Cross-polarization (CP) experiments employing frequency-swept radiofrequency (rf) pulses have been successfully used in static spin systems for obtaining broadband signal enhancements. These experiments have been recently extended to heteronuclear *I*, *S* = spin-1/2 nuclides under magic-angle spinning (MAS), by applying adiabatic inversion pulses along the *S* (low-γ) channel while simultaneously applying a conventional spin-locking pulse on the *I*-channel (^{1}H). This study explores an extension of this adiabatic frequency sweep concept to quadrupolar nuclei, focusing on CP from ^{1}H (*I* = 1/2) to ^{2}H spins (*S* = 1) undergoing fast MAS ($\nu r$ = 60 kHz). A number of new features emerge, including zero- and double-quantum polarization transfer phenomena that depend on the frequency offsets of the swept pulses, the rf pulse powers, and the MAS spinning rate. An additional mechanism found operational in the ^{1}H–^{2}H CP case that was absent in the spin-1/2 counterpart, concerns the onset of a pseudo-static zero-quantum CP mode, driven by a quadrupole-modulated rf/dipolar recoupling term arising under the action of MAS. The best CP conditions found at these fast spinning rates correspond to double-quantum transfers, involving weak ^{2}H rf field strengths. At these easily attainable (ca. 10 kHz) rf field conditions, adiabatic level-crossings among the {$|1\u27e9,|0\u27e9,|\u22121\u27e9$} *m*_{S} energy levels, which are known to complicate the CP MAS of quadrupolar nuclei, are avoided. Moreover, the CP line shapes generated in this manner are very close to the ideal ^{2}H MAS spectral line shapes, facilitating the extraction of quadrupolar coupling parameters. All these features were corroborated with experiments on model compounds and justified using numerical simulations and average Hamiltonian theory models. Potential applications of these new phenomena, as well as extensions to higher spins *S*, are briefly discussed.

## I. INTRODUCTION

Solid-state nuclear magnetic resonance is a powerful analytical tool for the study of biomolecules and materials.^{1–6} Enabling many of these studies is cross-polarization^{7} magic-angle spinning (CPMAS),^{8} which based on achieving the Hartmann-Hahn (HH) matching condition,^{7,9} has been widely used for increasing the sensitivity of both spin-1/2^{8,10–15} and quadrupolar^{16–19} nuclear magnetic resonance (NMR). While very effective under moderate MAS spinning rates and magnetic field strengths, the spin-lock pulses that have to be applied as part of CP’s matching condition are often unsuitable for covering wide spectral bandwidths, particularly as magnet technologies yield platforms operating at fields ≥20 T,^{20,21} and chemical shift dispersions become concomitantly larger. Further challenges arise for ultra-wideline NMR spectra of quadrupolar nuclei, which usually have patterns that are significantly broader than those of spin-1/2 nuclides, and are influenced by MAS-driven level crossings among the spin-locked states.^{22–26}

Frequency-swept radio-frequency (rf) pulses have often been utilized to extend the bandwidths of NMR experiments in power-limited excitation,^{27} inversion,^{28–30} or decoupling pulse sequences.^{31,32} For the solid-state NMR of both spin-1/2 and quadrupolar nuclei, swept rf pulses play important roles in the excitation of broad powder patterns,^{33,34} and in increasing the bandwidth of Carr-Purcell Meiboom-Gill acquisitions.^{33,35,36} Adiabatically swept radio-frequency (rf) pulses have been combined into the standard cross-polarization scheme for *I*(1/2)-*S*(1/2)^{37–41} and *I*(^{1}H)-*S*(^{2}H)^{42} spin systems under MAS, for the sake of improving HH matching conditions. In these methods, the amplitude and/or frequency of an adiabatic rf pulse were swept through the HH conditions for better polarization transfers. Recently, an improved technique that combines frequency-swept pulses together with CP is the broadband adiabatic inversion cross-polarization (BRAIN-CP) method.^{33} The BRAIN-CP method is capable of delivering HH matching over a wide range of isotropic and anisotropic frequency dispersions. While initially utilized under static conditions in combination with spin-echoes for obtaining wideline NMR spectra of both spin-1/2 and quadrupolar nuclei,^{43–48} BRAIN-CP has been recently extended to MAS NMR of spin-1/2 nuclides.^{34} In this MAS variant, BRAIN-CP proved suitable for uniform excitation and refocusing over wide frequency regions—particularly at the very fast (≥50 kHz) MAS spinning rates that are envisioned for routine use in ultrahigh field NMR experiments.

This manuscript explores the applicability of BRAIN-CPMAS to quadrupolar nuclei. Specifically, the aim of this work was to obtain CP-enhanced ^{1}H–^{2}H wideband MAS spectra.^{19,49–52} To this end, Wideband Uniform-Rate Smooth-Truncation (WURST)-shaped^{28} rf adiabatic inversion pulses were applied on the ^{2}H channel, while spin-locking rf pulses were applied on the ^{1}H channel under sample spinning conditions. Due to the frequency sweep on ^{2}H channel multiple HH matching conditions occur, including zero-quantum (ZQ) and double-quantum (DQ) CP modes akin to those observed in BRAIN-CPMAS; also observed were novel static-like matching conditions, arising from a MAS-driven, quadrupolar-enabled rf-dipolar recoupling. Rotary resonance (RR) interferences arising from the modulation of the *S*-spin shift anisotropy and/or the first-order quadrupolar interaction were also observed. Average Hamiltonian analyses were conducted to understand these phenomena, which were predicted by numerical simulations and corroborated experimentally by spectra of model amino acid compounds. Overall, it is found that BRAIN-CPMAS implemented under suitable conditions can deliver sensitivity-enhanced, distortion-free spectra that closely mimic the ideal ^{2}H line shapes expected under ultrafast spinning rates.

## II. THEORETICAL BACKGROUND

### A. Frequency-swept CPMAS Hamiltonian for an *I* = 1/2 −*S* = 1 (^{1}H − ^{2}H) spin pair

The frequency-modulated (FM) doubly rotating frame Hamiltonian that describes the *I* = 1/2 → *S* = 1 CP dynamics of an isolated *I*-*S* spin pair under MAS and subject to a frequency-swept adiabatic passage on the *S* spin is

Here $\omega 1I=\gamma IB1I$ represents an on-resonance irradiation of the *I*-spin, $\omega 1S=\gamma SB1S,$ and $\Omega S\u2032=\Omega S\u2212\omega p(t)$ is an effective offset frequency of the *S* spin as seen in the FM frame. In this frame, $\Omega S=\omega 0\u2212\omega rf$ describes the center offset of the sweep and $\omega p(t)=(\Delta \omega /tp)t\u2212\Delta \omega /2$ is the rf’s instantaneous offset frequency, where $\Delta \omega =2\pi \Delta \nu $ and $tp$ are the bandwidth of the adiabatic sweep and the duration of a WURST pulse whose rf amplitude envelope is given as *A*(*t*) = $(1\u2212cosn[\pi t/tp])$ (*n* = even number), respectively.^{28,30,34} Any potential homonuclear *I*-*I* and *S*-*S* dipolar coupling interactions and *I*-and *S*-spin chemical shift anisotropies (CSAs) are ignored for simplicity, and $2b(t)IzSz$ is a MAS-modulated heteronuclear dipolar coupling interaction that under MAS is modulated as^{15}

In Eq. (2), the orientation-dependent {*b*_{k}} coefficients correlate an *I*-*S* dipolar vector defined in its principal axis frame to the rotor frame,^{15} as related by $\beta d$ and $\gamma d$ Euler angles. The first-order quadrupolar interaction, $HQ(1)(t)$, is given under MAS by^{53}

where $CQ=e2qQ\u210f$ and

A detailed description of an *S*-spin behavior under the action of an adiabatic WURST pulse and MAS is provided in the supplementary material A. Using this single-spin description, physical insight into the nature of the CP process under similar conditions can be gathered by removing the dominant first-order quadrupolar term $HQ(1)(t)$ from Eq. (1). This requires transforming the overall Hamiltonian into a quadrupolar interaction frame defined as^{54–56}

where

When Eq. (5) is considered over a powder, the *γ*-angle averaged Coriolis term, $\u27e8i(dUQ\u22121/dt)UQ\u27e9\gamma ,$ vanishes.^{54} Since all terms in Eq. (1), with the exception the rf Hamiltonian, commute with $HQ(1)(t)$ at all times, only the rf Hamiltonian will be transformed by going into the quadrupolar interaction frame. This will result in

The spin-locking behavior of *S* over the sweep can be evaluated by inspecting the parameter $\alpha =\omega 1S2CQ\omega r$, which specifies the extent of adiabaticity in the level-crossings.^{22} When $\alpha =\omega 1S2CQ\omega r\u226a1$, the system is said to be in the sudden passage regime; under these conditions, the rf does not promote population redistributions among the $|1\u27e9,|0\u27e9$, and $|\u22121\u27e9$ eigenstates during these levels’ crossings. As discussed within the context of Fig. S1 of the supplementary material, $\nu 1S$ in the 5–16 kHz range fulfills this sudden-passage regime for prototypical ^{2}H sites (e.g., *C*_{Q} = 168 kHz) at high ($\nu r$ ≥ 40 kHz) spinning rates. Under such conditions, one can solely consider the term

in Eq. (7). Accounting in this accelerated frame for $HQ(1)(t)$ simplifies the description of the CP Hamiltonians for both $|1\u27e9\u2194|0\u27e9$ and $|0\u27e9\u2194|\u22121\u27e9$ subspaces, leading to identical spin-1/2-like expressions for both subspaces (refer to Eqs. (S9) and (S10) in the supplementary material):

Assuming further that the rf’s amplitude envelope $A(t)=(1\u2212cos40[\pi t/tp])$ ≈ 1 and defining $Sz=2Sz12$, $Sx=2Sx12$ as pseudo-fictitious spin-1/2 operators possessing the same coefficients as the original spin-1 angular momentum operators, Eq. (9) becomes

Applying a final transformation into a doubly tilted frame where all rf fields lie parallel to the *z*′ axes,^{15,57} the individual subspace’s CP Hamiltonians become

The effective resonance frequencies along these tilted *z*′-axes are

while the tilt angles $\theta I$ and $\theta S(t)$ that relate the *z*′-axes to the *z*-axes in the FM frame are given by

where the last relation uses the fact that $sin(\theta S[t])$ involves a Bessel integral that can be expanded as an infinite Fourier series. With these definitions and assumptions, and if using only the three initial CP-relevant terms in $H\u223cT\u2032(t),$ Eq. (11) is reduced to^{14,15,17}

The product of the $\omega rt$-dependent Fourier expansions arising from the modulation of the quadrupolar and dipolar terms in Eq. (17) can be separated into time-independent and time-dependent terms:

where

An explicit derivation of these $\Lambda k(k=0,\xb11,and\xb12)$ terms is summarized in the supplementary material. This supplementary material can be used to gauge the magnitude/relevance of these terms. For instance, for a general crystallite orientation $(\beta =45\xb0,\gamma =0\xb0)$ of a dipolar coupled ^{1}H–^{2}H pair collinear with the quadrupolar tensor possessing a ^{1}H–^{2}H dipolar coupling of 4 kHz and a *C*_{Q} = 168 kHz, the magnitude of the quadrupole-driven *rf*-dipolar recoupled term $\Lambda 0$ is about 680 Hz. This is a significant coupling strength, which will generate a static-like, *k* = 0, ZQ_{0} CP mode even under very fast MAS conditions (*vide infra*).

Utilizing the single-transition operator formalism usually employed to analyze CP transfers,^{15,58} Eq. (18) can be rewritten as follows:

where, $\omega \Delta (t)=\omega 1I\u2212\omega eS(t),\omega \Sigma (t)=\omega 1I+\omega eS(t)$, and the single-transition operators for the zero-quantum (ZQ) and double-quantum (DQ) coherences are defined by

Since the single transition operators defined in the ZQ and DQ subspaces commute with each other, their time evolutions can be solved separately

and

### B. Zero-order Hartmann-Hahn average Hamiltonians

The oscillating terms associated with $\Lambda \xb11$ and $\Lambda \xb12$ in Eqs. (24) and (25) are analogous to forms that we derived for swept CPMAS NMR for an *I* = *S* = 1/2 case,^{34} meaning that we can use our previous average Hamiltonian analysis to determine their averages over a single rotor period. Notice that Eqs. (24) and (25) possess dual time dependencies, associated to the adiabatic frequency sweeps (*I*_{z} terms) and to the MAS-driven modulations (*I*_{x} terms), respectively. Under the fast spinning rates and slow adiabatic sweeps of interest in this study, an average Hamiltonian over the time scale of a rotor period may be approximated by assuming quasi-constant offsets $\omega \Delta ,\Sigma (t0)$ over any given rotor period. This assumption is valid as the rotor periods to be considered (15-30 $\mu s$ range) will typically be orders-of-magnitude shorter than the adiabatic pulse duration ($tp=$ 5-10 ms). Approximating the $\omega eS(t)$ as quasi-constant for time intervals $t0\u2264t\u2264t0+2\pi \omega r$, the effective resonance frequency along the *S* channel can be averaged for each rotor period as

At a similar level of approximation, an evaluation of the $cos2(\u222b02\pi \omega r3CQ\omega Q(t\u2032)dt\u2032)$ term contributing to $\omega eS(t0)$ over a rotor period leads to a null average integral, and therefore to $cos2(0)=1$. With these assumptions, average Hamiltonians over a rotor period $\u27e8HT\zeta \u27e9av=1\tau r\u222b0\tau rHT\zeta (t\u2032)dt\u2032,\zeta =$ ZQ or DQ can be obtained from Eqs. (24) and (25) via the transformation $UpHT\zeta Up\u22121\u2212iUpddtUp\u22121,$ where $Up=exp(ik\omega rtIz\zeta )$ and $\zeta =$ DQ or ZQ refers to the zero- or double-quantum sub-spaces. This evaluation produces

where $k=\xb11$ and $\xb12,$$I+\zeta =Ix\zeta +iIy\zeta $, *m* is +1 for the ZQ-CP process, and −1 for the DQ-CP process. In the above equations, $t0$ has been generalized for simplicity to a suitable time $t$.

While the $k=\xb11,\xb12$ terms are as usual in CPMAS experiments involving polarization transfer to spin-1/2 nuclides, the static-like CPMAS transfer process involving the $\Lambda 0$ term is peculiar to CPMAS of quadrupoles and arises due to the interference between the periodic quadrupole-driven modulation of the rf, and the identically periodic MAS-driven modulation of the dipolar coupling. This recoupling effect can be separated from Eq. (24), and for a dipole-coupled *I* = 1/2 − *S* = 1 spin pair will be given by

When $\omega 1I=\omega eS(t),$ this CP mode will transfer *I*-magnetization to the *S*-spin. Notice that, as is the case for static solids, only a ZQ HH match is possible; a DQ-HH process is not allowed because there does not exist a condition nulling the coefficient of $IzDQ$ in Eq. (25), $\omega \Sigma (t),$ for any combination of $\omega 1I$ and $\omega eS(t).$

### C. Hartmann-Hahn matching and rotary resonance inversion conditions under MAS

As in the *I* = *S* = 1/2 case, time-progressive ZQ- and DQ-Hartmann-Hahn matching conditions, as well as rotary resonance (RR) inversions, may affect an *I* = 1/2−*S* = 1 system over the course of the frequency sweep. To visualize how these happen during BRAIN-CPMAS experiments on quadrupoles, we inspect the time-progressive conditions that make the $IzDQ$ and $IzZQ$ coefficients in Eqs. (27) and (28) zero—which will be the conditions enabling the departure of the system from pure *I*-spin polarization. A $ZQ\xb1k$ (*k* = 0, 1, or 2) HH matching condition will therefore be met when the instantaneous frequency offset, $\omega p(t)$, introduced by the adiabatic sweep, satisfies $\nu eI\u2212\nu eS(t)=\xb1k\nu r.$ In a similar manner, a DQ_{k} (*k* = 1 or 2) HH matching condition will be met when $\omega p(t)$ instantaneously satisfies $\nu 1H+\nu eS(t)=k\nu r$. A third potential phenomenon, a RR inversion, has been shown to arise driven by oscillations of either CSA or quadrupolar interactions, whenever a spin-locked magnetization passes through effective field conditions satisfying $\nu eS(t)=k\nu r$.^{26,55,59,60} These effects will revert the spin-locked state of the magnetization vis-à-vis the effective field. Given the low-γ nature of ^{2}H nuclei and their relatively small CSA spans (≤1 kHz), RR inversion phenomena of this kind will take place driven mainly by the first order quadrupole interaction $HQ(1)(t)$. To the best of our knowledge, this RR-driven inversion of spin-locked quadrupolar states has so far not been noted, even if a similar effect has been utilized to improve the excitation of multiple-quantum coherences in half-integer quadrupolar nuclei.^{26}

Figure 1 presents numerical simulations of ^{1}H–^{2}H BRAIN-CPMAS deriving from these considerations. These calculations incorporate different offset frequencies $\Omega S/2\pi $ (0, $\xb130,$ and $\xb160$ kHz), null and non-null quadrupole couplings, and varying swept offsets $\Delta \nu $ (60 kHz and 110 kHz), assuming a MAS rate of $\nu r$ = 60 kHz. Also illustrated in the figures are the fulfillment of DQ_{±k} (*k* = 1, 2), $ZQ\xb1k$ (*k* = 0, 1, or 2), and RR conditions, as well as the behavior of the spin-locked (*S*_{z}) and transverse (*S*_{x}) *S* magnetizations arising from a situation where these are null at the beginning of the swept pulse HH matching, and all polarization exists as *I*_{z}. Apart from the scaling effects coming from the 2 and $2$ coefficients that are present for the $Sz12$ and $Sx12$ operators, respectively, these predictions are similar to those made for the *S* = 1/2 case. The zero-crossings timings associated with the ZQ and DQ conditions are clearly connected with I → S transfers of spin-locked polarization, and these time points measured at any arbitrary $\xb1\Omega S/2\pi $ are clear mirror images to each other along *t* = *t*_{p}/2. Moreover, when the offset $\Omega S/2\pi $ = 0, the two time points that satisfy the same DQ_{1} condition become mirror imaged to each other along *t* = *t*_{p}/2. These two mirror-imaged processes are associated with similar enhancements but acting on *S*-polarizations of opposite sign, due to the sign change that the *S*_{z} magnetization component experiences when it crosses the equatorial line during the adiabatic sweep. These events move to earlier times during the 0–*t*_{p} window as the $\Omega S/2\pi $ offset becomes negative, and to later times during the adiabatic sweep as $\Omega S/2\pi $ increases. Still, it is noteworthy how the introduction of *C*_{Q} ≠ 0 breaks the symmetry of the aforementioned transfer efficiencies of the two DQ_{1} coherences. This helps to create a non-zero *S*_{z} polarization at the conclusion of the sweep. Two differences are also worth noting between the present *S* = 1 case and earlier spin-1/2 analyses. One concerns the RR inversions, which will now be driven by modulation of the first-order quadrupolar interaction rather than of the CSA—and hence they may arise even in the presence of ^{2}H nuclei with small CSAs. While this RR phenomenon does not play an important role if choosing sweeps $\Delta \nu \u2264\nu r$ and/or dealing with small $\Omega S/2\pi $ offsets, it becomes visible when this is not the case. The second phenomenon concerns the appearance of new, ZQ_{0}-derived HH transfers.

DQ_{1} effects dominate the HH transfers under conditions of small $\Omega S/2\pi $ offsets. For $\Delta \nu =\nu r$ (Fig. 1(a)), only the DQ_{1} time points satisfy a condition for CP polarization transfer at a small $\Omega S/2\pi (\u2264\xb1\nu r$/4). For instance, at $\Omega S/2\pi $ = 0, the DQ_{1} time points are visible in Fig. 1(a) at *t*_{1} = 2.27 ms and *t*_{2} = 5.73 ms ($tp=$ 8 ms). As $\Omega S/2\pi $ increases, the ZQ_{0}, DQ_{2}, RR_{1}, and ZQ_{−1} conditions also play roles during the 0–$tp$ time window. Among these events, a ZQ_{0}, $\nu r$-independent event originating from the quadrupole-driven rf/dipolar recoupling effect arises for *C*_{Q} ≠ 0, as predicted by Eq. (28). Thus, when $\Omega S/2\pi $ = $32$ kHz, a time point is found at *t* = 2.04 ms for a ZQ_{0} HH transfer, in addition to the time point found at 6.67 ms for the DQ_{1}—notice that only a single DQ_{1} time point appears in the 0–$tp$ time window at |$\Omega S/2\pi $| > 15 kHz. When $\Omega S/2\pi $ = $\u221232$ kHz, the ZQ_{0} condition found at *t* = 5.96 ms and the DQ_{1} time found at 1.33 ms are the mirror images of those arising when $\Omega S/2\pi $ = $32$ kHz. At a further increased offset frequency, $\Omega S/2\pi $ = $60$ kHz, both time points of the DQ_{1} condition disappear from the 0–*t*_{p} time window, but ZQ_{0} and DQ_{2} conditions arise. Additionally, a RR_{1} condition is found at 4.03 ms, at which a signal inversion occurs due to the presence of the first-order quadrupolar interaction. The phenomena observed at $\Omega S/2\pi $ = $60$ kHz and visible at 2.23 ms, 5.71 ms, and 3.97 ms for ZQ_{0}, DQ_{2}, and RR_{1} conditions, respectively, are then mirror imaged when $\Omega S/2\pi $ = $\u221260$ kHz.

For bigger $\Delta \nu $ sweeps exceeding the $\nu r$ value, for instance $\Delta \nu =$ 110 kHz (Fig. 1(b)), additional ZQ_{0}, DQ_{2}, RR_{1}, etc. conditions come into play within the 0–*t*_{p} time window, in addition to the main DQ_{1} events that dominated the CP events at a smaller $\Omega S/2\pi $. For instance, two ZQ_{0} time points that are mirror images of each other along *t*_{p}/2 are visible at 0.60 ms and 7.40 ms with $\Omega S/2\pi $ = 0. Time points of various other CP conditions arise when $\Omega S/2\pi =$ $\xb132$ kHz and $\xb160$ kHz for this $\Delta \nu =$ 110 kHz case, as compared to the previous $\Delta \nu =$ 60 kHz instance (Fig. 1(b)). Overall, the interaction between the various ZQ and DQ polarization transfer processes for these wide-sweep conditions—building magnetization with opposite spin temperatures and acting on a polarization that is being adiabatically swept over the course of the chirped pulse—is quite involved.

While the polarization transfers at time points that satisfy the DQ_{1} zero crossing at near on-resonance conditions are visible in the simulations, it is interesting to notice the differences arising if considering or disregarding the presence of sizable quadrupolar couplings. For the former case and $\Omega S$ = 0, DQ_{1}-driven transfers are clearly visible at the two symmetric zero-crossing points. For a *C*_{q} = 168 kHz, however, the latter DQ_{1}-driven transfer is attenuated. Notice that the sweep window width $\Delta \nu $ incorporated in Fig. 1(a) is matched to the MAS spinning rate $\nu r$ (= 60 kHz). Under this sweeping condition, a relatively small $\nu 1S$ (<12 kHz) provides a single, optimal DQ_{1}-driven transfer mode. Extending $\Delta \nu $ beyond $\nu r$ (Fig. 1(b)) provides multiple *S*_{z} inversions, creating the chance for undesirable RR inversion conditions due to the first-order quadrupolar interaction when $\Omega S\u2260$ 0. These cases are further analyzed in the Results section (vide infra).

### D. Analytical propagation of the spin ensemble throughout the frequency swept CPMAS transfer

The static-like Hamiltonian in Eq. (28) and the rotationally averaged Hamiltonians in Eq. (27) not only allow one to rationalize the behavior observed in numerical simulations but also to analytically propagate the spin density matrix throughout the whole course of the adiabatic frequency sweep. Starting from a $\rho (t=0)=Iz=IzDQ\u2212IzZQ$ state, the CP signals accrued along the spin locked *S*-channel at a time $tp$, given by *S*_{z} in the FM frame, is given as^{58}

with

where *ζ* stands for a DQ- or ZQ-coherence. As the $Iz\zeta $ and $I+\zeta $ terms in $\u27e8HT\zeta \u27e9av$ do not commute, Eq. (30) must, in principle, be evaluated numerically for an exact calculation. However, if the adiabaticity of the WURST pulse employed is sufficiently high and the $d\theta s(t)/dt\u226a1$ condition is satisfied, one can assume that perturbations in $\theta S(t)$ are negligible, and hence, the angle between the *S*-spin magnetization and the effective field remains constant. In such a case, an approximate average Hamiltonian over the whole mixing time, $\u27e8HT\zeta \u27e9av(t)\xaf=1tp\u222b0tp\u27e8HT\zeta \u27e9av(t)dt$ may be obtained, by integrating $Iz\zeta $ and $I+\zeta $ terms separately.^{30} In that case, the spins’ evolution operator, Eq. (30), can be approximated as

Eq. (31) represents a rotation in ${\zeta}$ subspace through an angle $\psi \zeta $ about an axis whose orientation is given with a polar angle set ($\theta \zeta ,\varphi \zeta $). Given these separate fictitious-spin-$12$ rotations, evaluation of Eq. (29) leads to

where

and

Under these simplifications, the behavior for a quadrupolar nucleus will be identical to that derived earlier for *S* = 1/2,^{34} except that the additional quadrupole-driven *k* = 0 ZQ-CP condition and the spin-dependent factors serve to scale the effects of the rf nutation rate.

## III. MATERIALS AND METHODS

### A. Numerical calculations

All the frequency-swept CPMAS simulations that are described consider Hamiltonians in their appropriate rotating frames. Their propagation was carried out utilizing in-house written Matlab$\xae$ (The Mathworks, Inc.) programs, where time evolutions were evaluated numerically by calculating the time propagation of density matrices using piecewise (2 $\mu s$) time increments. This temporal propagation accounted for variations in the rf waveforms’ amplitudes and phases (including the frequency sweeps), and for the MAS-driven rotational modulations of the quadrupolar, dipolar, and CS interactions. An isolated *S* = 1 (based on glycine’s ^{2}H parameters treated so far) site was considered for examining the inversion, adiabatic level-crossing, and RR effects that could result from the application of the frequency swept pulses. The actual CP process was examined by considering an isolated *I* = 1/2/*S* = 1 spin pair and a suitable MAS-modulated dipolar Hamiltonian (dipolar coupling constant = 8 kHz). Powder averaging calculations for all interactions were accounted for by considering 538 different crystal orientations subtending the ZCW Euler angle set,^{61} assuming for simplicity that all of the dipolar, quadrupolar, and CS tensors are axially symmetric and coincident along the directions of their components of largest absolute magnitude.

### B. Experimental

Conventional CPMAS and BRAIN-CPMAS experiments were carried out on powdered samples of glycine-2,2-d_{2} and L-tyrosine-(phenyl-3,5-d_{2})⋅HCl. Samples were purchased from Sigma-Aldrich (St. Louis, MO) and used without further treatment except for the L-tyrosine-(phenyl-3,5-d_{2}), which was dissolved in 1M hydrochloric acid and recrystallized by slow evaporation. About 3 mg of each sample was packed into a 1.3 mm Bruker rotor for obtaining MAS rates in the 40-65 kHz range. All experiments were carried out at room temperature in a 14.1 T magnet equipped with a Bruker Avance$\xae$ console operating at ^{1}H and ^{2}H frequencies of 600.92 and 92.1 MHz, respectively. The swept rf pulse shapes were constructed by utilizing the shaped pulse tool of the Bruker Topspin$\xae$ software. The WURST pulse employed in the BRAIN-CPMAS experiments utilized an amplitude-modulated profile $\nu 1s(1\u2212cos40[\pi t/\tau p])$ and a chirped frequency profile undergoing a linear sweep between 0 and $tp.$ 2000 data points were employed to digitize these WURST pulse shapes. Suitable frequency sweep windows $(\Delta \nu )$ and CP mixing times ($tp$) were sought over 40–400 kHz and 2–14 ms ranges, respectively. For $\nu r=$ 60 kHz, optimal BRAIN-CPMAS matching parameters were found for $\Delta \nu $ = $\nu r$ = 60 kHz and $tp$ = 8 ms. Optimal ^{2}H and ^{1}H rf power conditions were found experimentally by sweeping both rf channels independently. In accordance with Fig. 1, these parameters were usually found to satisfy the DQ_{1} condition $\nu eS=\nu r\u2212\nu 1H$ (where $\nu eS$ includes the instantaneous offset frequency coming from the adiabatic sweep); for instance, when $\nu r=$ 60 kHz, the optimal rf pulse powers were found at $\nu 1S=$ 5 $\xb1$ 15 kHz with $\nu 1I=$ 42 − 47 kHz. The performance of the BRAIN-CPMAS experiment was evaluated within a $\Omega s/2\pi $ = $\xb1$120 kHz^{2}H offset-frequency range. For comparison, conventional CPMAS NMR experiments were also carried out with independent rf pulse power optimizations, employing a ramped (90%-110%) spin-lock pulse^{62,63} along the ^{1}H channel while simultaneously applying a rectangular spin-lock pulse on the ^{2}H. The mixing time used in these CPMAS experiments ranged from 0.5 to 2 ms, also based on optimizations. All NMR spectra were acquired by co-adding 128 transients with recycle delays of ca. 4 s. The hard 90° ^{1}H and ^{2}H pulses were 2 $\mu s$ for all experiments. SPINAL-64^{64} proton decoupling was used during the acquisition period, with a 100 kHz decoupling rf field.

## IV. RESULTS

The objective of our experiments was to verify some of the basic derivations of Sec. II, as well as to explore BRAIN-CPMAS features arising upon enhancing ^{2}H NMR signals over a range of offsets at fast spinning conditions. Figure 2 shows the optimal ^{1}H and ^{2}H rf field strengthsdetermined experimentally for the BRAIN-CPMAS (a and b) and for ramped HH-CPMAS (c and d), using partially deuterated glycine as model. For the BRAIN-CPMAS case, an optimal ^{2}H rf field $\nu 1S$ was found at 8 ± 4 kHz (Fig. 2(b)), when $\nu 1H$ was set at 47 ± 5 kHz and $\Delta \nu $ = $\nu r=$ 60 kHz. This corresponds to the DQ_{1} condition; other modes visible in the rf pulse sweep profiles, such as DQ_{2} ($\nu 1H=$116 ± 5 kHz; $\nu 1S=$ 8 ± 4 kHz) and ZQ_{1} ($\nu 1H=$ 78 ± 5 kHz; $\nu 1S=$ 8 ± 4 kHz), are not as efficient as the DQ_{1} mode (Figs. 2(a) and 2(b)). For a ramped HH-CPMAS case incorporating a 90%-110% ramp on the ^{1}H channel, optimized ^{2}H and ^{1}H rf fields were found at the DQ_{1} ($\nu 1S$ = 11±4 kHz; $\nu 1H$ = 49 ± 5 kHz) and ZQ_{1} ($\nu 1S$ = 73 ± 4 kHz; $\nu 1H$ = 130 ± 5 kHz) conditions. The y-axes of the various plots are scaled relative to the largest absolute magnitude, which is normalized to one.

Figure 3 compares BRAIN-CPMAS, ramped HH-CPMAS, and direct-excitation ^{2}H spectra measured for glycine-2,2-*d*_{2}, which were acquired to evaluate the effectiveness of the various ^{1}H-^{2}H CPMAS methods. Spectra were acquired using MAS spinning rates of 40, 50, and 65 kHz; included for completion on the bottom panels are ideal ^{2}H MAS line shapes of a ^{2}H site simulated with *C*_{Q} = 168 kHz. Both BRAIN-CPMAS and ramped HH-CPMAS NMR spectra were acquired after optimizations of $\nu 1S$ and $\nu 1H$ for each spinning rate according to the strategy introduced in Figure 2. The number shown beside each spectrum represents the relative integrated intensity of all the sideband manifolds as compared to the DQ_{1} mode spectrum arising from HH-CPMAS whose centerband possesses the strongest intensity for every MAS rate and whose height has been used to normalize the intensities of all the remaining spectra. Notice that although both DQ_{1} and ZQ_{1} HH-CPMAS modes are included in this comparison, only the DQ_{1} BRAIN-CPMAS condition is presented, as this provides the highest intensity and best fidelity line shape, while employing the weakest ^{2}H rf pulse powers. Features to notice from these comparisons are as follows: (i) The centerband obtained from HH-CPMAS’ DQ_{1} mode is the highest at every $\nu r$, and although this does not hold when considering the overall spectral intensity if integrated among all sidebands plus centerband, its results resemble nearly isotropic spectra. (ii) The BRAIN-CPMAS DQ_{1} condition yields MAS sideband patterns at every spinning rate that are in excellent agreement with ideal simulations; the experimental conditions in each case feature relatively undemanding ^{1}H and ^{2}H rf fields (e.g., $\nu 1S$ = 8 ± 4 kHz and $\nu 1H$ = 47 ± 4 kHz for $\nu r=$ 60 kHz). (iii) Although the ZQ_{1} mode of HH-CPMAS method also provides an MAS sideband pattern that matches well with simulations, it does so at the expense highrf pulse power requirements for both ^{1}H and ^{2}H channels (e.g., $\nu 1S$ = 73 ± 4 kHz and $\nu 1H$ = 130 ± 5 kHz for $\nu r=$ 60 kHz).

Figure 4 shows the offset dependence of the BRAIN-CPMAS method, as explored by collecting a series of spectra under DQ_{1} matching conditions when $\Omega S$ = 0, as a function of the ^{2}H centerband’s offset vs. the width of the adiabatic sweep. Three different spinning rates ($\nu r$ = 40, 55, and 60 kHz) were utilized, while setting $\Delta \nu =\nu r$ in order to satisfy a “single MAS spinning-band sweep” case. Remarkably, for every $\nu r$, “modes” arise where peaks change their signs as the frequency is swept, with the width of each mode spanning ∼0.5$\nu r$, and the sign change influenced solely by the effective offset between the centerband and the center of the chirped pulse. All these offset dependencies are nearly symmetric with respect to the $\Omega S$ = 0 position. These periodic inversions cannot be entirely ascribed to RR phenomena, as no such effects are expected over the offset range $\u2212\nu r2\u2264\Omega S/2\pi $ ≤ $+\nu r2$ for the small values of $\nu 1S$, while the first inversions are observed at $|\Omega S/2\pi |$ ≈ $\nu r4$. The origin of these sign changes in the experimental offset-swept spectra can be heuristically understood by investigating the offset-swept profile of time-progressive ZQ- and DQ-CP matching conditions. For this purpose, Figure 5 illustrates the time-progressive HH matching conditions that are satisfied by the DQ_{±1}, DQ_{±2}, ZQ_{0}, ZQ_{±1}, and ZQ_{±2} BRAIN-CPMAS modes, during the course of an 8 ms long WURST-40 pulse. As derived in Eqs. (24)–(28), these conditions are given as $\nu 1I\u2212\nu eS(t)=0$ and $\nu 1I\u2213\nu eS(t)\xb1k\nu r=0$. Also shown in Fig. 5 are the positions where RR conditions will arise as a function of carrier offset $\Omega S/2\pi $ and of the contact time. Notice that for small offsets and for sweeps spanning a range $\u2264\Delta \nu $, two different lines for each CP mode are present due to the quadratic dependence of $veS(t)$ on the sweep’s offset—even if only one of these promoted a significant *I* → *S* polarization transfer, as discussed for Fig. 1. Figure 5(a) was derived assuming $\Delta \nu =\nu r=$ 60 kHz, $\nu 1S$ = 5 kHz, and $\nu 1H$ = 47 kHz; the experimental offset-swept spectrum shown in Fig. 4, collected employing the same experimental parameters, is placed on the top for purposes of comparing the frequency positions of the sign inversions. From this comparison between experiments and the various potential CP transfer modes, it appears that the changes in the *S*-spin polarization arising at $|\Omega s/2\pi |\u2248\nu r/4,3\nu r/4,$ and 5$\nu r/4$, coincide with a change in the mode that is affecting the transfer—from DQ to ZQ, and vice versa. Indeed, all spectral sign changes coincide with the starting or ending of one of these HH transfer modes. This is reasonable, as these modes will polarize *S*_{z} in opposite directions—cf. Eqs. (29)–(32). Furthermore, it appears that when multiple potential HH modes can become active over the course of the contact time for a particular $\Omega S$ offset value, one of them usually dominates the sign of the resulting *S*-magnetization. Moreover, much weaker effects are here observed upon traversing a RR condition than in the spin-1/2 BRAIN CPMAS case, apparently due to the higher efficiency of the shielding anisotropy to enable these processes.^{28}

Figure 5(b) illustrates a second example of this offset dependence, this time involving the more complicated case $\Delta \nu >\nu r,$ together with experimental BRAIN CPMAS data collected for $\nu r=$55 kHz, $\Delta \nu =$100 kHz, $\nu 1S$ = 5 kHz, and $\nu 1H$ = 47 kHz. As previously, the figure also indicates when the various HH matching modes are satisfied, as a function of the pulse’s sweep/contact time and as a function of the *S*-spin frequency offset $\Omega S$. The fact that the sweep now exceeds the spinning speed means that many more multiple modes can be active over the course of one CP contact period; still, as before, sign polarization inversions occur again at offsets intervals of ˜0.5$\nu r$, which coincide with the beginning or ending positions of the lines satisfying the DQ_{1} or DQ_{2} conditions.

Fig. 6 shows a practical application of this polarization enhancement approach, with a ^{1}H-^{2}H BRAIN-CPMAS spectrum of L-tyrosine-(phenyl-3,5-*d*_{2})⋅HCl collected at a spinning rate $\nu r$=50 kHz, compared against a directly acquired trace measured using a 90° pulse. The relative signal intensity gain provided by BRAIN-CPMAS is in this case ca. 11 times, much more than that achieved for glycine-2,2-*d*_{2} (ca. 2×). The reason for this improved CP performance is probably the larger number of rigid hydrogen atom positions in L-tyrosine-(phenyl-3,5-*d*_{2})⋅HCl than that in glycine-2,2-*d*_{2}. Glycine-2,2-*d*_{2}, in its zwitterionic form, only has its three mobile ammonium protons available for transferring their polarization to the deuterons. By contrast, there are 5 rigid hydrogen atoms bonded to carbons which produce much stronger dipolar couplings with deuterons in L-tyrosine-(phenyl-3,5-*d*_{2})⋅HCl, in addition to mobile ammonium protons. Therefore, it is expected that this form of CPMAS will be of great value in studies of biomolecules and materials featuring unreceptive nuclides like ^{2}H.

## V. DISCUSSION AND CONCLUSIONS

Classical fixed-frequency CPMAS ramped schemes produce good enhancements in solid-state NMR spectra of spin-1 nuclei like ^{2}H, provided that sufficiently strong rf-pulse powers of both ^{1}H and ^{2}H channels are applied. This enables one to bypass MAS-derived level crossing complications, but at the expense of demanding experimental conditions with the potential to damage probe hardware, and creating distortions in the MAS spinning sideband intensities away from their ideal line shapes, which make the accurate determination of quadrupolar and CSA parameters difficult. Herein, we introduce the use of BRAIN-CPMAS to deal with these issues in ^{1}H–^{2}H NMR experiments; the result is a robust method, which is easy to setup, utilizes low rf powers, and delivers undistorted MAS NMR line shapes with high gains in signal intensity compared to conventional methods. This latter feature probably reflects the larger range of coupling values with which a frequency-swept pulse can successfully fulfill the HH conditions for multiple crystallites in a powder, over a square-wave counterpart.

Despite these advantages, limitations and complications also arise in these experiments. One concern is the complex polarization behavior shown in Figs. 4 and 5, which involves multiple sign changes of the peaks as a function of carrier offset frequency, $\Omega S$. From a practical standpoint, however, these are not too relevant, given the relatively narrow chemical shift range and γ-values characterizing ^{2}H. These considerations may become more problematic if considering CP to other integer spin nuclides such as ^{14}N (*S* = 1) or ^{10}B (*S* = 3). Based on our repeated experiments, it appears that the optimal sweep width, $\Delta \nu ,$ for the WURST pulse scheme adopted in the BRAIN-CPMAS sequence is equal to the MAS spinning rate $\nu r$. Under such conditions, the BRAIN-CPMAS contact conditions become robust and are satisfied regardless of the values of the quadrupolar coupling constants. However, we have observed few competitive advantages of the ^{1}H–^{2}H BRAIN-CPMAS method over its conventional ramped HH-CPMAS method, when sample spinning happens at lower (e.g., 10-15 kHz) MAS rates.

Besides its practical utility, the occurrence of multiple concurrent time-dependent processes makes the spin physics of BRAIN-CPMAS remarkably interesting. Multiple new processes absent in static counterparts arise, including simultaneous ZQ and DQ HH transfer modes, quadrupolar-based rotary-resonance phenomena, and a static-like ZQ_{0} driven by a periodic modulation of *H*_{Q} and a concomitant quadrupole-modulated rf-dipolar recoupling effect. Other potential phenomena, like laboratory frame quadrupolar-dipolar cross terms arising due from perturbation of the Zeeman quantization axes by the quadrupolar interaction, were found to be negligible within the present context (see the supplementary material). It is thus intriguing, both in terms of potential applications and the new underlying physics, what kind of phenomena will arise if such cross-polarization schemes are extended to quadrupolar nuclei with higher nuclear spin numbers. Such investigations are under way.

## SUPPLEMENTARY MATERIAL

See supplementary material for (A) the effects of adiabatic pulses on the ^{2}H MAS NMR of rotating powders, (B) the origin of the static-like ZQ_{0} CP conditions formed by the quadrupole-driven rf-dipolar recoupling interaction, and (C) the effects of laboratory-frame quadrupole-dipole cross-coupling effects on the CP of the spin-1 species.

## ACKNOWLEDGMENTS

We are grateful to Dr. Zhehong Gan (National High Magnetic Field Laboratory, NHMFL) for insightful discussions. This work was supported by the NHMFL through the National Science Foundation Cooperative Agreement (No. DMR-0084173) and by the State of Florida. L.F. acknowledges support from the Israel Science Foundation Grant No. 795/13, the Kimmel Institute for Magnetic Resonance (Weizmann Institute), and the generosity of the Perlman Family Foundation. R.W.S. thanks NSERC for funding this research in the form of a Discovery Grant and Discovery Accelerator Supplement, and is also grateful for an Early Researcher Award from the Ontario Ministry of Research and Innovation, and for a 50th Anniversary Golden Jubilee Chair from the University of Windsor.