Approximations and Numerical Choices

StarkZee is designed to be self-contained and fast enough for exploratory spectroscopy studies. The main approximations are explicit in the solver rather than hidden behind external atomic-structure packages:

  1. Hydrogenic basis and analytic radial functions. Matrix elements are built from analytic hydrogenic wavefunctions [bethesalpeter],

    \[R_{nl}(r) = \sqrt{\left(\frac{2Z}{n}\right)^3 \frac{(n-l-1)!}{2n(n+l)!}} e^{-Zr/n} \left(\frac{2Zr}{n}\right)^l L_{n-l-1}^{2l+1}\left(\frac{2Zr}{n}\right),\]

    giving self-contained \(r\) and \(r^2\) matrix elements without requiring Cowan, FAC, or another external structure code. Empirical NIST energies can still be injected for field-free level positions when accurate line centers matter.

  2. Within-shell Stark mixing. The Stark operator is retained exactly inside each principal shell, but couplings to neighboring \(n\) manifolds are neglected. This is appropriate below the Inglis-Teller regime, where Stark shifts remain small compared with the shell spacing (see TODO item 1 for the case where it is not).

  3. Within-shell quadratic Zeeman only — no inter-n configuration interaction. Like the Stark operator above, the diamagnetic (quadratic) Zeeman term \(H_{QZ} = (e^2B^2/8m_e)\,r^2\sin^2\theta\) is diagonalized inside a single principal shell (\(\langle n,l_1|r^2|n,l_2\rangle\), same \(n\) on both sides); couplings to neighboring \(n\) manifolds are neglected. Ferri, Peyrusse & Calisti (2022) show this inter-\(n\) mixing is not negligible at the field strengths relevant to magnetized white-dwarf atmospheres (\(B \sim 10^2\)–\(10^3\) T): it is required to reproduce the red-shifting of the high-PQN Balmer lines their Fig. 1(c) shows, an effect StarkZee’s intra-shell treatment cannot produce (it blue-shifts instead — see Reproducing Figure 1 from Ferri et al. (2022) and TODO item 1). Implementing the multi-\(n\) configuration-interaction basis this requires is tracked as future work.

  4. Quasi-static ions with optional dynamics. The baseline profile treats ionic microfields as static during the radiative event and averages over their distribution. When ion motion is important, the FFM layer modifies the static profile through a fluctuation rate rather than rebuilding the atomic calculation.

  5. Electron-impact broadening. Fast electrons enter through a Lorentzian impact width. By default (frequency_de­pendent_width=True) the GBK width \(\gamma_e(\Delta E)\) is evaluated at each component detuning. Setting frequency_de­pendent_width=False fixes the width at the resonance-center value \(\gamma_e(0)\), which is faster and usually adequate when far wings are not the observable of interest.

  6. Microfield orientation quadrature. The angular integral is evaluated with an \(N_\mu\)-point Gauss-Legendre rule over \(\mu = \cos\theta \in [0,1]\), using the symmetry of the Hamiltonian under \(\mu\to-\mu\); see Section 3.7 .