  We present an ab initio theory of the spin-wave stiffness tensor for ordered
and disordered itinerant ferromagnets with pair exchange interactions derived
from a method of infinitesimal spin rotations. The resulting formula bears an
explicit form of a linear-response coefficient which involves one-particle
Green's functions and effective velocity operators encountered in a recent
theory of electron transport. Application of this approach to ideal metal
crystals yields more reliable values of the spin stiffness than traditional
ill-convergent real-space lattice summations. The formalism can also be
combined with the coherent potential approximation for an effective-medium
treatment of random alloys, which leads naturally to an inclusion of
disorder-induced vertex corrections to the spin stiffness. The calculated
concentration dependence of the spin-wave stiffness of random fcc Ni-Fe alloys
can be ascribed to a variation of the reciprocal value of alloy magnetization.
Calculations for random iron-rich bcc Fe-Al alloys reveal that their spin-wave
stiffness is strongly reduced owing to the atomic ordering; this effect takes
place due to weakly coupled local magnetic moments of Fe atoms surrounded by a
reduced number of Fe nearest neighbors.
