Introduction to average Hamiltonian theory. II. Advanced examples

Abstract

Where the first part of our tutorial Introduction to average Hamiltonian theory (Brinkmann, 2016) introduced in detail the basic concepts and demonstrated the application to two composite radio-frequency (rf) pulses in nuclear magnetic resonance (NMR) spectroscopy, this second part will present in a comprehensive but educational manner two, more advanced examples for the application of average Hamiltonian theory in solid-state NMR spectroscopy, both to analyse and design rf pulse sequences: (i) The Rotational-Echo Double Resonance (REDOR) sequence, which recouples the heteronuclear dipolar coupling during sample rotation around an axis at the magic-angle of $54.74^{\circ }$ with respect to the external static magnetic field. We will gradually increase the complexity of applying average Hamiltonian theory by first considering ideal, infinitesimally short rf pulses. Next, we will examine finite pulses with an rf phase of zero, and finally, we will explore finite pulses with arbitrary rf phases. In the latter case, if a first order average Hamiltonian proportional to heteronuclear longitudinal two-spin order ($2I_z{S}_z$) is desired, solutions for the choice of rf phases include the XY and MLEV type schemes. (ii) The Lee–Goldburg homonuclear dipolar decoupling sequence under static samples conditions and its improved successors, Flip-Flop Lee–Goldburg (FFLG) and Frequency-Switched Lee–Goldburg (FSLG).