DWBA

The Distorted Wave Born Approximation (DWBA) is a theoretical approach in scattering theory (or, more generally, in quantum mechanics). In the Born approximation (BA), when calculating the interaction between matter and incident radiation ('scattering'), the total field inside the material is assumed to simply be the incident field. In other words, the modification of the field due to scattering events is assumed to be negligible; this is valid in the limit of weak scattering. Every particle simply sees the incident field, and scatters independently (though these scattered fields interfere with one another, giving rise to the usual appearance of scattering features in the far-field).

The Born approximation is not valid in all cases; in particular, when scattering becomes very strong, the approximation is no longer valid. As one example, at some point one must consider multiple scattering: the scattered rays can, themselves, become sources of scattering. GISAXS and other grazing-incidence techniques typically probe regimes where the Born approximation is not valid. For instance, in GISAXS, the x-ray beam will be reflected by the film-substrate and film-ambient interfaces; and may in fact undergo multiple reflections (waveguiding). This, and other dynamical scattering effects need to be accounted for in order to understand GISAXS data.

The DWBA is an extension to the BA, which accounts for higher-order multiple scattering effects. Conceptually, it accounts for the fact that the radiation field in the material should be solved under the condition that the scattering entities introduce substantial perturbations to the field. Mathematically, DWBA uses the BA as an idealized case, to which successive perturbation terms are introduced.

Mathematical form

In the simplest DWBA formalism, the sample is assumed to be a flat interface, with perturbations thereof that encode the in-plane order (lateral electron-density distribution). The sample's structure can also be thought of as a variety of nanoscale scattering objects distributed over an otherwise flat substrate. As usual, the scattering intensity can be split into a Form Factor and Structure Factor contribution:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I(\mathbf{q}) = \left \langle \right |F|^2 \rangle S(q_{\parallel}) }

The structure factor (S) describes the spatial arrangement of the objects (Fourier transform of the position autocorrelation function), while the form factor (F) is the Fourier transform of the shape of the objects. In the BA, this simply:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(\mathbf{q}) = \Delta \rho \int \limits _{V} e^{ i \mathbf{q} \cdot \mathbf{r} } \mathrm{d}V }

However, to account for reflection and refraction effects, the DWBA introduces additional terms. The figure below shows four different scattering events which are considered:

The first term is the BA, while the next 3 terms describe various specific multiple-scattering events. In principle, an infinite series of different combinations of reflection and scattering need to be included. However, as will be seen, higher-order terms are progressively more and more unlikely (and thus weak, in terms of their contribution to the total scattering). The DWBA, by considering these 4 terms, can adequately reproduce much GISAXS data. The various terms described above interfere coherently; thus the final scattering that is measured on the detector comes from their combination:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} F_{\mathrm{DWBA}}(q_{\parallel},k_{iz},k_{fz}) = & \, F(q_{\parallel},k_{fz}-k_{iz}) \\ & + r_F(\alpha_i) F(q_{\parallel},k_{fz}+k_{iz}) \\ & + r_F(\alpha_f) F(q_{\parallel},-k_{fz}-k_{iz}) \\ & + r_F(\alpha_i) r_F(\alpha_f) F(q_{\parallel},-k_{fz}+k_{iz}) \end{alignat} }

It is important to notice that the higher-order contributions to F are multiplied by the Fresnel reflectivity (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle r_F} ). This occurs since these terms involve reflection events.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_F = \frac{k_z-\tilde{k}_z}{k_z+\tilde{k}_z} }

Where:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde{k}_z = - \sqrt{ n^2 k^2_0 - |k_{\parallel}|^2 } }

And n is the complex refractive index of the substrate.

Because the reflectivity falls off very rapidly with increasing angle (scaling of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle R \sim q^{-4}} ), these terms are only relevant at small angles. This has several implications:

The DWBA is only relevant at small incident angles (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \alpha_i} ); i.e. it matters for grazing-incidence experiments, but not transmission scattering (TSAXS).
The DWBA is only relevant at small exit angles (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \alpha_f} ); i.e. it matters for GISAXS but not GIWAXS, and is most pronounced at small angles (near the Yoneda and horizon).
Higher-order terms (beyond the 4 covered by the DWBA) are strongly suppressed because they involve even more multiplications by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle R_F} . These can thus be ignored, except when angles are very close to the critical angle (e.g. x-ray waveguiding).

Using the effective form factor (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle F_{\mathrm{DWBA}}} ), the total incoherent cross-section becomes:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\mathrm{d}\sigma}{\mathrm{d}\Omega} = \left \langle |F_{\mathrm{DWBA}}|^2 \right \rangle S(q_{\parallel}) }

While the coherent cross-section accounts for the specular rod.

Channels

The DWBA has four scattering terms, which can conceptually be divided into two channels: transmission channel (scattering direction consistent with incident beam) and reflection channel (scattering consistent with reflected beam).