Bloch equation, modified

For example, if the molecular structure of one or both members of the RP is unknown, the hyperfine coupling constants and -factors can be measured from the spectrum and used to characterize them, in a fashion similar to steady-state EPR. Sometimes there is a marked difference in spin relaxation times between two radicals, and this can be measured by collecting the time dependence of the CIDEP signal and fitting it to a kinetic model using modified Bloch equations [64]. 

Two different approaches have been followed to calculate the lineshapes within a relaxation model. According to a phenomenological approach based on the modified Bloch equations [154, 155], the intensity distribution of the theoretical Mossbauer spectrum may be written as [156] ... 

Fig. 25. Fe-M6ssbauer spectra of [Fe(acpa)2]BPli4 HjO between 189 and 286 K. Full curves have been calculated on the basis of the modified Bloch equations using the parameter values of Table 9. According to Ref. [156]...

The Mossbauer spectra of the complex [Fe(acpa)2]PF6 shown in Fig. 26 have also been interpreted on the basis of a relaxation mechanism [168]. For the calculations, the formalism using the modified Bloch equations again was employed. The resulting correlation times x = XlXh/(tl + Xh) are temperature dependent and span the range between 1.9 x 10 s at 110 K and 0.34 x 10 s at 285 K. Again the correlation times are reasonable only at low temperatures, whereas around 200 K increase of the population of the state contributes to... 

This approach works well for electron transfer reactions where the rate is simply related to the broadening, but to proceed further in kinetic applications of ESR spectroscopy we must deal with the Bloch equations and modified Bloch equations. 

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... 

The transition region between slow and fast exchange has been described (111, 112,162) for the above case by solution of modified Bloch equations that take into account the possibility of exchange between the various environments. It can be shown that as the lifetime increases, the original... 

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. 

Restricted rotation about the C—N bond causes exchange-modified proton spectra to occur in the methyl group at readily accessible temperatures. In this case the exact solution of the modified Bloch equations is further simplified because we have the restriction that the A and B sites are equally populated. In cases where the population of A and B sites is lopsided the exact solutions are complicated. In the case of symmetrical population one has also the subtle difference that the exchange A —>B is accompanied by a simultaneous mirror image... 

In the previous section we have been confronted with the results of coherent transient experiments which showed that the low-temperature homogeneous lineshape of electronic transitions of impurity molecules is exponentially activated. Next to the optical domain such an exponential activation was observed in the microwave domain by van t Hof and Schmidt and explained on the basis of the modified Bloch equations. 

Returning now to a discussion of the Redfield equations (40) we first note that, with our choice in (39), the only nondiagonal elements that are coupled to each other are p,2 and P34. Except for the driving-field terms these equations therefore have a form very similar to the modified Bloch equations postulated by McConnell. The quantum-mechanical derivation of (40), however, leads to new insight and restrictions in the use of these equations in the optical domain (Section IV). 

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. ... 

Conformations and Dynamic Situations.— The degenerate Cope rearrangement of bullvalene (35) is the subject of three papers which report values for the thermodynamic parameters (Table 1) obtained from lineshape analysis of pulsed Fourier Transform C spectra at different temperatures. The communication by Nakanishi and Yamamoto analyses data at 35 temperatures using modified Bloch equations. The margin of error stated is fairly high. A very thorough and well explained analysis... 

A chemical system related to that just discussed is the aqueous Cl + CI2 system. Also this system has been the subject of two independent, and partly contradictory, NMR studies. The earliest is that by Hall, Kydon, Richards and Sharp [135] who used the system as a model for multi-site exchange processes. The authors present a comprehensive discussion of the analytical and numerical features of the modified Bloch equations as applied to chemical exchange in a general n-site system. Hall et al, then apply their treatment to the exchange equilibrium... 

Analytical expressions based upon the modified Bloch equation... 

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