THE JOURNAL OF CHEMICAL PHYSICS 142, 064201 (2015)

Cross-polarization phenomena in the NMR of fast spinning solids subject to adiabatic sweeps Sungsool Wi,1,a) Zhehong Gan,1 Robert Schurko,2 and Lucio Frydman1,3,a)

1

National High Magnetic Field Laboratory, Tallahassee, Florida 32304, USA Department of Chemistry and Biochemistry, University of Windsor, 401 Sunset Avenue, Windsor N9B 3P4, Ontario, Canada 3 Department of Chemical Physics, Weizmann Institute of Sciences, 76100 Rehovot, Israel 2

(Received 24 November 2014; accepted 15 January 2015; published online 9 February 2015) Cross-polarization magic-angle spinning (CPMAS) experiments employing frequency-swept pulses are explored within the context of obtaining broadband signal enhancements for rare spin S = 1/2 nuclei at very high magnetic fields. These experiments employ adiabatic inversion pulses on the S-channel (13C) to cover a wide frequency offset range, while simultaneously applying conventional spin-locking pulse on the I-channel (1H). Conditions are explored where the adiabatic frequency sweep width, ∆ν, is changed from selectively irradiating a single magic-angle-spinning (MAS) spinning centerband or sideband, to sweeping over multiple sidebands. A number of new physical features emerge upon assessing the swept-CP method under these conditions, including multiple zero- and double-quantum CP transfers happening in unison with MAS-driven rotary resonance phenomena. These were examined using an average Hamiltonian theory specifically designed to tackle these experiments, with extensive numerical simulations, and with experiments on model compounds. Ultrawide CP profiles spanning frequency ranges of nearly 6 · γB1s were predicted and observed utilizing this new approach. Potential extensions and applications of this extremely broadband transfer conditions are briefly discussed. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4907206]

I. INTRODUCTION

Frequency-swept radiofrequency (rf) pulses have long been used in nuclear magnetic resonance (NMR) spectroscopy and imaging (MRI) for achieving uniform inversion or excitation profiles over a wide range of frequency offsets.1–13 Reasons driving the use of these schemes include decreasing the sensitivity to B1-inhomogeneity effects and achieving a more efficient coverage of wide bandwidths when faced with limited peak powers. Frequency sweeps are usually achieved by modulating the rf phase with time and are often implemented in an “adiabatic” mode, where the angle between the spin magnetization and the effective field remains constant. Such adiabatic pulse schemes have been utilized in solid-state NMR studies of nuclei possessing broad anisotropic frequency spans; adiabatic CPMG-like pulse schemes utilizing WURSTshaped sweeps have proven particularly useful for obtaining wideline NMR spectra of both spin-1/2 and quadrupolar nuclei.14–19 Based on these features, a technique dubbed broadband adiabatic-inversion cross-polarization (BRAINCP)20 has recently been developed. BRAIN-CP can cover a broad range of Hartman-Hahn (HH) matching conditions,21,22 and deliver superior wide-line spectra of rare nuclei under static conditions. Lying at the core of BRAIN-CP is an adiabatic inversion pulse, which replaces the monochromatic spinlock pulse of the rare spin S-channel in conventional CP. This leads to sequential fulfillment of the Hartman-Hahn a)Authors to whom correspondence should be addressed. Electronic ad-

dresses: [email protected] and [email protected] 0021-9606/2015/142(6)/064201/16/$30.00

matching conditions, as the effective field strength of the RF sweeping on the rare S-nuclei, matches the spin-lock field of the polarizing I-spin. BRAIN-CP can also be combined with WURST-based CPMG schemes for further improving the sensitivity; broadband CP NMR spectra of static powders, including 119Sn, 195Pt, 199Hg, 39K, 14N, and 35Cl species, were thus demonstrated.20,23–25 Under moderate field strengths, conventional CP utilizing rectangular26,27 or ramped28 pulses fulfills most of the needs for obtaining spectra from common spin-1/2 nuclei like 13 C, 15N, or 31P subject to magic-angle-spinning (MAS).29–32 However, the advent of new magnet technologies expected to yield platforms operating at fields ≥25 T,33,34 call for the development of CP methods that cover broader frequency bandwidths under fast MAS rate and reasonable rf power levels. For instance, the ∼250 ppm range characteristic of 13C chemical shifts will span in excess of 95 kHz at the 36 T magnet currently being built at the National High Magnetic Field Laboratory (NHMFL) in Tallahassee, Florida.34 This highlights the need to develop efficient, broadband CP techniques suitable for covering ultrawide spectral windows and accommodating high MAS spinning rates also for these spin1/2 species. The broadbandness of the BRAIN-CP method makes this a natural candidate to be explored. As will be discussed below, however, many of the simplifying assumptions underlying BRAIN-CP are no longer applicable under MAS, where interferences arise between the adiabatic frequency sweep, the spin-lock pulse, and the sample spinning modulation. Herein, an analysis of these effects is presented; in particular, the interferences arising between BRAIN-CPMAS

142, 064201-1

© 2015 AIP Publishing LLC

064201-2

Wi et al.

J. Chem. Phys. 142, 064201 (2015)

and a number of rotary resonance (RR) conditions are discussed in detail. New potential forms for eliminating or avoiding these interferences were explored both numerically and theoretically with average Hamiltonian analyses, and experimentally corroborated by 13C solid-state NMR experiments on model amino acid compounds. Overall, this development can deliver proton-enhanced, distortion-free spectra under ultrafast spinning rates, over a frequency-offset window that exceeds the low-γ rf field γB1, by nearly a factor of 10. II. THEORETICAL BACKGROUND A. Adiabatic inversions in static and spinning powders

An essential component of this study is the adiabatic inversion pulse. Although the detailed properties and working mechanisms of adiabatic inversion pulses have been amply described,1,4,6–8 we briefly summarize their properties using a WURST pulse as a prototype that is applied on an anisotropic spin-1/2 nucleus under static and spinning conditions. This pulse has an amplitude- and phase-modulated waveform that sweeps a range of frequencies linearly. In the usual rotating frame, its associated Hamiltonian will be given by1
Hrfs = −ω1s · A (t) · Sx cos ψ [t] + S y sin ψ [t] . (1) Here, ω1s = 2πν1s = −γs B1s is the rf-field maximum amplitude, A(t) is the
rf’s amplitude envelope that we assume as 1 − cos40 πt/t p (although other even powers ofthis WURST   modulation or odd powers with absolute value cosn πt/t p would be equally valid), Sx and Sy are the transverse spin angular momentum operators, and the time-dependent phase

 ωeS = 2πνeS =

Ωs + ωcsa −

profile ψ (t) is given by ψ (t) =

(2)

where ψ0, ∆ω = 2π∆ν, and t p are an arbitrary initial phase, the bandwidth of the adiabatic frequency sweep, and the duration of WURST pulse, respectively. The inversion properties of a frequency-swept adiabatic pulse can be conveniently viewed in the frequency-modulated (FM) frame that is rotating synchronously with ψ (t).1,4,8 The spin Hamiltonian H(t) defined in Eq. (1) can be transformed in this FM frame into a Hamiltonian H ′ (t) given by U H (t) U −1 d U −1, where U = exp [iΨ (t) Sz]. Thus, if a S-spin that + iU dt precesses at a Larmor frequency (+ isotropic chemical shift) ω0 = 2πν0 is irradiated by an adiabatic pulse, the resultant FM Hamiltonian is    
dΨ (t) Sz − ω1s 1 − cos40 πt/t p Sx , H ′(t) = − Ωs + ωcsa − dt (3) where Ωs = ω0 − ωrf is the center offset of the sweep, ωcsa is the chemical shift anisotropy (CSA), and dψ (t) /dt is the RF’s instantaneous offset frequency given in the FM frame as dψ (t) ∆ω ∆ω = ω p (t) = . t− dt tp 2

(4)

The non-secular, Coriolis term in Eq. (3) can be safely disregarded. It follows that a suitable value of ω1S then leads to the inversion of an initial state Sz, via a Sz → +Sx → −Sz trajectory along the perimeter of a semicircle drawn on the Sz − Sx plane of the FM frame. The effective frequency of nutation during this process is

∆ω ∆ω t+ tp 2

Finally, the effects of MAS can be accounted for in Eq. (5) by imparting onto the CSA a time dependence ωcsa  = 2k=−2,k,0 ak eikω r t Sz, where the {ak (δcsa, η)} coefficients have the usual dependence on the Euler angle set transforming the tensor parameters δcsa, η defined in the CSA principal axes system, into the rotor frame.35 Given the emphasis of this work on spin-1/2 nuclides at high fields, we consider the inversion properties of such swept pulse when applied on a site S subject to a CSA under fast MAS. Shown in Fig. 1 are the simulated inversion profiles of the Sz operator, arising for on-resonance sweep conditions for such isolated site as a function of different ∆ν values upon spinning at a MAS rate of νr = 60 kHz. The inversion behavior evidenced is unusual even for narrowband ∆ν < νr sweeps, in the sense that Sz inverts not only around the centerband (CB) frequency position being swept but also around sideband frequency positions separated by

∆ω 2 ∆ω t − t + ψ 0, 2t p 2

2  
2 + ω1S 1 − cos40 πt/t p .

(5)

±νr , ±2νr , etc. The inversion frequency width of each frequency band is identical to the sweep bandwidth as long as the centerband sweep and the pseudo-sweeps at the sidebands do not overlap—even if a complete inversion profile is obtained only for the centerband. A frequency span of maximum inversion is, therefore, obtained for ∆ν = νr (Fig. 1(b)). If νr < ∆ν < 2νr , an inverted region from the centerband will overlap with those of the neighboring sidebands, resulting in regions of double (i.e., no net) inversion due to the destructive interference among adjacent inversion band’s (Figs. 1(c) and 1(d)). When ∆ν ≥ 2νr (Figs. 1(e) and 1(f)), sweeps at sidebands positioned at multiple nνr harmonics (n = 0, ±1, ±2, . . .) will interfere with one another, and eventually the possibility of fully inverting Sz is lost throughout the entire frequency range. This ⟨Sz⟩ behavior is also reflected onto the spectra that can be collected after the application of adiabatically swept

064201-3

Wi et al.

J. Chem. Phys. 142, 064201 (2015)

FIG. 1. Sz inversion profiles calculated upon varying the span ∆ν of a WURST pulse for an anisotropic site under a MAS rate ν r = 60 kHz. An isolated ensemble of spin-1/2 nuclei (δ csa = 400 ppm, η = 1, ν 0 = 150 MHz) irradiated at on-resonance by a rf field of ν 1S = 20 kHz, was considered. ∆ν was varied as 30 (a), 60 (b), 80 (c), 110 (d), 200 (e), and 300 kHz (f), always for a t p = 10 ms sweep pulse length. Dashed vertical lines indicate the positions of the site’s CB and spinning sidebands. The red central double arrow (∆ν, CB) refers to the actual extent of the sweeps; horizontal blue arrows are virtual replicas centered at off-resonance positions separated by the spinning rate. Notice the differences arising when ∆ν is chosen to be smaller than νr; the shape and width of the inversion profile are identical to the isotropic Sz state expectation within the center-sweep region ((a) and (b)); the maximum width of the inversion profile is achieved when ∆ν = ν r (b). When ν r > ∆ν > 2ν r , interferences occur between the center and sideband inversion profiles, resulting in the decrease of the successfully inverted central region ((c) and (d)). When ∆ν ≫ 2ν r , the frequency regions of interference exceed the |2ν r | window, resulting in the disappearance of Sz’s central inversion profile ((e) and (f)).

pulses. Figure 2 shows such spectral simulations, assuming that after the adiabatic sweeps spectra were read by application of ideal 90◦ “read” pulses of infinite bandwidth. Under static conditions, the inversion frequency ranges that can be expected for on- or off-resonance adiabatic sweeps, reflect oneto-one correspondence with the frequency ranges that were swept (see Figs. 2(a) and 2(b) for spectra of a spin-1/2 nuclide with δcsa = 60 kHz). For the static cases, inverted spectral lineshapes are obtained for whichever ∆ν region is chosen: selective inversions for ∆ν < δcsa, and full powder inversions

if ∆ν ≥ δcsa. Upon subjecting the same powder to spinning (νr = 30 kHz), however, the response is different. When either the centerband (Fig. 2(c)) or a single spinning-sideband (Fig. 2(d)) is selectively irradiated with a narrow ∆ν value (∆ν < 2νr ), completely inverted, distortion-free MAS sideband manifolds are obtained even if ∆ν < δcsa. This is in agreement with previously reported results.14–16,36 Irradiation at the centerband behaves slightly more ideal than the sideband-irradiation case even if ∆ν > 2νr ; for instance, a reasonable inversion is obtained for the centerband-irradiation case in Fig. 2(c) even with

064201-4

Wi et al.

J. Chem. Phys. 142, 064201 (2015)

FIG. 2. Inverted powder spectra calculated for an anisotropic site of spin-1/2 nuclei (δ csa = 400 ppm; η = 1; ν 0 = 150 MHz) upon varying the ∆ν of the swept pulse under static ((a) and (b)) and MAS ((c) and (d)) conditions (ν r = 30 kHz)—in all cases followed by an ideal 90◦ “read” pulse. Adiabatic pulses were applied on-resonance ((a) and (c)) and +30 kHz off-resonance ((b) and (d)). The ∆ν values used are shown on the left and indicated by red horizontal lines. All pulses were swept for t p = 10 ms with ν 1S = 20 kHz. Following the hard read pulse, the free evolutions were digitized for 8.192 ms (data points: 4096; dwell time: 2 µs) and Fourier transformed to yield the spectra. The spectrum shown on the bottom of each column was computed without inversion.

∆ν = 100 kHz. Distortions clearly result when a moderate ∆ν value is used at a sideband irradiation at Ωs = +νr . Eventually, phase-distorted MAS spectra result when sufficiently large ∆ν values are employed regardless of centering the irradiation at zero or at νr , because of the multiple interferences that occur between the MAS sideband modulations and the frequency sweep of the adiabatic pulse. B. BRAIN cross polarization under MAS: Basic features

The capability demonstrated in Fig. 2 of obtaining distortion-free, broadband spectral inversion regions under MAS by incorporating an adiabatic pulse with a relatively narrowband (∆ν < 2νr ) swept irradiation, suggests that this low-power route could serve for developing a broadband version of the CPMAS NMR experiment. Shown in Fig. 3(a) is the prototype pulse sequence employed for exploring this. In this BRAIN CPMAS sequence, I-spin transverse magnetization is spinlocked during a time t p , while simultaneously irradiating the S-channel with a WURST pulse over this CP mixing time. The expectation is that, as in the static BRAIN case, any Smagnetization accrued over the course of this CP period will follow the adiabatic inversion that the swept pulse creates, ending up aligned along the Sz axis at the completion of the mixing time. A 90◦ “read” pulse would then convert this accrued Sz magnetization into a detectable transverse signal.

To simplify the description of this BRAIN-CPMAS method, we consider first the outcome of “brute force” numerical density matrix calculations for an isolated I (1H)-S (13C) spin pair under fast (65 kHz) MAS. In these simulations, the offset frequency of the I-channel was ignored (Ω I = 0), and that of the S-channel was set to either ΩS = 0, ΩS /2π = νr /2, or ΩS /2π = νr . The first aim of these calculations was identifying the optimal I and S rf-field strengths ν1I , ν1S enabling a transfer; to do so we followed a series of criteria that included fulfilling the so-called zero-quantum (ZQ) and double-quantum (DQ) Hartman-Hahn matching conditions29–32 arising under fast MAS. These are conditions satisfying νes ± ν1I = k νr (k = ±1 or ±2), where νeS and ν1I are the effective rf frequencies for the S- and I-channels, and the positive/negative combinations refer to the ZQ/DQ conditions, respectively. Among the many possibilities for satisfying these matching conditions, our choice was to select the smallest possible ν1I rf fields for each case, while employing a ν1S value that is larger than 0.675 ∆ν/t p to ensure that the Spowder undergoes an adiabatic passage.37 For instance, given our arbitrary choices νr = 65 kHz and ν1S = 45 kHz, optimal ν1I values were found at ν1I = 17 (DQ1: k = 1; ν1S + ν1I ≈ νr ), 76 (DQ2: k = 2; ν1S + ν1I ≈ 2νr ), 108 (ZQ−1: k = −1; ν1I − ν1S ≈ νr ), and 175 kHz (DQ−2: k = −2; ν1I − ν1S ≈ 2νr ), for the various DQ and ZQ matching conditions. The ranges of optimal ν1I and ν1S values found by this approach were found experimentally accurate within a ±5 kHz range, see

064201-5

Wi et al.

J. Chem. Phys. 142, 064201 (2015)

FIG. 3. (a) BRAIN-CPMAS scheme including a phase-modulated WURST pulse applied along the S-channel together with a standard spin-lock I scheme. A 90◦ “read” pulse is required after the CP mixing time to convert the Sz-polarization obtained from the CP process into detectable signal and I -decoupling is applied during the signal detection. The pulse phases used in the sequence were: φ 1 = x, x, y, y, −x, −x, −y, −y; φ 2 = y, y, −x, −x, −y, −y, x, x; φ R X = x, −x, y, −y, −x, x, −y, y. Illustrated in (b) are the x- (blue) and y-components (red), modulation phase [ψ (t)], pulse amplitude given as ω 1S 1 − cos40  
πt /t p , and instantaneous frequency [dΨ(t)/dt] of the rf-field all for parameters ∆ν = 10 kHz, t p = 10 ms, and Ψ0 = 0◦.

supplementary material.38 This robustness is not surprising as in swept CP experiments, it is not the ν1S that defines the HH transfer, but rather the changing effective rf frequency νeS given by the rf-field strength and by the instantaneous frequency offset dψ (t) /dt coming from the adiabatic sweep. With these assumptions set, an initial input density matrix of the simulation was provided as Ix, and the forward and backward polarization transfers between I- and S-channels were recorded by monitoring Ix (blue), Sx (green), and Sz (red) signals for the entire CP mixing time. Figures 4(a)–4(d) illustrate the behavior of these spin polarizations, calculated as a function of time for the aforementioned νr , ν1S , and ν1I values. I-S CP transfers clearly occur in most of these cases, as evidenced by signal losses of the spin-locked Ispin magnetization and by signal gains for the various S-spin polarization components. The transverse S-spin generated by the CP appears to oscillate strongly in the rotating frame, as they follow the adiabatic sweep pathway of the WURST pulse. Still, in all instances, Sx (and Sy) end up going to zero as the bulk of the S-magnetization ends up as ±Sz at the conclusion of the adiabatic sweep (t = t p ). CP polarization build-up along the S-channel can be viewed more clearly in the FM frame. All

of the S-spin magnetizations begin at zero, develop along the Sz − Sx plane, and end up at ±Sz at the end of the sweep as the I → S transfers take place in the FM frame. A spherical plot is included in each case in Fig. 4 to visualize the buildup of S-spin polarization accrued on the Sz − Sx plane in the FM frame. Analysis of Fig. 4 reveals that the I → S transfer occurs at specific times, when the ν1I and νeS fields satisfy a DQ or ZQ CP condition νeS ± ν1I = ±k νr (k = ±1 or ±2). Among the three offset values tested, the best CP efficiency was observed for ΩS /2π = νr /2, under the νeS + νe I = νr DQ matching condition. In the previously treated BRAIN CP static case,17 these MAS-derived conditions did not exist, and polarization transfer occurred only when |νeS − ν1I | = 0. Moreover, in those static cases, the transferred polarization would always end up along the −Sz axis at the conclusion of the sweep. This is different from the behavior shown in Fig. 4, which predicts that under fast MAS the net CP signal gain at t p is not necessarily aligned with −Sz, but can also be along +Sz—depending on the CP mode and the offset-frequency range considered. In the particular cases treated in Figs. 4(a)–4(d), the end-point magnetization is −Sz when the DQ mode is operational at

064201-6

Wi et al.

J. Chem. Phys. 142, 064201 (2015)

FIG. 4. Spin dynamics of BRAIN-CPMAS calculated numerically for an isolated, dipole-coupled I -S pair (I =1H; S=13C) for different ν 1I spin-locking fields. The dipolar coupling constant is 4 kHz; S-spin δ csa = 300 ppm (η = 1, ν 0 = 150 MHz). The rf pulse strength, frequency sweep width, and duration were ν 1S = 45 kHz, ∆ν = 110 kHz, and t p = 10 ms, respectively. Calculations were carried out for a MAS rate ν r = 65 kHz, starting from a transverse Ix magnetization and monitoring the rapid, time-dependent transients of Ix (blue), Sx (green), and Sz (red) in the rotating frame. The effective rf-field strengths of the spin-lock pulse employed were (a) ν 1I = 17 kHz (ν eS +ν e I = ν r ); (b) ν 1I = 76 kHz (ν eS +ν e I = 2ν r ); (c) ν 1I = 108 kHz (ν e I −ν eS = ν r ); (d) ν 1I = 175 kHz (ν e I −ν eS = 2ν r ), favoring, respectively, the DQ1 (a), DQ2 (b), ZQ−1 (c), and ZQ−2 (d) HH matching conditions. The carrier frequency of the S-channel was placed on-resonance (first column), off-resonance at +ν r /2 (second column), and off-resonance at +ν r (third column). In each case, the effective S-spin polarization formed during the CP process is also displayed in an accelerated FM frame, as spherical plots adjacent to each I x , S x , and Sz trajectories.

ΩS /2π = 0 and ΩS /2π = +νr /2, or when the ZQ mode is operational at ΩS /2π = +νr . The opposite is the case when the DQ mode is operational at ΩS /2π = +νr or when the ZQ mode is operational at ΩS /2π = 0 and ΩS /2π = +νr /2. This means that the sign of the S-spin magnetization at the end of the rf sweep acts as an indicator of the CP mechanism/mode, as well as of MAS-driven RR effects39,40 which, in the presence of CSA, will efficiently invert an Sz spin-locked magnetization when νeS matches ±νr or ±2νr . The extension of BRAIN CP to rotating solids has revealed some new and interesting behavior in terms of adiabatic inversion and spin polarization transfer; we now turn to a more thorough theoretical description of this swept-rf CPMAS experiment, and give careful consideration to the incorporation of the RR effects. III. ANALYTICAL DESCRIPTION OF THE BRAIN-CPMAS NMR UNDER ADIABATIC SWEEPS A. The frequency-swept CPMAS Hamiltonian

For the sake of clarifying the behavior summarized in Fig. 4, we consider a case where the I (1H) spins are locked on

resonance (i.e., ΩI = 0), and the S-spins are devoid of CSA. For the pulse scheme in Fig. 3(a), the Hamiltonian of an isolated I-S dipolar pair under MAS in the double-rotating frame is given by H = −ω1I I x − ΩS Sz + Hrfs + 2b (t) Iz Sz,

(6)

where ω1I (= γ I B1I ) is the rf field strength applied along Ispin, Hrfs is the adiabatic rf-Hamiltonian given in Eq. (1), and 2b (t) Iz Sz is a heteronuclear dipolar coupling interaction which under MAS is modulated as32 b (t) =

2 k=−2,k,0

bk eikω r t ,

(7)

where the {bk } coefficients have the known Euler angle dependencies in the I-S dipolar tensor defined in a principal axes system to the rotor frame.32 As mentioned, an adiabatic sweep can be visualized in a FM frame rotating synchronously with the instantaneous sweep phase ψ (t). Assuming for simplicity a square-shaped rf amplitude profile, ω1S , the corresponding rf Hamiltonian HrfS is

064201-7

Wi et al.

J. Chem. Phys. 142, 064201 (2015)

  −ω1S eiΨ(t)Sz e−iΨ(t)Sz Sx eiΨ(t)Sz e−iΨ(t)Sz

sin (θ S [t 0]) = ω1S /ωeS (t 0)

d  iΨ(t)Sz −iΨ(t)Sz e e = −ω1S Sx + ω p (t) Sz. (8) dt Since all other terms in Eq. (6) commute with Sz, the total Hamiltonian in the FM frame is −i

HFM = −ω1I I x − ΩFM S (t) Sz − ω1S Sx + 2b (t) Iz Sz,

(9)

where ΩFM S (t) = Ω S − ω p (t), and ω p (t) is the instantaneous frequency of the FM frame, as defined in Eq. (4). The Hamiltonian in Eq. (9) possesses dual time modulations, coming from the adiabatic frequency sweep and from the νr-driven MAS modulations. 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 a quasi-static offset value ωp for a rotor period. This assumption will be valid as the rotor periods considered are in the 10-30 µs ranges, whereas the adiabatic pulse duration t p ≈ 10 ms. Then, by approximating the ωp (t 0) term in ΩFM S as quasi-constant for time intervals t = t 0 to t = t 0 + 2π/ωr , a rotational transformation into a doubly tilted frame where all rf fields lie in parallel to the z ′ axes,32,41 leads to HT (t) = −ω1I Iz − ωeS (t 0) Sz + 2 sin θ S (t 0) b (t 0) I x Sx − 2 cos θ S (t 0) b (t) I x Sz.

(10)

In this equation, the effective rf field strength along the tilted z ′-axis of S is given by   2 2 + ΩFM ωeS (t 0) = ω1S S (t 0)   2 1  = ω1S *1 + 2 ΩS − ω p (t 0) + ω1S , ( ) 2 1/2   ∆ω ∆ω 1 − = ω1S 1 + 2 ΩS + t 0  , (11) 2 tp   ω1S and the tilt angle θ S (t 0) that relates the tilted z ′-axis to the original z-axes in the FM frame is given by
cos (θ S [t 0]) = ΩS − ω p [t 0] /ωeS (t 0) , (12)

( ) 2 −1/2   ∆ω ∆ω 1  − = 1 + 2 Ω S + t 0  . (13) 2 tp   ω1S The last term in Eq. (10), possessing spin operators I x Sz, can safely be dropped because it does not contribute to the CP signal transfer between I- and S-spin. Then, by utilizing a single-transition operator notation, the relevant terms in Eq. (10) can be rewritten into32,42 HT (t) = −ω∆ (t 0) Iz2,3 − ωΣ (t 0) Iz1,4
+ sin θ S (t 0) b (t) I x2,3 + I x1,4 ,

(14)

where ω∆ (t 0) = ω1I − ωeS (t 0), ωΣ (t 0) = ω1I + ωeS (t 0), and Iξs, t (ξ = x, y, or z; s, t = 2, 3 or 1, 4) are single-transition operators defined by Iz2,3 =

Iz − Sz , 2

I x2,3 =

I+ S− + I− S+ , 2

Iz1,4 =

Iz + Sz , 2

I x1,4 =

I+ S+ + I− S− , 2

I+ S− − I− S+ I+ S+ − I− S− , I y1,4 = (15) 2i 2i acting in either the {|2⟩ , |3⟩} or the {|1⟩ , |4⟩} subspaces. Given that the single transition operators commute with one another, HT (t) in Eq. (14) can be written as the direct sum of two separate pseudo 2 × 2 matrices I y2,3 =

HT2,3 (t) = −ω∆ (t 0) Iz2,3 + b (t) sin θ S (t 0) I x2,3

(16)

HT1,4 (t) = −ωΣ (t 0) Iz1,4 + b (t) sin θ S (t 0) I x1,4.

(17)

and

These last two equations will determine the ZQ and the DQ CP processes, respectively, for a rotor period. Obtaining an average Hamiltonian over a rotor period from these doubly tilted frame expressions will result by applying an additional
Up = exp ikωr t Izs, t transformation onto the MAS Hts, t Hamiltonians32 d H˜ Ts, t = Up HTs, t Up−1 − iUp Up−1 dt

(18)

producing

 ( ) 2  1/2    1 ∆ω ∆ω   Izs, t H˜ Ts, t = −(ω1I − kωr ) + mω1S  1 + Ω + − t s 0  2   2 t ω p  1S    +

2 k=−2

bk e

( ) 2 −1/2     1 ∆ω ∆ω × I xs, t cos(kωr t) + I ys, t sin(kωr t) , 1 + 2 Ωs + 2 − t t 0  ω1S p  

ikω r t  

where m is +1 for the ZQ process and −1 for the DQ CP process. Zeroth-order average Hamiltonians over a rotor period can finally be calculated from these expressions by taking an integral of a time t r = ω2πr ,

⟨ H˜ Ts, t ⟩av

1 = tr

t r H˜ Ts, t dt, 0

leading to

(19)

(20)

064201-8

Wi et al.

J. Chem. Phys. 142, 064201 (2015)

 ( ) 2 1/2 ( ) 2 −1/2     1 bk  ∆ω ∆ω  1 ∆ω ∆ω    s, t s, t   ˜ − − ⟨ HT ⟩av = −(ω1I − kωr ) + mω1S  1 + 2 Ωs + t   Iz + 1 + 2 Ωs + t  I+s, t , 2 t 2 2 t   ω ω p p   1S 1S     

(21)

where k = ±1 and ±2, I+s, t = I xs, t + iI ys, t , and t 0 has been generalized for simplicity to a suitable time t.

time points for different modes that fall into the time window of the swept S-pulse are indicated in Fig. 5.

B. Fulfilling the HH matching conditions over the course of a sweep

C. Swept CPMAS and the onset of CSA-driven rotary resonance effects

The time-progressive ZQ and DQ HH matching conditions of the swept CPMAS experiment can be determined by inspecting the coefficients of the Iz2,3 and Iz1,4 terms in Eq. (21). A ZQ−k HH transfer event will occur when the instantaneous S-pulse offset ω p (t) fulfills an “antimatching” CP condition, νeS (t) − ν1I + k νr = 0 (k = −1 or −2). Likewise, a DQk HH transfer occurs at a time t when a matching CP condition νeS (t) + ν1I − k νr = 0 (k = 1 or 2) is met. Because of the time-progressive nature of the ω p (t) term, only a few instants will satisfy these matching conditions during the entire CP mixing time. This is illustrated in Figure 5(a) which shows ZQ−k and DQk HH matching curves calculated for a variety of ΩS /2π offsets frequency under fast MAS (νr = 65 kHz). Highlighted arrows in these curves indicate the time points at which the various DQk and ZQ−k matchings are satisfied. Notice that, because of ωes (t)’s ∥ω p (t)∥ dependence (Eq. (11)), DQ- or ZQ-CP matching conditions can be met at two different transient times t 1 and t 2 throughout the course of the sweep. Depending on ∥Ωs ∥, these time points may or may not fall within the 0 ≤ t ≤ t p period and hence may or may not be physically relevant to the CP process. If ΩS /2π = 0, these t 1 and t 2 points will be mirror images of each other across the center point of the symmetric frequency sweep. If an off-resonance frequency contribution is considered, these t 1 and t 2 time points will shift uniformly for the DQ1 conditions to earlier times when ΩS < 0 and to later times when ΩS > 0. Conversely, if the frequency offset is in the range of |νr | /2 < |ΩS |/2π < |νr |, only a single contact time will be found in the time period 0 ≤ t ≤ t p —with the other appearing at either t > t p (ΩS > 0) or at t < 0 (ΩS < 0). For offset-frequencies in the |ΩS |/2π > |νr | range, both CP-contact time points of DQ1 disappear from the physically meaningful 0 ≤ t ≤ t p window. Moreover, if parameters are optimized to produce an efficient DQ1 mode in the central ΩS ≈ 0 spectral region, other modes such as DQ2, ZQ−1, and ZQ−2 will not contribute to the CP signal transfer within the physically meaningful 0 ≤ t ≤ t p mixing interval. Similar considerations can be derived for the remaining DQ±2, ZQ±1, and ZQ±2 transfer modes. One interesting property following from Eq. (11) is that if CP transfer times are met for ΩS /2π at certain t i s, those for −ΩS /2π will be met at t p − t i . For instance, the transfer times found for ΩS /2π = 32.5 kHz during a 10 ms sweep (Fig. 5) are 6.44 and 9.47 ms for DQ1 and 1.72 ms for ZQ−1; those found at ΩS /2π = −32.5 kHz are then at 3.56 and 0.53 ms for DQ1 and 8.28 ms for ZQ−1. Other relevant CP transfer

An important aspect mentioned earlier concerns the complex sign behavior of the S-polarizations arising at the conclusion of these swept CPMAS experiments. These complexities are illustrated again in Figs. 5(c) and 5(d), which show the time-dependent Ix (blue), Sx (green), and Sz (red) transients involved in the CPMAS signal transfer dynamics in the presence and absence of S-spin CSA. The effects of introducing this anisotropy are two-fold. On one hand, the powder-angle dependence introduced by the CSA originates a spread of the CP transfer instant around the various time points that were found analytically in Fig. 5(a) without considering CSA. This spread in the timings of a spin-packet’s actual HH transfer reflects the contributions of the S-spin anisotropic frequencies, which may add a shift to the effective offset-frequency term. Examination of Figs. 5(c) and 5(d), however, reveals that in the presence of CSA, a more significant phenomenon begins to affect the swept-CP MAS experiment: this is the presence of ±Sz → ∓Sz inversions, introducing in turn reversals in the outcome of the overall CP profile. These reversals are associated to MAS-driven RR effects.39,40 When a shift anisotropy is modulated by a MAS process of spinning rate νr, a phenomenon known as RR takes place, which dephases a magnetization that has been spin-locked by a field ∥ν1S ∥ = νr, 2νr.43–45 The reason for such dephasing can be visualized in a tilted rotating frame where the spinlocking rf takes the role of a Zeeman-like field, and the CSA time modulation would be akin to an oscillating transverse field tilting the magnetization away of the ν1s spin-lock. In the constant-rf scenario where all such RR effects have been analyzed so far, the strength of the CSA-imparted precession is orientation-dependent and leads, over a powdered sample, to a dephasing of the spin-locked magnetization. The frequency swept case hereby considered, by contrast, is different: given the relatively slow sweep rates being considered, the effective field that is now relevant for defining the rotary resonance effect will “sweep” adiabatically through the ∥νeS ∥ = νr, 2νr RR conditions. Because of this adiabatic character, the “transverse” CSA field will effectively be able to affect an inversion of the spin-locked magnetization for all orientations, leading to a homogeneous inversion for the full powder, devoid of dephasing. With RR as a new mechanism capable of morphing a spinlocked magnetization into an “anti-spin-locked” state (and vice versa), one needs to consider how will this factor into the signs of the S-spin polarizations that may have formed throughout the swept CPMAS process. Fig. 5(b) facilitates

064201-9

Wi et al.

J. Chem. Phys. 142, 064201 (2015)

FIG. 5. CP transfer and RR inversion during swept BRAIN CPMAS (νr = 65 kHz) experiments. (a) DQ1 (solid red line), DQ2 (dashed red line), ZQ−1 (solid magenta line), and ZQ−2 (dashed magenta line) matching conditions at the specified ΩS offsets are shown, by the time points (zero crossing conditions) at which CP signal transfers occur. (b) Curves showing the positions that satisfy the RR condition ν eS (t) = ν r , occurring during the CP mixing time in the presence of an S-spin (δ csa = 120 ppm, η = 0). ((c) and (d)) Time propagations of an initial Ix state (blue) and of the CP-enhanced Sx (green) and Sz (red) polarizations, recorded with and without the inclusion of CSA effects. Notice the sign inversion of the CP-enhanced S-components occurring at the time points at which a RR condition is satisfied—but only when CSA is included.

visualization, by illustrating the time points that satisfy the νeS(t) = νr RR condition for a series of frequency offsets and parameters like those hitherto utilized for calculating the swept CPMAS matching conditions. Highlighted in Fig. 5(b) are the RR time points that fall in the 0 ≤ t ≤ t p range. Notice then that if a S-spin component has been generated by CP, it will be inverted by the CSA if this creation happened prior to a RR-driven S-spin event, but will remain spin-locked otherwise. Notice that for some offsets (e.g., the fourth and fifth columns in Fig. 5), RR occurs before the HH transfer events, resulting in non-inverted CP signals. Thus, curves calculated with (Fig. 5(c)) and without (Fig. 5(d)) including the S-spin CSA are identical. For other ΩS/2π offsets, the RR inversion occurs after transfer events like DQ1, resulting in an inversion of the CP signals. These features are clearly visible from the timedependent Sx and Sz transients in Fig. 5(c). Moreover, for the on-resonance case, two time points may satisfy the RR condition; since two inversions are the same as no inversion, no RR-driven effects on the final CP signal. This RR inversion effect can be generalized to arbitrary offset-frequencies. It then follows that, in general, the sign of Sz obtained upon sweeping around a particular Ωs, will be 180◦ out-of-phase to that obtained when sweeping around −Ωs. Thus, in the presence of CSA, the Sz states calculated at ±Ωs would be placed on the opposite hemispheres in the FM frame. This RR inversion effect takes place even when a small magnitude of CSA is present—of the kinds that are effectively averaged out from a spectrum at the MAS rates here considered. Still, their effects on the final state of the spinlocked magnetizations can be remarkable.

D. Propagating the spin ensemble throughout the BRAIN-CPMAS transfer

The Hamiltonian in Eq. (21) enables the propagation of the spin density matrix throughout the course of the CP sweep time. In particular, the value taken by Sz in the FM frame for the whole time period 0 to t p , upon starting from a ρ (t = 0) = Iz = Iz14 − Iz23 state, is32,42 


 † ⟨Sz⟩ t p = trace Iz1,4U1,4 t p Iz1,4U1,4 tp 

 † −trace Iz2,3U2,3 t p Iz2,3U2,3 tp

(22)

(  Us, t (t p ) = Tˆ exp −i

(23)

with

0

tp

)   dt ⟨ H˜ Ts, t ⟩av (t) ,

where Tˆ is the Dyson time-ordering operator that determines the ordering of terms in the  integration. In principle, the Izs, t and I+s, t terms in H˜ Ts, t av do not commute, and therefore Eq. (23) should be evaluated numerically in a stepwise manner. However, considering the working mechanism of a sweep for which the adiabaticity of the pulse employed fulfills dθ s (t) /dt ≪ 1, perturbations in θ S (t) can be neglected and the angle between the spin magnetization and the effective field assumed constant. In such case, an approximate

 average Hamiltonian over the whole mixing time, H˜ Ts, t av 

 t = t1p 0 p H˜ Ts, t av (t)dt may be obtained by integrating Izs, t and

064201-10

Wi et al.

J. Chem. Phys. 142, 064201 (2015)

I+s, t terms separately.8 The whole CPMAS process then yields bk ⟨ H˜ Ts, t ⟩av = [kωr + m(Π0 + Π1 − ω1I )]Izs, t + Π1 I+s, t , ω1S

(24)

where m = −1 or +1 for ZQ- and DQ-HH processes, respectively, ( ) (  ∆ω ∆ω ) 2 1  2  ΩS + ω + Ω + Π0 = S 1S 2∆ω  2 2   ( (  ∆ω ) ∆ω ) 2  2  (25) − ΩS − ω1S + ΩS − 2 2   and
 2
2   ∆ω   2   Ω + ω1S + ΩS + ∆ω ω1S S   2 + 2 . Π1 = (26) ln  

2∆ω    2   Ω − ∆ω + ω2 + Ω − ∆ω  S 1S 2 2   S Equation (24) is the equivalent of a zeroth-order average Hamiltonian for the whole CP process. According to it, a ZQdriven signal transfer occurs if an anti-matching CP condition, Π0 + Π1 − ω1I + kωr = 0 (k = −1 or −2), is met, and a DQ-driven process occurs if a matching condition Π0 + Π1 + ω1I − kωr = 0 (k = 1 or 2) is satisfied. A description of how Eqs. (24)–(26) were obtained is summarized in the supplementary material.38 These expressions revert to the known ZQ-HH mode at ωeS − ω1I + kωr = 0 (k = −1 or −2) and to the DQ-HH mode at ωeS + ω1I − kωr = 0 (k = 1 or 2), if sweeps are not considered. Given the Hamiltonian in Eq. (24), the spins’ evolution operator can be evaluated as42 ( ) ( )
Us, t (t p )  exp −it p ⟨ H˜ Ts, t ⟩av = exp iφ s, t Izs, t exp iθ s, t I ys, t ( )

× exp −i β s, t Izs, t exp −iθ s, t I ys, t exp −iφ s, t Izs, t . (27) Equation (27) represents a rotation in {|s⟩ , |t⟩} subspace through β s, t about an axis with polar angles (θ s, t φ s, t ). Given this rotation, the S-spin polarization Sz in the FM frame is then  2 1,4



cos θ + sin2 θ 1,4 cos β 1,4 ⟨Sz⟩ t p = 2  2 2,3


 cos θ + sin2 θ 2,3 cos β 2,3 − , (28) 2 √ s, t where ωeff = β s, t /t p = Γ s, t describes the spin dynamics of the process, sin2θ s, t =

b 2k Π12 2 Γ s, t ω 1S

its geometry, and

Γ s, t = (Π0 + Π1 − mω1I )2 + 2mkωr (Π0 + Π1 − mω1I ) 2 + k 2ωr2 + b2k Π12/ω1S .

(29)

While encompassing the details of the CPMAS dynamics, these equations still do not factor in the rotary resonance effects described earlier and which, as mentioned, may significantly alter the final ⟨Sz⟩ expectation value. Consider, for instance, a sweep centered at an on-resonance Ωs = 0 position. In such case, a DQ1-derived contribution occurring at t 2 > t p /2 may have a phase that lags by π behind the polarization generated at t 1 < t p /2. These two DQ1 buildups will add destructively,

resulting in partial ⟨Sz⟩ cancellation. The extent of this destructive interference will be maximal for the ΩS = 0 on-resonance position, for which the CP events at t 1 and t 2 are placed symmetrically about the FM frame’s equator. By contrast, other Ωs offsets and CP modes, such as the ZQ ones, do not experience destructive signal interferences because they happen only once for the |ΩS | > 0. The frequency-swept CP enhancement will thus be most effective at off-resonance irradiation points. To account for these RR effects on the CP dynamics, one can write Eq. (28) as ) ( 

2 1 2,3 2 2,3 ⟨Sz⟩ t p = ε sin θ sin ω tp 2 eff ( )
1 1,4  − sin2 θ 1,4 sin2 ωeff (30) tp , 2 where ε is an a posteriori factor inserted to reflect the offsetdependent RR inversion effect; ε is +1 when Ωs ≥ 0, and −1 when Ωs < 0. Figure 6 shows the Ωs -dependent CP profile expected upon introducing this ε factor, within the average Hamiltonian framework. The same profile is also presented for conventional Hartmann-Hahn under the same νr = 65 kHz MAS rate for comparison; both of these plots are calculated for varying ΩS and ν1S , under a fixed ν1I (= 17 kHz) and νr . Under these conditions, the optimal CP modes that are operational in both approaches are the ZQ and the DQ ones. Notice how, as the offset-frequency increases from zero to |±ΩS|, the optimal ν1S of DQ and ZQ modes in each CP method moves so as to counterbalance the increase in the offset-frequency; eventually, for both methods, the matching conditions disappear from the plot window when the ν1S value becomes negative. Also worth remarking is the fact that, due to the occurrence of the RR inversion for ΩS ≤ 0, the lower half of the offset-frequency profiles attain reversed intensities in BRAIN-CPMAS, vis-àvis those in a conventional CPMAS experiment. Finally, notice that because the swept method attains matching conditions in a time-progressive manner, the optimal ν1S value for each CP mode contains a contribution from the instantaneous frequency offset dψ (t) /dt, and is therefore different from the optimal ν1S of the corresponding conventional CPMAS mode.

IV. MATERIALS AND METHODS A. Numerical simulations

All the frequency-swept simulations that were described above resulted from numerically considering the effects of the Hamiltonian in Eq. (1). Utilizing in-house programs written in Matlab® (The Mathworks, Inc.), the performance of the frequency-swept CPMAS scheme was examined numerically by calculating the time propagation of the density matrix using piecewise (2 µs) time increments in the RF amplitude, rf phase, and—when applicable—rotor position. An isolated anisotropic chemical site was considered for examining the inversions of longitudinal magnetizations, the lineshapes of static and spinning powders, as well as when considering the RR inversion effect. These simulations nulled the offset frequency of I (Ω I = 0), and ΩS was set to either 0, ΩS /2π = νr /2, or ΩS /2π = νr . The actual CP process was examined by considering

064201-11

Wi et al.

J. Chem. Phys. 142, 064201 (2015)

FIG. 6. Theoretical 2D-(Ω S, ν 1S ) maps of the DQ- and ZQ-CP modes of BRAIN-CPMAS and conventional CPMAS methods, calculated using the average Hamiltonian theory in the text at ν 1H (= 17 kHz) and ν r (= 65 kHz). The optimal ν 1C = 37.2 kHz and 48 kHz satisfying BRAIN- and HH-CPMAS’s DQ1 modes, respectively, were used for both plots for comparison. The frequency sweep width and the duration of the S-pulse are ∆ν = 110 kHz and t p = 10 ms. Calculations were carried out by considering a single S (13C) site (δ csa = 40 ppm, η = 0.3), and an isolated S-I (13C-1H) dipolar pair with bIS = 23 kHz.

an isolated I-S spin pair in the presence of a suitable MASmodulated dipolar Hamiltonian. Powder averaging calculations of the S-spin’s CSA as well as of the I-S dipolar coupling interactions were carried out by considering the 6044 and 1154 crystal orientations of the ZCW’s Euler angle sets46 for the static and spinning cases, respectively, assuming coincident dipolar and CSA tensors. B. Experimental

This study’s predictions were examined experimentally on powdered samples of [2-13C]Ala, [U-13C]Ala, and [U13 C]Gly, focusing on conventional and BRAIN 1H-13C CPMAS experiments. Amino acids were purchased from Cambridge Isotope Laboratories (Andover, MA) and used without further treatment except for [U-13C]Ala, that was dissolved in water and recrystallized by slow evaporation. About 2 mg and 7 mg of each sample were packed into 1.3 mm and 2.5 mm MAS rotors, respectively. All experiments were carried out at

room temperature, on 11.7 T and 14.1 T magnets equipped with Bruker Avance consoles operating at 1H frequencies of 500.23 MHz and 600.92 MHz, respectively (13C frequencies of 125.80 and 150.45 MHz, respectively). A 2.5 mm Bruker MAS NMR probe was used at 11.7 T for obtaining a moderate MAS rate, νr = 25 kHz; a 1.3 mm Bruker MAS NMR probe at 14.1 T provided an “ultrafast” MAS rate, νr = 65 kHz. The 1 H and 13C 90◦ pulses used were 2 µs for all experiments. The swept rf pulse shapes were constructed numerically on the basis of WURST utilizing Matlab scripts, and subsequently fed into the NMR console for out-clocking. The shape of these pulse schemes was akin to the ones employed in the 20 static BRAIN-CP experiments: an amplitude pulse profile  
40 ν1s 1 − cos πt/τp , and an offset-frequency profile undergoing a linear sweep with time between 0 to t p . The number of digital data points employed for encoding these WURST pulse shapes was 2000. Suitable frequency sweep windows (∆ν) and CP mixing times (t p ) for the duration of the WURST pulse were also used, after scouting the ranges of 40–400 kHz

064201-12

Wi et al.

J. Chem. Phys. 142, 064201 (2015)

FIG. 7. Experimental ΩC -dependent CP profiles of BRAIN- and conventional CPMAS methods measured on 2-13C Alanine at MAS rates of ν r = 25 kHz (a) and ν r = 65 kHz (b) with individually optimized ν 1C and ν 1H values: for BRAIN CPMAS ν 1H = 17 kHz and ν 1C = 45 kHz in the case of ν r = 65 kHz, and ν 1H = 42 kHz and ν 1C = 20 kHz in the case of ν r = 25 kHz. For CPMAS, ν 1H was 112 kHz (rectangular) and 148 kHz (ramped) in the case of ν r = 65 kHz, and 44 kHz (rectangular) and 82 kHz (ramped) in the case of ν r = 25 kHz. The same ν 1C (= 45 kHz) was used as the BRAIN-CPMAS method in each case of MAS rate, and spectra were collected in ∆Ω S = 1.5 kHz (10 ppm) increments to simulate the effects of sites with varying offsets. Included in (c) are expanded versions of (b) to better view the central ΩC range, with the BRAIN inset’s negative peaks multiplied by −1 in order to better appreciate the CP dip.

and 2–14 ms, respectively. For νr = 65 kHz, CP-optimal pulse parameters were found with ∆ν = 110 kHz (< 2νr ); ◦ t p = 10 ms; ψ0 = 0 . When νr = 25 kHz, these parameters were ∆ν = 48 kHz, t p = 10 ms, ψ0 = 0◦. The optimal 13C and 1 H rf power conditions of these swept experiment for a given MAS rate were found experimentally by sweeping both rf channels independently, while keeping the 13C offset slightly off-resonance (e.g., 20 ppm) to avoid the aforementioned RRdriven signal intensity cancellation effects occurring for an on-resonance sweep. When νr = 65 kHz, the optimal 13C rf frequency was found at 45 ± 5 kHz with ν1H = 17 kHz, as

predicted by the DQ1 condition ν1H = νr − νeC . In a separate set of experiments employing νr = 25 kHz, optimal 1H rf frequencies were set at ν1H = νr + νeC = 42 ± 5 kHz and ν1H = 2νr + νeC = 65 ± 5 kHz; the optimal 13C rf frequency was set at 20 ± 5 kHz as found from an independent frequency sweep. For comparison, conventional CPMAS experiments were also carried out with independent optimizations, employing either square-shaped or ramped (90%-110%) spin-lock pulses along the 1H channel while simultaneously applying a rectangular spin-lock pulse on the 13C. The mixing time used in

064201-13

Wi et al.

J. Chem. Phys. 142, 064201 (2015)

FIG. 8. Comparisons between the experimental and theoretical ΩC -dependent CP profiles observed for 2-13C Alanine from the BRAIN- (a) and conventional CPMAS (b) experiments measured at a MAS rate ν r = 65 kHz. The theoretical profiles in (a) and (b) show cross-sections taken at the optimal ν 1C values from Figs. 6(a) (ν 1C = 37.2 kHz) and 6(b) (ν 1C = 48 kHz) that match the DQ1 mode of each case at the central region, with ν 1H = 17 kHz and ν r = 65 kHz (tensor parameters employed in both simulations were: b IS = 23 kHz; δ csa = 40 ppm, ηCSA = 0.3). Experimental CPMAS profiles are taken from Fig. 7(b). The theoretical profile shown in (a) with a red line depicts the theoretical BRAIN-CPMAS profile expected when neglecting CSA-driven RR effects. Both the experimental and simulated spectra of the BRAIN-CPMAS case shown in (a) are the same DQ1 mode. However, the experimental HH CPMAS spectra that were compared to the simulated DQ1 mode in (b) correspond to ZQ1 (rectangular; ν 1H = 112 kHz, ν 1C = 48 kHz) and ZQ2 (ramped; ν 1H = 148 kHz, ν 1C = 48 kHz) modes that were found independently by experimental optimizations.

these CPMAS experiments was 2 ms for both modes, based on optimizations. All CP spectra were acquired by co-adding 4 transient signals with 5 s of recycling delay. SPINAL-6447 proton decoupling was used during the direct acquisition period, with a 100 kHz decoupling power. V. RESULTS

The main aim of the experiments performed was to verify the various phenomena introduced in Figs. 4–6, and in particular the offset dependence of the BRAIN CPMAS profile at the fast spinning rates and large range of offsets that would be relevant at high fields. Towards this end, the wide-band profiles of BRAIN and conventional CPMAS methods were measured for model amino acids while varying 13C’s offsetfrequency (Ωs ) under optimal ν1H and ν1C conditions. The CPMAS’ performances were evaluated between −400 and 500 ppm 13C offset ranges when νr = 25 kHz, and between −1000 and 1100 ppm when νr = 65 kHz. Frequency-swept spectra arising from BRAIN CPMAS experiments on 2-13C-Ala under moderate (νr = 25 kHz) and fast (νr = 65 kHz) MAS rates are shown in Figs. 7(a) and 7(b), respectively. A single vertical line in each profile corresponds to a CP spectrum measured at a particular 13C frequency offset

(spectra were in fact free from CSA-derived spinning sidebands because of the fast MAS rates employed). Also included in Figs. 7(a) and 7(b) for comparison are the offset dependent profiles of CPMAS experiments based on the conventional spin-lock pulse along 13C channel, while employing either a rectangular or a ramped pulse block on the 1H channel. In these HH experiments, the same ν1C value as in the swept CP experiment was used, while employing an optimal ν1H that was found by sweeping the latter in the 0–200 kHz range. It follows from Fig. 7 that at moderate MAS rates (νr = 25 kHz), the swept CP method does not possess an advantage over the conventional HHCP methods. When νr = 65 kHz, however, the swept CP method provides a superior capability for polarizing a far wider range of offsets—-over 285 kHz in this case. For the central frequency region, the best CP modes correspond to DQ1, ZQ1, and ZQ2 for the BRAIN-, rectangular HH-, and ramped HH-CPMAS experiments, respectively. The width of the DQ1 mode in the BRAIN-CPMAS profile is |2νr |, which reaches to 130 kHz when νr = 65 kHz. This is far wider than what is achieved by the ramped HH-CPMAS method. The signal intensity of the swept CP method is comparable to that of rectangular HH-CPMAS—even if for all cases, the ramped HH-CPMAS method provides the strongest signal intensity for frequencies that are close to the on-resonance position. An

064201-14

Wi et al.

J. Chem. Phys. 142, 064201 (2015)

FIG. 9. Influence of the MAS rate ν r (a) and of the adiabatic frequency sweep range ∆ν (b), on the ΩC -dependent 13C polarization profile in BRAIN-CPMAS. Experiments were carried out on U-13C Glycine and spectra show both the CO and Cα peaks, highlighted in red and blue colors, respectively. Variable spinning rates of ν r = 50, 55, 60, and 65 kHz were tested in (a) while maintaining ∆ν/t p at 70 kHz/10 ms. Variable ∆ν of 400, 110, 70, and 30 kHz were swept for t p = 10 ms in (b), while maintaining the MAS rate at ν r = 65 kHz. Notice that due to the fast spinning rates, the ΩC -dependent CP profiles are invariant to the variations in ∆ν/t p (b).

interesting feature displayed by all the swept CPMAS profiles is the presence of an overall anti-symmetry about Ωs ≡ 0; this is due to the afore-mentioned RR inversion effects that occur when the matching condition νeC = νr is met. Fig. 7(c) shows this with a magnified view of the central portion of Fig. 7(b), with the negative intensity portion in the BRAIN-CPMAS profile multiplied by −1 to better compare its overall intensity profile to the HH-CPMAS results. It is worth remarking that similar RR effects were also observed with other types of adiabatic pulse schemes, such as hyperbolic secants.8 It is also valuable to compare the experimental frequency profiles that arise from these swept CPMAS experiments, against the predictions stemming from analytical simulations based on using the average Hamiltonian theory developed in Sec. III D. For this we take 1D slices at specified ν1C values (i.e., along vertical) from the 2D ν1C − Ωs plots in Fig. 6, and compare these with experimental BRAIN- and HH-CPMAS profiles. Shown in Fig. 8(a) is the 1D slice calculated analytically by employing ν1H = 17 kHz, 37 kHz ≤ ν1C ≤ 42 kHz, and νr = 65 kHz, compared against the experimental BRAINCPMAS profile measured on 2-13C Ala and already introduced in Fig. 7(b). Shown in Fig. 8(b) against an experimental CP profile is the simulated Ωs -dependent CP profile expected under HH match an optimum ν1C = 48 kHz, calculated for ν1H = 17 kHz and νr = 65 kHz. As can be seen from Fig. 8(a), the CP model introduced earlier together with the RR inversion

explains the overall shape and frequency positions of the Ωs dependent CP profile. It is worth concluding by showing the dependences of the swept CP profile on the MAS rate, as well as on the width of the adiabatic frequency sweep ∆ν. U-13C Gly was used for monitoring these profiles for the CO and Cα sites, in the −500 ppm to +700 ppm frequency range. Figure 9(a) shows profiles measured for the sites by varying the MAS rate from 50 to 65 kHz in 5 kHz step under a fixed adiabatic sweep rate, ∆ν/tp = 70 kHz/10 ms. Marked by red arrows at ΩC/2π = 0 and at ±νr are the frequency positions where the RR inversion occurs and the width of the DQ1 mode ends for each site, respectively. The width of the DQ1 mode for each site increases linearly as the MAS rate increases from 50 kHz to 65 kHz. Figure 9(b) demonstrates the negligible effect of ∆ν on CP profiles at these fast MAS rates (νr = 65 kHz): the overall shape as well as signal intensity profile is identical in all cases. VI. DISCUSSION AND CONCLUSIONS

In principle, the classical square-pulse scheme can provide arbitrarily broadband CPMAS transfer profiles, provided sufficient rf power can be employed. However constraints of various kinds—arcing, sample power deposition, low-γ values, available rf field strength—-impose limitations to the spinlocking capabilities of the standard CPMAS method. The

064201-15

Wi et al.

present study explored the possibility to polarize large bandwidths at low rf amplitudes, by employing adiabatic passage schemes covering a wide frequency range of ΩS -offsets. While adiabatic passage variants of the Hartmann-Hahn experiment have been previously proposed,48,49 a distinction should be introduced between those experiments sweeping adiabatically through an optimum match via fixed-frequency amplitude modulations of the rf, and BRAIN-CP methods sweeping over wide ranges of rf frequencies. The latter not only modulate the effective rf field amplitude but also achieve far wider matching offsets than those dictated by rf strength considerations. On exploring such approach under MAS, similarities but also significant differences were noted vis-à-vis the static BRAIN CP approach that we have recently introduced. Most salient among the new, MAS-induced features, are the needs to contend with both νr -dependent DQ and ZQ matching conditions during a sweep, and with new rotary resonance phenomena. The matching conditions of the ZQ- and DQ-CPMAS modes in the swept rf scheme could be accurately predicted by an average Hamiltonian model specifically developed to analyze the BRAIN-CPMAS experiment. Under fast spinning rates, this formalism predicted the ZQ−1,−2 transfers at νeS − ν1I = −k νr , and the DQ+1,+2 transfers at νeS + ν1I = k νr , where k  2 + (ΩS /2π + ∆ν/2 − ∆νt/t p )2. Iden= 1 and 2 and νeS = ν1S tical transfer events could occur twice at t 1 and t 2 because of the quadratic dependence of νeS on the effective offset-frequency. S polarizations created at these different times, however, can be differently affected by RR effects. In particular, partial signal cancelations appeared if CP matching conditions appeared on both sides of the Bloch sphere’s equator. This phenomenon leads to substantial cancelations at Ωs offsets that are close to the zero frequency position; by contrast, CP modes obtained at other frequency offsets possess only a single CP contact point within the entire 0 ≤ t ≤ t p mixing time, and do not experience these partial signal cancelations. Apart from this distinction, it is interesting to note that the ensuing BRAINCPMAS provides signal enhancements that are as strong as those of the rectangular CPMAS—even though transfers in the former occur only at a few time points over the contact instead of during the entire CP mixing period, and even though they require nearly an order-of-magnitude lower powers. This efficiency may be partly due to the fact that in the swept CP method, polarizations generated at a time point t do not go back and forth between the two spin reservoirs as the matching condition is satisfied only at a transient time point. The various predictions and phenomena derived in this study on the basis of time-averaged Hamiltonian theories and of numerical simulations, coincided well with experiments on model samples. As part of these experiments, an enhancement profile spanning in excess of 285 kHz (1900 ppm at 14.1 T) was measured with the new swept CPMAS method on 13C-labeledAla—an unprecedented frequency span, particularly remarkable in light of the weak rf fields ν1I , ν1S that were involved. Even the width of the single DQ1 transfer mode, the most practically useful mode of the BRAIN-CPMAS method, reached ∼130 kHz; ca twice as wide as the whole window experimentally obtained by the ramped HH-CPMAS method (75 kHz). This transfer mode displays a dip in its polarization enhance-

J. Chem. Phys. 142, 064201 (2015)

ment at the central position of the sweep; although unwanted, this dip is fairly sharp and can either be placed at a position that is a priori known to be devoid of resonances (e.g., the ∼100 ppm region in a protein NMR experiment), or be compensated by the acquisition of multiple scans with slightly shifted frequency sweeps. Moreover, the magnitude of the central DQ1 mode transfer in the swept method becomes larger as the MAS rate increases. Thus, BRAIN-CPMAS promises to become an ever more attractive alternative over conventional HH approaches, for obtaining broadband CP transfers of nuclei possessing large chemical shift dispersions and anisotropies and subject to even higher MAS rates. In particular, for the magnetic field strength of 36 T being planned for the NHMFLs the series connected hybrid (SCH) magnet33,34 where bandwidth profiles of common nuclei such as 31P are expected to reach 370 kHz, BRAIN-CPMAS could become a method of choice. Further uses could arise in moderate-field cryogenic MAS experiments, where the use of Helium gas prevents the application of strong rf pulses without arcing. Further experiments aimed at exploiting these phenomena, including CPMAS NMR studies of rare nuclei possessing large chemical shift dispersions and anisotropies and methods relying on twodimensional correlations, will be reported in an upcoming publication. ACKNOWLEDGMENTS

This work was supported by the National High Magnetic Field Laboratory through the National Science Foundation Cooperative Agreement (No. DMR-0084173) and by the State of Florida. This research was also funded by the Israel Science Foundation Grant No. 795/13, by ERC Advanced Grant No. 246754, and by the generosity of the Perlman Family Foundation. 1E.

Kupce and R. Freeman, J. Magn. Reson., Ser. A 115, 273 (1995). R. Bendall, J. Magn Reson., Ser A. 112, 126 (1995).

2M.

3T.-L. Hwang, M. Garwood, A. Tannús, and P. C. M. van Zill, J. Magn. Reson.,

Ser. A 121, 221 (1996). Baum, R. Tycko, and A. Pines, Phys. Rev. A 32, 3435 (1985). J. Ordidge, M. Wylezinska, J. W. Hugg, E. Butterworth, and F. Franconi, Magn. Reson. Med. 36, 562 (1996). 6J. Shen, J. Magn Reson Ser B. 112, 131 (1996). 7D. Rosenfeld, S. L. Panfil, and Y. Zur, J. Magn. Reson. 129, 115 (1997). 8M. Garwood and L. delaBarre, J. Magn. Reson. 153, 155 (2001). 9M. S. Silver, R. I. Joseph, and D. I. Hoult, J. Magn. Reson. 59, 347 (1984). 10M. S. Silver, R. I. Joseph, and D. I. Hoult, Phys. Rev. A 31, 2753 (1985). 11J. Pfeuffer, I. Tkac, I.-Y. Choi, H. Merkle, K. Ugurbil, M. Garwood, and R. Gruetter, Magn. Reson. Med. 41, 1077 (1999). 12H. Barfuss, H. Fischer, D. Hentschel, R. Ladebeck, A. Oppelt, and R. Wittig, NMR Biomed. 3, 31 (1990). 13A. Abragam, Principles of Nuclear Magnetism (Oxford University Press, Oxford, 1961). 14K. Riedel, C. Herbst, J. Leppert, O. Ohlenschläger, M. Görlach, and R. Ramachandran, J. Biomol. NMR 35, 275 (2006). 15G. Kervern, G. Pintacuda, and L. Emsey, Chem. Phys. Lett. 435, 157 (2007). 16A. Pell, G. Kervern, L. Emsley, M. Deschamps, D. Massiot, P. J. Grandinetti, and G. Pintacuda, J. Chem. Phys. 134, 024117 (2011). 17L. A. O’Dell and R. W. Schurko, Chem. Phys. Lett. 464, 97 (2008). 18A. W. MacGregor, L. A. O’Dell, and R. W. Schurko, J. Magn. Reson. 208, 103 (2011). 19L. A. O’Dell and R. W. Schurko, J. Am. Chem. Soc. 131, 6658 (2009). 20K. J. Harris, A. Lupulescu, B. E. G. Lucier, L. Frydman, and R. W. Schurko, J. Magn. Reson. 224, 38 (2012). 21S. R. Hartmann and E. L. Hahn, Phys. Rev. 128, 2042 (1962). 4J.

5R.

064201-16

Wi et al.

J. Chem. Phys. 142, 064201 (2015)

22A.

35M.

23K.

36R.

Pines, M. G. Gibby, and J. S. Waugh, J. Chem. Phys. 59, 569 (1973). J. Harris, S. L. Veinberg, C. R. Mireault, A. Lupulescu, L. Frydman, and R. W. Schurko, Chem. Eur. J. 19, 16469 (2013). 24R. W. Schurko, Acc. Chem. Res. 46, 1985 (2013). 25B. E. G. Lucier, K. E. Johnston, W. Xu, J. C. Hanson, S. D. Senanayake, S. Yao, M. W. Bourassa, M. Srebro, J. Autschbach, and R. W. Schurko, J. Am. Chem. Soc. 136, 1333 (2014). 26E. O. Stejskal, J. Schaefer, and J. S. Waugh, J. Magn. Reson. 28, 105 (1977). 27J. Schaefer and E. O. Stejskal, J. Am. Chem. Soc. 98, 1031 (1976). 28R. L. Cook, C. H. Langford, R. Yamdagni, and C. M. Preston, Anal. Chem. 68, 3979 (1996). 29M. Sardashti and G. E. Maciel, J. Magn. Reson. 72, 467 (1987). 30R. Wind, S. F. Dec, H. Lock, and G. E. Maciel, J. Magn. Reson. 79, 136 (1988). 31B. H. Meier, Chem. Phys. Lett. 188, 201 (1992). 32X. Wu and K. W. Zilm, J. Magn. Reson., Ser. A 104, 154 (1993). 33I. R. Dixon, M. D. Bird, and J. R. Miller, IEEE Trans. Appl. Supercond. 16, 981 (2006). 34I. R. Dixon, T. Adkins, H. Ehmler, W. S. Marshall, and M. D. Bird, IEEE Trans. Appl. Supercond. 23, 4300204 (2013).

M. Maricq and J. S. Waugh, J. Chem. Phys. 70, 3300 (1979). Siegel, T. T. Nakashima, and R. E. Wasylishen, J. Magn. Reson. 184, 85 (2007). 37A. Tal and L. Frydman, Prog. Nucl. Magn. Reson. Spectrosc. 57, 241 (2010). 38See supplementary material at http://dx.doi.org/10.1063/1.4907206 for brief description. 39A. G. Redfield, Phys. Rev. 98, 1787 (1955). 40Z. Gan, D. M. Grant, and R. R. Ernst, Chem. Phys. Lett. 254, 349 (1996). 41D. E. Demco, J. Tegenfelt, and J. S. Waugh, Phys. Rev. B 11, 4133 (1975). 42M. H. Levitt, J. Chem. Phys. 94, 30 (1991). 43F. G. Oas, R. G. Griffin, and M. H. Levitt, J. Chem. Phys. 89, 692 (1988). 44M. H. Levitt, T. G. Oas, and R. G. Griffin, Israel J. Chem. 28, 271 (1988). 45Z. Gan and D. M. Grant, Chem. Phys. Lett. 168, 304 (1990). 46V. B. Cheng, H. Henry, J. Suzukawa, and M. Wolfsberg, J. Chem. Phys. 59, 3992 (1973). 47B. M. Fung, A. K. Khitrin, and K. Ermolaev, J. Magn. Reson. 142, 97 (2000). 48S. Hediger, B. H. Meier, N. D. Kurur, G. Bodenhausen, and R. R. Ernst, Chem. Phys. Lett. 223, 283 (1994). 49S. Hediger, B. H. Meier, and R. R. Ernst, Chem. Phys. Lett. 240, 449 (1995).

Journal of Chemical Physics is copyrighted by AIP Publishing LLC (AIP). Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. For more information, see http://publishing.aip.org/authors/rights-and-permissions.