Concepts¶

The model¶

For photon energy \(E\) in eV the model returns the complex quantity

\[ \delta(E) + i\,\beta(E) \;=\; s\Big[\;\sum_k f_k(E+\Delta) \;+\; i\,\beta_\text{cont}(E+\Delta) \;+\; \delta_\text{cont}(E+\Delta)\Big] \]

with the refractive index \(n = 1 - \delta + i\beta\). Here \(f_k\) are the peaks, \(\beta_\text{cont}\) is a parametric continuum in \(\beta\), \(\delta_\text{cont}\) is a fixed continuum \(\delta\) table, \(s\) is an overall scale (typically a density) and \(\Delta\) is an energy shift.

Peaks come in Kramers-Kronig pairs¶

The real and imaginary parts of each peak are a Hilbert-transform pair, \(\delta = \mathcal{H}[\beta]\) with \(\mathcal{H}f(x) = \frac{1}{\pi}\,\mathrm{PV}\!\!\int \frac{f(t)}{x-t}\,dt\). So \(\delta < 0\) below a resonance and \(\delta > 0\) above it. Resonances sit far from \(0\,\text{eV}\), so no mirror term at \(-E_0\) is used. The amplitude amp is the height of \(\beta\) at the peak center for every lineshape.

Lineshape	Parameters	\(\beta\) and \(\delta\)
`gaussian`	`e0`, `width`, `amp`	\(x=\frac{E-E_0}{w}\); \(\beta=A e^{-x^2}\), \(\delta=\frac{2A}{\sqrt{\pi}}\,D(x)\) with \(D\) Dawson's function
`lorentzian`	`e0`, `gamma`, `amp`	\(\delta+i\beta=\dfrac{A\gamma}{E-E_0-i\gamma}\)
`voigt`	`e0`, `sigma`, `gamma`, `amp`	\(z=\frac{E-E_0+i\gamma}{\sqrt2\sigma}\); \(\beta=\frac{A}{N}\mathrm{Re}\,w(z)\), \(\delta=\frac{A}{N}\mathrm{Im}\,w(z)\), \(N=\mathrm{Re}\,w\!\left(\tfrac{i\gamma}{\sqrt2\sigma}\right)\), \(w\) the Faddeeva function
`pseudo_voigt`	`e0`, `fwhm`, `eta`, `amp`	\(\eta\,L+(1-\eta)\,G\) with one shared FWHM; exact by linearity

Width conventions, since they differ between programs:

Gaussian width is the \(1/e\) half-width: \(\beta\) falls to \(A/e\) at \(E_0\pm w\), so \(\sigma=w/\sqrt2\) and \(\mathrm{FWHM}=2\sqrt{\ln 2}\,w\).
Lorentzian gamma is the half width at half maximum (HWHM).
Voigt takes the Gaussian standard deviation sigma and the Lorentzian HWHM gamma.
Pseudo-Voigt uses a single fwhm for both parts and the Lorentzian fraction eta.

The continuum¶

Real spectra sit on a smooth background with an ionization step. The library models it in \(\beta\): continuum(bkg, slope) gives \(\beta_\text{bkg}+\text{slope}\cdot E\), and each step(e0, width, height, decay) adds \(\tfrac{h}{2}\left[1+\mathrm{erf}\!\left(\tfrac{E-E_0}{w}\right)\right]+\text{decay}\cdot\max(E-E_0,0)\).

The \(\delta\) that goes with a continuum and step has no simple closed form (a step does not decay, so its Hilbert transform needs a reference energy). Instead, the continuum \(\delta\) is fixed ahead of time with a full numerical KK transform (kkcalc2, which also folds in the bare-atom behavior far from the edge) and supplied as a table with delta_table(energy, delta). Only the peaks are fit; the continuum, and so its \(\delta\), stays fixed.

kkcalc2 and density

With the pinned kkcalc2 version the computed \(\delta\) scales linearly with its density argument. The shipped tables are per unit density (density=1.0) and the film density enters as the model scale.

Units¶

The example tables are per unit density (1 g/cm³). Absorption data given as a mass absorption coefficient \(\mu\) (cm²/g) convert to \(\beta\) with

\[ \beta = \frac{\mu\,\rho\,\lambda}{4\pi}, \qquad \lambda = \frac{hc}{E}, \]

available as lineshapes.mu_to_beta(energy_ev, mu, density) (and beta_to_mu).

What is not in the model¶

No fitting, no reflectivity, no tensor or orientation algebra. The library builds the optical-constant callable and gets out of the way: pair it with scipy.optimize for fitting (see Fitting without rebuilding) and with a reflectivity code that takes \(\delta\) and \(\beta\) per energy.