Line Bloch equations

The spin-spin relaxation time, T, defined in the Bloch equations, is simply related to the width Av 2 Lorentzian line at the half-height T. Thus, it is in principle possible to detennine by measuring... 

The relationship between the line shape of an NMR spectrum and the lifetime of chemical processes is provided by the Bloch equations. Let us imagine that there is a chemical equilibrium... 

The phenomenological equations proposed by Felix Bloch in 19462 have had a profound effect on the development of magnetic resonance, both ESR and NMR, on the ways in which the experiments are described (particularly in NMR), and on the analysis of line widths and saturation behavior. Here we will describe the phenomenological model, derive the Bloch equations and solve them for steady-state conditions. We will also show how the Bloch equations can be extended to treat inter- and intramolecular exchange phenomena and give examples of applications. 

A very similar application of the modified Bloch equations was based in the work of Adams and Connelly.4 ESR spectra (Figure 5.8) of [Mo P(0 Me)3 2(MeC = CMc)Cp] show the expected triplet (two equivalent 31P nuclei) at 280 K, but only a doublet at 160 K. At intermediate temperatures, the lines broaden. The interpretation is that the alkyne undergoes a pendulum oscillation, which in the extrema diverts spin density from one or the other phosphite. Interestingly, the diamagnetic cation undergoes a similar motion on the NMR time scale, but then the alkyne undergoes a complete rotation. Thus, analysis of the effect leads to a measure of the rate of the oscillation. The... 

For biased equilibria, Anet10 5 developed a method of estimating k and the population of the least stable conformer from line-broadening measurements. This method uses the expression developed by Gutowsky and Holm106 from the Bloch equations... 

By numerical integration of the Bloch equations describing the classical trajectory of an atom from the nozzle through the standing wave field at 243 nm to the detector and integration over all possible trajectories and over the velocity distribution of the atoms, a theoretical line shape is deduced which is then fitted to the experimental data. The solid lines in Fig. 2 are obtained from this fitting procedure. 

The Bloch equations can be solved analytically under the condition of slow passage, for which the time derivatives of Eq. 2.48 are assumed to be zero to create a steady state. The nuclear induction can be shown to consist of two components, absorption, which is 90° out of phase with B, and has a Lorentzian line shape, and dispersion, which is in phase with B,. The shapes of these signals are shown in Fig. 2.10. By appropriate electronic means (see Section 3.3), we can select either of these two signals, usually the absorption mode. 

The simplest theoretical approach to exchange is via the Bloch equations, to which terms are added to reflect the rate phenomena. The spectra shown in Fig. 2.14 are obtained from such a treatment. It is apparent that the line shapes depend on the ratio R/ vA — vB), where the exchange rate R = 1/t. Thus fast and slow are measured with respect to differences in the nuclear precession frequencies in the two sites. Exchange rates can be measured by analysis of line shapes and by certain pulse experiments, as described in later chapters. 

The third and most recent theory comes from Matsumoto and Giese, who suggested that the observed steady-state (SSEPR) spectrum of radical c is due to a superposition of two conformations of the same radical, and that one of these structures has a pyramidalized center. Iwasaki et al. invoked a fourth model, hyperfine modulation, to explain the spectra of the propagating radical. They were able to simulate the observed 9- and 13-line SSEPR spectra using a set of modified Bloch equations for a two-site exchange model between two conformations. 

Basic quantum mechanical calculations (i.e., the Bloch equations) show that the general EPR resonance absorption (T) will have a Eorentzian line shape... 

Fig. 5.3.4 [Houl] Linear (broken lines) and nonlinear (continuous lines) response spectra calculated from the Bloch equations for a 30° pulse (a) and a 90° pulse (b) by the siun of first- and third-order responses. The ticks mark the values obtained for the full, untruncated response.

The AlP nuclear magnetic resonance spectrum of a corundum or a-AlgOg single crystal is an orientation-dependent quintet arising from the quadrupole moment = - - 0.149 (136-139). O Reilly (140) obtained the Al dispersion mode envelope (141) powder pattern spectrum which results when y-alumina is heated to 1400°, and thereby eonverted to a-AlaOs. He interpreted the line shape in terms of the Redfield modification (142,143) of the Bloch equations. 

We will analyze the SM spectra and their fluctuations semiclassically using the stochastic Bloch equation in the limit of a weak laser field. The Kubo-Anderson sudden jump approach [58-61] is used to describe the spectral diffusion process. For several decades, this model has been a useful tool for understanding line shape phenomena, namely, of the average number of counts < > per measurement time T, and has found many applications mostly in ensemble measurements, for example, NMR [60], and nonlinear spectroscopy [62]. More recently, it was applied to model SMS in low-temperature glass systems in order to describe the static properties of line shapes [14-16, 63] and also to model the time-dependent fluctuations of SMS [64-66]. 

Now, we consider the important limit of weak laser intensity. In this limit, the Wiener-Khintchine theorem relating the line shape to the one-time correlation function holds. As we shall show now, a three-time correlation function is the central ingredient of the theory of fluctuations of SMS in this limit. In Appendix B, we perform a straightforward perturbation expansion with respect to the Rabi frequency Q in the Bloch equation, Eq. (4.6), to find... 

Electron exchange reactions may also be followed using line-width measurements. The relaxation of a diamagnetic nucleus will be modified if the species picks up or loses an electron, the magnitude of the effect depending upon ai and the time the unpaired electron spends in the ion or radical, Such exchange reactions have been analysed using both modified Bloch equations and a density matrix formalism. ... 

If the theoretical line shape is known for a peak, the function can be fitted to the data, and the intensity, width, and integral determined. In practice, however, experimental line shapes can be too complex to characterize conveniently. NMR lines, for example, have been shown by the Bloch equations to be Lorentzian in shape. A number of experimental factors, however, can contribute non-Lorent-zian components to the observed line shape. Furthermore, unresolved peaks will have to be fitted to a sum of individual Lorentzians rather than to a single curve. 

The Bloch equations (Eq. 5) can be solved under different conditions. The transient solution yields an expression for 0-22 (0> time-dependent population of the excited singlet state S. It will be discussed in detail in Section 1.2.4.3 in connection with the fluorescence intensity autocorrelation function. Here we are interested in the steady state solution (an = 0-22 = < 33 = di2 = 0) which allows to compute the line-shape and saturation effects. A detailed description of the steady state solution for a three level system can be found in [35]. From those the appropriate equations for the intensity dependence of the excitation linewidth Avfwhm (FWHM full width at half maximum) and the fluorescence emission rate R for a single absorber can be easily derived [10] ... 

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