Information of spin space group $C\ce{^{1}{c}}\ce{^{\text{-}1}{c}}\ce{^{\text{-}1}{e}}\ce{^{\infty m}{1}}$ ( nSSG:15.68.1.1 )

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

Wyckoff Positions

WP	Site symmetry	Representative
4a	$\ce{^{1}{2}}\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{\infty m}{1}} $	(0,1/4,1/4 | 0,0,0)
4b	$\ce{^{1}{2}}\ce{^{-1}{2}}\ce{^{-1}{2}}\ce{^{\infty m}{1}} $	(0,1/4,3/4 | 0,0,0)
8c	$\ce{^{1}{-1}}\ce{^{\infty m}{1}} $	(1/4,3/4,0 | 0,0,z)
8d	$\ce{^{1}{-1}}\ce{^{\infty m}{1}} $	(0,0,0 | 0,0,z)
8e	$\ce{^{1}{2}}..\ce{^{\infty m}{1}} $	(a,1/4,1/4 | 0,0,z)
8f	$.\ce{^{-1}{2}}.\ce{^{\infty m}{1}} $	(0,b,1/4 | 0,0,0)
8g	$..\ce{^{-1}{2}}\ce{^{\infty m}{1}} $	(0,1/4,c | 0,0,0)
8h	$..\ce{^{-1}{2}}\ce{^{\infty m}{1}} $	(1/4,0,c | 0,0,0)
16i	$\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:(1,0,1/2)	$\ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
Y:(1,0,0)	$\ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
Z:(0,0,1/2)	$\ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
R:(1/2,1/2,1/2)	$\ce{^{2}{2/}}\ce{^{2}{m}}\ce{^{\infty m}{1}} $
S:(1/2,1/2,0)	$\ce{^{2}{2/}}\ce{^{2}{m}}\ce{^{\infty m}{1}} $
A:(u,0,1/2)	$\ce{^{m}{m}}\ce{^{2}{m}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
B:(0,v,1/2)	$\ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
Δ:(0,v,0)	$\ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
H:(1,0,w)	$\ce{^{1}{m}}\ce{^{2}{m}}\ce{^{-1}{m}}\ce{^{\infty}{1}} $
Λ:(0,0,w)	$ \ce{^{1}{m}}\ce{^{2}{m}}\ce{^{-1}{m}}\ce{^{\infty}{1}} $
Σ:(u,0,0)	$\ce{^{m}{m}}\ce{^{2}{m}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
D:(1/2,1/2,w)	$\ce{^{2}{2/}}\ce{^{-1}{m}}\ce{^{\infty}{1}} $
K:(0,v,w)	$\ce{^{m}{2/}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
M:(u,0,w)	$\ce{^{-1}{2/}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
P:(u,v,0)	$\ce{^{-1}{2/}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
Q:(u,v,1/2)	$\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)

8c

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

8d

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

8e

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

16i

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) $
R:(-1/2,1/2,1/2)	L:(1/2,1/2,1/2)	$2(R)L_{1}^{S,+}(2)⊕2(R)L_{1}^{S,-}(2) $
S:(-1/2,1/2,0)	V:(1/2,1/2,0)	$2(S)V_{1}^{S,+}(1)⊕2(S)V_{1}^{S,-}(1) $
T:(-1,0,1/2)	M:(0,1,1/2)	$2(T)M_{1}^{S}(4) $
Y:(-1,0,0)	Y:(0,1,0)	$Y_{1}^{S,+}(2)⊕Y_{1}^{S,-}(2)⊕Y_{2}^{S,+}(2)⊕Y_{2}^{S,-}(2) $
Z:(0,0,1/2)	A:(0,0,1/2)	$2(Z)A_{1}^{S}(4) $
A:(u,0,1/2)	U:(0,-u,1/2)	$2(A)U_{1}^{S}U_{2}^{S}(4) $
B:(0,v,1/2)	B:(v,0,1/2)	$2B_{1}^{S}(2)⊕2B_{2}^{S}(2) $
D:(1/2,1/2,w)	GP:(1/2,-1/2,w)	$4(D)GP_{1}^{S}(2) $
Δ:(0,v,0)	B:(v,0,0)	$2(Δ)B_{1}^{S}(2)⊕2(Δ)B_{2}^{S}(2) $
H:(0,1,w)	B:(1,0,w)	$2(H)B_{1}^{S}(2)⊕2(H)B_{2}^{S}(2) $
Λ:(0,0,w)	B:(0,0,w)	$2(Λ)B_{1}^{S}(2)⊕2(Λ)B_{2}^{S}(2) $
Σ:(u,0,0)	Λ:(0,-u,0)	$2(Σ)Λ_{1}^{S}(2)⊕2(Σ)Λ_{2}^{S}(2) $
K:(0,v,w)	B:(v,0,w)	$4(K)B_{1}^{S}(1)⊕4(K)B_{2}^{S}(1)$	Spin Splitting
M:(u,0,w)	GP:(0,-u,w)	$4(M)GP_{1}^{S}(2) $
P:(u,v,0)	GP:(v,-u,0)	$4(P)GP_{1}^{S}(2) $
Q:(u,v,1/2)	GP:(v,-u,1/2)	$4(Q)GP_{1}^{S}(2) $
GP:(u,v,w)	GP:(v,-u,w)	$8GP_{1}^{S}(1)$	Spin Splitting