Information of spin space group $I\ce{^{\text{-}1}{4}}\ce{^{\infty m}{1}}$ ( nSSG:5.79.1.1 )

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 }
-b, a, c | -x, -y, -z	{ -1 ‖ 4+001 | 0 }
b, -a, c | -x, -y, -z	{ -1 ‖ 4-001 | 0 }
a+1/2, b+1/2, c+1/2 | x, y, z	{ 1 ‖ 1 | 1/2 1/2 1/2 }
-a+1/2, -b+1/2, c+1/2 | x, y, z	{ 1 ‖ 2001 | 1/2 1/2 1/2 }
-b+1/2, a+1/2, c+1/2 | -x, -y, -z	{ -1 ‖ 4+001 | 1/2 1/2 1/2 }
b+1/2, -a+1/2, c+1/2 | -x, -y, -z	{ -1 ‖ 4-001 | 1/2 1/2 1/2 }
a, b, c | -x, y, z	{ m ‖ 1 | 0 }
-a, -b, c | -x, y, z	{ m ‖ 2001 | 0 }
-b, a, c | x, -y, -z	{ 2 ‖ 4+001 | 0 }
b, -a, c | x, -y, -z	{ 2 ‖ 4-001 | 0 }
a+1/2, b+1/2, c+1/2 | -x, y, z	{ m ‖ 1 | 1/2 1/2 1/2 }
-a+1/2, -b+1/2, c+1/2 | -x, y, z	{ m ‖ 2001 | 1/2 1/2 1/2 }
-b+1/2, a+1/2, c+1/2 | x, -y, -z	{ 2 ‖ 4+001 | 1/2 1/2 1/2 }
b+1/2, -a+1/2, c+1/2 | x, -y, -z	{ 2 ‖ 4-001 | 1/2 1/2 1/2 }

Wyckoff Positions

WP	Site symmetry	Representative
2a	$\ce{^{-1}{4}}..\ce{^{\infty m}{1}} $	(0,0,c | 0,0,0)
4b	$\ce{^{1}{2}}..\ce{^{\infty m}{1}} $	(0,1/2,c | 0,0,z)
8c	$\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}{4}}\ce{^{\infty m}{1}} $
M:(1,1,1)	$\ce{^{-1}{4}}\ce{^{\infty m}{1}} $
P:(1/2,1/2,1/2)	$\ce{^{-1}{4}}\ce{^{\infty}{1}} $
X:(1/2,1/2,0)	$\ce{^{1}{2}}\ce{^{\infty m}{1}} $
N:(1/2,0,1/2)	$\ce{^{1}{1}}\ce{^{\infty m}{1}} $
Λ:(0,0,w)	$\ce{^{2}{4}}\ce{^{\infty}{1}} $
W:(1/2,1/2,w)	$\ce{^{1}{2}}\ce{^{\infty}{1}} $
Q:(1/2,v,1/2)	$\ce{^{1}{1}}\ce{^{\infty}{1}} $
Σ:(u,0,0)	$\ce{^{m}{2}}\ce{^{\infty}{1}} $
Y:(u,1-u,0)	$\ce{^{m}{2}}\ce{^{\infty}{1}} $
Δ:(u,u,0)	$\ce{^{m}{2}}\ce{^{\infty}{1}} $
A:(u,u,w)	$\ce{^{1}{1}}\ce{^{\infty}{1}} $
B:(u,0,w)	$\ce{^{1}{1}}\ce{^{\infty}{1}} $
C:(u,v,0)	$\ce{^{m}{2}}\ce{^{\infty}{1}} $
GP:(u,v,w)	$\ce{^{1}{1}}\ce{^{\infty}{1}} $

Spin Brillouin Zone

4/mI

Band Representations (Magnon)

4b

k-vector	k-vector-G↑	A↑G(2)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2) $
M:(0,0,-1)	Y:(0,1,0)	$(M)Y_{1}^{S}(2) $
N:(-1/2,1,-1/2)	L:(1/2,1/2,1/2)	$2(N)L_{1}^{S}(1)$	Spin Splitting
P:(-1/2,1/2,-1/2)	U:(0,1/2,1/2)	$(P)U_{2}^{S}(2) $
X:(-1/2,1/2,-1)	M:(0,1,1/2)	$2(X)M_{2}^{S}(1)$	Spin Splitting
A:(u,u,w)	GP:(2*u,-w,-u)	$2(A)GP_{1}^{S}(1)$	Spin Splitting
B:(u,0,w)	GP:(u,-w,-u)	$2(B)GP_{1}^{S}(1)$	Spin Splitting
C:(u,v,0)	B:(u+v,0,-u)	$2(C)B_{1}^{S}(1)$	Spin Splitting
Δ:(u,u,0)	B:(2*u,0,-u)	$2(Δ)B_{1}^{S}(1)$	Spin Splitting
Λ:(0,0,w)	Λ:(0,-w,0)	$Λ_{1}^{S}(2) $
Q:(1/2,v,1/2)	GP:(1/2+v,-1/2,-1/2)	$2(Q)GP_{1}^{S}(1)$	Spin Splitting
Σ:(u,0,0)	B:(u,0,-u)	$2(Σ)B_{1}^{S}(1)$	Spin Splitting
W:(-1/2,1/2,-1+w)	U:(0,1-w,1/2)	$2(W)U_{2}^{S}(1)$	Spin Splitting
Y:(u,1-u,0)	B:(1,0,-u)	$2(Y)B_{1}^{S}(1)$	Spin Splitting
GP:(u,v,w)	GP:(u+v,-w,-u)	$2GP_{1}^{S}(1)$	Spin Splitting

8c

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