= Solution
The <addition energy> has both charging and orbital terms,
$$
E_{\rm add}(N)=\frac{e^2}{C_\Sigma}+\epsilon_{N+1}-\epsilon_N.
$$
The larger diamonds indicate <quantum dot shell closure> after \b[2, 6 and 12 <electrons>]. A <two-dimensional isotropic harmonic oscillator> confinement, $V(x,y)=m^*\omega_0^2(x^2+y^2)/2$, explains that sequence: shell $q=n_x+n_y$ has $q+1$ orbital states and capacity $2(q+1)$ after <spin degeneracy>. The cumulative capacities are $(q+1)(q+2)=2,6,12,20,\ldots$. Within a degenerate shell, the extra orbital term vanishes to this approximation; crossing a shell boundary costs an extra $\hbar\omega_0$.
The <gate capacitance> lever arm is $\alpha_g=C_g/C_\Sigma=1/2$. Large and small gate widths differ by $7-5=2\ \mathrm{mV}$, so \b[$\hbar\omega_0\simeq e\alpha_g(2\ \mathrm{mV})=1.0\ \mathrm{meV}$]. Equivalently the tip energies differ by $3.5-2.5=1.0\ \mathrm{meV}$. Confinement anisotropy, exchange and disorder would split the ideal degeneracies; the diagram supports approximate shell symmetry rather than proving a perfectly parabolic potential.
\Image[/past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2021/paper-2-shell-filling.png]
{title=Original spin-degenerate two-dimensional oscillator shell capacities and closed-shell electron counts}
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