Information of spin space group $I\ce{^{\text{-}1}{4_{1}}}\ce{^{\text{-}1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}}$ ( nSSG:22.98.1.1 )

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 ‖ 2110 | 0 }
-b, -a, -c | x, y, z	{ 1 ‖ 21-10 | 0 }
-b, a+1/2, c+1/4 | -x, -y, -z	{ -1 ‖ 4+001 | 0 1/2 1/4 }
b, -a+1/2, c+1/4 | -x, -y, -z	{ -1 ‖ 4-001 | 0 1/2 1/4 }
a, -b+1/2, -c+1/4 | -x, -y, -z	{ -1 ‖ 2100 | 0 1/2 1/4 }
-a, b+1/2, -c+1/4 | -x, -y, -z	{ -1 ‖ 2010 | 0 1/2 1/4 }
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 ‖ 2110 | 1/2 1/2 1/2 }
-b+1/2, -a+1/2, -c+1/2 | x, y, z	{ 1 ‖ 21-10 | 1/2 1/2 1/2 }
-b+1/2, a, c+3/4 | -x, -y, -z	{ -1 ‖ 4+001 | 1/2 0 3/4 }
b+1/2, -a, c+3/4 | -x, -y, -z	{ -1 ‖ 4-001 | 1/2 0 3/4 }
a+1/2, -b, -c+3/4 | -x, -y, -z	{ -1 ‖ 2100 | 1/2 0 3/4 }
-a+1/2, b, -c+3/4 | -x, -y, -z	{ -1 ‖ 2010 | 1/2 0 3/4 }
a, b, c | -x, y, z	{ m ‖ 1 | 0 }
-a, -b, c | -x, y, z	{ m ‖ 2001 | 0 }
b, a, -c | -x, y, z	{ m ‖ 2110 | 0 }
-b, -a, -c | -x, y, z	{ m ‖ 21-10 | 0 }
-b, a+1/2, c+1/4 | x, -y, -z	{ 2 ‖ 4+001 | 0 1/2 1/4 }
b, -a+1/2, c+1/4 | x, -y, -z	{ 2 ‖ 4-001 | 0 1/2 1/4 }
a, -b+1/2, -c+1/4 | x, -y, -z	{ 2 ‖ 2100 | 0 1/2 1/4 }
-a, b+1/2, -c+1/4 | x, -y, -z	{ 2 ‖ 2010 | 0 1/2 1/4 }
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	{ m ‖ 2110 | 1/2 1/2 1/2 }
-b+1/2, -a+1/2, -c+1/2 | -x, y, z	{ m ‖ 21-10 | 1/2 1/2 1/2 }
-b+1/2, a, c+3/4 | x, -y, -z	{ 2 ‖ 4+001 | 1/2 0 3/4 }
b+1/2, -a, c+3/4 | x, -y, -z	{ 2 ‖ 4-001 | 1/2 0 3/4 }
a+1/2, -b, -c+3/4 | x, -y, -z	{ 2 ‖ 2100 | 1/2 0 3/4 }
-a+1/2, b, -c+3/4 | x, -y, -z	{ 2 ‖ 2010 | 1/2 0 3/4 }

Wyckoff Positions

WP	Site symmetry	Representative
4a	$\ce{^{1}{2}}.\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,0,0 | 0,0,z)
4b	$\ce{^{1}{2}}.\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(0,0,1/2 | 0,0,z)
8c	$\ce{^{1}{2}}..\ce{^{\infty m}{1}} $	(0,0,c | 0,0,z)
8d	$..\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(a,a,0 | 0,0,z)
8e	$..\ce{^{1}{2}}\ce{^{\infty m}{1}} $	(-a,a,0 | 0,0,z)
8f	$.\ce{^{-1}{2}}.\ce{^{\infty m}{1}} $	(a,1/4,1/8 | 0,0,0)
16g	$\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{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $
M:(1,1,1)	$\ce{^{-1}{4}}\ce{^{-1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $
P:(1/2,1/2,1/2)	$\ce{^{-1}{4}}\ce{^{2}{2}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
X:(1/2,1/2,0)	$\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{1}{2}}\ce{^{\infty m}{1}} $
N:(1/2,0,1/2)	$\ce{^{2}{2}}\ce{^{\infty m}{1}} $
Λ:(0,0,w)	$\ce{^{2}{4}}\ce{^{-1}{2}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
Δ:(u,u,0)	$\ce{^{m}{2}}\ce{^{1}{2}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
Q:(1/2,v,1/2)	$\ce{^{2}{2}}\ce{^{\infty}{1}} $
Σ:(u,0,0)	$\ce{^{2}{2}}\ce{^{-1}{2}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
W:(1/2,1/2,w)	$\ce{^{m}{2}}\ce{^{m}{2}}\ce{^{1}{2}}\ce{^{\infty}{1}} $
Y:(u,1-u,0)	$\ce{^{1}{2}}\ce{^{m}{2}}\ce{^{m}{2}}\ce{^{\infty}{1}} $
A:(u,u,w)	$\ce{^{m}{2}}\ce{^{\infty}{1}} $
B:(u,0,w)	$\ce{^{-1}{2}}\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

Band Representations (Magnon)

16g

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) $
M:(0,0,1)	Z:(0,0,1)	$(M)Z_{1}^{S}(2)⊕(M)Z_{2}^{S}(2)⊕(M)Z_{3}^{S}(2)⊕(M)Z_{4}^{S}(2) $
N:(1/2,0,1/2)	L:(1/2,1/2,1/2)	$4 (N)L_{1}^{S}(2) $
P:(1/2,1/2,1/2)	H:(0,1,1/2)	$2 (P)H_{1}^{S}(2)⊕2 (P)H_{2}^{S}(2) $
X:(1/2,1/2,0)	Y:(0,1,0)	$2(X)Y_{1}^{S}(1)⊕2(X)Y_{2}^{S}(1)⊕2(X)Y_{3}^{S}(1)⊕2(X)Y_{4}^{S}(1)$	Spin Splitting
Δ:(u,u,0)	Δ:(0,2*u,0)	$4Δ_{1}^{S}(1)⊕4Δ_{2}^{S}(1)$	Spin Splitting
Λ:(0,0,w)	Λ:(0,0,w)	$2 Λ_{1}^{S}(2)⊕2 Λ_{2}^{S}(2) $
Q:(1/2,v,1/2)	GP:(1/2-v,1/2+v,1/2)	$4 (Q)GP_{1}^{S}(2) $
Σ:(u,0,0)	M:(u,u,0)	$4 (Σ)M_{1}^{S}(2) $
W:(1/2,1/2,w)	H:(0,1,w)	$4(W)H_{1}^{S}(1)⊕4(W)H_{2}^{S}(1)$	Spin Splitting
Y:(u,-u,1)	A:(2*u,0,1)	$4 (Y)A_{1}^{S}(1)⊕4 (Y)A_{2}^{S}(1)$	Spin Splitting
A:(u,u,w)	E:(0,2*u,w)	$8(A)E_{1}^{S}(1)$	Spin Splitting
B:(u,0,w)	GP:(u,u,w)	$4 (B)GP_{1}^{S}(2) $
C:(u,v,0)	M:(u-v,u+v,0)	$8(C)M_{1}^{S}(1)$	Spin Splitting
GP:(u,v,w)	GP:(u-v,u+v,w)	$8GP_{1}^{S}(1)$	Spin Splitting

4a

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

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)	Z:(0,0,1)	$(M)Z_{1}^{S}(2) $
N:(1/2,0,1/2)	L:(1/2,1/2,1/2)	$(N)L_{1}^{S}(2) $
P:(1/2,1/2,1/2)	H:(0,1,1/2)	$(P)H_{1}^{S}(2) $
X:(1/2,1/2,0)	Y:(0,1,0)	$2(X)Y_{1}^{S}(1)$	Spin Splitting
Δ:(u,u,0)	Δ:(0,2*u,0)	$2Δ_{1}^{S}(1)$	Spin Splitting
Λ:(0,0,w)	Λ:(0,0,w)	$Λ_{1}^{S}(2) $
Q:(1/2,v,1/2)	GP:(1/2-v,1/2+v,1/2)	$(Q)GP_{1}^{S}(2) $
Σ:(u,0,0)	M:(u,u,0)	$(Σ)M_{1}^{S}(2) $
W:(1/2,1/2,w)	H:(0,1,w)	$2(W)H_{1}^{S}(1)$	Spin Splitting
Y:(u,-u,1)	A:(2*u,0,1)	$2(Y)A_{1}^{S}(1)$	Spin Splitting
A:(u,u,w)	E:(0,2*u,w)	$2(A)E_{1}^{S}(1)$	Spin Splitting
B:(u,0,w)	GP:(u,u,w)	$(B)GP_{1}^{S}(2) $
C:(u,v,0)	M:(u-v,u+v,0)	$2(C)M_{1}^{S}(1)$	Spin Splitting
GP:(u,v,w)	GP:(u-v,u+v,w)	$2GP_{1}^{S}(1)$	Spin Splitting

8e

k-vector	k-vector-G↑	A↑G(4)
Γ:(0,0,0)	Γ:(0,0,0)	$Γ_{1}^{S}(2)⊕Γ_{3}^{S}(2) $
M:(0,0,1)	Z:(0,0,1)	$(M)Z_{1}^{S}(2)⊕(M)Z_{3}^{S}(1) $
N:(1/2,0,1/2)	L:(1/2,1/2,1/2)	$2 (N)L_{1}^{S}(2) $
P:(1/2,1/2,1/2)	H:(0,1,1/2)	$(P)H_{1}^{S}(2)⊕(P)H_{2}^{S}(2) $
X:(1/2,1/2,0)	Y:(0,1,0)	$2(X)Y_{1}^{S}(1)⊕2(X)Y_{3}^{S}(1)$	Spin Splitting
Δ:(u,u,0)	Δ:(0,2*u,0)	$2Δ_{1}^{S}(1)⊕2Δ_{2}^{S}(1)$	Spin Splitting
Λ:(0,0,w)	Λ:(0,0,w)	$Λ_{1}^{S}(2)⊕Λ_{2}^{S}(2) $
Q:(1/2,v,1/2)	GP:(1/2-v,1/2+v,1/2)	$2 (Q)GP_{1}^{S}(2) $
Σ:(u,0,0)	M:(u,u,0)	$2 (Σ)M_{1}^{S}(2) $
W:(1/2,1/2,w)	H:(0,1,w)	$2(W)H_{1}^{S}(1)⊕2(W)H_{2}^{S}(1)$	Spin Splitting
Y:(u,-u,1)	A:(2*u,0,1)	$4 (Y)A_{1}^{S}(1)$	Spin Splitting
A:(u,u,w)	E:(0,2*u,w)	$4(A)E_{1}^{S}(1)$	Spin Splitting
B:(u,0,w)	GP:(u,u,w)	$2 (B)GP_{1}^{S}(2) $
C:(u,v,0)	M:(u-v,u+v,0)	$4(C)M_{1}^{S}(1)$	Spin Splitting
GP:(u,v,w)	GP:(u-v,u+v,w)	$4GP_{1}^{S}(1)$	Spin Splitting

8d

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