Unique Definition of the Zeeman-Splitting $g$ Tensor of a Kramers Doublet

It has been widely accepted in the past that the $g$ tensor describing the Zeeman splitting of a Kramers doublet is not uniquely defined. Here we show that with two basic requirements, (i) the invariance of the Zeeman Hamiltonian under symmetry transformations and (ii) its continuous change with the variations of the parameters of the system (geometry and crystal field), a unique determination of the elements of the $g$ tensor is achieved.

Figure 1

Calculated $g$ factor for the ground state Kramers doublet of cubic actinide complexes by ab initio [16] (circles) and by crystal field model (line) as function of the ratio between the cubic crystal field and the spin-orbit coupling parameters of the actinide ion.

Figure 2

Ab initio calculated multiplets of tetragonally distorted octahedral complex ${\mathrm{PaCl}}_6^{2-}$. Continuous lines correspond to ${\Gamma }_7(E_{3/2})$ and dashed lines to ${\Gamma }_6(E_{1/2})$ irreps of the $D_{4h}$ group. $R_0$ is the equilibrium Pa-Cl distance in the octahedral geometry and $\Delta R$ is the deviation of equatorial (left) and axial (right) bond lengths.

Figure 3

Ab initio calculated $g$ factors for the ground ${\Gamma }_7$ (a), the first excited ${\Gamma }_7$ (b), and the ground ${\Gamma }_6$ (c) states (the three lowest states of Fig. 2) as functions of the tetragonal distortion (see Fig. 2 for the details). Continuous lines correspond to $g_{\parallel }$ and dashed lines to $g_{\perp }$.