Zeeman Splitting of Vibronic Levels for Octahedral Actinide and Lanthanide Complexes, Free and in Crystals

The splitting of vibronic levels in the presence of a magnetic field is determined in terms of a few parameters for an octahedral complex in the limit of Zeeman interaction (A) small compared to vibronic splitting, (B) comparable to vibronic interaction, and (C) large compared to vibronic interaction. The relative density of phonon states controls the size of the vibronic splitting in crystals, and is influential in determining whether case A, B, or C above applies. Data for cubic ${\mathrm{Cs}}_2$U${\mathrm{Br}}_6$ show that vibronic levels are split according to the parent electronic level, illustrating case C, because the vibronic splitting is much smaller than that which would have occurred for a local-mode or free-ion complex. The free U${\mathrm{Br}}_6^{2-}$ belongs to case A; the reduction to case C is due to the effect of the phonon relative density of states. The vibronic Zeeman effect is thus a tool for learning about the relative density of phonon states, particularly in localized modes. Similar conclusions apply to vibronic shifts, so that for nonlocalized phonons the vibronic spectra should reflect the density of states of the phonons without the necessity of correction for vibronic shifts or splittings. For case A, particularly applicable to the free-ion complex, the Zeeman splitting of a vibronic level is isotropic and consists of three equally spaced levels with a $g$ value that is one-half that of the parent electronic state. Some general relations between vibronic level differences are derived which are independent of particular models and values of matrix elements.