Magnetic frustration in the spinel compounds Ge Co_2 O_4 and Ge Ni_2 O_4

In both spinel compounds GeCo${}_2$O${}_4$ and GeNi${}_2$O${}_4$ which order antiferromagnetically (at $T_N=23.5K$ and $T_{N_1}=12.13K$, $T_{N_2} = 11.46 K$)withdifferentCurieWeisstemperatures($T_{CW}$=80.5 K and -15 K), the usual magnetic frustration criterion $f=\mathrm{\mid }{T}_{CW}\mathrm{\mid }\mathrm{/}{T}_N>>1$ is not fulfilled. Using neutron powder diffraction and magnetization measurements up to 55 T, both compounds are found with a close magnetic ground state at low temperature and a similar magnetic behavior (but with a different energy scale), even though spin anisotropy and first neighbor exchange interactions are quite different. This magnetic behavior can be understood when considering the main four magnetic exchange interactions. Frustration mechanisms are then enlightened.