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The energy between the magnetic energy levels at 3000 G, gjH, is only 10-3 of kT at 300 K. At thermal equilibrium the Boltzmann factor, exp(-gjH/kt), gives the population ratio of the two levels, so the levels are almost equally populated. The application of microwave energy causes transitions between the magnetic levels. The microwave field stimulates transitions in both directions with a probability that depends on the microwave power and on the number of spins in each level. Transitions from the lower to upper levels absorb energy, while upper- to lower-level transitions emit energy. Since the population is slightly greater in the lower level there will be a net absorption of microwave energy; this provides the observed ESR signal. Under steady application of the microwave field with no other interactions, the populations in the magnetic energy levels would soon become equal; there would then be no net absorption of microwave energy and no ESR signal.

However, the spin system is subject to other interactions, the very interactions that bring about thermal equilibrium. These interactions can be collectively called spinlattice interactions. They comprise radiationless interactions between the spin system and the thermal motion of the "lattice" or surroundings. The inverse of the rate of spin-lattice induced transitions is described by a characteristic time called the spin-lattice relaxation time and is denoted by the symbol T1.

At sufficiently low microwave powers, the spin-lattice relaxation processes are fast enough to maintain a thermal equilibrium population between magnetic energy levels. As the microwave power is increased the net upward rate of microwave-induced spin transitions from the lower to upper states is increased and eventually competes with the spin-lattice induced net downward rate. The spin populations in the two magnetic states become more equal and the ESR signal intensity decreases; this is known as power saturation. Normally, one wants to use low enough microwave power to avoid power saturation.

In addition to spin-lattice relaxation, in which energy is transferred from the spin system to the lattice, there exist spin-spin relaxation mechanisms, in which energy is redistributed within the spin system. One may think of this redistribution as a modulation of the spin energy levels. In both fluid and solid phases, the net local magnetic fields are rapidly varying due to different types of molecular motion, and a given spin level at mSgPH is therefore modulated. At high spin concentrations, direct spin-spin exchange and dipolar interaction can also occur. The characteristic time for spin-spin relaxation within a single spin system is symbolized by T2.

In a single spin system the spin-lattice (Ti) and spinspin (T2) relaxation times can be given a precise classical and quantum-mechanical description. A collection of spins has a magnetic moment vector M, which can be resolved into three components, Mx, My, and Mz. Before a magnetic field is applied, the number of spins in the two magnetic energy states is equal; after the field is applied, some of the spins begin flipping to achieve a thermal equilibrium distribution between the two states. For an applied magnetic field in the z-direction the spin flips cause Mz to change toward a steady value M0, which is proportional to the measured static magnetic susceptibility. Mz approaches M0 with a time constant T1 such that Mz = e-1 M0 = 63%M0 in time T1. So that resonance can be observed, the microwave magnetic field H1 is applied perpendicular to Hz. If the intensity of H1 is increased greatly with a pulse of microwaves, the spin system saturates. This means the populations in the upper and lower spin states are equalized, Mz = 0, and the resonance absorption disappears. After the pulse, the recovery of Mz toward M0 with a time constant T1 can be observed by the growth of the resonance line. The term T1 is also called the longitudinal relaxation time, because it refers to relaxation along the magnetic-field axis.

The Mx and My components of M are not changed by a spin flip. The mx and my components of each individual spin are randomly oriented before and after the magnetic field Hz is applied. However, application of H1 in the x -y plane can produce a net phase alignment of the mx and my components to give Mx and My. When H1 is removed, the phase coherence of the spins decays by 63% in time T2. The term T2 is also called the transverse relaxation time because it refers to relaxation of magnetization components transverse to the external magnetic field.

An ESR line is not infinitely sharp; it has a shape and width due to spin relaxation. The equations of motion for Mx, My, and Mz in the presence of an applied field H0 and including the spin relaxation processes discussed above are called the Bloch equations. The solution to these equations predicts a Lorentzian line with a halfwidth at halfheight of T2-1. Lorentzian lineshapes are indeed often found for free radicals in liquids. In this case T2 can be determined from the linewidth. The Bloch equations also predict how the ESR signal intensity will vary with increasing microwave power. The ESR signal increases, reaches a maximum, and then decreases with increasing microwave power; this behavior is called power saturation. From an analysis of the power saturation curve of ESR intensity versus microwave power, it is possible to determine T1.

In solids, typical ESR lineshapes are Gaussian instead of Lorentzian. One common interpretation of the Gaussian lineshape is that it is composed of a distribution of Lorentzian lineshapes, each of which corresponds to a group of spins forming a "spin packet" which "see" the same local magnetic environment. If these spin packets are randomly distributed in intensity they will superimpose to give a Gaussian lineshape. Note that for Gaussian lines T2 cannot be determined from the linewidth. Gaussian lines still undergo microwave power saturation, but very careful and sometimes complex analysis is required to extract values of T1 and T2.

A more direct method to obtain values of the spin-lattice and spin-spin relaxation times is to use time-domain ESR methods, which are briefly described next.

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