The Reflectivity of an Interface

The reflectivity of an interface is calculated from Fresnel s equations, which are derived from the boundary conditions that the tangential (t) component of the electric field vector, and the normal (n) component of the electric displacement vector, D = e - E , are continuous across the inter-face. We define the so-called Fresnel reflection coefficient r as the ratio of the complex amplitudes of the electric field vectors of the reflected and the incident waves  

FIGURE 5. Frequency dependence of real and imaginary parts of the dielectric constant of Pt. The normal-incidence reflectivity of Pt (in air) is shown in the inset. After Ref 15. 

The Fresnel coefficients are a function of the polarization and the angle of incidence of the incident beam. We commonly work with linearly polarized light, where the electric field vector is either perpendicular (5, J.) or parallel (a id to the plane of incidence. For 5-polarization, the electric field vector E is parallel to the surface at all angles of incidence, while for p-polarized light at oblique angles of incidence, the electric field vector has a component normal to the surface. At normal incidence, the physical difference between 5- and p-polarization vanishes. 

For oblique angles of incidence, it is useful to define for each phase j an angle-dependent quantity, as  

FIGURE 6. Schematic of an obliquely damped electromagnetic wave. Note that the planes of constant amplitude and constant phase do not coincide. 

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Whilst ellipsometry is undoubtedly the technique to use to obtain accurate optical constants for surfaces and surface films, it suffers from being both expensive and slow, and consequently has only been of limited use to electrochemists. As a result of these difficulties, several techniques based on simply monitoring of the reflectivity of an interface have been developed. Whilst in theory these are less informative, in practice they have proved much more useful. The major advantages are lower cost and greater speed, making them ideal for kinetic studies. 

The use of evanescent waves is very valuable to the study of interfacial properties. Techniques such as total internal reflection fluorescence (TIRF) and attenuated transmitted reflectance (ATR) use the energy of evanescent waves to probe thin regions in the vicinity of an interface to determine surface concentrations of interfacial species. 

The nature of an interface (liquid/liquid, solid/liquid, air/liquid and so on) should reflect on the relevant interfacial phenomena, so that detailed understandings of chemical and structural characteristics of the interface at a microscopic level are of primary importance for further advances in various sciences. In practice, solid/liquid and air/liquid interfacial systems have been studied widely by various experimental techniques, and the knowledges about the characteristics of the interfaces have been accumulated. However, very little is known about the chemical and structural characteristics of a liquid/liquid interface at a microscopic level. So far, thermodynamic and electrochemical techniques have been applied to study liquid/liquid interfacial chemistry. Nonetheless, its dynamic aspects have rarely been explored. 

Using Fresnel s equations (Eqs. 6.4-4) the reflectance at an interface between two homogeneous media can be calculated for all angles of incidence and azimuths of the plane of polarization. As a prerequisite, the compound specific optical functions e or or... 

Another method evolved from the work of Fresnel. The relationship between the reflectance from an interface between two transparent media and their respective refractive indices is given by Fresnel s equation ... 

In the case of an interface between two immiscible liquids due to the presence of interfacial tension, the interface tends to contract. The magnitude of interfacial tension is always lower than the surface tension of the liquid with the higher tension. The liquid-liquid interface has been investigated by specular reflection of X rays to gain structural information at angstrom (A = 10 cm = 0.10 nm) resolution. " ... 

In this expression Rp is the Fresnel reflectivity of the water-air interface, i.e., the reflectivity of an infinitely sharp interface between homogeneous media, p is the scattering-length density of the subphase and p(z) is that of the monolayer. In the measurements, the ratio R/Rp is determined as a function of... 

In the preceding section we examined the reflectivity from an interface between two media. Next we consider the case (see Heavens9) in which a thin film (medium 1) of thickness t is now interposed between a vacuum (medium 0) and a substrate (medium 2) as in Figure 7.6. A thin film deposited on a thick, flat substrate will correspond to such a system. In this case the reflected radiation that is observed will consist not only of rays reflected at the 0-1 interface but will also of rays transmitted from medium 1 to medium 0 after having been reflected at the 1-2 interface once, twice, etc. 

In the homogeneous solution, nucleation is accompanied by the formation of an interface between solution and crystal at the cost of an increase in the free energy. On the other hand, the monomer in crystal has a smaller free energy than that in solution if the solution is supersaturated (S > 1). Therefore, there are two opposite tendencies in the nucleation reaction. The first is the increase in the free energy caused by the formation of the interface, which is reflected in the first term on the right-hand side of Equation (6.2). This term is always positive. The other tendency is the decrease in the free energy caused by the formation of the crystal, which is shown in the second term, which is negative when 5 > 1. 

The actual structure at a vapor-liquid interface can be probed with x-rays. Rice and co-workers [72,73,117] use x-ray reflection to determine the composition perpendicular to the surface and grazing incidence x-ray diffraction to study the transverse structure of an interface. In a study of bismuth gallium mixtures. 

The detailed examination of the behavior of light passing through or reflected by an interface can, in principle, allow the determination of the monolayer thickness, its index of refiraction and absorption coefficient as a function of wavelength. The subjects of ellipsometry, spectroscopy, and x-ray reflection deal with this goal we sketch these techniques here. 

We have seen that partial reflection of an X-ray beam occurs from a surface at angles of incidence above (j> with the intensity dropping off rapidly with (j>. In fact partial or total reflection occurs at the interface between any two materials with different re-... 

As long as 9 > 9, and the sample is not absorbing, the reflectivity of the prism/sample interface will be complete. However, at any wavelength where the sample is absorbing, the reflectivity will be attenuated or less than complete. Thus, an absorption spectrum that is similar to that obtained in transmission can be produced in ATR. 

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