Past exam of the condensed matter physics major option course of the University of Oxford / 2015 / 3 / b / b / ii

For $c<0$, maximize $P$ to lower the quartic coefficient. The identity $P=[1-(n_x^4+n_y^4+n_z^4)]/2$ shows that its maximum $1/3$ occurs for $n_x^2=n_y^2=n_z^2=1/3$. ThereforeThere are eight symmetry-related easy directions. This is a cubic easy-axis anisotropy even though the selected directions are body diagonals; it is not a continuous easy plane. At $c=0$ the quartic free energy instead has a sphere of degenerate orientations to this order.