The same questions asked of three inverse problems: what is the configuration space, what does the experiment project it onto, what is folded in the projection and of which of the three kinds, what external tool is used today, and what remains open. One page per case follows; this is the comparison. First version (L005).
| Rietveld refinement and mode crystallography | Drude parameters from extinction (Daun et al. 2019) | Dielectric functions from infrared spectra (FOCUS, difes) | |
|---|---|---|---|
| The system | A crystalline powder; a distorted structure of pseudo-symmetric type. | Liquid germanium nanoparticles in an aerosol, ~38 nm, in a plasma reactor. | A solid, opaque or thin, at a temperature; its lattice and free-carrier response. |
| Configuration space | Atomic coordinates, occupancies, displacement parameters, cell, profile (widths, shape, asymmetry), zero shift, background, scale, corrections: tens to hundreds of parameters. | \( \omega_p \), \( \tau \), a scale \( C \) (number density along the path), the size distribution, the temperature; the choice of model (Rayleigh, Mie, approximate Mie). | \( \epsilon_\infty \) and a list of oscillators with three or four parameters each (Drude, Lorentz, four-parameter semi-quantum, Gaussian); the number of oscillators itself. |
| The forward map | Structure factors, peak positions from the cell, a profile convolution, a background: the Rietveld model. | Extinction cross-section from the dielectric function (Drude) through Rayleigh or Mie theory, integrated along the line of sight: \( T_\lambda = C\,Q_{\text{ext}}(x, \epsilon) \). | \( \epsilon(\omega) \) from the oscillators; \( \tilde n = \sqrt{\epsilon} \); Fresnel reflectance \( R = |\tilde r|^2 \) or emittance \( 1 - R \); for thin samples the transmittance with the thickness. |
| What the data are | Intensity against \( 2\theta \): thousands of points, a few hundred resolved peaks, each with a position, an area, a width. | Transmittance against wavelength, 410–700 nm (halogen lamp) or 205–585 nm (laser-driven source): a smooth decay with a plasmon shoulder in the ultraviolet. | Reflectance or emittance against wavenumber on a finite range, typically 400–4000 \(\mathrm{cm}^{-1}\); a series in temperature. |
| Folded exactly (structural) | Orientation: the powder average keeps only \( |\mathbf{Q}| \). Phases of the structure factors. Homometric structures. Light atoms next to heavy ones with X-rays. | At long wavelengths only \( \omega_p^2\tau \) (the dc conductivity) enters; \( C \) against the overall magnitude of \( Q_{\text{ext}} \) when the model's magnitude is wrong (the Rayleigh case). | The phase of \( \tilde r \) from \( |\tilde r| \) alone. The decomposition of a smooth \( \epsilon(\omega) \) into overlapping Lorentzians. Anything outside the measured range. |
| Folded at the noise (practical) | Correlated displacements of atoms related by pseudo-symmetry; occupancy against displacement parameter; background against peak tails; strain against size broadening. | The curved valley in \( (\omega_p, \tau) \) of Fig. 9(a) with the halogen lamp; much shorter with the ultraviolet source (Fig. 9(c)). | Widths against strengths of neighbouring modes; \( \epsilon_\infty \) against the high-frequency tail; the Drude term against the lowest range measured. |
| Model inadequacy seen | Residual shapes that no profile parameter removes (anisotropic broadening, stacking faults); a wrong space group that still fits. | Rayleigh against Mie; the free-electron model itself: the extinction-derived values exclude those from density and resistivity at 90 % credibility (Table 3, Fig. 9). | Lorentzian shapes for anharmonic modes; a finite-range Kramers–Kronig without its correction; a bulk model on a sample with a surface layer. |
| External tool used today | Symmetry-adapted modes against a chosen parent (AMPLIMODES, ISODISTORT); restraints from bond valence and distances; neutrons next to X-rays; a series in temperature. | Ellipsometry; bulk density and resistivity through \( \omega_p = \sqrt{N_e e^2/\epsilon_0 m_e} \), \( \tau = m_e\sigma_{\text{dc}}/N_e e^2 \); a second light source; Bayesian inference with a fitted noise model and a model-error term. | Kramers–Kronig with finite-range corrections; sum rules; continuity across the temperature series; transmission on thin samples; ellipsometry. |
| What a script carries | The decomposition given the parent; the release strategy by amplitude; the automatic symmetry constraints; the checks. | The forward model; the noise model from the smoothed residuals; the priors; the MCMC. All of it is in the paper; none of it is scripted for the next sample. | difes recipes: the bench verbs of a fit in a text file, run on every spectrum of a series. |
| What the human still brings | The parent structure. The space group. Which atoms are allowed to move. | That the metal is Drude. The size from TEM. The temperature. | The number and kind of oscillators. The range. Whether the sample is opaque. |
| Open, for the programme | The move done over a corpus with the tube kept (P2, P3, P4). | A Mie surrogate instead of the approximation; the design computation of which band pins \( (\omega_p, \tau) \) (P6). | Identifiability of oscillator parameters against the range measured, over the difes and FOCUS libraries (P5). |
Each row of the table is a claim; the case pages give its source. Rows marked by the paper's table and figure numbers are Daun et al. 2019; the infrared column rests on the difes documentation and on practice, and is the column most in need of Iñigo's correction.