Vacancy as a Phonon Field Perturbation. I. Lattice Distorsion Around a Vacancy

Abstract

The second quantization of the phonon field allows a theory to be developed which gives the reticular centre displacements, produced by vacancy formation. The theory is based on Bravais lattices, and a detailed calculation is carried out for the Debye approximation.Numerical results for argon at 50 "K are compared with those of previous theories.Par un calcul de seconde quantization on dtudie la distorsion d'un rbseau cristallin autour d'une vacance. La thhorie est appliquire aux cristaux de Bravais en employant l'approximation de Debye. Les resultats numdriques pour un cristal d'argon B 50 "K sont cornpards B ceux des theories prhckdentes.where r is the distance of the atom from the defect point, v the atomic volume, and a a dimensionless constant, related to the material properties. However, equation (1) is an unsatisfactory solution, because elasticity theory fails in the immediate neighbourhood of the vacancy, i.e. just in the region which gives the most important contribution to scatterings and relaxation energy. The first attempt to account for the lattice discrete nature, in a static model, was made by KANZAKI [6], for a f.c.c. lattice. His method, based on expansion of the atom displacements in normal lattice coordinates, can be used both for vacancies and interstitials. An iterative procedure, leading to successive solutions of linear algebraic equation systems, has been formulated by TEWORDT [ 7 ] . It requires the use of a large electronic computer, and, moreover, may be regarded as an attempt to give numerical rather than analytical solutions.

Numerical investigation of heat transfer and solidification in a cavity filled with molten metal alloys in the presence of thermodiffusion

“…Later, Galina and Omini [33] reconsidered Lodding's theory and expanded it to include expressions that could be used to estimate the thermodiffusion factor in metal isotopic…”

“…For isotopes, however, there is no need to calculate γ ij , as γ 11 = γ 21 . Using the Lamb formula (similar to the Stokes formula in fluid dynamics) and considering ions as spheres flowing in a stream, the drift velocity w i was predicted as a function of viscosity η and the ion radius r i as below [33]:…”

“…On the other hand, the assumption that the thermal forces act as = − ∇ in this approach is not without controversy, and Eq. 2.12 includes many parameters that may not be easily available for binary molten metals [33].…”

Numerical investigation of heat transfer and solidification in a cavity filled with molten metal alloys in the presence of thermodiffusion

“…For binary molten metals, their equations are more complex, since numerous geometric and lattice factors were included. For a dilute solution, in which the concentration of one component was small, their equations were simplified and decoupled from the enhancement factor [33]:…”

Accomplishment of the Analysis of an Edge Dislocation for a Simple Cubic Lattice

“…In the Debye model, let us call SI and st the longitudinal and transversal velocities respectively, and transform the sum 2 into an integral over the Debye sphere with radius Q, which is related to the volume v of the elementary cell by Q vQS = 6 n a . Equation (2) …”