(Solved): 4.10 Polarization due to impurity atoms in a solid The following represents a simple two-dimension ...

4.10 Polarization due to impurity atoms in a solid The following represents a simple two-dimensional model of a situation of actual physical interest. A solid at absolute temperature \( T \) contains \( N_{0} \) negatively charged impurity ions per unit volume, these ions replacing some of the ordinary atoms of the solid. The solid as a whole is, of course, electrically neutral. This is so because each negative ion with charge \( -e \) has in its vicinity one positive ion with charge \( +e \). The positive ion is small and thus free to move between lattice sites. In the absence of an external electric field it will, therefore, be found with equal probability in any one of the four equidistant sites surrounding the stationary negative ion. (See Fig. 4.16; the lattice spacing is \( a \).) If a small electrical field \( \varepsilon \) is applied along the \( x \) direction, calculate the electric polarization, i.e., the mean electric dipole moment per unit volume along the \( x \) direction.