Information of spin space group $P\ce{^{\text{-}1}{n}}\ce{^{1}{m}}\ce{^{\text{-}1}{a}}\ce{^{\infty m}{1}}$ ( nSSG:11.62.1.1 )

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

Wyckoff Positions

WP	Site symmetry	Representative
4a	$\ce{^{1}{-1}}\ce{^{\infty m}{1}} $	(0,0,0 | 0,0,z)
4b	$\ce{^{1}{-1}}\ce{^{\infty m}{1}} $	(0,0,1/2 | 0,0,z)
4c	$.\ce{^{1}{m}}.\ce{^{\infty m}{1}} $	(a,1/4,c | 0,0,z)
8d	$\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}} $
R:(1/2,1/2,1/2)	$ \ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
S:(1/2,1/2,0)	$ \ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
T:(0,1/2,1/2)	$ \ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
U:(1/2,0,1/2)	$ \ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
X:(1/2,0,0)	$ \ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{\infty m}{1}} $
Y:(0,1/2,0)	$ \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}} $
A:(u,0,1/2)	$ \ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
B:(0,v,1/2)	$ \ce{^{2}{m}}\ce{^{m}{m}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
C:(u,1/2,0)	$ \ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
D:(1/2,v,0)	$ \ce{^{2}{m}}\ce{^{m}{m}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
Δ:(0,v,0)	$ \ce{^{2}{m}}\ce{^{m}{m}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
E:(u,1/2,1/2)	$ \ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
G:(1/2,0,w)	$ \ce{^{2}{m}}\ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{\infty}{1}} $
H:(0,1/2,w)	$ \ce{^{2}{m}}\ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{\infty}{1}} $
Λ:(0,0,w)	$ \ce{^{2}{m}}\ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{\infty}{1}} $
P:(1/2,v,1/2)	$ \ce{^{2}{m}}\ce{^{m}{m}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
Q:(1/2,1/2,w)	$ \ce{^{2}{m}}\ce{^{1}{m}}\ce{^{-1}{m}}\ce{^{\infty}{1}} $
Σ:(u,0,0)	$ \ce{^{-1}{m}}\ce{^{1}{m}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
K:(0,v,w)	$ \ce{^{-1}{2/}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
L:(1/2,v,w)	$ \ce{^{-1}{2/}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
M:(u,0,w)	$ \ce{^{m}{2/}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
N:(u,1/2,w)	$ \ce{^{m}{2/}}\ce{^{1}{m}}\ce{^{\infty}{1}} $
V:(u,v,0)	$ \ce{^{-1}{2/}}\ce{^{2}{m}}\ce{^{\infty}{1}} $
W:(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)

4a

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)	E:(1/2,1/2,1/2)	$(R)E_{1}^{S}(4) $
S:(1/2,1/2,0)	C:(1/2,1/2,0)	$(S)C_{1}^{S}(4) $
T:(0,1/2,1/2)	D:(0,1/2,1/2)	$(T)D_{1}^{S}(4) $
U:(1/2,0,1/2)	A:(1/2,0,1/2)	$(U)A_{1}^{S,+}(2)⊕(U)A_{2}^{S,+}(2) $
X:(1/2,0,0)	Y:(1/2,0,0)	$(X)Y_{1}^{S,+}(2)⊕(X)Y_{2}^{S,+}(2) $
Y:(0,1/2,0)	Z:(0,1/2,0)	$(Y)Z_{1}^{S}(4) $
Z:(0,0,1/2)	B:(0,0,1/2)	$(Z)B_{1}^{S,+}(2)⊕(Z)B_{2}^{S,+}(2) $
A:(u,0,1/2)	F:(u,0,1/2)	$(A)F_{1}^{S}(2)⊕(A)F_{2}^{S}(2) $
B:(0,v,1/2)	V:(0,v,1/2)	$(B)V_{1}^{S}(2)⊕(B)V_{2}^{S}(2) $
C:(u,1/2,0)	G:(u,1/2,0)	$(C)G_{1}^{S}G_{2}^{S}(4) $
D:(1/2,v,0)	W:(1/2,v,0)	$(D)W_{1}^{S}(2)⊕(D)W_{2}^{S}(2) $
Δ:(0,v,0)	Λ:(0,v,0)	$(Δ)Λ_{1}^{S}(2)⊕(Δ)Λ_{2}^{S}(2) $
E:(u,1/2,1/2)	G:(u,1/2,1/2)	$(E)G_{1}^{S}G_{2}^{S}(4) $
G:(1/2,0,w)	F:(1/2,0,w)	$(G)F_{1}^{S}(2)⊕(G)F_{2}^{S}(2) $
H:(0,1/2,w)	G:(0,1/2,w)	$(H)G_{1}^{S}G_{2}^{S}(4) $
Λ:(0,0,w)	F:(0,0,w)	$(Λ)F_{1}^{S}(2)⊕(Λ)F_{2}^{S}(2) $
P:(1/2,v,1/2)	U:(1/2,v,1/2)	$(P)U_{1}^{S}(2)⊕(P)U_{2}^{S}(2) $
Q:(1/2,1/2,w)	G:(1/2,1/2,w)	$(Q)G_{1}^{S}G_{2}^{S}(4) $
Σ:(u,0,0)	F:(u,0,0)	$(Σ)F_{1}^{S}(2)⊕(Σ)F_{2}^{S}(2) $
K:(0,v,w)	GP:(0,v,w)	$2(K)GP_{1}^{S}(2) $
L:(1/2,v,w)	GP:(1/2,v,w)	$2(L)GP_{1}^{S}(2) $
M:(u,0,w)	F:(u,0,w)	$2(M)F_{1}^{S}(1)⊕2(M)F_{2}^{S}(1)$	Spin Splitting
N:(u,1/2,w)	G:(u,1/2,w)	$2(N)G_{1}^{S}G_{2}^{S}(2)$	Spin Splitting
V:(u,v,0)	GP:(u,v,0)	$2(V)GP_{1}^{S}(2) $
W:(u,v,1/2)	GP:(u,v,1/2)	$2(W)GP_{1}^{S}(2) $
GP:(u,v,w)	GP:(u,v,w)	$4GP_{1}^{S}(1)$	Spin Splitting

4b

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)	E:(1/2,1/2,1/2)	$(R)E_{1}^{S}(4) $
S:(1/2,1/2,0)	C:(1/2,1/2,0)	$(S)C_{1}^{S}(4) $
T:(0,1/2,1/2)	D:(0,1/2,1/2)	$(T)D_{1}^{S}(4) $
U:(1/2,0,1/2)	A:(1/2,0,1/2)	$(U)A_{1}^{S,-}(2)⊕(U)A_{2}^{S,-}(2) $
X:(1/2,0,0)	Y:(1/2,0,0)	$(X)Y_{1}^{S,+}(2)⊕(X)Y_{2}^{S,+}(2) $
Y:(0,1/2,0)	Z:(0,1/2,0)	$(Y)Z_{1}^{S}(4) $
Z:(0,0,1/2)	B:(0,0,1/2)	$(Z)B_{1}^{S,-}(2)⊕(Z)B_{2}^{S,-}(2) $
A:(u,0,1/2)	F:(u,0,1/2)	$(A)F_{1}^{S}(2)⊕(A)F_{2}^{S}(2) $
B:(0,v,1/2)	V:(0,v,1/2)	$(B)V_{1}^{S}(2)⊕(B)V_{2}^{S}(2) $
C:(u,1/2,0)	G:(u,1/2,0)	$(C)G_{1}^{S}G_{2}^{S}(4) $
D:(1/2,v,0)	W:(1/2,v,0)	$(D)W_{1}^{S}(2)⊕(D)W_{2}^{S}(2) $
Δ:(0,v,0)	Λ:(0,v,0)	$(Δ)Λ_{1}^{S}(2)⊕(Δ)Λ_{2}^{S}(2) $
E:(u,1/2,1/2)	G:(u,1/2,1/2)	$(E)G_{1}^{S}G_{2}^{S}(4) $
G:(1/2,0,w)	F:(1/2,0,w)	$(G)F_{1}^{S}(2)⊕(G)F_{2}^{S}(2) $
H:(0,1/2,w)	G:(0,1/2,w)	$(H)G_{1}^{S}G_{2}^{S}(4) $
Λ:(0,0,w)	F:(0,0,w)	$(Λ)F_{1}^{S}(2)⊕(Λ)F_{2}^{S}(2) $
P:(1/2,v,1/2)	U:(1/2,v,1/2)	$(P)U_{1}^{S}(2)⊕(P)U_{2}^{S}(2) $
Q:(1/2,1/2,w)	G:(1/2,1/2,w)	$(Q)G_{1}^{S}G_{2}^{S}(4) $
Σ:(u,0,0)	F:(u,0,0)	$(Σ)F_{1}^{S}(2)⊕(Σ)F_{2}^{S}(2) $
K:(0,v,w)	GP:(0,v,w)	$2(K)GP_{1}^{S}(2) $
L:(1/2,v,w)	GP:(1/2,v,w)	$2(L)GP_{1}^{S}(2) $
M:(u,0,w)	F:(u,0,w)	$2(M)F_{1}^{S}(1)⊕2(M)F_{2}^{S}(1)$	Spin Splitting
N:(u,1/2,w)	G:(u,1/2,w)	$2(N)G_{1}^{S}G_{2}^{S}(2)$	Spin Splitting
V:(u,v,0)	GP:(u,v,0)	$2(V)GP_{1}^{S}(2) $
W:(u,v,1/2)	GP:(u,v,1/2)	$2(W)GP_{1}^{S}(2) $
GP:(u,v,w)	GP:(u,v,w)	$4GP_{1}^{S}(1)$	Spin Splitting

4c

k-vector	k-vector-G↑	A'↑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)	E:(1/2,1/2,1/2)	$(R)E_{1}^{S}(4) $
S:(1/2,1/2,0)	C:(1/2,1/2,0)	$(S)C_{1}^{S}(4) $
T:(0,1/2,1/2)	D:(0,1/2,1/2)	$(T)D_{1}^{S}(4) $
U:(1/2,0,1/2)	A:(1/2,0,1/2)	$(U)A_{1}^{S,+}(2)⊕(U)A_{2}^{S,-}(2) $
X:(1/2,0,0)	Y:(1/2,0,0)	$(X)Y_{1}^{S,+}(2)⊕(X)Y_{2}^{S,-}(2) $
Y:(0,1/2,0)	Z:(0,1/2,0)	$(Y)Z_{1}^{S}(4) $
Z:(0,0,1/2)	B:(0,0,1/2)	$(Z)B_{1}^{S,+}(2)⊕(Z)B_{2}^{S,-}(2) $
A:(u,0,1/2)	F:(u,0,1/2)	$2 (A)F_{1}^{S}(2) $
B:(0,v,1/2)	V:(0,v,1/2)	$(B)V_{1}^{S}(2)⊕(B)V_{2}^{S}(2) $
C:(u,1/2,0)	G:(u,1/2,0)	$(C)G_{1}^{S}G_{2}^{S}(4) $
D:(1/2,v,0)	W:(1/2,v,0)	$(D)W_{1}^{S}(2)⊕(D)W_{2}^{S}(2) $
Δ:(0,v,0)	Λ:(0,v,0)	$(Δ)Λ_{1}^{S}(2)⊕(Δ)Λ_{2}^{S}(2) $
E:(u,1/2,1/2)	G:(u,1/2,1/2)	$(E)G_{1}^{S}G_{2}^{S}(4) $
G:(1/2,0,w)	F:(1/2,0,w)	$2 (G)F_{1}^{S}(2) $
H:(0,1/2,w)	G:(0,1/2,w)	$(H)G_{1}^{S}G_{2}^{S}(4) $
Λ:(0,0,w)	F:(0,0,w)	$2 (Λ)F_{1}^{S}(2) $
P:(1/2,v,1/2)	U:(1/2,v,1/2)	$(P)U_{1}^{S}(2)⊕(P)U_{2}^{S}(2) $
Q:(1/2,1/2,w)	G:(1/2,1/2,w)	$(Q)G_{1}^{S}G_{2}^{S}(4) $
Σ:(u,0,0)	F:(u,0,0)	$2 (Σ)F_{1}^{S}(2) $
K:(0,v,w)	GP:(0,v,w)	$2(K)GP_{1}^{S}(2) $
L:(1/2,v,w)	GP:(1/2,v,w)	$2(L)GP_{1}^{S}(2) $
M:(u,0,w)	F:(u,0,w)	$4(M)F_{1}^{S}(1)$	Spin Splitting
N:(u,1/2,w)	G:(u,1/2,w)	$2(N)G_{1}^{S}G_{2}^{S}(2)$	Spin Splitting
V:(u,v,0)	GP:(u,v,0)	$2(V)GP_{1}^{S}(2) $
W:(u,v,1/2)	GP:(u,v,1/2)	$2(W)GP_{1}^{S}(2) $
GP:(u,v,w)	GP:(u,v,w)	$4GP_{1}^{S}(1)$	Spin Splitting

8d

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)	E:(1/2,1/2,1/2)	$2(R)E_{1}^{S}(4) $
S:(1/2,1/2,0)	C:(1/2,1/2,0)	$2(S)C_{1}^{S}(4) $
T:(0,1/2,1/2)	D:(0,1/2,1/2)	$2(T)D_{1}^{S}(4) $
U:(1/2,0,1/2)	A:(1/2,0,1/2)	$(U)A_{1}^{S,+}(2)⊕(U)A_{1}^{S,-}(2)⊕(U)A_{2}^{S,+}(2)⊕(U)A_{2}^{S,-}(2) $
X:(1/2,0,0)	Y:(1/2,0,0)	$(X)Y_{1}^{S,+}(2)⊕(X)Y_{1}^{S,-}(2)⊕(X)Y_{2}^{S,+}(2)⊕(X)Y_{2}^{S,-}(2) $
Y:(0,1/2,0)	Z:(0,1/2,0)	$2(Y)Z_{1}^{S}(4) $
Z:(0,0,1/2)	B:(0,0,1/2)	$(Z)B_{1}^{S,+}(2)⊕(Z)B_{1}^{S,-}(2)⊕(Z)B_{2}^{S,+}(2)⊕(Z)B_{2}^{S,-}(2) $
A:(u,0,1/2)	F:(u,0,1/2)	$2(A)F_{1}^{S}(2)⊕2(A)F_{2}^{S}(2) $
B:(0,v,1/2)	V:(0,v,1/2)	$2(B)V_{1}^{S}(2)⊕2(B)V_{2}^{S}(2) $
C:(u,1/2,0)	G:(u,1/2,0)	$2(C)G_{1}^{S}G_{2}^{S}(4) $
D:(1/2,v,0)	W:(1/2,v,0)	$2(D)W_{1}^{S}(2)⊕2(D)W_{2}^{S}(2) $
Δ:(0,v,0)	Λ:(0,v,0)	$2(Δ)Λ_{1}^{S}(2)⊕2(Δ)Λ_{2}^{S}(2) $
E:(u,1/2,1/2)	G:(u,1/2,1/2)	$2(E)G_{1}^{S}G_{2}^{S}(4) $
G:(1/2,0,w)	F:(1/2,0,w)	$2(G)F_{1}^{S}(2)⊕2(G)F_{2}^{S}(2) $
H:(0,1/2,w)	G:(0,1/2,w)	$2(H)G_{1}^{S}G_{2}^{S}(4) $
Λ:(0,0,w)	F:(0,0,w)	$2(Λ)F_{1}^{S}(2)⊕2(Λ)F_{2}^{S}(2) $
P:(1/2,v,1/2)	U:(1/2,v,1/2)	$2(P)U_{1}^{S}(2)⊕2(P)U_{2}^{S}(2) $
Q:(1/2,1/2,w)	G:(1/2,1/2,w)	$2(Q)G_{1}^{S}G_{2}^{S}(4) $
Σ:(u,0,0)	F:(u,0,0)	$2(Σ)F_{1}^{S}(2)⊕2(Σ)F_{2}^{S}(2) $
K:(0,v,w)	GP:(0,v,w)	$4(K)GP_{1}^{S}(2) $
L:(1/2,v,w)	GP:(1/2,v,w)	$4(L)GP_{1}^{S}(2) $
M:(u,0,w)	F:(u,0,w)	$4(M)F_{1}^{S}(1)⊕4(M)F_{2}^{S}(1)$	Spin Splitting
N:(u,1/2,w)	G:(u,1/2,w)	$4(N)G_{1}^{S}(1)⊕4(N)G_{2}^{S}(1)$	Spin Splitting
V:(u,v,0)	GP:(u,v,0)	$4(V)GP_{1}^{S}(2) $
W:(u,v,1/2)	GP:(u,v,1/2)	$4(W)GP_{1}^{S}(2) $
GP:(u,v,w)	GP:(u,v,w)	$8GP_{1}^{S}(1)$	Spin Splitting