Information of spin space group $P\ce{^{1}{a}}\ce{^{1}{\text{-}3}}\ce{^{\text{-}1}{(1/2 \ 1/2 \ 1/2)}}\ce{^{\infty m}{1}}$ ( nSSG:205.206.2.1 )

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, -b, -c | x, y, z	{ 1 ‖ -1 | 0 }
-a+1/2, b+1/2, c | x, y, z	{ 1 ‖ m100 | 1/2 1/2 0 }
a, -b+1/2, c+1/2 | x, y, z	{ 1 ‖ m010 | 0 1/2 1/2 }
a+1/2, b, -c+1/2 | x, y, z	{ 1 ‖ m001 | 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+1/2, b+1/2, c+1/2 | -x, -y, -z	{ -1 ‖ 1 | 1/2 1/2 1/2 }
a, -b, -c+1/2 | -x, -y, -z	{ -1 ‖ 2100 | 0 0 1/2 }
-a+1/2, b, -c | -x, -y, -z	{ -1 ‖ 2010 | 1/2 0 0 }
-a, -b+1/2, c | -x, -y, -z	{ -1 ‖ 2001 | 0 1/2 0 }
c+1/2, a+1/2, b+1/2 | -x, -y, -z	{ -1 ‖ 3+111 | 1/2 1/2 1/2 }
b+1/2, c+1/2, a+1/2 | -x, -y, -z	{ -1 ‖ 3-111 | 1/2 1/2 1/2 }
-b+1/2, c, -a | -x, -y, -z	{ -1 ‖ 3-1-1-1 | 1/2 0 0 }
-c, -a+1/2, b | -x, -y, -z	{ -1 ‖ 3+1-1-1 | 0 1/2 0 }
-b, -c+1/2, a | -x, -y, -z	{ -1 ‖ 3--11-1 | 0 1/2 0 }
c, -a, -b+1/2 | -x, -y, -z	{ -1 ‖ 3+-11-1 | 0 0 1/2 }
b, -c, -a+1/2 | -x, -y, -z	{ -1 ‖ 3--1-11 | 0 0 1/2 }
-c+1/2, a, -b | -x, -y, -z	{ -1 ‖ 3+-1-11 | 1/2 0 0 }
-a+1/2, -b+1/2, -c+1/2 | -x, -y, -z	{ -1 ‖ -1 | 1/2 1/2 1/2 }
-a, b, c+1/2 | -x, -y, -z	{ -1 ‖ m100 | 0 0 1/2 }
a+1/2, -b, c | -x, -y, -z	{ -1 ‖ m010 | 1/2 0 0 }
a, b+1/2, -c | -x, -y, -z	{ -1 ‖ m001 | 0 1/2 0 }
-c+1/2, -a+1/2, -b+1/2 | -x, -y, -z	{ -1 ‖ -3+111 | 1/2 1/2 1/2 }
-b+1/2, -c+1/2, -a+1/2 | -x, -y, -z	{ -1 ‖ -3-111 | 1/2 1/2 1/2 }
b+1/2, -c, a | -x, -y, -z	{ -1 ‖ -3-1-1-1 | 1/2 0 0 }
c, a+1/2, -b | -x, -y, -z	{ -1 ‖ -3+1-1-1 | 0 1/2 0 }
b, c+1/2, -a | -x, -y, -z	{ -1 ‖ -3--11-1 | 0 1/2 0 }
-c, a, b+1/2 | -x, -y, -z	{ -1 ‖ -3+-11-1 | 0 0 1/2 }
-b, c, a+1/2 | -x, -y, -z	{ -1 ‖ -3--1-11 | 0 0 1/2 }
c+1/2, -a, b | -x, -y, -z	{ -1 ‖ -3+-1-11 | 1/2 0 0 }
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, -b, -c | -x, y, z	{ m ‖ -1 | 0 }
-a+1/2, b+1/2, c | -x, y, z	{ m ‖ m100 | 1/2 1/2 0 }
a, -b+1/2, c+1/2 | -x, y, z	{ m ‖ m010 | 0 1/2 1/2 }
a+1/2, b, -c+1/2 | -x, y, z	{ m ‖ m001 | 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+1/2, b+1/2, c+1/2 | x, -y, -z	{ 2 ‖ 1 | 1/2 1/2 1/2 }
a, -b, -c+1/2 | x, -y, -z	{ 2 ‖ 2100 | 0 0 1/2 }
-a+1/2, b, -c | x, -y, -z	{ 2 ‖ 2010 | 1/2 0 0 }
-a, -b+1/2, c | x, -y, -z	{ 2 ‖ 2001 | 0 1/2 0 }
c+1/2, a+1/2, b+1/2 | x, -y, -z	{ 2 ‖ 3+111 | 1/2 1/2 1/2 }
b+1/2, c+1/2, a+1/2 | x, -y, -z	{ 2 ‖ 3-111 | 1/2 1/2 1/2 }
-b+1/2, c, -a | x, -y, -z	{ 2 ‖ 3-1-1-1 | 1/2 0 0 }
-c, -a+1/2, b | x, -y, -z	{ 2 ‖ 3+1-1-1 | 0 1/2 0 }
-b, -c+1/2, a | x, -y, -z	{ 2 ‖ 3--11-1 | 0 1/2 0 }
c, -a, -b+1/2 | x, -y, -z	{ 2 ‖ 3+-11-1 | 0 0 1/2 }
b, -c, -a+1/2 | x, -y, -z	{ 2 ‖ 3--1-11 | 0 0 1/2 }
-c+1/2, a, -b | x, -y, -z	{ 2 ‖ 3+-1-11 | 1/2 0 0 }
-a+1/2, -b+1/2, -c+1/2 | x, -y, -z	{ 2 ‖ -1 | 1/2 1/2 1/2 }
-a, b, c+1/2 | x, -y, -z	{ 2 ‖ m100 | 0 0 1/2 }
a+1/2, -b, c | x, -y, -z	{ 2 ‖ m010 | 1/2 0 0 }
a, b+1/2, -c | x, -y, -z	{ 2 ‖ m001 | 0 1/2 0 }
-c+1/2, -a+1/2, -b+1/2 | x, -y, -z	{ 2 ‖ -3+111 | 1/2 1/2 1/2 }
-b+1/2, -c+1/2, -a+1/2 | x, -y, -z	{ 2 ‖ -3-111 | 1/2 1/2 1/2 }
b+1/2, -c, a | x, -y, -z	{ 2 ‖ -3-1-1-1 | 1/2 0 0 }
c, a+1/2, -b | x, -y, -z	{ 2 ‖ -3+1-1-1 | 0 1/2 0 }
b, c+1/2, -a | x, -y, -z	{ 2 ‖ -3--11-1 | 0 1/2 0 }
-c, a, b+1/2 | x, -y, -z	{ 2 ‖ -3+-11-1 | 0 0 1/2 }
-b, c, a+1/2 | x, -y, -z	{ 2 ‖ -3--1-11 | 0 0 1/2 }
c+1/2, -a, b | x, -y, -z	{ 2 ‖ -3+-1-11 | 1/2 0 0 }

Wyckoff Positions

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

Spin Brillouin Zone

Band Representations (Magnon)

8b

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

16c

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

48e

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