## 2 Classical Physical Model for the Hydrogen Isotopic Ratio Determination
In addition to the co-existence of the neutral populations described above for each hydrogen isotope, the profile of the emitted Balmer-\(\alpha\) lines are also subject to the Zeeman effect caused by the magnetic field. It is out of the scope of this paper to describe in detail the effect of an external magnetic field on an emission line, as one can refer to standard textbooks treating this topic. Instead, we refer the readers to our previous paper [10] and the references therein for a brief description. However, it is widely accepted that for magnetic fields with strengths higher than about 1 T, one can use the strong magnetic field approximation, as the energy-level Zeeman splitting is much larger than the fine structure level splitting. Therefore, in the following, we use the strong magnetic field approximation in the computation of the perturbed energies of the hydrogen isotopes and Zeeman-Doppler broadening when calculating theoretical or synthetic H/D Balmer-\(\alpha\) line profiles under different conditions, i.e., population fractions and isotopic rations, mainly. As the energy levels involved in the Balmer \(\text{H}\alpha/\text{D}\alpha\) lines have low principal quantum numbers and the electron densities we deal with in tokamaks are relatively low, we will simply ignore the Stark broadening, which is due to the electric micro-field created by the plasma-charged particles. Another crucial point is that the presence of an external magnetic field introduces anisotropy for the emitted photons. In fact, the isotropy of spontaneously emitted photons is no longer valid when considering spectral profiles of lines emitted in tokamaks, especially the divertor region. The shape of the \(\text{H}\alpha/\text{D}\alpha\) line depends strongly on the angle \(\theta\) between the direction of the magnetic field and the observation direction, i.e.,  
Figure 1: Zeeman–Doppler profiles of the Balmer-\(\alpha\) lines (H\(\alpha\)/D\(\alpha\)) computed for a gas mixture composed of 10% of hydrogen and 90% of deuterium with a temperature of 2 eV in the presence of a magnetic field of 2 T. (**a**) Perpendicular observation (\(\theta=90^{\circ}\)): for each hydrogen isotope, the total line profile (red solid line) is a sum of a central \(\pi\) component (blue solid line) located respectively at \(\lambda_{D_{a}}=6561.04\) Å for D\(\alpha\) and \(\lambda_{H_{a}}=6562.8\) for H\(\alpha\) lines and two lateral \(\sigma^{+}\) and \(\sigma^{-}\) components (black solid line). (**b**) Parallel observation: the spectrum was calculated for a parallel observation with respect to the magnetic field direction, i.e., for an angle \(\theta=0^{\circ}\) or \(180^{\circ}\). The method termed here as the classical method is based on the fit of the experimental H\(\alpha\)/D\(\alpha\) line spectra by modeling their theoretical profiles as explained previously in this section. In the following, to illustrate this physical model, we consider a synthetic profile instead of an experimental spectrum. The advantage of using a synthetic line profile is that there is no instrumental function to account for and as all the parameters are known, it is easy to assess the optimization technique allowing to fit the synthetic line profile. An example is illustrated in Figure 4. Synthetic data (synthetic spectrum calculated with  
Figure 3: A synthetic D\(\alpha\)/H\(\alpha\) line profile calculated for a parallel observation with respect to the magnetic field direction whose intensity is 2 T. A 90%D + 10%H gas mixture was assumed for this calculation. The Balmer-\(\alpha\) line emission profile (red solid line) is assumed resulting from the combination of two neutral populations having Maxwellian VDFs with temperatures of 2 eV (blue dashed line) and 20 eV (black dotted line). The relative fractions or concentrations of each population are as follows: 75% of cold neutrals and 25% of warm neutrals for each hydrogen isotope.