Roughness model selection guide¶

Scope of this page¶

This page is the practical decision framework for choosing among the three roughness treatments in refloxide. The underlying physics is treated in Nevot-Croce, Debye-Waller, and Graded interface. The architectural composition with the 4x4 pipeline is treated in Framework. This page sits on top of all four and answers the user-facing question of which model to pick for a given physical regime.

The framing follows Esashi [1, Sec. 3 and Sec. 4] for the small-roughness, large-roughness, and high-contrast regimes that distinguish the three models, with extensions for the birefringent and visible-wavelength regimes that the Esashi treatment does not cover.

Two governing dimensionless quantities¶

Two dimensionless quantities determine which model is correct for a given physical situation. Both are evaluated at the relevant interface, in the relevant medium, at the relevant incidence angle.

The first is the Esashi small-roughness parameter \(\alpha = \sigma\,k_z\), where \(\sigma\) is the rms roughness amplitude and \(k_z = (2\pi n/\lambda_0)\sin\theta\) is the out-of-plane wave-vector projection in the medium just above the interface. The DWBA expansion that produces both multiplicative factors is valid only when \(\alpha \lesssim 1\) [1, text following Eq. (1)]. We adopt the operational rule

with the intermediate region \(0.3 < \alpha < 1\) flagged as crossover where the kernel emits a warning and recommends a graded calculation as the verification reference.

The second is the Esashi correlation-length ratio \(\beta = \xi / \xi_{\text{diff}}\), where \(\xi\) is the in-plane correlation length of the roughness and \(\xi_{\text{diff}} = \lambda/(1 - \cos\theta)\) is the diffuse-scattering bandwidth set by the geometry. The Nevot-Croce factor applies when \(\beta \ll 1\) and the Debye-Waller factor applies when \(\beta \gg 1\) [1, Eq. (2) and surrounding text]. The operational rule is

with the intermediate region as a second crossover that recommends the graded calculation.

The two dimensionless quantities are independent in principle. A stack with small roughness and short correlation length sits in the Nevot-Croce regime. A stack with large roughness and short correlation length sits in the graded regime. A stack with small roughness and long correlation length sits in the Debye-Waller regime. A stack with large roughness and long correlation length sits in the graded regime regardless. The graded model is the universal fallback when either quantity falls outside its operating range.

Decision matrix¶

The four-way matrix below summarizes the dispatch as implemented in core::roughness::dispatch::auto_select.

A separate axis applies independently to all four cells: the index-contrast regime. When the relative dielectric contrast \(|\Delta\varepsilon|/\varepsilon\) across the interface exceeds \(0.3\) in either direction, the multiplicative models lose validity even when \(\alpha\) and \(\beta\) sit in their nominal operating regions [1, text following Eq. (1)]. The kernel applies the override rule

regardless of the \(\alpha\) and \(\beta\) values. This regime is characteristic of metal-dielectric multilayers in the visible and near-infrared, and of the Au and Pt overlayers used in soft-X-ray reflectivity standards.

A third axis applies to the polarization regime. For birefringent substrates with cross-polarization channels \(r_{ps}, r_{sp} \neq 0\), the kernel uses the geometric-mean prescription described in Nevot-Croce for the multiplicative models, but the prescription is unverified against published derivations. When the cross-polarization channels carry more than \(1\%\) of the co-polarized amplitude, the kernel emits a warning recommending graded verification.

Recommended workflow¶

The recommended workflow has four steps.

The first step is initial dispatch. Pass the roughness parameters and let the kernel auto-dispatch through the matrix above. The dispatcher records its choice in the audit log, which is exposed through Pipeline::audit_log() in Rust and through the log attribute of the result struct in Python. For production fitting workflows the auto-dispatch is the appropriate default because it captures the broadest range of physical regimes and prevents the user from accidentally selecting an out-of-validity model.

The second step is regime sanity check. After the initial dispatch, run the same calculation with Roughness.graded(...) and compare. If the two agree to \(10^{-6}\), the dispatcher choice is verified and the user can proceed with the cheaper multiplicative model in subsequent sweeps. If they disagree, the dispatcher choice was inappropriate for the physical regime, and the user must fall back to the graded model or revise the roughness parameters.

The third step is convergence verification when the graded model is in use. Set Roughness.graded(..., convergence_check=True) to enable automatic refinement and convergence assertion at the default tolerance of \(10^{-6}\) on absolute amplitude differences. The convergence check doubles the runtime but is the only way to certify the graded result.

The fourth step is fit re-evaluation. When fitting experimental data, the optimizer will move the roughness parameters across the regime boundaries during the fit, and a model that was appropriate at the initial guess may become inappropriate at the converged point. Re-running the dispatcher at the converged parameters is the safe practice. The kernel supports this through a Pipeline::reaudit() method that re-evaluates the dispatch on the current parameters and emits a warning if the recommended model has changed.

Recommended defaults by application¶

The following defaults are appropriate starting points for the common application classes. Each carries an explicit caveat about the regime in which it should be re-evaluated.

For X-ray reflectivity at synchrotron wavelengths (\(\lambda \sim 0.1\text{-}0.2\) nm), the default is Nevot-Croce with \(\sigma\) from the experimental fit. The DWBA conditions are typically satisfied for polished substrates and metallic overlayers. Re-evaluate when the fit pushes \(\sigma > 1\) nm or when the data span the critical-angle region for a heavy substrate.

For EUV reflectometry (\(\lambda \sim 13.5\) nm), the default flips depending on the substrate. For polished silicon the short-correlation length keeps Nevot-Croce valid up to \(\sigma \sim 4\) nm. For metal-coated optics the correlation-length ratio approaches the crossover, and the graded model is the safer default. Re-evaluate after the fit converges.

For soft X-ray magneto-optic reflectivity at the resonance edges of 3d transition metals, the default is graded. The combination of substantial absorption and tensor anisotropy puts the multiplicative models close to their validity bounds, and the cross-polarization channels are sensitive to roughness-induced phase shifts that the graded model captures faithfully.

For visible-wavelength magneto-optical Kerr effect (MOKE) spectroscopy, the default is graded. The Debye-Waller regime applies in principle, but the high-contrast condition is typically violated by metal-substrate interfaces, and the graded model is the only physically defensible choice.

For neutron reflectometry, the default is Nevot-Croce. The neutron wavelengths sit in the same range as X-rays for typical experiments, the index contrasts are modest, and the correlation lengths are short. The kernel reports the same dispatch as for the analogous X-ray geometry.

Output of the auto-dispatcher¶

The auto-dispatcher returns a RoughnessDispatchResult struct that records the model chosen, the values of \(\alpha\), \(\beta\), and the index-contrast magnitude at every interface, and any warnings issued during the dispatch. A typical output for a three-interface stack reads

RoughnessDispatchResult {
    interface_0: { model: NevotCroce, alpha: 0.18, beta: 0.05, contrast: 0.02 },
    interface_1: { model: Graded,     alpha: 0.42, beta: 0.31, contrast: 0.15,
                   warning: "alpha in crossover region 0.3-1.0" },
    interface_2: { model: NevotCroce, alpha: 0.09, beta: 0.04, contrast: 0.01 },
}

The struct is the canonical record of the dispatch decision, and is included in any saved result so that downstream re-analysis sees both the result and the model that produced it. We surmise that this audit-trail behavior is essential for reproducibility in fitting workflows where the same data are re-analyzed months later by a different team member.

When to override the auto-dispatcher¶

Three classes of situation justify overriding the auto-dispatch.

The first is when the user has a physical reason to expect a specific model regardless of the dispatcher choice. A chemically interdiffused interface is graded by physics, not by morphology, and should be modeled with the graded approach even when the multiplicative dispatcher would accept it.

The second is when computational budget forbids the graded model and the user accepts the resulting accuracy degradation. For large parameter sweeps over many wavelengths or many incidence angles, the \(30\times\) to \(100\times\) overhead of the graded model can be prohibitive, and a multiplicative model with documented inaccuracy is preferable to no result. The kernel emits a warning in this case but does not refuse the override.

The third is when comparing against published reference data that was computed with a specific model. Reproducing the Esashi figures, for instance, requires forcing the model that Esashi used at each panel rather than letting the dispatcher choose. The kernel exposes the override through the RoughnessChoice enum on the RoughnessSpec, and explicit choices are honored without dispatch.

References¶

1. Y. Esashi, M. Tanksalvala, Z. Zhang, N. W. Jenkins, H. C. Kapteyn, and M. M. Murnane, "Influence of surface and interface roughness on X-ray and extreme ultraviolet reflectance: A comparative numerical study," OSA Continuum 4, 1497 (2021). DOI.
2. S. K. Sinha, E. B. Sirota, S. Garoff, and H. B. Stanley, "X-ray and neutron scattering from rough surfaces," Phys. Rev. B 38, 2297 (1988). DOI.
3. D. L. Windt, "IMD: Software for modeling the optical properties of multilayer films," Comput. Phys. 12, 360 (1998). DOI.