Attenuation function

The use of fundamental parameters is attractive for various reasons. They impose fewer restrictions on the number of standards required for analysis. This simplifies the standardisation protocol for maintaining a XRF system, and permits greater flexibility in dealing with different types of materials. Inten-sity/concentration algorithms of the fundamental type, i.e. without recourse to the use of standards, have gradually developed [238-240] and are now widely available [241]. Functionality and quality of XRF software have reached a very high level, with a large variety of evaluation procedures and correction models for quantitative analysis, and calculation of fundamental parameter coefficients for effective matrix corrections. Nevertheless, there is still a need for accuracy improvement of fundamental parameters, such as the attenuation functions. 

H(u) is the Fourier Transform of h(r) and is called the contrast transfer function (CTF). u is a reciprocal-lattice vector that can be expressed by image Fourier coefficients. The CTF is the product of an aperture function A(u), a wave attenuation function E(u) and a lens aberration function B(u) = exp(ix(u)). Typically, a mathematical description of the lens aberration function to lowest orders builds on the Weak Phase Approximation and yields the expression ... 

As a result of these very general considerations, one expects the dielectric response function, as expressed by the complex permittivity, k (oj), or the attenuation function, a(oi), of ordinary molecular fluids to be characterized, from zero frequency to the extreme far-infrared region, by a relaxation spectrum. To first order, k (co) may be represented by a sum of terms for individual relaxation processes k, each given by a term of the form ... 

The Attenuation Function. The complex permittivity k (w) and the attenuation function a(ui) are related by basic electromagnetic theory, and independently of any molecular model, as follows ... 

For fundamental physical reasons, the attenuation function for any process must vanish as to - °°. This expectation is borne out by far-infrared measurements of ct(u>) for a variety of molecular systems exhibiting a relaxation-type absorption in the microwave and millimeter-wave region (17-23). While H2O as a solute in nonhydrogenbonding solvents also shows this behavior (35), the millimeter-wave and far-infrared spectrum of 0(2.) is complicated by contributions to a (10) due to intermolecular vibrations involving a cluster of H2O molecules (libration and translation), in addition to the high-frequency tail of the relaxation absorption. A heuristic treatment of the general problem (30) makes the relaxation time,... 

In summary, it is expected that the bulk attenuation function for ordinary molecular fluids is reasonably well represented by relaxation-type processes in the microwave region. At high frequencies, in the region of the extreme far-infrared, deviations from Eq. (4) will occur, even for a process with a single relaxation time. Phenomenologically, as to- , the efficiency with which a representative collision interrupts the absorption or emission of radiation in a molecular fluid must decrease toward zero, and the relaxation time in Eq. (4) must become frequency-dependent. 

H2O The Pure Liquid. Because of the large magnitude of the attenuation of H20(H) and the ubiquitous presence of water in biological systems, the bulk attenuation function of most biological fluids is dominated by the attenuation due to Until re-... 

Figure 2. Attenuation function for HiO(l). aKF.t.(a>) is the contribution to the total attenuation function due to the rotational relaxation, as calculated from the model in Ref. 30. The circles are the experimental points from Refs. 42 and 43.

Figure 3. Schematic of the total attenuation function arorf ). and its contributions, for H20(l) at millimeter and far-infrared wavelengths.

In spite of the complexity of the analysis of the millimeter-wave and far-infrared spectrum of l CKO, the principal contributor to the bulk attenuation function of typical biological systems in this frequency range, no compelling experimental evidence or theoretical constructs exist to disqualify the assumption that this system is a complex, but typical, collision-broadened system whose fundamental features (1) are well understood. 

As for all the systems relegated to Section 2 the attenuation function for structural H2O in the microwave and far-infrared region, as well as that for free H2O, can be understood in terms of collision-broadened, equilibrium systems. While the average values of the relaxation times, distribution parameters, and the features of the far-infrared spectra for these systems clearly differ, the physical mechanisms descriptive of these interactions are consonant. The distribution of free and structural H2O molecules over molecular environments is different, and differs for the latter case with specific systems, as are the rotational dynamics which govern the relaxation responses and the quasi-lattice vibrational dynamics which determine the far-infrared spectrum. Evidence for resonant features in the attenuation function for structural H2O, which have sometimes been invoked (24-26,59) to play a role in the microwave and millimeter-wave region, is tenuous and unconvincing. 

Figure 4. Interactions among the free HaO, the structural H20, the cooperative system, and an external EM field. a(

Figure 5. Schematic of an external EM field interaction with the equilibrium system and the dissipative subsystem. a(w) and are the attenuation function and the biological-response function for the equilibrium system, respectively aBIsM and [hns(i ) are the same functions, respectively, for the dissipative subsystem,- m is the frequency of the EM field.

The Attenuation Function. In contrast to the existing exper-mental evidence which would suggest that the biological response fucntions 8- Dis(to) may have sharply frequency-selective features at millimeter wavelengths (145-14 7), M. cm- -, and that the Raman... 

The preceeding discussion has dealt with the qualitative effects of the Bose-condensation on the attenuation function, ctDig(m)- In general, we require the frequencies and matrix elements p and PmnI for all the possible transitions, m ->-n, among the vibrational states postulated by FrBhlich, in order to characterize the resulting absorption spectra, Eq. (27),... 

---