Bloch longitudinal relaxation time

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 ... 

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... 

A real two-level particle is inevitably subject to the decay of its excited state, owing to at least spontaneous emission (Fig. 2.4(b)). Giving consideration to the decay of the excited state immediately makes the absorption spectral line of the transition 1 2 have a finite width even in the absence of an external field. In actual fact, the interaction of the particle with its surroundings, which can be considered as a thermal bath, can additionally shorten the lifetime of the excited state. In the general case, the relaxation of the population of the excited level to its equilibrium state is called longitudinal relaxation and is characterized by a longitudinal relaxation time Ti (Bloch 1946). 

Bloch defined T, the longitudinal relaxation time, by the equation (in the laboratory frame ) ... 

Longitudinal and transverse relaxations have been assumed by Bloch et al. [6] to be first-order rate processes. Following this assumption, the increase of Mz to M0 and the decay of Mx and My to zero may be expressed in terms of spin-lattice and spin-spin relaxation times, T, and T2 ... 

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. 

The signal intensities A correspond to the transverse magnetization after the 180°, t, 90° sequence. The transverse magnetization, in turn, arises from the partially relaxed longitudinal magnetization, given by integration of the Bloch equation (1.17 a) between AT, = — Mq at time t = 0 and Mz = Mz at t = v. 

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)),... 

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