Past exam of the chemistry A course of the University of Cambridge / 2022 / Paper 2 / A10 / b / iii / Solution

The extra electron is bound to an approximately singly charged donor through a dielectric-screened Coulomb attraction. In the isotropic hydrogenic donor state model, replace the vacuum electron mass by $m^{\ast }$ and the vacuum permittivity by ${ϵ}_r{ϵ}_0$:At low temperature, the donor is mainly in its $1s$ state. Allowed electric-dipole transitions to $np$ states haveAssign the three printed lines to $n=2,3,4$. Their independently inferred binding energies areThey are mutually close, giving $R^{\ast }\simeq 10.63\mathrm{meV}$. The hydrogenic impurity absorption series accumulates at the donor-ionization limit. Above about $10.6\mathrm{meV}$, absorption can place the electron in the conduction-band continuum, explaining the continuous spectrum.

Using the supplied $m^{\ast }=0.2m_e$ and $R_H=13.61\mathrm{eV}$,The corresponding effective radius isThese are the parameters inferred from the printed lines within the stipulated scalar model, rather than substituted tabulated material constants. Real impurity spectra can depart from a single hydrogenic ladder through anisotropy and short-distance impurity effects.^[ref]