## Fundamentals of NMR Relaxation

Relaxation is a general term used to describe the movement of a system toward equilibrium after an injection of energy. When atomic nuclei in a molecule are excited in a pulsed nuclear magnetic resonance (NMR) experiment, the excited state may exist for a rather long time, often seconds to minutes , Characteristic relaxation times estimated in an NMR experiment can be used to provide information about molecular motion. In the following paragraphs, we provide a simplified summary of pulsed NMR relaxation. The reader may consult texts on this subject for further details [1, 2],

At a magnetic field strength of zero, a diamagnetic sample is not magnetized. Suppose we now apply a magnetic field to this sample. A magnetic field will be induced in the sample, but this induced field will take some time to reach equilibrium. In the Bloch theory of NMR , the approach to equilibrium is assumed to be exponential. This assumption leads to the following expression for the time dependence of the magnetization Mz in the z direction, usually taken as the direction of the applied field:

where M0 is the magnetization at thermal equilibrium, and T\ is a characteristic relaxation time called the longitudinal relaxation time. Longitudinal denotes that the z component of the magnetization is of importance here. When the magnetization is moved away from the z direction, it will return to the 2 axis with a time constant Tx.

Different T\ values may apply for the various atoms and nuclear environments in a molecule. In the limit of rapid isotropic motion in solution, the widths of the individual resonance peaks are inversely related to Tx values.

We shall see that analysis of the NMR peak shape is one way to estimate T\. Another way is by the so called inversion-recovery pulse sequence , In this multiple-pulse NMR experiment, the z magnetization is inverted with a it pulse, so-named because it is ir radians removed from the +z axis. Thus, a 7T pulse is in the —z direction. A variable time period is allowed for the magnetization to return partly to the z direction, and then a ttH pulse is used for measurement ,

Results from an inversion-recovery pulse sequence (Figure 8.1) show that the peaks at short measurement times are inverted, and at longer times they eventually return to the usual positive peaks. Nonlinear regression analysis of each peak height vs. time profile using the relevant model for peak height Ait) at delay time t, provides the set of T\ values characteristic of the different nuclear environments in the sample. The model is given as

where A* is the limiting peak height at long times and Rt = 1/Tj is the longitudinal relaxation rate.

As the z magnetization reappears with time constant T\, the magnetization that has been induced in the x-y plane disappears. This process constitutes another relaxation. Called the transverse relaxation, it is assumed to be exponential and characterized by decay time T2. If the only mechanism for transverse relaxation is the return of magnetization to the z direction, then Ti = T2. However, a number of factors can cause T2 to be smaller

Figure B.I Example of NMR spectra resulting from an inversion-recovery pulse sequence. (Reprinted with permission from , copyright by Pergammon Press.)

than Ty. One is the inhomogeneity of the magnetic field, which depends on instrumental conditions and can be separated from other effects on T2. Therefore, the symbol T2 usually refers to longitudinal relaxation in the absence of inhomogeneous field effects.

Local magnetic fields characteristic of the sample may also cause T2 to be smaller than 7\. Values of T2 may provide information about molecules in the sample. In solids, for example, varying environments in the sample make T2 very small, although 7, may be very large bacause of the lack of motion , We shall discuss examples of solid state NMR relaxation in Section C of this chapter.

The overall transverse relaxation time includes both T2 and the influence of inhomogeneity of the magnetic field. This combined-effect relaxation is given the symbol T2. It can be obtained from the free induction decay, which decreases exponentially with time constant T2. However, the free induction decay is often more characteristic of the solvent , Therefore, it is more reliable to obtain T2 from the width (W) of the NMR lines at half-height, using the relation

Nonlinear regression onto the appropriate Gaussian, Lorentzian, or mixed Gaussian-Lorentzian model (see Section 3.B.2) can be used to obtain accurate estimates of the line width Wand thereby T'2. According to theoretical predictions, the limiting line shape is Lorentzian when Tx = T2, and Gaussian when 7\ §> T2.

Spin-echo pulse NMR or spin-locking experiments  can also be used to obtain T2. In the latter case, the model is a simple exponential decay in peak amplitude A(t) at delay time /:

where A0 is the peak intensity at t = 0, and R2 = UT2 is the transverse relaxation rate. 