Nuclear Cross-Effect NMR

Zhenfeng Pang, Jake Lumsden, Kong Ooi Tan

► To cite this version:

Zhenfeng Pang, Jake Lumsden, Kong Ooi Tan. Nuclear Cross-Effect NMR. Journal of Physical Chemistry Letters, 2025, pp.10568-10574. 10.1021/acs.jpclett.5c02565. http://onlinelibrary.wiley.com/doi/abs/10.1021/acs.jpclett.5c02565

HAL Id: hal-05296802

https://hal.science/hal-05296802v1

Submitted on 3 Oct 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
This document is confidential and is proprietary to the American Chemical Society and its authors. Do not
copy or disclose without written permission. If you have received this item in error, notify the sender and
delete all copies.

Nuclear Cross Effect NMR

<table><tr><td>Journal:</td><td>The Journal of Physical Chemistry Letters</td></tr><tr><td>Manuscript ID</td><td>jz-2025-02565y.R3</td></tr><tr><td>Manuscript Type:</td><td>Letter</td></tr><tr><td>Date Submitted by the Author:</td><td>n/a</td></tr><tr><td>Complete List of Authors:</td><td>Pang, Zhenfeng; École Normale Supérieure, Chemistry<br>Lumsden, Jake; École Normale Supérieure, Chemistry<br>Tan, Kong Ooi; École Normale Supérieure, Chemistry</td></tr></table>
Nuclear Cross Effect NMR

Zhenfeng Pang*, Jake Lumsden, Kong Ooi Tan*

Chimie Physique et Chimie du Vivant, CPCV, CNRS UMR 8228, Sorbonne Université, Ecole normale supérieure, PSL

University, F-75005, Paris, France

*Corresponding Authors:

Zhenfeng Pang

E-Mail: zhenfeng.pang@ens.psl.eu

Kong Ooi Tan

E-Mail: kong-ooi.tan@ens.psl.eu

Keywords: Dynamic Nuclear Polarization, Cross effect, Solid-State NMR

ZP's ORCID: 0000-0003-1566-5454

JL's ORCID: 0009-0000-8316-0234

KOT's ORCID: 0000-0002-3094-3398
**Abstract:**

Dynamic nuclear polarization (DNP) has significantly enhanced the sensitivity of nuclear magnetic resonance (NMR) experiments. While Overhauser DNP has a well-established nuclear counterpart, the nuclear Overhauser effect (NOE), the cross effect has traditionally been considered as an exclusively electron-mediated process. Here, we present the first evidence of a nuclear counterpart to the cross effect, termed the *nuclear cross effect*, which mediates polarization transfer in a three-spin nuclear system. The effect was experimentally verified using single-crystal ¹³C-¹⁵N-labeled glycine, with results consistent with both analytical theory and numerical simulations. This discovery deepens our understanding of spin dynamics and opens avenues for the development of new NMR or DNP methodologies. In particular, *pulsed cross-effect DNP* has demonstrated better performance than the conventional continuous wave (CW) cross-effect (Pang *et al.*, *J. Chem. Phys.* **2022**, 156, 244109), although prior validation was limited to numerical simulations. The experimental realization of the pulsed nuclear cross effect in this work provides new insights into its theoretical foundations, addressing previous challenges posed by hardware limitations and the absence of a suitable model system. Furthermore, the pulsed cross-effect mechanism has the potential to advance enhanced-NMR applications in various fields.

**Table of Contents (TOC) Graphic**

Nuclear Cross Effect (NCE)

<figure><img src="image_2.png" /></figure>

$$ \Delta\Omega = \omega_1(\text{Rabi Field}) $$

<figure><img src="image_3.png" /></figure>
Dynamic nuclear polarization (DNP) significantly enhances the sensitivity of nuclear magnetic resonance (NMR) spectroscopy by transferring polarization from electrons to nuclei.¹⁻³ Due to the large gyromagnetic ratio of electrons, hyperpolarized NMR allows the study of structures and dynamics of molecules that were previously inaccessible due to low sensitivity. In magic angle spinning (MAS) DNP experiments, biradicals that mediate cross effect (CE)⁴⁻⁶ are commonly employed because they yield robust DNP performances.⁷,⁸ In CE DNP, it is theoretically known that the efficiency of polarization transfer depends on the polarization difference between the two unpaired electrons,⁹,¹⁰ and the maximum achievable polarization difference between the two electrons is limited to the thermal polarization of one electron in continuous-wave (CW) DNP, i.e., the maximum enhancement $\varepsilon$ is $\gamma_e/\gamma_{1H} \sim 658$ for ¹H-DNP. Nevertheless, it was theoretically discussed and shown in our recent work that the numerically simulated DNP enhancement factor is doubled when one of the electrons is inverted by a $\pi$ pulse rather than CW saturation.¹¹ In this work—our thought experiment employing a rotating-static switch setup (vide infra) and an ideal biradical—we demonstrate that the maximum $\varepsilon$ can reach $2 \times \gamma_e/\gamma_{1H} \sim 1316$ if one of the electrons is fully inverted using pulsed microwaves, i.e., a *pulsed* cross effect DNP.

Demonstrating *pulsed* cross effect DNP experimentally presents significant challenges due to hardware limitations and lack of an ideal biradical that perfectly satisfies the cross-effect condition—where resonances of two electrons have narrow lines and are separated exactly by the nuclear Larmor frequency. This prompted us to test our hypothesis on a purely nuclear system (without unpaired electrons), where the NMR hardware and model spin system are readily available. Inspired by the nuclear Overhauser effect (NOE), a nuclear counterpart of the Overhauser effect DNP,¹²⁻¹⁴ we extended the concept of CE DNP to a three-spin nuclear system and called it *nuclear cross effect* (NCE). NCE occurs when the chemical shift difference between the two homonuclear nuclear spins matches the Rabi field applied to a third heteronuclear spin (e.g., a ¹³C-¹³C-¹⁵N system). In this work, we experimentally verified NCE on a glycine single crystal, which yields narrow ¹³C and ¹⁵N lines under static (non-spinning) conditions. The exploration of nuclear counterparts of DNP mechanisms not only aids the development of innovative DNP techniques and enhances the understanding of magnetic resonance phenomena but also fosters connections between various subfields, including NMR, DNP, and EPR. The insights gained from this study contribute to refining DNP methodologies and lay the groundwork for future advancements in boosting polarization efficiency in both electron-nuclear and purely nuclear systems.
<figure><img src="image_4.png" /><figcaption>**Figure 1:** Simulated CW and pulsed CE experiments. (a) Pulse sequence for CW CE MAS DNP. (b) Pulse sequence for pulsed CE MAS DNP. For both CW and pulsed cases, the microwave frequency is fixed to match the isotropic g value of the first electron. The simulated EPR spectrum is shown in **Fig. S1**. (c) DNP buildup curves of CW (saturation, solid line) and pulsed CE (inversion, dashed line) cases on systems with different electron $T_1$ relaxation times under the magic angle spinning frequency of 12.5 kHz (rotor period ~ 80 µs). Schematic diagram of the rotating-static switch experiment with rotor events (d) and energy-level diagram (e). DNP occurs only when the cross-effect matching condition is satisfied (at **②** and **③**). The polarization of electron recovers to thermal equilibrium without nuclear depolarization when the rotor is static (interval **④**). (f) The simulated DNP buildup curves of powder (solid line) and single crystal (dashed line) with duration of one complete rotor rotation of 80 µs (blue) and 5 ms (red).</figcaption></figure>

It is known in literature that the cross-effect (CE) DNP enhancement depends on the polarization difference between two electrons, $\Delta P_e = P_{1e} - P_{2e}$, whose resonance frequencies differ by the nuclear Larmor frequency ($|\omega_{0e1} - \omega_{0e2}| \approx |\omega_{0n}|$). In CW-DNP, the maximum $\Delta P_e$ equals the thermal polarization of electrons when one of the electrons is fully saturated ($|\Delta P_e| = |P_{2e}|$ if $P_{1e} = 0$). In principle, we can double the polarization difference ($|\Delta P_e| = 2 \times |P_{2e}|$) by applying a microwave $\pi$ pulse to one of the electrons ($P_{2e} = -P_{1e}$), significantly increasing DNP performance. Such an idea was proposed and investigated in literature,¹¹,¹⁵ but only in static case. Here, we will revisit such an experiment under MAS condition but with either CW (or saturation $P_{1e} = 0$) or pulsed (or inversion $P_{1e} = -P_{2e}$) $\mu$W irradiation cases with different $T_{1e}$ values.

For simplicity, numerical simulation was performed on a model single-crystal three-spin electron-electron-¹H system where the g values were chosen such that the electron lines are well-isolated (see simulated EPR spectrum in **Fig. S1**) for easier selective saturation (CW case) or inversion (pulsed case)
and satisfy the CE condition at certain rotor angles. In CW case, µw is applied continuously (Fig. 1a) throughout the experiment. In the pulsed case, an electron is inverted by a π pulse at the beginning of each rotor period (Fig. 1b). The simulated results show that the pulsed CE DNP yields over 50% higher enhancement than CW when T₁e is comparable to the delay between π pulses (or rotor period ~ 80 µs; Fig. 1c). The benefit diminishes when T₁e is long (1 ms), where π pulses act like saturation pulses; or when T₁e is short (10 µs), where infrequent pulses fail to maintain the polarization difference (Fig. S2b-d).

Although pulsed CE DNP offers faster buildup (buildup time <0.1 s in Fig. 1c), the steady-state enhancement is not doubled and even underperforms the CW case with T₁e = 10 µs. This results from the interplay between DNP and depolarization, which depends on several factors, including ΔPₑ and Pₙ steady state (Fig. S2d), and T₁e. We emphasize that DNP occurs when Pₙ < ΔPₑ, and depolarization occurs when Pₙ > ΔPₑ.¹⁶⁻¹⁹ To obtain a higher net DNP performance, one needs to suppress or avoid depolarization. Note that the CE DNP and depolarization are essentially governed by the same process but will lead to opposite outcomes (higher Pₙ in DNP or lower Pₙ in depolarization) depending on the initial condition (relative magnitudes of Pₙ and ΔPₑ). Since it is challenging to avoid depolarization while sustaining DNP at the same time in the conventional MAS DNP setup, we propose a thought experiment in which we match the CE condition when ΔPₑ >> Pₙ (to favor DNP), and mismatch it when ΔPₑ << Pₙ (to avoid depolarization). Under ideal conditions, such setup should push ε closer towards the theoretical maximum, i.e., ~658 for the CW or ~1316 for the pulsed regime, respectively.

In order to push the DNP enhancement ε closer towards the theoretical limit, where depolarization should be minimized, we propose a thought or *Gedanken* experiment in which the sample rotates and stops repeatedly at regular intervals (Fig. 1d). Note that a similar approach was proposed and implemented for DNP on quadrupolar nuclei, where the sample rotates for better DNP, and stops for NMR acquisition.²⁰

The sequence of our pulse-rotation-static protocol (Fig. 1d) is given by: (1) apply a selective π pulse (①) to invert one electron, (2) rotate for one rotor period (τr), (3) stop rotation to allow full electron relaxation (④). The cycle is then repeated for DNP buildup. For efficient DNP, the rotor period (τr) and the switch time between rotating and static state (Δt) should satisfy the inequality Δt << τr < T₁e. The energy levels of the |αβα⟩ and |βαβ⟩ states (Fig. 1e) indicate that the cross-effect matching condition (E|αβα⟩ ≈ E|βαβ⟩, e.g. ②) is only satisfied during rotation. Consequently, both DNP and depolarization
occur exclusively during rotation. Since the rotation period is shorter than $T_{1e}$, $\Delta P_e$ remains close to $2P_e^{eq}$ during the rotation, enabling efficient DNP. Then, the electron polarization recovers to the thermal equilibrium without nuclear spins losing polarization (depolarization) during the static phase. As expected, simulations show $\epsilon \sim 1200$ for single crystal and $\sim 1000$ for powders with a spinning frequency of 12.5 kHz (Fig. 1f), nearly twice the theoretical limit under saturation conditions ($\epsilon \sim 658$).

According to the energy level diagrams (Fig. S4), the maximum nuclear polarization of pulsed CE experiments could reach $\frac{p_{\alpha,e}-p_{\beta,e}}{p_{\alpha,e}^2+p_{\beta,e}^2}$, in which $p_{\alpha,e}$ and $p_{\beta,e}$ are the probability of $|\alpha\rangle$ and $|\beta\rangle$ states of electrons at equilibrium. Under the high-temperature approximation, $p_{\alpha,e}$ and $p_{\beta,e}$ are close to 0.5. The maximum nuclear polarization is around $2(p_{\alpha,e} - p_{\beta,e})$ which is twice the electron thermal polarization. To achieve high enhancement in pulsed cross-effect DNP, three key factors must be considered:

1. Relaxation times and rotor period: much shorter cross effect buildup time ($\tau_{CE} = 1/\omega_{CE}$)¹¹ than electron $T_1$ relaxation time ($\tau_{CE} \ll T_{1e}$); shorter rotating period than $T_{1e}$ ($\tau_f < T_{1e}$); much slower nuclear relaxation rate than electron relaxation rate ($T_{1n} \gg T_{1e}$).

2. A highly selective pulse with proper duration and Rabi field to invert one of the two electrons.

3. The ability to immediately and precisely match/mismatch the CE condition to favour DNP over depolarization according to the polarization difference between the two electrons.

Experimentally, the relative magnitudes of relaxation and cross-effect buildup rates are satisfied in most of DNP samples. Selective inversion of electron polarization can potentially be achieved using a Trinityl-Nitroxide biradical,²¹,²² where the narrower trityl linewidth allows selective inversion with a pulsed EPR setup.

To practically implement the rotating-switch experiment, we propose using a high-speed servo motor (e.g., 200 Hz), which enables a rapid alternation between rotation and static conditions. For example, a 40 mM SA-BDPA sample in DNP juice at 80 K and ~0.35 T has a reported $T_{1e}$ of ~10 ms,²³ which exceeds the 5 ms rotation period at 200 Hz. If monoradicals are chemically tethered with resonance offsets matching the ¹H Larmor frequency at certain orientations, the simulated enhancements (~1200) could be tested experimentally. However, integrating such a motor into a cryogenic NMR probe would require advanced engineering. On the other hand, we also explored rf and/or µw strategies to suppress depolarization by modulating the CE matching condition electronically. While no viable solution was identified with current hardware, we hypothesized that CE-like polarization transfer can occur in a purely nuclear system—without electrons—and the samples and rf hardware are readily
available.

Next, we will build an analytical description based on the 'unified' theoretical framework we recently formalized,¹¹ and readers are encouraged to consult this work for a comprehensive background. Such an approach also aligns with the idea of unifying polarization-transfer theory in magnetic resonance that has been developed in the field recently.²⁴,²⁵ We consider a three-nucleus system, two ¹³C (spin S) and one ¹⁵N (spin I), and we aim to transfer the polarization difference between the ¹³C spins to ¹⁵N, i.e., $\Delta P_C \to P_N$. The Hamiltonian in the double rotating frame with an RF spinlock on ¹⁵N is given by:

$$ \hat{\mathcal{H}} = \Omega_{S1}\hat{S}_{1z} + \Omega_{S2}\hat{S}_{2z} + d(3\hat{S}_{1z}\hat{S}_{2z} - \hat{S}_1 \cdot \hat{S}_2) + d_{\text{IS}}^{(1)}\hat{S}_{1z}\hat{I}_z + d_{\text{IS}}^{(2)}\hat{S}_{2z}\hat{I}_z + \omega_{1I}\hat{I}_x, \quad (1) $$

where $\Omega_{S1}$, $\Omega_{S2}$ are the chemical shift of two S spins, $d$ is the homonuclear dipolar coupling between two S spins, $d_{\text{IS}}^{(1)}$ and $d_{\text{IS}}^{(2)}$ are the dipolar coupling in between I spin and the two respective S spins, $\omega_{1I}$ is the ¹⁵N nutation frequency during spinlock. We neglected the J couplings for simplicity. Then, we rotate the frame such that the ¹⁵N rf pulse is along the z axis using a propagator $\hat{U}_1 = \exp(i \cdot \frac{\pi}{2} \hat{I}_y)$:

$$ \hat{\mathcal{H}}_t = \Omega_{S1}\hat{S}_{1z} + \Omega_{S2}\hat{S}_{2z} + d(3\hat{S}_{1z}\hat{S}_{2z} - \hat{S}_1 \cdot \hat{S}_2) + d_{\text{IS}}^{(1)}\hat{S}_{1z}\hat{I}_x + d_{\text{IS}}^{(2)}\hat{S}_{2z}\hat{I}_x - \omega_{1,I}\hat{I}_z. \quad (2) $$

Similar to the derivation of the normal CE DNP in our previous work,¹¹ the Hamiltonian could be block diagonalized, where the matrix representation of the middle 4×4 block matrix, called nuclear cross-effect $\hat{\mathcal{H}}^{\text{NCE}}$ subspace here, is given by

$$ \hat{\mathcal{H}}^{\text{NCE}} = -\frac{d}{2}\hat{\mathbb{E}} + \Delta\omega\hat{S}_z^{\text{NCE}} - d\hat{S}_x^{\text{NCE}} - \omega_{1I}\hat{I}_z^{\text{NCE}} + \Delta D_{zx}\hat{S}_z^{\text{NCE}}\hat{I}_x^{\text{NCE}}, \quad (3) $$

where $\hat{\mathbb{E}}$ is an identity matrix, $\Delta\omega = \Omega_{S1} - \Omega_{S2}$, and $\Delta D_{zx} = d_{\text{IS}}^{(1)} - d_{\text{IS}}^{(2)}$. Following that, we apply a propagator $\hat{U}_2 = \exp(-i\theta_2\hat{S}_y^{\text{NCE}})\exp(-i\theta_12\hat{S}_z^{\text{NCE}}\hat{I}_y)$, where $\tan\theta_1 = -\frac{\Delta D_{zx}}{2\omega_{1I}}$ and $\tan\theta_2 = -\frac{d\cos\theta_1}{\Delta\omega}$, so that the resulting Hamiltonian is mathematically similar to that of a standard cross-polarization (see Eq. S1-S3 for full derivation):¹¹

$$ \hat{\mathcal{H}}_t^{\text{NCE}} = \omega_{1I,\text{eff}}\hat{I}_z^{\text{NCE}} + \omega_{1S,\text{eff}}\hat{S}_z^{\text{NCE}} + 2d\hat{S}_y^{\text{NCE}}\hat{I}_y^{\text{NCE}} \sin\theta_1, \quad (4) $$
where $\omega_{11,eff} = \frac{1}{2}\sqrt{\Delta D_{zx}^2 + 4\omega_{11}^2}$ and $\omega_{1S,eff} = \sqrt{\frac{4\omega_{11}^2 d^2}{\Delta D_{zx}^2 + 4\omega_{11}^2} + \Delta\omega^2}$. We have neglected the identity part ($-\frac{d}{2}\hat{\mathbb{E}}$) because it commutes with all other Hamiltonians in the subspace. Note that the initial density operator in the tilted frame here is no longer the simple polarization difference between the two S spins, but $\hat{\rho}^{\text{NCE}}(0) = \hat{U}\hat{S}_z^{\text{NCE}}\hat{U}^{-1} = \hat{S}_z^{\text{NCE}}\cos\theta_2 - \hat{S}_x^{\text{NCE}}\sin\theta_2$, where $\hat{U} = \hat{U}_2\hat{U}_1$. Nevertheless, we consider only the $\hat{S}_z^{\text{NCE}}\cos\theta_2$ term because the $\hat{S}_x^{\text{NCE}}$ (lab-frame zero-quantum coherence of two carbons) term eventually loses its coherence during the $^{15}$N spinlock time. Note that $\hat{\mathcal{H}}_t^{\text{NCE}}$ (Eq. 4) has a mathematical similar expression as the Hamiltonian for cross polarization.¹¹ Thus, we can directly quote the well-known Hartmann-Hahn matching condition, $\omega_{11,eff} = \omega_{1S,eff}$, and in the interaction frame of $\omega_{11,eff}\hat{I}_z^{\text{NCE}} + \omega_{1S,eff}\hat{S}_z^{\text{NCE}}$ the first-order average Hamiltonian is,

$$ \hat{\mathcal{H}}^{\text{NCE}} = \frac{d \sin \theta_1}{2} (\hat{S}_+^{\text{NCE}}\hat{I}_-^{\text{NCE}} + \hat{S}_-^{\text{NCE}}\hat{I}_+^{\text{NCE}}). \quad (5) $$

with the matching condition given by (see Eq. S4):

$$ \omega_{11}^2 = \Delta\omega^2 + d^2 - \frac{1}{4}\Delta D_{zx}^2. \quad (6) $$

If all the dipolar couplings are small, it can be approximated that $\omega_{11} \sim \Delta\omega$, i.e., the spin-locking Rabi field ($\omega_{11}$) applied on the target ($I$) nuclei should match the chemical-shift difference ($\Delta\omega$) between the two source ($S$) nuclei—an NMR equivalent of the well-known CE DNP condition. The similarity is not surprising as they were anticipated from the unified theoretical framework developed in our previous work.¹¹ If the CE condition is satisfied, $\hat{\rho}^{\text{NCE}}$ will evolve under Eq. 5 to become:

$$ \frac{\hat{\rho}^{\text{NCE}}(t)}{\cos \theta_2} = \hat{S}_z^{\text{NCE}} \cos^2(\omega_{\text{NCE}}t) + \hat{I}_z^{\text{NCE}} \sin^2(\omega_{\text{NCE}}t) \\ + (\hat{S}_x^{\text{NCE}}\hat{I}_y^{\text{NCE}} - \hat{S}_y^{\text{NCE}}\hat{I}_x^{\text{NCE}}) \sin(2\omega_{\text{NCE}}t), \quad (7) $$

where the build-up rate $\omega_{\text{NCE}} = d \sin \theta_1 / 2$. Note that the detection operator changes from $\hat{I}_z^{\text{NCE}}$ to $\cos\theta_1 \hat{I}_z^{\text{NCE}} + \sin\theta_1(\sin\theta_2 \hat{S}_x^{\text{NCE}}\hat{I}_x^{\text{NCE}} + \cos\theta_2 \hat{S}_z^{\text{NCE}}\hat{I}_x^{\text{NCE}})$ in the tilted-frame transformation by $\hat{U} = \hat{U}_2\hat{U}_1$ (vide supra). Nevertheless, we consider only $\cos\theta_1 \hat{I}_z^{\text{NCE}}$ and neglected the latter terms ($\hat{S}_x^{\text{NCE}}\hat{I}_x^{\text{NCE}}$ and $\hat{S}_z^{\text{NCE}}\hat{I}_x^{\text{NCE}}$) because its projection to the $\hat{\rho}^{\text{NCE}}(t)$ (Eq. 7) is zero.
<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /><figcaption>**Figure 2: Nuclear cross effect.** (a) Pulse sequence of NCE. (b) Buildup curves of <sup>15</sup>N calculated from analytical theory (Eq. 7) and numerical simulation without relaxation.</figcaption></figure>

Similar to CE DNP, to facilitate NCE transfer from <sup>13</sup>C to <sup>15</sup>N, one of the <sup>13</sup>C spins is selectively inverted after the <sup>1</sup>H-<sup>13</sup>C CP block (Fig. 2a) to prepare the Ŝ<sub>Z</sub><sup>NCE</sup> state that depends on the polarization difference |ΔP<sub>C</sub>| = |P<sub>C1</sub> - P<sub>C2</sub>|~2|P<sub>C1</sub>| assuming uniform CP transfer and near 100 % inversion efficiency. Then, polarization transfer ΔP<sub>C</sub> → P<sub>N</sub> occurs when the <sup>15</sup>N spin-lock field strength matches the chemical-shift difference between the <sup>13</sup>C spins (Eq. 6). To verify our theory, we have performed numerical simulations at the predicted matching condition (Eq. 6) and the results (Fig. 2b) yield excellent agreement with the analytical theory (Eq. 7). In conclusion, we can now turn ‘on’ or ‘off’ NCE (or tactically avoiding <sup>15</sup>N depolarization) with just RF pulses—more practical than physically spinning/stopping the sample (supra infra) in CE DNP.
<figure><img src="image_7.png" /><figcaption>**Figure 3:** Experimental NMR results on single-crystal glycine. (a) Images of glycine single crystal (b) $^{13}$C (blue) and $^{15}$N (red) 1D spectra (c) Dependence of $^{13}$C and $^{15}$N intensity on the power of spin lock in nuclear cross effect experiments. (d) $^{13}$C 1D spectra with matched and mismatched nuclear cross effect. (e) Dependence of $^{15}$N intensity and $\Delta I_C$ on the frequency of the selective pulse. (f) $^{15}$N spectrum using various $^{13}$C selective pulses for NCE where inversion: single selective $\pi$ pulse, excitation: single selective $\pi/2$ pulse, and saturation: a long duration selective pulse which is as long as the spin lock on $^{15}$N in **Fig. 2a**.</figcaption></figure>

To experimentally verify our theory and simulations, we used a single crystal of $^{13}$C-$^{15}$N α-glycine. $^{13}$C-$^{15}$N labelled glycine (CAS: 211057-02-2) was recrystallized from its powder. 1 ml of saturated aqueous glycine solution was placed in a 10 ml glass vial, sealed with parafilm, and punctured with a needle syringe to allow slow evaporation. After a few days, crystals formed at the bottom of the vial; the good-quality crystals were collected and transferred to another 1 ml saturated solution. This solution was again left for several days until complete solvent evaporation, yielding larger high-quality crystals. The single crystal was then packed into a 2.5 mm rotor (**Fig. 3a**) for NMR experiments using a Bruker Avance III HD 9.4 T/ 400 MHz spectrometer. Note that the MAS rotor served solely as a sample container and it was not spun during the experiment, i.e., a static experiment. Besides that, the MAS stator of the 2.5 mm triple channel MAS probe was adjusted to 90° with respect to the static magnetic field, as this crystal orientation yields stronger dipolar couplings that favor a faster NCE buildup rate (**Fig. S7**). The Rabi frequencies were 88 kHz at 30 W for $^{1}$H, 56 kHz at 60 W for $^{13}$C, and 35 kHz at 100 W for $^{15}$N. The cross-polarization contact time used in all experiments was 2.8 ms, with a spin-lock nutation frequency of 47 kHz. The selective inversion pulse on $^{13}$C was sinc-shaped with a duration of 1 ms. The spin-lock duration on the $^{15}$N channel was 100 ms for the NCE buildup. The pulse sequence (TopSpin 3.5 pl 7) and raw data are available on Zenodo (https://zenodo.org/records/14908066). The doublet at ~186 ppm in the $^{13}$CO 1D spectrum (**Fig. 3b**) corresponds to the $^{13}$C$_{α}$-$^{13}$CO homonuclear
dipolar splitting, while the Cα quartet at ~42 ppm arises from the combined influence of ¹³C-¹³C (1843 Hz) and ¹³C-¹⁵N (1074 Hz) dipolar couplings.

We actively searched for the NCE by monitoring the ¹⁵N peak intensity while sweeping the ¹⁵N spin-lock Rabi field ω₁₁ (Fig. 2a). Indeed, ¹⁵N signal was obtained when ω₁₁/2π ~ 13 kHz (~8 W, Fig. 3c), which is close to the predicted ω₁₁/2π ~14.3 kHz (~9.6 W; Eq. 6) using the spin parameters determined earlier (Δ ~ 14.412 kHz, d ~ 1876 Hz, and ΔDₓₓ ~ 1058 Hz). On the other hand, we also performed the ¹³C-detected NCE experiment (Fig. S6) to verify that the increase in ¹⁵N signal during NCE originates from the ¹³C spins. We measured the integrated signal difference between the ¹³C spins (ΔP<sub>C</sub> = P<sub>Cα</sub> - P<sub>CO</sub>), and the spectra (Fig. 3d) clearly show that the difference between the Cα and CO signal is indeed smaller at the NCE condition (¹⁵N spin-lock power ~9 W) than at the mismatched condition (without ¹⁵N rf irradiation or 0 W). Following that, the observation that the CO signal becomes less positive while the Cα signal becomes less negative in the matched condition proves again that the CE transfers the polarization difference between the two ¹³C spins, not the polarization of a single spin. The ¹³C-detected experiments were then repeated for different ¹⁵N rf power (ω₁₁) and it showed that the broad dip in the ¹³C polarization difference (Fig. 3c) corresponds well to the peak in the ¹⁵N signal. However, we don't expect them to match exactly because of relaxation and RF inhomogeneity, etc.
<figure><img src="image_8.png" /><figcaption>**Figure 4:** Enhancement of nuclear cross effect. (a) Pulse sequence used to accumulate <sup>15</sup>N polarization. (b) <sup>15</sup>N intensity of <sup>1</sup>H-<sup>13</sup>C-<sup>15</sup>N double CP (HCN CP), direct polarization (DP), and nuclear cross effect (NCE). Given that NCE and DP experiments both yield similar signal intensity, and that the measured <sup>1</sup>H-<sup>13</sup>C CP enhancement factor is around &epsilon; ~ 3, we determine that the NCE efficiency is 1 / (3 &times; &gamma;<sub>13C</sub>/&gamma;<sub>15N</sub> &times; 2) &asymp; 6.7 %. (c) Simulated buildup curves of nuclear cross effect with different T<sub>2</sub> relaxation times of <sup>15</sup>N.</figcaption></figure>

Additionally, we also repeated the NCE experiments as a function of the RF carrier frequency of the
<sup>13</sup>C π pulse. This yields an NCE frequency profile (Fig. 3e) that closely resembles a typical CE DNP
field profile.<sup>3,26,27</sup> We also compared the <sup>15</sup>N signal intensity of experiments with selective π pulse,
π/2 pulse, and saturation on <sup>13</sup>C (Fig. 3f). Again, the excellent agreement between the <sup>15</sup>N and ΔP<sub>C</sub>
intensities verifies that ΔP<sub>C</sub> → P<sub>N</sub> polarization transfer has occurred. To our knowledge, such
methodology comparing ΔP<sub>e</sub> and <sup>1</sup>H polarization for the DNP case has never been experimentally
demonstrated,<sup>28</sup> since it is not trivial to perform EPR detection in the same setup that enables CE DNP
experiments. We expect such an experiment for the DNP counterpart will help understand and design
better DNP radicals—if we can measure electron polarization under MAS DNP condition.<sup>29</sup> As the
experimental results, numerical simulations, and analytical theory are in good agreement. We have
therefore proven a new DNP-inspired NMR experiment called *nuclear cross effect*, which emanated
from the unified theoretical framework that we developed.

Following that, we can continue building up <sup>15</sup>N polarization by periodically replenishing the <sup>13</sup>C
polarization from ¹H via CP before repeating each NCE cycle (n) until the ¹⁵N polarization reaches a steady state (Fig. 4).²⁶⁻²⁸ In between each NCE cycle, the ¹⁵N polarization is momentarily stored in the z-axis via two 90° pulses, and a delay of five T₁H was used to allow full ¹H relaxation. ³⁰⁻³² Note that further accumulation of ¹⁵N polarization beyond the first n = 1 cycle was not possible because the relaxation time ¹⁵N (426 ms) is shorter than the ¹H spins (T₁H ~ 436 ms). Note that phase cycling was implemented to disregard the influence of thermal polarization of ¹⁵N on the measured NCE efficiency. By comparing the maximum obtainable ¹⁵N NCE signal with directly excited ¹⁵N thermal polarization (Fig. 4b), we conclude that the NCE enhancement factor is ε ~ 0.33, which corresponds to ~ 6.7 % of the theoretical maximum of 2γ₁₃C/γ₁₅N ~ 5. On the other hand, ¹³C-¹⁵N CP yields ε ~ 1.17, which corresponds to ~ 47 % of the theoretical maximum of γ₁₃C/γ₁₅N ~ 2.5. In other words, NCE is only 0.33/1.17 (~28 %) as efficient as CP.

The inefficient NCE performance could be also caused by the short ¹⁵N T₁ρ (224 ms), a common issue where conventional CP experiments also encounters for certain samples. The hypothesis is verified by numerical simulation (Fig. 4c) using different T₂ values (which resembles T₁ρ in the absence of slow dynamics that interferes with the spin dynamics induced by the RF pulses). The results show a higher ¹⁵N signal for longer T₂, and it could achieve near the theoretical limit of 2γC/γN ~ 5 if T₂ ≫ T₁NCE. In principle, this can be experimentally realized by repeating the experiments at lower temperatures, which often lengthen the relaxation times. Although all experiments shown here were performed on single crystals for simplicity, preliminary experiments show that NCE also works on a powder sample (Fig. S8), despite a much weaker NCE signal since only a small fraction of the crystallites satisfies the orientation-dependent (due to chemical-shift anisotropy and dipolar couplings) NCE matching condition.

Numerical simulations demonstrate that pulsed CE DNP using µw π pulses can achieve higher enhancement than conventional CW DNP experiments. To prove that one can obtain a DNP enhancement higher than the CW DNP limit of γ<sub>e</sub>/γ<sub>1H</sub> ~658, we simulated a *Gedanken* experiment where different rotor angles are intermittently switched and held static to avoid nuclear depolarization. Although a model sample and complete instrumentation required to experimentally verify our simulation results is not readily available, we envision that, high-power pulsed microwave devices such as a high-power extended interaction klystron amplifier (EIKA) or a pulsed gyrotron can be employed to selectively invert a single electron spin.<sup>15,33,34</sup> For the latter strategy, we propose that an existing commercial CW gyrotron can be readily converted to a pulsed gyrotron by replacing the CW power supply (Spellman) with a pulsed power supply (ScandiNova), which can generate ~MW µw pulses. Prior to the high capital investment and effort to experimentally verify pulsed CE DNP, we explored
pulsed *nuclear* cross effect (NCE) on a three-spin nuclear system using a commercial NMR spectrometer.
We have successfully demonstrated NCE by transferring the polarization difference between the ¹³C
spins to the ¹⁵N spin on a glycine single crystal at static. The experimental results were verified by ¹³C-
detected experiments, analytical theory, and numerical simulations. While the nuclear cross effect is
exploited here to simulate the electron-nucleus process, we believe the findings in this work provide a
foundation for future developments in advanced magnetic resonance (NMR, EPR, or DNP) techniques.

Acknowledgements

We thank Dr. Kent Thurber (NIH/NIDDK) and Dr. Frédéric Mentink-Vigier (National High Magnetic Field Laboratory) for their insightful discussions on this work.

**Funding:** This work was supported by funding obtained from the French National Research Agency: PulsedDNP (ANR-20-ERC9-0008) and HFPulsedDNP (ANR-21-CE29-0019).

**Author contributions:** K.O.T. and Z.P. designed the research. Z.P. performed the experiments. Z.P. and K.O.T. analyzed the data. Z.P. and J.L. prepared the single crystal. K.O.T. acquired fundings required for the research. All authors contributed to writing the paper.

**Competing interests:** The authors declare that they have no competing financial interests.

**Supporting Information:**

Additional experimental details, numerical simulations, NMR samples, and derivation of analytical
equations (PDF)

Transparent Peer Review report available (PDF)

References

(1) Abragam, A.; Goldman, M. Principles of Dynamic Nuclear Polarisation. *Reports on Progress in Physics* **1978**, *41* (3), 395–467. https://doi.org/10.1088/0034-4885/41/3/002.

(2) Wenckebach, T. *Essentials of Dynamic Nuclear Polarization*, Spindrift Publications, 2016.

(3) Lilly Thankamony, A. S.; Wittmann, J. J.; Kaushik, M.; Corzilius, B. Dynamic Nuclear Polarization for Sensitivity Enhancement in Modern Solid-State NMR. *Prog Nucl Magn Reson Spectrosc* **2017**, *102–103*, 120–195. https://doi.org/10.1016/j.pnmrs.2017.06.002.

(4) Hwang, C. F.; Hill, D. A. New Effect in Dynamic Polarization. *Phys Rev Lett* **1967**, *18* (4), 110–112. https://doi.org/10.1103/PhysRevLett.18.110.

(5) Hwang, C. F.; Hill, D. A. Phenomenological Model for the New Effect in Dynamic Polarization. *Phys Rev Lett* **1967**, *19* (18), 1011–1014. https://doi.org/10.1103/PhysRevLett.19.1011.

(6) Kessenikh, A. V; Manenkov, A. A.; Pyatnitskii, G. I. On Explanation Of Experimental Data On Dynamic Polarization Of Protons In Irradiated Polyethylenes. *Soviet Phys. -Solid State* **1964**, *6*, 641–643.
(7) Hu, K.-N.; Yu, H.; Swager, T. M.; Griffin, R. G. Dynamic Nuclear Polarization with Biradicals. *J Am Chem Soc* **2004**, *126* (35), 10844–10845. https://doi.org/10.1021/ja039749a.

(8) Hu, K.-N.; Song, C.; Yu, H.; Swager, T. M.; Griffin, R. G. High-Frequency Dynamic Nuclear Polarization Using Biradicals: A Multifrequency EPR Lineshape Analysis. *J Chem Phys* **2008**, *128* (5), 052302. https://doi.org/10.1063/1.2816783.

(9) Thurber, K. R.; Tycko, R. Theory for Cross Effect Dynamic Nuclear Polarization under Magic-Angle Spinning in Solid State Nuclear Magnetic Resonance: The Importance of Level Crossings. *Journal of Chemical Physics* **2012**, *137* (8), 0–14. https://doi.org/10.1063/1.4747449.

(10) Kundu, K.; Dubroca, T.; Rane, V.; Mentink-Vigier, F. Spinning-Driven Dynamic Nuclear Polarization with Optical Pumping. *J Phys Chem A* **2022**, *126* (16), 2600–2608. https://doi.org/10.1021/acs.jpca.2c01559.

(11) Pang, Z.; Jain, S.; Yang, C.; Kong, X.; Tan, K. O. A Unified Description for Polarization-Transfer Mechanisms in Magnetic Resonance in Static Solids: Cross Polarization and DNP. *J Chem Phys* **2022**, *156* (24), 244109. https://doi.org/10.1063/5.0092265.

(12) Carver, T. R.; Slichter, C. P. Experimental Verification of the Overhauser Nuclear Polarization Effect. *Physical Review* **1956**, *102* (4), 975–980. https://doi.org/10.1103/PhysRev.102.975.

(13) Overhauser, A. W. Polarization of Nuclei in Metals. *Physical Review* **1953**, *92* (2), 411–415. https://doi.org/10.1103/PhysRev.92.411.

(14) Kaiser, R. Use of the Nuclear Overhauser Effect in the Analysis of High-Resolution Nuclear Magnetic Resonance Spectra. *J Chem Phys* **1963**, *39* (10), 2435–2442. https://doi.org/10.1063/1.1734045.

(15) Zhao, Y.; El Mkami, H.; Hunter, R. I.; Casano, G.; Ouari, O.; Smith, G. M. Large Cross-Effect Dynamic Nuclear Polarisation Enhancements with Kilowatt Inverting Chirped Pulses at 94 GHz. *Commun Chem* **2023**, *6* (1), 171. https://doi.org/10.1038/s42004-023-00963-w.

(16) Equbal, A.; Leavesley, A.; Jain, S. K.; Han, S. Cross-Effect Dynamic Nuclear Polarization Explained: Polarization, Depolarization, and Oversaturation. *J Phys Chem Lett* **2019**, *10* (3), 548–558. https://doi.org/10.1021/acs.jpclett.8b02834.

(17) Hediger, S.; Lee, D.; Mentink-Vigier, F.; Paëpe, G. De. MAS-DNP Enhancements: Hyperpolarization, Depolarization, and Absolute Sensitivity. *eMagRes* **2018**, *7*, 1–13. https://doi.org/10.1002/9780470034590.emrstm1559.

(18) Mentink-Vigier, F.; Mathies, G.; Liu, Y.; Barra, A.-L.; Caporini, M. A.; Lee, D.; Hediger, S.; G. Griffin, R.; De Paëpe, G. Efficient Cross-Effect Dynamic Nuclear Polarization without Depolarization in High-Resolution MAS NMR. *Chem Sci* **2017**, *8* (12), 8150–8163. https://doi.org/10.1039/C7SC02199B.

(19) Mentink-Vigier, F.; Paul, S.; Lee, D.; Feintuch, A.; Hediger, S.; Vega, S.; De Paëpe, G. Nuclear Depolarization and Absolute Sensitivity in Magic-Angle Spinning Cross Effect Dynamic Nuclear Polarization. *Physical Chemistry Chemical Physics* **2015**, *17* (34), 21824–21836. https://doi.org/10.1039/C5CP03457D.

(20) Hirsh, D. A.; Rossini, A. J.; Emsley, L.; Schurko, R. W. 35Cl Dynamic Nuclear Polarization Solid-State NMR of Active Pharmaceutical Ingredients. *Physical Chemistry Chemical Physics* **2016**, *18* (37), 25893–25904. https://doi.org/10.1039/C6CP04353D.

(21) Liu, Y.; Villamena, F. A.; Rockenbauer, A.; Song, Y.; Zweier, J. L. Structural Factors Controlling the Spin-Spin Exchange Coupling: EPR Spectroscopic Studies of Highly Asymmetric Trinityl–Nitroxide Biradicals. *J Am Chem Soc* **2013**, *135* (6), 2350–2356. https://doi.org/10.1021/ja311571v.

(22) Mathies, G.; Caporini, M. A.; Michaelis, V. K.; Liu, Y.; Hu, K. N.; Mance, D.; Zweier, J. L.; Rosay, M.; Baldus, M.; Griffin, R. G. Efficient Dynamic Nuclear Polarization at 800 MHz/527 GHz with Trinityl–Nitroxide Biradicals. *Angewandte Chemie – International Edition* **2015**, *54* (40), 11770–11774. https://doi.org/10.1002/anie.201504292.

(23) Can, T. V.; Weber, R. T.; Walsh, J. J.; Swager, T. M.; Griffin, R. G. Ramped-Amplitude NOVEL. *J Chem Phys* **2017**, *146* (15), 154204. https://doi.org/10.1063/1.4980155.

(24) Korzeczek, M. C.; Dagys, L.; Müller, C.; Tratzmiller, B.; Salhov, A.; Eichhorn, T.; Scheuer, J.; Knecht, S.;
Plenio, M. B.; Schwartz, I. Towards a Unified Picture of Polarization Transfer — Pulsed DNP and Chemically Equivalent PHIP. *Journal of Magnetic Resonance* **2024**, *362*, 107671. https://doi.org/10.1016/j.jmr.2024.107671.

(25) Sabba, M.; Wili, N.; Bengs, C.; Whipham, J. W.; Brown, L. J.; Levitt, M. H. Symmetry-Based Singlet–Triplet Excitation in Solution Nuclear Magnetic Resonance. *J Chem Phys* **2022**, *157* (13), 134302. https://doi.org/10.1063/5.0103122.

(26) Michaelis, V. K.; Smith, A. A.; Corzilius, B.; Haze, O.; Swager, T. M.; Griffin, R. G. High-Field 13C Dynamic Nuclear Polarization with a Radical Mixture. *J Am Chem Soc* **2013**, *135* (8), 2935–2938. https://doi.org/10.1021/ja312265x.

(27) Hu, K.-N.; Bajaj, V. S.; Rosay, M.; Griffin, R. G. High-Frequency Dynamic Nuclear Polarization Using Mixtures of TEMPO and Trityl Radicals. *J Chem Phys* **2007**, *126* (4), 044512. https://doi.org/10.1063/1.2429658.

(28) Hovav, Y.; Shimon, D.; Kaminker, I.; Feintuch, A.; Goldfarb, D.; Vega, S. Effects of the Electron Polarization on Dynamic Nuclear Polarization in Solids. *Phys. Chem. Chem. Phys.* **2015**, *17* (8), 6053–6065. https://doi.org/10.1039/C4CP05625F.

(29) Nir-Arad, O.; Fialkov, A. B.; Shlomi, D. H.; Manukovsky, N.; Mentink-Vigier, F.; Kaminker, I. High-Field Pulsed EPR Spectroscopy under Magic Angle Spinning. *Sci Adv* **2024**, *10* (35). https://doi.org/10.1126/sciadv.adq6073.

(30) Mathies, G.; Jain, S.; Reese, M.; Griffin, R. G.; Shankar Palani, R.; Mardini, M.; Quan, Y.; Griffin, R. G. Pulsed Dynamic Nuclear Polarization with Trityl Radicals. *J Phys Chem Lett* **2016**, *7* (1), 111–116. https://doi.org/10.1021/acs.jpclett.5b02720.

(31) Bertarello, A.; Berruyer, P.; Skantze, U.; Sardana, S.; Sardana, M.; Elmore, C. S.; Schade, M.; Chiarparin, E.; Schantz, S.; Emsley, L. Quantification of Magic Angle Spinning Dynamic Nuclear Polarization NMR Spectra. *Journal of Magnetic Resonance* **2021**, *329*, 107030. https://doi.org/10.1016/j.jmr.2021.107030.

(32) Johnson, R. L.; Schmidt-Rohr, K. Quantitative Solid-State 13C NMR with Signal Enhancement by Multiple Cross Polarization. *Journal of Magnetic Resonance* **2014**, *239*, 44–49. https://doi.org/10.1016/j.jmr.2013.11.009.

(33) Tan, K. O.; Jawla, S.; Temkin, R. J.; Griffin, R. G. Pulsed Dynamic Nuclear Polarization. *eMagRes* **2019**, *8* (3), 339–352. https://doi.org/10.1002/9780470034590.emrstm1551.

(34) Pagonakis, I. Gr.; Genoud, J.; Gao, C.; Sirigiri, J. R.; Alaniva, N. H.; Alberti, S.; Angehn, F. N.; Bjorgvinsdottir, S.; Chen, P.-H.; Dapp, A.; Gunzenhauser, R.; Hogge, J.-P.; Kaser, T.; Millen, M.; Urban, M.; Barnes, A. B. Development of Kilowatt THz Gyrotrons for Pulsed Dynamic Nuclear Polarization - Cavity Design Code Phaedra - In *2022 47th International Conference on Infrared, Millimeter and Terahertz Waves (IRMMW-THz)*, IEEE, 2022; pp 1–2. https://doi.org/10.1109/IRMMW-THz50927.2022.9895907.