X, K relations

Calculations of vibrational spectra of bent triatomic molecules with second order Hamiltonians produce results with accuracies of the order of 1-5 cm-1. An example is shown in Table 4.9. These results should again be compared with those of a Dunham expansion with cubic terms [Eq. (0.1)]. An example of such an expansion for the bent S02 molecule is given in Table 0.1. Note that because the Hamiltonian (4.96) has fewer parameters, it establishes definite numerical relations between the many Dunham coefficients similar to the so-called x — K relations (Mills and Robiette, 1985). For example, to the lowest order in l/N one has for the symmetric XY2 case the energies E(vu v2, V3) given by... 

As already discusf d for the single-oscillator case in Section II.C.2, it is possible to obtain Dunham-like form for the local-mode eigenspectrum (4.33). Given the anharmonic character of the algebraic formulation, we expect to find a certain number of relations between the Dunham coefficients similar to those of the x-K relations of Lehmann [73, 74]. In fact, we can compare the traditional Dunham series for a triatomic molecule... 

Finally it should be emphasized that the two ways of setting up the hamiltonian matrix, in either local mode basis functions or normal mode basis functions, are mathematically exactly equivalent they are two different representations of the same problem. This is emphasized by Figure 7, which shows as an example the two matrices for V s 4 (with the x,K relations already imposed in the normal mode matrix). The only parameters appearing are O), x, and X. The two matrices have identical eigenvalues. Their... 

The quantity x k) in Equation (8.20) is the experimentally observed absorption, like that in Figure 8.32, after subtraction of the smoothly declining background. What is left is a sum of sine waves of which we require the wavelengths which can be related to Rj, provided the phase factor 6j k) is known. This process of obtaining wavelengths from a superposition of... 

As in the case of the characteristic, theoretical values of characteristic K2 depend upon supply conditions. According to Shepelev," in the case of air supply through a nozzle with a uniform outlet velocity profile, K X-K = (1 +Pr)/ (2ir0.082 ). Thus, when = 6.88 and Pr = 0.7, A, = 5.85. Grimitlyn suggests the following relation between K2 and coefficients ... 

However, mathematics is essential to explain how structural data are derived from EXAFS. The EXAFS function, x(k), is extracted from the X-ray absorption spectrum in Fig. 4.10 by removing the approximately parabolic background and the step, i.e. the spectrum of the free atom. As in any scattering experiment, it is customary to express the signal as a function of the wavenumber, k, rather than of energy. The relation between k and the kinetic energy of the photoelectron is ... 

In an isotropic medium, vectors such as stress S and strain X are related by vector equations such as, S = kX, where S and X have the same direction. If the medium is not isotropic the use of vectors to describe the response may be too restrictive and the scalar k may need to be replaced by a more general operator, capable of changing not only the magnitude of the vector X, but also its direction. Such a construct is called a tensor. 

The kernel K(t-x) is related to the dissipation process. The quantum fluctuation-dissipation relation yields... 

The force constant K relates with the strength of the bond. For a single bond, it is approximately 5 x 105 dynes cm . It becomes double and triple for double and triple bonds respectively. 

It has been shown (74, 8S) that for such a system the expansion constant X is related to the bulk modulus K by... 

We thus have [Vx] = KJR from equation (5.3) and [n] = [V ] = 7 vx[ x] = K /xiRJRV from equation (5.6). Since the Schottky equilibrium constants are known for ionized defects, it is profitable to express in terms of using the relation... 

A solution to equation (16a) having the form = Tk X is called a harmonic of the attached number k. Clearly, this function satisfies problem (16a) with the initial condition u0(x) = Tk X k x). Let us find out the conditions under which every harmonic y, k = 1,2,..., N — 1, is stable. From the recurrence relations... 

Equation 7-18 relates the volume of added M+ to [M+], [X-], and the constants V7, C, and CjJ,. To use Equation 7-18 in a spreadsheet, enter values of pM and compute corresponding values ofVM, as shown in Figure 7-10 for the iodide titration of Figure 7-7. This is backward from the way you normally calculate a titration curve in which VM would be input and pM would be output. Column C of Figure 7-10 is calculated with the formula [M+l = 10 pm, and column D is given by [X-] = k sp/[M+]. Column E is calculated from Equation 7-18. The first input value of pM (15.08) was selected by trial and error to produce a small V You can start wherever you like. If your initial value of pM is before the true starting point, then VM in column E will be negative. In practice, you will want more points than we have shown so that you can plot an accurate titration curve. 

If phase-sensitive detection is used then x Ab can be found from the part of applied field (dispersion) and x" a b can be found from the part of that oscillates 90° out of phase with the applied field (absorption). In practice it is unnecessary to measure both X and x because there exists a theoretical relationship between x and x" so that a determination of one member of the pair uniquely determines the other member of the pair. It follows from this fact that the complex function, %(k, oo) — x"(k, oo), is analytic and vanishes on an infinite semicircle in the lower half of the complex frequency plane (z-plane) and that x a b and X" ab are related through the Kramer s Kronig relations... 

Figure 5. Rate of formation of crosslinks dX/da in relation to fractional conversion a according to equation (6) no modifier, O modifier at initial concentration 0.1 phm, X, K = 0.5 A, K=1.0 Q K=2.0...

A back Fourier transform on individual peak can be then applied to isolate individual contributions. The so-obtained filtered EXAFS signal is analyzed in order to determine the related structural parameters N, R and a. A common method consists to build a theoretical model from the relation (2) using known electronic parameters for X(k), So2, Aj(ir,k) and Oy (k), and, by treating N, R, and a as free parameters, to minimize the difference between the theoretical and experimental curves. The electronic parameters can be evaluated from EXAFS signals recorded for references of known structures or from theoretical calculations using the efficient ab initio FeFF codes [7, 8],... 

Spatial dispersion effects are usually considered separately from time dependences and correspond to static limit to = 0. Consequently s(k, 0) = s(k) and x(k, 0) = x(k) are basic susceptibility functions. Within the LRA the relation similar to Equation (1.131) is valid. It formally represents a solution to the nonlocal Poisson equation with a -dependent susceptibility. 

The transient effect is accomplished within this interval when k t) reaches its stationary value k = I i m, x k(t), which is approximately /tD. Relating the fluorescence intensity to its asymptotic value, Nickel et al. [262] introduced the normalized value which serves as a characteristic of the transient effect ... 

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