Information of spin space group $R_{I}\ce{^{1}{\text{-}3}}\ce{^{1}{c}}\ce{^{\infty m}{1}}$ ( nSSG:167.166.2.1 )

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+1/2 | x, y, z	{ 1 ‖ 2100 | 0 0 1/2 }
b, a, -c+1/2 | x, y, z	{ 1 ‖ 2110 | 0 0 1/2 }
-a, -a+b, -c+1/2 | x, y, z	{ 1 ‖ 2010 | 0 0 1/2 }
-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+1/2 | x, y, z	{ 1 ‖ m100 | 0 0 1/2 }
-b, -a, c+1/2 | x, y, z	{ 1 ‖ m110 | 0 0 1/2 }
a, a-b, c+1/2 | x, y, z	{ 1 ‖ m010 | 0 0 1/2 }
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+5/6 | x, y, z	{ 1 ‖ 2100 | 2/3 1/3 5/6 }
b+2/3, a+1/3, -c+5/6 | x, y, z	{ 1 ‖ 2110 | 2/3 1/3 5/6 }
-a+2/3, -a+b+1/3, -c+5/6 | x, y, z	{ 1 ‖ 2010 | 2/3 1/3 5/6 }
-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+5/6 | x, y, z	{ 1 ‖ m100 | 2/3 1/3 5/6 }
-b+2/3, -a+1/3, c+5/6 | x, y, z	{ 1 ‖ m110 | 2/3 1/3 5/6 }
a+2/3, a-b+1/3, c+5/6 | x, y, z	{ 1 ‖ m010 | 2/3 1/3 5/6 }
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+1/6 | x, y, z	{ 1 ‖ 2100 | 1/3 2/3 1/6 }
b+1/3, a+2/3, -c+1/6 | x, y, z	{ 1 ‖ 2110 | 1/3 2/3 1/6 }
-a+1/3, -a+b+2/3, -c+1/6 | x, y, z	{ 1 ‖ 2010 | 1/3 2/3 1/6 }
-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+1/6 | x, y, z	{ 1 ‖ m100 | 1/3 2/3 1/6 }
-b+1/3, -a+2/3, c+1/6 | x, y, z	{ 1 ‖ m110 | 1/3 2/3 1/6 }
a+1/3, a-b+2/3, c+1/6 | x, y, z	{ 1 ‖ m010 | 1/3 2/3 1/6 }
a, b, c+1/2 | -x, -y, -z	{ -1 ‖ 1 | 0 0 1/2 }
-b, a-b, c+1/2 | -x, -y, -z	{ -1 ‖ 3+001 | 0 0 1/2 }
-a+b, -a, c+1/2 | -x, -y, -z	{ -1 ‖ 3-001 | 0 0 1/2 }
a-b, -b, -c | -x, -y, -z	{ -1 ‖ 2100 | 0 }
b, a, -c | -x, -y, -z	{ -1 ‖ 2110 | 0 }
-a, -a+b, -c | -x, -y, -z	{ -1 ‖ 2010 | 0 }
-a, -b, -c+1/2 | -x, -y, -z	{ -1 ‖ -1 | 0 0 1/2 }
b, -a+b, -c+1/2 | -x, -y, -z	{ -1 ‖ -3+001 | 0 0 1/2 }
a-b, a, -c+1/2 | -x, -y, -z	{ -1 ‖ -3-001 | 0 0 1/2 }
-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+5/6 | -x, -y, -z	{ -1 ‖ 1 | 2/3 1/3 5/6 }
-b+2/3, a-b+1/3, c+5/6 | -x, -y, -z	{ -1 ‖ 3+001 | 2/3 1/3 5/6 }
-a+b+2/3, -a+1/3, c+5/6 | -x, -y, -z	{ -1 ‖ 3-001 | 2/3 1/3 5/6 }
a-b+2/3, -b+1/3, -c+1/3 | -x, -y, -z	{ -1 ‖ 2100 | 2/3 1/3 1/3 }
b+2/3, a+1/3, -c+1/3 | -x, -y, -z	{ -1 ‖ 2110 | 2/3 1/3 1/3 }
-a+2/3, -a+b+1/3, -c+1/3 | -x, -y, -z	{ -1 ‖ 2010 | 2/3 1/3 1/3 }
-a+2/3, -b+1/3, -c+5/6 | -x, -y, -z	{ -1 ‖ -1 | 2/3 1/3 5/6 }
b+2/3, -a+b+1/3, -c+5/6 | -x, -y, -z	{ -1 ‖ -3+001 | 2/3 1/3 5/6 }
a-b+2/3, a+1/3, -c+5/6 | -x, -y, -z	{ -1 ‖ -3-001 | 2/3 1/3 5/6 }
-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+1/6 | -x, -y, -z	{ -1 ‖ 1 | 1/3 2/3 1/6 }
-b+1/3, a-b+2/3, c+1/6 | -x, -y, -z	{ -1 ‖ 3+001 | 1/3 2/3 1/6 }
-a+b+1/3, -a+2/3, c+1/6 | -x, -y, -z	{ -1 ‖ 3-001 | 1/3 2/3 1/6 }
a-b+1/3, -b+2/3, -c+2/3 | -x, -y, -z	{ -1 ‖ 2100 | 1/3 2/3 2/3 }
b+1/3, a+2/3, -c+2/3 | -x, -y, -z	{ -1 ‖ 2110 | 1/3 2/3 2/3 }
-a+1/3, -a+b+2/3, -c+2/3 | -x, -y, -z	{ -1 ‖ 2010 | 1/3 2/3 2/3 }
-a+1/3, -b+2/3, -c+1/6 | -x, -y, -z	{ -1 ‖ -1 | 1/3 2/3 1/6 }
b+1/3, -a+b+2/3, -c+1/6 | -x, -y, -z	{ -1 ‖ -3+001 | 1/3 2/3 1/6 }
a-b+1/3, a+2/3, -c+1/6 | -x, -y, -z	{ -1 ‖ -3-001 | 1/3 2/3 1/6 }
-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+1/2 | -x, y, z	{ m ‖ 2100 | 0 0 1/2 }
b, a, -c+1/2 | -x, y, z	{ m ‖ 2110 | 0 0 1/2 }
-a, -a+b, -c+1/2 | -x, y, z	{ m ‖ 2010 | 0 0 1/2 }
-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+1/2 | -x, y, z	{ m ‖ m100 | 0 0 1/2 }
-b, -a, c+1/2 | -x, y, z	{ m ‖ m110 | 0 0 1/2 }
a, a-b, c+1/2 | -x, y, z	{ m ‖ m010 | 0 0 1/2 }
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+5/6 | -x, y, z	{ m ‖ 2100 | 2/3 1/3 5/6 }
b+2/3, a+1/3, -c+5/6 | -x, y, z	{ m ‖ 2110 | 2/3 1/3 5/6 }
-a+2/3, -a+b+1/3, -c+5/6 | -x, y, z	{ m ‖ 2010 | 2/3 1/3 5/6 }
-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+5/6 | -x, y, z	{ m ‖ m100 | 2/3 1/3 5/6 }
-b+2/3, -a+1/3, c+5/6 | -x, y, z	{ m ‖ m110 | 2/3 1/3 5/6 }
a+2/3, a-b+1/3, c+5/6 | -x, y, z	{ m ‖ m010 | 2/3 1/3 5/6 }
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+1/6 | -x, y, z	{ m ‖ 2100 | 1/3 2/3 1/6 }
b+1/3, a+2/3, -c+1/6 | -x, y, z	{ m ‖ 2110 | 1/3 2/3 1/6 }
-a+1/3, -a+b+2/3, -c+1/6 | -x, y, z	{ m ‖ 2010 | 1/3 2/3 1/6 }
-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+1/6 | -x, y, z	{ m ‖ m100 | 1/3 2/3 1/6 }
-b+1/3, -a+2/3, c+1/6 | -x, y, z	{ m ‖ m110 | 1/3 2/3 1/6 }
a+1/3, a-b+2/3, c+1/6 | -x, y, z	{ m ‖ m010 | 1/3 2/3 1/6 }
a, b, c+1/2 | x, -y, -z	{ 2 ‖ 1 | 0 0 1/2 }
-b, a-b, c+1/2 | x, -y, -z	{ 2 ‖ 3+001 | 0 0 1/2 }
-a+b, -a, c+1/2 | x, -y, -z	{ 2 ‖ 3-001 | 0 0 1/2 }
a-b, -b, -c | x, -y, -z	{ 2 ‖ 2100 | 0 }
b, a, -c | x, -y, -z	{ 2 ‖ 2110 | 0 }
-a, -a+b, -c | x, -y, -z	{ 2 ‖ 2010 | 0 }
-a, -b, -c+1/2 | x, -y, -z	{ 2 ‖ -1 | 0 0 1/2 }
b, -a+b, -c+1/2 | x, -y, -z	{ 2 ‖ -3+001 | 0 0 1/2 }
a-b, a, -c+1/2 | x, -y, -z	{ 2 ‖ -3-001 | 0 0 1/2 }
-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+5/6 | x, -y, -z	{ 2 ‖ 1 | 2/3 1/3 5/6 }
-b+2/3, a-b+1/3, c+5/6 | x, -y, -z	{ 2 ‖ 3+001 | 2/3 1/3 5/6 }
-a+b+2/3, -a+1/3, c+5/6 | x, -y, -z	{ 2 ‖ 3-001 | 2/3 1/3 5/6 }
a-b+2/3, -b+1/3, -c+1/3 | x, -y, -z	{ 2 ‖ 2100 | 2/3 1/3 1/3 }
b+2/3, a+1/3, -c+1/3 | x, -y, -z	{ 2 ‖ 2110 | 2/3 1/3 1/3 }
-a+2/3, -a+b+1/3, -c+1/3 | x, -y, -z	{ 2 ‖ 2010 | 2/3 1/3 1/3 }
-a+2/3, -b+1/3, -c+5/6 | x, -y, -z	{ 2 ‖ -1 | 2/3 1/3 5/6 }
b+2/3, -a+b+1/3, -c+5/6 | x, -y, -z	{ 2 ‖ -3+001 | 2/3 1/3 5/6 }
a-b+2/3, a+1/3, -c+5/6 | x, -y, -z	{ 2 ‖ -3-001 | 2/3 1/3 5/6 }
-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+1/6 | x, -y, -z	{ 2 ‖ 1 | 1/3 2/3 1/6 }
-b+1/3, a-b+2/3, c+1/6 | x, -y, -z	{ 2 ‖ 3+001 | 1/3 2/3 1/6 }
-a+b+1/3, -a+2/3, c+1/6 | x, -y, -z	{ 2 ‖ 3-001 | 1/3 2/3 1/6 }
a-b+1/3, -b+2/3, -c+2/3 | x, -y, -z	{ 2 ‖ 2100 | 1/3 2/3 2/3 }
b+1/3, a+2/3, -c+2/3 | x, -y, -z	{ 2 ‖ 2110 | 1/3 2/3 2/3 }
-a+1/3, -a+b+2/3, -c+2/3 | x, -y, -z	{ 2 ‖ 2010 | 1/3 2/3 2/3 }
-a+1/3, -b+2/3, -c+1/6 | x, -y, -z	{ 2 ‖ -1 | 1/3 2/3 1/6 }
b+1/3, -a+b+2/3, -c+1/6 | x, -y, -z	{ 2 ‖ -3+001 | 1/3 2/3 1/6 }
a-b+1/3, a+2/3, -c+1/6 | x, -y, -z	{ 2 ‖ -3-001 | 1/3 2/3 1/6 }
-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
6a	$\ce{^{1}{-3}}\ce{^{-1}{m}}.\ce{^{\infty m}{1}} $	(0,0,0 | 0,0,0)
6b	$\ce{^{-1}{-3}}\ce{^{-1}{m}}.\ce{^{\infty m}{1}} $	(0,0,1/4 | 0,0,0)
12c	$\ce{^{1}{3}}\ce{^{-1}{m}}.\ce{^{\infty m}{1}} $	(0,0,c | 0,0,0)
18d	$.\ce{^{1}{2}}/\ce{^{-1}{m}}.\ce{^{\infty m}{1}} $	(1/2,0,1/4 | 0,0,0)
18e	$.\ce{^{-1}{2}}/\ce{^{-1}{m}}.\ce{^{\infty m}{1}} $	(1/2,0,0 | 0,0,0)
36f	$.\ce{^{-1}{2}}.\ce{^{\infty m}{1}} $	(a,0,0 | 0,0,0)
36g	$.\ce{^{1}{2}}.\ce{^{\infty m}{1}} $	(a,0,1/4 | 0,0,z)
36h	$.\ce{^{-1}{m}}.\ce{^{\infty m}{1}} $	(a,-a,c | 0,0,0)
72i	$\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 /mm}{1}} $
T:(0,0,3/2)	$\ce{^{1}{-3}}\ce{^{1}{m}}\ce{^{\infty /mm}{1}} $
F:(0,1/2,1)	$\ce{^{1}{2/}}\ce{^{1}{m}}\ce{^{\infty /mm}{1}} $
L:(-1/2,1/2,1/2)	$\ce{^{1}{2/}}\ce{^{1}{m}}\ce{^{\infty /mm}{1}} $
Λ:(0,0,w)	$\ce{^{m}{-3}}\ce{^{1}{m}}\ce{^{\infty 2}{1}} $
Σ:(u,-2*u,0)	$\ce{^{1}{2/}}\ce{^{m}{m}}\ce{^{\infty 2}{1}} $
Y:(u,u,3/2)	$\ce{^{1}{2/}}\ce{^{m}{m}}\ce{^{\infty 2}{1}} $
C:(u,-u,w)	$\ce{^{m}{2/}}\ce{^{1}{m}}\ce{^{\infty 2}{1}} $
GP:(u,v,w)	$\ce{^{m}{-1}}\ce{^{\infty 2}{1}} $

Spin Brillouin Zone

Band Representations (Magnon)

36g

k-vector	A↑G(12)
Γ:(0,0,0)	$Γ_{1}^{S,+}(2)⊕Γ_{1}^{S,-}(2)⊕Γ_{3}^{S,+}(4)⊕Γ_{3}^{S,-}(4) $
T:(0,0,3/2)	$T_{1}^{S}T_{2}^{S}(8)⊕T_{3}^{S}(4) $
F:(0,1/2,1)	$2F_{1}^{S,+}(2)⊕2F_{1}^{S,-}(2)⊕F_{2}^{S,+}(2)⊕F_{2}^{S,-}(2) $
L:(-1/2,1/2,1/2)	$3L_{1}^{S}(4) $
Λ:(0,0,w)	$Λ_{1}^{S}(2)⊕Λ_{2}^{S}(2)⊕2Λ_{3}^{S}(4) $
Σ:(u,-2*u,0)	$4Σ_{1}^{S}(2)⊕2Σ_{2}^{S}(2) $
Y:(u,u,3/2)	$3Y_{1}^{S}Y_{2}^{S}(4) $
C:(u,-u,w)	$3C_{1}^{S}(2)⊕3C_{2}^{S}(2) $
GP:(u,v,w)	$6GP_{1}^{S}(2) $

72i

k-vector	A↑G(24)
Γ:(0,0,0)	$Γ_{1}^{S,+}(2)⊕Γ_{1}^{S,-}(2)⊕Γ_{2}^{S,+}(2)⊕Γ_{2}^{S,-}(2)⊕2Γ_{3}^{S,+}(4)⊕2Γ_{3}^{S,-}(4) $
T:(0,0,3/2)	$2T_{1}^{S}T_{2}^{S}(8)⊕2T_{3}^{S}(4) $
F:(0,1/2,1)	$3F_{1}^{S,+}(2)⊕3F_{1}^{S,-}(2)⊕3F_{2}^{S,+}(2)⊕3F_{2}^{S,-}(2) $
L:(-1/2,1/2,1/2)	$6L_{1}^{S}(4) $
Λ:(0,0,w)	$2Λ_{1}^{S}(2)⊕2Λ_{2}^{S}(2)⊕4Λ_{3}^{S}(4) $
Σ:(u,-2*u,0)	$6Σ_{1}^{S}(2)⊕6Σ_{2}^{S}(2) $
Y:(u,u,3/2)	$6Y_{1}^{S}Y_{2}^{S}(4) $
C:(u,-u,w)	$6C_{1}^{S}(2)⊕6C_{2}^{S}(2) $
GP:(u,v,w)	$12GP_{1}^{S}(2) $