Random-phase approximation for critical temperatures of collinear magnets with multiple sublattices: $\mathrm{Gd}X$ compounds $(X=\mathrm{Mg},\mathrm{Rh},\mathrm{Ni},\mathrm{Pd})$

An efficient scheme for evaluating the critical temperatures of ferromagnetic, antiferromagnetic, and ferrimagnetic crystals with multiple sublattices is presented. The approach is based on a pairwise Heisenberg Hamiltonian and a random-phase approximation (Tyablikov’s decoupling) for magnon Green’s functions. The pair exchange interactions are derived from self-consistent electronic structure calculations using a magnetic force theorem. The developed technique is applied to hexagonal gadolinium and its selected intermetallic compounds $\mathrm{Gd}X$ $(X=\mathrm{Mg},\mathrm{Rh},\mathrm{Ni},\mathrm{Pd})$ with CsCl and CrB structures. The calculated critical temperatures are quite sensitive to a neglect of the nonmagnetic $\left(X\right)$ element; their values are in a fair agreement with experiment.