The spin-wave energy in nickel is calculated along the principal axis directions from the center to the surface of the Brillouin zone, the calculation being based upon the theoretical energy bands of nickel as given by Hanus. The splitting of the up- and down-spin $d$ bands is treated as a parameter, adjusted to give the experimentally observed long-wavelength spin-wave energy. The temperature dependence of the magnetization is calculated on the basis of the spin-wave energies. It is found that the temperature analogous to the Debye temperature of lattice vibrations is of the order of ${10}^3$ °K, rather than ${10}^4$ °K, as simple models yield, that the number of Fourier coefficients (in real space) required to fit the dispersion curve corresponds to the six nearest atomic planes, that the energy gap for single-particle excitations is of the order of 1000°K (although sensitive to the parameters assumed), and that no cutoff momentum is indicated for the strong ferromagnetic assumptions made here. Furthermore, the coefficient of the $q^4$ term in the dispersion curve and the corresponding coefficient of $T^{\frac{5}{2}}$ in the magnetization law is about a factor of 5-10 larger than that of a nearest-neighbor Heisenberg ferromagnet. The range of validity of $T^{\frac{3}{2}}$ and $T^{\frac{5}{2}}$ terms in the magnetization-temperature law is restricted to temperatures less than 80°K.

• Received 28 July 1966

DOI:https://doi.org/10.1103/PhysRev.153.574