Past exam of the chemistry course of the year 4 of the natural sciences course of the University of Cambridge / 2022 / Paper 3 / Section B / 39

A step elastically reflects a surface-state wave. Interference between $e^{ikx}$ and $r{e}^{-ikx}$ produces a local density oscillation at $2k$, so the measured spatial period is ${\lambda }_{\mathrm{osc}}=\pi /k$, not $2\pi /k$. Thus $k_1=\pi /20=0.15708{\mathring{\mathrm{A}}}^{-1}$ and $k_2=\pi /15=0.20944{\mathring{\mathrm{A}}}^{-1}$. Write $E=E_0+A{k}^2$ with $A={\hslash }^2/(2m^{\ast })$. The two points giveHere ${\hslash }^2/(2m_e)=3.80998\mathrm{eV}{\mathring{\mathrm{A}}}^2$. The fitted minimum is $E_0=1.00-A(\pi /20{)}^2=\boxed{0.35714\mathrm{eV}}$ above the Fermi level. This assumes the bias measures sample excitation energy, parabolic dispersion over the measured interval, and no wave-vector-dependent phase contribution that changes the fitted period.

inverse photoemission spectroscopy is appropriate: the fitted minimum lies above the Fermi level, so the state is unoccupied in equilibrium. Injection of electrons followed by photon emission probes unoccupied states. Ordinary ultraviolet photoemission removes electrons from occupied states and cannot map this empty minimum without an additional population mechanism.

The same wave vectors now differ in energy by only 0.070 eV, so the curvature is $A^{\prime }=0.14A=3.64756\mathrm{eV}{\mathring{\mathrm{A}}}^2$. ThereforeThe nearly free-electron in-plane mass and high unoccupied energy suggest an image-potential-derived state, with an electron confined outside the film by its image attraction and surface barrier but relatively free parallel to it. Relative to the supplied clean-platinum vacuum level, its minimum is 0.84 eV below vacuum, a plausible scale for an image state. gold adsorption can change the work function, so 0.84 eV is conditional on using the supplied clean-surface value; an adsorbate work-function measurement is needed for a quantitative assignment. The dispersion alone supports the interpretation without proving it uniquely.

An Au(111) Shockley state requires the near-surface band structure and projected bulk gap of a sufficiently thick gold region. In an ultrathin film, its wavefunction reaches the platinum interface, where hybridization and leakage into platinum states can shift, broaden or destroy the bound surface state. As the gold thickness exceeds the relevant decay length, the top surface becomes electronically decoupled from the substrate and the gold-like state emerges. A monolayer's atomic identity is insufficient to guarantee a bulk-gold surface electronic structure.

When $|eV|$ reaches a vibrational quantum, a tunnelling electron can lose energy by exciting that vibration. Opening an inelastic channel changes $dI/dV$ at both voltage polarities. If the conductance increases in each direction, $d^{2}I/d{V}^2$ has a positive feature at positive threshold and a negative feature at the corresponding negative threshold. Approximate antisymmetry follows from approximate symmetry of the conductance; unequal tip and sample electronic structures can make amplitudes unequal.

The three energy scales are 8, 48 and 375 meV, corresponding to about 65, 387 and 3025 cm${}^{-1}$. A plausible assignment is a low-energy frustrated lateral translation or rotation, an adsorbate–metal stretch or hindered molecular mode at 48 meV, and a C–H stretching vibration at 375 meV. The terminal carbon of methyl isocyanide supplies a plausible adsorption contact, but exact mode assignment requires calculated normal modes or independent vibrational measurements. In particular the 48 meV band should not be labelled a C–N stretching vibration simply because the molecule contains that bond: its frequency is far too low for the usual internal stretching mode.

The 375 meV pair should shift most fractionally, towards zero bias, because C–H stretching has strong hydrogen displacement. A reduced-mass estimate givesso the pair would move approximately from $\pm 375$ to $\pm 275$ mV. A frozen-carbon estimate gives $1/\sqrt{2}$, or about 265 mV; the actual normal mode includes coupled motion. Whole-molecule modes change much less: the molecular mass rises only from approximately 41 to 44, giving a translational estimate $\sqrt{41/44}\simeq 0.965$.