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We show that the magnetism of double perovskite AFe_{1/2}M_{1/2}O_3 systemsmay be described by the Heisenberg model on the simple cubic lattice, whereonly half of sites are occupied by localized magnetic moments. Thenearest-neighbor interaction J_1 is more than 20 times the next-nearestneighbor interaction J_2, the third-nearest interaction along the spacediagonal of the cube being negligible. We argue that the variety of magneticproperties observed in different systems is connected with the variety ofchemical ordering in them. We analyze six possible types of the chemicalordering in 2x2x2 supercell, and argue that the probability to find them in areal compound does not correspond to a random occupation of lattice sites bymagnetic ions. The exchange J_2 rather than J_1 define the magnetic energyscale of most double perovskite compounds that means the enhanced probabilityof 1:1 short range ordering. Two multiferroic compounds PbFe_{1/2}M_{1/2}O_3(M=Nb, Ta) are exceptions. We show that the relatively high temperature ofantiferromagnetic transition is compatible with a layered short-range chemicalorder, which was recently shown to be most stable for these two compounds [I.P. Raevski, {\em et al.}, Phys.\ Rev.\ B \textbf{85}, 224412 (2012)]. We showalso that one of the types of ordering has ferrimagnetic ground state. Theclusters with short-range order of this type may be responsible for aroom-temperature superparamagnetism, and may form the cluster glass at lowtemperatures.

Magnetic interactions in disordered perovskite PbFe_{1/2}Nb_{1/2}O_3 and related compounds. Dominance of nearest-neighbor interaction