Standing wave models

The mathematical derivation of the analytical expressions for standing waves in resist films disposed over appropriate substrates is presented in Chapter 9. In this chapter, we will make use of the relevant standing wave expressions for the normal and nonincidence illumination conditions. 

For the specific case of normal incidence illumination of a homogenous resist film (see Fig. 12.3), the electric field inside the resist 2(2) is given by  

It should he noted that absorption is accounted for in Eq. (12.13) through the imaginary part of the index of refraction. Absorption coefficient a is related to the imaginary part of the refractive index by 

In the special case where the resist film is not homogenous, such that there is a small variation in the imaginary part of the refractive index along the z direction, with the real part constant, the anisotropic absorbance in that case can be expressed as 

the electric field intensity can be determined as a function of z by using 

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Abstract The Thomas-Fermi and Hartree-Fock calculations of non-hydrogen atomic structure rely on complicated numerical computations without a simple visualizable physical model. A new approach, based on a spherical wave structure of the extranuclear electron density on atoms, self-similar to prominent astronomical structures, simplifies the problem by orders of magnitude. It yields a normalized density distribution which is indistinguishable from the TF function and produces radial disuibutions, equivalent to HF results. Extended to calculate atomic ionization radii, it yields more reliable values than SCF simulation of atomic compression. All empirical parameters used in the calculation are shown to be consistent with the spherical standing-wave model of atomic electron density. 

In order to estimate atomic polarizabilities, it is noted that the inverse of charge density at the crests of the spherical-wave representation of atoms, in units of a je, should be such a measure. This quantity has been calculated before [6] from a spherical standing-wave model of the atom, shown schematically as a radial projection in Fig. 7. 

The Pythagorean theory of music, for lack of a more convincing explanation, became firmly associated with celestial events and structures, to such an extent that initiates like Johannes Kepler could actually hear the music of the spheres . Although this sound is less audible today, the original theory has survived in the form of the modern standing-wave model of harmonic vibrations. Many other number patterns of the same type have also survived into the modern era as the basis of erudite new theories. 

The second model is a quantum mechanical one where free electrons are contained in a box whose sides correspond to the surfaces of the metal. The wave functions for the standing waves inside the box yield permissible states essentially independent of the lattice type. The kinetic energy corresponding to the rejected states leads to the surface energy in fair agreement with experimental estimates [86, 87],... 

Whereas the XSW technique takes advantage of the standing wave established on the total reflection of X-rays from a mirror surface, a conceptually more straightforward approach is that of simply specularly reflecting an X-ray beam from an electrode coated with the film of interest, measuring the ratio of the intensities of the incident and reflected rays, and fitting the data, using the Fresnel equations, to a suitable model an approach similar to optical ellipsometry. 

Whereas the profile in linear wave equations is usually arbitrary it is important to note that a nonlinear equation will normally describe a restricted class of profiles which ensure persistence of solitons as t — oo. Any theory of ordered structures starts from the assumption that there exist localized states of nonlinear fields and that these states are stable and robust. A one-dimensional soliton is an example of such a stable structure. Rather than identify elementary particles with simple wave packets, a much better assumption is therefore to regard them as solitons. Although no general formulations of stable two or higher dimensional soliton solutions in non-linear field models are known at present, the conceptual construct is sufficiently well founded to anticipate the future development of standing-wave soliton models of elementary particles. 

Macroscopic experiments allow determination of the capacitances, potentials, and binding constants by fitting titration data to a particular model of the surface complexation reaction [105,106,110-121] however, this approach does not allow direct microscopic determination of the inter-layer spacing or the dielectric constant in the inter-layer region. While discrimination between inner-sphere and outer-sphere sorption complexes may be presumed from macroscopic experiments [122,123], direct determination of the structure and nature of surface complexes and the structure of the diffuse layer is not possible by these methods alone [40,124]. Nor is it clear that ideas from the chemistry of isolated species in solution (e.g., outer-vs. inner-sphere complexes) are directly transferable to the surface layer or if additional short- to mid-range structural ordering is important. Instead, in situ (in the presence of bulk water) molecular-scale probes such as X-ray absorption fine structure spectroscopy (XAFS) and X-ray standing wave (XSW) methods are needed to provide this information (see Section 3.4). To date, however, there have been very few molecular-scale experimental studies of the EDL at the metal oxide-aqueous solution interface (see, e.g., [125,126]). 

In order to describe microscopic systems, then, a different mechanics was required. One promising candidate was wave mechanics, since standing waves are also a quantized phenomenon. Interestingly, as first proposed by de Broglie, matter can indeed be shown to have wavelike properties. However, it also has particle-Uke properties, and to properly account for this dichotomy a new mechanics, quanmm mechanics, was developed. This chapter provides an overview of the fundamental features of quantum mechanics, and describes in a formal way the fundamental equations that are used in the construction of computational models. In some sense, this chapter is historical. However, in order to appreciate the differences between modem computational models, and the range over which they may be expected to be applicable, it is important to understand the foundation on which all of them are built. Following this exposition. Chapter 5 overviews the approximations inherent... 

Therefore, oscillations of K (t) result in the transition of the concentration motion from one stable trajectory into another, having also another oscillation period. That is, the concentration dynamics in the Lotka-Volterra model acts as a noise. Since along with the particular time dependence K — K(t) related to the standing wave regime, it depends also effectively on the current concentrations (which introduces the damping into the concentration motion), the concentration passages from one trajectory onto another have the deterministic character. It results in the limited amplitudes of concentration oscillations. The phase portrait demonstrates existence of the distinctive range of the allowed periods of the concentration oscillations. 

The behaviour of the correlation functions shown in Fig. 8.5 corresponds to the regime of unstable focus whose phase portrait was earlier plotted in Fig. 8.1. For a given choice of the parameter k = 0.9 the correlation dynamics has a stationary solution. Since a complete set of equations for this model has no stationary solution, the concentration oscillations with increasing amplitude arise in its turn, they create the passive standing waves in the correlation dynamics. These latter are characterized by the monotonous behaviour of the correlations functions of similar and dissimilar particles. Since both the amplitude and oscillation period of concentrations increase in time, the standing waves do not reveal a periodical motion. There are two kinds of particle distributions distinctive for these standing waves. Figure 8.5 at t = 295 demonstrates the structure at the maximal concentration... 

This statement could be proved in the manner similar to that used in Section 8.2. It is important to note that the correlation dynamics of the Lotka and Lotka-Volterra model do not differ qualitatively. A stationary solution exists for d = 3 only. Depending on the parameter k, different regimes are observed. For k kq the correlation functions are changing monotonously (a stable solution) but as k < o> the spatial oscillations of the correlation functions (unstable solution) are observed. In the latter case a solution of non-steady-state equations of the correlation dynamics has a form of the non-linear standing waves. In one- and two-dimensional cases there are no stationary solutions of the Lotka model. 

A little-known paper of fundamental importance to modern atomic theory was published by Hantaro Nagaoka in 1904 [10]. Apart from oblique citation, it was soon buried and forgotten. With hindsight it deserved better than that. It contained the seminal ideas underlying the nuclear model of the atom, the standing-wave nature of orbital electrons and radiationless stationary states. It was so far ahead of contemporary thinking that later imitators either failed to appreciate its significance, or pretended to be unaware of it. 

The nuclear concentration of mass anticipated Rutherford s model of the atom, and Bohr s planetary model by a decade. The spectral integers, linked to a standing-wave pattern, predates de Broglie s proposal by two decades. 

Figure 12. Axial temperature profiles, one-dimensional model. Standing wave situation. E...

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