Relaxation Effects in Nuclear Magnetic Resonance Absorption

The exchange of energy between a system of nuclear spins immersed in a strong magnetic field, and the heat reservoir consisting of the other degrees of freedom (the "lattice") of the substance containing the magnetic nuclei, serves to bring the spin system into equilibrium at a finite temperature. In this condition the system can absorb energy from an applied radiofrequency field. With the absorption of energy, however, the spin temperature tends to rise and the rate of absorption to decrease. Through this "saturation" effect, and in some cases by a more direct method, the spin-lattice relaxation time $T_1$ can be measured. The interaction among the magnetic nuclei, with which a characteristic time $T_2^{}{}_{}{}^{\prime }$ is associated, contributes to the width of the absorption line. Both interactions have been studied in a variety of substances, but with the emphasis on liquids containing hydrogen.

Magnetic resonance absorption is observed by means of a radiofrequency bridge; the magnetic field at the sample is modulated at a low frequency. A detailed analysis of the method by which $T_1$ is derived from saturation experiments is given. Relaxation times observed range from ${10}^{-4\mathrm{}}$ to ${10}^2$ seconds. In liquids $T_1$ ordinarily decreases with increasing viscosity, in some cases reaching a minimum value after which it increases with further increase in viscosity. The line width meanwhile increases monotonically from an extremely small value toward a value determined by the spin-spin interaction in the rigid lattice. The effect of paramagnetic ions in solution upon the proton relaxation time and line width has been investigated. The relaxation time and line width in ice have been measured at various temperatures.

The results can be explained by a theory which takes into account the effect of the thermal motion of the magnetic nuclei upon the spin-spin interaction. The local magnetic field produced at one nucleus by neighboring magnetic nuclei, or even by electronic magnetic moments of paramagnetic ions, is spread out into a spectrum extending to frequencies of the order of $\frac{1}{{\tau }_c}$, where ${\tau }_c$ is a correlation time associated with the local Brownian motion and closely related to the characteristic time which occurs in Debye's theory of polar liquids. If the nuclear Larmor frequency $\omega$ is much less than $\frac{1}{{\tau }_c}$, the perturbations caused by the local field nearly average out, $T_1$ is inversely proportional to ${\tau }_c$, and the width of the resonance line, in frequency, is about $\frac{1}{T_1}$. A similar situation is found in hydrogen gas where ${\tau }_c$ is the time between collisions. In very viscous liquids and in some solids where $\omega {\tau }_c>1\mathrm{}$, a quite different behavior is predicted, and observed. Values of ${\tau }_c$ for ice, inferred from nuclear relaxation measurements, correlate well with dielectric dispersion data.

Formulas useful in estimating the detectability of magnetic resonance absorption in various cases are derived in the appendix.