NMR multiple echoes observed in solidHe3

A large number of echoes-called multiple echoes-following two isolated NMR radio-frequency pulses separated by a time $\tau$, 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 $\tau$, the nuclear-spin transverse magnetization has a helical configuration of pitch $\gamma G\tau =k_0$, 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 $p\tau$. A detailed comparison with the experimental results obtained at high temperatures ($300 \mathrm{mK}<T<\mathrm{}1\mathrm{} \mathrm{K}$) and low temperatures ($1<T<\mathrm{}20\mathrm{}$ 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 $T_c$.