A Pedestrian Guide to DWBA

Detlef-M. Smilgies, CHESS


Distorted Wave Born Approximation (DWBA) is already quite a mouthful as a term, let alone the 4-letter abbreviation. Here a brief introduction to basic ideas shall be given.

Let's take a brief look at the Born approximation, the standard scattering theory. In a very simplified way we have an incoming plane wave field Φi(k_i,r) that scatters off an electron density distribution ρ(r) and produces an outgoing plane wave Φf with wave vector k_f and ||k_k||=||k_i||. Then the scattering intensity is given as

I(k) = |< Φf | ρ(r) | Φi>|2

using the elegant matrix element notation from quantum mechanics. The matrix element is easy to rewrite as an integral, and we end up that the scattering is given by the Fourier transform of the electron density, with the scattering vector give as q=kf-ki. But for now we will maintain the matrix element, as it greatly simplifies the next step.

Now, GISAXS operates in reflection mode, so obviously the Born approximation will give poor results. However, thing don't have to much get more difficult: the theory and wave functions of x-ray reflectivity are well known for stratified media, such as thin films. The reflectivity wave functions are given by

Ψi,f = t(kz) Φi,f + r(kz) Φi,f

t(kz) is the transmission amplitude into the layer and r(kz) is the reflection from the bottom of the layer. Here we have already separated of the part of the plane wave travelling parallel to the interface (x and y directions) and are only concerned with the part of the wave travelling perpendicular to the interface. In the semi-infinite substrate there is just a transmitted wave, and in air |r(kz)|2 yields the x-ray reflectivity signal.

This ansatz can be solved for each layer and a laterally averaged electron density analytically using the matrix method - brace yourself, even for a simply thin film this is already quite a lot of stuff to get the complex reflection amplitude r(k_r). But let's not get distracted by the technical demands - the physical interpretation of this formula is quite straight forward: for each layer we will have an incoming plane wave and a reflected wave that are superimposed with a fixed phase relation given by r(k_r).

If we have only one interface, we will retrieve the well-known Fresnel formula describing the reflection and refraction of a plane wave at a planar inteface. Introducing another layer such as a thin film, makes the whole phenomenology a lot richer: First of all we will get the Kiessig fringes due to the interference of the waves reflected from the film surface and the substrate, when the incident angle is above the critical angles. Inbetween the critical angles and when the film medium is of lower electron density than the substrate, thing will get even more interesting: The wave is still strongly reflected by the substrate, so a standing wave field develops above the substrate. If the film thickness coincides with a node of the standing wave field, we obtain a resonance. Under such conditions part of the wave bounces back and forth inside the film like a wave in a waveguide. Finally, if we stay below the critical angle of the film, we will have an evanescent wave in the film, i.e. the amplitude is exponentially attenuated as we go deeper. In case the resulting "penetration depth" is a good deal smaller than the film thickness,we can consider the film as effectivly semi-infinite, and obtain depth-sensitive information of the surface-near region.

Back to the original task: We need to ask where in the wave field scattering will occur. If we study surfaces, we may have regular deposits on the surface - this is the case for which the popular DWBA software IsGISAXS was originally developed for.If we study thin films, the waves will get scattered by the density fluctuations del-rho in the film - remember, in our ansatz we already solved the exact solution for the laterally averaged electron density. So for the latter case we can now simply replace the plan waves with the wave function inside the thin film:

I(k_f) = |<Psi_f | del-rho(r) | Psi_i>|2

This is already the essence of DWBA, and we see that it isw completely analogous to the original Born approximation, on that we picked wave functions that are better matched to the problem. Instead of the full electron density, we now have the density fluctuations. The density fluctuations may be very regular - for instance domains in a microphase-separated block copolymer. A regular arrangement of objects will give rise to Bragg reflections The objects will often also have regular shapes - here enters the form factor from transmission SAXS.

Now the next step is to consider, whether we can solve the DWBA formula, and yes, it's going to be messy, but let's keep in mind that the distorted waves are merely the sum of 2 superimposed plane waves, so just one step more complex. Substituting the plane waves for the DWs and  performing the  | |2 will result in 16 terms. Using the rules for complex numbers, the mixed terms can by combined, leaving us with 10 terms as shown in detail in a variety of papers. However, again let's get back to the original matrix element and let's see whether we can learn something without going whole hog into the algebra.

I(k_f) = |<t(kz) Phi_f + r(kz) Phi_f | del-rho(r) | t(kz) Phi_i + r(kz) Phi_i>|2

If the incident angle is about two times the critical angle of the substrate, r(kz) will be lower than 0.1. At this point the interference of incident and reflected wave can be neglected, as the reflected wave is too weak to have much of an effect. This does not hold for the outgoing waves: if we use an area detector we will see the outgoing scattering intensity in the region of the critical angles of film and substrate which will create a nicely shaped Yoneda band with various fringes, depending on the film thickness. So this we'll get even if there is nothing to scatter from inside the film! On the other hand, if we chose the incident angle to be right at a waveguide resonance, the overall scattering signal from the film will be enhanced. This is particularly useful at the TE0 resonance where we don't have a node inside the film, i.e. the whole film gets illuminated. The higher resonances can be used for more sophisticated analysis, however, if we have a weak scatterer and we want to get the most of it, TE0 is the way to go! The other appealing thing about TE0 is that it is just a tad above the critical angle of the film. Right at the critical angle the wave field inside the film is parallel to the substrate, and at TE0 this is still approximately the case. Hence we are safe from the notorious "double vision" at somewhat higher incident angles, where incoming and reflected wave scatter independently giving rise to a doubling of scattering features along the surface normal.

This is about as far as we can go without torturing the pencil and comprises my usual introduction to GISAXS for my users. Having a bit of a SAXS background is essential - remember: the SAXS signal is just the Fourier transform of the electron density! (But it helps to know a bunch of examples in the corner  of one's mind how simple things scatter - see Lazzari's manual.) To understand the sometimes confusing dynamic scattering regime inbetween critical angles, it is essential to have a good grasp on reflectivity theory. Here we can simplify matters by staying above the critical angle and go quasi-kinematic. Or hit the TE0 resonance - as long as we are careful to interpret the intensity in the vertical direction with due care.