UltraHigh-ResolutionNMR: Sustained InductionDecays of Long-Lived Coherences

UltraHigh-ResolutionNMR: Sustained InductionDecays of Long-Lived Coherences Aur elien Bornet, Sami Jannin,* J. A. (Ton) Konter, Patrick Hautle, Ben van den Brandt, and Geoffrey Bodenhausen Institut des Sciences et Ing enierie Chimiques, Ecole Polytechnique F ed erale de Lausanne, EPFL, Batochime, 1015 Lausanne, Switzerland Paul Scherrer Institute, CH-5232 Villigen, Switzerland D epartement de Chimie, Ecole Normale Sup erieure, 24 Rue Lhomond, 75231, Paris Cedex 05, France Universit e Pierre-et-Marie Curie, Paris, France CNRS, UMR 7203, Paris, France


' INTRODUCTION
Most nuclear magnetic resonance (NMR) methods employ Fourier transformations of free induction decays (FIDs). 1 Although widely used, this approach suffers from homogeneous decay and imperfect homogeneity of the static magnetic field, so that it is challenging to achieve line-widths below 1 Hz. 2 Sophisticated NMR pulse sequences have been developed to achieve reasonable line-widths (1 < Δν < 50 Hz) in moderately inhomogeneous fields, exploiting cross relaxation effects, 3 observation in the earth's magnetic field, 4 or a spatial correlation between the static and radio frequency (rf ) field profiles. 5 By combining refocusing and coherence transfer through couplings, one can obtain acceptable line-widths (1 < Δν < 50 Hz) even in very inhomogeneous fields (Δν > 2 kHz). 6 In systems with two scalar-coupled homonuclear spins I = 1/2 and S = 1/2, one can excite long-lived coherences (LLCs) that can have very long lifetimes T LLC and hence very narrow line-widths Δν LLC = 1/(πT LLC ). 7À9 Their precession frequency is independent of offset (and hence of chemical shifts and inhomogeneous broadening) and is only determined by the sum of scalar and residual dipolar couplings (T IS = 2J IS + D IS ). So far, LLCs have only been observed indirectly in the manner of two-dimensional (2D) spectroscopy, either in combination with field cycling 7 or in high field. 8,9 This work describes a one-dimensional (1D) "on-the-fly" method where the rf irradiation required to sustain the LLCs in high magnetic field is briefly interrupted at regular intervals, so that the LLCs can be temporarily observed as single-quantum coherences (SQCs). In contrast with conventional FIDs, the "sustained induction decays" (SIDs) that are observed in this manner provide line-widths as narrow as Δν LLC = 14 mHz even in poorly shimmed magnets (Δν > 20 Hz). This technique is fully compatible with signal enhancement by "dissolution" dynamic nuclear polarization (DNP). 10 ' PRINCIPLES Long-lived coherences (LLCs) constitute a class of zeroquantum coherences that can be excited by extremely low frequency fields (ELFs) in a vanishing static field. 7 LLCs can also be excited in high fields by creating a state where the coherences I x and ÀS x have opposite phases, so that they can be locked by a continuous "sustaining" rf field. 8,9 This rf field in effect suppresses the chemical shifts, thus rendering the spins magnetically equivalent, so that their eigenstates can be classified according to "symmetrical" and "antisymmetrical" irreducible representations of the spin permutation group. LLCs span zeroquantum transitions between states of different symmetry. Their oscillatory decays can be subjected to a Fourier transformation, yielding doublets that are reminiscent of "J-spectroscopy". 11À13 The lifetimes T LLC of LLCs can be a factor k longer than the transverse relaxation times T 2 = T SQC of ordinary singlequantum coherences (T LLC = kT 2 ), so that the line-widths Δν LLC = 1/(πT LLC ) can be narrower by a factor Δν LLC /Δν SQC = 1/k. Depending on the role of extraneous relaxation mechanisms, 9 one can expect k e 3 in small molecules in the extreme ABSTRACT: Long-lived coherences (LLCs) in homonuclear pairs of chemically inequivalent spins can be excited and sustained during protracted radio frequency irradiation periods that alternate with brief windows for signal observation. Fourier transformation of the sustained induction decays recorded in a single scan yields NMR spectra with line-widths in the range 10 < Δν < 100 mHz, even in moderately inhomogeneous magnetic fields. The resulting doublets, which are reminiscent of J-spectra, allow one to determine the sum of scalar and residual dipolar interactions in partly oriented media. The signal intensity can be boosted by several orders of magnitude by "dissolution" dynamic nuclear polarization (DNP).
narrowing limit, and k e 9 in the slow 14 motion limit typical of large molecules. In practice, we have observed 2.5 < k < 4.3 over a range of correlation times. 15 The effects exploited in this work are closely related to those described by Hartmann and Hahn, 16 by Chingas et al., 17 by Levitt, 18 and by Konrat et al. 19 In all cases (regardless of whether the transfer occurs via dipolar interactions in solids or via scalar couplings in liquids), there is an oscillatory to-and-fro motion between coherences associated with individual spins.
Generally speaking, LLCs should not be confused with long-lived states (LLSs), also known as singlet states (SS) if there are only two spins in the system. LLSs refer to populations of antisymmetric singlet states. 20À24 LLSs have lifetimes that can be much longer than LLCs (T LLS . T LLC ), but do not have any oscillatory character and cannot give rise to J-spectra in the manner of LLCs. LLSs can be enhanced by "dissolution" DNP. 10,25 If the oscillatory decays of LLCs are observed point-by-point in the manner of two-dimensional (2D) spectroscopy, they cannot be enhanced ("hyperpolarized") by "dissolution" DNP. Recently, several 2D experiments have been successfully converted into "ultrafast" versions that can be combined with "dissolution" DNP. 26,27 However, the continuous rf field required to sustain LLCs is not compatible with current "ultrafast" schemes. In this work, we describe a novel one-dimensional (1D) method that is fully compatible with hyperpolarization techniques. In our experiment, the signals are observed during brief interruptions of the sustaining rf field (scheme B1 in Figure 1). In the simplest version of the experiments, the windows are kept short, so that the evolution under chemical shifts, couplings, and transverse relaxation can be neglected (Figure 1, B1). In more sophisticated experiments, the sensitivity can be improved by making the windows somewhat longer, and by inserting π refocusing pulses in these windows to refocus chemical shifts (scheme B2 in Figure 1).
The initial Boltzmann equilibrium populations, described by the density operator σ = I z + S z (which may be enhanced by DNP), must be transformed into σ = I x À S x . Scheme A1 of Figure 1 starts with a nonselective (π/2) x pulse to excite the state σ = ÀI y À S y followed by a delay τ 1 = 1/(2|ΔΩ IS |), where ΔΩ IS = Ω I À Ω S . Because the rf carrier is positioned halfway between the two chemical shifts at ω rf = (Ω I + Ω S )/2, σ = ÀI y À S y is transformed into σ = I x À S x during the delay τ 1 . The precession under J IS in the interval τ 1 can be neglected because 2πJ IS , ΔΩ IS . In scheme A2, a semiselective π pulse applied to either I or S to invert the populations across either of the two doublets is immediately followed by a nonselective (π/2) y pulse to excite σ = I x À S x . 8,9 In aqueous solutions, it may be necessary to suppress the intense HDO peak. Hence, scheme A3 uses an echo sequence (π/2) x ÀτÀ(π) x (I,S) Àτ with a bandselective refocusing pulse that acts on spins I and S but is too weak to refocus the solvent resonance. The two pulsed field gradients (PFGs) lead to dephasing of all magnetization components with offsets that lie outside the range of the band-selective refocusing pulse. Like in scheme A1, σ = I x À S x is created after a delay 2τ + τ 1 = 2τ + 1/(2|ΔΩ IS |). Finally, scheme A4 uses a "long-lived state filter" as explained elsewhere. 15 The two latter schemes also have the advantage of avoiding possible radiation damping induced by large HDO signals, especially when enhanced by DNP.
Both schemes B1 and B2 in Figure 1 rely on a continuouswave (CW) rf field for "sustaining" or "locking" the LLC, to suppress the chemical shifts of spins I and S, with a carrier ω rf = (Ω I + Ω S )/2 and an rf amplitude that must be larger than the offsets ω 1 > |Ω I À Ω S |/2. More sophisticated methods may also be used to sustain LLCs over greater bandwidths. 28 During rf irradiation, the eigenstates are better described in the singletÀtriplet base. 9 In this base, the density operator σ = I x À S x can be written σ = (|S 0 aeAET 0 | + |T 0 aeAES 0 |), that is, a zero-quantum coherence spanning the central triplet state T 0 = N(|Rβae + AEβR|) and the singlet state S 0 = N(|Rβae À AEβR|) where N = 2 À1/2 . In the windows τ 3 or τ 3 /2 where the rf field is switched off, the density operator, described in the product base as single-quantum coherences σ = I x À S x , gives rise to signals that can be observed. In both schemes B1 and B2 in Figure 1, the LLCs are sustained during the intervals τ 2 , while the signals are detected in the windows τ 3 or τ 3 /2. The sustainÀ observe cycles are repeated n times, resulting in "sustained induction decays" (SIDs) with a total length t max = nΔt digitized at intervals Δt = τ 2 + τ 3 in schemes B1 or B2. The Δt intervals are equivalent to the "dwell times" of ordinary free induction decays. The signals can be readily Fourier transformed, giving a frequency domain spectrum with a digital resolution that is determined by 1/t max and a spectral width 1/Δt that should be larger than the total coupling T = 2J + D if one wishes to avoid aliasing. This type of "windowed acquisition" is reminiscent of solid-state NMR methods such as WAHUHA, MREV, and their numerous variants. 29,30 Recently, "windowed acquisition" has also been used for homonuclear dipolar decoupling with shaped rf pulses in the manner of DUMBO. 31 If the windows are too short, the signals can be perturbed by transient effects due to transmitter breakthrough, bearing in mind that the preamplifier must be protected during rf irradiation, and that this protection must be removed in the windows. On the other hand, if the windows are too long, the single-quantum coherences σ = I x À S x will decay through transverse T 2 relaxation, dephase in the inhomogeneous static field, and evolve under the chemical shifts and scalar couplings. With an analogue-to-digital converter (ADC) running at 500 kHz, we can acquire a sample point every 2 μs, and take averages over all points recorded in each interval τ 3 of scheme B1 or in the first τ 3 /2 interval of scheme B2. Reducing the number of sampling points leads to a loss in signal-to-noise ratio. In practice, the dead time between the point where the CW rf field is switched off, and where the signal can be observed is typically 3 μs, so that eight sampling points can be taken in each window if τ 3 = 20 μs, or 498 points in each window if τ 3 = 1000 μs. If the sustaining intervals in scheme B1 of Figure 1 are adjusted to keep a constant dwell time Δt = τ 2 + τ 3 = 50 ms so that a spectral width is 1/Δt = 20 Hz or (10 Hz, windows τ 3 = 20 or 1000 μs lead to rf duty cycles of 99.96% or 98%, respectively. Pulse sequences for exciting and sustaining LLCs with "on the fly" observation of the magnetization in brief windows. One of the four preparation sequences A1ÀA4 allows one to create a density operator σ = I x À S x (see text). Either of the two sequences B1 and B2 can be used to sustain the LLCs by CW irradiation and to acquire signals in the windows τ 3 or τ 3 /2. The sustainingÀacquisition blocks must be repeated n times.

ARTICLE
During the irradiation intervals τ 2 , the coherence Q LLC = (|S 0 aeAET 0 | + |S 0 aeAET 0 |) evolves under the effect of the total coupling T IS = 2J IS + D IS and decays with the relaxation rate R LLC = 1/T LLC : In terms of the usual Cartesian product operators, this leads to: This is consistent with our recent work, 9 but our initial paper 8 underestimated the effect of the couplings by a factor 2. During each window τ 3 in the scheme B1 of Figure 1, the density operator evolves under the chemical shifts and again under the total coupling constant T IS , albeit attenuated by a factor 2, and decays with the single-quantum relaxation rate R 2 = 1/T 2 . The overall effect for each sustain-and-observe cycle Δt = τ 2 + τ 3 in scheme B1 can be written: where Δt 0 = τ 2 + τ 3 /2 = Δt À τ 3 /2, reflecting the scaling of the total coupling constant when the rf field is switched off. So we can define an apparent total coupling constant: Using the notation R 2 = R SQC = kR LLC with k e 9, the average decay rate AERae in eq 3 is For k = 3, τ 2 = 49.98 ms, and τ 3 = 20 μs, this amounts to a mere 0.08% increase in the average relaxation rate and hence to a negligible contribution to the line-width. When the CW rf field along the x-axis is switched on again, the differences I x À S x and 2I y S z À 2I z S y resume their identity as LLCs, while the sum I y + S y is spin-locked and decays with R 1F , and the sum 2I x S z + 2I z S x is dephased under the effect of the rf field inhomogeneity. With a chemical shift difference ΔΩ IS /(2π) = 300 Hz, we have cos(ΔΩ IS τ 3 ) = 0.9993∼1. This infinitesimal "leakage" of the LLC seems negligible, but it is amplified as the sustainÀobserve sequence is repeated n times with cos(ΔΩ IS τ 3 ) n so that cos(ΔΩ IS τ 3 ) 100 = 0.936, thus affecting the decay of the LLC. The resulting time domain signals sampled at intervals Δt are However, when τ 3 is long, the "leakage" can become significant: for τ 3 = 1 ms, cos(ΔΩ IS τ 3 ) = 0.9553, and hence cos(ΔΩ IS τ 3 ) 100 = 0.01, so that the scheme described below is to be preferred.
To suppress contributions from I y + S y and 2I x S z + 2I z S x , scheme B2 uses a π pulse in the middle of each window to refocus the chemical shifts. As a result, the density operator at the end of (e) Real part of the "sustained induction decay" (SID) acquired "on the fly" in a single scan (note that the time scale was expanded by a factor 100 with respect to (a)), arising from an LLC excited in the same sample with sequence A3 of Figure 1, sustained and observed with sequence B2. The parameters were τ 3 /2 = 100 μs, Δt = τ 2 + τ 3 = 50 ms, rf amplitude of the CW sustaining field γB 1 /(2π) = 4.5 kHz, offsets Ω I /(2π) = ÀΩ S /(2π) = 145 Hz, the rf carrier being set halfway between the two chemical shifts. (f) Spectrum obtained by a real Fourier transformation of the SID of (e), showing a doublet with linewidths AEΔνae ≈ 16.4 mHz and a splitting 2J IS ≈ 11.5286 Hz. If undesirable spin-locked I x + S x terms had not been suppressed, they would give rise to peak at ν = 0. The narrowest linewidths AEΔνae = 14 mHz (not shown) were observed with scheme B1, τ 3 = 30 μs, and Δt = τ 2 + τ 3 = 50 ms. Journal of the American Chemical Society ARTICLE each window τ 3 in scheme B2 is: The resulting time domain signals sampled at intervals Δt are: ' EXPERIMENTAL EVIDENCE Figure 2e shows an SID that can be compared to the FID presented in Figure 2a and to the modulated echo decay of Figure 2c. The three signals stem from the two protons in an isotropic solution of 2,3-dibromothiophene (20 mM in DMSO-d 6 with 30 mM ascorbic acid 33 to scavenge paramagnetic oxygen), recorded with a simple π/2 pulse (Figure 2a), in a J-resolved 2D manner 32 (Figure 2c), and with "on the fly" LLCs in windows τ 3 /2 = 100 μs with scheme B2 (Figure 2e). Their Fourier transforms are presented in Figure 2b, d, and f, respectively. The LLCs "SID" signal is described by eq 8 and slowly decays with a time-constant AETae = 1/AERae = 19.9 s. Its Fourier transforms (Figures 2f and 3a) show two lines at ν = (J IS separated by 2J IS with line-widths AEΔνae = 1/(πAETae) = 16.4 mHz (resolution enhanced by a factor ε Δ = ν/AEΔνae ≈ 180 and 8.5 with respect to conventional FID and echo modulation, respectively). The fact that the couplings are twice as effective in the rotating frame than in the laboratory frame is reminiscent of total correlation spectroscopy ("TOCSY"). 34 Note that the antiphase terms 2I y S z À 2I z S y cannot induce any signals in the orthogonal channel, so that we have a case of pure amplitude (rather than phase) modulation. The "on the fly" LLC spectrum of the two diastereotopic protons of glycine in L-Ala-Gly is shown in Figure 3b. Figure 4a shows how the insertion of refocusing pulses in the windows allows one to eliminate the effects of chemical shifts. For longer windows 100 μs < τ 3 < 2 ms, scheme B2 provides longer decays and hence narrower line-widths. Note that the narrowest lines are obtained, albeit at the expense of sensitivity, with scheme B1 with very short windows (typically τ 3 = 20 μs). Figure 4b shows how refocusing allows one to obtain an accurate measurement of scalar couplings J IS (or total couplings T IS in anisotropic media) even for long windows τ 3 . (The slight decrease in J IS app for long τ 3 is described by eq 4). Finally, Figure 4c shows how longer windows τ 3 , which allow one to average over a larger number of data points in each window, result in improved signal-to-noise ratios, which are proportional to τ 3 1/2 as expected. "On the Fly" LLCs in Inhomogeneous B 0 . In principle, the evolution of LLCs is immune to the inhomogeneity of the magnetic field if one uses scheme B2 of Figure 1. We should remember however that all excitation schemes A1ÀA4 of Figure 1 require one to distinguish the chemical shifts of the two spins I and S, but not their mutual coupling constant. The methods can thus tolerate a moderate inhomogeneity of the static field, as long as the line-widths fulfill the condition Δν* = 1/(πT 2 *) < ΔΩ I . Figure 5 shows how a deliberate missetting of the shim currents (z 1 , z 2 , z 3 , x, y, z 0 x, and z 0 y) to broaden the line-widths to about Δν* = 20 Hz has little effect on the linewidths AEΔνae of the LLCs and the apparent scalar couplings J IS app (+5.3 and +3 mHz, respectively). Ex-situ NMR 4,5 and MRI in moderately inhomogeneous fields (e.g., in the vicinity of   ( 0.2 mHz). In a poorly shimmed magnetic field, some broadening (+5.3 mHz) and a slight error in J IS app (+3 mHz) are thus observed. The areas of the peaks are identical. The LLCs were excited with sequence A3 and sustained and observed with sequence B2 of Figure 1 with the following parameters: τ 4 = 500 μs, Δt = τ 2 + τ 3 = 50 ms, rf amplitude of CW sustaining field γB 1 /(2π) = 4.5 kHz.
Journal of the American Chemical Society ARTICLE discontinuities of the magnetic susceptibility) may benefit from this property.
"On the Fly" LLCs To Measure Very Weak Alignments. Very weak molecular alignments, yielding minute residual dipolar couplings (RDC's) in the millihertz range, can be readily resolved with our method. Figure 6 shows the "on the fly" LLC spectra of two solutions of 3-bromothiophene-2-carboxylic acid in (1:1) D 2 O/DMSO-d 6 , with and without addition of a 0.25% pentaethylene glycol monododecyl ether (C 12 E 5 ). The very weak alignment of the solute gives rise to a net RDC with D IS app = À21.4 ( 0.8 mHz. The order parameter of the r HH vector can be estimated to be S e (1.01 ( 0.04) 10 À5 (assuming r HH = 2.662 Å like in thiophene, 35 with the average HÀH vector oriented along B 0 , i.e., θ = 0). This is the first time residual dipolar couplings are considered in LLC experiments, in addition to scalar couplings. In high field, the corresponding Hamiltonians have the same form. Further work is in progress to determine the effects of RDCs on the homogeneous and inhomogeneous linewidths of LLCs.
Hyperpolarized LLCs. Because LLC spectra can be recorded in a single scan, they can be boosted by "dissolution" DNP. Spectra of a 20 μL solution of 50 mM 2,3-dibromothiophene dissolved in a 3:2 DMSO-d 6 /D 2 O (v/v) mixture doped with 30 mM TEMPOL are compared in Figure 7 with and without hyperpolarization by "dissolution" DNP (see Methods). The dissolution, transfer, and injection required 3.2 s. After an additional 3 s of settling time in the NMR tube, some bubbles and convection cannot be ruled out. These tend to broaden ordinary (single-quantum) line-widths, but have little effect on LLC spectra. The LLCs were excited, sustained, and observed with sequences A3 and B2 of Figure 1. The enhancement was ε DNP ≈ 300. It may be possible to improve this performance by preventing losses of the proton polarization due to relaxation in low fields during the voyage. 36 ' CONCLUSION Ultra high-resolution spectra of long-lived coherences (LLCs) can be obtained "on the fly" in one-dimensional fashion by timeshared "windowed acquisition". This allows one to determine very accurate total couplings T = 2J + D. The method can be applied to either isotropic or anisotropic phases, providing ultra high resolution even in moderately inhomogeneous magnetic fields, and the signals can be enhanced by "dissolution" DNP. The technique has been applied to pairs of spins in this study, but we intend to extend the scope of application of "on the fly" LLCs in the near future to multiple spin systems (N > 2) with broadband excitation and detection (replacing CW by composite pulses) of several LLCs in the same molecule or in mixtures. Because inhomogeneous fields are not detrimental to LLCs, ex situ or in-cell studies should be readily feasible with unprecedented line-widths, and because the long lifetimes of LLCs are exquisitely sensitive to the presence of paramagnetic species, 33 we believe they should be a sensitive probe for the detection paramagnetic oxygen.
Journal of the American Chemical Society ARTICLE Hyperpolarization. DNP was performed by thermal mixing at 1.2 K and 3.35 T in a home-built "dissolution" DNP polarizer 37À39 by applying a CW microwave irradiation at f μw = 93.89 GHz and P μw = 30 mW for 5 min. The DNP buildup of 1 H magnetization is fast (τ DNP ≈ 120 s) and yields high proton spin polarization P( 1 H) ≈ 20À40% depending on sample composition. 40 The 20 μL of frozen beads of the polarized sample, together with 90 μL of frozen beads of a 3 M D 2 O solution of sodium ascorbate, were rapidly dissolved with 3 mL of preheated D 2 O (T = 440 K and P = 1.2 MPa) and intimately mixed within 700 ms, transferred in 1.5 s to a 11.7 T Bruker magnet through a 1.5 mm inner diameter PTFE tube pressurized with helium gas at 0.6 MPa, and allowed to settle for 0.5 s, prior to injection into a prelocked NMR tube, which required another 0.5 s. After a further 3 s settling time in the NMR tube to allow turbulences to slow, the LLC was excited, sustained, and observed with the sequences A3 and B2 of Figure 1. ' REFERENCES