Define what is meant by an isotropic tensor. By considering a rotation of a second rank isotropic tensor $B_{i j}$ by $90^{\circ}$ about the $z$-axis, show that its components must satisfy $B_{11}=B_{22}$ and $B_{13}=B_{31}=B_{23}=B_{32}=0$. Now consider a second and different rotation to show that $B_{i j}$ must be a multiple of the Kronecker delta, $\delta_{i j}$.

Suppose that a homogeneous but anisotropic crystal has the conductivity tensor

$$\sigma_{i j}=\alpha \delta_{i j}+\gamma n_{i} n_{j}$$

where $\alpha, \gamma$ are real constants and the $n_{i}$ are the components of a constant unit vector $\mathbf{n}$ $(\mathbf{n} \cdot \mathbf{n}=1)$. The electric current density $\mathbf{J}$ is then given in components by

$$J_{i}=\sigma_{i j} E_{j}$$

where $E_{j}$ are the components of the electric field $\mathbf{E}$. Show that

(i) if $\alpha \neq 0$ and $\gamma \neq 0$, then there is a plane such that if $\mathbf{E}$ lies in this plane, then $\mathbf{E}$ and $\mathbf{J}$ must be parallel, and

(ii) if $\gamma \neq-\alpha$ and $\alpha \neq 0$, then $\mathbf{E} \neq 0$ implies $\mathbf{J} \neq 0$.

If $D_{i j}=\epsilon_{i j k} n_{k}$, find the value of $\gamma$ such that

$$\sigma_{i j} D_{j k} D_{k m}=-\sigma_{i m}$$