Microscopic origins of the zero-field splitting parameters for 6S(3d5) state ions at tetragonal symmetry crystal field

The microscopic origin of the zero-field splitting (ZFS) parameters including D and (a+2F/3), for 6S(3d5) state ion at tetragonal symmetry crystal field (CF), taking into account the electronic magnetic interactions, i.e. the spin-spin (SS), the spin-other-orbit (SOO), and the orbit-orbit (OO) interactions, besides the well-known spin-orbit (SO) interaction, have been investigated using the complete diagonalization method (CDM) and the microscopic spin Hamiltonian theory. Although the SO mechanism is the most important one, the contributions to the ZFS parameters D and (a+2F/3) due to the four additional mechanisms: SS, SOO, OO, and the combined SO∼SS∼SOO∼OO coupling mechanism, are appreciable and shall not be omitted. The individual contributions to the ZFS parameters arising from the spin quartet states and the spin doublet states have been analyzed. It is shown that the ZFS parameters D and (a+2F/3) arise from the spin quartet states as well as the combined effects between the spin doublet states and the spin quartets states, whereas the contributions to D and (a+2F/3) from the spin doublet states are zero. Our investigations show that the rank-2 ZFS parameter D primarily results from the spin quartet states, whereas the rank-4 ZFS parameter (a+2F/3) primarily results from the combined effects between the spin doublet states and the spin quartet states. The contributions to the rank-2 ZFS parameter D from the net spin quartet states exceed 95% and the contributions to the rank-4 ZFS parameter (a+2F/3) from the combined effects between the spin doublet states and the spin quartet states exceed 88.2% for the selected ranges of the crystal field parameters. The dependence of the ZFS parameters D and (a+2F/3) on the CF parameters Dq, B20, and Bl40 for 6S(3d5) state ions at tetragonal symmetry have been studied. It is found that the relations hold: |D(-Dq)|∼-|D(Dq)|, (a+2F/3)(-Dq)∼(a+2F/3)(-Dq) for |Dq|>800 cm-1. The illustrative evaluation is performed for typical crystal materials: Mn2+: Rb2CdF4, Mn2+: K2MgF4, and Mn2+: K2ZnF4 crystals. The good agreements between the theoretical values and the experimental finding are obtained.