w)	Λ:(0,0,w)	$Λ_{1}^{S}(2)⊕Λ_{2}^{S}(2)⊕Λ_{3}^{S}(2) $
Σ:(u,-2*u,0)	Σ:(u,-2*u,0)	$3Σ_{1}^{S}(2) $
Y:(u,u,3/2)	Y:(u,u,3/2)	$3Y_{1}^{S}(2) $
C:(u,-u,w)	C:(u,-u,w)	$3C_{1}^{S}(2) $
GP:(u,v,w)	GP:(u,v,w)	$6GP_{1}^{S}(1)$	Spin Splitting