Drude parameters from extinction spectra

The paper in the project (T11): K. J. Daun, J. Menser, M. Asif, S. Musikhin, T. Dreier, C. Schulz, J. Quant. Spectrosc. Radiat. Transfer 226 (2019) 146–156, read in full (L003). An example, Iñigo says, not the subject; it is a good one because the authors do by hand, on one spectrum, almost everything the programme proposes to do by script over many, and because their honest result is a diagnosis rather than a value.

The experiment and the configuration space

Germanium nanoparticles are made in a microwave plasma reactor from \(\mathrm{GeH_{4}}\) in Ar/H₂ at 100 mbar; they are liquid at the measurement point, where the temperature is taken as 1550 \(\pm\) 35 K from earlier silicon work. Transmission electron microscopy gives spheres of 38.3 nm mean diameter and 5.7 nm standard deviation, treated as monodisperse. Line-of-sight attenuation is measured through the aerosol with two sources: a halogen lamp, 410–700 nm, and a laser-driven xenon-plasma source, 205–585 nm, the second much brighter and reaching the ultraviolet (Section 2, Figs. 1–3, 7).

The forward map: the extinction spectrum \( T_\lambda = C\,Q_{\text{ext},\lambda}(x, \epsilon) \), with \( x = \pi d_p/\lambda \) the size parameter, \( C \) the path-integrated number density times the geometric cross-section, and \( \epsilon \) the Drude dielectric function of liquid germanium (Eq. 4), \( \epsilon_{\text{I}} = 1 - \omega_p^2\tau^2/(1 + \omega^2\tau^2) \), \( \epsilon_{\text{II}} = \omega_p^2\tau/\omega(1 + \omega^2\tau^2) \). Three candidate expressions for \( Q_{\text{ext}} \): Rayleigh (Eq. 10), exact Mie (Eq. 9, Riccati–Bessel sums) and a fourth-order approximation to Mie (Eqs. 11–14). The configuration inferred is \( \theta = [\omega_p, \tau, C] \); \( C \) is a nuisance parameter and is marginalised out (Eq. 23). The temperature, the size and the choice of model are fixed from outside.

The inference, done by hand

Bayesian, by Markov-chain Monte Carlo with \(10^{6}\) samples (Section 4). The noise model is fitted to the data themselves: a Savitzky–Golay smooth of the spectrum, the residuals against the smooth, and a variance law \( \delta b = \phi_1 b + \phi_2\exp(\phi_3 b) \) (Eq. 17, Fig. 6). For the approximate Mie model a model-error term is added after Kaipio and Somersalo: the difference between the approximation and the exact Mie result, sampled over the prior range of \( (\omega_p, \tau) \), gives a mean and a covariance that enter the likelihood (Eq. 18); a 5 % white-noise term is added on top as a conservative allowance for vignetting and misalignment. The priors are uniform on \( \omega_p \in [1.5, 4.5]\times10^{16} \) rad/s and \( \tau \in [1.5, 4.5]\times10^{-16} \) s (Eq. 22). The result is the marginal posterior of \( (\omega_p, \tau) \) and its 90 % highest-posterior-density region.

What comes out: the collapse, drawn

Table 3 and Fig. 8: every model reproduces the measured spectrum, and the Drude parameters differ.

Fig. 9 is the figure this project would put on its front page. With the halogen lamp and the Rayleigh model, the 90 % credible region is a long curved band across the whole prior range of \( \omega_p \): the visible spectrum, modelled this way, sees one combination of the two parameters and nothing of the other. With the approximate Mie model and the same spectrum it is a narrow, tilted region. With the laser-driven source, which reaches the surface-plasmon resonance near 190 nm, it is narrower still. And none of the three overlaps the small region computed from bulk density and resistivity, nor the ellipsometry point. The authors' sentence: "the fact that different Drude parameters can produce similar extinction spectra highlights the ill-posedness of this problem" (Section 5).

Read against the three kinds

• Practical non-identifiability is the band of Fig. 9(a). Its remedy in the paper is better data in the right place: the ultraviolet, where the plasmon shoulder sits. The authors found that by using two lamps; the Fisher information per wavelength band would have said it in advance.
• Model inadequacy appears twice. Rayleigh against Mie: the Rayleigh cross-section has the right spectral shape and the wrong magnitude (the inset of Fig. 8), so the fit compensates with a larger \( C \) and an unphysical \( \tau \) of \( 4.5\times10^{-16} \) s; the shape fits, the parameters are wrong. Then the free-electron model against liquid germanium: the extinction-derived region excludes the values from density and resistivity, and the authors' reading is that about three electrons per atom, not four, are in the conduction band, or that electron scattering at the surface matters at 40 nm — either way a model that is not Drude with bulk values.
• Structural non-identifiability is present in the limit the paper avoids: a spectrum confined to long wavelengths, where only \( \omega_p^2\tau \) enters, could not separate the two at all.

What it gives the programme

Three things, each a proposal (P6). The model-error term that the approximate Mie model needs would be unnecessary with a forward surrogate learned once from the exact Mie sums; the MCMC would then run on the exact theory. The choice of spectral range can be computed as an experimental design instead of found by trying lamps. And the conclusion of the paper — that the extinction-derived parameters are inconsistent with the bulk ones — is a model-inadequacy signal of exactly the kind a corpus is meant to detect: one such finding is a paper, the same finding across every liquid metal measured this way is a theory. The paper's own closing sentence asks for "a theory that draws on first-principle predictions of electron transport and density-of-states", that is, for a different forward map.