NMR multiple echoes observed in solid He 3

TLDR: In this paper, the effect of the nuclear demagnetizing field in the equation of motion of the magnetization has been investigated in the case of multiple echoes, where a large number of echoes are observed following two isolated NMR radio-frequency pulses separated by a time.

Abstract: A large number of echoes-called multiple echoes-following two isolated NMR radio-frequency pulses separated by a time $\ensuremath{\tau}$, have been observed in bcc $^{3}\mathrm{He}$. A quantitative theory of this phenomenon is given, taking into account the nonlinear effect of the nuclear demagnetizing field in the equation of motion of the magnetization. In a magnetic-field gradient $G$, just before the second pulse at time $\ensuremath{\tau}$, the nuclear-spin transverse magnetization has a helical configuration of pitch $\ensuremath{\gamma}G\ensuremath{\tau}={k}_{0}$, which is converted by the second radio-frequency pulse into a sinusoidal modulation of the magnetization along the magnetic field gradient. The corresponding sinusoidal demagnetizing field modulates spatially the NMR resonance frequency so that the transverse magnetization is a superposition of helical configurations with pitch $p{k}_{0}$ (multiple of ${k}_{0}$), yielding multiple echoes at times $p\ensuremath{\tau}$. A detailed comparison with the experimental results obtained at high temperatures ($300 \mathrm{mK}lTl1 \mathrm{K}$) and low temperatures ($1lTl20$ mK), leads to a satisfactory agreement with the experiments. The main practical application of the multiple-echo study is to provide a method of measurement of the absolute value of the nuclear-spin susceptibility without knowing the shape or the number of spins of the sample. The transverse relaxation time ${T}_{2}$ and the diffusion coefficient $D$ are also obtained with this analysis. When applied to the low-temperature measurements, the multiple-echo analysis has pointed out, for the first time, the increase of the magnetic susceptibility of solid $^{3}\mathrm{He}$ in the paramagnetic phase near the ordering temperature ${T}_{c}$, while in contrast the relaxation times ${T}_{1}$ and ${T}_{2}$ and the diffusion coefficient $D$ do not change appreciably, even just above ${T}_{c}$.