Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2018 / Paper 2 / Section A(i) / 12

Read the displayed axes as horizontal $[001]$ and vertical $[010]$; the viewing direction is therefore consistent with $[100]$. crystal facets edge-on in this projection have normals in the $yz$ plane. The flat upper/lower boundaries are approximately horizontal, the end face vertical, and the bevels approximately $45^{\circ }$ to both.

A probable low-index assignment is upper face $(010)$, end face $(001)$ and lower-right bevel $(0\bar{1}1)$, with the upper-right bevel $(011)$ if it is also indexed. Thus the flat faces belong to $\{100\}$ and the diagonal faces to $\{110\}$.

For example,giving $90^{\circ }$ and $45^{\circ }$ between normals. The upper $(010)$ normal and lower bevel $(0\bar{1}1)$ normal instead make $135^{\circ }$; their acute unoriented plane angle is $45^{\circ }$. Adjacent polygon corners between a flat edge and bevel have internal angles near $135^{\circ }$.

The distinction between a normal angle, an acute plane angle and an internal polygon angle avoids a supplementary-angle error. This is a probable projected indexation, not a unique full three-dimensional crystal facet reconstruction. The associated work uses tomography precisely to recover information absent from one projection.^[ref] The original facet-normal diagram in the growth-and-LED figure shows the $45^{\circ }$ relation independently of the source micrograph.

The Wulff construction minimizes total orientation-dependent surface free energy at fixed crystal volume:Planes at distances proportional to their surface energies bound the equilibrium shape. The construction is thermodynamic; observed growth rates need not select the same shape.

For a mixed-metal particle, minimize a more complete free energy including bulk mixing/ordering, Au–Ag interface energy, exposed-surface composition, adsorbates and elastic strain, while conserving the amount of each metal. Important parameters include orientation-dependent surface energies, metal chemical potentials, interface coherency, lattice mismatch, temperature and ligand coverage. surface segregation can change the exposed composition and hence the relevant ${\gamma }_i$.

A generalized free-energy minimization can address a multicomponent particle; the elementary Wulff construction alone does not predict its core-shell composition. Sequential Au-core formation and Ag deposition can preserve a kinetically trapped interface when diffusion is too slow to reach equilibrium. If shape relaxation is fast but interdiffusion slow, an external shape may approach a constrained minimum even though the complete particle is not at global equilibrium. Growth kinetics, ligand stabilization and finite-size effects limit a bare-surface equilibrium model. A faceted image is therefore insufficient proof that either shape or composition is an equilibrium state.

Since energy and angular frequency are related by $E=\hslash \omega$, the given eigenvalue relation yieldsWith $E_P=4.3\mathrm{eV}$,The ${\lambda }_i$ are dimensionless modal eigenvalues, not optical wavelengths despite the symbol. The same $\hslash$ multiplies both mode and bulk angular frequencies and cancels in the energy ratio. These are bare-core predictions of the stipulated lossless Drude model; adding a dielectric/metallic shell, interband response or damping changes the actual resonances.