Hyperfine interactions in non-relativistic approximation

The hyperfine structure (splitting) of energy levels is mainly caused by electric and magnetic multipole interactions between the atomic nucleus and electronic shells. From the known data on hyperfine structure we can determine the electric and magnetic multipole momenta of the nuclei, their spins and other parameters. 

In a non-relativistic approximation the usual fine structure (splitting) of the energy terms is considered as a perturbation whereas the hyperfine splitting - as an even smaller perturbation, and they both are calculated as matrix elements of the corresponding operators with respect to the zero-order wave functions. 

As usual, the operators describing hyperfine interactions are to be expressed in terms of irreducible tensors. Then we are in a position to find formulas for their matrix elements. The corresponding operator, caused 

Hyperfine interactions control the splitting of the usual atomic level with given J into a number of components, each of them corresponding to a certain value of vectorial sum J +1 (I is the angular momentum of the nucleus), describing total momentum of atom F, i.e. 

Due to the presence of the interaction between J and I they will not be exact quantum numbers, only their vectorial sum (total momentum F) will give the exact quantum number. However, this interaction is fairly small, therefore, the hyperfine splitting may be considered separately for each level via calculation of the corresponding matrix elements. The irreducible form of operator (22.1) is as follows  

The representation of hyperfine interactions in the form of multipoles follows from expansion of the potentials of the electric and magnetic fields of a nucleus, conditioned by the distribution of nuclear charges and currents, in a series of the corresponding multipole momenta. It follows from the properties of the operators obtained with respect to the inversion operation that the nucleus can possess non-zero electric multipole momenta of the order k = 0,2,4. and magnetic ones with 

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