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Homonuclear J-Coupling Spectroscopy using J-Synchronized Echo Detection

In the strong coupling regime with J-coupling much larger than chemical shift differences, J-coupling spectroscopy enables spectral identification of molecules even when conventional NMR fails. While this classically required the presence of a heteronucleus, we recently showed that J-coupling spectra can be acquired in many homonuclear systems using spin-lock induced crossing (SLIC). Here, we present an alternative method using a spin echo train in lieu of a spin-locking SLIC pulse, which has a number of advantages. In particular, spin echo acquisition within the pulse train enables simultaneous collection of time and frequency data. The resulting 2D spectrum can be used to study dynamic spin evolution, and the time domain data can be averaged to create a 1D J-coupling spectrum with increased signal-to-noise ratio.


Introduction
To create easily interpreted spectra, NMR spectroscopy is typically performed under weak coupling conditions where the chemical shift frequency differences ∆ are significantly larger than the scalar couplings, . However, magnetic fields >1 T are typically required to create sufficient frequency dispersion. The ability to acquire high-resolution spectra at lower magnetic fields would be advantageous in many situations, such as for benchtop and educational instruments [1][2][3], portable operations for oil-field exploration [4][5], spectroscopy in the presence of ferromagnetic and paramagnetic substances [6], optically-detected NMR with nitrogen vacancies as sensors [7][8], and chip-scale spectrometers [9][10][11]. Unfortunately, as field strength is reduced spectral information becomes difficult to interpret and is eventually lost, because in the strong-coupling regime (when ≫ ∆ ) the NMR spectrum of any homonuclear spin system exhibits only a single dominant spectral line. Although adding a heteronucleus such as 13 C or 31 P can enable J-coupling spectroscopy in this regime, it is not a practical solution in most situations, and the signal from natural-abundance isotopomers is small [12][13][14][15][16].
We previously showed that homonuclear J-coupling spectroscopy at low field strengths can be implemented using the spin-lock induced crossing (SLIC) sequence [17][18][19][20][21]. SLIC detects the level anti-crossings created whenever there is at least one chemical shift difference among the spins. In SLIC, very weak spin-locking is applied to the system, followed by acquisition of the free-induction decay (FID) signal. Measurements are repeated for a series of spin-lock amplitudes to create a J-coupling spectrum by plotting FID intensity as a function of spin-lock nutation frequency. Dips in FID intensity indicate the positions of level anti-crossings, which in turn are a reflection of the J-couplings within the molecule. Unfortunately, this procedure is slow and suffers from drawbacks such as sensitivity to B0 and B1 offsets and instability. It is also difficult to extend the technique to two dimensions, for example to simultaneously measure the dependence on spin-locking time and nutation frequency, since the acquisitions must be completely repeated along the time dimension.
In this work, we introduce an alternative method for homonuclear J-coupling spectroscopy based on a train of spin echoes rather than spin-locking. Since the energy levels are detected whenever the pulse timing is J-synchronized, rather than via level anti-crossings, we follow the nomenclature of Pileio, et al. and refer to this technique as synchronized echo (abbreviated SyncE in this work). It has a number of significant advantages, most importantly the ability to perform echo acquisitions within the sequence, thereby producing a 2D spectrum in the same amount of time as a typical 1D spectrum with SLIC. The echo acquisitions can be averaged to produce more than a 30-fold improvement in the contrast-to-noise ratio compared to the original SLIC sequence. Importantly, using either SLIC or synchronized echoes detects the same spin energy levels and produces equivalent J-coupling spectra.
To summarize, magnetization is first rotated onto the x-axis using a 90˚ excitation pulse (Fig.   1a). In the rotating frame, the energy levels in the strong-coupling regime consist of sets of degenerate states separated by gaps of the order J with eigenstates defined by their total spin quantum number F. Spin-locking is then applied, which splits these degenerate states proportionally to their magnetic quantum number mF. At particular nutation frequencies, , some of these states exhibit level anti-crossings, and x-axis magnetization (Mx) coherently evolves into dark quantum states that do not contribute to the FID signal (Fig. 2a). These crossings are detected as dips in the FID intensity. The frequency and intensity of the dips reflect J-couplings and chemical shift differences in the system, respectively.
The Fourier transform for a train of appropriately spaced pulses is similar to that of CW spinlocking, albeit with additional harmonic components, so one might suppose the response of the spin system to the pulse train should be similar, if not identical. It would therefore be possible to substitute a spin echo train for spin-locking in the SLIC sequence and obtain the same J-coupling spectrum, for example using the sequence shown in Fig. 1b. Analysis of the quantum mechanics shows that this is only partially true, and that in fact synchronized echoes create an equivalent Jcoupling spectrum but with dips of twice the intensity.
To understand why, we analyze the sequence following the method of Pileio, et al. and Tayler, et al., but generalizing to systems with more than two spins [17,25]. In their M2S/S2M pulse sequence for converting magnetization to singlet order in a pair of spin-1/2 nuclei, they used a series of two spin echo trains. The first stage of M2S is a J-synchronized echo train that converts x-axis magnetization into a singlet-triplet coherence and is equivalent to that in Fig. 1b. The sequence begins with a 90˚ pulse that moves longitudinal magnetization into transverse magnetization along the x-axis (Mx). In terms of a density matrix, Mx is a coherence composed of all combinations of matrices with a +/-1 difference in magnetic spin quantum numbers, i.e.  For a given pair of states, we can create a Bloch sphere and analyze whether the combination of Hamiltonian and pulse sequence can move the state around the Bloch sphere. We find that the pulse train interconverts any two states separated by the selection rule Δ = ±1, ∆ = 0. The simplest example following this rule is the transition between | ⟩ and | ⟩, or in more explicit form |1,0⟩ and |0,0⟩. These are connected by the interaction term − ) = , where = − , the difference in resonance frequencies between the two spins. In a Bloch sphere defined by the states | ⟩, | ⟩, the interaction term driven by attempts to move the state around the sphere from | ⟩ to | ⟩. However, at the same time the quantum state precesses about the longitudinal axis at the frequency gap between these states, Δ = . Because is significantly larger than , only a small amount of | ⟩ makes it to | ⟩ before precession reverses the process and it returns to | ⟩. The state thus makes a small circle near the top of the sphere.
To enable full interconversion between these states, either the energy gap can be removed (as in SLIC), or the effect of precession can be eliminated (using a J-synchronized echo train). In the latter, the spacing between refocusing pulses is chosen so that after a half-cycle of precession the state is brought to the opposite side of the sphere, from where the interaction term can continue the evolution from | ⟩ to | ⟩. The state is thus kept on the side of the sphere in which evolution proceeds in the correct direction ( Fig. 2b). Figure 2: (a) The synchronized echo (SyncE) sequence converts Mx, a coherence among triplet states (red), into a singlet-triplet coherence (blue), which is NMR silent. Mx can be completely depleted as it is fully transformed to the singlet-triplet coherence. On the other hand, in SLIC Mx is represented as a population of two spin-locked triplet states (red circles). During spin-locking, the population of one of these states is converted to singlet, so at the end of the sequence 50% of Mx is depleted (blue circles). The other half of Mx can also be depleted via a second block of spin-locking with the opposite phase. (b) Spin echo converts the | ⟩ component to | ⟩ in a series of stages around the Bloch sphere.
After a full evolution from | ⟩ to | ⟩, the density matrix has the form ∑ | ± 1, ⟩⟨ , ± 1|, in this example consisting of coherences between | ⟩ and ± |, plus their complex conjugate. At this point, the M2S sequence continues with a second pulse train to convert the coherence to a singlet-triplet population difference. We instead perform an FID acquisition to measure the remaining x-axis magnetization Mx. The singlet-triplet coherence terms do not appear in Mx, so at this point Mx has been depopulated and the spectral line has a lower intensity as a result.
For more complex systems, this same analysis can be performed between every combination of states separated by Δ = ±1, ∆ = 0. For example, in a three spin-1/2 system, the combinations would be: , with , and , − with , − . For these transition to be detected, the resulting coherences do not need to contain singlet states or be long-lived in order to be dark states, they simply need to be missing from the definition of Mx.
Two differences from SLIC are apparent. First, synchronized echo results in the creation of coherences rather than population differences, and these may have different relaxation properties resulting in different dynamics. Second, synchronized echo creates coherences between | ⟩ and both ⟨ | and ⟨ |, whereas with SLIC the singlet interacts with only one of the three triplet states at a time. Therefore, twice as much magnetization is removed from Mx using the synchronized echo sequence.

Multi-acquisition Scheme
In the most straightforward replacement of spin-locking with a pulse train, the remaining x-axis magnetization would be detected following the full evolution period. However, the synchronized echo technique has the additional benefit of gaps between the 180˚ pulses in which echo acquisitions can be collected (Fig. 1c). By measuring each echo, we can track the dynamics at each step in the evolution around the Bloch sphere while magnetization transfer is still occurring.
This has two benefits. First, by measuring Mx over time, the rate of magnetization transfer can be measured and possibly used to determine chemical shift differences and relaxation rates. In contrast, to perform the same type of measurement with SLIC would require repeated experiments with different evolution times. For 2D experiments, echo acquisition therefore greatly decreases the total experiment time. Second, the echo acquisitions can be averaged over time to create a 1D projection of the J-coupling spectrum. Averaging significantly reduces the noise compared with taking a single measurement at the end of the sequence. Moreover, the specific time points to average can be windowed to choose the periods with strongest signals, thereby optimizing the contrast-to-noise ratio.
To create a J-coupling spectrum with synchronized echoes, data are acquired with sequences shown in Fig. 1b or 1c The drawback of using a constant T is that nutation frequency points (and the associated frequency resolution) cannot be chosen arbitrarily as they can be for SLIC.

Results and Discussion
To compare techniques, J-coupling spectra of neat ethyl acetate were first measured with SLIC and synchronized echo (SyncE) sequences shown in Figures 1a and 1b respectively. The SLIC pulse and total echo train length was T = 1 second, and the FID was acquired following the complete series of pulses. Results are shown in Fig. 3a and 3b along with spectra predicted via simulation. The dips for SyncE were about twice as deep as for SLIC, as predicted by theory (blue curves), while the noise level for SyncE was about 1/3 of the level for SLIC, resulting in a contrast-to-noise ratio (CNR) about 6 times higher using SyncE (Table 1). However, a higher resolution (0.3 vs. 0.5 Hz) was possible with SLIC because the SyncE resolution was constrained by the 1 s sequence duration.
We then implemented the multi-acquisition SyncE scheme (Fig. 1c), collecting data during each echo. We extended the pulse train to 3 seconds to provide higher resolution (0.167 Hz). Figure   3c shows results when the echo data are averaged to create a 1D J-coupling spectrum. The dips are slightly deeper than using the 1 second sequence, and the single -scan noise is about 3 times smaller, giving another factor-of-4 improvement in CNR. This results in a 20 to 30-fold improvement compared with the original SLIC sequence. A spectrum was also acquired using composite 180˚ pulses (dashed red spectrum), which increased both depth and noise slightly, providing the same CNR results as simple 180˚ hard pulses. All spectra acquired with synchronized echoes clearly exhibit harmonic dips near 3.     We also performed simulations to evaluate the sensitivity of the sequence to B0 and B1 errors.
For SLIC, even a small offset in B0 causes shifts in dip location as well as baseline oscillations, so B0 must be carefully locked (Fig. 5a). The synchronized echo sequence is less sensitive to these errors, particularly if sufficiently short pulse lengths are used so that the flip angle maintains the necessary accuracy. Fig. 5b shows the effects of B0 offsets when using 2 ms refocusing pulses.
Much larger offsets are tolerated, before the dips eventually exhibit broadening and splitting.
B1 offsets for spin-locking in SLIC lead to shifts in the dip locations (Fig. 6a). Since dip locations for synchronized echo are determined by delay times, which can be made highly accurate, it is more robust against such shifts. Instead, equivalent errors in the echo train pulse angles lead to broadening and splitting of the dips, similar to that observed for a B0 offset.
Figures 6b-c show that an accuracy of at least ±2˚ is needed for simple refocusing pulses, although this can be extended to ±20˚ or more with composite refocusing pulses. SyncE is also easier to calibrate than SLIC, as it requires only one B1 calibration for the 180˚ pulse, rather than calibration for a series of spin-locking amplitudes with SLIC. Finally, although our synchronized echo technique appears superficially similar to other CPMG pulse sequences, the strong-coupling regime leads to a very different response of the spin system at low field. For example, Freeman and Hill used a similar sequence to determine J-coupling at high field with high accuracy [26]. In that method, J-coupling leads to oscillations in echo intensity during the pulse train, which are converted to the spectral domain with a Fourier transform to find J-coupling values. These oscillations occur for any pulse spacing. In our technique, dynamics in echo intensity are instead driven by chemical shift differences, and the J-coupling values determine the pulse spacings at which these dynamics occur. This produces a spectrum of dips very similar to those acquired with quantum magnetometry using dynamical decoupling sequences [27][28][29], but again in those techniques dynamics are driven by dipolar or scalar couplings, and the frequencies at which dips occur correspond to the Larmor frequencies as spins precess about B0.

Conclusion
Synchronized echo detection creates homonuclear J-coupling spectra with twice the signal strength of SLIC, and when combined with a multi-acquisition scheme increases the sensitivity by more than an order of magnitude. The reduced sensitivity to B0 and B1 errors helps make these synchronized echo approaches very robust even at the ultra-low (6.5 mT) field strengths used in the present work. Comparisons of dynamics between the synchronized echo and SLIC methods might also provide insights into different relaxation rates between dressed state populations versus coherences.

Materials and Methods
SLIC and synchronized echo measurements were performed on neat ethyl acetate, anhydrous ethanol, and a mixture of 46% ethanol with 54% deionized water (%w/w). All solvents were purchased from Sigma Aldrich.
Spectra at 276 kHz (6.5 mT) were measured in a custom-built high-homogeneity electromagnetbased MRI scanner with a Tecmag Redstone TM console described previously [30]. For the presently described work, a solenoidal sample coil was used, designed to hold 10 mm NMR tubes, and a B0 field-frequency lock was used to maintain the resonance frequency within ±0.25 Hz. The scanner was shimmed to achieve a linewidth of deionized water of better than 0.5 Hz.
RF pulses directly from the synthesizer were used, resulting in a 90˚ pulse length of 1 ms using about 4 µW. In most cases, a simple 2 ms 180˚ hard pulse was used for the spin echo pulse train, The actual pulse delays were adjusted appropriately for the pulse width so that the time between the 180˚ pulse centers was 2τ. We used a total pulse train time T = 1 s with n = 2 to 60 loops or T where is the spin-lock nutation frequency, and A, B, C, and D are constants. Normally A was between 0 and 1, B was between 1 and 10, ≈ 0, and D was between 0 and 1.5.
To construct a 2D spectrum from echo acquisitions, the magnitude of each echo was determined From the 2D dataset, a 1D J-coupling spectrum was created by averaging across the time domain dimension, then dividing by the maximal signal from the whole set of averages, in the same manner as for FID acquisitions. The T2 background was also removed in the same way.
Contrast-to-noise ratio (CNR) was calculated for ethyl acetate to compare the quality of the