Exponential function transform

The Fourier transform of a pure Lorentzian line shape, such as the function equation (4-60b), is a simple exponential function of time, the rate constant being l/Tj. This is the basis of relaxation time measurements by pulse NMR. There is one more critical piece of information, which is that in the NMR spectrometer only magnetization in the xy plane is detected. Experimental design for both Ti and T2 measurements must accommodate to this requirement. 

From Eq. (2-18) on page 2-7, the Laplace transform of a time delay is an exponential function. For example, first and second order models with dead time will appear as... 

Sometimes the FID doesn t behave as we would like. If we have a truncated FID, Fourier transformation (see Section 4.4) will give rise to some artefacts in the spectrum. This is because the truncation will appear to have some square wave character to it and the Fourier transform of this gives rise to a Sine function (as described previously). This exhibits itself as nasty oscillations around the peaks. We can tweak the data to make these go away by multiplying the FID with an exponential function (Figure 4.1). 

When a reaction which transforms the reactants into nonchiral products is started, AH and AD, owing to the KIE, are consumed at different rates and an optical rotation is induced in the mixture. For a reaction that follows first-order or pseudo-first-order kinetics with the rate constants kH and kD, the time dependence of the optical rotation, a, is described by a two-exponential function (64). 

Platinum and palladium porphyrins in silicon rubber resins are typical oxygen sensors and carriers, respectively. An analysis of the characteristics of these types of polymer films to sense oxygen is given in Ref. 34. For the sake of simplicity the luminescence decay of most phosphorescence sensors may be fitted to a double exponential function. The first component gives the excited state lifetime of the sensor phosphorescence while the second component, with a zero lifetime, yields the excitation backscatter seen by the detector. The excitation backscatter is usually about three orders of magnitude more intense in small optical fibers (100 than the sensor luminescence. The use of interference filters reduce the excitation substantially but does not eliminate it. The sine and cosine Fourier transforms of/(f) yield the following results ... 

Since we found in Chap. 6 that the responses of linear systems are a series of exponential terms, the Laplace transformation of the exponential function is the most important of any of the functions. 

Remember that the Laplace transformation of the exponential was K/(s + a). So the (s -I- a) term in the denominator of a Laplace transformation is similar to the (z — term in a z transformation. Both indicate an exponential function. In the s plane, we have a pole at s = —a. In the z plane we will find later in this chapter that we have a pole at z = So we can imme-... 

Fick s second law (Eq. 18-14) is a second-order linear partial differential equation. Generally, its solutions are exponential functions or integrals of exponential functions such as the error function. They depend on the boundary conditions and on the initial conditions, that is, the concentration at a given time which is conveniently chosen as t = 0. The boundary conditions come in different forms. For instance, the concentration may be kept fixed at a wall located atx0. Alternatively, the wall may be impermeable for the substance, thus the flux at x0 is zero. According to Eq. 18-6, this is equivalent to keeping dC/dx = 0 at x0. Often it is assumed that the system is unbounded (i.e., that it extends from x = - °o to + °°). For this case we have to make sure that the solution C(x,t) remains finite when x -a °°. In many cases, solutions are found only by numerical approximations. For simple boundary conditions, the mathematical techniques for the solution of the diffusion equation (such as the Laplace transformation) are extensively discussed in Crank (1975) and Carslaw and Jaeger (1959). 

This expansion may be compared with an expansion of the exponential function, exp icot, in the inverse Fourier transform of the spectral density,... 

Both forms have a distribution of relaxation times about t0 which contribute to s(Laplace transform can be obtained numerically, and in this case C(t) can be well fit to a stretched exponential function [46] ... 

This expansion is valid to second order with respect to St. This is a convenient and practical method for computing the propagation of a wave packet. The computation consists of multiplying X t)) by three exponential operators. In the first step, the wave packet at time t in the coordinate representation is simply multiplied by the first exponential operator, because this operator is also expressed in coordinate space. In the second step, the wave packet is transformed into momentum space by a fast Fourier transform. The result is then multiplied by the middle exponential function containing the kinetic energy operator. In the third step, the wave packet is transformed back into coordinate space and multiplied by the remaining exponential operator, which again contains the potential. 

Most substances intermix rapidly within their distribution spaces, and the rate-limiting step in their removal from the system is biochemical transformation or renal excretion. Substances of this nature are best described by compartmental models and exponential functions. 

The signal-to-noise ratio (S/N) can be improved by multiplying the FID by a decaying exponential function, exp(—nt LB), which attenuates the noise at the end of the FID but broadens peaks in the transformed spectrum by the amount LB (Hz). 

Symmetiy Relations. Each normal coordinate and every wavefunction involving products of the normal coordinates, must transform under the symmetry operations of the molecule as one of the symmetry species of the molecular point group. The ground-state function in Eq. (3 a) is a Gaussian exponential function that is quadratic in Q, and examination shows that this is of Xg symmetry for each normal coordinate, since it is unchanged by any of the symmetry operations. From group theory the symmetry of a product of two functions is deduced from the symmetry species for each function by a systematic procedure discussed in detail in Refs. 4, 5,7, and 9. The results for the D i, point group apphcable to acetylene can be summarized as follows ... 

The same result follows when the average of the exponential function given in Eq. (51) is transformed using the cumulant expansion theorem and, assuming a Gaussian process, all correlations higher than second order (Stepisnik, 1981, 1985) are neglected. The particle velocity autocorrelations form a tensor... 

Exchange frequency, 167 Exchange integral, 194 Exergonic reaction, 223 Exponential function, Laplace transform of, 83... 

Note that expression (15.90) has an appearance similar to a Fourier transform from the spatial z domain to the frequency w domain. Of course, this is only an apparent similarity, because we have a travel time t in the argument of the exponential function, instead of the depth, z. However, we can exploit this apparent similarity and construct an expression similar to the inverse Fourier transform (15.62) ... 

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