Paramagnetic Relaxation and Electron Spin Echo

Let us begin by outlining the theory on paramagnetic relaxation. Although it has repeatedly been a subject of previous excellent treatises [9, 12], we dare to repeat it again here because it is an essential basis for using the ESE method to study the nature and behavior of free radicals. 

In a static magnetic field, each unpaired electron-spin of a free radical precesses about the axis parallel or antiparallel to the magnetic field (z-direction). The quantum state of the electron spin is expressed by a or p corresponding to these precessions. The population of the P spin state is larger than that of the a state by a Boltzman factor under the thermal equilibrium condition. so that the spin system has a total magnetization along the z-axis. The x-and y-components of the magnetic moments of electron spins are cancelled out under the thermal equilibrium condition because of the incoherence of the precession motion. 

The restoration process of the spin magnetization toward the thermal equilibrium is called paramagnetic relaxation. It can be divided into two categories longitudinal relaxation and transverse relaxation. The z-component of the total magnetization is restored by the former relaxation, while the coherence in precessing on the xy-plane is destroyed by the latter relaxation. 

Application of a second microwave pulse with a proper duration (180° pulse) along the x-direction at time t after the 90° pulse rotates each magnetization about the x-axis by 180°. Suppose the precession frequency of an A spin is to, the phase angle of the A spin with respect to the y-axis is (to — co0)t and 

The ESE intensity does not depend on t if the precession frequency of each electron spin does not fluctuate during the time interval 2t. However it does more or less fluctuate under the influence of surrounding spins etc., the precession of the spins looses coherence. The refocussing becomes more and more imperfect and the ESE intensity becomes weaker and weaker as the time elapses after the 90° pulse. This behavior of the spin magnetization is called phase relaxation. Both the free induction decay and the phase relaxation are called transverse relaxation. 

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