Electron microscopy
 
eg & t2g Symmetries/States
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Figure 3137a shows the molecular orbital energy structure for an octahedral TMO6 cluster. The split t2g and eg orbitals can only mix with symmetry adapted linear combinations of the 6 oxygen (O) 2p orbitals having the same symmetry. The separation between the t2g and eg levels is called the LF (ligand field) splitting parameter ΔO. σ-overlap is larger than π-overlap so that the upward shift of the eg level is larger than the upward shift of the t2g level.

Molecular orbital energy structure for an octahedral TMO6 cluster

Figure 3137a. Molecular orbital energy structure for an octahedral TMO6 cluster.

As shown in Figure 3137b, in a cubic transition-metal (TM) oxide crystal, the divalent TM ion is in a site that has octahedral Oh symmetry and the d-levels split into threefold degenerate (lower energy) t2g states and twofold degenerate (higher energy) eg states that can accommodate six and four electrons, respectively (including spin states). The tetragonal symmetry splits the levels further. The t2g states split into a singlet, dxy, and a doublet dxz and dyz. The eg states split into d3z2-r2 and dx2-y2 levels.

Ligand field splitting of d orbitals in an octahedral ligand field

Figure 3137b. Ligand field splitting of d orbitals in an octahedral ligand field.

Figure 3137c shows the ligand field splitting of d orbitals for Fe in an octahedral ligand field, resulting in lower energy t2g and higher energy eg orbitals and presenting the spin states of Fe in octahedral field. Two cases are show on the right of the figure. If an exchange energy larger than the t2g - eg splitting favors parallel spin alignment, then a high spin ground state is obtained. If the exchange energy is smaller than the t2g - eg splitting, then a low spin ground state is obtained. In this case, the fourth and fifth electrons are not in the high-energy eg state but are located in the lower t2g state.

Ligand field splitting of d orbitals for Fe in an octahedral ligand field

Figure 3137c. Ligand field splitting of d orbitals for Fe in an octahedral ligand field.

In some TM (transition metal) cases, the filling of orbitals with electrons may affect the local structure and thus induce geometrical distortion around the TM ion. The Jahn–Teller effect, also called Jahn–Teller distortion, describes this type of distortions. A typical Jahn-Teller ion is Mn3+ as shown in Figure 3137d. The ion in the high-spin configuration contains a single electron in the upper eg state when it is placed in an octahedral LF (ligand field). A tetragonal distortion can lower the energy of the system. The lowering in total energy is due to the lowering of one of the eg orbitals by lengthening the bond along the z axis. Note that the overall energy of the system is not further lowered by splitting the t2g state because the center of gravity is retained.

John-Teller effect for Mn3+ (3d4)

Figure 3137d. John-Teller effect for Mn3+ (3d4).

 

 

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