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A possibility to describe magnetism in the iron pnictide parent compounds interms of the two-dimensional frustrated Heisenberg $J_1$-$J_2$ model has beenactively discussed recently. However, recent neutron scattering data has shownthat the pnictides have a relatively large spin wave dispersion in thedirection perpendicular to the planes. This indicates that the third dimensionis very important. Motivated by this observation we study the $J_1$-$J_2$-$J_c$model that is the three dimensional generalization of the $J_1$-$J_2$Heisenberg model for $S = 1/2$ and S = 1. Using self-consistent spin wavetheory we present a detailed description of the staggered magnetization andmagnetic excitations in the collinear state. We find that the introduction ofthe interlayer coupling $J_c$ suppresses the quantum fluctuations andstrengthens the long range ordering. In the $J_1$-$J_2$-$J_c$ model, we findtwo qualitatively distinct scenarios for how the collinear phase becomesunstable upon increasing $J_1$. Either the magnetization or one of the spinwave velocities vanishes. For $S = 1/2$ renormalization due to quantumfluctuations is significantly stronger than for S=1, in particular close to thequantum phase transition. Our findings for the $J_1$-$J_2$-$J_c$ model are ofgeneral theoretical interest, however, the results show that it is unlikelythat the model is relevant to undoped pnictides.

Three dimensional generalization of the $J_1$-$J_2$ Heisenberg model on a square lattice and role of the interlayer coupling $J_c$