Dynamic nuclear polarization by two-pulse phase modulation Open Access

Two recent reviews, covering applications in materials science and structural biology, leave no doubt about the added value of combining high-resolution magic-angle spinning (MAS) nuclear magnetic resonance (NMR) with dynamic nuclear polarization (DNP).^1,2 MAS NMR provides detailed information on the local chemical environment of observed nuclei in solids, even if they are amorphous or disordered. DNP adds the extra sensitivity needed to investigate species that inherently occur at low concentrations such as surface sites, intermediates, and (transient) molecular structures embedded in a large protein matrix. DNP has drastically broadened the scope of MAS NMR, and the list of new applications is growing steadily.

Current commercial MAS DNP instrumentation, up to 900 MHz/21.1 T/592 GHz, has been designed around classical DNP mechanisms like the solid-effect³ and the cross-effect.^4,5 Transfer of electron spin polarization, usually to protons, relies on continuous high-power microwave irradiation at or near the electron Larmor frequency, which is generated by dedicated gyrotrons.^6–8 Output power is tens of watts, which converts into an electron nutation frequency of an MHz or so in the MAS rotor.^9,10 This means that compared to the other terms in the spin Hamiltonian, the term describing microwave irradiation is weak or of the same order.

An alternate approach to DNP is to transfer electron polarization to nuclei by means of a pulse sequence. With the help of spin dynamics and optimal control, the polarization transfer can be made more efficient, and higher gains in sensitivity can be obtained at reduced average microwave power. During the pulse sequence, however, strong microwave irradiation is needed. Perturbation theories that describe the transfer of polarization by a DNP pulse sequence are only effective when the electron nutation frequency is large (5–10 MHz at least) compared to the electron–nuclear spin interactions. Above 95 GHz, generating high-power microwave pulses with a well-defined frequency and phase is technically challenging, but research is progressing^11,12 and prototypes^13,14 exist. Still, for the time being, pulsed DNP experiments are restricted to low magnetic fields.

Considerable progress has been made in the development of DNP pulse sequences in recent years. In the original DNP pulse sequence NOVEL (Nuclear-spin Orientation Via Electron-spin Locking),¹⁵ a spin lock is applied to the electrons at a nutation frequency that matches the proton Larmor frequency, $ω1S=ω0I$⁠. The sequence is very efficient at low magnetic fields, e.g., at 15 MHz/0.34 T/9.7 GHz (X band),^16,17 but at fields relevant for high-resolution MAS NMR, the NOVEL matching condition demands extremely high microwave power during the lock pulse. This problem was first addressed with the off-resonance NOVEL sequence.¹⁸ By considering the generalized matching condition $ω1S2+Ω2=ω0I$⁠, a low nutation frequency can be compensated by off-resonance irradiation of the electron spins (Ω = ω_0S − ω_μw ≠ 0). Experiments at 0.34 T showed that NOVEL remains efficient with this relaxed requirement for the electron nutation frequency.

An alternative strategy, which provides further flexibility in the choice of the nutation frequency, was introduced with the TOP (Time-Optimized Pulsed) DNP sequence (Fig. 1).¹⁹ This sequence induces a transfer of polarization when a multiple of the modulation frequency of the sequence (ω_m = 2π/t_m) plus or minus the effective frequency (the effective rotation angle of the electron spin magnetization imposed by each block of the sequence, ω_eff = β_eff/t_m) matches the proton Larmor frequency, i.e., nω_0I + kω_m + lω_eff = 0 with n, l = ±1, k = ±1, …, ±∞. In experiments at 51 MHz/1.2 T/34 GHz (Q band), an enhancement factor of 100 was obtained with the TOP DNP sequence at a nutation frequency of 20 MHz. Last year, we introduced the XiX (X-inverse-X) DNP sequence (Fig. 1).²⁰ The matching condition for XiX DNP has the same form as for TOP DNP, but in experiments at 1.2 T, again at a nutation frequency of 20 MHz, XiX produced a two-times higher gain in sensitivity than TOP DNP. We attributed this to a better preservation of the electron–nuclear dipolar coupling in the effective Hamiltonian, which leads to a faster electron–proton polarization transfer under the influence of the XiX sequence. More recently, an adiabatic implementation of the BEAM (Broadband-Excitation by Amplitude Modulation, see Fig. 1) DNP sequence resulted in an enhancement factor of 360 at 0.34 T.²¹ The BEAM sequence follows the same matching condition as TOP and XiX but, nevertheless, a high nutation frequency of 32 MHz (relative to a proton Larmor frequency of 15 MHz) was needed in these experiments. This makes it clear that the suitability of a DNP pulse sequence for enhancing the sensitivity of high-resolution MAS NMR is not determined by the matching condition alone. Instead, it is necessary to investigate the efficiencies of the DNP conditions of a sequence and their dependence on the nutation frequency individually.

Here, we introduce a new member of the TOP/XiX/BEAM family of DNP pulse sequences. The sequence is termed Two-Pulse Phase Modulation (TPPM) DNP (Fig. 1) after the analogous heteronuclear decoupling sequence that is ubiquitously used in MAS NMR.²² We give a general theoretical description for electron–proton polarization transfer by periodic DNP pulse sequences using operator-based, triple-mode Floquet theory^23,24 and investigate in detail the DNP conditions of the TOP, XiX, and TPPM sequences, including their nutation frequency dependence. In experiments at 1.2 T, the highest gain in sensitivity is obtained with TPPM DNP, but at a relatively high nutation frequency of 33 MHz. Interestingly, XiX DNP performs very well at a nutation frequency of only 7 MHz and hardly improves as the nutation frequency is increased. Experiments also show a correlation between a theoretically predicted fast electron–proton polarization transfer and a short build-up time of dynamic nuclear polarization in bulk. The factor by which the thermal nuclear polarization is finally enhanced, however, is determined by other properties of the sequence. We find, for example, that the concentration dependence of the enhancement factor is different for XiX and TOP DNP.

In this section, we give a (semi-)analytical description of electron–nuclear polarization transfer induced by a periodic microwave pulse sequence and provide specifics on the numerical simulations of this process. The analytical description is an adaptation of the operator-based triple-mode Floquet theory in combination with the Van Vleck–Primas perturbation approach,²⁵ developed by Scholz et al. for de- and recoupling experiments in MAS NMR,^23,24 and specifically the TPPM decoupling sequence.²⁶ This adaptation was first suggested to describe the TOP DNP sequence¹⁹ and is completed here. We emphasize that we consider for analysis only a single instance of the transfer of electron polarization to a nearby, directly dipolar coupled nuclear spin. Neither the repeated application of the microwave sequence that is required to build up dynamic nuclear polarization of the bulk nor the associated spin diffusion process are included in the modeling. This is an oversimplification, but it is a good starting point, as the efficiency of this initial step plays a crucial role in the pulsed DNP process.

The basic spin system for numerical simulations consists of an electron and a proton 3.5 Å apart, corresponding to a dipolar coupling of 1.8 MHz. The strength of the coupling was adjusted to reproduce the dependence of the enhanced ¹H NMR signal on the contact time (“contact curves”) in (off-resonance) NOVEL experiments with trityl OX063 in DNP juice at 0.34 T^17,18 and is considerably stronger than the individual electron–proton couplings observed in electron paramagnetic resonance (EPR) experiments on Finland trityl.^30,31 The reason is that the unpaired electron of trityl OX063 is surrounded by tens of protons at similar distances, which are simultaneously available for polarization transfer. Analogous to the intensity of the spin–flip lines in the EPR spectrum of trityl, this can be modeled by an increased effective dipolar coupling.^31–33 

To predict where the best pulsed DNP conditions will occur in the experiment, g-anisotropy (g_x = g_y = 2.003 19, g_z = 2.002 58) and a second proton, located at 3.5 Å from the electron and at 2.7 Å from the first proton, are included in the numerical simulations. In this case, $Iz$ with I_z = I_z,1 + I_z,2 (summing over the two ¹H spins) is calculated. To translate this to the enhancement factor ɛ_B (see Sec. III), $Iz(t)$ is divided by $Iz,1(0)$⁠, which is the proton z-magnetization at thermal equilibrium (note that in our spin system, the two protons are equivalent). Simulations were run in Hilbert space, and relaxation was not included unless noted otherwise. Typical two-dimensional parameter scans run readily on a workstation. However, extended sets of simulations of TOP/XiX/TPPM DNP were performed on the Scientific Compute Cluster (SCC) of the University of Konstanz.

Pulsed DNP experiments were performed at 34 GHz/1.2 T/51 MHz on a Bruker Elexsys E580 pulsed EPR spectrometer with Q-band extension (SuperQ-FT-u bridge). The pulse programmer was a PatternJet II with a 2 ns resolution. Pulses were amplified with a prototype BLA50 ESR 33–35 GHz 50 W amplifier (Bruker BioSpin GmbH). The microwave power is attenuated after amplification and only a maximum of about 20 W reaches the resonator. This typically generates a maximum electron nutation frequency of 40 MHz in the microwave resonator of the EN 5107D2 EPR/ENDOR probe when it is fully overcoupled. A few experiments reported herein were performed with an HA8019 33–35 GHz 10 W amplifier (HBH Microwave GmbH). With this amplifier, the maximum power at the resonator is about 4 W, and the maximum nutation frequency is about 19 MHz. Each second pulse in TPPM DNP was generated with an MPFU (microwave pulse forming unit), whose relative phase was calibrated against the x-channel of the spectrometer.

The EPR/ENDOR probe was made suitable for ¹H NMR at 51.7 MHz with a detachable tuning/matching box. NMR experiments were controlled by an iSpin-NMR console (SpinCore Technologies, Inc.), and RF pulses were amplified using a refurbished LPI-10 amplifier (E&I). The ¹H NMR signal was detected with a Hahn echo, $(t90)x−τ−(t180)x−τ−$⁠, with t₉₀ = 3.5 µs and the EXORCYCLE phase cycle.³⁵ For quantitative comparisons, τ was set to 40 µs, as this suppresses the proton background signal from the probe relative to the proton signal from the sample. A recycle delay of 44 s was used for these measurements. All other experiments were performed with a τ of 20 µs and a recycle delay of 24 s.

Samples consisted of a frozen glassy matrix of d₈-glycerol:D₂O:H₂O 60:30:10 v:v:v (a.k.a. “DNP juice”) doped with trityl OX063 (Polarize ApS).¹⁷ Echo-detected EPR spectra are shown in Fig. S1. Quartz sample tubes had an outer diameter of 1.6 mm and were fused on one end. The sample volume was 1–2 µl. During the experiments, the sample temperature was kept at 80 K using a CF 935 flow cryostat (Oxford Instruments) and He as a cryogen. The concentration of trityl OX063 was verified for all samples by UV/Vis absorption at 266 nm (molar absorption coefficient 35 000 M⁻¹ cm⁻¹)³⁶ using a Cary 60 spectrophotometer (Agilent Technologies).

The experimentally relevant parameter space of TPPM DNP at 1.2 T was explored in full by means of numerical simulations. ɛ_B was evaluated after a contact time of 800 ns in two-dimensional parameter scans in which the microwave resonance offset was varied from −110 to 110 MHz and the nutation frequency from 0 to 40 MHz. These parameter scans were performed for pulse lengths of 4, 6, 8, …, 82 ns and phase differences between the pulses of 5°, 10°, 15°, …, 180°. Conditions of interest, i.e., a particularly high $εB$⁠, are shown together in Fig. S2. Of the experimentally investigated conditions, the combination of t_p = 16 ns and ϕ = 120°, see the simulation in Fig. 2, was found to produce the highest enhancement factor.

Experimental investigation of conditions of interest involved an iterative process in which the nutation frequency, resonance offset, phase, and repetition time of the pulse sequence were subsequently and repeatedly optimized to produce the highest ¹H NMR intensity. For the 16 ns/120° condition, the results of the optimizations of the resonance offset and the phase are shown in Fig. 3, for a nutation frequency of 33 MHz. Upon optimization, this condition close to resonance (2–4 MHz) fared better than the condition further off resonance (15–25 MHz) at a nutation frequency of 38 MHz (see Fig. 2). The optimal phase difference was found to be 115°, not 120°, but this may be due to uncertainty in the phase calibration. The 16 ns/120° condition, after optimization, was better than the 14 ns/80° and 18 ns/155° conditions, even though the simulations suggest these are of similar or better quality (Fig. S2).

In a series of experiments on a sample containing 6.4 mM trityl [Fig. S1(a)], the gain in sensitivity obtained with TPPM DNP at the 16 ns/120° condition was compared to the gains in sensitivity obtained with XiX and TOP DNP. For the latter two sequences, the conditions were chosen as previously published: the nutation frequency was 18–19 MHz; for XiX,^20, t_p = 48 ns; and for TOP,¹⁹  t_p = 10 ns, d = 14 ns. The experimental field profile of XiX DNP was remeasured and shown in Fig. 4. Contrary to our earlier observation at 2.5 mM,²⁰ the Ω/2π = ±60 MHz conditions produce higher enhancement factors than the Ω/2π = ±39 MHz conditions. The optimized microwave-on signals of TOP, XiX, and TPPM DNP are shown together in Fig. S4. The spin-lattice relaxation time of the proton spins was also measured (Fig. S5), as were the bulk build-up times, T_BU, at each condition (Fig. S6). Enhancement factors, ɛ_B, can now be determined following Eq. (17) and are shown in Table I. Compared to XiX and TOP, TPPM DNP produces a 20% higher enhancement of proton polarization. The short T_BU of TPPM DNP (15 s) makes this improvement more pronounced when the practical sensitivity gain, $εBT1n/TBU$⁠, is considered (see Table I).

Figure 5 shows $Ion$ as a function of the repetition time of the TOP, XiX, and TPPM sequences. The optimal repetition time is considerably longer for TPPM (1020 µs) than for XiX and TOP DNP (102 and 204 µs). As a consequence, even though the nutation frequency required for the TPPM sequence to induce an e⁻–¹H polarization transfer is higher than for XiX and TOP DNP, the average applied microwave power is comparable (120 vs 72 and 93 mW). Interestingly, for TPPM DNP, choosing a repetition time shorter than 1020 µs reduces the build-up time of the bulk polarization (see Fig. S9). The decrease in $Ion$ is, however, not fully compensated, and the practical sensitivity gain is still reduced.

Parameter scans of XiX and TOP DNP show nearly vertical, narrow regions of positive and negative enhancements, which come together at high nutation frequencies [see the simulations in Figs. 6(a) and 6(b)]. This makes it possible to experimentally investigate the nutation frequency dependence of the DNP efficiency for these sequences in a straightforward manner, i.e., without changing other parameters. Enhanced ¹H NMR signals were measured at the conditions indicated by the arrows, following the Ω/2π = −60 MHz condition of XiX and the Ω/2π = +95 MHz condition of TOP. At each nutation frequency, the microwave resonance offset was optimized, and the resulting mini-field profiles are plotted in Figs. 6(c) and 6(d). Repetition times were optimized at each nutation frequency, and bulk build-up times were measured (see Table S1). In the case of XiX, the intensity of the enhanced ¹H NMR signal barely changes, while in the case of TOP, the intensity increases significantly with the nutation frequency. Table S1 makes clear that this increase is mostly due to a decrease in T_BU, from 65 to 29 s, at a constant optimal repetition time. For XiX, the decrease in T_BU is much less pronounced, from 29 down to 20 s, but the optimal repetition time is lengthened.

Figures 7(a)–7(c) show representative trajectories of S_z in the modulating interaction frame for TOP, XiX, and TPPM DNP. The scaling factors are calculated by double-mode Fourier analysis of these trajectories (see Sec. II) and are listed in Table I, along with the alignments. Corresponding semi-analytical contact curves [solid lines, Eqs. (14) and (15), followed by powder averaging] are plotted in Fig. 7(d). The initial rise of $Iz(t)$ and the time of the first maximum are determined by the strength of the e⁻–¹H dipolar coupling times the scaling factor. Therefore, the curves show the faster transfer of polarization induced by the TPPM sequence (κ = 0.34) compared to XiX (κ = 0.21) and TOP (κ = 0.10). After this, the curves oscillate around $Iz$ = 0.5 (corresponding to an enhancement factor ɛ_B of 329), as expected after powder averaging and for alignments very close to 1 (Table I). The agreement with numerical propagation [dashed lines in Fig. 7(d)] is excellent, showing that the perturbation theory approach works well.

A comparison of the TOP, XiX, and TPPM sequences at different conditions in Table I shows that a larger scaling factor leads to a shorter T_BU. The investigation of the nutation dependence essentially shows the same (see Table S1). For both XiX and TOP DNP, the scaling factor increases with the nutation frequency, with the scaling factor of the former being approximately twice as large. For both sequences, this leads to a reduction of T_BU, which is, as noted earlier, more drastic for TOP than for XiX. In the case of XiX, the increase in the optimal repetition time likely mitigates the positive effect of the larger scaling factor.

Numerically simulated contact curves, which take into account g-anisotropy, a second proton, as well as electron and nuclear spin relaxation, are shown for TOP, XiX, and TPPM DNP in Fig. 8(a). The fast initial increase of ɛ_B is again determined by the scaling factor, with TPPM being faster than XiX and XiX being faster than TOP. The oscillations are less pronounced than in Fig. 7(d) and fade at longer contact times. Due to relaxation, for TPPM DNP, the enhancement factor decays with the contact time, whereas for XiX, it remains roughly constant, and for TOP, it even goes up slightly. This can be qualitatively understood from the modulating frame trajectories in Figs. 7(a)–7(c). These trajectories show $U1S−1SzU1S$ [U_1S is defined in Eq. (3)] but also offer a visualization of the evolution of the electron spin magnetization in the electron rotating frame under the influence of the microwave pulse sequences [$U1S−SzU1S−1$]. For TOP and XiX, the magnetization vector stays close to the z-axis, meaning that the contributions of S_x and S_y are small at all times. For TPPM, however, the magnetization traverses the entire unit sphere, and the contributions of S_x and S_y are large. This makes TPPM, more than XiX and TOP, susceptible to (fast) transverse relaxation.

Figure 8(a) reveals another characteristic of the TPPM DNP sequence. Compared to Fig. 7(d), the final enhancements, i.e., at long contact times, are reduced for all three sequences, but particularly severely for XiX (ɛ_B is about 80 at 20 µs) and TOP (ɛ_Bs are about 60 and 80 at 20 µs). This reduction is mostly due to line-broadening caused by the g-anisotropy of trityl, to which TPPM is much less sensitive than the other two sequences. The reason is apparent when the effective frequency is plotted as a function of the resonance offset for the 16 ns/120° TPPM DNP condition (see Fig. S10). The matching condition [Eq. (9)] is met for k = 2 and l = −1, and the curve 2ν_1S − ν_eff sheers along the ¹H Larmor frequency, meaning that the mismatch is small across a large offset range. A similar phenomenon has been observed for the BEAM sequence at 0.34 T.²¹ 

Contact curves were also experimentally investigated, and the results are plotted in Fig. 8(b). For TPPM, XiX, and high-nutation TOP, the characteristic fast rise and plateau are observed.¹⁸ For XiX and high-nutation TOP, the rise times are considerably faster than expected from the scaling factor and numerical simulations. For TPPM, the expected decay of the enhancement at longer contact times due to transverse relaxation is not observed. The XiX contact curve, however, does decay slightly. For low-nutation TOP, the shape of the contact curve has become exponential. Together with the quantitative experiments summarized in Table I, Figs. 8(a) and 8(b) suggest that the simulations overestimate the performance of TPPM but underestimate it for XiX and TOP.

When we first introduced the XiX DNP sequence,²⁰ we reported a maximum enhancement factor of 58 and a $εB/TBU$ of 12. For TOP DNP, in the same series of experiments, we obtained a maximum enhancement factor of 40 and a $εB/TBU$ of 6. Thus, in these experiments, XiX fared much better than TOP, while in the current set of experiments, the difference in performance between XiX and TOP is not as pronounced (Table I). The earlier experiments were performed on a sample containing 2.5 mM trityl and at a nutation frequency of about 18 MHz, which was generated by operating a 10 W amplifier at full power. The results reported in Table I were obtained from a sample containing 6.4 mM trityl and with a 50 W amplifier, whose output power was attenuated. All other experimental conditions were the same as before. To investigate this discrepancy, we performed a new series of pulsed DNP experiments with a sample containing 2.1 mM trityl [Fig. S1(b)]. ¹H NMR spectra enhanced by XiX and TOP DNP induced with the 10 W or the 50 W amplifier are plotted together in Fig. S11. These experiments reproduce the high gain in sensitivity achieved by XiX DNP relative to TOP, as we observed previously, and show that the choice of the amplifier has only a minor effect. Hence, we conclude that the change in relative performance cannot be attributed to subtle differences in the microwave irradiation by the different amplifiers but instead arises from the difference in the polarizing agent concentrations.

We have investigated the conditions under which the TPPM sequence generates DNP both with numerical simulations and experimentally at 34 GHz/1.2 T/51 MHz. For a pulse length t_p of 16 ns, a phase difference ϕ of 120°, and a nutation frequency ν_1S of 33 MHz, the TPPM sequence provides a practical sensitivity gain of $εBT1n/TBU$ = 316 in a sample of 6.4 mM trityl OX063 in DNP juice. This is more than could be obtained with XiX (⁠$εBT1n/TBU$ = 238) or TOP DNP (⁠$εBT1n/TBU$ = 226) under the same conditions. We attribute the good performance of the TPPM sequence to a high offset tolerance (insensitivity to line-broadening by g-anisotropy) and a large scaling factor, which leads to a fast build-up of the bulk nuclear polarization.

The four DNP pulse sequences TOP, XiX, TPPM, and BEAM all follow the flexible, high-field compatible matching condition of Eq. (9). However, that matching can be accomplished does not guarantee that polarization transfer is efficient or that high enhancement factors can be obtained. This is determined by the pulse (and delay) lengths, the phases of the pulses, the nutation frequency, and the microwave resonance offset in a different way for each sequence. An important point to keep in mind is that maintaining the DNP efficiency requires linear scaling of the pulse (and delay) lengths, the nutation frequency, and the resonance offset with the magnetic field. Thus, when appropriately scaled, the simulated parameter scans in Figs. 2, 6, and S2 also give a good indication of the efficiency of TOP, XiX, and TPPM at higher magnetic fields. This means that the good performance of TPPM DNP at 1.2 T, which required a high nutation frequency of 33 MHz (relative to 51 MHz), will be difficult to reproduce at magnetic fields suitable for high-resolution MAS NMR, even though, due to the long optimal repetition time, a high average microwave power is not needed (see Table I). On the other hand, this makes the efficiency of XiX DNP at a nutation frequency of 7 MHz, which was not predicted by our simple modeling (see below), all the more interesting.

By means of the operator-based, triple mode Floquet theory, we could derive the DQ and ZQ effective Hamiltonians of Eqs. (11) and (12). They determine the evolution of the state of the spin system under the influence of the DNP pulse sequence at the matching condition [Eq. (9)]. The analytical expression for the time-dependence of the longitudinal magnetization of the nuclear spins, $Iz(t)$ [Eqs. (14) and (15)], has just two parameters: the scaling factor, $κ=a+(k)a−(−k)$⁠, and the alignment, $ρ′(0)Sz′$⁠. Once these parameters are determined, contact curves are in excellent agreement with numerical simulations [Fig. 7(d)]. Wili et al.²¹ derived the effective Hamiltonian by means of average Hamiltonian theory. To be able to obtain a time-independent effective Hamiltonian by averaging, they performed an additional frame transformation that removes the oscillation at the effective frequency. The agreement with numerical simulations in this case is also excellent. The advantage of their approach is that polarization transfer is also described when the matching condition is not perfectly met. This is cumbersome with Floquet theory. However, a strategy to deal with this issue has recently been devised.³⁹ 

The plots of $S̃z(t)$ in the modulating frame in Figs. 7(a)–7(c) help to gain intuitive insight into the TOP, XiX, and TPPM DNP sequences. For example, from these trajectories, it is immediately clear why neither TOP nor XiX nor TPPM (with t_p = 16 ns and ϕ = 120°) benefit from a preparation pulse. In all three cases, $ω⃗eff$ is pretty much along the z-axis, and the alignment is close to ±1. In the case of TPPM, which works best close to resonance, the gradual precession at ω_eff (approximately) around the z-axis is made possible by the phase difference, ϕ, between the pulses. The case of high-nutation frequency BEAM at 0.34 T (X band),²¹ which works best with a 90° preparation pulse along y, is visualized in Fig. S12. The pulses of unequal lengths, applied close to resonance and alternating between x and −x, cause a precession around (approximately) the −x-axis. The preparation pulse is thus mandatory to obtain good alignment. Since the z-axis and $ω⃗eff$ approximately align, the trajectories in Figs. 7(a)–7(c) also visualize the large scaling factor of XiX compared to TOP DNP and the still larger scaling factor of TPPM. As $S̃z(t)$ moves further away from the z-axis, the x- and y-components of $S̃z(t)$ are accordingly larger.

The extensive set of experimental data at 1.2 T provides new, practical insight into pulsed DNP. An important observation is that a larger scaling factor leads to a faster build-up time of the bulk polarization, T_BU (Tables I and S1). A shorter repetition time also leads to a shorter T_BU (Fig. S9) and, in addition, to a higher $εB$ (Fig. 5); however, for XiX and TPPM, only down to an optimum value (see below). These behaviors resemble those of the classical solid-effect, where a higher applied microwave power leads to a shorter T_BU and a higher enhancement factor.⁴⁰ In line with intuition, the more vigorously the electron polarization is pumped into the nuclear spin bath, the higher the maximum enhancement factor and the faster it is reached. More vigorous pumping can be accomplished by applying small strokes as often as possible, i.e., by continuous microwave irradiation, or by applying highly efficient strokes at a slower, optimized pace, i.e., by a well-chosen and well-timed microwave pulse sequence.

While TPPM provides a higher gain in sensitivity than XiX and TOP, the improvement is not as large as expected from the simulations [Fig. 8(a)]. This is likely related to the long optimal repetition rate of TPPM DNP (Fig. 5). Before another effective transfer of polarization can take place, the electron spin magnetization must have returned to thermal equilibrium, i.e., to −z. This means that the pace at which a DNP pulse sequence is best repeated depends on the longitudinal relaxation rate of the electron spins and the final direction of the magnetization at the end of the contact time. With this in mind, the trajectory in Fig. 7(c) offers no obvious explanation for the long optimal repetition time of TPPM DNP. At the end of each block, the magnetization is brought back close to where it started. Possibly, after many cycles of transfer, due to EPR line width, pulse imperfections, and an inhomogeneous B₁-field, the magnetization fans out. When the TPPM sequence is applied again, many spin packets no longer follow the ideal trajectory shown in Fig. 7(c). They may end up close to the x, y-plane, from where it takes a long time to get back to −z. A potential solution to this problem would be to design a TPPM sequence consisting of composite pulses⁴¹ or make use of phase or super cycles,⁴² taking care to keep all spin packets together even after many cycles of transfer.

The rise time in the contact curves in Figs. 8(a) and 8(b) is well predicted by the simulations for TPPM, but for XiX and TOP, the prediction is too slow. The rise time depends on the scaling factor and on the effective electron–proton dipolar coupling. As described in Sec. II, the value of the dipolar coupling used in the simulations has been adjusted empirically to reproduce the rise time in experimental contact curves of (off-resonance) NOVEL with trityl OX063 in DNP juice at the X band.^17,18 If we assume that this parameterization of the dipolar coupling is generally applicable to pulsed DNP experiments with trityl OX063 in DNP juice, there must be something unusual about XiX and TOP DNP. An explanation that comes to mind is that also for TOP and XiX, a fanning out of the magnetization occurs after repeated cycles of polarization transfer. As this happens, trajectories will likely obtain larger x- and y-components, which means the scaling factor effectively increases and the transfer of polarization speeds up. In the case of TOP and XiX, where the calculated trajectories are rather confined [Figs. 7(a) and 7(b)], it is an advantageous effect.

Finally, the fanning out may also offer an explanation for the very good performance of XiX DNP at 7 MHz and, subsequently, flat nutation frequency dependence [Fig. 6(c) and Table S1]. Numerical simulations predict a 6-fold increase of the enhancement factor going from ɛ_B = −19 at 7 MHz to ɛ_B = −121 at 36 MHz [see Fig. 6(a)]. In addition, the theoretical scaling factor increases with the nutation frequency (Table S1). This means that at low nutation frequency, the fanning out is advantageous, leading to a higher effective scaling factor. However, at higher nutation frequencies, where the scaling factor of XiX is already large, it becomes detrimental. Like in TPPM, it leads to an increase in the optimal repetition time (Table S1) and a reduced gain in sensitivity.

Previously, in NOVEL experiments at X band, we investigated the dependence of the enhancement factor on the concentration of trityl OX063 in DNP juice.¹⁷ The enhancement factor increased approximately linearly with the concentration up to 7–8 mM, where an abrupt deterioration of the enhancement as well as drastic changes in the echo-detected EPR spectrum were observed. In experiments at Q band, we also observe a general increase of the enhancement factors with the polarizing agent concentration, up to 7–8 mM. However, how much is gained at a higher concentration seems to depend on the sequence. In particular, we find that TOP DNP benefits more from a higher polarizing agent concentration than XiX. At the moment, we have no good explanation for this phenomenon, but increased interactions between the unpaired electrons of trityl likely play a role. At 6.4 mM, the average strength of the electron–electron dipolar coupling is 200 kHz, which is significant compared to the electron–proton dipolar coupling. This may cause interference effects that vary from sequence to sequence. Electron–electron interactions, in any case, reduce the phase memory time. Although not fully consistent with experimental observations [Fig. 8(b)], theory and numerical simulations indicate that the effect of fast transverse relaxation on the transfer of polarization depends on the sequence [compare Figs. 7(d) and 8(a)].

The gain in sensitivity one eventually obtains with pulsed DNP for a given average microwave power follows from a subtle interplay between the electron and nuclear relaxation rates and the properties of the sequence such as scaling factor, response to line-broadening, optimal repetition rate, and dependence on polarizing agent concentration. Some of these properties (scaling factor, response to line-broadening) are readily found from the analysis of a single cycle of polarization transfer. Others (optimal repetition rate, concentration dependence) require the modeling of many transfer cycles as well as the diffusion of the enhanced polarization into the sample. We did not pursue this here because spin dynamics simulations of large solid-state systems are still computationally very costly. This would make scanning large numbers of DNP conditions by means of numerical simulations cumbersome. Moreover, setting the parameters of such a complex spin system must be performed with care. Extensive testing against experiments will be necessary before larger models can be used effectively to design DNP pulse sequences. Our investigations, however, show that such simulations will add value.

At 1.2 T, the new TPPM (two-pulse phase modulation) DNP pulse sequence efficiently polarizes bulk proton spins. It, however, requires a relatively high nutation frequency, i.e., 33 MHz, compared to a proton Larmor frequency of 51 MHz. The XiX (X-inverse-X) DNP sequence is less efficient but already performs well at nutation frequencies as low as 7 MHz. The theoretical description of the transfer of polarization induced by TPPM, or any other periodic DNP pulse sequence, in a dipolar-coupled electron–nuclear spin system is under control. Numerical simulations of this process give an indication of the gain in sensitivity that can be generated for NMR of the bulk proton spins, but they do not account for all experimental observations. Augmented modeling will, therefore, be important for the design of new and improved DNP pulse sequences, for example, with the help of optimal control. Moreover, it will offer better guidance for the development of dedicated instruments, in particular suitable microwave sources, for pulsed DNP at high magnetic fields.

The supplementary material includes the EPR characterization of samples used for pulsed DNP experiments; simulated parameter scans of all TPPM DNP conditions of interest; supporting experimental data for Table I, namely, 1H NMR spectra with and without DNP, measurements of the 1H longitudinal relaxation time and build-up times of the bulk 1H polarization; investigation of the DNP conditions close to resonance for XiX DNP; investigation of the dependence of the build-up time of bulk 1Hs on the repetition time for TPPM DNP; supporting experimental data for the investigation of the nutation frequency dependence for XiX and TOP DNP; offset dependence of the mismatch for TPPM DNP; 1H NMR spectra of the 2.1 mM trityl OX063 sample enhanced by XiX and TOP DNP; and the trajectory S̃z(t) in the modulating frame for BEAM at 0.34 T.

This research was supported by the Deutsche Forschungsgemeinschaft through the Emmy Noether Program (Project No. 321027114) and SFB 1527 (Project No. 454252029), and by the Young Scholar Fund of the University of Konstanz. The authors thank Bruno Erne for his help with the construction of the tuning/matching box, Mykhailo Azarkh for experimental support, and Ilya Kuprov and Matthias Ernst for fruitful discussion.

The authors have no conflicts to disclose.

Venkata SubbaRao Redrouthu: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). Sanjay Vinod-Kumar: Data curation (equal); Formal analysis (equal); Investigation (equal); Resources (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). Guinevere Mathies: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (lead); Investigation (equal); Methodology (equal); Project administration (lead); Resources (equal); Software (equal); Supervision (lead); Validation (equal); Visualization (equal); Writing – original draft (lead); Writing – review & editing (equal).

The data that support the findings of this study as well as the Matlab scripts that were used to analyze the DNP pulse sequences presented here are openly available in Zenodo at https://zenodo.org/record/8061361 and https://zenodo.org/record/8061384 . Scripts used for numerical simulations have been made available in the Spinach library at https://spindynamics.org.