Longitudinal and transverse relaxation
المؤلف:
Peter Atkins، Julio de Paula
المصدر:
ATKINS PHYSICAL CHEMISTRY
الجزء والصفحة:
ص536-538
2025-12-11
63
Longitudinal and transverse relaxation
At thermal equilibrium the spins have a Boltzmann distribution, with more α spins than βspins; however, a magnetization vector in the xy-plane immediately after a 90° pulse has equal numbers of α and β spins. Now consider the effect of a 180° pulse, which may be visualized in the rotating frame as a flip of the net magnetization vector from one direction along the z-axis to the opposite direction. That is, the 180° pulse leads to population inversion of the spin system, which now has more β spins than α spins. After the pulse, the populations re vert to their thermal equilibrium values exponentially. As they do so, the z-component of magnetization reverts to its equilibrium value M0 with a time constant called the longitudinal relaxation time, T1 (Fig. 15.34):
Mz(t) − M0∝e−t/T1
Because this relaxation process involves giving up energy to the surroundings (the ‘lattice’) as β spins revert to α spins, the time constant T1 is also called the spin–lattice relaxation time. Spin–lattice relaxation is caused by local magnetic fields that fluctuate at a frequency close to the resonance frequency of the α → β transition. Such fields can arise from the tumbling motion of molecules in a fluid sample. If molecular tum bling is too slow or too fast compared to the resonance frequency, it will give rise to a fluctuating magnetic field with a frequency that is either too low or too high to stimulate a spin change from β to α, so T1 will be long. Only if the molecule tumbles at about the resonance frequency will the fluctuating magnetic field be able to induce spin changes effectively, and only then will T1 be short. The rate of molecular tumbling increases with temperature and with reducing viscosity of the solvent, so we can expect a dependence like that shown in Fig. 15.35. A second aspect of spin relaxation is the fanning-out of the spins in the xy-plane if they precess at different rates (Fig. 15.36). The magnetization vector is large when all the spins are bunched together immediately after a 90° pulse. However, this orderly bunching of spins is not at equilibrium and, even if there were no spin–lattice relaxation, we would expect the individual spins to spread out until they were uniformly distributed with all possible angles around the z-axis. At that stage, the component of magnetization vector in the plane would be zero. The randomization of the spin directions occurs exponentially with a time constant called the transverse relaxation time, T2:
My(t) ∝e−t/T2
Because the relaxation involves the relative orientation of the spins, T2 is also known as the spin–spin relaxation time. Any relaxation process that changes the balance betweenαandβspins will also contribute to this randomization, so the time constant T2 is almost always less than or equal to T1. Local magnetic fields also affect spin–spin relaxation. When the fluctuations are slow, each molecule lingers in its local magnetic environment and the spin orientations randomize quickly around the applied field direction. If the molecules move rapidly from one magnetic environment to another, the effects of differences in local magnetic field average to zero: individual spins do not precess at very different rates, they can remain bunched for longer, and spin–spin relaxation does not take place as quickly. In other words, slow molecular motion corresponds to short T2 and fast motion corresponds to long T2 (as shown in Fig. 15.35). Calculations show that, when the motion is fast, T2 ≈ T1. If the y-component of magnetization decays with a time constant T2, the spectral line is broadened (Fig. 15.37), and its width at half-height becomes
Typical values of T2 in proton NMR are of the order of seconds, so linewidths of around 0.1 Hz can be anticipated, in broad agreement with observation. So far, we have assumed that the equipment, and in particular the magnet, is perfect, and that the differences in Larmor frequencies arise solely from interactions within the sample. In practice, the magnet is not perfect, and the field is different at different locations in the sample. The inhomogeneity broadens the resonance, and in most cases this inhomogeneous broadening dominates the broadening we have dis cussed so far. It is common to express the extent of inhomogeneous broadening in terms of an effective transverse relaxation time, T2*, by using a relation like eqn 15.33, but writing
where ∆ν1/2 is the observed width at half-height of a line with a Lorenztian shape of the form I ∝ 1/ (1 + ν2). As an example, consider a line in a spectrum with a width of 10 Hz. It follows from eqn 15.34 that the effective transverse relaxation time is
Fig. 15.34 In longitudinal relaxation the spins relax back towards their thermal equilibrium populations. On the left we see the precessional cones representing spin-1–2 angular momenta, and they do not have their thermal equilibrium populations (there are more α-spins than β-spins). On the right, which represents the sample a long time after a time T1 has elapsed, the populations are those characteristics of a Boltzmann distribution (see Molecular interpretation 3.1). In actuality, T1 is the time constant for relaxation to the arrangement on the right and T1 ln 2 is the half-life of the arrangement on the left.
Fig. 15.36 The transverse relaxation time, T2, is the time constant for the phases of the spins to become randomized (another condition for equilibrium) and to change from the orderly arrangement shown on the left to the disorderly arrangement on the right (long after a time T2 has elapsed). Note that the populations of the states remain the same; only the relative phase of the spins relaxes. In actuality, T2 is the time constant for relaxation to the arrangement on the right and T2 ln 2 is the half-life of the arrangement on the left.
Fig. 15.35 The variation of the two relaxation times with the rate at which the molecules move (either by tumbling or migrating through the solution). The horizontal axis can be interpreted as representing temperature or viscosity. Note that, at rapid rates of motion, the two relaxation times coincide.
Fig. 15.37 A Lorentzian absorption line. The width at half-height is inversely proportional to the parameter T2 and the longer the transverse relaxation time, the narrower the line.
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