X-ray reflectivity
The deviations can then be analyzed to obtain the density profile of the interface normal to the surface.
The earliest measurements of X-ray reflectometry were published by Heinz Kiessig in 1931, focusing mainly on the total reflection region of thin nickel films on glass.
[7] Parratt's work explored the surface of copper-coated glass, but since that time the technique has been extended to a wide range of both solid and liquid interfaces.
When an interface is not perfectly sharp, but has an average electron density profile given by
is the X-ray wavelength (e.g. copper's K-alpha peak at 0.154056 nm),
Typically one can then use this formula to compare parameterized models of the average density profile in the z-direction with the measured X-ray reflectivity and then vary the parameters until the theoretical profile matches the measurement.
For films with multiple layers, X-ray reflectivity may show oscillations with Q (angle/wavelength), analogous to the Fabry-Pérot effect, here called Kiessig fringes.
[8] The period of these oscillations can be used to infer layer thicknesses, interlayer roughnesses, electron densities and their contrasts, and complex refractive indices (which depend on atomic number and atomic form factor), for example using the Abeles matrix formalism or the recursive Parratt-formalism as follows: where Xj is the ratio of reflected and transmitted amplitudes between layers j and j+1, dj is the thickness of layer j, and rj,j+1 is the Fresnel coefficient for layers j and j+1 where kj,z is the z component of the wavenumber.
With conditions RN+1 = 0 and T1 = 1 for an N-interface system (i.e. nothing coming back from inside the semi-infinite substrate and unit amplitude incident wave), all Xj can be calculated successively.
as follows: X-ray reflectivity measurements are analyzed by fitting to the measured data a simulated curve calculated using the recursive Parratt's formalism combined with the rough interface formula.
The fitting parameters are typically layer thicknesses, densities (from which the index of refraction
Additional fitting parameters may be background radiation level and limited sample size due to which beam footprint at low angles may exceed the sample size, thus reducing reflectivity.
Due to the curve having many interference fringes, it finds incorrect layer thicknesses unless the initial guess is extraordinarily good.
Unfortunately, simulated annealing may be hard to parallelize on modern multicore computers.
In 1998,[10] it was found that genetic algorithms are robust and fast fitting methods for X-ray reflectivity.
When fitting, the measurement and the best simulation are typically represented in logarithmic space.
fitting error function takes into account the effects of Poisson-distributed photon counting noise in a mathematically correct way: However, this
If high-intensity regions are important (such as when finding mass density from critical angle), this may not be a problem, but the fit may not visually agree with the measurement at low-intensity high-angle ranges.
It is defined in the following way: Needless to say, in the equation data points with zero measured photon counts need to be removed.
The drawback of this 2-norm in logarithmic space is that it may give too much weight to regions where relative photon counting noise is high.
The application of neural networks (NNs) in X-ray reflectivity (XRR) has gained attention for its ability to offer high analysis speed, noise tolerance and its ability to find global optima.
Neural networks offer a fast and robust alternative to fit programs by learning from large synthetic datasets that are easy to calculate in the forward direction and providing quick predictions of material properties, such as layer thickness, roughness, and density.
The first application of neural networks in XRR was demonstrated in the analysis of thin film growth,[11] and a wide range of publications has explored the possibilities offered by neural networks, including free form fitting, fast feedback loops for autonomous labs and online expeirmnet control.
One of the main challenges in XRR is the non-uniqueness of the inverse problem, where multiple Scattering Length Density (SLD) profiles can produce the same reflectivity curve.
Recent advances in neural networks have focused on addressing this by designing architectures that explore all possible solutions, providing a broader view of potential material profiles.
[12] An up to date overview over current analysis software can be found in the following link.
[13] Diffractometer manufacturers typically provide commercial software to be used for X-ray reflectivity measurements.
However, several open source software packages are also available: Refnx and Refl1D for X-ray and neutron relectometry,[14][15] and GenX[16][17] are commonly used open source X-ray reflectivity curve fitting software.
They are implemented in the Python programming language and runs therefore on both Windows and Linux.
Reflex[18][19] is a standalone software dedicated to the simulation and analysis of X-rays and neutron reflectivity from multilayers.
