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Electron Spin Resonance 341

other interactions, the populations in the magnetic energy spin states are equalized, M z = 0, and the resonance ab-
levels would soon become equal; there would then be no sorption disappears. After the pulse, the recovery of M z
net absorption of microwave energy and no ESR signal. toward M 0 with a time constant T 1 can be observed by the
However, the spin system is subject to other interac- growth of the resonance line. The term T 1 is also called the
tions, the very interactions that bring about thermal equi- longitudinal relaxation time, because it refers to relaxation
librium.Theseinteractionscanbecollectivelycalledspin– along the magnetic-field axis.
lattice interactions. They comprise radiationless interac- The M x and M y components of M are not changed by
tions between the spin system and the thermal motion of a spin flip. The m x and m y components of each individual
the “lattice” or surroundings. The inverse of the rate of spin are randomly oriented before and after the magnetic
spin–lattice induced transitions is described by a charac- field H z is applied. However, application of H 1 in the x–y
teristic time called the spin–lattice relaxation time and is plane can produce a net phase alignment of the m x and m y
denoted by the symbol T 1 . components to give M x and M y . When H 1 is removed, the
At sufficiently low microwave powers, the spin–lattice phase coherence of the spins decays by 63% in time T 2 .
relaxation processes are fast enough to maintain a thermal The term T 2 is also called the transverse relaxation time
equilibrium population between magnetic energy levels. because it refers to relaxation of magnetization compo-
As the microwave power is increased the net upward rate nents transverse to the external magnetic field.
of microwave-induced spin transitions from the lower to An ESR line is not infinitely sharp; it has a shape and
upper states is increased and eventually competes with width due to spin relaxation. The equations of motion
the spin–lattice induced net downward rate. The spin pop- for M x , M y , and M z in the presence of an applied field
ulations in the two magnetic states become more equal H 0 and including the spin relaxation processes discussed
and the ESR signal intensity decreases; this is known as above are called the Bloch equations. The solution to these
power saturation. Normally, one wants to use low enough equations predicts a Lorentzian line with a halfwidth at
microwave power to avoid power saturation. halfheight of T −1 . Lorentzian lineshapes are indeed of-
2
In addition to spin–lattice relaxation, in which energy ten found for free radicals in liquids. In this case T 2 can
is transferred from the spin system to the lattice, there be determined from the linewidth. The Bloch equations
exist spin–spin relaxation mechanisms, in which energy is also predict how the ESR signal intensity will vary with
redistributed within the spin system. One may think of this increasing microwave power. The ESR signal increases,
redistribution as a modulation of the spin energy levels. In reaches a maximum, and then decreases with increasing
bothfluidandsolidphases,thenetlocalmagneticfieldsare microwave power; this behavior is called power satura-
rapidly varying due to different types of molecular motion, tion. From an analysis of the power saturation curve of
and a given spin level at m S gβH is therefore modulated. ESR intensity versus microwave power, it is possible to
At high spin concentrations, direct spin–spin exchange determine T 1 .
and dipolar interaction can also occur. The characteristic In solids, typical ESR lineshapes are Gaussian in-
time for spin–spin relaxation within a single spin system stead of Lorentzian. One common interpretation of the
is symbolized by T 2 . Gaussian lineshape is that it is composed of a distribution
In a single spin system the spin–lattice (T 1 ) and spin– of Lorentzian lineshapes, each of which corresponds to a
spin (T 2 ) relaxation times can be given a precise classi- group of spins forming a “spin packet” which “see” the
cal and quantum-mechanical description. A collection of same local magnetic environment. If these spin packets are
spins has a magnetic moment vector M, which can be randomly distributed in intensity they will superimpose to
resolved into three components, M x , M y , and M z . Be- give a Gaussian lineshape. Note that for Gaussian lines T 2
fore a magnetic field is applied, the number of spins in cannot be determined from the linewidth. Gaussian lines
the two magnetic energy states is equal; after the field is still undergo microwave power saturation, but very care-
applied, some of the spins begin flipping to achieve a ther- ful and sometimes complex analysis is required to extract
mal equilibrium distribution between the two states. For values of T 1 and T 2 .
an applied magnetic field in the z-direction the spin flips Amoredirectmethodtoobtainvaluesofthespin–lattice
cause M z to change toward a steady value M 0 , which is and spin–spin relaxation times is to use time-domain ESR
proportional to the measured static magnetic susceptibil- methods, which are briefly described next.
ity. M z approaches M 0 with a time constant T 1 such that
M z = e −1 M 0 = 63%M 0 in time T 1 . So that resonance can IX. DOUBLE-RESONANCE
be observed, the microwave magnetic field H 1 is applied AND TIME-DOMAIN ESR
perpendicular to H z . If the intensity of H 1 is increased
greatly with a pulse of microwaves, the spin system satu- Double-resonance experiments are usually carried out in
rates. This means the populations in the upper and lower spectroscopy to increase spectral resolution. There are