Relaxation Times via General Solution of Blochs Equations

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. 

This first-order differential model suffices to describe the dynamical behavior of M. From a different perspective the applied field torques the microscopic magnetic moments, causing them to process about the z-axis with an angular velocity defined as co= (g HQ)/h. The resultant equation of motion for the magnetization of a system of free spins in a static magnetic field can be expressed as 

The spectroscopic dynamics is induced by subjecting the sample to a second oscillatory field. Hi, with angular velocity ca This field affects the phase angle of 

Vector quantity M may now be resolved into its three components — M, My, and Mz— incorporating both time constants, 7) and Ti. When the amplitude of the radio-frequency field, H, is much smaller than ftiat of static field Hq, Bloch s differential equations may be written in tire laboratory frame of reference (x,y,z)  

Time constants T] and T2 appearing here in the relaxation terms are ealled the longitudinal and transverse relaxation times. 

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