Calculating the two-photon continuum emission

This topic guide provides a detailed description of the two-photon continuum emission calculation as implemented in fiasco.Ion.two_photon().

In hydrogen-like ions, the transition \(2S_{1/2} \rightarrow 1S_{1/2} + h\nu\) cannot occur as an electric dipole transition, but only as a much slower magnetic dipole transition. The dominant transition then becomes \(2S_{1/2} \rightarrow 1S_{1/2} + h\nu_{1} + h\nu_{2}\).

In helium-like ions, the transition from \(1s2s ^{1}S_{0} \rightarrow 1s^{2}\ ^{1}S_{0} + h\nu\) is forbidden under quantum selection rules since \(\Delta J = 0\). Similarly, the dominant transition becomes \(1s2s ^{1}S_{0} \rightarrow 1s^{2}\ ^{1}S_{0} + h\nu_{1} + h\nu_{2}\).

In both cases, the energy of the two photons emitted equals the energy difference of the two levels. As a consequence, no photons can be emitted beneath the rest wavelength for a given transition. See the introduction of Drake [Dra86] for a concise description of the process.

The emission is given by

\[C_{2p}(\lambda, T, n_{e}) = hc \frac{n_{j}(X^{+m}) A_{ji} \lambda_{0} \psi(\frac{\lambda_{0}}{\lambda})}{\psi_{\text{norm}}\lambda^{3}}\]

where \(\lambda_{0}\) is rest wavelength of the (disallowed) transition, \(A_{ji}\) is the Einstein spontaneous emission coefficient, \(\psi\) is so-called spectral distribution function, given approximately by

\[\psi(y) \approx 2.623 \sqrt{\cos{\Big(\pi(y-\frac{1}{2})\Big)}}\]

according to [GM78] and \(\psi_{\text{norm}}\) is a normalization factor such that

\[\frac{1}{\psi_{\text{norm}}} \int_{0}^{1} \psi(y) dy = 2\]

for hydrogen-like ions and \(1\) for helium-like ions. Finally, \(n_{j}(X^{+m})\) is the density of ions with charge state \(m\) of element \(X\) in excited state \(j\) and is given by

\[n_{j}(X^{+m}) = \frac{n_{j}(X^{+m})}{n(X^{+m})} \frac{n(X^{+m})}{n(X)} \frac{n(X)}{n_{H}} \frac{n_{H}}{n_{e}} n_{e}.\]