Hyperfine-changing transitions in $^3$He II and other one-electron ions by electron scattering

We consider the spin-exchange (SE) cross section in electron scattering from $^3$He\,{\scriptsize II}, which drives the hyperfine-changing \hbox{3.46 cm} (8.665 GHz) line transition. Both the analytical quantum defect method --- applicable at very low energies --- and accurate R-matrix techniques for electron-He$^+$ scattering are employed to obtain SE cross sections. The quantum defect theory is also applied to electron collisions with other one-electron ions in order to demonstrate the utility of the method and derive scaling relations. At very low energies, the hyperfine-changing cross sections due to e$-$He$^+$ scattering are much larger in magnitude than for electron collisions with neutral hydrogen, hinting at large rate constants for equilibration. Specifically, we obtain rate coefficients of $K(10\,{\rm K}) = 1.10 \times 10^{-6}\,\rm cm^3/s$ and $K(100\,{\rm K}) = 3.49\times 10^{-7}\,\rm cm^3/s$.