We obtain the energy spectrum and calculate the partition function for Zener's $s-d$ exchange-interaction model in the strong-coupling limit and with constant interaction-$\frac{J}{N}$ between impurity ions and conduction electrons. The system is shown to undergo a transition to an ordered state independently of the sign of $J$ and for arbitrarily small impurity concentration $c$. The critical temperature $T_c$ is given by $8k{T}_c=J{\mathrm{}\left(2c\right)\mathrm{}}^{\frac{1}{2}}$ for ferromagnetic coupling ($J>\mathrm{}0\mathrm{}$) and by $8k{T}_c=\left|J\right|[1-{\mathrm{}(1-2c)\mathrm{}}^{\frac{1}{2}}]$ for antiferromagnetic coupling ($J<\mathrm{}0\mathrm{}$) when $0\le c\le \frac{1}{2}$. When $c>\mathrm{}\frac{1}{2}$ and $J<\mathrm{}0\mathrm{}$ our procedure for evaluating the statistical sum becomes invalid.

• Received 23 August 1965

DOI:https://doi.org/10.1103/PhysRev.141.479