The two-spin solid effect (2SSE) is one of the established continuous wave dynamic nuclear polarization mechanisms that enables enhancement of nuclear magnetic resonance signals. It functions via a state-mixing mechanism that mediates the excitation of forbidden transitions in an electron–nuclear spin system. Specifically, microwave irradiation at frequencies ωμwω0S ± ω0I, where ω0S and ω0I are electron and nuclear Larmor frequencies, respectively, yields enhanced nuclear spin polarization. Following the recent rediscovery of the three-spin solid effect (3SSE) [Tan et al., Sci. Adv. 5, eaax2743 (2019)], where the matching condition is given by ωμw = ω0S ± 2ω0I, we report here the first direct observation of the four-spin solid effect (4SSE) at ωμw = ω0S ± 3ω0I. The forbidden double- and quadruple-quantum transitions were observed in samples containing trityl radicals dispersed in a glycerol–water mixture at 0.35 T/15 MHz/9.8 GHz and 80 K. We present a derivation of the 4SSE effective Hamiltonian, matching conditions, and transition probabilities. Finally, we show that the experimental observations agree with the results from numerical simulations and analytical theory.

Dynamic nuclear polarization (DNP) is a technique that enhances nuclear magnetic resonance (NMR) signal intensities by transferring polarization from unpaired electrons to nearby nuclei. It requires microwave irradiation of electron–nuclear transitions and can yield a theoretical maximum enhancement factor of ɛ ∼658 for electron and 1H spin pairs.1–3 The excellent sensitivity bestowed by DNP results not only in a significant saving in measurement cost and time but also allows important structural information4,5 to be extracted from samples that were previously inaccessible. Hence, DNP has been widely applied in studying membrane proteins,6–8 amyloid fibrils,9–11 inorganic materials, and medical imaging.2,12 Despite the abundant successful applications, the details of the complete polarization-transfer steps and the factors affecting the DNP efficiency are still not yet fully understood. For instance, the observation of the Overhauser effect DNP in insulating solids, the attenuation of the cross effect (CE) DNP enhancements at higher fields, and the absence of a complete quantum-mechanical description involving multiple electrons in thermal mixing (TM) DNP are examples of the complexity of DNP that remains poorly understood.13–19 

In addition to the DNP mechanisms mentioned earlier, the two-spin solid effect (2SSE) is another DNP process involving a single electron and nuclear spin. It mediates polarization transfer via excitation of forbidden electron–nuclear transitions when the microwave frequency, ωμw, is set to the sum or difference of the electron and nuclear Larmor frequencies, i.e., ω0S ± ω0I.20–23 One of the reasons that the 2SSE is relatively less exploited than other DNP mechanisms is the modest DNP enhancement factor ɛ. A perturbation-theory-based calculation shows that the enhancement goes as ω1S/ω0I2 and, in the last decade, ω0I/2π has increased from 200 to 800 MHz, whereas ω1S/2π has remained constant at ∼1 MHz. Thus, improved microwave sources, such as gyroamplifiers, are required to bring the 2SSE into more general usage. Furthermore, as higher Rabi fields become available, higher order solid effects could likely emerge as viable nuclear signal enhancement mechanisms.

However, at low fields, where large electron Rabi fields, ω1S, are currently available, we have recently reported efficient 2SSE performance with > 250 at 0.35 T with ω1S/2π ∼ 3.2 MHz using the normal SE condition (ωμw = ω0S ± ω0I). Additionally, the microwave field profile also shows significant peaks with ɛ > 100 at the ω0S ± 2ω0I positions, which we attributed to the three-spin solid effect (3SSE).24 The phenomenon was then theoretically analyzed for a three-spin system comprised of one electron and two nuclei. The analytical theory predicted that the effect is dependent on the number of 1H spins surrounding the unpaired electron—a discovery that we exploited to probe the size of the spin diffusion barrier exhibited by the trityl radicals.24–27 The finding was recently corroborated by other approaches.28,29 The observation of the 3SSE prompted us to search for other higher-order transitions, and we report here our recent observation of the four-spin solid effect (4SSE), i.e., a forbidden transition present only in a four-spin system comprised of one electron and three nuclei.

Using a critically coupled microwave cavity and a long DNP buildup time τ ∼ 20 s,24 we observed excellent enhancement factors for the 2SSE (ɛ ∼300) and 3SSE (ɛ ∼150). Accordingly, we actively searched for the presence of the 4SSE DNP by setting the microwave offset frequencies, Ω, at three times the 1H Larmor frequency away from the EPR line, i.e., at Ω = ω0Sωμw = ±3ω0I. Figure 1 shows the full DNP field profile of trityl radicals dispersed in d8-glycerol/D2O/H2O (6:3:1 by volume) performed with a 0.35 T/9.8 GHz/15 MHz pulsed EPR/DNP spectrometer. After optimizing the microwave coupling and the NMR detection circuitry (see the “Experimental Methods” for details), we recorded DNP profiles for OX063 and Finland trityl samples. The experimental data shown in the region of Ω/2π35 MHz were obtained by sweeping the microwave frequency using the ELDOR channel while maintaining the magnetic field constant, and eight scans were accumulated per data point. To ensure the optimum DNP performance, the data points in Ω/2π35 MHz region were acquired by sweeping the magnetic field while keeping the microwave cavity critically coupled, and 64 scans were collected at each field position. Moreover, we used long DNP polarization times—20 s for OX063 and 33 s for Finland—to ensure that the DNP-enhanced NMR signal is saturated.

FIG. 1.

(a) The DNP field profiles of the OX063 (blue) and Finland (red) trityl DNP samples. Note that the EPR spectrum of Finland trityl (black) has an arbitrary unit on the y-axis, and it has a similar spectrum as trityl OX063. (b) shows an enlarged scale of the Ω/2π35 MHz region shown in (a). The vertical dashed lines show the theoretically predicted 4SSE conditions at Ω = ±3ω0I.

FIG. 1.

(a) The DNP field profiles of the OX063 (blue) and Finland (red) trityl DNP samples. Note that the EPR spectrum of Finland trityl (black) has an arbitrary unit on the y-axis, and it has a similar spectrum as trityl OX063. (b) shows an enlarged scale of the Ω/2π35 MHz region shown in (a). The vertical dashed lines show the theoretically predicted 4SSE conditions at Ω = ±3ω0I.

Close modal

The DNP sample, instrumentation, and NMR pulse sequence [Fig. 2(a)] described here are similar to those used previously.24 The DNP sample is comprised of 5 mM OX063 or Finland trityl dispersed in a mixture of d8-glycerol/D2O/H2O (6:3:1 by volume), i.e., “DNP juice,” at 80 K. The experiments were performed on an X-band spectrometer (0.35 T/9.8 GHz/15 MHz for 1H) using an ENDOR probe (Bruker EN 4118X-MD4) with a home-built tuning-and-matching circuit. Note that it is crucial to obtain the highest Q value on the microwave channel via critical coupling so that the electron Rabi field can efficiently excite the strongly forbidden 4SSE transitions.30 We obtained an electron Rabi field of ω1S/2π ∼ 3 MHz using ∼10 W from a Bruker AmpX10 amplifier, and the DNP field profiles were obtained by sweeping the static B0 field while keeping the microwave irradiation ωμw frequency constant. In our original instrument arrangement, where an over coupled configuration was used, the electron Rabi field was lower (ω1S/2π ∼2 MHz), and we could not observe the 4SSE. Moreover, a significant improvement in NMR sensitivity and stability was achieved by terminating the EPR modulation coil in the ENDOR probe with an electrical short. This effectively minimizes the crosstalk between the NMR coil and the field modulation coil, suppressing the noise level by a factor of ∼20. Subsequently, a μw-off signal could be obtained within ∼10 minutes (32 scans with a recycle delay of 20 s). We also measured the T1n of the bulk 1H in the OX063 and Finland trityl DNP samples to be 26 and 22 s, respectively.

FIG. 2.

(a) Schematic diagram of DNP pulse sequence used in this work. The 4SSE performance at the DQ peak Ω/2π ∼ 43 MHz was measured as a function of the (b) microwave Rabi field, (c) polarization time τ, and (d) delay d. All experiments were performed on Finland trityl, and the data were fitted with simple single exponential curves (red) to guide the eye. The exponential rate constants obtained are 2.3 MHz, T1DNP ∼ 22 s, and T1n ∼ 20 s, respectively. The Rabi field was measured by performing EPR nutation experiments.

FIG. 2.

(a) Schematic diagram of DNP pulse sequence used in this work. The 4SSE performance at the DQ peak Ω/2π ∼ 43 MHz was measured as a function of the (b) microwave Rabi field, (c) polarization time τ, and (d) delay d. All experiments were performed on Finland trityl, and the data were fitted with simple single exponential curves (red) to guide the eye. The exponential rate constants obtained are 2.3 MHz, T1DNP ∼ 22 s, and T1n ∼ 20 s, respectively. The Rabi field was measured by performing EPR nutation experiments.

Close modal

We will analytically derive the effective Hamiltonian, matching condition, and transition probability of 4SSE DNP. We begin by writing down the lab-frame Hamiltonian of a four-spin e–1H–1H–1H system,

Ĥ=ΩŜz+ω0IÎ1z+Î2z+Î3z+ŜzA1Î1z+A2Î2z+A3Î3z+ŜzB1Î1x+B2Î2x+B3Î3x,
(1)

where ω0I = −γB0 is the nuclear Larmor frequency, γ is the gyromagnetic ratio (γ1H/2π ∼ 42.5775 MHz/T for 1H), and Ω is the microwave offset frequency. A and B represent the secular and pseudo-secular hyperfine coupling of the respective e–1H pair. We have neglected the 1H–1H dipolar couplings here because their magnitude is small (∼10 kHz) compared to the hyperfine interactions (1 MHz), and their absence simplifies the theoretical treatment. Next, we diagonalize the Hamiltonian Ĥ [Eq. (1)] to determine the energy eigenstates of the four-spin system. As the off-diagonal term B1ŜzÎx does not commute with the nuclear Larmor term ω0IÎz, we apply a unitary transformation on the basis set following standard procedures (see Appendixes B.2.5 of Schweiger and Jeschke31), i.e., we state the ansatz that Ĥ can be diagonalized by performing a unitary transformation ĤΛ=ÛĤÛ1, where Û is given by

Û=expik=1,2,3ηkαŜαÎky+ηkβŜβÎky,Ŝα=1̂2+Ŝz;Ŝβ=1̂2Ŝz.
(2)

Then, it can be shown that the ansatz is correct if the angles ηkα/kβ fulfill the following conditions:

ηkα=tan1BkAk+2ω0I,ηkβ=tan1BkAk2ω0I,
(3)

where 1̂ denotes the identity operator. We implemented and expanded the diagonalization procedure to the four-spin system using Mathematica (Wolfram Research) to obtain the energy eigenvalues. Moreover, we also neglected the secular hyperfine couplings Ak to simplify the calculations. These terms can be safely ignored because they have negligible influence if the condition Akω0I, ω1S is satisfied.15 Following that, we obtained the matching conditions for the two quantum (2Q) αβββΛβαααΛ and four quantum (4Q) ααααΛββββΛ 4SSE transitions,

Ω=±k=1,2,3ω0I1+Bk2/4ω0I2±3ω0IifBkω0I,
(4)

and the corresponding 4Q and 2Q transition amplitudes upon microwave excitation with Rabi field, ω1S, are

ai,j=ψiΛ|Ûω1SŜxÛ1|ψjΛ,
(5)
a1,16=a8,9=B1B2B3ω1S24+B12/ω0I24+B22/ω0I24+B32/ω0I2ω0I3B1B2B3ω1S16ω0I3ifBω0Iandω1Sω0I,

where ψΛ are the eigenstates in ĤΛ. Next, we apply Fermi’s golden rule, which states that the transition rate between the energy eigenstates (in the case of weak perturbation) is proportional to the square of the matrix element,22 

PDQa8,92=B12B22B32ω1S2256ω0I6,PQQa1,162=B12B22B32ω1S2256ω0I6.
(6)

Hence, the results show that the 4SSE DNP transfer efficiency has a ω0I6 dependence compared to the εω0I2 relation exhibited by the conventional 2SSE. Besides that, Eq. (5) reveals that 4SSE depends on the product of three e–1H dipolar couplings, i.e., its DNP efficiency is exceptionally sensitive to the number of 1H spin close to the unpaired electrons.

To confirm our hypothesis that the observed DNP peaks at Ω/2π 45 MHz are indeed due to the 4SSE, we performed numerical simulations on a four-spin e–1H–1H–1H system using a customized DNP simulation package in MATLAB.24,30,32 Indeed, the simulated DNP field profile (Fig. 3) shows that two peaks are observed at Ω/2π 45 MHz. We note that the two peaks are not necessarily symmetric, a feature that was also observed in experimental results. Moreover, the widths of the simulated peaks are narrower than the experimental results [Fig. 1(b)], probably because we have not included a sufficient number of 1H spins and 1H–1H dipolar couplings in the simulations. Additionally, we intentionally set one of the e–1H hyperfine couplings to zero in the simulations and observed that the peaks at Ω/2π 45 MHz vanish (data not shown). This confirms that at least three 1H spins nearby the electron are required to yield the 4SSE.

FIG. 3.

Simulated (a) quadruple-quantum (4Q) ααααΛββββΛ and (b) double-quantum (2Q) 2SSE αβββΛβαααΛ DNP using the custom MATLAB program. The simulations were performed using 144 powder orientations, γB1/2π ∼ 4.2 MHz, T1e ∼ 1 ms, T2e ∼ 5 µs, T1n ∼ 13 s, T2n ∼ 1 ms, and DNP polarization time τ ∼ 41 ms. All three e–1H distances were set to 4 Å, but with different sets of Euler angles in the principal-axis system (PAS), i.e., [α, β, γ] = [0, 0, 0]°, [5, 20, 5]°, and [10, 10, 10]°. The 4 Å distance was chosen as it represents the effective e–1H distance used in the literature.24,33,34 The electron g-tensor used is [2.0039, 2.0038, 2.0037] with Euler angles of [0, 40, 0]°. Note that 1H–1H couplings are not considered here.

FIG. 3.

Simulated (a) quadruple-quantum (4Q) ααααΛββββΛ and (b) double-quantum (2Q) 2SSE αβββΛβαααΛ DNP using the custom MATLAB program. The simulations were performed using 144 powder orientations, γB1/2π ∼ 4.2 MHz, T1e ∼ 1 ms, T2e ∼ 5 µs, T1n ∼ 13 s, T2n ∼ 1 ms, and DNP polarization time τ ∼ 41 ms. All three e–1H distances were set to 4 Å, but with different sets of Euler angles in the principal-axis system (PAS), i.e., [α, β, γ] = [0, 0, 0]°, [5, 20, 5]°, and [10, 10, 10]°. The 4 Å distance was chosen as it represents the effective e–1H distance used in the literature.24,33,34 The electron g-tensor used is [2.0039, 2.0038, 2.0037] with Euler angles of [0, 40, 0]°. Note that 1H–1H couplings are not considered here.

Close modal

Figure 1(a) shows that the normal 2SSE and 3SSE peaks are very well reproduced at the Ω = ±ω0I and Ω = ±2ω0I positions. Moreover, the DNP enhancement on OX063 is significantly higher than our previously reported results,24 most likely due to a longer DNP polarization time of 20 s instead of 8 s. The microwave offset frequency is referenced to be Ω = 0 at the peak EPR intensity of the Finland trityl radical. More importantly, we observed two weak and broad peaks at microwave offset frequencies of Ω/2π ∼ 43 and −45 MHz [Fig. 1(b)] on both DNP samples. These peak positions are indeed very close to the predicted 4SSE peaks at Ω = ±3ω0I, where ω0I/2π ∼ 14.8 MHz. The peak at the 4Q position, Ω/2π ∼ –45 MHz, matches well with the theoretical prediction, and a slight offset of 1.5 MHz was observed for the 2Q peak at Ω/2π ∼ 43 MHz. We suspect that the origin of the small offset—relative to the width of the EPR spectrum of ∼5 MHz—could be due to the g-anisotropy, shift induced by higher-order Hamiltonians, or large electron–electron couplings in trityl aggregates. Although this effect is beyond the scope of this manuscript, we are planning further investigations of its origin.

We also notice that the Finland trityl has a significantly stronger 4SSE enhancement relative to OX063, despite a weaker enhancement for the 3SSE and the 2SSE conditions. This is an intriguing result given that the experiments were performed using the same experimental conditions, solvent, and spectrometer. The larger enhancement exhibited by the Finland sample at the 4SSE condition implies that the high density of nearby 1H nuclei (methyl groups) on the trityl molecule might be crucial in mediating the e1H–1H–1H SE DNP mechanism. Although the OX063 trityl molecule has more 1H nuclei than Finland, the methylene groups are more distant from the central carbon atom—where the unpaired electron primarily resides.33,34 Hence, overall electron–nuclear dipolar couplings could be smaller for OX063 and explains its weaker 4SSE performance, as suggested in Eq. (5).

To further characterize the 4SSE DNP mechanism, we measured the DNP enhancement as a function of the microwave Rabi field γB1 [Fig. 2(b)]. As expected from the low 4SSE transition probability [Eq. (5)], we observed that the DNP performance is not yet saturated despite applying γB1/2π ∼ 3 MHz—a field strength sufficient to saturate both the 2SSE and 3SSE DNP.24 Besides that, we also characterized the DNP buildup time T1DNP of the 4SSE to be ∼22 s [Fig. 2(c)], which is only slightly longer than the T1n ∼ 20 s [Fig. 2(d)] observed on the bulk 1H nuclei. These data imply that the 4SSE DNP process likely falls in the “relaxation-limited regime” where the DNP polarization could not build up further because any further polarization gain is offset by the faster 1H relaxation process.35 In such situations, what we measured is an apparent T1DNP value that is unsurprisingly similar to the T1n value. We expect the 4SSE DNP performance could be improved by increasing the electron Rabi field and/or lengthening the relaxation times T1n by lowering the temperature. This will allow us to determine the non-relaxation-limited T1DNP value that better matches the theoretically predicted transition probabilities.

In conclusion, we have experimentally observed the 4SSE mediated by an e1H–1H–1H spin system. The observation is explained with an analytical theory and numerical simulations. To the best of our knowledge, this is the first report of such an effect in the DNP-NMR literature, and it illustrates that the “solid effect” consists of a family of matching conditions spaced by the nuclear Larmor frequency, ω0I. We note a similar four-spin mechanism reported on a system involving two electrons, a proton, and a carbon spin, i.e., a four-spin cross effect.36 On the other hand, Agarwal has reported recently an observation of negative cross peaks in a 2D spin diffusion MAS NMR experiment on histidine, which was attributed to a four-spin effect in a third-order Hamiltonian.37 As DNP/NMR technology continues to develop and improve, we expect higher sensitivity and resolution NMR spectra will become more readily available, and additional similar higher-order effects might be observed. We anticipate these findings could potentially assist in explaining anomalous experimental observations.

K.O.T. acknowledges the support from the French National Research Agency (Grant No. ANR-20- ERC9-0008) and RESPORE (Grant No. 339299). This work was supported by the National Institutes of Health through grants to R.G.G. (Grant Nos. GM132997, AG058504, and GM132079).

The authors have no conflicts to disclose.

The data that support the findings of this study are available within the article.

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