Multi-Electron Crystal Field Theory Weak Field Scheme ...ecee.colorado.edu/~wpark/class/Crystal/2005 ecen 5005 class 18.pdf · Tanabe-Sugano Diagram • The methodology developed

ECEN 5005

Crystals, Nanocrystals and Device Applications

Class 18

Group Theory For Crystals

• Multi-Electron Crystal Field Theory

• Weak Field Scheme

• Strong Field Scheme

• Tanabe-Sugano Diagram

1

Notation Convention for Spectroscopic Terms

• Russell-Saunders coupling scheme

- A state is specified by a set of quantum numbers, (L, ML, S, MS).

- Excluding spin-orbit interaction, the states having the same L and S

are usually degenerate. Thus, a term is conventionally represented

by L and S only.

- L is denoted by a capital letter,

i.e. L = 0 → S, L = 1 → P, L = 2 → D, L = 3 → F, etc.

- S is represented by adding (2S+1) as a superscript in front of L.

- Example:

The ground term for a free V3+ ion has L = 3 and S = 1. → 3F

• J-J coupling scheme

- A state is specified by a set of quantum numbers, (L, S, J, MJ).

- Spin-orbit interaction is not ignored and thus the states with

different J can have different energies even though they have the

same L and S. Thus, a term needs to be represented by L, S and J.

- L and S are denoted by the same convention as above.

- J is represented as a subscript after L.

- Example:

The ground term for a free Pr3+ ion has

L = 5, S = 1 and J = 4. → 3H4

2

Application of Russell-Saunders Coupling Scheme

• Consider a V3+ ion which has 2 electrons in 3d shell.

- Each electron has l=2 and s=1/2.

- They can have any values of ml = -2, -1, 0, 1, 2 and ms = ±1/2.

- However, they cannot have the same values of ml and ms, as

prohibited by the Pauli exclusion principle.

• In order to find multi-electron states, we need to obtain L by adding l1

and l2 and S by adding s1 and s2.

• Recall the angular momentum addition rule:

- L is the sum of l1 and l2, therefore allowed values of L are

L = |l1 - l2|, |l1 - l2| + 1, … , l1 + l2 - 1, l1 + l2

- Once L is determined, then the allowed values of ML are

ML = -L, -L + 1, … , L - 1, L

- Same principle applies to S and J.

• For V3+ ion, l1 = 2 and l2 = 2.

- The allowed values of L are 0, 1, 2, 3, 4.

- Similarly, S = 0, 1 as s1 = 1/2 and s2 = 1/2.

• Pauli exclusion principle prohibits (L=0, S=1), (L=2, S=1), (L=4,

S=1), (L=1, S=0) and (L=3, S=0).

• Thus the allowed terms are (L=0, S=0), (L=1, S=1), (L=2, S=0), (L=3,

S=1) and (L=4, S=0), or 1S, 3P, 1D, 3F and 1G.

3

Application of Russell-Saunders Coupling Scheme

• If the hydrogen-like atom model is valid, then all of the terms

obtained before must have the same energy, because we are dealing

with a fixed n (principal quantum number) for all electrons.

• However, in multi-electron systems, there is an important term that

was not included in the hydrogen-like atom problem. That is the

Coulomb repulsion between electrons.

• The Coulomb interaction between electrons split the energy levels

according to L and S, which is why we denote a term with L and S in

the Russell-Saunders coupling scheme.

• An empirical rule is

- The term with the largest spin quantum number has the lowest

energy.

- Among the terms with the same spin quantum number, largest

angular momentum quantum number gives the lowest energy.

- This rule is very effective in finding the ground level.

- This is the celebrated Hund’s rule.

• Applying the Hund’s rule to the case of V3+ ion, we find the ground

term is 3F.

4

Multi-Electron Crystal Field Theory

• When an ion with many electrons is placed in a crystal field, the

crystal field will shift and split the energy levels of the multi-electron

system.

• In addition to the simple hydrogen-like atom Hamiltonian, there are

three main interactions that need be included, the Coulomb

interaction between electrons, spin-orbit interaction, and the crystal

field.

• In general, it is impossible to obtain exact solutions. Thus, we solve

for the problem including only the largest interactions and then add

the smaller terms later as small perturbations.

• Weak field scheme: The strength of the crystal field is small

compared to electron-electron interaction and spin-orbit interaction.

- First, obtain the energy levels of free ion without the crystal field.

- Then, include the crystal field effect and investigate the splitting of

free ion energy levels due to the crystal field.

• Strong field scheme: The strength of the crystal field is much larger

than the electron-electron interaction and spin-orbit interaction.

- First, obtain the energy levels and wavefunctions of one electron

system under the crystal field.

- Then, include the electron-electron interaction and spin-orbit

interaction and investigate the splitting of the one-electron crystal

field levels due to the additional interactions.

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Application of weak field scheme for d2 system

• As shown before, the allowed terms for free V3+ ion are 1S, 3P, 1D, 3F

and 1G. For each term, there are (2L+1) allowed values of ML. That is,

each term has a degeneracy of (2L+1). These degeneracy is partly

lifted by crystal field.

• The reduction scheme may be obtained by using the reduction

formula just as we did for the single-electron case.

• In an octahedral crystal field,

free ion term (degeneracy)

terms in an octahedral field (degeneracy)

1S (1) 1A1 (1) 3P (3) 3T1 (3) 1D (5) 1E (2) + 1T2 (3) 3F (7) 3A2 (1) + 3T1 (3) + 3T2 (3) 1G (9) 1A1 (1) + 1E (2) + 1T1 (3) + 1T2 (3)

free ion weak crystal field

3F

1D

3P

1G

1S

3T1

3T2

3A2

1T2

1E

3T1

1A1

1T2

1T1

1E

1A1

free ion weak crystal field

3F

1D

3P

1G

1S

3T1

3T2

3A2

1T2

1E

3T1

1A1

1T2

1T1

1E

1A1

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Strong Field Scheme

• Start with the one-electron energy

levels and wavefunctions determined

by the single-electron crystal field

theory.

• As shown previously, the single d-

electron states split into two levels in

an octahedral crystal field.

• Let us denote the triply degenerate

lower level (E = −4Dq) as t2g and the

doubly degenerate upper level (E = 6Dq) as eg.

4Dq

6Dq

free ion octahedral field

φξ, φζ, φη

φu, φv

5-fold degenerated-shell

t2

e

E = 0

4Dq

6Dq

free ion octahedral field

φξ, φζ, φη

φu, φv

5-fold degenerated-shell

t2

e

E = 0

• There are three possible configurations, (1) (t2g)2 - both electrons in

t2g, (2) t2geg - one in t2g and one in eg, (3) eg2 - both electrons in eg.

4Dq

6Dq

t2

e

E = 0

4Dq

6Dq

t2

e

E = 0

• Ignoring the electron-electron interaction, the energies of the three

configurations are E = -8Dq for (t2g)2, 2Dq for t2geg and 12Dq for eg2.

7

Reduction of Direct-Product Representation

• The two-electron states are represented by the direct-product

representations.

- (t2g)2 = T2g × T2g, t2geg = T2g × Eg, and eg2 = Eg × Eg

- These direct-product representations are reducible.

• Recall the discussion on direct-product representation in Class 9. The

character of the direct-product matrix is the product of characters of

individual matrices.

( ) ( ) ( )BABA χ⋅χ=×χ

• The above equation allows us to determine the characters of the

direct-product representation. Then, we can apply the usual reduction

formula, as done formally in Class 9.

• For example, the characters for T2g × T2g are

E 8C3 3C2 6C'2 6C4 i 8iC3 3iC2 6iC'2 6iC4T2g 3 0 -1 1 -1 3 0 -1 1 -1

T2g × T2g 9 0 1 1 1 9 0 1 1 1

• Now use the reduction formula to find

gggggg TTEATT 21111 +++=×

gggg TTTE 211 +=×

ggggg EAAEE ++=× 21

8

Strong Field Scheme

• For each irreducible representation, we may have spin singlet (S = 0)

and triplet (S = 1) states. However, some of these states are forbidden

by the Pauli exclusion principle.

strong field configuration (degeneracy, energy)

terms in an octahedral field (degeneracy)

(t2)2 (9, E = -8Dq) 1A1 (1) + 1E (2) + 1T2 (3) + 3T1 (3)

t2e (12, E = 2Dq) 1T1 (3) + 1T2 (3) + 3T1 (3) + 3T2 (3)

e2 (4, E = 12Dq) 1A1 (1) + 1E (2) + 3A2 (2)

strong field configuration strong crystal field

(t2)2

3T1

3T2

3A2

1T2

1E

3T1

1A1

1T2

1T1

1E

1A1

t2e

e2

strong field configuration strong crystal field

(t2)2

3T1

3T2

3A2

1T2

1E

3T1

1A1

1T2

1T1

1E

1A1

t2e

e2

9

Correlation Diagram

• Both weak field scheme and strong field scheme yield the same set of

final terms. However, their order in energy is very different. So in

order to get the correct energy levels, one must use the appropriate

scheme that is right for the system of interest.

free ion level

weak field energy level

strong field energy level

strong field configuration

10

Tanabe-Sugano Diagram

• The methodology developed for two-electron system may be

extended for multi-electron systems. Energy levels of a multi-electron

system in a crystal field, calculated by using the strong field scheme,

are expressed as a function of crystal field strength.

- First devised by Tanabe and Sugano, 1954.

- Excellent description for transition metal ions in solids.

• Tanabe-Sugano diagram for Cr3+ ion in octahedral field

11

Tanabe-Sugano Diagram

12

Tanabe-Sugano Diagram for Mn2+ (5 d-electrons)

• When crystal field is moderate, the ground term is 6A1 which has a

strong field configuration of -

according to Hund’s rule.

232et

10Dq

t2

e

6A12T2

10Dq

10Dq

t2

e

6A12T2

10Dq• At extremely strong crystal field, 2T2

term which has a strong field

configuration, t , becomes the ground

term. – crystal field energy becomes

greater than spin pairing energy.

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