Critical angle

Attenuated total reflection, on which atr—ftir is based, occurs when the rarer medium is absorbing and is characterized by a complex refractive index (40). The absorbing characteristics of this medium allow coupling to the evanescent field such that this field is attenuated to an extent dependent on k. The critical angle in the case of attenuated total reflection loses its meaning, but internal reflection still occurs. Thus, if the internally reflected beam is monitored, its intensity will reflect the loss associated with the internal reflection process at the interface with an absorbing medium. 

Attenuated total reflection (ATR), also called internal reflection, is based on the phenomenon of total internal reflection. In ATR the infrared beam is directed into an infrared-transmitting crystal so that it strikes the crystal surface at less than the critical angle and undergoes total internal 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 primary X-ray beam is directed onto the solid surface in grazing incidence. The angle of incidence is kept below the critical angle at which total reflection occurs. The critical angle is given by... 

The angular dependence of the fluorescence yield in the ne borhood of the critical angle should be considered in detail to establish the chemical nature of surface impurities, as well as for quantitation in terms of their concentrations (Figure 1). 

Agglomerated impurities, such as particles or droplet residues, do not participate in the interference phenomenon leading to total reflection their fluorescence intensity is independent of the angle of incidence below the critical angle, and drops by a factor of 2 if the critical angle is surpassed due to the disappearance of the reflected component in the exciting beam nonreflecting impurities and residues). 

In this list, p is the mass density, X b is the sum of scattering lengths of the atoms con rising the molecule, 8 is the real part of the refractive index, Gq is the critical angle, and is the critical neutron momentum. 

Fig. 3.61. Schematic illustration of projectile trajectories, showing focusing collisions when the projectiles impinge under a critical angle (5 c [3.150].

Fig. 4.4. Penetration depth z ofX-rays striking silicon at a variable glancing angle d>i. The curves were calculated for three different photon energies. The dashed vertical line signifies the respective critical angle [4.21],...

Fig. 4.10. Fluorescence signal from small particles or thin films deposited on a silicon substrate used as sample carrier. The intensity was calculated for particles, thin films, or sections ofdiffe-rent thickness but equal mass of analyte, and plotted against the glancing angle f. A Mo-Ka beam was assumed for excitation. Particles or films more than 100 nm thick show double intensity below the critical angle of0.1° [4.21].

---