Spin-Noise-Detected Two-Dimensional Fourier-Transform NMR Spectroscopy.

We introduce two-dimensional NMR spectroscopy detected by recording and processing the noise originating from nuclei that have not been subjected to any radio frequency excitation. The method relies on cross-correlation of two noise blocks that bracket the evolution and mixing periods. While the sensitivity of the experiment is low in conventional NMR setups, spin-noise-detected NMR spectroscopy has great potential for use with extremely small numbers of spins, thereby opening a way to nanoscale multidimensional NMR spectroscopy.

Felix Bloch predicted the existence of spin-noise in 1946.[1] Experimental verification had to wait until 1985[2] due to the low amplitude of the phenomenon. Recent progress in magnetic resonance instrumentation, in particular, cryogenically cooled probes[3] and force-detected magnetic resonance as well as extensive work on one-dimensional spin-noise spectroscopy,[4−8] has opened new possibilities for in-depth investigation of the physical phenomenon and the exploration of its application potential. For a recent review, see ref (9).

In the present context, we focus on the transverse components of spin-noise as opposed to the longitudinal component, which is exploited in the force-detected magnetic resonance experiments.[5] The fluctuating transverse spin-noise exhibits random phase contributions, which average to zero with the transverse relaxation time constant of T2 (or T2* if inhomogeneous broadening prevails). Noise blocks therefore would have an expectation value of zero due to cancellations, but averaging over the signal magnitude or power can be used for signal accumulation. The earlier described spin-noise spectroscopy[10−16] and imaging experiments[17] were obtained by Fourier-transforming individual noise blocks and co-adding the magnitude or power representations of these data. This is equivalent to computing the Fourier-transform of the autocorrelation of the spin-noise as described by the Wiener–Khintchine theorem (WKT).[18,19] Both procedures can be used for signal accumulation by avoiding cancellations due to the random phases that are exhibited by the (uncorrelated) noise signals.

To obtain two-dimensional Fourier-transform spin-noise NMR spectra, we use the following basic concept, illustrated in Figure 1.

Figure 1

Acquisition scheme for noise-detected two-dimensional NMR. The general scheme consists of an evolution time t1 sandwiched between two mixing periods τ1 and τ2 and two noise acquisition periods t0 and t2, during which the noise blocks m0(t0) and m2(t2) are acquired in an identical manner. In the spin-noise-HMQC (snHMQC) pulse sequence, used to demonstrate and test the concept of spin-noise-detected two-dimensional NMR spectra, black rectangular bars represent 90° hard pulses on the 13C channel. The hatched rectangles represent periods of heteronuclear decoupling (WALTZ). No pulses are applied on the 1H channel.

Two noise blocks, m0(t0) and m2(t2), are recorded in identical fashion similar to the CONQUEST paradigm.[5,20] Between the two noise blocks, one places an evolution period, as usual in multidimensional NMR, bracketed by two mixing periods. Cross-correlating the time domain noise blocks m0(t0) with m2(t2) for each t1 value yields a conventional two-dimensional time domain NMR data set that can be processed in the usual manner. It is however crucial that the cross-correlation (either in the time or frequency domain) is performed prior to signal averaging. The experiment depicted in Figure 1, as a first demonstration of the principle of indirect detection by spin-noise, correlates spin-noise-detected 1H chemical shifts with heteronuclear multiple-quantum coherence without applying any rf pulses on the 1H channel, henceforth called a 1H spin-noise-HMQC (snHMQC) experiment. We use the symbols I and S for the nuclei 1H and 13C, respectively, in the following analysis. During the first acquisition period t0, a 1H noise block m0 is recorded, while decoupling the S spins, and the recorded signal can be described aswhere a0(ta) describes the complex-valued random emission amplitude of an emission event (we do not distinguish between induced and spontaneous emission in the present context) at time point ta, ΩI is the I-spin resonance frequency, R2 = 1/T2 or 1/T2* depending on which regime applies, and n0(t0) is the background broad-band noise (which may be assumed to be white noise over the observed spectral width) amplitude from the instrument or other sources. Heteronuclear decoupling of spin-noise signals works as expected; therefore, there is no coupling modulation in eq 1. It was shown previously that heteronuclear decoupling does not cause spectral interference with the hardware used.[16]

At the end of this first acquisition interval, t0 decoupling is turned off, and the surviving coherences arising from the incomplete cancellation of spin-noise transverse components start evolving under heteronuclear coupling constant JIS. At the end of the τ1 = (2JIS)−1 period, heteronuclear antiphase coherence is generated. The 90° pulse on the S channel transforms it into a superposition of two spin coherences (double- and zero-quantum coherences), which evolve during t1. The second 90° pulse regenerates antiphase I-spin single-quantum coherence, which is refocused to in-phase transverse magnetization in the second delay τ2 = (2JIS)−1. Focusing on the pathways of interest, the measured signal during the t2 period can be written aswhere t0max is the maximum acquisition time, which is equal for the periods t0 and t2 in this experiment, and ΩS is the Larmor frequency of spin S. The transfer coefficient f(τ1, τ2) includes the effects of deviations from ideally matched delays. RDQ and RZQ are the double- and zero-quantum relaxation rates, respectively, n2(t2) is the background noise amplitude from the instrument or other sources, b represents the contributions to the signal from the emission events occurring in the period between the two direct acquisition periods t0 and t2, and the integral encompassing c represents contributions to the signal from emission events originating at time tc within the t2 period. Decoupling of the S spins is switched on during t0 and t2. Hence, there is no coupling evolution during this period.

The desired signals can now be “distilled” from the data in the two measurement blocks by calculating the correlation function m0(t0) ⊗ m2(t2). It is convenient to calculate the Fourier-transform of this correlation function because by the WKT, it is equal to the product of the Fourier-transforms of the time domain functionswhere FT symbolizes the Fourier-transform, and we use the common convention that capital letter variables with angular frequency arguments represent the Fourier-transforms of the corresponding lowercase quantities.

Because m0 and the desired “a” signals in m2 are themselves convolutions (WKT), their Fourier-transforms are found readily asandwith L being the complex Lorentzian line shape functionWe then obtain for the Fourier-transform of the correlation functionThe cross-terms between the A, b, C, and the different N terms are completely uncorrelated and can be averaged out to arbitrary precision by accumulating the cross-correlation function (or its Fourier-transform) over many acquisitions. Only the first summand in eq 7, which contains the square of the random amplitude |A0(Ω)|2 and represents the correlated signals, increases linearly with the number of co-added cross-correlated data blocks. The uncorrelated terms (the other summands) only grow with the square root of this number. A further Fourier-transform with respect to t1 gives the correlation peaks of interest in the formwhere we have neglected the cross-terms and used the correspondence Ω2 = Ω to conform with multidimensional NMR conventions. It is seen here that heteronuclear cross-peaks are obtained at the coordinates (ΩI ± ΩS, ΩI), as expected.

In Figure 2, we show an experimental two-dimensional snHMQC spectrum of 13C-enriched glucose acquired with the scheme of Figure 1 and processed using a TopSpin 3.1 “au”-program (written in C) and a Matlab script implementing eqs 7 and 8, all available in the Supporting Information.

Figure 2

(a) Spin-noise-detected two-dimensional HMQC (snHMQC) spectrum of 99% 13C-enriched glucose in 2H2O. For the spectrum shown here, 6000 passes were co-added. Processing is described in the text. The dashed vertical line indicates the position of the 1HO2H-t1 noise artifacts. The two circled regions illustrate the positions of the zero- and double-quantum coherence cross-peaks of the anomeric C–H positioned at f1 = ΩH-1 ± ΩC-1, f2 = ΩH, respectively. Red and blue contour lines represent positive and negative levels, respectively. The residual solvent signal gives rise to a peak at zero frequency in f1, and several truncation artifacts are visible at that frequency due to the short maximum evolution time. (b) A 13C decoupled 1H single pulse spectrum is shown for reference.

The spectrum has been acquired with low resolution in both dimensions in order to reduce relaxation losses. As predicted above, two cross-peaks are observed for each 1H resonance, corresponding to the heteronuclear zero- and double-quantum frequencies. The spectrum is phase-sensitive in the indirect dimension, while in the direct dimension, only a real part exists. In the f1 dimension, peak splitting occurs due to homonuclear carbon and proton couplings. Due to the evolution of chemical shifts and spin–spin couplings prior to the indirect detection period t1, a mixed phase spectrum is obtained with the pulse scheme in Figure 1.

The fact that spin-noise can have an effect strong enough to drive a two-dimensional coherence transfer experiment will come as a surprise for many readers. Therefore, in the Supporting Information, we give an estimate of the magnitude of the spin-noise signal under the conditions of our experiment in Figure 2, following the basic ideas by Guéron and Leroy[11] as well as Hoult and Ginsberg.[21] It should also be mentioned that stochastic excitation for obtaining two-dimensional correlation NMR spectra, albeit using a pseudorandom number generator to drive the rf excitation pulses, has been introduced by Blümich and co-workers earlier.[22,23]

Because the repetition interval of spin-noise-detected experiments is completely independent of the 1H longitudinal relaxation time T1, the acquisition schemes can be repeated as fast as the hardware allows. Still, the per root-of-transient-number signal-to-noise ratio in the spectra obtained is very low compared to pulse excitation spectra because the separation of the small amount of correlated noise from uncorrelated noise depends on the amount of signal averaging. Several enhancements to the efficiency of the acquisition scheme are currently under investigation. However, one should bear in mind that by extrapolating to lower spin numbers, noise power amplitudes will exceed pulse excitation amplitudes originating from natural polarization.[17,24]

On the basis of the noise-correlation principle outlined here, a large range of multidimensional magnetic resonance experiments becomes feasible. In principle, any coherence transfer pulse sequence can be modified accordingly as long as the relaxation times permit. In a potential application at the nanoscale, that is, below ∼108 nuclear spins, application of rf pulses, in particular, refocusing pulses, on the detection channel will also be possible because the magnitude of the residual coherence generated will be below the thermal Curie law polarization. Thus, we expect substantial future developments of spin-noise-detected MR techniques at the nanoscale, which will not be restricted to heteronuclear correlation. The paradigm of detection through correlated spin-noise may very likely also find useful applications in optically detected NMR spectroscopy, in particular, diamond-NV-center experiments,[25−28] and could also be implemented in optical multidimensional noise spectroscopy.[29,30]

Experimental Methods

The experiments were performed on a 700 MHz Bruker Avance III system equipped with a TCI cryo-probe. The spin-noise-detected two-dimensional HMQC (snHMQC) spectrum shown was recorded on 99% 13C-enriched glucose in 2H2O (0.648 mol L–1) in a 5 mm NMR tube using the scheme of Figure 1, with the following parameters: t0 = t2 = 27 ms, 1H spectral width 9.5 kHz, maximum t1 = 3.78 ms, 13C spectral width 10.6 kHz, 90°(13C) pulse 12.6 μs, repetition delay 250 ms. The mixing times τ were 1.72 ms. One pass through all 80 t1 values of the sequence thus takes 27 s. Because no refocusing pulses could be used on the 1H channel (to avoid generation of spurious coherence), all acquisition and evolution times are generally short as T2* rather than T2 determines the loss of coherence in this particular experiment. For the spectrum shown in Figure 2, 6000 passes through the pulse sequence were co-added. Processing is described in the text. The 2H2O deuterium signal was used for field frequency locking.

To process the data, TopSpin 3.1 C-programs and Matlab scripts, which are available in the Supporting Information, were used. First, Fourier-transformation along the direct dimension was performed in TopSpin 3.1 (by the command xf2). Then, the cross-correlation of the two noise blocks, m0 and m2, for each t1 time point was achieved by an in-house written “au”-program multiplying the transformed data blocks, point by point (see eq 3). Fourier-transformation along the indirect t1 dimension and addition of 6000 different experiments were done using a Matlab script.