Information of spin space group $F\ce{^{\text{-}1}{2}}\ce{^{\text{-}1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}}$ ( nSSG:5.22.1.1 )

General Positions

The continuously infinite group ^∞1 is omitted.

Coordinates	Seitz symbol
a, b, c | x, y, z	{ 1 ‖ 1 | 0 }
-a, -b, c | x, y, z	{ 1 ‖ 2001 | 0 }
a, -b, -c | -x, -y, -z	{ -1 ‖ 2100 | 0 }
-a, b, -c | -x, -y, -z	{ -1 ‖ 2010 | 0 }
a, b+1/2, c+1/2 | x, y, z	{ 1 ‖ 1 | 0 1/2 1/2 }
-a, -b+1/2, c+1/2 | x, y, z	{ 1 ‖ 2001 | 0 1/2 1/2 }
a, -b+1/2, -c+1/2 | -x, -y, -z	{ -1 ‖ 2100 | 0 1/2 1/2 }
-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 ‖ 1 | 1/2 0 1/2 }
-a+1/2, -b, c+1/2 | x, y, z	{ 1 ‖ 2001 | 1/2 0 1/2 }
a+1/2, -b, -c+1/2 | -x, -y, -z	{ -1 ‖ 2100 | 1/2 0 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 ‖ 1 | 1/2 1/2 0 }
-a+1/2, -b+1/2, c | x, y, z	{ 1 ‖ 2001 | 1/2 1/2 0 }
a+1/2, -b+1/2, -c | -x, -y, -z	{ -1 ‖ 2100 | 1/2 1/2 0 }
-a+1/2, b+1/2, -c | -x, -y, -z	{ -1 ‖ 2010 | 1/2 1/2 0 }
a, b, c | -x, y, z	{ m ‖ 1 | 0 }
-a, -b, c | -x, y, z	{ m ‖ 2001 | 0 }
a, -b, -c | x, -y, -z	{ 2 ‖ 2100 | 0 }
-a, b, -c | x, -y, -z	{ 2 ‖ 2010 | 0 }
a, b+1/2, c+1/2 | -x, y, z	{ m ‖ 1 | 0 1/2 1/2 }
-a, -b+1/2, c+1/2 | -x, y, z	{ m ‖ 2001 | 0 1/2 1/2 }
a, -b+1/2, -c+1/2 | x, -y, -z	{ 2 ‖ 2100 | 0 1/2 1/2 }
-a, b+1/2, -c+1/2 | x, -y, -z	{ 2 ‖ 2010 | 0 1/2 1/2 }
a+1/2, b, c+1/2 | -x, y, z	{ m ‖ 1 | 1/2 0 1/2 }
-a+1/2, -b, c+1/2 | -x, y, z	{ m ‖ 2001 | 1/2 0 1/2 }
a+1/2, -b, -c+1/2 | x, -y, -z	{ 2 ‖ 2100 | 1/2 0 1/2 }
-a+1/2, b, -c+1/2 | x, -y, -z	{ 2 ‖ 2010 | 1/2 0 1/2 }
a+1/2, b+1/2, c | -x, y, z	{ m ‖ 1 | 1/2 1/2 0 }
-a+1/2, -b+1/2, c | -x, y, z	{ m ‖ 2001 | 1/2 1/2 0 }
a+1/2, -b+1/2, -c | x, -y, -z	{ 2 ‖ 2100 | 1/2 1/2 0 }
-a+1/2, b+1/2, -c | x, -y, -z	{ 2 ‖ 2010 | 1/2 1/2 0 }

Wyckoff Positions

WP	Site symmetry	Representative
4a	$\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,0,0 | 0,0,0)
4b	$\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,0,1/2 | 0,0,0)
4c	$\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(1/4,1/4,1/4 | 0,0,0)
4d	$\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(1/4,1/4,3/4 | 0,0,0)
8e	$\ce{^{-1}{2}}..\ce{^{\infty m}{1}} $	(a,0,0 | 0,0,0)
8f	$.\ce{^{-1}{2}}.\ce{^{\infty m}{1}} $	(0,b,0 | 0,0,0)
8g	$..\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,0,c | 0,0,z)
8h	$..\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(1/4,1/4,c | 0,0,z)
8i	$.\ce{^{-1}{2}}.\ce{^{\infty m}{1}} $	(1/4,b,1/4 | 0,0,0)
8j	$\ce{^{-1}{2}}..\ce{^{\infty m}{1}} $	(a,1/4,1/4 | 0,0,0)
16k	$\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}{2}}\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $
T:(1,0,0)	$\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $
Y:(0,1,0)	$\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $
Z:(0,0,1)	$\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $
L:(1/2,1/2,1/2)	$\ce{^{1}{1}}\ce{^{\infty m}{1}} $
A:(u,0,1)	$\ce{^{2}{2}}\ce{^{-1}{2}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
B:(0,v,1)	$\ce{^{-1}{2}}\ce{^{2}{2}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
Δ:(0,v,0)	$\ce{^{-1}{2}}\ce{^{2}{2}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
H:(0,1,w)	$\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty}{1}} $
Λ:(0,0,w)	$\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty}{1}} $
Σ:(u,0,0)	$\ce{^{2}{2}}\ce{^{-1}{2}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
E:(0,v,w)	$\ce{^{-1}{2}}\ce{^{\infty}{1}} $
J:(u,0,w)	$\ce{^{-1}{2}}\ce{^{\infty}{1}} $
M:(u,v,0)	$\ce{^{m}{2}}\ce{^{\infty}{1}} $
GP:(u,v,w)	$\ce{^{1}{1}}\ce{^{\infty}{1}} $

Spin Brillouin Zone

Band Representations (Magnon)

8g

k-vector	k-vector-G↑	A↑G(2)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2) $
L:(1/2,1/2,-1/2)	V:(1/2,1/2,0)	$2 (L)V_{1}^{S}(1)$	Spin Splitting
T:(0,1,-1)	M:(0,1,1/2)	$(T)M_{1}^{S}(2) $
Y:(0,1,0)	A:(0,0,1/2)	$(Y)A_{1}^{S}(2) $
Z:(0,0,-1)	Y:(0,1,0)	$(Z)Y_{1}^{S}(2) $
A:(1+u,1,0)	B:(1+u,0,-u/2)	$(A)B_{1}^{S}(2) $
B:(1,1+v,0)	B:(1,0,v/2)	$B_{1}^{S}(2) $
Δ:(0,v,0)	B:(0,0,v/2)	$(Δ)B_{1}^{S}(2) $
H:(0,1,w)	U:(0,-w,1/2)	$(H)U_{1}^{S}(2) $
Λ:(0,0,w)	Λ:(0,-w,0)	$Λ_{1}^{S}(2) $
Σ:(u,0,0)	B:(u,0,-u/2)	$(Σ)B_{1}^{S}(2) $
E:(0,v,w)	GP:(0,-w,v/2)	$(E)GP_{1}^{S}(2) $
J:(u,0,w)	GP:(u,-w,-u/2)	$(J)GP_{1}^{S}(2) $
M:(u,v,0)	B:(u,0,-u/2+v/2)	$2(M)B_{1}^{S}(1)$	Spin Splitting
GP:(u,v,w)	GP:(u,-w,-u/2+v/2)	$2GP_{1}^{S}(1)$	Spin Splitting

8h

k-vector	k-vector-G↑	A↑G(2)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2) $
L:(1/2,1/2,-1/2)	V:(1/2,1/2,0)	$2 (L)V_{1}^{S}(1)$	Spin Splitting
T:(0,1,-1)	M:(0,1,1/2)	$(T)M_{2}^{S}(2) $
Y:(0,1,0)	A:(0,0,1/2)	$(Y)A_{2}^{S}(2) $
Z:(0,0,-1)	Y:(0,1,0)	$(Z)Y_{1}^{S}(2) $
A:(1+u,1,0)	B:(1+u,0,-u/2)	$(A)B_{1}^{S}(2) $
B:(1,1+v,0)	B:(1,0,v/2)	$B_{1}^{S}(2) $
Δ:(0,v,0)	B:(0,0,v/2)	$(Δ)B_{1}^{S}(2) $
H:(0,1,w)	U:(0,-w,1/2)	$(H)U_{2}^{S}(2) $
Λ:(0,0,w)	Λ:(0,-w,0)	$Λ_{1}^{S}(2) $
Σ:(u,0,0)	B:(u,0,-u/2)	$(Σ)B_{1}^{S}(2) $
E:(0,v,w)	GP:(0,-w,v/2)	$(E)GP_{1}^{S}(2) $
J:(u,0,w)	GP:(u,-w,-u/2)	$(J)GP_{1}^{S}(2) $
M:(u,v,0)	B:(u,0,-u/2+v/2)	$2(M)B_{1}^{S}(1)$	Spin Splitting
GP:(u,v,w)	GP:(u,-w,-u/2+v/2)	$2GP_{1}^{S}(1)$	Spin Splitting

16k

k-vector	k-vector-G↑	A↑G(4)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{2}^{S}(2) $
L:(1/2,1/2,-1/2)	V:(1/2,1/2,0)	$4(L)V_{1}^{S}(1)$	Spin Splitting
T:(0,1,-1)	M:(0,1,1/2)	$(T)M_{1}^{S}(2)⊕(T)M_{2}^{S}(2) $
Y:(0,1,0)	A:(0,0,1/2)	$(Y)A_{1}^{S}(2)⊕(Y)A_{2}^{S}(2) $
Z:(0,0,-1)	Y:(0,1,0)	$(Z)Y_{1}^{S}(2)⊕(Z)Y_{2}^{S}(2) $
A:(1+u,1,0)	B:(1+u,0,-u/2)	$2(A)B_{1}^{S}(2) $
B:(1,1+v,0)	B:(1,0,v/2)	$2B_{1}^{S}(2) $
Δ:(0,v,0)	B:(0,0,v/2)	$2(Δ)B_{1}^{S}(2) $
H:(0,1,w)	U:(0,-w,1/2)	$(H)U_{1}^{S}(2)⊕(H)U_{2}^{S}(2) $
Λ:(0,0,w)	Λ:(0,-w,0)	$Λ_{1}^{S}(2)⊕Λ_{2}^{S}(2) $
Σ:(u,0,0)	B:(u,0,-u/2)	$2(Σ)B_{1}^{S}(2) $
E:(0,v,w)	GP:(0,-w,v/2)	$2(E)GP_{1}^{S}(2) $
J:(u,0,w)	GP:(u,-w,-u/2)	$2(J)GP_{1}^{S}(2) $
M:(u,v,0)	B:(u,0,-u/2+v/2)	$4(M)B_{1}^{S}(1)$	Spin Splitting
GP:(u,v,w)	GP:(u,-w,-u/2+v/2)	$4GP_{1}^{S}(1)$	Spin Splitting