X-ray reflectivity (sometimes known as X-ray specular reflectivity, X-ray reflectometry, or XRR) is a surface-sensitive analytical technique used in

_{j} is the ratio of reflected and transmitted amplitudes between layers j and j+1, d_{j} is the thickness of layer j, and r_{j,j+1} is the for layers j and j+1
:$r\_\; =\; \backslash frac$
where k_{j,z} is the z component of the _{z} because $Q\; =\; k\_\backslash text\; +\; k\_\backslash text$. With conditions R_{N+1} = 0 and T_{1} = 1 for an N-interface system (i.e. nothing coming back from inside the semi-infinite substrate and unit amplitude incident wave), all X_{j} can be calculated successively. Roughness can also be accounted for by adding the factor
:$r\_\; =\; r\_\; e^$
where $\backslash sigma$ is a standard deviation (aka roughness).
Thin film thickness and critical angle can also be approximated with a linear fit of squared incident angle of the peaks $\backslash theta^2$ in rad^{2} vs unitless squared peak number $N^2$ as follows:
:$\backslash theta^2\; =\; \backslash left(\backslash frac\backslash right)^2\; N^2\; +\; \backslash theta\_c^2$.

chemistry
Chemistry is the scientific
Science () is a systematic enterprise that builds and organizes knowledge
Knowledge is a familiarity or awareness, of someone or something, such as facts
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, physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

, and materials science
The interdisciplinary
Interdisciplinarity or interdisciplinary studies involves the combination of two or more academic disciplines into one activity (e.g., a research project). It draws knowledge from several other fields like sociology, a ...

to characterize surfaces
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A surface, as the term is most gener ...

, thin film
A thin film is a layer of material ranging from fractions of a nanometer
one nanometric Scanning_Tunneling_Microscope.html"_;"title="carbon_nano_tube,_photographed_with_Scanning_Tunneling_Microscope">carbon_nano_tube,_photographed_with_Scannin ...

s and multilayers.J. Als-Nielsen, D. McMorrow, ''Elements of Modern X-Ray Physics'', Wiley, New York, (2001). It is a form of reflectometry
Reflectometry uses the reflection of wave
In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Mot ...

based on the use of X-ray
An X-ray, or, much less commonly, X-radiation, is a penetrating form of high-energy electromagnetic radiation
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Moti ...

s and is related to the techniques of neutron reflectometry
Neutron reflectometry is a neutron diffraction technique for measuring the structure of thin films, similar to the often complementary techniques of X-ray reflectivity and ellipsometry. The technique provides valuable information over a wide variet ...

and ellipsometry
Ellipsometry is an optical
Optics is the branch of physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and ...

.
The basic principle of X-ray reflectivity is to reflect a beam of X-rays from a flat surface and to then measure the intensity of X-rays reflected in the specular direction (reflected angle equal to incident angle). If the interface is not perfectly sharp and smooth then the reflected intensity will deviate from that predicted by the law of Fresnel reflectivity. The deviations can then be analyzed to obtain the density profile of the interface normal to the surface.
History

The technique appears to have first been applied to X-rays by Lyman G. Parratt in 1954. Parratt's initial 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.Approximation

When an interface is not perfectly sharp, but has an average electron density profile given by $\backslash rho\_e(z)$, then the X-ray reflectivity can be approximated by: :$R(Q)/R\_F(Q)\; =\; \backslash left,\; \backslash frac\; \backslash ^2$ Here $R(Q)$ is the reflectivity, $Q\; =\; 4\; \backslash pi\; \backslash sin\; (\; \backslash theta\; )\; /\; \backslash lambda$, $\backslash lambda$ is the X-ray wavelength (typically copper'sK-alpha Characteristic X-rays are emitted when outer-shell
Shell may refer to:
Architecture and design
* Shell (structure)A shell is a type of structural element which is characterized by its geometry, being a three-dimensional solid whose thickness is v ...

peak at 0.154056 nm), $\backslash rho\; \_\backslash infty$ is the density deep within the material and $\backslash theta$ is the angle of incidence. Below the critical angle $Q\; <\; Q\_c$ (derived from Snell's law
of light at the interface between two media of different refractive index, refractive indices, with n2 > n1. Since the velocity is lower in the second medium (v2 < v_{1}), the angle of refraction θ_{2} is less than the angle of in ...

), 100% of incident radiation is reflected, $R=1$. For $Q\; \backslash gg\; Q\_c$, $R\; \backslash sim\; Q^$. 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.
Oscillations

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. 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 onatomic number
The atomic number or proton number (symbol ''Z'') of a chemical element
In chemistry, an element is a pure Chemical substance, substance consisting only of atoms that all have the same numbers of protons in their atomic nucleus, nuclei. ...

and atomic form factor
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succe ...

), for example using the Abeles matrix formalism
through a layer
The transfer-matrix method is a method used in optics
Optics is the branch of physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , i ...

or the recursive Parratt-formalism as follows:
:$X\_j\; =\; \backslash frac\; =\; \backslash frac\; e^$
where Xwavenumber
In the physical science
Physical science is a branch of natural science that studies abiotic component, non-living systems, in contrast to life science. It in turn has many branches, each referred to as a "physical science", together called the ...

. For specular reflection where the incident and reflected angles are equal, Q used previously is two times kCurve fitting

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 $n$ and eventually the wavevector z component $k\_$ is calculated) and interfacial roughnesses. Measurements are typically normalized so that the maximum reflectivity is 1, but normalization factor can be included in fitting, as well. 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. Several fitting algorithms have been attempted for X-ray reflectivity, some of which find a local optimum instead of the global optimum. The Levenberg-Marquardt method finds a local optimum. Due to the curve having many interference fringes, it finds incorrect layer thicknesses unless the initial guess is extraordinarily good. The derivative-freesimplex method
In optimization (mathematics), mathematical optimization, George Dantzig, Dantzig's simplex algorithm (or simplex method) is a popular algorithm for linear programming.
The name of the algorithm is derived from the concept of a simplex and was su ...

also finds a local optimum. In order to find global optimum, global optimization algorithms such as simulated annealing are required. Unfortunately, simulated annealing may be hard to parallelize on modern multicore computers. Given enough time, simulated annealing
Simulated annealing (SA) is a probabilistic technique for approximating the global optimum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema ( ...

can be shown to find the global optimum with a probability approaching 1, but such convergence proof does not mean the required time is reasonably low. In 1998, it was found that genetic algorithm
In computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application.
Computer science is the study of c ...

s are robust and fast fitting methods for X-ray reflectivity. Thus, genetic algorithms have been adopted by the software of practically all X-ray diffractometer manufacturers and also by open source fitting software.
Fitting a curve requires a function usually called fitness function, cost function, fitting error function or figure of merit (FOM). It measures the difference between measured curve and simulated curve, and therefore, lower values are better. When fitting, the measurement and the best simulation are typically represented in logarithmic space.
From mathematical standpoint, the $\backslash chi^2$ fitting error function takes into account the effects of Poisson-distributed photon counting noise in a mathematically correct way:
:$F\; =\; \backslash sum\_i\; \backslash frac$.
However, this $\backslash chi^2$ function may give too much weight to the high-intensity regions. 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.
Another popular fitting error function is the 2-norm in logarithmic space function. It is defined in the following way:
:$F\; =\; \backslash sqrt$.
Needless to say, in the equation data points with zero measured photon counts need to be removed. This 2-norm in logarithmic space can be generalized to p-norm in logarithmic space. 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.
Open source software

Diffractometer manufacturers typically provide commercial software to be used for X-ray reflectivity measurements. However, several open source software packages are also available: GenX is a commonly used open source X-ray reflectivity curve fitting software. It is implemented in thePython programming language
Python is an interpreted high-level
High-level and low-level, as technical terms, are used to classify, describe and point to specific Objective (goal), goals of a systematic operation; and are applied in a wide range of contexts, such as, for ...

and runs therefore on both Windows and Linux. Motofit runs in the IGOR Pro
IGOR Pro is a Science, scientific data analysis software, numerical computing environment and programming language that runs on Microsoft Windows, Windows or Mac OS X, Mac operating systems. It is developed by WaveMetrics, WaveMetrics Inc., and was ...

environment, and thus cannot be used in open-source operating systems such as Linux. Micronova XRR runs under Java
Java ( id, Jawa, ; jv, ꦗꦮ; su, ) is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea to the north. With a population of 147.7 million people, Java is the world's List of ...

and is therefore available on any operating system on which Java is available.
Reflex is a standalone software dedicated to the simulation and analysis of X-rays and neutron reflectiity from multilayers. REFLEX is a user-friendly freeware program working under Windows, Mac and Linux platforms.
References

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