On the application of magic echo cycles for quadrupolar echo spectroscopy of spin-1 nuclei

Applications of Floquet–Magnus expansion, average Hamiltonian theory and Fer expansion to study interactions in solid state NMR when irradiated with the magic-echo sequence

“…This characteristic can be explained by considering the convergence of the Magnus expansion with strong RF power and short pulse spacings. The Magnus expansion converges faster with the magic echo sequence compared to the conventional two-pulse quadrupolar echo cycle as in the article by Mananga et al 11 . More details on the efficiency of the magic echo cycle compared to the simple two-pulse quadrupolar echo sequence can be found in the literature of solid-state NMR spectroscopy 9,10,11,34,35,36 .…”

“…The zero-order term of the average Hamiltonian was calculated with the magic echo pulse sequence for each interaction 9,11 : $${\overline{\mathrm{H}}}_{\Delta \omega }^0=\frac{1}{3\tau }\frac{\Delta \omega }{{\omega }_{\mathrm{italicRF}}}{I}_y[1-\cos {\omega }_{\mathrm{italicRF}}(2\tau -\alpha \left)\right]-\frac{1}{3\tau }\frac{\Delta \omega }{{\omega }_{\mathrm{italicRF}}}{I}_x\left[\sin {\omega }_{\mathrm{italicRF}}\right(2\tau -\alpha )+\frac{4}{\pi }{\omega }_{\mathrm{italicRF}}\alpha ]$$ $${\overline{\mathrm{H}}}_D^0={\omega }_D{I}_z{I}_z-{\omega }_D\mathrm{I}.\mathrm{I}+\frac{1}{4\tau }\frac{{\omega }_D}{{\omega }_{\mathrm{italicRF}}}{I}_x{I}_x\left[2{\omega }_{\mathrm{italicRF}}\right(2\tau +\alpha \left)\right]+\sin 2{\omega }_{\mathrm{italicRF}}(2\tau -\alpha )]++\frac{1}{4\tau }\frac{{\omega }_D}{{\omega }_{\mathrm{italicRF}}}{I}_y{I}_y[2{\omega }_{\mathrm{italicRF}}(2\tau -\alpha )]-\sin 2{\omega }_{\mathrm{italicRF}}(2\tau -\alpha \left)\right]-\frac{1}{2\tau }\frac{{\omega }_D}{{\omega }_{\mathrm{italicRF}}}[I_x{I}_y+I_y{I}_x]{\sin }^2{\omega }_{\mathrm{italicRF}}(2\tau -\alpha )$$ $${\overline{\mathrm{H}}}_{\omega Q}^0=\frac{1}{7\tau }\frac{12\alpha {\omega }_Q}{\pi }{I}_{x,2}+\frac{4(3\alpha -\tau ){\omega }_Q}{7\tau }true[I_{x,1}{I}_{x,...}$$…”

“…While the well-known theory of average Hamiltonian theory (AHT) 4 is a powerful analytical tool, it is not always applicable or sufficiently accurate. The technique of AHT is more appropriated for static experiments and stroboscopic observation, and has been used to study all types of interactions encountered in NMR, such as chemical shift anisotropy 9-11 , dipolar 9,10,12 , and quadrupolar 11,13,14 interactions. While this technique remains the most popular theoretical technique used so far in NMR because of its intuitive way of obtaining effective time-independent Hamiltonians, it is limited to a single time-dependent perturbation 15 .…”

On the Fer expansion: Applications in solid-state nuclear magnetic resonance and physics

“…This characteristic can be explained by considering the convergence of the Magnus expansion with strong RF power and short pulse spacings. The Magnus expansion converges faster with the magic echo sequence compared to the conventional two-pulse quadrupolar echo cycle as in the article by Mananga et al [59]. More details on the efficiency of the magic echo cycle compared to the simple two-pulse quadrupolar echo sequence can be found in the literature of solid-state NMR spectroscopy [59,121,123,135,246].…”

Investigation of the effect of finite pulse errors on the BABA pulse sequence using the Floquet–Magnus expansion approach

“…The situation with quadrupolar nuclei is much worse, with resonances hundreds of kHz off-resonance. Therefore, accounting for finite pulse width is important due to the evolution of the spin system under the dipolar or quadrupolar interaction [4, 18, 19, 20, 21, 22]. Hence, a good understanding of the spin system behavior during the finite pulses and theory therein is important.…”