Bloch equation derivation

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. 

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 ... 

Expressions for determining rate constants from exchange contributions to observed linewidth for unequally populated systems in the fast exchange limit have been derived from the formal solutions to the Bloch equations modified for chemical exchange [3, 127-129]. These equations relate each rate constant to the site populations, chemical shift difference between sites, and spin relaxation times T and T2. For example, the forward rate A i 2 is given by [3, 127] ... 

The Lorentzian lineshape is obtained for liquid samples under ideal high-resolution NMR conditions10 and is readily derived from the Bloch equations.31 The normalized Lorentzian lineshape function in the frequency domain is given by... 

The Bloch equation gives the time derivative of the density matrix p in terms of its commutator with the Hamiltonian for the system, and the decay rate matrix T. Each of the matrices, p, H, and T are n x n matrices if we consider a molecule with n vibration-rotation states. We so ve this equation by rewriting the n x n square matrix p as an n -element column vector. Rgwrit ng p in this way transforms the H and V matrices into an n x n complex general matrix R. We obtain... 

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. 

For a given sequence, Bloch equations give the relationship between the explanatory variables, x, and the true response, i]. The / -dimensional vector, 0, corresponds to the unknown parameters that have to be estimated x stands for the m-dimensional vector of experimental factors, i.e., the sequence parameters, that have an effect on the response. These factors may be scalar (m — 1), as previously described in the TVmapping protocol, or vector (m > 1) e.g., the direction of diffusion gradients in a diffusion tensor experiment.2 The model >](x 0) is generally non-linear and depends on the considered sequence. Non-linearity is due to the dependence of at least one first derivative 5 (x 0)/50, on the value of at least one parameter, 6t. The model integrates intrinsic parameters of the tissue (e.g., relaxation times, apparent diffusion coefficient), and also experimental nuclear magnetic resonance (NMR) factors which are not sufficiently controlled and so are unknown. 

To gain some insight into the shape of the solution for the canonical density matrix, let us consider first a one-dimensional problem of electrons moving in a potential V(x). Writing the Bloch equation (Eq. (6)) for the Hamiltonians and and subtracting them to remove the derivative, one obtains the so-called equation of motion of the density matrix as... 

The Bloch equations and related approximate models derived using similar principles are very useful as simple frameworks for analyzing optical response of material systems. Some examples for their use are provided in Chapter 18. 

Fourier transformation of a time-domain signal f(jt) decaying with time constant T-i in an exponential fashion Such a signal is the NMR impulse-response function which can be derived from the Bloch equations (Fig. 4.1.1(b), cf. eqn. 2.2.19),... 

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). 

These fundamental equations which govern all of the transient phenomena that we will discuss in this chapter are the electric dipole analogs of the Bloch equations in NMR.IO The derivation of... 

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] ... 

The spin dynamics of the deuteron (spin /= 1) are more complex than those of the spin 1/2 nuclei, and the simple vector model used in other chapters, derived from the Bloch equations, provides no particular insight into deuteron spin dynamics. However, some of the geometric simplicity of the Bloch equations is present in a product-operator formalism, used to describe spin 1 NMR [117]. This formalism can provide a visual understanding of the deuteron pulse sequences in terms of simple precession and pulse rotations, albeit among a greater number of coordinate axes. The formalism can be used to understand the production of quadrupole order and the T q relaxation time (Figure 8.2(b)) and the two-dimensional deuteron exchange experiment (section 8.5). 

---