Broadband excitation in solid-state NMR using interleaved DANTE pulse trains with N pulses per rotor period

We analyze the direct excitation of wide one-dimensional spectra of nuclei with spin I=1/2 or 1 in rotating solids submitted to pulse trains in the manner of Delays Alternating with Nutations for Tailored Excitation (DANTE), either with one short rotor-synchronized pulse of duration τp in each of K rotor periods (D1(K)) or with N interleaved equally spaced pulses τp in each rotor period, globally also extending over K rotor periods (D(N)(K)). The excitation profile of D(N)(K) scheme is a comb of rf-spikelets with Nν(R)=N/T(R) spacing from the carrier frequency, and a width of each spikelet inversely proportional to the length, KT(R), of D(N)(K) scheme. Since the individual pulse lengths, τp, are typically of a few hundreds of ns, D(N)(K) scheme can readily excite spinning sidebands families covering several MHz, provided the rf carrier frequency is close enough to the resonance frequency of one the spinning sidebands. If the difference of isotropic chemical shifts between distinct chemical sites is less than about 1.35/(KT(R)), D(N)(K) scheme can excite the spinning sidebands families of several sites. For nuclei with I=1/2, if the homogeneous and inhomogeneous decays of coherences during the DANTE sequence are neglected, the K pulses of a D1(K) train have a linearly cumulative effect, so that the total nutation angle is θ(tot)=K2πν1τp, where ν1 is the rf-field amplitude. This allows obtaining nearly ideal 90° pulses for excitation or 180° rotations for inversion and refocusing across wide MAS spectra comprising many spinning sidebands. If one uses interleaved DANTE trains D(N)(K) with N>1, only spinning sidebands separated by intervals of Nν(R) with respect to the carrier frequency are observed as if the effective spinning speed was Nν(R). The other sidebands have vanishing intensities because of the cancellation of the N contributions with opposite signs. However, the intensities of the remaining sidebands obey the same rules as in spectra obtained with νR. With increasing N, the intensities of the non-vanishing sidebands increase, but the total intensity integrated over all sidebands decreases. Furthermore, the NK pulses in a D(N)(K) train do not have a simple cumulative effect and the optimal cumulated flip angle for optimal excitation, θ(tot)(opt)=NK2πν1τp, exceeds 90°. Such D(N)(K) pulse trains allow achieving efficient broadband excitation, but they are not recommended for broadband inversion or refocusing as they cannot provide proper 180° rotations. Since D(N)(K) pulse trains with N>1 are shorter than basic D1(K) sequences, they are useful for broadband excitation in samples with rapid homogeneous or inhomogeneous decay. For nuclei with I=1 (e.g., for (14)N), the response to basic D1(K) pulse train is moreover affected by inhomogeneous decay due to 2nd-order quadrupole interactions, since these are not of rank 2 and therefore cannot be eliminated by spinning about the magic angle. For large quadrupole interactions, the signal decay produced by second-order quadrupole interaction can be minimized by (i) reducing the length of D(N)(K) pulse trains using N>1, (ii) fast spinning, (iii) large rf-field, and (iv) using high magnetic fields to reduce the 2nd-order quadrupole interaction.