Direct determination of weak exchange interactions in ${\mathrm{Cr}}^{3+}$-doped spinel Zn${\mathrm{Ga}}_2$${\mathrm{O}}_4$

Low-temperature electron-spin-resonance spectra of ${\mathrm{Cr}}^{3+}$-doped Zn${\mathrm{Ga}}_2$ ${\mathrm{O}}_4$ exhibit numerous weak lines, which are due to isolated non-nearest-neighbor ${\mathrm{Cr}}^{3+}$-${\mathrm{Cr}}^{3+}$ pairs. From an analysis of these spectra it is possible to determine the magnetic exchange interactions up to the sixth-neighbor shell. The assignment of the lines to definite types of pairs is facilitated by using the strain-modulation technique. If the interaction is written as $H_{\mathrm{ex}}=-J_i{\overrightarrow{\mathrm{S}}}_0·{\overrightarrow{\mathrm{S}}}_i+H_{\mathrm{an}}$ the exchange constants $J_i$ are found to be: $J_2=-0.94\mathrm{}$ ${\mathrm{cm}}^{-1\mathrm{}}$, $J_3=-1.22\mathrm{}$ ${\mathrm{cm}}^{-1\mathrm{}}$, $J_4=-0.80\mathrm{}$ ${\mathrm{cm}}^{-1\mathrm{}}$, $J_5=-0.45\mathrm{}$ ${\mathrm{cm}}^{-1\mathrm{}}$, and $J_6=-0.55\mathrm{}$ ${\mathrm{cm}}^{-1\mathrm{}}$. The error margin is ±0.01 ${\mathrm{cm}}^{-1\mathrm{}}$ in all cases. The anisotropic part of the interaction $H_{\mathrm{an}}$ can be fully accounted for by magnetic dipole-dipole interaction. The results are discussed in terms of a simple exchange-path model. It turns out that anion-anion interaction is comparable in strength with anion-cation interaction. Using the above-given set of $J_i$ values, together with the previously determined value of $J_1=-22.2\mathrm{}\pm 0.5$ ${\mathrm{cm}}^{-1\mathrm{}}$, it is possible to reproduce the asymptotic Curie temperature of Zn${\mathrm{Cr}}_2$ ${\mathrm{O}}_4$ to within an accuracy of 10%.