Critical angle analysis

In Total Reflection X-Ray Fluorescence Analysis (TXRF), the sutface of a solid specimen is exposed to an X-ray beam in grazing geometry. The angle of incidence is kept below the critical angle for total reflection, which is determined by the electron density in the specimen surface layer, and is on the order of mrad. With total reflection, only a few nm of the surface layer are penetrated by the X rays, and the surface is excited to emit characteristic X-ray fluorescence radiation. The energy spectrum recorded by the detector contains quantitative information about the elemental composition and, especially, the trace impurity content of the surface, e.g., semiconductor wafers. TXRF requires a specular surface of the specimen with regard to the primary X-ray light. 

For the selected experimental examples discussed within the next two sections the conditions of a broad size and distance distribution of the surface structures are well supported by the real space analysis. Thus in the following experimental sections instead of the full measured 2d intensity, only one selected horizontal slice at constant qz (of the critical angle of the polymer under investigation) will be shown [20], From a comparison of the Figs. 2-4 it is obvious, that in addition to the full 2d intensity, the split Yoneda peak is visible in a single horizontal slice at constant qz as well. 

The stress distribution analysis shows that maximum stress concentration develops in the radial direction at the pole of the particle, and shear yielding is initiated at around 45° on the surface of rigid particles. Debonding occurs at the pole of the particle, and extends to a critical angle [27]. In case of total adhesion, debonding does not occur and there is cavitation in the matrix, at some distance from the particle pole (not at the interface). 

One of the optical properties used in X-ray analysis is total reflection of X-rays at an optically flat surface. This phenomenon occurs when photons impinge on a surface below the critical angle 6c of total reflection. Defined in practical units 6c (in minutes of arc) can be expressed as a function of atomic number Z ... 

In Table 1, critical angles, reflectivity, and penetration depth of some common materials used in total reflection X-ray analysis are summarized for an excitation energy of 17.44keV, corresponding to Mo K excitation. The table shows that the reflectivity for materials composed of light elements is high, but decreases for heavier elements. In the latter case, absorption effects become more prominent and affect the reflectivity negatively. 

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