Asymptotic Lattice Displacements about Point Defects in Cubic Metals

Abstract

We have calculated the asymptotic displacements (a) about a single vacancy in Al, Na, K, and Li, (b) about a single interstitial Cu atom in a Cu host lattice, and (c) about a unit single double force along a $〈100〉$ direction in Al and Cu. These calculations were made using the asymptotic equations of the method of lattice statics which, in its full form, is based on the Fourier transformation of the direct-space force equations between the detect and the host atoms in a large "supercell" of the lattice. Results were also obtained for each of the defect types in Al and Cu by means of an alternative approach proposed by Lie and Koehler. The asymptotic displacements around the spherical defects in Al, Cu, Na, and K were compared with corresponding results obtained using the exact method of lattice statics. From this comparison it appears that elasticity theory cannot be justifiably applied closer than the (4,4,4) neighbor to the defect in Al, the (5,4,4) neighbor in Cu, and the 26th or the 27th neighbors in Na and K. We also find large displacements along $〈110〉$ in Al and along $〈111〉$ in Cu, Na, and K. The displacements obtained using a Green's-function technique developed by Lie and Koehler agree well with our results for the spherical detect in Al and for the unit single double-force defects in Al and Cu. There are significant differences in the $〈100〉$ directions between the two sets of results for a spherical detect in Cu. In general, the asymptotic method of lattice statics appears to be more exact and to involve fewer computational manipulations than the method of Lie and Koehler.