The Bloch Equations

The Bloch equations provide a mathematical expression for the magnetization based on classical vector mechanics. The overall magnetization M is the sum of three Cartesian components  

The magnetic field Bq that gives rise to the Zeeman splitting is in the z direction, and the applied magnetic field B operates in the xy plane. 

When perturbed from equilibrium in the z direction, the system returns to equilibrium in a first-order rate process with a time constant T the spin-lattice, or longitudinal, relaxation time. Consequently, the rate of return from a perturbed value to the equilibrium value Mq is given by 

Similarly, relaxation processes in the xy plane are controlled by the spin-spin, or transverse, relaxation time T2, so that the equations 

The various magnetization components are influenced by the magnetic fields Bq and B. The forces between the M and B vectors provide the mechanism to move the magnetization out of equilibrium. Eqs. A2-2, A2-3, and A2-4 describe only the return to equilibrium. To include the interaction of the magnetic fields on M, Bloch followed a purely phenomenological 

The NMR phenomenon can be quantitatively described in classical terms. This was first done by Bloch.The approach is helpful in developing an understanding of nuclear relaxation processes. 

Consider a nucleus with magnetic moment pi in a magnetic field Ho- According to classical mechanics the rate of change of the angular momentum G is the torque T. 

The torque is given by the vector cross product of the vectors pi and Hq. 

The angular momentum is related to the moment by the magnetogyric ratio. 

If the nuclei in unit volume are summed, the result is the magnetization M, so 

In Bloch s original treatment of NMR,23 he postulated a set of phenomenological equations that accounted successfully for the behavior of the macroscopic magnetization M in the presence of an rf field. These relations are based on Eq. 2.41, where M replaces X, and B is any magnetic field—static (B0) or rotating (B,). By expanding the vector cross product, we can write a separate equation for the time derivative of each component of AT  

By the usual convention, the component Bz = B0, the static field, while Bx and By represent the rotating rf field, as expressed in Eq. 2.41, when it is applied, and are normally zero otherwise. To account for relaxation, Bloch assumed that Mz would decay to its equilibrium value of M0 by a first-order process characterized by a time T, (called the longitudinal relaxation time, because it covers relaxation along the static field), while Mx and AT, would decay to their equilibrium value of zero with a first-order time constant T2 (the transverse relaxation time). Overall, then the Bloch equations become 

The Bloch equations can be solved analytically under certain limiting conditions, as described in Section 2.9. 

The longitudinal relaxation time T, measures the restoration of M, to its equilibrium value M0, which is the same process that we discussed from the perspective of populations in energy levels in Section 2.5, spin—lattice relaxation. The factors that are important in determining the rate at which energy from the spin system flows to its surroundings will be discussed in Chapter 8. 

In terms of the Bloch equations as given in the preceding text, both of these processes result in decrease in T2, but only the first is considered as contributing to a fundamental determination of T2 as a molecular parameter. The second process is an artifact of the experimental measurement (inhomogeneity in B0).The term T2 is used to denote the effective T2, that is, to encompass both processes  

As before we limit ourselves to near resonance processes involving weak radiation fields and model the molecule as a two-state system the ground state and an excited state selected by the frequency of the incident radiation. Hence, our starting point aretheBlochequations,Eqs.(10.181),forthereduceddensitymatrixaj, i,j = 1,2) of such systems 

E )/h = CO — co2 correspond to an implementation of the dressed-state picture given the molecular states 1) and 2) with energy spacing E2 = E2 — E, Eqs (18.43) are written for the density matrix elements in the representation of the dressed states l,co) (the molecular state 1 plus a photon of frequency co) and 2,0) (molecular state 2 with no photons) whose spacing is rj. 

It should be kept in mind that, as already noted, the 2 states system is greatly oversimplified as a molecular model. We will nevertheless see that it provides important insight that remains useful also for realistic molecular applications. In what follows we considerthe implications ofEqs (18.43) on the optical observables that were discussed in Sections 18.2 and 18.3. 

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Comparing this with the Bloch equation establishes a correspondence between t and /p/j. Putting t = ip/j, one finds... 

To evaluate the density matrix at high temperature, we return to the Bloch equation, which for a free particle (V(x) = 0) reads... 

The solution to this is a Gaussian function, which spreads out in time. Hence the solution to the Bloch equation for a free particle is also a Gaussian ... 

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 classical description of magnetic resonance suffices for understanding the most important concepts of magnetic resonance imaging. The description is based upon the Bloch equation, which, in the absence of relaxation, may be written as... 

The Bloch equation is simplified, and the experiment more readily understood, by transfonnation into a frame of reference rotating at the frequency ciDq=X Bq about die z-axis whereupon ... 

With this definition, the Bloch equations can be written as in equation (B2.4.4)). 

The Bloch equation approach (equation (B2.4.6)) calculates the spectrum directly, as the portion of the spectrum that is linear in a observing field. Binsch generalized this for a frilly coupled system, using an exact density-matrix approach in Liouville space. His expression for the spectrum is given by equation (B2.4.42). Note that this is fomially the Fourier transfomi of equation (B2.4.32). so the time domain and frequency domain are coimected as usual. 

Reeves L W and Shaw K N 1970 Nuclear magnetic resonance studies of multi-site chemical exchange. I. Matrix formulation of the Bloch equations Can. J. Chem. 48 3641-53... 

The quantitative formulation of chemical exchange involves modification of the Bloch equations making use of Eq. (4-67). We will merely develop a qualitative view of the result." We adopt a coordinate system that is rotating about the applied field Hq in the same direction as the precessing magnetization vector. Let and Vb be the Larmor precessional frequencies of the nucleus in sites A and B. Eor simplicity we set ta = tb- As the frequency Vq of the rotating frame of reference we choose the average of Va and Vb, thus. 

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... 

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 motion of the neutron polarization P(t) - the quantum mechanical expectancy value of neutron spin - is described by the Bloch equation... 

According to the solution of the Bloch equations (Chapter 5), the magnetic resonance absorption, sometimes called the v-mode signal , v, is given by eqn (1.10). 

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. 

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. 

Equations (5.12a-c) are the Bloch equations in the rotating coordinate frame. 

Suppose we have a system in which a spin can exist in either of two different sites, A or B, and that these are distinguished by different resonant frequencies, coa and coB, and/or by different relaxation times, T2a and T2b If there is no exchange between sites, site A spins and site B spins can be described separately and independently by sets of Bloch equations. When exchange takes place, however, additional rate terms - completely analogous to terms in chemical rate equations - must be added to the Bloch equations. 

These are Green s functions diffusing in what is interpreted as an imaginary time [3, according to the Bloch equation, dp/8(3 = JYp (a diffusion-type partial differential equation). These Green s functions satisfy the equation... 

Since Aa> is proportional to the magnetic field, higher fields allow for exploiting faster exchanges which consequently leads to a more important CEST effect. More detailed explanation of the CEST theory (52,159) and numerical solution of the Bloch equations describing the CEST effect (160) can be found in the literature. 

When an NMR experiment is performed, the application of a RFpulse orthogonal to the axis of the applied magnetic field perturbs the Boltzmann distribution, thereby producing an observable event that is governed by the Bloch equations [3]. Using a vector representation, the... 

Substituting in Eq. (38) and taking the trace over the variables of the dipole-dipole system, we get a set of equations analogous to the Bloch equations (18)... 

All the spin manipulations are done by a tailored magnetic field that the neutron spin feels during its passage through the instrument. The motion of the expectation value of the neutron spin (s) obeys the Bloch equation ... 

In the first approach, we shall limit ourselves to the hypothesis where nuclear magnetization conforms to the Bloch equations. Concerning the... 

We now intend to derive the Bloch equations in order to express Ti and T2 according to spectral densities at appropriate frequencies. The starting point is the evolution equation of an elementary magnetic moment p subjected to a random field b... 

For both processes approximate equations were derived from the exact solution of the Bloch equations for the longitudinal relaxation time of a system in which water protons undergo chemical exchange between two magnetically distinct environments A and B ... 

Combination of Equation (6) with (7) and (8) gives a set of differential equations which may be solved 4 ) to give the steady state behavior of the nuclear magnetization as the frequency of Hi or the field H is varied. These differential equations are known as the Bloch equations and are... 

If the exchange rate, k, is much smaller than the frequency difference between the signals for the exchanging sites, then the NMR spectrum will exhibit well separated peaks for these resonances. Based on the Bloch equation [146], it is possible to find relationships connecting the shape of the NMR signal, Avii2, the lifetime, T, for a nucleus in different positions of a molecule and the rate constant, k. The lifetime is related to the rate constant by Eq. (4). 

Various theoretical formalisms have been used to describe chemical exchange lineshapes. The earliest descriptions involved an extension of the Bloch equations to include the effects of exchange [1, 2, 12]. The Bloch equations formalism can be modified to include multi-site cases, and the effects of first-order scalar coupling [3, 13, 24]. As chemical exchange is merely a special case of general relaxation theories, it may be compre-... 

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