## Abstract

The long wavelength non-interacting spin wave energy for metals at low temperatures is expressed as $\hslash\omega_q$ = Dq$^2$ = (D$_0$ + D$_1$T$^2$)q$^2$. The dependence of the coefficient D$_1$ on the density of states function, the number of electrons per atom, n, and the effective short range interaction energy, I, is discussed. The variation of D with T$^2$ comes from the change with temperature of the relative occupation $\zeta$ and the chemical potentials of the $\pm$ spin sub-bands as well as from the direct asymptotic expansion of the Fermi distribution functions occurring in the expression for D. Although the calculation is based essentially on the random phase approximation this is here used in an improved version due to Izuyama & Kubo (1964) where these distribution functions replace the thermal averages $\langle a^*_{k\sigma}a_{k\sigma}\rangle$ which would occur in a strict random phase approximation formulation. The general relation for D$_1$ which is obtained reduces to a very simple form giving D = D$_0\zeta$/$\zeta_0$ in the limit $\zeta_0\rightarrow$0, where $\zeta_0$ is the value of $\zeta$ at 0 $^\circ$K. The general relation for D$_1$ is evaluated numerically for a single tightly bound energy band corresponding to a simple cubic lattice structure. Although the values of D$_1$ obtained are in general negative as normally expected, positive values may also occur for some values of the parameters n and I. The experimental value of D$_1$ for iron (Phillips 1966) is in good agreement with the calculated values for reasonable values of the parameters. It is suggested that other experimental data should also be analysed in terms of D$_1$.