Saturation Bloch equations

It is important to avoid saturation of the signal during pulse width calibration. The Bloch equations predict that a delay of 5 1] will be required for complete restoration to the equilibrium state. It is therefore advisable to determine the 1] values an approximate determination may be made quickly by using the inversion-recovery sequence (see next paragraph). The protons of the sample on which the pulse widths are being determined should have relaxation times of less than a second, to avoid unnecessary delays in pulse width calibration. If the sample has protons with longer relaxation times, then it may be advisable to add a small quantity of a relaxation reagent, such as Cr(acac) or Gkl(FOD)3, to induce the nuclei to relax more quickly. 

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 stacked set of saturation recovery spectra is given in Fig. 2.29, which shows that the signal intensity approaches an equilibrium value Arjo related to the fully relaxed magnetization M0. The intensity Az as a function of r is the integral of the Bloch equation (1.17 a) between M., = Mz and M0, or in terms of intensities, from Ar at time t to A f at time go ... 

In order to see if such a value of B, would cause spectral saturation, we may estimate the corotating component of the rf magnetization from the Bloch equations (Abragam and Bleaney, 1970, pp. 115-119), whence... 

Instead of tR for the recycle delay in partial saturation experiments, the generic symbol t is used in the following to denote an adjustable filter time. Within the validity of the Bloch equations, the longitudinal magnetization M tt,r) of a pixel corresponding to position r, which has been partially saturated, relaxes according to (cf. eqn (2.2.36)),... 

The simple T2 weight (7.2.4) introduced in Hahn-echo measurements is to be modified if partial saturation of the sample magnetization results from repetition times tR = tto that are short compared to 5 Ti [Pop3]. The associated reduction in thermodynamic equilibrium magnetization can be described within the validity of the Bloch equations hy replacing Mo(r) in (7.2.4) by the saturated magnetization (7.2.1) (cf. eqn (6.2.5)). For perfect pulses one obtains... 

The e.s.r. signal, proportional to S, is irradiated with a large oscillating magnetic field at 0) sufficient to cause partial saturation of the electron transitions. The dependence of on the strength Hij of an oscillating magnetic field applied exactly at resonance is found from the Bloch equations to be—... 

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] ... 

Lapert et al described the time-optimal control of a spin-1/2 particle whole dynamics is described by Bloch equations with both Ty and Tj terms. They demonstrated the use of the technique by solving the saturation problem as an example. 

The contributions to the fifth-order nonlinear optical susceptibility of dense medium have been theoretically estimated by using both the local-field-corrected Maxwell-Bloch equations and Bloembergen s approach. In addition to the obvious fifth-order hyperpolarizability contribution, the fifth-order NLO susceptibility contains an extra term, which is proportional to the square of the third-order hyperpolarizability and which originates purely from local-field effects, as a cascaded contribution. Using as model the sodium 3s 3p transition system, it has been shown that the relative contribution of the cascaded term to the fifth-order NLO susceptibility grows with the increase of the atomic density and then saturates. 

An ESR line is not infinitely sharp it has a shape and width due to spin relaxation. The equations of motion for Mx, My, and M in the presence of an applied field Ho and including the spin relaxation processes discussed above are called the Bloch equations. The solution to these equations predicts a Lorentzian line with a halfwidth at halfheight of Lorentzian lineshapes are indeed often found for free radicals in liquids. In this case T2 can be determined from the linewidth. The Bloch equations also predict how the ESR signal intensity will vary with increasing microwave power. The ESR signal increases, reaches a maximum, and then decreases with increasing microwave power this behavior is called power saturation. From an analysis of the power saturation curve of ESR intensity versus microwave power, it is possible to determine Ti. 

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