Critical angle for total reflection

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. 

The luminous nonpolarized wings are due to part of the emission built up in the middle of the crystal. They can be understood when taking total internal reflection into account. Refraction of the emission from the object O at the zeolite L-air interface observed at an angle 2 is explained in Fig. 16a. The emission can appear between 0° and the critical angle for total reflection which is... 

As implied by the discussion above craze fibril extension ratio or its inverse the fibril volume fraction of the craze is an important parameter of the microstructure. Fibril volume fractions can be measured by several different methods. The refractive index n of the craze can be measured by measuring the critical angle for total reflection of light by the craze surface. Using the Lorentz-Lorenz equation Vf then can be computed from The method is difficult because small variations... 

Figure 4. Two configurations for evanescent wave optics, (a) Top total internal reflection of a plane wave at the base of a glass prism. Bottom the reflectivity R recorded by a detector as a function of the angle of incidence shows the increase to unity at 6, the critical angle for total reflection, (b) ATR setup for the excitation of surface plasmons (PSPs) in Kretschmann geometry. Top a thin metal film (thickness 50 nm) is evaporated onto the base of the prism and acts as resonator driven by the photon field. Bottom the resonant excitation of the PSP wave is seen in the reflectivity curve as a sharp dip at coupling angle 6g.

Recently Wright et al. have obtained room temperature values of all five elastic constants for uniaxially drawn polymethylmethacrylate (Perspex) and polystyrene (Carinex) by measuring the critical angle for total reflection of an ultrasonic beam incident on immersed samples. The specimens were stretched by similar extents at different temperatures, with orientation assessed by measurements of optical birefringence. 

Neutron reflection from fluid interfaces has revealed detailed arrangements of molecules in adsorbed layers (Lu et al., 2000). Basically neutrons can be totally reflected from interfaces in a manner similar to the total internal reflection of light at interfaces where a suitable refractive index distribution exists. Qose to the critical angle for total reflection, the intensity of the weak scattered beam provides information on the refractive index distribution normal to the interface. For neutrons, a refractive index can be defined in terms of the density of the various nuclei present and their scattering cross sections. As in neutron scattering, the use of ordinary and deuterated compounds is a powerful tool in identifying the location and arrangement of molecules. 

The structure of crazes in bulk specimens was studied by Kambour [15], who used the critical angle for total reflection at the craze/polymer interface to determine the reliactive index of the craze, and showed that the craze was roughly 50 per cent polymer and 50 per cent void. Another investigation involved transmission electron microscopy of polystyrene crazes impregnated with an iodine-sulphur eutectic to maintain the craze in its extended state [33, 34]. The structure of the craze was clearly revealed as fibrils separated by the voids that are responsible for the overall low density. 

Figure 2. X-ray reflectivity curves as a fimction of the toluene partial pressure for a porous HSQ low-k dielectric film. As the toluene partial pressure increases, the critical angle for total reflection also increases (moves to higher 0) in response to the toluene condensing in progressively larger pores. The standard uncertainty in log(R) is less than the line width...

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