Maheswar Maji
Int. Ph.D. 2009
Hamiltonian of an impurity ion in crystal potential
Octahedron symmetry & impurity ion
Overview
Some group theory concepts
Characters for Full Rotation group
Character table for Octahedral (O) group
Details of splitting of orbitals
Further splitting due to lowering the symmetry
𝐻 = { 𝑝𝑖
2
2𝑚−
𝑍𝑒2
𝑟𝑖𝜇𝑖
+ 𝑒2
𝑟𝑖𝑗𝑗
+ 𝜁𝑖𝑗 𝒍𝑖 . 𝒔𝑗 +
𝑗
𝛾𝑖𝜇 𝒋𝑖 . 𝑰𝜇 } + 𝑉𝑐𝑟𝑦𝑠𝑡𝑎𝑙
𝐻 = { 𝑝𝑖
2
2𝑚−
𝑍𝑒2
𝑟𝑖𝜇𝑖
+ 𝑒2
𝑟𝑖𝑗𝑗
+ 𝜁𝑖𝑗 𝒍𝑖 . 𝒔𝑗 +
𝑗
𝛾𝑖𝜇 𝒋𝑖 . 𝑰𝜇 } + 𝑉𝑐𝑟𝑦𝑠𝑡𝑎𝑙
Electronic
Hamiltonian without
any coupling (Ho)
𝐻 = { 𝑝𝑖
2
2𝑚−
𝑍𝑒2
𝑟𝑖𝜇𝑖
+ 𝑒2
𝑟𝑖𝑗𝑗
+ 𝜁𝑖𝑗 𝒍𝑖 . 𝒔𝑗 +
𝑗
𝛾𝑖𝜇 𝒋𝑖 . 𝑰𝜇 } + 𝑉𝑐𝑟𝑦𝑠𝑡𝑎𝑙
Electronic
Hamiltonian without
any coupling (Ho)
Spin-orbit coupling &
Hyperfine interaction
b/w electrons &
impurity ion
𝐻 = { 𝑝𝑖
2
2𝑚−
𝑍𝑒2
𝑟𝑖𝜇𝑖
+ 𝑒2
𝑟𝑖𝑗𝑗
+ 𝜁𝑖𝑗 𝒍𝑖 . 𝒔𝑗 +
𝑗
𝛾𝑖𝜇 𝒋𝑖 . 𝑰𝜇 } + 𝑉𝑐𝑟𝑦𝑠𝑡𝑎𝑙
Electronic
Hamiltonian without
any coupling (Ho)
Spin-orbit coupling &
Hyperfine interaction
b/w electrons &
impurity ion
Crystal potential of
Host ion acts on
impurity ion
Competition b/w two perturbations..
SO int >> Vcrys
Vcrys as additional correction
Rare earth ions Yb,Nd..
Vcrys >> SO int
Vcrys as major correction to Ho
Transition metal ion Fe, Ni,..
Cube has same set of symmetries as of a
regular octahedron (cube is the dual of an
octahedron)
Free atom Full rotational symmetry
Full rotational group
Atom in cubic crystal Octahedron symmetry
Octahedral Group(O)
Irreps of Higher Symm. group generally forms
Reducible reps of lower symmetry group O
Reducible reps of O can be uniquely
decomposed in it’s Irreps: Decomposition
theorem for Reducible reps
Reducible reps always results in splitting
𝑆𝑎𝑦 𝒳 𝒞𝑘 𝑐𝑎𝑟𝑎𝑐𝑡𝑒𝑟 𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝑐𝑙𝑎𝑠𝑠 𝑖𝑛 𝑎 𝑟𝑒𝑑𝑢𝑐𝑖𝑏𝑙𝑒 𝑟𝑒𝑝𝑠
𝒳 𝒞𝑘 = 𝑎𝑖𝒳 𝛤𝑖 (𝒞𝑘)𝛤𝑖
𝑎𝑖 =1
𝑁𝑘 𝒳
𝛤𝑖 𝒞𝑘 ∗ 𝒳(𝒞𝑘)
𝑘
𝑎𝑖 =1
𝑁𝑘 𝒳
𝛤𝑖 𝒞𝑘 ∗ 𝒳(𝒞𝑘)
𝑘
Character of
reducible reps
𝑎𝑖 =1
𝑁𝑘 𝒳
𝛤𝑖 𝒞𝑘 ∗ 𝒳(𝒞𝑘)
𝑘
Character of
reducible reps
Characters of irreps
Of lower sym Gr
No of
elements in
Ck
𝑎𝑖 =1
𝑁𝑘 𝒳
𝛤𝑖 𝒞𝑘 ∗ 𝒳(𝒞𝑘)
𝑘
Character of
reducible reps
Characters of irreps
Of lower sym Gr
No of
elements in
Ck
•If dimensionality of an irreps j>1 , then that energy
level is j fold degenerate
Basis Function of full rotation group:
Spherical Harmonic
𝑌𝑙𝑚 𝜃, 𝜑 = 2𝑙 + 1
4𝜋 𝑙 − 𝑚 !
𝑙 + 𝑚 !
12
𝑃𝑙𝑚 cos𝜃 𝑒−𝑖𝑚𝜑
ℙ𝑅𝑌𝑙𝑚 𝜃′, 𝜑′ = 𝐷 𝑙 (𝑅)𝑚′𝑚𝑌𝑙𝑚 ′ 𝜃, 𝜑
𝑚′
ℙ𝛼𝑌𝑙𝑚 𝜃, 𝜑 = 𝑒−𝑖𝑚𝛼 𝑌𝑙𝑚 𝜃, 𝜑
𝐷 𝑙 (𝛼)𝑚 ′ 𝑚 = 𝑒−𝑖𝑚𝛼 𝛿𝑚 ′ 𝑚 −𝑙 ≤ 𝑚 ≤ 𝑙
𝐷 𝑙 𝛼 = 𝑒−𝑖𝑙𝛼 ⋯ 𝒪⋮ ⋱ ⋮𝒪 ⋯ 𝑒𝑖𝑙𝛼
𝒳 𝑙 𝛼 = 𝑡𝑟𝑎𝑐𝑒 𝐷 𝑙 𝛼 =sin[(𝑙+
1
2)𝛼]
sin[𝛼
2]
8𝐶3: ±120° 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝑠 𝑎𝑏𝑜𝑢𝑡 𝑎𝑛 𝑎𝑥𝑒𝑠 𝑡𝑟𝑜𝑢𝑔 𝑡𝑒 𝑡𝑟𝑖𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑓𝑎𝑐𝑒 𝑐𝑒𝑛𝑡𝑟𝑜𝑖𝑑𝑠 𝑜𝑓 𝑡𝑒 𝑜𝑐𝑡𝑎𝑒𝑑𝑟𝑜𝑛
6𝐶4 ∶ ±90° 𝑎𝑏𝑜𝑢𝑡 𝑡𝑒 𝑐𝑜𝑟𝑛𝑒𝑟𝑠 𝑜𝑓 𝑡𝑒 𝑜𝑐𝑡𝑎𝑒𝑑𝑟𝑜𝑛
3𝐶2 = 3𝐶42 ∶ 180° 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛 𝑎𝑏𝑜𝑢𝑡 𝑡𝑒 𝑐𝑜𝑟𝑛𝑒𝑟𝑠 𝑜𝑓 𝑡𝑒 𝑜𝑐𝑡𝑎𝑒𝑑𝑟𝑜𝑛
6𝐶2
′
∶ 180° 𝑡𝑤𝑜 𝑓𝑜𝑙𝑑 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛 𝑎𝑏𝑜𝑢𝑡 110 𝑎𝑥𝑖𝑠 𝑝𝑎𝑠𝑠𝑖𝑛𝑔 𝑡𝑟𝑜𝑢𝑔 𝑡𝑒 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝑡𝑒 𝑒𝑑𝑔𝑒𝑠.
O 𝑬
𝟖𝑪𝟑 𝟑𝑪𝟐
= 𝟑𝑪𝟒𝟐
𝟔𝑪𝟐′
𝟔𝑪𝟒
(𝒙𝟐 + 𝒚𝟐 + 𝒛𝟐)
𝐴1
1 1 1 1 1
xyz 𝐴2
1 1 1 -1 -1
(𝒙𝟐 − 𝒚𝟐,𝟑𝒛𝟐 − 𝒓𝟐)
𝐸
2 -1 2 0 0
(𝒙, 𝒚, 𝒛) 𝑇1
3 0 -1 -1 1
(𝒙𝒚, 𝒚𝒛, 𝒛𝒙) 𝑇2
3 0 -1 1 -1
8𝐶3: ±120° 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝑠 𝑎𝑏𝑜𝑢𝑡 𝑎𝑛 𝑎𝑥𝑒𝑠 𝑡𝑟𝑜𝑢𝑔 𝑡𝑒 𝑡𝑟𝑖𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑓𝑎𝑐𝑒 𝑐𝑒𝑛𝑡𝑟𝑜𝑖𝑑𝑠 𝑜𝑓 𝑡𝑒 𝑜𝑐𝑡𝑎𝑒𝑑𝑟𝑜𝑛
3𝐶2 = 3𝐶42
∶ 180° 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛 𝑎𝑏𝑜𝑢𝑡 𝑡𝑒 𝑐𝑜𝑟𝑛𝑒𝑟𝑠 𝑜𝑓 𝑡𝑒 𝑜𝑐𝑡𝑎𝑒𝑑𝑟𝑜𝑛
𝒙′ 𝒚′ 𝒛′ 𝒙′ 𝟐 𝒚′ 𝟐 𝒛′ 𝟐
𝑬
𝑥 𝑦 𝑧 𝑥2 𝑦2 𝑧2
𝟖𝑪𝟑 𝑦 𝑧 𝑥 𝑦2 𝑧2 𝑥2
𝟑𝑪𝟐
= 𝟑𝑪𝟒𝟐
𝑥 −𝑦 −𝑧 𝑥2 𝑦2 𝑧2
𝟔𝑪𝟐′ 𝑦 𝑥 −𝑧 𝑦2 𝑥2 𝑧2
𝟔𝑪𝟒 𝑥 𝑧 −𝑦 𝑥2 𝑧2 𝑦2
𝑇1 ∶ 𝐵𝑎𝑠𝑖𝑠 𝑥, 𝑦, 𝑧
8𝐶3 ∶ 0 1 00 0 11 0 0
𝒳 = 0
3𝐶2 = 3𝐶42 ∶
1 0 00 −1 00 0 −1
𝒳 = −1
6𝐶2′ ∶
0 1 01 0 00 0 −1
𝒳 = −1
6𝐶4 ∶ 1 0 00 0 10 −1 0
𝒳 = 1
𝐸 8𝐶3 3𝐶2 = 3𝐶4
2 6𝐶2′ 6𝐶4
𝑻𝟏 3 0 −1 −1 1
𝑇2: 𝐵𝑎𝑠𝑖𝑠(𝑥𝑦, 𝑦𝑧, 𝑧𝑥)
8𝐶3 ∶ 𝑥𝑦 → 𝑦𝑧, 𝑦𝑧 → 𝑥𝑧, 𝑧𝑥 → 𝑥𝑦
0 1 00 0 11 0 0
𝒳 = 0
3𝐶2 = 3𝐶42 ∶ 𝑥𝑦 → −𝑥𝑦,𝑦𝑧 → 𝑦𝑧,𝑧𝑥 → −𝑥𝑧
−1 0 00 1 00 0 −1
𝒳 = −1
6𝐶2′ ∶ 𝑥𝑦 → 𝑥𝑦, 𝑦𝑧 → −𝑥𝑧, 𝑧𝑥 → −𝑦𝑧
1 0 00 0 −10 −1 0
𝒳 = 1
6𝐶4 ∶ 𝑥𝑦 → 𝑥𝑧, 𝑦𝑧 → −𝑦𝑧, 𝑧𝑥 → −𝑥𝑦
0 0 10 −1 0−1 0 0
𝒳 = −1
𝐸 8𝐶3 3𝐶2 = 3𝐶4
2 6𝐶2′ 6𝐶4
𝑻𝟐 3 0 −1 1 −1
𝐸: 𝐵𝑎𝑠𝑖𝑠(𝑥2 − 𝑦2 , 3𝑧2 − 𝑟2)
8𝐶3 ∶ 𝑥2 − 𝑦2 → 𝑦2 − 𝑧2 , 3𝑧2 − 𝑟2
→ (2𝑥2 − 𝑦2 − 𝑧2)
−
1
2−
1
23
2−
1
2
𝒳 = −1
3𝐶2 = 3𝐶4
2 ∶ 𝑥2 −𝑦2 → 𝑥2 −𝑦2, 3𝑧2 −𝑟2 → 3𝑧2 −𝑟2
1 00 1
𝒳 = 2
6𝐶2′ ∶ 𝑥2 − 𝑦2 → 𝑦2 − 𝑥2 , 3𝑧2 − 𝑟2 → 3𝑧2 − 𝑟2
1 00 −1
𝒳 = 0
6𝐶4 ∶ 𝑥2 − 𝑦2 → 𝑥2 − 𝑧2 , 3𝑧2 − 𝑟2
→ 2𝑦2 − 𝑥2 − 𝑧2
1
2−
1
2
−3
2−
1
2
𝒳 = 0
𝐸 8𝐶3 3𝐶2 = 3𝐶42 6𝐶2
′ 6𝐶4
𝑬 2 −1 2 0 0
Reducible Reps of O group
𝒳 𝑙 𝛼 = 𝑡𝑟𝑎𝑐𝑒 𝐷 𝑙 𝛼 =sin[(𝑙+
1
2)𝛼]
sin[𝛼
2]
8𝐶3 ∶2𝜋3
∶sin 𝑙 +
12
2𝜋3
sin 2𝜋6
= −1
3𝐶2 = 3𝐶42 = 6𝐶2
′ ∶ 𝒳 2
𝜋 = 1
6𝐶4: 𝒳 2
𝜋2 = −1
𝑬 𝟖𝑪𝟑 𝟑𝑪𝟐 = 𝟑𝑪𝟒
𝟐 𝟔𝑪𝟐′ 𝟔𝑪𝟒
𝜞𝒓𝒐𝒕𝟐 5 −1 1 1 −1
𝑎𝑖 =1
𝑁𝑘 𝒳
𝛤𝑖 𝒞𝑘 ∗ 𝒳(𝒞𝑘)
𝑘
𝐸 8𝐶3 3𝐶2 = 3𝐶42 6𝐶2
′ 6𝐶4
𝜞𝒓𝒐𝒕𝟐 5 −1 1 1 −1
𝐸 8𝐶3 3𝐶2 = 3𝐶4
2 6𝐶2′ 6𝐶4
𝑨𝟐 1 1 1 −1 −1 𝐸 8𝐶3 3𝐶2 = 3𝐶4
2 6𝐶2′ 6𝐶4
𝑬 2 −1 2 0 0
𝑎𝐴2=
1
24 1.1.5 + 8.1. −1 + 3.1.1 + 6. −1.1
+ 6. −1. −1 = 0
𝑎𝐸 =1
24 1.2.5 + 8. −1. −1 + 3.2.1 + 6.0.1
+ 6.0. −1 = 1
𝐸 8𝐶3 3𝐶2 = 3𝐶42 6𝐶2
′ 6𝐶4
𝜞𝒓𝒐𝒕𝟐 5 −1 1 1 −1
𝐸 8𝐶3 3𝐶2 = 3𝐶4
2 6𝐶2′ 6𝐶4
𝑻𝟏 3 0 −1 −1 1 𝐸 8𝐶3 3𝐶2 = 3𝐶4
2 6𝐶2′ 6𝐶4
𝑻𝟐 3 0 −1 1 −1
𝑎𝑇1=
1
24 1.3.5 + 8.0. −1 + 3. −1.1 + 6. −1.1
+ 6.1. −1 = 0
𝑎𝑇2=
1
24 1.3.5 + 8.0. −1 + 3. −1.1 + 6.1.1
+ 6. −1. −1 = 1
𝛤𝑟𝑜𝑡2 = 𝐸 + 𝑇2
The splitting is affected by following facts
Nature of metal ion : depends on the value of l
Arrangement of ligands around the metal ion
Nature of the ligands surrounding the metal ion
𝛤𝑟𝑜𝑡2 = 𝐸 + 𝑇2 , 𝛤𝑟𝑜𝑡
3 = 𝐴2 + 𝑇1 + 𝑇2
𝐼− < 𝐵𝑟− < 𝑆2− < 𝐶𝑙− < 𝑁𝑂3− < 𝑂𝐻−
References:
Group Theory- Application to the physics of
Condensed matter
M.S. Dresselhaus et al.
Fundamentals of Semiconductors: Physics
and Materials Properties
By Peter Y. Yu, Manuel Cardona
Thanks…
Subroto Mukerjee
Ananyo Moitra
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