static_profile

starkzee.static_profile.build_stark_matrix(n, Z, Fz, Fx)

Build the (2n²) × (2n²) Stark electric-field perturbation matrix in eV.

The linear Stark interaction for an electron in an external electric field F = Fz ẑ + Fx x̂ is:

V_E = −e (z Fz + x Fx)

In the hydrogenic basis the matrix elements reduce to products of a radial element and an angular element. The within-shell (Δn = 0) radial element ⟨n, l | r | n, l±1⟩ is given analytically:

⟨n, l | r | n, l−1⟩ = (3n/2Z) √(n² − l²) [a₀]

The angular elements ⟨l, m_l | cos θ | l±1, m_l⟩ (for Fz) and the combinations for Fx are provided by angular_dipole_element().

The x-component is constructed as:

x/r = (T_{−1} + T_{+1}) / √2

where T_q are the spherical tensor components of r̂.

Parameters:

• n (int) – Principal quantum number.

• Z (int) – Nuclear charge.

• Fz (float) – Electric field component along B (z-axis) [V m⁻¹].

• Fx (float) – Electric field component perpendicular to B (x-axis) [V m⁻¹].

Returns:

Hermitian Stark perturbation matrix in eV.

Return type:

ndarray, shape (2n², 2n²), dtype complex

Notes

This matrix operates within a single n-shell; the quadratic Stark effect (coupling to n ± 1, ±2 shells) is neglected, which is valid when the Stark shift ≪ the shell spacing Z² Ry (1/n² − 1/(n+1)²).

starkzee.static_profile.solve_starkzee(n, Z, B, Fz, Fx, quadratic_zeeman=True, fine_structure=True, A=1)

Diagonalize the combined Stark + Zeeman Hamiltonian for shell n.

Adds the Stark perturbation build_stark_matrix() to the atomic/magnetic Hamiltonian build_hamiltonian() and diagonalizes the sum with numpy.linalg.eigh:

H = H_atom(B) + V_E(Fz, Fx)

This is the inner-loop solver called for every (microfield magnitude, microfield angle) quadrature point during profile integration.

Parameters:

• n (int) – Principal quantum number.

• Z (int) – Nuclear charge.

• B (float) – Magnetic field [T].

• Fz (float) – Electric field component along B [V m⁻¹].

• Fx (float) – Electric field component perpendicular to B [V m⁻¹].

• quadratic_zeeman (bool, optional) – Include the diamagnetic quadratic Zeeman term (default True).

• fine_structure (bool, optional) – Include MV + Darwin corrections (default True).

Returns:

• eigenvalues (ndarray, shape (2n²,)) – Energy eigenvalues in ascending order [eV].

• eigenvectors (ndarray, shape (2n², 2n²)) – Orthonormal eigenstates as columns, in the canonical |n, l, m_l, m_s⟩ basis.

starkzee.static_profile.calculate_static_profile(n_u, n_l, Z, B, Ne_m3, Te_ev, energies_ev, num_f=20, num_mu=6, use_screening=True, quadratic_zeeman=True, fine_structure=True, frequency_dependent_width=True, A=1, Ti_ev=None, species='H')

Compute the static-ion Stark-Zeeman line profile for n_u → n_l.

Integrates the Stark-Zeeman Hamiltonian over the plasma microfield distribution using Gauss-Legendre quadrature for both the field magnitude and the angle μ = cos θ between the microfield and B. For each quadrature point the combined Hamiltonian H = H_atom(B) + V_E(Fz, Fx) is diagonalized and the transition intensities accumulated into three polarization components.

Parameters:

• n_u (int) – Upper and lower principal quantum numbers.

• n_l (int) – Upper and lower principal quantum numbers.

• Z (int) – Nuclear charge (1 for hydrogen).

• B (float) – Magnetic field [T]. B=0 is fully supported; at zero field the π/σ decomposition is physically meaningless (no preferred axis) and all three returned components are equal by spherical symmetry.

• Ne_m3 (float) – Electron density [m⁻³].

• Te_ev (float) – Electron temperature [eV].

• energies_ev (array-like) – Photon energies at which to evaluate the profile [eV].

• num_f (int, optional) – Number of microfield quadrature points (default 20).

• num_mu (int, optional) – Number of Gauss-Legendre angle points (default 6).

• use_screening (bool, optional) – Use Hooper screened microfield distribution (default True).

• quadratic_zeeman (bool, optional) – Include diamagnetic (quadratic) Zeeman term (default True).

• fine_structure (bool, optional) – Include mass-velocity and Darwin corrections (Dirac fine structure) so that 2s_{1/2} = 2p_{1/2} (default True).

• frequency_dependent_width (bool, optional) – When True (default), evaluate the GBK electron-impact width once per discrete transition at its detuning from the gross-structure line center dE_i − E0. This is the physically correct interpretation: each Stark-Zeeman component sitting far from line center (e.g. a σ± component shifted by strong Zeeman) receives a reduced width, consistent with the breakdown of the impact approximation in the far wings. When False, use the single on-resonance value w(0) for every transition (faster; valid when Zeeman splitting ≪ ω_c).

• Ti_ev (float, optional) – Ion temperature [eV]. When supplied, Doppler broadening is folded into the Lorentzian accumulation as a Voigt profile, eliminating the need for a separate post-processing convolution. Default is None (bare Lorentzian, no Doppler).

• species (str, optional) – Emitting species ('H', 'D', or 'T'); used to determine the ion mass for the Doppler width. Only relevant when Ti_ev is set. Default is 'H'.

• approximations (Notes on)

• -----------------------

• **sigma_D** (the Doppler width is computed as)

• mean(energies_ev). (σ_D = E_mean × sqrt(Ti / mc²) where E_mean =)

• Strictly

• σ_D(dE_i) (each transition at energy dE_i should use)

• but

• H-alpha (the fractional error is ΔE/E ≈ Δ_Zeeman/E0 — about 0.03 % for)

• practice. (at 10 T — and is negligible in)

• step** (**Lorentzian FFT)

• both (convolution uses w_resonance (the on-resonance GBK width) for)

• of (frequency_dependent_width settings. The per-transition variation)

• σ_D. (w is ~4 % across the line and is negligible relative to)

Returns:

• profile_pi (ndarray) – π (Δm = 0) polarization component.

• profile_sig_plus (ndarray) – σ+ (Δm = +1) polarization component.

• profile_sig_minus (ndarray) – σ− (Δm = −1) polarization component.

Notes

Observable intensity at angle θ to B:

I(θ) = I_π sin²θ + ½(I_σ+ + I_σ−)(1 + cos²θ)

Transverse (θ = 90°): I_π + ½(I_σ+ + I_σ−). Along B (θ = 0°): I_σ+ + I_σ−. Angle-averaged: ⅔ I_π + ⅓(I_σ+ + I_σ−).

starkzee.static_profile.discrete_transitions(n_u, n_l, Z, B, Fz=0.0, Fx=0.0, quadratic_zeeman=True, fine_structure=True, min_strength=0.0, A=1)

Return all discrete Stark-Zeeman dipole transitions at a single field configuration.

Diagonalizes the Stark-Zeeman Hamiltonian for both shells and enumerates every (upper eigenstate i, lower eigenstate j, polarization q) triplet with non-zero dipole matrix element squared.

Parameters:

• n_u (int) – Upper and lower principal quantum numbers.

• n_l (int) – Upper and lower principal quantum numbers.

• Z (int) – Nuclear charge.

• B (float) – Magnetic field [T].

• Fz (float, optional) – Electric field component along B [V m⁻¹] (default 0).

• Fx (float, optional) – Electric field component perpendicular to B [V m⁻¹] (default 0).

• quadratic_zeeman (bool, optional) – Include diamagnetic Zeeman term (default True).

• fine_structure (bool, optional) – Include mass-velocity + Darwin corrections (default True).

• min_strength (float, optional) – Discard transitions with dipole strength abs(d_q)² < min_strength [a₀²] (default 0).

• A (int, optional) – Atomic mass number of the emitter (1 = H, 2 = D, 3 = T). Sets the reduced-mass Rydberg used for the absolute level energies (default 1).

Returns:

• dict with five equal-length arrays sorted by transition energy

• energy_ev – Transition energy E_upper_i − E_lower_j [eV].

• q – Polarization integer: 0 = π, −1 = σ−, +1 = σ+.

• strength – abs(d_q(i→j))² [a₀²]. Summed over all transitions equals line_strength() (unitary invariance).

• upper_idx – Upper eigenstate index (0 … 2n_u²−1).

• lower_idx – Lower eigenstate index (0 … 2n_l²−1).