Cartesian form of the Hamiltonian operator

We now return to the spherical harmonic form of the quadrupole Hamiltonian, equation (4.29), which is 

The dominant term is that with q = 0 it may be rewritten in the form 

Equation (4.35) can now expanded in the Cartesian coordinate system defined in figure 4.1. We replace RpRi cos0ip by J]XPIXII where the X, represent X, Y, Z. Hence equation (4.35) becomes j 

Interactions arising from nuclear magnetic and electric moments 

The Qjk are components of a second-rank cartesian tensor called the nuclear quadrupole tensor, and the Vjk are components of the electric field gradient tensor. Both Q and V are traceless, symmetric tensors of the second rank. The electric field gradient tensor components can be written in a more compact form by noting that 

This relationship may be demonstrated considering the two possible cases, j = k and j k. First we let Xj = X = X then 

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