X-ray reflectivity (sometimes known as X-ray specular reflectivity, X-ray reflectometry, or XRR) is a surface-sensitive analytical technique used in chemistry, physics, and materials science to characterize surfaces, thin films and multilayers. It is a form of reflectometry based on the use of X-rays and is related to the techniques of neutron reflectometry and ellipsometry.
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 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. First calculations of XRR curves were performed by Lyman G. Parratt in 1954. 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.
Approximation When an interface is not perfectly sharp, but has an average electron density profile given by ρ e ( z ) {\displaystyle \rho _{e}(z)} , then the X-ray reflectivity can be approximated by the so called Master formula:
R ( Q ) / R F ( Q ) = | 1 ρ ∞ ∫ − ∞ ∞ e i Q z ( d ρ e d z ) d z | 2 {\displaystyle R(Q)/R_{F}(Q)=\left|{\frac {1}{\rho _{\infty }}}{\int \limits _{-\infty }^{\infty }{e^{iQz}\left({\frac {d\rho _{e}}{dz}}\right)dz}}\right|^{2}}
Here R ( Q ) {\displaystyle R(Q)} is the reflectivity, Q = 4 π sin ( θ ) / λ {\displaystyle Q=4\pi \sin(\theta )/\lambda } , λ {\displaystyle \lambda } is the X-ray wavelength (e.g. copper's K-alpha peak at 0.154056 nm), ρ ∞ {\displaystyle \rho _{\infty }} is the density deep within the material and θ {\displaystyle \theta } is the angle of incidence. The Fresnel reflectivity, R F ( Q ) {\displaystyle R_{F}(Q)} , in the limit of small angles where polarization can be neglected, is given by:
R F ( Q ) = | Q − Q ′ Q + Q ′ | 2 {\displaystyle R_{F}(Q)=\left|{\frac {Q-Q'}{Q+Q'}}\right|^{2}}
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