Bloch equations spin 1 dynamics

Remarkably, when our general ME is applied to either AN or PN in Section 4.4, the resulting dynamically controlled relaxation or decoherence rates obey analogous formulae provided the corresponding density matrix (generalized Bloch) equations are written in the appropriate basis. This underscores the universality of our treatment. It allows us to present a PN treatment that does not describe noise phenomenologically, but rather dynamically, starting from the ubiquitous spin-boson Hamiltonian. 

A complex dynamical behavior was experimentally and numerically found in a system of spin- atoms in an optical resonator with near-resonant cw laser light and external static magnetic field [69]. Three-dimensional Bloch equations were solved, and a chaotic motions was found and compared with experiment. 

Their theory, based on the classical Bloch equations, (31) describes the exchange of non-coupled spin systems in terms of their magnetizations. An equivalent description of the phenomena of dynamic NMR has been given by Anderson and by Kubo in terms of a stochastic model of exchange. (32, 33) In the latter approach, the spectrum of a spin system is identified with the Fourier transform of the so-called relaxation function. 

The population difference results in macroscopic magnetization that is measurable. It is not possible to measure the magnetization of an individual nuclear spin in the experiment. Hence, we must treat the dynamics either of the macroscopic magnetization, which is described by Bloch equations, or by an ensemble of nuclear spins, which require a Master equation [5]. 

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). 

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 spectroscopic dynamics problem was examined mathematically for the case of the (two-level) magnetic resonance transition by Bloch, who described the temporal evolution of the magnetization in terms of a first-order differential equation analogous to dnidt = -k n—n, where n represents a time-dependent function that, in this case, represents a spin-state population difference. (In a two-level system and in the form written, n would represent the population difference between the ground and excited states and the solution of the differential equation would correspond to tire time course of the decay to the ground state.) The solution to this first-order differential equation is an exponential function in which a time constant is introduced and attributed to a characteristic relaxation time that is denoted by T]. In other words, k is proportional toTi. This time constant T is called the spin-lattice relaxation time, and is defined as the rate at which the electrons return to thermal equihbrium due to coupling with the lattice. 

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