Invariants in the paramagnetic resonance spectra of impurity-doped crystals

We show that in cubic crystals with anisotropic impurity centers the sum of squares of the magnetic resonance [electron paramagnetic resonance (EPR)] frequencies is invariant with respect to the magnetic field direction. The connection between such an invariant and the $g$-tensor components of the impurity is derived for different types of centers. The established regularity is confirmed experimentally for the spin-noise spectra of a cubic $\mathrm{Ca}{\mathrm{F}}₂\ensuremath{-}{\mathrm{Nd}}³⁺$ crystal. We show how this property of the EPR spectra can be efficiently used for the assignment of paramagnetic centers in cubic crystals.