Motional Averaging of Nuclear Resonance in a Field Gradient

The traditional view of nuclear-spin decoherence in a field gradient due to molecular self-diffusion is challenged on the basis of temperature dependence of the linewidth, which demonstrates different behaviors between liquids and gases. The conventional theory predicts that in a fluid, linewidth should increase with temperature; however, in gases we observed the opposite behavior. This surprising behavior can be explained using a more detailed theoretical description of the dephasing function that accounts for position autocorrelation effects.

Figure 1

Measurement of molecular self-diffusion in a constant field gradient. A 90° radio-frequency (rf) pulse tips the magnetization. This resulting nuclear induction signal is measured in a constant gradient of amplitude $g$. The time evolution of magnetization, as described by the textbook [10, 11, 12] expression (1), interrogates the molecular self-diffusion process.

Figure 2

Validation of the $T^{-1/2}$ law in the high-temperature regime ($T>C$) for gases. The values of $\ln T$ shown correspond to the temperature range $T=180–490\mathrm{K}$. Three different gases were investigated: methane, acetylene, and propylene. The temperature dependence on linewidth (scaled to gradient strength) was found to be $\mathrm{\Delta }f\propto T^{-0.47\pm 0.04}$ (averaged over the three gases). Although different values of the applied gradient $g$ are shown here to avoid overlapping of the curves (methane, $g=0.15\mathrm{G}/\mathrm{cm}$; acetylene, $g=0.07\mathrm{G}/\mathrm{cm}$; propylene, $g=0.1\mathrm{G}/\mathrm{cm}$), the scaled linewidth is independent of $g$.

Figure 3

Temperature dependence of linewidth ($\mathrm{\Delta }f$) in liquids. Nine different liquids were investigated, as shown by the different symbols. The values of $\ln (T)$ shown span the temperature range $T=180–450\mathrm{K}$. (a) For a fixed gradient strength of $g=0.05\mathrm{G}/\mathrm{cm}$, all linewidths were broadened by a similar amount, hence the overlap in the data. At fixed $g$, the linewidth did not exhibit any detectable dependence on temperature. (b) Increasing applied gradient strength did not alter the independence of linewidth on temperature. Applied gradient strengths were nitromethane ($1\mathrm{G}/\mathrm{cm}$), dichloromethane (dcm, $0.5\mathrm{G}/\mathrm{cm}$), acetonitrile ($0.4\mathrm{G}/\mathrm{cm}$), chloroform ($0.3\mathrm{G}/\mathrm{cm}$), benzene ($0.3\mathrm{G}/\mathrm{cm}$), water ($0.2\mathrm{G}/\mathrm{cm}$), trifluoroacetic acid (tfa, $0.1\mathrm{G}/\mathrm{cm}$), dimethyl sulfoxide (dmso, $0.1\mathrm{G}/\mathrm{cm}$), acetone ($0.05\mathrm{G}/\mathrm{cm}$). The symbol in (a) refers to the same liquid as in (b).

Figure 4

Dependence of linewidth ($\mathrm{\Delta }f$) on applied gradient strength ($g$) for gases. Three different gases were investigated. Two different regimes are found: in the limit of strong applied gradients, $\mathrm{\Delta }f$ scales as $g^{1.0\pm 0.1}$ whereas for weak gradients $\mathrm{\Delta }f$ scales as $g^{1.8\pm 0.2}$. All data were acquired at ambient temperature.

Figure 5

Dependence of linewidth ($\mathrm{\Delta }f$) on gradient strength ($g$) for liquids. Nine different liquids were investigated. Two different regimes are found: in the limit of strong applied gradients, $\mathrm{\Delta }f$ scales as $g^{1.1\pm 0.1}$, whereas for weak gradients $\mathrm{\Delta }f$ scales as $g^{2.0\pm 0.2}$. All data were acquired at ambient temperature.