Strain field due to substitutional transition-metal impurities in bcc metals: Application to dilute vanadium alloys.

The formalism for the generalized Kanzaki lattice static method is developed for a metallic crystal with bcc symmetry. The interactions up to second-nearest neighbors are included in the derivation of the dynamical matrix, impurity-induced force, and the atomic displacements. The formalism is applied to calculate the strain field due to 3d, 4d, and 5d substitutional transition-metal impurities (Ti, Cr, Mn, Fe, Nb, Mo, Ta, and W) in the vanadium metal, the only bcc systems in which the study of electric field gradients has been performed. The effective ion-ion interaction potential for the transition metals, proposed by Wills and Harrison, which includes properly the d-band effects, is used in the numerical calculations. In all the systems the atomic displacements are calculated up to 21 nearest neighbors and these show oscillatory behavior. The maximum displacement (or strain field) is caused by a Ta impurity and is about 3.5% of the first-nearest-neighbor distance. The strain field is found to depend, both in strength and range, on the excess ion-ion interaction potential due to the impurity. The atomic displacements exhibit the same trend, as shown by x-ray studies of the fractional change in the lattice parameter.