Imperfections in Crystalline Solids

A qualitative understanding of the behaviors of point defects can be established by considering atoms as hard spheres packed together to form the crystal. Crude as the hard sphere model may seem, it can be used to explain many of the observations made about point defects. In Section 4.1, we define the hard sphere radius of an atom and show its influence on the site preference of solute atoms. In Section 4.2, we use the hard sphere model to show the type of the distortions (spherically symmetric or not) in the host crystal around a solute atom. This allows us to explain why certain solutes have a much stronger solid solution hardening effect than others.

We then need to go beyond the hard sphere model in order to be more quantitative. In Section 4.3, we define the Seitz radius, which is more useful than the hard sphere radius for keeping track of the volume occupied by atoms of different kinds in solid solutions. We will see that atoms often appear to take on a different radius as a solute atom in another crystal compared to the radius it takes in its own crystal. In Section 4.4, we apply elasticity theory to predict the elastic fields around a solute atom. For simplicity, the size of the point defect is shrunk to zero and is modeled as force dipoles acting on a point in an elastic medium. In Section 4.5, a more realistic model is developed, in which the solute atom is modeled as an elastic sphere to be inserted into a hole inside an elastic medium. Elastic fields arise because the initial size of the sphere is larger than the initial size of the hole. Even though many atomistic and electronic details concerning point defects are ignored, the models developed in this chapter are increasingly more quantitative and can be used to explain a large number of behaviors of point defects.

Hard sphere model

Hard sphere radius

It is common to treat atoms in a crystal as undeformable spheres and to calculate the atomic sizes from the lattice parameters (measured using X-ray diffraction).We call this the hard sphere approach.