[B._Beckhoff,_et_al.]_Handbook_of_Practical_X-Ray_(b-ok.org).pdf

As the primary field varies with the angle of

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at an angle higher than the critical angle. As the primary field varies with the angle of incidence also the intensity of the fluorescence signal shows this variation. From the obtained shape it is possible to distinguish between film type samples, residue samples, thin layers,
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512 C. Streli et al. 0.9mrad 1.8mrad 3.6mrad Intensity (a.u.) - 50 - 60 -4 0 - 20 - 30 - 10 0 60 50 40 30 20 10 Inside Si reflector Distance from surface (nm) Vacuum 4 3.5 3 2.5 2 1.5 1 0.5 0 Fig. 7.45. Intensity above and below the Si surface for Mo-K α radiation (17.5 keV) for various angles of incidence, 1.8 mrad is the critical angle and buried layers. The theory behind this is discussed in detail in [145–155] as well as in [113]. The differences have been shown in Fig. 7.43. The step-like function (Sc) is obtained, if the contamination forms par- ticles on the surface of the wafer (equivalent chemical analysis TXRF). The peaking curve (Ni) is obtained, when the atoms are evenly distributed within a layer of a few nanometers thickness placed on the wafer surface. Most of the real samples do not show one of these extreme cases. If one does not want to measure the complete angle-dependent behavior, only a single measure- ment performed at the operating angle, where the two curves (Sc, Ni) are crossing, allows already accurate quantification. For completeness, the typical curve for the bulk material (silicon of the wafer) or bulk contamination is given. Usually the bulk Si signal is used for control of the angular adjustment. Quantification differs for the types of contaminants. For the case of granular residues on a substrate (particulate type)—which is equivalent to chemical analysis by using TXRF—the intensity above the critical angle is constant, because the thin, small “sample” is completely ex- cited. The intensity doubles at the critical angle in a step-like fashion and remains at the twofold value down to very small angles due to total reflection: I particle i ( ϕ 0 ) = k particle · I 0 · c i · (1 + R ( ϕ 0 )) , (4a) following the angular behavior of the reflection coefficient R ( ϕ 0 ). This inten- sity is proportional to the primary intensity I 0 and the interesting area-related concentration c i . For incidence angles ϕ w below ϕ crit , (4a) can be simplified to I particle i = k particle · I 0 · c i · 2 . (4b)
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Methodological Developments and Applications 513 For buried layers (film like) in a substrate the intensity far above the crit- ical angle becomes constant. The asymptotic behavior of the intensities for the particulate and film like contaminations can even be equal, if both con- centration values are the same and the appropriate scaling factors k particle and k film are chosen. But the intensity for the buried layers steadily increases with decreasing angle and can reach (theoretically) the fourfold value at the critical angle. For the smaller incidence angles the intensity approaches zero, according to I film i ( ϕ 0 ) = k film · I 0 · c i · (1 R ( ϕ 0 )) · ϕ z P = k film · I 0 · c i · T ( ϕ 0 ) · ϕ 0 ϕ T ( ϕ 0 ) · ρ · τ m , (7.5) where Z P is the penetration depth z p = 1 ρτ m ϕ T .
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