convolutions

starkzee.convolutions.calculate_doppler_width_ev(E0_ev, Ti_ev, A_emitter)

Return the thermal Doppler 1/e half-width ΔE_D [eV].

For a Maxwellian ion distribution at temperature T_i the emission line is Doppler broadened into a Gaussian with 1/e half-width:

ΔE_D = E₀ × v_th / c

where the thermal velocity is

v_th = √(2 k_B T_i / m) → v_th/c = √(2 T_i / (m c² / e))

and m c² is the emitter rest energy in eV. The resulting FWHM is

FWHM = 2 √(ln 2) × ΔE_D

Parameters:

• E0_ev (float) – Line-center photon energy [eV].

• Ti_ev (float) – Ion temperature [eV].

• A_emitter (float) – Atomic mass number of the emitting species (e.g. 1 for H, 2 for D, 12 for C). Used to compute the emitter mass m = A × m_p.

Returns:

Gaussian 1/e half-width ΔE_D [eV]. The full Gaussian profile is exp(−(E − E₀)² / ΔE_D²).

Return type:

float

Notes

The formula uses the non-relativistic approximation v_th ≪ c, valid for all plasma temperatures relevant to magnetic fusion diagnostics.

starkzee.convolutions.convolve_fft(grid, profile, kernel)

Convolve profile with kernel on a uniform grid using FFT.

Both profile and kernel must be sampled on the same uniform grid. The convolution is computed via the convolution theorem:

(f * g)[n] = IFFT(FFT(f) × FFT(g))

The input arrays are zero-padded by half their length on each side to suppress the wrap-around artefacts of the circular FFT convolution. The kernel is area-normalized before application so that the integrated intensity of the profile is conserved.

Parameters:

• grid (array-like, shape (N,)) – Uniform coordinate grid (wavelengths or energies). Used only to determine array length; the actual coordinate values are not used.

• profile (array-like, shape (N,)) – Spectral profile to be convolved.

• kernel (array-like, shape (N,)) – Broadening kernel (not necessarily normalized). Its center must coincide with the center of grid; fftshift is applied internally to place the peak at the origin for correct phase.

Returns:

Convolved profile, trimmed back to length N.

Return type:

ndarray, shape (N,)

Notes

Edge-padding (mode='edge') is used for the profile to reduce ringing at the spectrum boundaries. Zero-padding is used for the kernel because the kernel is assumed to be compact relative to the grid.

starkzee.convolutions.apply_doppler_broadening(wavelengths_nm, profile, Ti_ev, species='H')

Apply thermal Doppler broadening to a spectrum on a uniform wavelength grid.

Constructs a Gaussian kernel with 1/e width

Δλ_D = λ₀ × v_th / c, v_th = √(2 T_i / m c²)

centered at the grid midpoint λ₀ = mean(wavelengths_nm), and convolves it with profile via convolve_fft().

Parameters:

• wavelengths_nm (ndarray, shape (N,)) – Uniform wavelength grid [nm].

• profile (ndarray, shape (N,)) – Input spectral profile (arbitrary units).

• Ti_ev (float) – Ion temperature [eV].

• species (str, optional) – Emitting species: 'H' / 'hydrogen', 'D' / 'deuterium', or 'T' / 'tritium'. Default is 'H'.

Returns:

Doppler-broadened profile (same units as input).

Return type:

ndarray, shape (N,)

Notes

The wavelength grid must be uniform (constant spacing). If the grid is non-uniform, resample before calling this function.

starkzee.convolutions.apply_instrument_broadening(wavelengths_nm, profile, fwhm_nm)

Apply Gaussian instrumental slit broadening to a spectrum.

Models the finite spectral resolution of the spectrometer as a Gaussian instrumental profile with the given FWHM. The standard deviation is

σ = FWHM / (2 √(2 ln 2))

and the kernel is exp(−(λ − λ̄)² / (2σ²)).

Parameters:

• wavelengths_nm (ndarray, shape (N,)) – Uniform wavelength grid [nm].

• profile (ndarray, shape (N,)) – Input spectral profile (arbitrary units).

• fwhm_nm (float) – Instrumental FWHM [nm]. If ≤ 0 the profile is returned unchanged.

Returns:

Instrument-broadened profile (same units as input).

Return type:

ndarray, shape (N,)