Distortion of a Crystal by Point Imperfections

TLDR

The displacement field around a point defect is often modeled by \(u=c\,r/r^{3}\), which yields nonzero surface stress unless an image displacement is added, and this image term can significantly affect measured volume changes, as noted by Miller and Russel for defects near a sphere’s center. The study investigates how incorporating the image displacement term alters the interpretation of volume changes and X‑ray diffraction patterns caused by point imperfections. Including the image term shows it contributes 30–50 % of the volume change, reverses the center‑surface discrepancy, makes geometrical and X‑ray expansions equal for uniformly distributed defects, and proves that a statistically uniform distribution of point defects uniformly expands any body, producing the expected alteration in its diffraction pattern.

Abstract

The expression u=cr/r3, (where c is a constant) sometimes assumed for the displacement around a point imperfection (interstitial or substitutional impurity, lattice vacancy) gives a nonzero stress at the surface of the solid. The additional ``image displacement'' necessary to insure that this stress vanishes is usually neglected, but may be important. For example, it accounts for from 30 to 50 percent of the volume change produced by such defects. This and other effects of the image term are discussed. Miller and Russel have pointed out that for a point imperfection near the center of a sphere the apparent volume change deduced from measurements of the x-ray lattice constant is greater than the geometrical volume change. It is shown that the reverse is true when the defect is near the surface, and that for a large number of defects scattered uniformly through the sphere the geometrical and x-ray expansions are equal. It can be shown quite generally that a body of arbitrary shape is expanded uniformly by a statistically uniform distribution of point imperfections, and that the x-ray diffraction pattern is altered in the way to be expected for such an expansion. To establish this, however, it is essential to take the image terms into account.