The irreducible representations associated with states of dipole symmetry have
been calculated for the space group
Oh3, the space group
with the correct symmetry for A-15 phase compounds. Also
assembled are the character tables of the
Oh3 group. Thus all
the direct interband dipole-transition selection rules for A-15
phase compounds can easily be determined.
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The 48 operations of
Oh3 include
the 24 listed above along with an additional 24 obtained by including an
inversion with the first 24. Of these 24 operations listed, the first 12
operations are rotations only (τ = 0),
whereas the second 12 include the fractional translation
τ =
(a/2)(î +
ĵ + ).
Table 2
Various Dipole Irreps Associated with the Symmetry Points and Lines of the
Oh3 Group
Dipole Transitions
Symmetry Point or Line
Γ15
Γ
Δ1,
Δ5
Δ
X1,
X3,
X4
X
Z1,
Z3,
Z4
Z
M1,
M2, M5,
M6,
M10
M
∑1,
∑3, ∑4
∑
Λ1,
Λ3
λ
R4
R
S1
S
T1,
T2, T3,
T4,
T5
T
Table 3
Character Table and Some Selection Rules for the M Symmetry
Point
R1
R2
R3
R4
R17
R18
R23
R24
R1′
R2′
R3′
R4′
R17′
R18′
R23′
R24′
M1
1
1
−1
−1
−1
2
2
−1
−1
−1
2
2
2
−1
−1
1
M2
1
−1
1
−1
1
−1
1
−1
−1
1
−1
1
−1
1
−1
1
M3
1
−1
1
−1
1
−1
1
−1
1
−1
1
−1
1
−1
1
−1
M4
1
1
−1
−1
−1
1
1
−1
1
1
−1
−1
−1
1
1
−1
M5
1
1
−1
−1
1
−1
−1
1
−1
−1
1
1
−1
1
1
−1
M6
1
−1
1
−1
−1
1
−1
1
−1
1
−1
1
1
−1
1
−1
M7
1
−1
1
−1
−1
1
−1
1
1
−1
1
−1
−1
1
−1
1
M8
1
1
−1
−1
1
−1
−1
1
1
1
−1
−1
1
−1
−1
1
M9
2
0
0
2
0
0
0
0
2
0
0
2
0
0
0
0
M10
2
0
0
2
0
0
0
0
−2
0
0
−2
0
0
0
0
Some allowed dipole
transitions
Some forbidden dipole
transitions
M8 ↮
M9
M10 ↮
M6
M10 ↮
M5
M5,6 ↮
M10,2
M2 ↮
M10
Table 4
Character Table and All the Selection Rules of the Symmetry Point
R and X
R1
R2–R4
R5–R8
R9–R12
R13–R18
R19–R24
R1′
R2′–R4′
R5′–R8′
R9′–R12′
R13′–R18′
R19′–R24′
R1
2
2
−1
−1
0
0
0
0
0
0
R2
2
2
−1
−1
0
0
0
0
0
0
R3
2
2
2
2
0
0
0
0
0
0
0
0
R4
6
−2
0
0
0
0
0
0
0
0
0
0
Allowed dipole
transitions
R1
R2
R3
R4
R13–R14
R19–R20
R1′
R2′
R3′
R4′
R13′–R14′
R19′–R20′
X1
2
2
0
0
0
0
0
0
2
2
0
0
X2
2
2
0
0
0
0
0
0
−2
−2
0
0
X3
2
−2
−2
2
0
0
0
0
0
0
0
0
X4
2
−2
2
−2
0
0
0
0
0
0
0
0
Allowed dipole
transitions
Forbidden dipole
transition
Xi ↮
X1
Table 5
Character Tables and All the Selection Rules for the Symmetry Lines
T, Z, and S
R1
R4
R23
R24
R2′
R3′
R17′
R18′
T1
1
−1
−a
a
1
−1
a
−a
T2
1
−1
−a
a
−1
1
−a
a
a =
−exp(iπp)
T3
1
−1
a
−a
1
−1
−a
a
0 ≤ p ≤
½
T4
1
−1
a
−a
−1
1
a
−a
T5
2
2
0
0
0
0
0
0
Allowed dipole
transitions
R1
R3
R2′
R4′
R1
R19
R2′
R20′
Z1
1
1
1
1
S1
2
0
0
0
Z2
1
1
−1
−1
Z3
1
−1
−1
1
Allowed dipole
transition
Z4
1
−1
1
−1
Allowed dipole
transitions
Forbidden dipole
transition
Z4 ↮
Z3
Table 6
Character Tables and Some of the Selection Rules for the Γ Symmetry
Point
R1
R2–R4
R5–R12
R13–R18
R19–R24
R1′
R2′–R4′
R5′–R12′
R13′–R18′
R19′–R24′
Γ1
1
1
1
1
1
1
1
1
1
1
Γ2
1
1
1
−1
−1
1
1
1
−1
−1
Γ12
2
2
−1
0
0
2
2
−1
0
0
Γ15′
3
−1
0
1
−1
3
−1
0
1
−1
Γ25′
3
−1
0
−1
1
3
−1
0
−1
1
Γ1′
1
1
1
1
1
−1
−1
−1
−1
−1
Γ2′
1
1
1
−1
−1
−1
−1
−1
1
1
Γ12′
2
2
−1
0
0
−2
−2
1
0
0
Γ15
3
−1
0
1
−1
−3
1
0
−1
1
Γ25
3
−1
0
−1
1
−3
1
0
1
−1
Some allowed dipole
transitions
Some forbidden dipole
trasmsitions
Γ25′ ↮
Γ12
Γ1
↮ Γ12
Γ25′ ↮
Γ1
Γ1
↮ Γ1
Γ12
↮ Γ12
Table 7
Character Tables and Some Selection Rules for the Symmetry Lines Δ
and Λa
R1
R2
R19
R20
R3′
R4′
R13′
R14′
Δ1
1
1
ω
ω
1
1
ω
ω
Phase factor ω
= exp(iπp)
Δ1′
1
1
−ω
−ω
−1
−1
ω
ω
0 ≤ p ≤
½
Δ2
1
1
−ω
−ω
1
1
−ω
−ω
Δ2′
1
1
ω
ω
−1
−1
−ω
−ω
Δ5
2
−2
0
0
0
0
0
0
Some allowed dipole
transitions
Some forbidden dipole
transitions
Δ2′ ↮
Δ2
Δ2′ ↮
Δ1
R1
R5
R9
R19
R21
R23
Λ1
1
1
1
v
v
v
Phase factor v =
exp(iπ3p)
Λ2
1
1
1
−v
−v
−v
0 ≤ p ≤
½
Λ3
2
−1
−1
0
0
0
All the allowed dipole
transitions
Parts of the character tables must be multiplied by the given phase
factor.
Table 8
Character Tables and Some Selection Rules for the Symmetry Line ∑
The 48 operations of
Oh3 include
the 24 listed above along with an additional 24 obtained by including an
inversion with the first 24. Of these 24 operations listed, the first 12
operations are rotations only (τ = 0),
whereas the second 12 include the fractional translation
τ =
(a/2)(î +
ĵ + ).
Table 2
Various Dipole Irreps Associated with the Symmetry Points and Lines of the
Oh3 Group
Dipole Transitions
Symmetry Point or Line
Γ15
Γ
Δ1,
Δ5
Δ
X1,
X3,
X4
X
Z1,
Z3,
Z4
Z
M1,
M2, M5,
M6,
M10
M
∑1,
∑3, ∑4
∑
Λ1,
Λ3
λ
R4
R
S1
S
T1,
T2, T3,
T4,
T5
T
Table 3
Character Table and Some Selection Rules for the M Symmetry
Point
R1
R2
R3
R4
R17
R18
R23
R24
R1′
R2′
R3′
R4′
R17′
R18′
R23′
R24′
M1
1
1
−1
−1
−1
2
2
−1
−1
−1
2
2
2
−1
−1
1
M2
1
−1
1
−1
1
−1
1
−1
−1
1
−1
1
−1
1
−1
1
M3
1
−1
1
−1
1
−1
1
−1
1
−1
1
−1
1
−1
1
−1
M4
1
1
−1
−1
−1
1
1
−1
1
1
−1
−1
−1
1
1
−1
M5
1
1
−1
−1
1
−1
−1
1
−1
−1
1
1
−1
1
1
−1
M6
1
−1
1
−1
−1
1
−1
1
−1
1
−1
1
1
−1
1
−1
M7
1
−1
1
−1
−1
1
−1
1
1
−1
1
−1
−1
1
−1
1
M8
1
1
−1
−1
1
−1
−1
1
1
1
−1
−1
1
−1
−1
1
M9
2
0
0
2
0
0
0
0
2
0
0
2
0
0
0
0
M10
2
0
0
2
0
0
0
0
−2
0
0
−2
0
0
0
0
Some allowed dipole
transitions
Some forbidden dipole
transitions
M8 ↮
M9
M10 ↮
M6
M10 ↮
M5
M5,6 ↮
M10,2
M2 ↮
M10
Table 4
Character Table and All the Selection Rules of the Symmetry Point
R and X
R1
R2–R4
R5–R8
R9–R12
R13–R18
R19–R24
R1′
R2′–R4′
R5′–R8′
R9′–R12′
R13′–R18′
R19′–R24′
R1
2
2
−1
−1
0
0
0
0
0
0
R2
2
2
−1
−1
0
0
0
0
0
0
R3
2
2
2
2
0
0
0
0
0
0
0
0
R4
6
−2
0
0
0
0
0
0
0
0
0
0
Allowed dipole
transitions
R1
R2
R3
R4
R13–R14
R19–R20
R1′
R2′
R3′
R4′
R13′–R14′
R19′–R20′
X1
2
2
0
0
0
0
0
0
2
2
0
0
X2
2
2
0
0
0
0
0
0
−2
−2
0
0
X3
2
−2
−2
2
0
0
0
0
0
0
0
0
X4
2
−2
2
−2
0
0
0
0
0
0
0
0
Allowed dipole
transitions
Forbidden dipole
transition
Xi ↮
X1
Table 5
Character Tables and All the Selection Rules for the Symmetry Lines
T, Z, and S
R1
R4
R23
R24
R2′
R3′
R17′
R18′
T1
1
−1
−a
a
1
−1
a
−a
T2
1
−1
−a
a
−1
1
−a
a
a =
−exp(iπp)
T3
1
−1
a
−a
1
−1
−a
a
0 ≤ p ≤
½
T4
1
−1
a
−a
−1
1
a
−a
T5
2
2
0
0
0
0
0
0
Allowed dipole
transitions
R1
R3
R2′
R4′
R1
R19
R2′
R20′
Z1
1
1
1
1
S1
2
0
0
0
Z2
1
1
−1
−1
Z3
1
−1
−1
1
Allowed dipole
transition
Z4
1
−1
1
−1
Allowed dipole
transitions
Forbidden dipole
transition
Z4 ↮
Z3
Table 6
Character Tables and Some of the Selection Rules for the Γ Symmetry
Point
R1
R2–R4
R5–R12
R13–R18
R19–R24
R1′
R2′–R4′
R5′–R12′
R13′–R18′
R19′–R24′
Γ1
1
1
1
1
1
1
1
1
1
1
Γ2
1
1
1
−1
−1
1
1
1
−1
−1
Γ12
2
2
−1
0
0
2
2
−1
0
0
Γ15′
3
−1
0
1
−1
3
−1
0
1
−1
Γ25′
3
−1
0
−1
1
3
−1
0
−1
1
Γ1′
1
1
1
1
1
−1
−1
−1
−1
−1
Γ2′
1
1
1
−1
−1
−1
−1
−1
1
1
Γ12′
2
2
−1
0
0
−2
−2
1
0
0
Γ15
3
−1
0
1
−1
−3
1
0
−1
1
Γ25
3
−1
0
−1
1
−3
1
0
1
−1
Some allowed dipole
transitions
Some forbidden dipole
trasmsitions
Γ25′ ↮
Γ12
Γ1
↮ Γ12
Γ25′ ↮
Γ1
Γ1
↮ Γ1
Γ12
↮ Γ12
Table 7
Character Tables and Some Selection Rules for the Symmetry Lines Δ
and Λa
R1
R2
R19
R20
R3′
R4′
R13′
R14′
Δ1
1
1
ω
ω
1
1
ω
ω
Phase factor ω
= exp(iπp)
Δ1′
1
1
−ω
−ω
−1
−1
ω
ω
0 ≤ p ≤
½
Δ2
1
1
−ω
−ω
1
1
−ω
−ω
Δ2′
1
1
ω
ω
−1
−1
−ω
−ω
Δ5
2
−2
0
0
0
0
0
0
Some allowed dipole
transitions
Some forbidden dipole
transitions
Δ2′ ↮
Δ2
Δ2′ ↮
Δ1
R1
R5
R9
R19
R21
R23
Λ1
1
1
1
v
v
v
Phase factor v =
exp(iπ3p)
Λ2
1
1
1
−v
−v
−v
0 ≤ p ≤
½
Λ3
2
−1
−1
0
0
0
All the allowed dipole
transitions
Parts of the character tables must be multiplied by the given phase
factor.
Table 8
Character Tables and Some Selection Rules for the Symmetry Line ∑