General laws of X-ray reflection from rough surfaces: I. Optical theorem and energy conservation law
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RACTION AND SCATTERING OF IONIZING RADIATIONS
General Laws of XRay Reflection from Rough Surfaces: I. Optical Theorem and Energy Conservation Law I. V. Kozhevnikov Shubnikov Institute of Crystallography, Russian Academy of Sciences, Leninskiі pr. 59, Moscow, 119333 Russia email: [email protected] Received July 16, 2009
Abstract—The optical theorem is generalized for the general case of Xray wave reflection from an arbitrary layered rough medium. It is proven strictly that in the absence of absorption the energy conservation law is exactly satisfied within the distorted wave Born approximation if the angular scattering distribution is calcu lated taking into account only the single wave scattering from the inhomogeneities of the medium and if the specular reflectance and transmittance are calculated with allowance for both single and double scattering. DOI: 10.1134/S1063774510040012
INTRODUCTION Beginning with classical studies [1, 2], X rays are widely used in science and technology, being a unique tool for precisely analyzing the surface and interface structure both over depths (stratified inhomogene ities) and over surfaces (roughness). At the same time, all Xray methods are mediated; i.e., informa tion about the internal sample structure is derived from the measured reflection curves and angular scat tering distributions. Thus, the problem of developing a correct theory to describe the interaction of an Xray wave with a rough medium becomes especially impor tant for the further development of Xray methods. As was noted in [3], the energy conservation law can serve as a touchstone for a particular theory: if this law is not satisfied, all conclusions of the theory can be called into question. Currently, one of the most widespread approaches in the theory of Xray scattering from rough surfaces is the distortedwave Born approximation (DWBA), where the scattering amplitude is presented as a series in the multiplicity of wave interaction with a rough surface [3–5]. However, the validity of the energy con servation law within this approximation has not been carefully analyzed. One exception is [3], where this problem was analyzed, but only for a particular case of scattering from a single surface in the limits of large and small roughness correlation lengths and on the assumption of the normal distribution of roughness heights. Moreover, it was concluded in [3] that the energy conservation law is satisfied approximately in DWBA. This conclusion does not stand from the physical point of view. The generality and fundamental character of the energy conservation law means that it is either sat
isfied or not satisfied within a particular theory, but this law in no way can be approximately satisfied. We will show below that the energy conservation law is satisfied exactly within DWBA if the angular dis tribution scattering is calculated taking into account only the single scattering of a wave from the inhomo geneities of the medium and if the specular reflectance and transmittance are calculated with all
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