Information of spin space group $P\ce{^{1}{n}}\ce{^{1}{\text{-}3}}\ce{^{\text{-}1}{n}}\ce{^{\infty m}{1}}$ ( nSSG:201.222.1.1 )

General Positions

The continuously infinite group ^∞1 is omitted.

Coordinates	Seitz symbol
a, b, c | x, y, z	{ 1 ‖ 1 | 0 }
a, -b+1/2, -c+1/2 | x, y, z	{ 1 ‖ 2100 | 0 1/2 1/2 }
-a+1/2, b, -c+1/2 | x, y, z	{ 1 ‖ 2010 | 1/2 0 1/2 }
-a+1/2, -b+1/2, c | x, y, z	{ 1 ‖ 2001 | 1/2 1/2 0 }
c, a, b | x, y, z	{ 1 ‖ 3+111 | 0 }
b, c, a | x, y, z	{ 1 ‖ 3-111 | 0 }
-b+1/2, c, -a+1/2 | x, y, z	{ 1 ‖ 3-1-1-1 | 1/2 0 1/2 }
-c+1/2, -a+1/2, b | x, y, z	{ 1 ‖ 3+1-1-1 | 1/2 1/2 0 }
-b+1/2, -c+1/2, a | x, y, z	{ 1 ‖ 3--11-1 | 1/2 1/2 0 }
c, -a+1/2, -b+1/2 | x, y, z	{ 1 ‖ 3+-11-1 | 0 1/2 1/2 }
b, -c+1/2, -a+1/2 | x, y, z	{ 1 ‖ 3--1-11 | 0 1/2 1/2 }
-c+1/2, a, -b+1/2 | x, y, z	{ 1 ‖ 3+-1-11 | 1/2 0 1/2 }
-a, -b, -c | x, y, z	{ 1 ‖ -1 | 0 }
-a, b+1/2, c+1/2 | x, y, z	{ 1 ‖ m100 | 0 1/2 1/2 }
a+1/2, -b, c+1/2 | x, y, z	{ 1 ‖ m010 | 1/2 0 1/2 }
a+1/2, b+1/2, -c | x, y, z	{ 1 ‖ m001 | 1/2 1/2 0 }
-c, -a, -b | x, y, z	{ 1 ‖ -3+111 | 0 }
-b, -c, -a | x, y, z	{ 1 ‖ -3-111 | 0 }
b+1/2, -c, a+1/2 | x, y, z	{ 1 ‖ -3-1-1-1 | 1/2 0 1/2 }
c+1/2, a+1/2, -b | x, y, z	{ 1 ‖ -3+1-1-1 | 1/2 1/2 0 }
b+1/2, c+1/2, -a | x, y, z	{ 1 ‖ -3--11-1 | 1/2 1/2 0 }
-c, a+1/2, b+1/2 | x, y, z	{ 1 ‖ -3+-11-1 | 0 1/2 1/2 }
-b, c+1/2, a+1/2 | x, y, z	{ 1 ‖ -3--1-11 | 0 1/2 1/2 }
c+1/2, -a, b+1/2 | x, y, z	{ 1 ‖ -3+-1-11 | 1/2 0 1/2 }
a, -c+1/2, b | -x, -y, -z	{ -1 ‖ 4+100 | 0 1/2 0 }
a, c, -b+1/2 | -x, -y, -z	{ -1 ‖ 4-100 | 0 0 1/2 }
c, b, -a+1/2 | -x, -y, -z	{ -1 ‖ 4+010 | 0 0 1/2 }
-c+1/2, b, a | -x, -y, -z	{ -1 ‖ 4-010 | 1/2 0 0 }
-b+1/2, a, c | -x, -y, -z	{ -1 ‖ 4+001 | 1/2 0 0 }
b, -a+1/2, c | -x, -y, -z	{ -1 ‖ 4-001 | 0 1/2 0 }
b, a, -c+1/2 | -x, -y, -z	{ -1 ‖ 2110 | 0 0 1/2 }
-b+1/2, -a+1/2, -c+1/2 | -x, -y, -z	{ -1 ‖ 21-10 | 1/2 1/2 1/2 }
-a+1/2, c, b | -x, -y, -z	{ -1 ‖ 2011 | 1/2 0 0 }
-a+1/2, -c+1/2, -b+1/2 | -x, -y, -z	{ -1 ‖ 201-1 | 1/2 1/2 1/2 }
c, -b+1/2, a | -x, -y, -z	{ -1 ‖ 2101 | 0 1/2 0 }
-c+1/2, -b+1/2, -a+1/2 | -x, -y, -z	{ -1 ‖ 2-101 | 1/2 1/2 1/2 }
-a, c+1/2, -b | -x, -y, -z	{ -1 ‖ -4+100 | 0 1/2 0 }
-a, -c, b+1/2 | -x, -y, -z	{ -1 ‖ -4-100 | 0 0 1/2 }
-c, -b, a+1/2 | -x, -y, -z	{ -1 ‖ -4+010 | 0 0 1/2 }
c+1/2, -b, -a | -x, -y, -z	{ -1 ‖ -4-010 | 1/2 0 0 }
b+1/2, -a, -c | -x, -y, -z	{ -1 ‖ -4+001 | 1/2 0 0 }
-b, a+1/2, -c | -x, -y, -z	{ -1 ‖ -4-001 | 0 1/2 0 }
-b, -a, c+1/2 | -x, -y, -z	{ -1 ‖ m110 | 0 0 1/2 }
b+1/2, a+1/2, c+1/2 | -x, -y, -z	{ -1 ‖ m1-10 | 1/2 1/2 1/2 }
a+1/2, -c, -b | -x, -y, -z	{ -1 ‖ m011 | 1/2 0 0 }
a+1/2, c+1/2, b+1/2 | -x, -y, -z	{ -1 ‖ m01-1 | 1/2 1/2 1/2 }
-c, b+1/2, -a | -x, -y, -z	{ -1 ‖ m101 | 0 1/2 0 }
c+1/2, b+1/2, a+1/2 | -x, -y, -z	{ -1 ‖ m-101 | 1/2 1/2 1/2 }
a, b, c | -x, y, z	{ m ‖ 1 | 0 }
a, -b+1/2, -c+1/2 | -x, y, z	{ m ‖ 2100 | 0 1/2 1/2 }
-a+1/2, b, -c+1/2 | -x, y, z	{ m ‖ 2010 | 1/2 0 1/2 }
-a+1/2, -b+1/2, c | -x, y, z	{ m ‖ 2001 | 1/2 1/2 0 }
c, a, b | -x, y, z	{ m ‖ 3+111 | 0 }
b, c, a | -x, y, z	{ m ‖ 3-111 | 0 }
-b+1/2, c, -a+1/2 | -x, y, z	{ m ‖ 3-1-1-1 | 1/2 0 1/2 }
-c+1/2, -a+1/2, b | -x, y, z	{ m ‖ 3+1-1-1 | 1/2 1/2 0 }
-b+1/2, -c+1/2, a | -x, y, z	{ m ‖ 3--11-1 | 1/2 1/2 0 }
c, -a+1/2, -b+1/2 | -x, y, z	{ m ‖ 3+-11-1 | 0 1/2 1/2 }
b, -c+1/2, -a+1/2 | -x, y, z	{ m ‖ 3--1-11 | 0 1/2 1/2 }
-c+1/2, a, -b+1/2 | -x, y, z	{ m ‖ 3+-1-11 | 1/2 0 1/2 }
-a, -b, -c | -x, y, z	{ m ‖ -1 | 0 }
-a, b+1/2, c+1/2 | -x, y, z	{ m ‖ m100 | 0 1/2 1/2 }
a+1/2, -b, c+1/2 | -x, y, z	{ m ‖ m010 | 1/2 0 1/2 }
a+1/2, b+1/2, -c | -x, y, z	{ m ‖ m001 | 1/2 1/2 0 }
-c, -a, -b | -x, y, z	{ m ‖ -3+111 | 0 }
-b, -c, -a | -x, y, z	{ m ‖ -3-111 | 0 }
b+1/2, -c, a+1/2 | -x, y, z	{ m ‖ -3-1-1-1 | 1/2 0 1/2 }
c+1/2, a+1/2, -b | -x, y, z	{ m ‖ -3+1-1-1 | 1/2 1/2 0 }
b+1/2, c+1/2, -a | -x, y, z	{ m ‖ -3--11-1 | 1/2 1/2 0 }
-c, a+1/2, b+1/2 | -x, y, z	{ m ‖ -3+-11-1 | 0 1/2 1/2 }
-b, c+1/2, a+1/2 | -x, y, z	{ m ‖ -3--1-11 | 0 1/2 1/2 }
c+1/2, -a, b+1/2 | -x, y, z	{ m ‖ -3+-1-11 | 1/2 0 1/2 }
a, -c+1/2, b | x, -y, -z	{ 2 ‖ 4+100 | 0 1/2 0 }
a, c, -b+1/2 | x, -y, -z	{ 2 ‖ 4-100 | 0 0 1/2 }
c, b, -a+1/2 | x, -y, -z	{ 2 ‖ 4+010 | 0 0 1/2 }
-c+1/2, b, a | x, -y, -z	{ 2 ‖ 4-010 | 1/2 0 0 }
-b+1/2, a, c | x, -y, -z	{ 2 ‖ 4+001 | 1/2 0 0 }
b, -a+1/2, c | x, -y, -z	{ 2 ‖ 4-001 | 0 1/2 0 }
b, a, -c+1/2 | x, -y, -z	{ 2 ‖ 2110 | 0 0 1/2 }
-b+1/2, -a+1/2, -c+1/2 | x, -y, -z	{ 2 ‖ 21-10 | 1/2 1/2 1/2 }
-a+1/2, c, b | x, -y, -z	{ 2 ‖ 2011 | 1/2 0 0 }
-a+1/2, -c+1/2, -b+1/2 | x, -y, -z	{ 2 ‖ 201-1 | 1/2 1/2 1/2 }
c, -b+1/2, a | x, -y, -z	{ 2 ‖ 2101 | 0 1/2 0 }
-c+1/2, -b+1/2, -a+1/2 | x, -y, -z	{ 2 ‖ 2-101 | 1/2 1/2 1/2 }
-a, c+1/2, -b | x, -y, -z	{ 2 ‖ -4+100 | 0 1/2 0 }
-a, -c, b+1/2 | x, -y, -z	{ 2 ‖ -4-100 | 0 0 1/2 }
-c, -b, a+1/2 | x, -y, -z	{ 2 ‖ -4+010 | 0 0 1/2 }
c+1/2, -b, -a | x, -y, -z	{ 2 ‖ -4-010 | 1/2 0 0 }
b+1/2, -a, -c | x, -y, -z	{ 2 ‖ -4+001 | 1/2 0 0 }
-b, a+1/2, -c | x, -y, -z	{ 2 ‖ -4-001 | 0 1/2 0 }
-b, -a, c+1/2 | x, -y, -z	{ 2 ‖ m110 | 0 0 1/2 }
b+1/2, a+1/2, c+1/2 | x, -y, -z	{ 2 ‖ m1-10 | 1/2 1/2 1/2 }
a+1/2, -c, -b | x, -y, -z	{ 2 ‖ m011 | 1/2 0 0 }
a+1/2, c+1/2, b+1/2 | x, -y, -z	{ 2 ‖ m01-1 | 1/2 1/2 1/2 }
-c, b+1/2, -a | x, -y, -z	{ 2 ‖ m101 | 0 1/2 0 }
c+1/2, b+1/2, a+1/2 | x, -y, -z	{ 2 ‖ m-101 | 1/2 1/2 1/2 }

Wyckoff Positions

WP	Site symmetry	Representative
2a	$\ce{^{-1}{4}}\ce{^{1}{3}}\ce{^{-1}{2}}\ce{^{\infty m}{1}} $	(1/4,1/4,1/4 | 0,0,0)
6b	$\ce{^{-1}{4}}\ce{^{1}{2}}.\ce{^{-1}{2}}\ce{^{\infty m}{1}} $	(3/4,1/4,1/4 | 0,0,0)
8c	$.\ce{^{1}{-3}}.\ce{^{\infty m}{1}} $	(0,0,0 | 0,0,z)
12d	$\ce{^{-1}{-4}}..\ce{^{\infty m}{1}} $	(0,3/4,1/4 | 0,0,0)
12e	$\ce{^{-1}{4}}..\ce{^{\infty m}{1}} $	(a,1/4,1/4 | 0,0,0)
16f	$.\ce{^{1}{3}}.\ce{^{\infty m}{1}} $	(a,a,a | 0,0,z)
24g	$\ce{^{1}{2}}..\ce{^{\infty m}{1}} $	(a,3/4,1/4 | 0,0,z)
24h	$..\ce{^{-1}{2}}\ce{^{\infty m}{1}} $	(1/4,b,b | 0,0,0)
48i	$\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}{m}}\ce{^{1}{-3}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
R:(1/2,1/2,1/2)	$\ce{^{1}{m}}\ce{^{1}{-3}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
M:(1/2,1/2,0)	$\ce{^{-1}{4/}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
X:(0,1/2,0)	$\ce{^{-1}{4/}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
Δ:(0,v,0)	$\ce{^{2}{4/}}\ce{^{m}{m}}\ce{^{1}{m}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
T:(1/2,1/2,w)	$\ce{^{2}{4/}}\ce{^{m}{m}}\ce{^{1}{m}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
Λ:(u,u,u)	$\ce{^{m}{-3}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
S:(u,1/2,u)	$\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
Σ:(u,u,0)	$\ce{^{2}{m}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
Z:(u,1/2,0)	$\ce{^{m}{m}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
A:(u,v,0)	$\ce{^{m}{2/}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
B:(u,1/2,w)	$\ce{^{m}{2/}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
C:(u,u,w)	$\ce{^{-1}{2/}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
GP:(u,v,w)	$\ce{^{m}{-1}}\ce{^{\infty}{1}} $

Spin Brillouin Zone

Band Representations (Magnon)

24g

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

48i

k-vector	k-vector-G↑	A↑G(48)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S,+}(2)⊕Γ_{1}^{S,-}(2)⊕Γ_{2}^{S,+}Γ_{3}^{S,+}(4)⊕Γ_{2}^{S,-}Γ_{3}^{S,-}(4)⊕3Γ_{4}^{S,+}(6)⊕3Γ_{4}^{S,-}(6) $
M:(1/2,1/2,0)	M:(1/2,1/2,0)	$6M_{1}^{S}(4)⊕6M_{2}^{S}(4) $
R:(1/2,1/2,1/2)	R:(1/2,1/2,1/2)	$R_{1}^{S,+}(2)⊕R_{1}^{S,-}(2)⊕R_{2}^{S,+}R_{3}^{S,+}(4)⊕R_{2}^{S,-}R_{3}^{S,-}(4)⊕3R_{4}^{S,+}(6)⊕3R_{4}^{S,-}(6) $
X:(0,1/2,0)	X:(0,1/2,0)	$6X_{1}^{S}(4)⊕6X_{2}^{S}(4) $
Δ:(0,v,0)	Δ:(0,v,0)	$6Δ_{1}^{S}(2)⊕6Δ_{2}^{S}(2)⊕6Δ_{3}^{S}(2)⊕6Δ_{4}^{S}(2) $
Λ:(u,u,u)	Λ:(u,u,u)	$8Λ_{1}^{S}(2)⊕8Λ_{2}^{S}Λ_{3}^{S}(4) $
S:(u,1/2,u)	S:(u,1/2,u)	$12S_{1}^{S}(2)⊕12S_{2}^{S}(2) $
Σ:(u,u,0)	Σ:(u,u,0)	$12Σ_{1}^{S}(2)⊕12Σ_{2}^{S}(2) $
T:(1/2,1/2,w)	T:(1/2,1/2,w)	$6T_{1}^{S}(2)⊕6T_{2}^{S}(2)⊕6T_{3}^{S}(2)⊕6T_{4}^{S}(2) $
Z:(u,1/2,0)	Z:(u,1/2,0)	$24Z_{1}^{S}(2)$	Spin Splitting
A:(u,v,0)	A:(u,v,0)	$24A_{1}^{S}(1)⊕24A_{2}^{S}(1)$	Spin Splitting
B:(u,1/2,w)	B:(u,1/2,w)	$24B_{1}^{S}(1)⊕24B_{2}^{S}(1)$	Spin Splitting
C:(u,u,w)	C:(u,u,w)	$24C_{1}^{S}(2) $
GP:(u,v,w)	GP:(u,v,w)	$48GP_{1}^{S}(1)$	Spin Splitting

8c

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

16f

k-vector	k-vector-G↑	A1↑G(16)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S,+}(2)⊕Γ_{1}^{S,-}(2)⊕Γ_{4}^{S,+}(6)⊕Γ_{4}^{S,-}(6) $
M:(1/2,1/2,0)	M:(1/2,1/2,0)	$2M_{1}^{S}(4)⊕2M_{2}^{S}(4) $
R:(1/2,1/2,1/2)	R:(1/2,1/2,1/2)	$R_{1}^{S,+}(2)⊕R_{1}^{S,-}(2)⊕R_{4}^{S,+}(6)⊕R_{4}^{S,-}(6) $
X:(0,1/2,0)	X:(0,1/2,0)	$2X_{1}^{S}(4)⊕2X_{2}^{S}(4) $
Δ:(0,v,0)	Δ:(0,v,0)	$2Δ_{1}^{S}(2)⊕2Δ_{2}^{S}(2)⊕2Δ_{3}^{S}(2)⊕2Δ_{4}^{S}(2) $
Λ:(u,u,u)	Λ:(u,u,u)	$4Λ_{1}^{S}(2)⊕2Λ_{2}^{S}Λ_{3}^{S}(4) $
S:(u,1/2,u)	S:(u,1/2,u)	$4S_{1}^{S}(2)⊕4S_{2}^{S}(2) $
Σ:(u,u,0)	Σ:(u,u,0)	$4Σ_{1}^{S}(2)⊕4Σ_{2}^{S}(2) $
T:(1/2,1/2,w)	T:(1/2,1/2,w)	$2T_{1}^{S}(2)⊕2T_{2}^{S}(2)⊕2T_{3}^{S}(2)⊕2T_{4}^{S}(2) $
Z:(u,1/2,0)	Z:(u,1/2,0)	$8Z_{1}^{S}(2)$	Spin Splitting
A:(u,v,0)	A:(u,v,0)	$8A_{1}^{S}(1)⊕8A_{2}^{S}(1)$	Spin Splitting
B:(u,1/2,w)	B:(u,1/2,w)	$8B_{1}^{S}(1)⊕8B_{2}^{S}(1)$	Spin Splitting
C:(u,u,w)	C:(u,u,w)	$8C_{1}^{S}(2) $
GP:(u,v,w)	GP:(u,v,w)	$16GP_{1}^{S}(1)$	Spin Splitting