Information of spin space group $F\ce{^{\text{-}1}{m}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{\infty m}{1}}$ ( nSSG:42.69.1.1 )

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 ‖ 2100 | 0 }
a, -b, c | x, y, z	{ 1 ‖ m010 | 0 }
a, b, -c | x, y, z	{ 1 ‖ m001 | 0 }
-a, b, -c | -x, -y, -z	{ -1 ‖ 2010 | 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 ‖ m100 | 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 ‖ 2100 | 0 1/2 1/2 }
a, -b+1/2, c+1/2 | x, y, z	{ 1 ‖ m010 | 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 ‖ 2010 | 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 ‖ m100 | 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 ‖ 2100 | 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, -c+1/2 | x, y, z	{ 1 ‖ m001 | 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 ‖ 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 ‖ m100 | 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 ‖ 2100 | 1/2 1/2 0 }
a+1/2, -b+1/2, c | x, y, z	{ 1 ‖ m010 | 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 ‖ 2010 | 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 ‖ m100 | 1/2 1/2 0 }
a, b, c | -x, y, z	{ m ‖ 1 | 0 }
a, -b, -c | -x, y, z	{ m ‖ 2100 | 0 }
a, -b, c | -x, y, z	{ m ‖ m010 | 0 }
a, b, -c | -x, y, z	{ m ‖ m001 | 0 }
-a, b, -c | x, -y, -z	{ 2 ‖ 2010 | 0 }
-a, -b, c | x, -y, -z	{ 2 ‖ 2001 | 0 }
-a, -b, -c | x, -y, -z	{ 2 ‖ -1 | 0 }
-a, b, c | x, -y, -z	{ 2 ‖ m100 | 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 ‖ 2100 | 0 1/2 1/2 }
a, -b+1/2, c+1/2 | -x, y, z	{ m ‖ m010 | 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 ‖ 2010 | 0 1/2 1/2 }
-a, -b+1/2, c+1/2 | x, -y, -z	{ 2 ‖ 2001 | 0 1/2 1/2 }
-a, -b+1/2, -c+1/2 | x, -y, -z	{ 2 ‖ -1 | 0 1/2 1/2 }
-a, b+1/2, c+1/2 | x, -y, -z	{ 2 ‖ m100 | 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 ‖ 2100 | 1/2 0 1/2 }
a+1/2, -b, c+1/2 | -x, y, z	{ m ‖ m010 | 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 ‖ 2010 | 1/2 0 1/2 }
-a+1/2, -b, c+1/2 | x, -y, -z	{ 2 ‖ 2001 | 1/2 0 1/2 }
-a+1/2, -b, -c+1/2 | x, -y, -z	{ 2 ‖ -1 | 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+1/2, c | -x, y, z	{ m ‖ 1 | 1/2 1/2 0 }
a+1/2, -b+1/2, -c | -x, y, z	{ m ‖ 2100 | 1/2 1/2 0 }
a+1/2, -b+1/2, c | -x, y, z	{ m ‖ m010 | 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 ‖ 2010 | 1/2 1/2 0 }
-a+1/2, -b+1/2, c | x, -y, -z	{ 2 ‖ 2001 | 1/2 1/2 0 }
-a+1/2, -b+1/2, -c | x, -y, -z	{ 2 ‖ -1 | 1/2 1/2 0 }
-a+1/2, b+1/2, c | x, -y, -z	{ 2 ‖ m100 | 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 | 0,0,0)
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,z)
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 | 0,0,0)
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,z)
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,z)
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{^{2}{-1}}\ce{^{\infty m}{1}} $
A:(u,0,1)	$\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
B:(0,v,1)	$\ce{^{2}{m}}\ce{^{m}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
Δ:(0,v,0)	$\ce{^{2}{m}}\ce{^{m}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
H:(0,1,w)	$\ce{^{2}{m}}\ce{^{1}{m}}\ce{^{m}{m}}\ce{^{\infty}{1}} $
Λ:(0,0,w)	$ \ce{^{2}{m}}\ce{^{1}{m}}\ce{^{m}{m}}\ce{^{\infty}{1}} $
Σ:(u,0,0)	$\ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
E:(0,v,w)	$\ce{^{m}{2/}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
J:(u,0,w)	$\ce{^{-1}{2/}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
M:(u,v,0)	$\ce{^{-1}{2/}}\ce{^{1}{m}}\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↑	A1↑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)	$L_{1}^{S}(2) $
T:(-1,0,0)	Z:(0,0,1)	$(T)Z_{1}^{S}(2) $
Y:(0,1,0)	Y:(0,1,0)	$Y_{1}^{S}(2) $
Z:(0,0,1)	T:(1,0,0)	$(Z)T_{1}^{S}(2) $
A:(-1+u,1,0)	H:(0,1,1-u)	$(A)H_{1}^{S}(2) $
B:(-1,1+v,0)	B:(0,1+v,1)	$B_{1}^{S}(2) $
Δ:(0,v,0)	Δ:(0,v,0)	$Δ_{1}^{S}(2) $
H:(-1,0,1+w)	A:(1+w,0,1)	$(H)A_{1}^{S}(2) $
Λ:(0,0,w)	Σ:(w,0,0)	$(Λ)Σ_{1}^{S}(2) $
Σ:(u,0,0)	Λ:(0,0,-u)	$(Σ)Λ_{1}^{S}(2) $
E:(0,v,w)	M:(w,v,0)	$(E)M_{1}^{S}(2) $
J:(u,0,w)	J:(w,0,-u)	$J_{1}^{S}(2) $
M:(u,v,0)	E:(0,v,-u)	$(M)E_{1}^{S}(2) $
GP:(u,v,w)	GP:(w,v,-u)	$GP_{1}^{S}(2) $

16l

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)	$2L_{1}^{S}(2) $
T:(-1,0,0)	Z:(0,0,1)	$(T)Z_{1}^{S}(2)⊕(T)Z_{2}^{S}(2) $
Y:(0,1,0)	Y:(0,1,0)	$Y_{3}^{S}(2)⊕Y_{4}^{S}(2) $
Z:(0,0,1)	T:(1,0,0)	$(Z)T_{3}^{S}(2)⊕(Z)T_{4}^{S}(2) $
A:(-1+u,1,0)	H:(0,1,1-u)	$(A)H_{3}^{S}(2)⊕(A)H_{4}^{S}(2) $
B:(-1,1+v,0)	B:(0,1+v,1)	$B_{1}^{S}(2)⊕B_{2}^{S}(2) $
Δ:(0,v,0)	Δ:(0,v,0)	$Δ_{1}^{S}(2)⊕Δ_{2}^{S}(2) $
H:(-1,0,1+w)	A:(1+w,0,1)	$(H)A_{1}^{S}(2)⊕(H)A_{2}^{S}(2) $
Λ:(0,0,w)	Σ:(w,0,0)	$(Λ)Σ_{1}^{S}(2)⊕(Λ)Σ_{2}^{S}(2) $
Σ:(u,0,0)	Λ:(0,0,-u)	$(Σ)Λ_{1}^{S}(2)⊕(Σ)Λ_{2}^{S}(2) $
E:(0,v,w)	M:(w,v,0)	$2(E)M_{1}^{S}(2) $
J:(u,0,w)	J:(w,0,-u)	$J_{1}^{S}(2)⊕J_{2}^{S}(2) $
M:(u,v,0)	E:(0,v,-u)	$(M)E_{1}^{S}(2)⊕(M)E_{2}^{S}(2) $
GP:(u,v,w)	GP:(w,v,-u)	$2GP_{1}^{S}(2) $

16n

k-vector	k-vector-G↑	A'↑G(4)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{4}^{S}(2) $
L:(-1/2,1/2,1/2)	L:(1/2,1/2,1/2)	$2L_{1}^{S}(2) $
T:(-1,0,0)	Z:(0,0,1)	$(T)Z_{1}^{S}(2)⊕(T)Z_{4}^{S}(2) $
Y:(0,1,0)	Y:(0,1,0)	$Y_{1}^{S}(2)⊕Y_{4}^{S}(2) $
Z:(0,0,1)	T:(1,0,0)	$(Z)T_{1}^{S}(2)⊕(Z)T_{4}^{S}(2) $
A:(-1+u,1,0)	H:(0,1,1-u)	$(A)H_{1}^{S}(2)⊕(A)H_{4}^{S}(2) $
B:(-1,1+v,0)	B:(0,1+v,1)	$B_{1}^{S}(2)⊕B_{2}^{S}(2) $
Δ:(0,v,0)	Δ:(0,v,0)	$Δ_{1}^{S}(2)⊕Δ_{2}^{S}(2) $
H:(-1,0,1+w)	A:(1+w,0,1)	$2(H)A_{1}^{S}(2) $
Λ:(0,0,w)	Σ:(w,0,0)	$2(Λ)Σ_{1}^{S}(2) $
Σ:(u,0,0)	Λ:(0,0,-u)	$(Σ)Λ_{1}^{S}(2)⊕(Σ)Λ_{4}^{S}(2) $
E:(0,v,w)	M:(w,v,0)	$2(E)M_{1}^{S}(2) $
J:(u,0,w)	J:(w,0,-u)	$2J_{1}^{S}(2) $
M:(u,v,0)	E:(0,v,-u)	$(M)E_{1}^{S}(2)⊕(M)E_{2}^{S}(2) $
GP:(u,v,w)	GP:(w,v,-u)	$2GP_{1}^{S}(2) $

16o

k-vector	k-vector-G↑	A'↑G(4)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{3}^{S}(2) $
L:(-1/2,1/2,1/2)	L:(1/2,1/2,1/2)	$2L_{1}^{S}(2) $
T:(-1,0,0)	Z:(0,0,1)	$(T)Z_{1}^{S}(2)⊕(T)Z_{3}^{S}(2) $
Y:(0,1,0)	Y:(0,1,0)	$Y_{1}^{S}(2)⊕Y_{3}^{S}(2) $
Z:(0,0,1)	T:(1,0,0)	$(Z)T_{1}^{S}(2)⊕(Z)T_{3}^{S}(2) $
A:(-1+u,1,0)	H:(0,1,1-u)	$(A)H_{1}^{S}(2)⊕(A)H_{3}^{S}(2) $
B:(-1,1+v,0)	B:(0,1+v,1)	$2B_{1}^{S}(2) $
Δ:(0,v,0)	Δ:(0,v,0)	$2Δ_{1}^{S}(2) $
H:(-1,0,1+w)	A:(1+w,0,1)	$(H)A_{1}^{S}(2)⊕(H)A_{2}^{S}(2) $
Λ:(0,0,w)	Σ:(w,0,0)	$(Λ)Σ_{1}^{S}(2)⊕(Λ)Σ_{2}^{S}(2) $
Σ:(u,0,0)	Λ:(0,0,-u)	$(Σ)Λ_{1}^{S}(2)⊕(Σ)Λ_{3}^{S}(2) $
E:(0,v,w)	M:(w,v,0)	$2(E)M_{1}^{S}(2) $
J:(u,0,w)	J:(w,0,-u)	$J_{1}^{S}(2)⊕J_{2}^{S}(2) $
M:(u,v,0)	E:(0,v,-u)	$2(M)E_{1}^{S}(2) $
GP:(u,v,w)	GP:(w,v,-u)	$2GP_{1}^{S}(2) $

32p

k-vector	k-vector-G↑	A↑G(8)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{2}^{S}(2)⊕Γ_{3}^{S}(2)⊕Γ_{4}^{S}(2) $
L:(-1/2,1/2,1/2)	L:(1/2,1/2,1/2)	$4L_{1}^{S}(2) $
T:(-1,0,0)	Z:(0,0,1)	$(T)Z_{1}^{S}(2)⊕(T)Z_{2}^{S}(2)⊕(T)Z_{3}^{S}(2)⊕(T)Z_{4}^{S}(2) $
Y:(0,1,0)	Y:(0,1,0)	$Y_{1}^{S}(2)⊕Y_{2}^{S}(2)⊕Y_{3}^{S}(2)⊕Y_{4}^{S}(2) $
Z:(0,0,1)	T:(1,0,0)	$(Z)T_{1}^{S}(2)⊕(Z)T_{2}^{S}(2)⊕(Z)T_{3}^{S}(2)⊕(Z)T_{4}^{S}(2) $
A:(-1+u,1,0)	H:(0,1,1-u)	$(A)H_{1}^{S}(2)⊕(A)H_{2}^{S}(2)⊕(A)H_{3}^{S}(2)⊕(A)H_{4}^{S}(2) $
B:(-1,1+v,0)	B:(0,1+v,1)	$2B_{1}^{S}(2)⊕2B_{2}^{S}(2) $
Δ:(0,v,0)	Δ:(0,v,0)	$2Δ_{1}^{S}(2)⊕2Δ_{2}^{S}(2) $
H:(-1,0,1+w)	A:(1+w,0,1)	$2(H)A_{1}^{S}(2)⊕2(H)A_{2}^{S}(2) $
Λ:(0,0,w)	Σ:(w,0,0)	$2(Λ)Σ_{1}^{S}(2)⊕2(Λ)Σ_{2}^{S}(2) $
Σ:(u,0,0)	Λ:(0,0,-u)	$(Σ)Λ_{1}^{S}(2)⊕(Σ)Λ_{2}^{S}(2)⊕(Σ)Λ_{3}^{S}(2)⊕(Σ)Λ_{4}^{S}(2) $
E:(0,v,w)	M:(w,v,0)	$4(E)M_{1}^{S}(2) $
J:(u,0,w)	J:(w,0,-u)	$2J_{1}^{S}(2)⊕2J_{2}^{S}(2) $
M:(u,v,0)	E:(0,v,-u)	$2(M)E_{1}^{S}(2)⊕2(M)E_{2}^{S}(2) $
GP:(u,v,w)	GP:(w,v,-u)	$4GP_{1}^{S}(2) $