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 Greens 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.
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