Amplitude Term

For a given shell, the maximum amplitude is given by the product of the number (N) of the j type of scatterer times its respective backscattering amplitude, F (fc). This maximum amplitude is then reduced by a series of amplitude reduction factors which are considered below. 

The Si(k) term takes into account amplitude reduction due to many-body effects and includes losses in the photoelectron energy due to electron shake-up (excitation of other electrons in the absorber) or shake-off (ionization of low-binding-energy electrons in the absorber) processes. 

Photoionization and therefore EXAFS takes place on a time scale that is much shorter than that of atomic motions so the experiment samples an average configuration of the neighbors around the absorber. Therefore, we need to consider the effects of thermal vibration and static disorder, both of which will have the effect of reducing the EXAFS amplitude. These effects are considered in the so-called Debye-Waller factor which is included as 

This can be separated into static disorder and thermal vibrational components  

It is generally assumed that the disorder can be represented by a symmetric Gaussian-type pair distribution function and that the thermal vibration will be harmonic in nature. 

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Where, /(k) is the sum over N back-scattering atoms i, where fi is the scattering amplitude term characteristic of the atom, cT is the Debye-Waller factor associated with the vibration of the atoms, r is the distance from the absorbing atom, X is the mean free path of the photoelectron, and is the phase shift of the spherical wave as it scatters from the back-scattering atoms. By talcing the Fourier transform of the amplitude of the fine structure (that is, X( )> real-space radial distribution function of the back-scattering atoms around the absorbing atom is produced. 

For the high resolution case, the phase-contrast effects are automatically introduced owing to the combined effect of defocus and spherical aberration, which gives rise to an image of a structure complicated by the fact that also the amplitude term, resulting from the propagation process, interacts in a non-linear way with the phase term [16,89,90,96]. 

Figure 5.6 Biphasic concentration-response plot for an enzyme displaying a high- and low-affinity binding interaction with an inhibitor. In panel A, the data are fit to Equation (5.4) and the best fit suggests a Hill coefficient of about 0.46. In panel B, the data are fitted to an equation that accounts for two, nonidentical binding interactions Vj/v0 = (a/(l + ([/]/ICs0))) + ((1 - a)/(l+([t]/IC(o)))> where a is an amplitude term for the population with high binding affinity, reflected by IC , and IC 0 is the IC50 for the lower affinity interaction. (See Copeland, 2000, for further details.)...

This is called a point-spread function, because it describes how what should be a point focus by geometrical optics is spread out by diffraction. The expression in the curly brackets is the one that is of interest. The other terms are phase and overall amplitude terms, as are usual with Fraunhofer diffraction expressions. The function Ji is a Bessel function of the first kind of order one, whose values can be looked up in mathematical tables. 2Ji(x)/x, the function in the curly brackets, is known as jinc(x). It is the axially symmetric equivalent of the more familiar sinc(x) = sin(x)/x (Hecht 2002), the diffraction pattern of a single slit, usually plotted in its squared form to represent intensity. Just as sinc(x) has a large central maximum, and then a series of zeros, so does jinc(x). Ji(x) = 0, but by L Hospital s rule the value of Ji(x)/x is then the ratio of the gradients, and jinc(0) = 1. The next zero in Ji(x) occurs when x = 3.832, and so that gives the first zero in jinc(x). This occurs at r = (3.832/n) x (q/2a)Xo in (3.2), which is the origin of the numerical factor in (3.1). 

We remark that an expression like (4.13) can no longer be derived in general because of the interference between the two amplitude terms in (4.16). This is the general feature of a nonseparable regime where the spectrum of resonances loses its regularity. When there exist two fundamental periodic orbits, we may expect that the spectrum of resonances still displays quasiperiodic regularities, as is the case for the three-disk scatterer [33]. 

The characteristics of the two sets of trajectories and their weights differ considerably. Trajectories riding initially on the lower left diabat, the aa fifi ) = (fill) class, will climb the wall in this surface and, as they enter the coupling region, they will be additionally accelerated and decelerated by the off-diagonal forces whose effects are modulated by the time dependent amplitude term pistPat + / p pt + For this class of trajectories,... 

On the other hand, the first factor in equation (8.55) is an amplitude term, which depends only on 6 and A, i.e. it contains no information on the Rydberg series as such, but only on the giant resonance and background phases. In fact, it is the profile of the giant resonance itself in absence of any series. In model calculations for A, we use the simple formula (5.20) for the phase shift of a giant resonance (see fig. 8.11). 

The preexponential amplitude terms represent the concentration of enzyme in each form at equilibrium for example, [EX]/[E]o = /( X 2[S]/(1 -t- ATi[S] -I-A i 2[S]). The concentration dependence of the rate of reaction to form E-X follows a hyperbola, which is a function of the saturation of the initial collision complex ... 

Ellipticity can be induced by reflection because of the fact that the reflection coefficients for the two components in the p and s directions are different. A reflection coefficient comprises two parts, an amplitude term and a phase term, and for this reason complex number representation is used to describe it. The "phasor notation is a convention that provides a compact way of representing reflection coefficients... 

If the solution contains the complex amplitude term from the physical point of view this means that there is a phase shift and thus the field can be represented as being the sum of the quadrature (Q) and the inphase (In) components. We will have ... 

The Debve-Waller factor is an amplitude term in any scattering experiment that takes account of the movements of the scatterers about their average positions. This results in attenuation of the scattering which increases with scattering vector. For EXAFS analysis the appropriate Debye-Waller factor takes account of variations in the absorber-scatterer distance, and thus depends on how much the motion of this pair of atoms is correlated. 

Although there is no clamping or growth of the wave, it is dispersive, so the frequency depends on wave number. To determine the dispersion relation, we simplify the momentum equation for a wave whose disturbance amplitude is small compared with the wavelength. Because of the small amplitude, terms involving squares and products may be neglected and the linearized inviscid, constant-density form becomes (Lighthill 1978)... 

The first term in equation (9-132) represents the time displacement of the input function and the second is the amplitude attenuation of the input. As can be seen, the amplitude term contains all of the kinetic and phase-distribution parameters. The exit concentration is also given by equation (9-132), but now with... 

Thus, the numerator terms are amplitude factors and the denominator polynomial represents the total nutrient processed by the organism. This allows partitioning the nutrient processed into a number of grossly defined pools. The number of pools is determined by the number of the denominator terms. This is entirely analogous to the rational polynomial which describes the steady state rate of an enzyme-catalyzed reaction in which the denominator consists of the enzyme species and the numerator contains amplitude terms. 

Here, the new quantity X, is a complex amplitude and describes both the original amplitude and the phase. The purely sinusoid part, is now free of phase information. This is highly significant, because it turns out that in general, we want to describe, modify and calculate amplitude and phase together. (From here on, amplitude terms in complex e q)onential expressions will generally be complex - if the pure amplitude is required, it will be denoted. )... 

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