by Nikolai V. Shokhirev
Prof. F. Ann Walker Research Group, Department of Chemistry, University of Arizona, Tucson, Arizona 85721, USA
A molecule (radical, complex) with electron spin S interacts with the external magnetic field according to the following Hamiltonian (the electron Zeeman interaction)
Here
is the electron magnetic moment,
= 9.274015 4 x 10^{-24} J/T is the Bohr magneton, |
m_{e} = 9.109 389 7 x 10^{-31} kg,
h = 6.626 075 5 x 10^{-34} J s , | 1.054 572 66 x 10^{-34} J s , |
S is the dimensionless electron spin (so that the electron angular moment is ),
is the electron g-tensor, that depends on the molecule and its electronic state.
If the g-tensor is proportional to the identity matrix then it is called isotropic (or scalar) and g is called the g-factor. For example, for the free electron g = g_{e} = 2.002319304386. Assuming that the g-tensor is isotropic and the external magnetic field is directed along the z-axis, the z-projections of the spin are
It gives the following energy levels
The above energies can be used for calculation of the average spin projection
For typical experimental magnetic field strength and temperature the value of << 1 and the exponents can be expanded
The simplified formula gives
Here were used
Consequently, the magnetic moment per molecule is
Here is the magnetic susceptibility (per molecule).
These results can be easy generalized to the case of arbitrary g-tensor.
The electron Zeeman Hamiltonian can be rewritten in terms of the effective magnetic field
In the expression for energy the term g B_{0} should be substituted with . The direction of the effective field is the spin quantization axis:
The magnetic susceptibility is also replaced with this tensor
Remark. If the average spin further average over orientations (see Averaging of vectors and tensors ) then
where
Back to Chemical Shifts.
©Nikolai V. Shokhirev, F. Ann Walker, 2002