Part 1 (Introduction. Crystal Field splitting of the d-type Atomic Orbitals. Taylor's notation
Part 2 (Energy Plane. Energy Surfaces. g-Tensor Surfaces
The g-value is a measure of energy splitting of electronic states in the external magnetic field. Both in experiment and in theory only the energy gap (the absolute value) can be determined. In other words, the g-values are defined only up to the sign. Below are the g-values vs. at V^{*} = 0 (D_{4} symmetry):
The above | g_{z} | plot is a cross-section of the 3D | g_{z} | surface along the - axis (V^{*} = 0):
The g_{z} dependence can be made smooth if we change the sign of one of its branches, for example like this:
This is Oosterhuis-Lang sign choice. The above picture is the cross-section along V^{*} = 0 of the three g-value surfaces in the previous section if the sign of g_{z} is changed to the opposite.
g_{z} | g_{y} | g_{x} |
For this sign choice in the pure octahedral field ( = 0, V^{*} = 0) g_{x} = g_{y} = 2 and g_{z} = -2.
There are two other popular choices.
Taylor's sign choice: the g_{x} surface is taken with the sign opposite to the images in Part 2. At ( = 0, V^{*} = 0) g_{x} = g_{y} = 2 and g_{x} = -2.
McGarvey's sign choice: all g-value surfaces are taken with the signs opposite to the images in Part 2. For the pure octahedral field g_{x} = g_{y} = g_{z} = -2. For this choice all three g-values tend to 2 (free electron value) when the upper level is well separated from the others.
The g-value signs can only be determined with the use of additional information (either theoretical or experimental).
Note that the three-level model cannot be continuously reduced to the case of the free electron because it is already assumed that the crystal field is strong enough so that the e_{g} doublet is shifted high upward.
There are two sign-independent quantities: absolute value and zero.
Absolute value. For each g-tensor component 0 | g | 4. At the origin of the energy plane | g_{x} | = | g_{y} | = | g_{z} | = 2. The | g | levels divide the energy plane into three narrow (1, 2 and 3) and three wide (4, 5 and 6) regions:
| g |-value regions
The table below summarizes the qualitative properties of the regions. The asymptotic values along the six semi-axes are presented in the last column.
Region | MO position | |g|-pattern | Semi-axis asymptote |
1 (narrow) | d_{xy} is the lowest | |g_{z}| > 2 > |g_{x}|, |g_{y}| | |g_{z}| = 4, |g_{x}|, |g_{y}| = 0 |
2 (narrow) | d_{yz} is the lowest | |g_{x}| > 2 > |g_{y}|, |g_{z}| | |g_{x}| = 4, |g_{y}|, |g_{z}| = 0 |
3 (narrow) | d_{xz} is the lowest | |g_{y}| > 2 > |g_{x}|, |g_{z}| | |g_{y}| = 4, |g_{x}|, |g_{z}| = 0 |
4 (wide) | d_{xz} is the highest | |g_{y}| < 2 < |g_{x}|, |g_{z}| | |g_{y}|, |g_{x}|, |g_{z}| = 2 |
5 (wide) | d_{xy} is the highest | |g_{z}| < 2 < |g_{x}|, |g_{y}| | |g_{z}|, |g_{x}|, |g_{y}| = 2 |
6 (wide) | d_{yz} is the highest | |g_{x}| < 2 < |g_{y}|, |g_{z}| | |g_{x}|, |g_{y}|, |g_{z}| = 2 |
The mechanical angular momentum is the sum of the spin ( S ) and orbit ( L ) momentums: . If J = 0 then the system displays no rotation. However, it can still interact with the external magnetic field because in the Hamiltonian they are combined with different weights ( g 2 ).
It can also be that for non-zero S, L and J the combination ( L + g_{e} S ) is equal to zero. This is the case for the third level:
g_{x} = 0 | J_{x} = 0 |
g_{y} = 0 | J_{y} = 0 |
g_{z} = 0 | J_{z} = 0 |
The above results were obtained under the assumption that g_{e} = 2 and the orbital reduction factor k = 1 (no reduction due to the presence of a covalent bond). For non-zero orbital reduction ( k < 1 ) g g (1 + 2 k )/3.
Part 4 (Calculations. Programs. References)
Part 5 (Experimental data processing)
Please e-mail me at nikolai@shokhirev.com |
©Nikolai Shokhirev, 2002-2003.