NMR Tutorial Spherical Tensors

An alternative way to calculate product operators from the density matrix is to use spherical harmonics for the operator basis instead of the cartesian basis matrices E/2, Iz, Ix and Iy. As with the cartesian basis 4n basis matrices are needed for a system of n spins. For one spin the basis matrices are the identity matrix

the Zeeman order population with the coherence order 0

the positive shift operator with the coherence order +1

and the negative shift operator with the coherence order -1

Products of operators like I1+I2+, I1+I2-, I1+I2z or I1zI2z are used to represent interactions between spins.

The spherical tensor basis does not have a geometric interpretation like the vector model. They are based on the spherical harmonics Yl,m that are known from the quantum mechanical rotator and the angular components of the hydrogen atom orbitals. Their usefulness comes from the fact that the coherence order can be easily calculated as the sum of all coherence orders of the basis operators in the product operator. Of the examples above the operator I1+I2+ is of the order 2, I1+I2z is of the order 1 and I1+I2- is of the order 0.

All base matrices are real, the orientation of the magnetic moments must therefore be given by a complex factor A⋅e with the magnitude A and the imaginary phase φ.