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Choose a basis whose first axis is $\mathbf{n}$. The tensor with uniaxial nematic order then has eigenvalues $2Q,-Q,-Q$. Hence $\mathrm{Tr}{S}^2=6Q^2$, $\mathrm{Tr}{S}^3=6Q^3$ and $\mathrm{Tr}{S}^4=18Q^4$. Substitution givesFor $a,c>0$, $Q=0$ remains a local minimum with curvature $12a$. Its nonzero stationary points satisfy $2a+3bQ+12c{Q}^2=0$, soThey first appear at $b_s=-\sqrt{32ac/3}$ as a stationary inflection point; for more negative $b$, $Q_{-}$ is a local maximum and $Q_{+}$ a local minimum. The latter is initially metastable, becomes degenerate with $Q=0$ at $b_t=-\sqrt{12ac}$, and then becomes the global minimum as $b$ decreases further. This is a first-order transition in a cubic-quartic Landau potential: the order changes discontinuously. The saddle-node bifurcation creating a local minimum is not the coexistence condition. The isotropic minimum never loses local stability in this fixed positive-$a$ scan.

The illustrative plots set $a=c=1$ and show $f/6$. A cubic invariant is permitted because head-tail reversal leaves $S$ unchanged; it does not implement $S\mapsto -S$. The physical range of a uniaxial nematic order parameter imposes $-1/2\le Q\le 1$; the usual small-order-parameter Landau result assumes the displayed minimum lies in that interval. The unrestricted quartic graphs express the intended Landau model, with the effect of enforcing the bound stated in part (f).^[ref]

At coexistence a nonzero $Q_t$ must be stationary and have the same free energy as $Q=0$. Dividing $f(Q_t)=0$ by $6Q_t^2$ and $f^{\prime }(Q_t)=0$ by $6Q_t$ givesEliminating $a$ yields $Q_t=-b/(6c)$ and then $b^2=12ac$. The allowed nonpositive root therefore givesAs $b$ is decreased through $b_t$, the equilibrium scalar order jumps from $0$ for $b>b_t$ to $Q_t$ just on the $b<b_t$ side. At $b=b_t$ both values coexist; the stable aligned branch below it is $Q_{+}$ from part (e). As a global-minimum check, at coexistenceThere are two equal minima and a barrier at $Q_t/2$, of height $a^2/(8c)$. Comparing these minima, rather than finding only the stationary-point discriminant, determines the transition.

This is the unrestricted Landau answer, compatible with the physical range of a uniaxial nematic order parameter when $a\le 3c$ so that $Q_t\le 1$. If one instead enforces $Q\in [-1/2,1]$ for arbitrary positive coefficients, negative $Q$ cannot win for $b\le 0$, and bounded quartic nematic coexistence minimizes $a/Q+3cQ$ over $0<Q\le 1$. For $a>3c$ its minimum is at the boundary, giving $b_t=-(a+3c)$ and a jump $0\to 1$. No coefficients were supplied to choose this boundary case; the near-transition Landau approximation normally uses the interior result.