 Blume–Capel mean-field tricritical parameters

For ferromagnetic exchange $J$ and coordination $z$, the mean-field approximation has $f=zJ{M}^2/2-k_{B}T\log [1+2e^{-g/(k_{B}T)}\cosh (zJM/(k_{B}T))]$. With $K=zJ/(k_{B}T)$ and $q=2e^{-g/(k_{B}T)}/[1+2e^{-g/(k_{B}T)}]$, its quadratic and quartic coefficients vanish at $q=1/3$, $K=3$. The sextic coefficient of $f/(k_{B}T)$ is $9/40>0$. Hence $T_{\mathrm{TCP}}=zJ/(3k_B)$ and $g_{\mathrm{TCP}}=k_B{T}_{\mathrm{TCP}}\log 4$; these are not exact finite-dimensional lattice values.