Theory of Spin Waves in Rare Earth Metals with Non-Linear <i>c</i>-<i>f</i> Exchange Interaction Effect

Using a simple band model which produces a helical spin ordering, we show the magnetic phase diagram by minimizing the unperturbed energy which includes the cf exchange interaction by replacing the f -spin operator Sn with the expectation value 〈 Sn〉. As the cf exchange interaction increases, the helix and the cone structures appears without any crystal fields, and then the ferromagnetic structure becomes more stable. Secondly, we obtain formulae of spin-wave dispersions and show their instabilities on the second-order transition boundaries. Near the ferro-helix boundary, the spin-wave constant of the ferromagnetic spin-wave vanishes, while in the helical phase the whole region of the wave-number 0 < qz < Q shows softening where the helical wave-number Q decreases continuously. Thirdly, by the method of the double-time Green function, we derive the spin-wave dispersion at finite temperatures. Finally, anomalous properties in magnon dispersions at finite temperatures for Gd, Ho and those for diluted Tb-Y alloys are explained by use of numerical calculations.