¹H–²H cross-polarization NMR in fast spinning solids by adiabatic sweeps

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-polarization⁷ 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(¹H)-S(²H)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 ¹H–²H wideband MAS spectra.^19,49–52 To this end, Wideband Uniform-Rate Smooth-Truncation (WURST)-shaped²⁸ rf adiabatic inversion pulses were applied on the ²H channel, while spin-locking rf pulses were applied on the ¹H channel under sample spinning conditions. Due to the frequency sweep on ²H 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 ²H line shapes expected under ultrafast spinning rates.

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 $ω1I=γIB1I$ represents an on-resonance irradiation of the I-spin, $ω1S=γSB1S,$ and $ΩS′=ΩS−ωp(t)$ is an effective offset frequency of the S spin as seen in the FM frame. In this frame, $ΩS=ω0−ωrf$ describes the center offset of the sweep and $ωp(t)=(Δω/tp)t−Δω/2$ is the rf’s instantaneous offset frequency, where $Δω=2πΔν$ 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) = $(1−cosn[πt/tp])$ (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 as¹⁵ 

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 $βd$ and $γd$ Euler angles. The first-order quadrupolar interaction, $HQ(1)(t)$⁠, is given under MAS by⁵³ 

where $CQ=e2qQℏ$ 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, $⟨i(dUQ−1/dt)UQ⟩γ,$ 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 $α=ω1S2CQωr$⁠, which specifies the extent of adiabaticity in the level-crossings.²² When $α=ω1S2CQωr≪1$⁠, the system is said to be in the sudden passage regime; under these conditions, the rf does not promote population redistributions among the $|1⟩,|0⟩$⁠, and $|−1⟩$ eigenstates during these levels’ crossings. As discussed within the context of Fig. S1 of the supplementary material, $ν1S$ in the 5–16 kHz range fulfills this sudden-passage regime for prototypical ²H sites (e.g., C_Q = 168 kHz) at high (⁠$νr$ ≥ 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 $|1⟩↔|0⟩$ and $|0⟩↔|−1⟩$ 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 $A(t)=(1−cos40[πt/tp])$ ≈ 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 $θI$ and $θS(t)$ that relate the z′-axes to the z-axes in the FM frame are given by

where the last relation uses the fact that $sin(θS[t])$ 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 $H∼T′(t),$ Eq. (11) is reduced to^14,15,17

The product of the $ωrt$-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 $Λk(k=0,±1,and±2)$ 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 $(β=45°,γ=0°)$ of a dipolar coupled ¹H–²H pair collinear with the quadrupolar tensor possessing a ¹H–²H dipolar coupling of 4 kHz and a C_Q = 168 kHz, the magnitude of the quadrupole-driven rf-dipolar recoupled term $Λ0$ is about 680 Hz. This is a significant coupling strength, which will generate a static-like, k = 0, ZQ₀ 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, $ωΔ(t)=ω1I−ωeS(t),ωΣ(t)=ω1I+ωeS(t)$⁠, 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

The oscillating terms associated with $Λ±1$ and $Λ±2$ 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 $ωΔ,Σ(t0)$ over any given rotor period. This assumption is valid as the rotor periods to be considered (15-30 $μs$ range) will typically be orders-of-magnitude shorter than the adiabatic pulse duration (⁠$tp=$ 5-10 ms). Approximating the $ωeS(t)$ as quasi-constant for time intervals $t0≤t≤t0+2πωr$⁠, 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 $cos2(∫02πωr3CQωQ(t′)dt′)$ term contributing to $ωeS(t0)$ 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 $⟨HTζ⟩av=1τr∫0τrHTζ(t′)dt′,ζ=$ ZQ or DQ can be obtained from Eqs. (24) and (25) via the transformation $UpHTζUp−1−iUpddtUp−1,$ where $Up=exp(ikωrtIzζ)$ and $ζ=$ DQ or ZQ refers to the zero- or double-quantum sub-spaces. This evaluation produces

where $k=±1$ and $±2,$$I+ζ=Ixζ+iIyζ$⁠, 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 $k=±1,±2$ terms are as usual in CPMAS experiments involving polarization transfer to spin-1/2 nuclides, the static-like CPMAS transfer process involving the $Λ0$ 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 $ω1I=ωeS(t),$ 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), $ωΣ(t),$ for any combination of $ω1I$ and $ωeS(t).$

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 $ZQ±k$ (k = 0, 1, or 2) HH matching condition will therefore be met when the instantaneous frequency offset, $ωp(t)$⁠, introduced by the adiabatic sweep, satisfies $νeI−νeS(t)=±kνr.$ In a similar manner, a DQ_k (k = 1 or 2) HH matching condition will be met when $ωp(t)$ instantaneously satisfies $ν1H+νeS(t)=kνr$⁠. 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 $νeS(t)=kνr$⁠.^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 ²H 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 ¹H–²H BRAIN-CPMAS deriving from these considerations. These calculations incorporate different offset frequencies $ΩS/2π$ (0, $±30,$ and $±60$ kHz), null and non-null quadrupole couplings, and varying swept offsets $Δν$ (60 kHz and 110 kHz), assuming a MAS rate of $νr$ = 60 kHz. Also illustrated in the figures are the fulfillment of DQ_±k (k = 1, 2), $ZQ±k$ (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 $±ΩS/2π$ are clear mirror images to each other along t = t_p/2. Moreover, when the offset $ΩS/2π$ = 0, the two time points that satisfy the same DQ₁ 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 $ΩS/2π$ offset becomes negative, and to later times during the adiabatic sweep as $ΩS/2π$ increases. Still, it is noteworthy how the introduction of C_Q ≠ 0 breaks the symmetry of the aforementioned transfer efficiencies of the two DQ₁ 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 ²H nuclei with small CSAs. While this RR phenomenon does not play an important role if choosing sweeps $Δν≤νr$ and/or dealing with small $ΩS/2π$ offsets, it becomes visible when this is not the case. The second phenomenon concerns the appearance of new, ZQ₀-derived HH transfers.

DQ₁ effects dominate the HH transfers under conditions of small $ΩS/2π$ offsets. For $Δν=νr$ (Fig. 1(a)), only the DQ₁ time points satisfy a condition for CP polarization transfer at a small $ΩS/2π(≤±νr$/4). For instance, at $ΩS/2π$ = 0, the DQ₁ time points are visible in Fig. 1(a) at t₁ = 2.27 ms and t₂ = 5.73 ms (⁠$tp=$ 8 ms). As $ΩS/2π$ increases, the ZQ₀, DQ₂, RR₁, and ZQ₋₁ conditions also play roles during the 0–$tp$ time window. Among these events, a ZQ₀, $νr$-independent event originating from the quadrupole-driven rf/dipolar recoupling effect arises for C_Q ≠ 0, as predicted by Eq. (28). Thus, when $ΩS/2π$ = $32$ kHz, a time point is found at t = 2.04 ms for a ZQ₀ HH transfer, in addition to the time point found at 6.67 ms for the DQ₁—notice that only a single DQ₁ time point appears in the 0–$tp$ time window at |$ΩS/2π$| > 15 kHz. When $ΩS/2π$ = $−32$ kHz, the ZQ₀ condition found at t = 5.96 ms and the DQ₁ time found at 1.33 ms are the mirror images of those arising when $ΩS/2π$ = $32$ kHz. At a further increased offset frequency, $ΩS/2π$ = $60$ kHz, both time points of the DQ₁ condition disappear from the 0–t_p time window, but ZQ₀ and DQ₂ conditions arise. Additionally, a RR₁ 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 $ΩS/2π$ = $60$ kHz and visible at 2.23 ms, 5.71 ms, and 3.97 ms for ZQ₀, DQ₂, and RR₁ conditions, respectively, are then mirror imaged when $ΩS/2π$ = $−60$ kHz.

For bigger $Δν$ sweeps exceeding the $νr$ value, for instance $Δν=$ 110 kHz (Fig. 1(b)), additional ZQ₀, DQ₂, RR₁, etc. conditions come into play within the 0–t_p time window, in addition to the main DQ₁ events that dominated the CP events at a smaller $ΩS/2π$⁠. For instance, two ZQ₀ time points that are mirror images of each other along t_p/2 are visible at 0.60 ms and 7.40 ms with $ΩS/2π$ = 0. Time points of various other CP conditions arise when $ΩS/2π=$ $±32$ kHz and $±60$ kHz for this $Δν=$ 110 kHz case, as compared to the previous $Δν=$ 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 DQ₁ 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 $ΩS$ = 0, DQ₁-driven transfers are clearly visible at the two symmetric zero-crossing points. For a C_q = 168 kHz, however, the latter DQ₁-driven transfer is attenuated. Notice that the sweep window width $Δν$ incorporated in Fig. 1(a) is matched to the MAS spinning rate $νr$ (= 60 kHz). Under this sweeping condition, a relatively small $ν1S$ (<12 kHz) provides a single, optimal DQ₁-driven transfer mode. Extending $Δν$ beyond $νr$ (Fig. 1(b)) provides multiple S_z inversions, creating the chance for undesirable RR inversion conditions due to the first-order quadrupolar interaction when $ΩS≠$ 0. These cases are further analyzed in the Results section (vide infra).

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 $ρ(t=0)=Iz=IzDQ−IzZQ$ 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 as⁵⁸ 

with

where ζ stands for a DQ- or ZQ-coherence. As the $Izζ$ and $I+ζ$ terms in $⟨HTζ⟩av$ 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 $dθs(t)/dt≪1$ condition is satisfied, one can assume that perturbations in $θS(t)$ 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, $⟨HTζ⟩av(t)¯=1tp∫0tp⟨HTζ⟩av(t)dt$ may be obtained, by integrating $Izζ$ and $I+ζ$ terms separately.³⁰ In that case, the spins’ evolution operator, Eq. (30), can be approximated as

Eq. (31) represents a rotation in ${ζ}$ subspace through an angle $ψζ$ about an axis whose orientation is given with a polar angle set (⁠$θζ,ϕζ$⁠). 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.

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$®$ (The Mathworks, Inc.) programs, where time evolutions were evaluated numerically by calculating the time propagation of density matrices using piecewise (2 $μs$⁠) 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 ²H 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.

Conventional CPMAS and BRAIN-CPMAS experiments were carried out on powdered samples of glycine-2,2-d₂ and L-tyrosine-(phenyl-3,5-d₂)⋅HCl. Samples were purchased from Sigma-Aldrich (St. Louis, MO) and used without further treatment except for the L-tyrosine-(phenyl-3,5-d₂), 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$®$ console operating at ¹H and ²H 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$®$ software. The WURST pulse employed in the BRAIN-CPMAS experiments utilized an amplitude-modulated profile $ν1s(1−cos40[πt/τp])$ 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 $(Δν)$ and CP mixing times (⁠$tp$⁠) were sought over 40–400 kHz and 2–14 ms ranges, respectively. For $νr=$ 60 kHz, optimal BRAIN-CPMAS matching parameters were found for $Δν$ = $νr$ = 60 kHz and $tp$ = 8 ms. Optimal ²H and ¹H 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 DQ₁ condition $νeS=νr−ν1H$ (where $νeS$ includes the instantaneous offset frequency coming from the adiabatic sweep); for instance, when $νr=$ 60 kHz, the optimal rf pulse powers were found at $ν1S=$ 5 $±$ 15 kHz with $ν1I=$ 42 − 47 kHz. The performance of the BRAIN-CPMAS experiment was evaluated within a $Ωs/2π$ = $±$120 kHz²H 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 ¹H channel while simultaneously applying a rectangular spin-lock pulse on the ²H. 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° ¹H and ²H pulses were 2 $μs$ for all experiments. SPINAL-64⁶⁴ proton decoupling was used during the acquisition period, with a 100 kHz decoupling rf field.

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 ²H NMR signals over a range of offsets at fast spinning conditions. Figure 2 shows the optimal ¹H and ²H 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 ²H rf field $ν1S$ was found at 8 ± 4 kHz (Fig. 2(b)), when $ν1H$ was set at 47 ± 5 kHz and $Δν$ = $νr=$ 60 kHz. This corresponds to the DQ₁ condition; other modes visible in the rf pulse sweep profiles, such as DQ₂ (⁠$ν1H=$116 ± 5 kHz; $ν1S=$ 8 ± 4 kHz) and ZQ₁ (⁠$ν1H=$ 78 ± 5 kHz; $ν1S=$ 8 ± 4 kHz), are not as efficient as the DQ₁ mode (Figs. 2(a) and 2(b)). For a ramped HH-CPMAS case incorporating a 90%-110% ramp on the ¹H channel, optimized ²H and ¹H rf fields were found at the DQ₁ (⁠$ν1S$ = 11±4 kHz; $ν1H$ = 49 ± 5 kHz) and ZQ₁ (⁠$ν1S$ = 73 ± 4 kHz; $ν1H$ = 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 ²H spectra measured for glycine-2,2-d₂, which were acquired to evaluate the effectiveness of the various ¹H-²H CPMAS methods. Spectra were acquired using MAS spinning rates of 40, 50, and 65 kHz; included for completion on the bottom panels are ideal ²H MAS line shapes of a ²H site simulated with C_Q = 168 kHz. Both BRAIN-CPMAS and ramped HH-CPMAS NMR spectra were acquired after optimizations of $ν1S$ and $ν1H$ 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 DQ₁ 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 DQ₁ and ZQ₁ HH-CPMAS modes are included in this comparison, only the DQ₁ BRAIN-CPMAS condition is presented, as this provides the highest intensity and best fidelity line shape, while employing the weakest ²H rf pulse powers. Features to notice from these comparisons are as follows: (i) The centerband obtained from HH-CPMAS’ DQ₁ mode is the highest at every $νr$⁠, 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 DQ₁ 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 ¹H and ²H rf fields (e.g., $ν1S$ = 8 ± 4 kHz and $ν1H$ = 47 ± 4 kHz for $νr=$ 60 kHz). (iii) Although the ZQ₁ 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 ¹H and ²H channels (e.g., $ν1S$ = 73 ± 4 kHz and $ν1H$ = 130 ± 5 kHz for $νr=$ 60 kHz).

Figure 4 shows the offset dependence of the BRAIN-CPMAS method, as explored by collecting a series of spectra under DQ₁ matching conditions when $ΩS$ = 0, as a function of the ²H centerband’s offset vs. the width of the adiabatic sweep. Three different spinning rates (⁠$νr$ = 40, 55, and 60 kHz) were utilized, while setting $Δν=νr$ in order to satisfy a “single MAS spinning-band sweep” case. Remarkably, for every $νr$⁠, “modes” arise where peaks change their signs as the frequency is swept, with the width of each mode spanning ∼0.5$νr$⁠, 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 $ΩS$ = 0 position. These periodic inversions cannot be entirely ascribed to RR phenomena, as no such effects are expected over the offset range $−νr2≤ΩS/2π$ ≤ $+νr2$ for the small values of $ν1S$⁠, while the first inversions are observed at $|ΩS/2π|$ ≈ $νr4$⁠. 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, ZQ₀, 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 $ν1I−νeS(t)=0$ and $ν1I∓νeS(t)±kνr=0$⁠. Also shown in Fig. 5 are the positions where RR conditions will arise as a function of carrier offset $ΩS/2π$ and of the contact time. Notice that for small offsets and for sweeps spanning a range $≤Δν$⁠, 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 $Δν=νr=$ 60 kHz, $ν1S$ = 5 kHz, and $ν1H$ = 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 $|Ωs/2π|≈νr/4,3νr/4,$ and 5$νr/4$⁠, 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 $ΩS$ 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 $Δν>νr,$ together with experimental BRAIN CPMAS data collected for $νr=$55 kHz, $Δν=$100 kHz, $ν1S$ = 5 kHz, and $ν1H$ = 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 $ΩS$⁠. 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$νr$⁠, which coincide with the beginning or ending positions of the lines satisfying the DQ₁ or DQ₂ conditions.

Fig. 6 shows a practical application of this polarization enhancement approach, with a ¹H-²H BRAIN-CPMAS spectrum of L-tyrosine-(phenyl-3,5-d₂)⋅HCl collected at a spinning rate $νr$=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-d₂ (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-d₂)⋅HCl than that in glycine-2,2-d₂. Glycine-2,2-d₂, 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-d₂)⋅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 ²H.