Ferroelectric materials exhibit so-called spontaneous polarization, when their polarization $P$ is different from zero even no external electric field is applied. Let's try to investigate this effect quantitatively. To do this, one must first determine how the internal energy per unit volume $W$ depends on polarization $P$. According to Landau phase transition theory, we need expand this dependency into a Taylor series around $P=0$. Terms with $P$ in odd power get cancelled due to inversion symmetry, so:$W=\frac{1}{2}\alpha (T-T_0) P^2+\frac{1}{4}\beta P^4+\frac{1}{6}\gamma P^6. \tag{1}$Here $\alpha,\beta,\gamma > 0$, $T$ is the temperature of the ferroelectric, and $T_0$ is the temperature at which the ferroelectric undergoes a phase transition changing its ability to spontaneously polarize.

In the equilibrium state, the energy must have a local minimum.

A1 Write down the conditions for a local energy minimum, if energy is given by equation $(1)$. For which values of $T$ the spontaneous polarization would be possible?

A2 Derive an equation for equilibrium values of spontaneous polarization, $P_0$.

A3 Expression from the previous task simplifies significantly in the limit $T\to T_0$. Find the values of $P_0$ in this limit. Express your answer in terms of $\alpha$, $\beta$ and $\Delta T=T_0-T$.

Use the resulting expression later in this problem.

If an external electric field $E$ is applied to a spontaneously polarized ferroelectric, its polarization will change slightly. It can be shown that for a sufficiently small external field this change can be considered linear, and one can define electric susceptibility as:$\chi\equiv \lim_{E\to 0}\frac{P_0(E)-P_0(0)}{E}\equiv \left.\frac{\partial P_0(E)}{\partial E}\right|_{E=0}.$

B1 Derive an expression for the electric susceptibility of a spontaneously polarized ferroelectric.