Physics Explanation of Use Cases and Operational Limits

The High-Magnetic-Field Regime

In laboratory magnetic confinement fusion devices (tokamaks, stellarators), the magnetic field reaches \(B \approx 1\)–\(10\) T:

  • The Zeeman splitting \(\Delta E_Z \approx 0.1\) meV is comparable to the fine-structure splitting of lower shells, requiring the full coupled Hamiltonian diagonalization.

  • Typical microfield strengths \(F \sim 10^5\)–\(10^6\) V/m represent a weak-Stark regime where Stark broadening acts as a symmetric perturbation on the Zeeman triplet structure.

For astrophysical compact objects (magnetized white dwarfs with \(B \sim 10^3\)–\(10^5\) T, neutron stars with \(B \sim 10^8\) T):

  • The quadratic Zeeman term \(H_Z^{(2)} \propto B^2\) dominates, causing significant blue-shifting and highly asymmetric splitting. The quadratic_zeeman=True flag enables exact numerical treatment of the \(\Delta l = \pm 2\) coupling within a single principal shell (see the Radiator Hamiltonian section).

  • StarkZee’s exact numerical computation of \(\langle n, l_1 | r^2 | n, l_2\rangle\) avoids the geometric-mean overestimation of up to 41% for \(n=5\), but this is still an intra-shell (same-\(n\)) matrix element — inter-:math:`n` configuration-interaction mixing driven by the quadratic term is not included. Ferri, Peyrusse & Calisti (2022) show this coupling is crucial precisely in this regime (\(B \sim 10^2\)–\(10^3\) T, white-dwarf-like conditions): it produces a red shift of the high-PQN Balmer lines that grows with the field, which StarkZee’s intra-shell treatment cannot reproduce. See Approximations and Numerical Choices and TODO item 1 for details.

Density Limits

  • Low density (\(N_e \lesssim 10^{18}\) m-3): The profile is dominated by thermal Doppler broadening; microfield weights converge to a delta function at \(F=0\).

  • High density (\(N_e \gtrsim 10^{24}\) m-3): The inter-particle spacing \(r_e\) approaches the atomic radius \(\langle r\rangle_n\). The static and \(\Delta n = 0\) approximations break down as the Stark shift exceeds the Rydberg shell spacing (Inglis-Teller limit).