 Solution

Along the stipulated fixed-aspect-ratio pathway, substitute $h=d/k$ before differentiating:The nonzero stationary diameter isComparison with the stated diameter gives $k{\gamma }_f+2{\gamma }_e=6{\gamma }_f+2{\gamma }_e$, hence $k=d/h=6$, provided ${\gamma }_f$ is nonzero. This is a plate-like shape, qualitatively consistent with the cheaper broad coherent faces. For negative $\Delta G$, substitution shows the stationary point is a maximum along this size pathway, as required for a nucleation barrier.

A useful distinction is that this calculation holds shape fixed. Minimizing the interface energy at fixed volume separately gives $d/h={\gamma }_e/{\gamma }_f$ for the cylindrical model; if the selected aspect ratio is also an equilibrium shape, it would require ${\gamma }_e=6{\gamma }_f$. The given critical-diameter relation alone does not otherwise determine that energy ratio. In the degenerate zero-face-energy limit the coefficient comparison cannot determine $k$.