Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 35C / d / Solution

Writing $x=m^2$, nonzero equilibria satisfy $a(T-T_c)+bx+c{x}^2=0$. For $b<0$, coexistence occurs at $x=-3b/(4c)$ and $T-T_c=3b^2/(16ac)$, so magnetization jumps and the transition is first order.

For $b=0$, $m=0$ above $T_c$ and $m^4=a(T_c-T)/c$ below: this is the tricritical continuous case. Below $T_c$,so the extra heat capacity is $a^{3/2}T/[4\sqrt{c(T_c-T)}]$ and diverges on approach from below; above it is absent.