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

Write $\mathbf{M}=M\mathbf{n}$ with $|\mathbf{n}|=1$, and let $P=n_x^2{n}_y^2+n_x^2{n}_z^2+n_y^2{n}_z^2$. Since the nonnegative numbers $n_i^2$ sum to one,The quartic coefficient is $b+cP$. If $c\ge 0$ it is at least $b>0$. If $c<0$, the given condition $2b+c>0$ implies $b+c/3>b/3>0$. For $T>T_c$ the quadratic coefficient $a(T-T_c)$ is also positive. Hence every nonzero $M$ raises the free energy:The stability argument must apply to all orientations, rather than checking just one cubic direction. The stated inequality is sufficient for that purpose, although the sharp quartic stability condition for negative $c$ is the weaker $b+c/3>0$.

For $c>0$, minimize the nonnegative anisotropy invariant $P$. Its minimum is zero, achieved only when one squared component of the unit vector is one and the others vanish. Thus the easy directions areThe six signs and axes are symmetry-related ferromagnetic domains. Allowing the magnetization amplitude to relax does not change the result: its minimum free-energy reduction is proportional to $-1/(b+cP)$, which is most negative for the smallest positive quartic coefficient. A $⟨110⟩$ direction has $P=1/4$ and is not an equally stable easy direction.

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.

Along any $⟨111⟩$ direction the quartic term is $(b+c/3)M^4$. Minimizing the amplitude below $T_c$ givesFor the ordered solution,Its free energy is $F_{\min }=F_0-a^2(T_c-T{)}^2/[4(b+c/3)]$. The denominator is positive by the supplied stability assumptions. The square-root onset is the Landau theory exponent; sufficiently close to a three-dimensional critical point, fluctuations can change the actual critical exponent without changing this saddle-point calculation.