The Thin Film Approximation

The Thin Film Approximation. If absorption and X-ray fluorescence are neglected (i.e. the film is assumed to be infinitely thin ), then the measured X-ray intensity from element A in a thin specimen of unit thickness, generated by a current i is given by 

Cliff and Lorimer (1975) used this equation to form the basis for X-ray microanalysis of thin foils, where the constant kAB contains all the factors needed to correct for atomic number differences. kAB varies with operating voltage, but is independent of sample thickness and composition if the two intensities are measured simultaneously. Its value can be determined experimentally with accuracy, using specimens of known composition. The value of kAB can be determined by calculation more rapidly, but with less accuracy. 

On the assumption that the background bremsstrahlung has been removed, and that the data are in the form of the measured intensities at full width half maximum (FWHF), equation (1) can be used to calculate the ratio of the number of atoms of A to the number of atoms of B, i.e. the concentrations in the binary material. 

Standard thin films of known composition may not always be available, and in multi-component systems many k factors should be determined, which is a time-consuming process. A new quantitative procedure for thin specimens has been developed to overcome these limitations  

In a thin specimen, the magnitude of Ia is proportional to the mass thickness, pt  

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A special case of the thin-film approximation considers that the substrate can diffuse within the film with negligible depletion. This corresponds to both Case V (unsaturated enzyme). Case I (saturated enzyme) and the border between them. A thin film is the only situation where we can fit experimental data to a theoretical calibration curve for the whole range of glucose concentrations with ... 

The thin film approximation assumes that absorption of X-rays within the sample (and any second order efifeet ensuing from absorption) is negligible. It is a good approximation for many of the TEM samples. Within this approximation the detected intensity (Iac) for the analytical line (a) of element A is proportional to the number of generated X-rays (Gao) and the detection efficiency (Pao) for this line. 

Following the principles of the Petrie model, and recalling that the film thickness <5 is much smaller than the radius S/R field equations are averaged over the thickness and that there are no shear stresses and moments in the film. The film is regarded, in fact, as a thin shell in tension, which is supported by the longitudinal force Fz in the bubble and by the pressure difference between the inner and outer surfaces, AP. We further assume steady state, a clearly defined sharp freeze line above which no more deformation takes place and an axisymmetric bubble. Bubble properties can therefore be expressed in terms of a single independent spatial variable, the (upward) axial position from the die exit,2 z. The object... 

We shall see that there are certain generic features of the thin-film approximation that have already been illustrated by the journal-bearing analysis, which will appear whenever the thin-film approximation is used. The most significant feature of the journal-bearing analysis, which is shared by all thin-film theories, is the assumption that velocity gradients in the direction parallel to the boundaries are asymptotically small compared with those... 

In the case of a unidirectional flow, the dominant balance in (5-62) and (5-63) involved only the pressure gradient Vsj and the largest viscous term 92u s/dz2. If we adopt the intuitively reasonable assumption that the same two terms should remain in balance in the thin-film approximation, we are led immediately to the result... 

Hence, the thin-film approximation will hold provided that... 

Figure 6-2. A sketch of the initial configuration of spreading drop on a solid substrate. The drop may be either a two-dimensional ridge of fluid (with the sketch showing the cross-sectional view), or it may be an axisymmetric drop. The thin film approximation requires that the radial breadth of the drop be much larger than its height. The initial values of these quantities are denoted as Rq and H0.

Hence we seek an asymptotic solution for the leading-order terms in the thin-film approximation, i.e., u<(>>p<(l>w<(>>, as a regular perturbation in 8, i.e.,... 

The shear-stress condition requires a bit more care, as indicated earlier. We can revert to the dimensional form, (6 12), with rs = 0 (we assume there are no wind-driven shear-stress contributions), and the unit normal and tangent vectors given by the thin-film approximations (6 17). The first term in (6 12) can be shown to take the form... 

The degree of deviation of the thin-film approximation from the exact spectral simulations depends upon the refractive indices of the immersion medium and the substrate 113, the oscillator strength of the film, and on the angle of incidence (Pi. The deviation can be more than 10% for a 50-nm film and several percent for a 1-nm film (Sections 2.2, 2.5, 3.3, and 3.10). 

Let us compare the thin-film approximation formulas for (1) the transmissivity (1.98) (2) the reflectivities for the external reflection from this film deposited onto a metallic substrate (1.82) (3) the internal reflection at (p > (pc (1.84) and (4) the external reflection from this film deposited on a transparent substrate (dielectric or semiconducting) (1.81) (Table 1.2). In all cases s-polarized radiation is absorbed at the frequencies of the maxima of Im( 2), vroi (1.1.18°), whereas the jo-polarized external reflection spectrum of a layer on a metallic substrate is influenced only by the LO energy loss function Im(l/ 2) (1.1.19°). The /7-polarized internal and external reflection spectra of a layer on a transparent substrate has maxima at vro as well as at vlq. Such a polarization-dependent behavior of an IR spectrum of a thin film is manifestation of the optical effect (Section 3.1). 

A comparison between band intensities obtained with the exact formulas and those obtained by the thin-film approximation (Fig. 3.21c, d) reveals that the highest deviation from linearity is observed for the modes polarized in the film plane. 

However, the thin-fihn approximation may yield significant errors even for films a few tenths of nanometers thick [7]. Yamamoto and Ishida [44], based on the KK technique of Hansen and Abdou [465] for a multi-interface system, and Buffeteau and Desbat [452], based on the Abeles matrix method and the fast Fourier transform for the KK relations, suggested procedures for measuring the optical constants from IRRAS without employing the thin-film approximation. These techniques can yield the optical constants of both organic and inorganic ultrathin films with high accuracy if the film thickness and are known. 

For thin-walled injection-molded parts, the calculation of thermally and pressure-induced stresses can be simplified considerably by using the thin-film approximation. This enables us to apply the following assumptions ... 

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