The low-energy excitations of the magnets are important for understanding the stability of the ferromagnetism and their thermal excitations. We describe in this chapter the spin-wave excitations in both the insulator and the metallic magnets on the basis of the random phase approximation (RPA). The spin waves in the insulator are useful for understanding those in the itinerant electron system. In the metallic system, there are individual spin-flip electron excitations called the Stoner excitations apart from the spin waves. We also deal with such excitations. Next, we introduce the dynamical spin susceptibility and explain its physical meaning. With use of the transverse dynamical spin susceptibility, we rederive the spin wave excitations as well as the Stoner excitations. This chapter also provides us with a basic knowledge of the dynamical susceptibility which will be used in the spin fluctuation theory for the weak magnets given in the next chapter.
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