Bloch equation

In 1946, Bloch [ 16] presented a theory for nuclear spin relaxation in which he derived a set of equations of motion to predict the behaviour of an ensemble of isolated spins interacting weakly with the lattice. In brief, the Bloch equations describe the evolution of the longitudinal (diagonal elements of the density matrix) and transverse (off-diagonal elements of the density matrix) spin magnetisation to their respective equilibrium values phenomenologically, with the first order rate of both processes 

The 1952 Nobel Prize in Physics was awarded to Felix Bloch for results that were published in 1946 [1]. The implementation of what are now the famous Bloch Equations led to a major advance in NMR spectroscopy and magnetic resonance imaging (MRI) which is now in common 

The first step, which is unfamiliar to chemists, is that — = T but we should recall Newton s 

Now we assume Bj,=By = Q (field in the z-direction) and B = Bq with My t = Q) = Q as a future condition. Those conditions knock out parts of the cross-product as we see in 

So simply using the conditions of the magnet geometry leads us to the key step of the derivation. Next, take the second derivative of the My component, this is the key stepl 

So My = Cl cos(yBoO + C2 sinCyBoO = 0atf = 0 4-Ci = 0 and so My = C2 sin(yBoO-Here we have used the sin and cos solution form with as-yet-unknown coefficients. 

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Equation (A1.6.64) describes the relaxation to equilibrium of a two-level system in tenns of a vector equation. It is the analogue of tire Bloch equation, originally developed for magnetic resonance, in the optical regime and hence is called the optical Bloch equation. 

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 ... 

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]. 

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

In chemical exchange, tire two exchanging sites, A and B, will have different Lannor frequencies, and cOg. Assuming equal populations in the two sites, and the rate of exchange to be k, the two coupled Bloch equations for the two sites are given by equation (B2.4.5)). 

This Liouville-space equation of motion is exactly the time-domain Bloch equations approach used in equation (B2.4.13). The magnetizations are arrayed in a vector, and anything that happens to them is represented by a matrix. In frequency units (1i/2ti = 1), the fomial solution to equation (B2.4.26) is given by equation (B2.4.27) (compare equation (B2.4.14H. 

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... 

Biexponential kinetics, 72-76 Biphasic kinetics, 72-76 Bloch equations, 261 Branching reactions, 189 Brpnsted-Bjerrum equation, 204... 

Theoretical level populations. Sinee there are population variations on time seale shorter than some level lifetimes, a complete description of the excitation has been modeled solving optical Bloch equations Beacon model, Bellenger, 2002) at CEA. The model has been compared with a laboratory experiment set up at CEA/Saclay (Eig. 21). The reasonable discrepancy when both beams at 589 and 569 nm are phase modulated is very likely to spectral jitter, which is not modeled velocity classes of Na atoms excited at the intermediate level cannot be excited to the uppermost level because the spectral profile of the 569 nm beam does not match the peaks of that of the 589 nm beam. 

In the DC-biased structures considered here, the dynamics are dominated by electronic states in the conduction band [1]. A simplified version of the theory assumes that the excitation occurs only at zone center. This reduces the problem to an n-level system (where n is approximately equal to the number of wells in the structure), which can be solved using conventional first-order perturbation theory and wave-packet methods. A more advanced version of the theory includes all of the hole states and electron states subsumed by the bandwidth of the excitation laser, as well as the perpendicular k states. In this case, a density-matrix picture must be used, which requires a solution of the time-dependent Liouville equation. Substituting the Hamiltonian into the Liouville equation leads to a modified version of the optical Bloch equations [13,15]. These equations can be solved readily, if the k states are not coupled (i.e., in the absence of Coulomb interactions). 

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. 

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... 

Here m(r, t) is the relative difference of the longitudinal magnetization M and its equilibrium value Mo m = (M — Mo)/Mo, D is the bulk diffusion coefficient and p is the bulk relaxation rate. The general solution to the Torrey-Bloch equation can be written as... 

In a one-dimensional pore, the Torrey-Bloch equation can be solved analytically and with fast diffusion and weak relaxation, it can be shown that the eigenvalues are [46]... 

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

As we shall see, all relaxation rates are expressed as linear combinations of spectral densities. We shall retain the two relaxation mechanisms which are involved in the present study the dipolar interaction and the so-called chemical shift anisotropy (csa) which can be important for carbon-13 relaxation. We shall disregard all other mechanisms because it is very likely that they will not affect carbon-13 relaxation. Let us denote by 1 the inverse of Tt. Rt governs the recovery of the longitudinal component of polarization, Iz, and, of course, the usual nuclear magnetization which is simply the nuclear polarization times the gyromagnetic constant A. The relevant evolution equation is one of the famous Bloch equations,1 valid, in principle, for a single spin but which, in many cases, can be used as a first approximation. 

Whenever the system is no longer constituted by single non-interacting spins, the simple Bloch Equation (2) must be completed by additional coupling terms. Let us consider the dipolar interaction between two spins... 

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. 

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