Information of spin space group $F\ce{^{\text{-}1}{m}}\ce{^{\text{-}1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}}$ ( nSSG:12.69.1.1 )

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

Wyckoff Positions

WP	Site symmetry	Representative
4a	$\ce{^{-1}{m}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $	(0,0,0 | 0,0,0)
4b	$\ce{^{-1}{m}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $	(0,0,1/2 | 0,0,0)
8c	$\ce{^{-1}{2}}/\ce{^{-1}{m}}..\ce{^{\infty m}{1}} $	(0,1/4,1/4 | 0,0,0)
8d	$.\ce{^{-1}{2}}/\ce{^{-1}{m}}.\ce{^{\infty m}{1}} $	(1/4,0,1/4 | 0,0,0)
8e	$..\ce{^{1}{2}}/\ce{^{1}{m}}\ce{^{\infty m}{1}} $	(1/4,1/4,0 | 0,0,z)
8f	$\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(1/4,1/4,1/4 | 0,0,0)
8g	$\ce{^{-1}{2}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $	(a,0,0 | 0,0,0)
8h	$\ce{^{-1}{m}}\ce{^{-1}{2}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $	(0,b,0 | 0,0,0)
8i	$\ce{^{-1}{m}}\ce{^{-1}{m}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,0,c | 0,0,0)
16j	$..\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(1/4,1/4,c | 0,0,z)
16k	$.\ce{^{-1}{2}}.\ce{^{\infty m}{1}} $	(1/4,b,1/4 | 0,0,0)
16l	$\ce{^{-1}{2}}..\ce{^{\infty m}{1}} $	(a,1/4,1/4 | 0,0,0)
16m	$\ce{^{-1}{m}}..\ce{^{\infty m}{1}} $	(0,b,c | 0,0,0)
16n	$.\ce{^{-1}{m}}.\ce{^{\infty m}{1}} $	(a,0,c | 0,0,0)
16o	$..\ce{^{1}{m}}\ce{^{\infty m}{1}} $	(a,b,0 | 0,0,z)
32p	$\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}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $
T:(1,0,0)	$\ce{^{-1}{m}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $
Y:(0,1,0)	$\ce{^{-1}{m}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $
Z:(0,0,1)	$\ce{^{-1}{m}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}} $
L:(1/2,1/2,1/2)	$\ce{^{1}{-1}}\ce{^{\infty m}{1}} $
A:(u,0,1)	$\ce{^{-1}{m}}\ce{^{2}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
B:(0,v,1)	$\ce{^{2}{m}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
Δ:(0,v,0)	$\ce{^{2}{m}}\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
H:(0,1,w)	$\ce{^{2}{m}}\ce{^{2}{m}}\ce{^{m}{m}}\ce{^{\infty}{1}} $
Λ:(0,0,w)	$ \ce{^{2}{m}}\ce{^{2}{m}}\ce{^{m}{m}}\ce{^{\infty}{1}} $
Σ:(u,0,0)	$\ce{^{-1}{m}}\ce{^{2}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
E:(0,v,w)	$\ce{^{-1}{2/}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
J:(u,0,w)	$\ce{^{-1}{2/}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
M:(u,v,0)	$\ce{^{m}{2/}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
GP:(u,v,w)	$\ce{^{m}{-1}}\ce{^{\infty}{1}} $

Spin Brillouin Zone

Band Representations (Magnon)

8e

k-vector	k-vector-G↑	Ag↑G(2)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S,+}(2) $
L:(1/2,1/2,1/2)	L:(1/2,1/2,1/2)	$2 L_{1}^{S,+}(1)$	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

16j

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

16o

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)	L:(1/2,1/2,1/2)	$2 L_{1}^{S,+}(1)⊕2 L_{1}^{S,-}(1)$	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)	$2 B_{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

32p

k-vector	k-vector-G↑	A↑G(8)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S,+}(2)⊕Γ_{1}^{S,-}(2)⊕Γ_{2}^{S,+}(2)⊕Γ_{2}^{S,-}(2) $
L:(1/2,1/2,1/2)	L:(1/2,1/2,1/2)	$4L_{1}^{S,+}(1)⊕4L_{1}^{S,-}(1)$	Spin Splitting
T:(0,-1,1)	M:(0,1,1/2)	$(T)M_{1}^{S,+}(2)⊕(T)M_{1}^{S,-}(2)⊕(T)M_{2}^{S,+}(2)⊕(T)M_{2}^{S,-}(2) $
Y:(0,-1,0)	A:(0,0,1/2)	$(Y)A_{1}^{S,+}(2)⊕(Y)A_{1}^{S,-}(2)⊕(Y)A_{2}^{S,+}(2)⊕(Y)A_{2}^{S,-}(2) $
Z:(0,0,1)	Y:(0,1,0)	$(Z)Y_{1}^{S,+}(2)⊕(Z)Y_{1}^{S,-}(2)⊕(Z)Y_{2}^{S,+}(2)⊕(Z)Y_{2}^{S,-}(2) $
A:(1+u,-1,0)	B:(1+u,0,-u/2)	$2(A)B_{1}^{S}(2)⊕2(A)B_{2}^{S}(2) $
B:(1,-1+v,0)	B:(1,0,-v/2)	$2B_{1}^{S}(2)⊕2B_{2}^{S}(2) $
Δ:(0,v,0)	B:(0,0,-v/2)	$2(Δ)B_{1}^{S}(2)⊕2(Δ)B_{2}^{S}(2) $
H:(0,-1,w)	U:(0,w,1/2)	$2(H)U_{1}^{S}(2)⊕2(H)U_{2}^{S}(2) $
Λ:(0,0,w)	Λ:(0,w,0)	$2Λ_{1}^{S}(2)⊕2Λ_{2}^{S}(2) $
Σ:(u,0,0)	B:(u,0,-u/2)	$2(Σ)B_{1}^{S}(2)⊕2(Σ)B_{2}^{S}(2) $
E:(0,v,w)	GP:(0,w,-v/2)	$4(E)GP_{1}^{S}(2) $
J:(u,0,w)	GP:(u,w,-u/2)	$4(J)GP_{1}^{S}(2) $
M:(u,v,0)	B:(u,0,-u/2-v/2)	$4(M)B_{1}^{S}(1)⊕4(M)B_{2}^{S}(1)$	Spin Splitting
GP:(u,v,w)	GP:(u,w,-u/2-v/2)	$8GP_{1}^{S}(1)$	Spin Splitting