Heteronuclear DNP of 1 H and 19 F nuclei using BDPA as a polarizing agent

This work explores the dynamic nuclear polarization (DNP) of 1 H and 19 F nuclei in a sample of 25/75 (% v/v) fluorobenzene/toluene containing the radical 1,3-bisphenylene-2-phenylallyl radical (BDPA) as a polarizing agent. Previously, heteronuclear effects in DNP were studied by analysing the shapes of DNP spectra, or by observing cross-relaxation between nuclei of different types. In this work, we report a rather specific DNP spectrum, where 1 H and 19 F nuclei obtain polarizations of opposite signs upon microwave (MW) irradiation. In order to explain this observation, we introduce a novel mechanism called heteronuclear thermal mixing (hn-TM). Within this mechanism the spectra of opposite signs can then be explained due to the presence of four-spin systems, involving a pair of dipolar coupled electron spins and hyperfine coupled nuclear spins of 1 H and 19 F, such that a condition relating their Larmor frequencies |𝜔 1𝑒 − 𝜔 2𝑒 | ≈ 𝜔 𝐻 − 𝜔 𝐹 is satisfied. Under this condition, a strong mixing of electron and nuclear states takes place, enabling simultaneous four-spin flip-flops. Irradiation of electron spin transitions with MW followed by such four-spin flip-flops produces non-equilibrium populations of |𝛼 𝐻 𝛽 𝐹 ⟩ and |𝛽 𝐻 𝛼 𝐹 ⟩ states, thus leading to the enhancements of opposite signs for 1 H and 19 F. Signal enhancements, build-up times and DNP-spectra as a function of MW power and polarizing agent concentration, all provide additional support for assigning the observed DNP mechanism as hn-TM and distinguishing it from other possible mechanisms. We also develop a quantum mechanical model of hn-TM based on averaging of spin Hamiltonians. Simulations based on this model show very good qualitative agreement with experimental data. In addition, the system exhibits cross-relaxation between 1 H and 19 F induced by the presence of BDPA, which was detected by measuring the 19 F signal build-up upon saturation of 1 H nuclei with a train of radio-frequency pulses. We demonstrate that such cross-relaxation most likely originates due to the same electron and nuclear states mixing in the four-spin systems.


Introduction
Dynamic Nuclear Polarization (DNP) allows increasing the nuclear magnetic resonance (NMR) signals by transferring large polarization of electron spins onto coupled nuclear spins via microwave (MW) irradiation. Signal enhancements due to DNP are widely used to improve the sensitivity of solid state and solution NMR spectroscopy 1,2 , as well as medical magnetic resonance imaging 3 . In many of these applications, the studied samples contain more than one type of polarizable nuclei, further referred to as heteronuclei. The presence of heteronuclei affects the DNP spectra, where the signal enhancements are recorded as a function of MW frequency applied to the system. Furthermore, in systems with heteronuclei the presence of unpaired electrons induces a polarization exchange between nuclei of different types, thereby indirectly affecting the dynamics of their polarization build-up in the DNP experiments.
Understanding the mechanisms of these heteronuclear effects is therefore important for the ultimate goal of obtaining the optimal conditions for DNP.
Lots of insights into these heteronuclear effects were previously obtained by extending the main physical mechanisms known to produce DNP in non-conducting solids, such as solid effect (SE) [4][5][6][7] , cross-effect (CE) 5,8 and thermal mixing (TM) 4,5 . One such mechanism involving two types of polarizable nuclei is a double-solid effect (double-SE), which can be explained using a simple quantum mechanical model involving three spins: one electron and two nuclei.
There, MW irradiation applied to the relevant "forbidden" transitions can simultaneously flip all three spins. Such phenomenon has been first observed experimentally by de Boer in a sample of deuterated m-xylene-d6 doped with 1,3-bisphenylene-2-phenylallyl radical (BDPA) as a polarizing agent in a magnetic field of 2.5 T 9 . In addition to conventional SE, the DNP spectrum there had features centred at the MW irradiation frequencies = + ± and = − ± , where , and are the Larmor frequencies of 1 H, 2 H and electrons respectively.
The theory of thermal mixing uses thermodynamic approach to explain DNP. In this mechanism, MW irradiation lowers the temperature of the energy reservoir formed by the electron spin dipolar interactions. DNP arises as a result of the energy exchange between this dipolar reservoir with nuclear Zeeman energy reservoirs. Under conditions of slow energy exchange with the lattice, this process produces the same temperature across all nuclear reservoirs, as can be experimentally confirmed by similar shapes of their DNP spectra [9][10][11][12][13][14][15] . On the other hand, nuclear Zeeman reservoirs are coupled to the electron spin dipolar reservoir regardless of the applied MW irradiation. Therefore, thermodynamically speaking, the two types of nuclei are in the indirect thermal contact with one another, leading to an observable polarization exchange between the two 11,16 . Recently, however, thermal mixing was also treated using models based on density matrix formalism 17,18 . In particular, the emergence of a common spin temperature in the dipolar reservoir of strongly coupled electron spins has been confirmed by the simulations 17 , which provides support for this concept, crucial to the thermodynamic description of TM-mechanism. However, no effective quantum mechanical treatment of TM in a heteronuclear system has been presented so far.
The effect of two types of polarizable nuclei has been considered quantum mechanically in a system containing two electrons, one coupled 1 H and one coupled 13 C nuclei 19 . In addition to the conventional SE and CE mechanisms expected for this system, that work predicted an existence of a four-spin mechanism called { 1 H, 13 C}-heteronuclear-cross-effect (hn-CE). There, a pair of levels becomes degenerate when the difference of the two electron Larmor frequencies (denoted by 1 and 2 ) is matched by a sum or difference of the two nuclear Larmor frequencies, i.e. | 1 − 2 |≈ | ± |, where and are the Larmor frequencies of 1 H and 13 C respectively. In particular, in a hypothetic system with two narrow electron lines centred at frequencies 1 and 2 , such that | 1 − 2 | ≈ − and 1 < 2 , irradiation at ≈ 1 produces positive enhancement for 1 H nuclei and negative enhancement for 13 C nuclei. In turn, irradiation at ≈ 2 produces negative enhancement for 1 H and positive enhancement for 13 C nuclei. However, to the best of our knowledge no experimental data clearly showing such DNP spectrum have been presented so far.
Kaminker et al. have discovered that DNP spectra of 2 H nuclei have the same shape as the DNP spectra of 1 H-nuclei arising due to the CE mechanism in a system with a nitroxide radical as a polarizing agent (magnetic field ~3.4 T, temperature 6 K) 20 . In addition, they observed a polarization transfer between 1 H and 2 H nuclei in an experiment which follows the recovery of 2 H signals, starting with saturated 2 H nuclei and highly polarized 1 H nuclei. The recovery curves show a characteristic overshoot, where 2 H nuclei quickly achieve polarization larger than thermal due to their cross-relaxation with 1 H. This high polarization then slowly decays towards the thermal equilibrium. TM-mechanism cannot provide an adequate explanation of these experiments, because as shown previously, the DNP in such systems arises due to the SE and CE 21 . The observed nuclear cross-relaxation can instead be explained using a four-spin model developed in ref. 19 by pointing out that the electron and nuclear state mixing in a four-spin system takes place regardless of the applied MW irradiation. Therefore the nonequilibrium polarization in one type of nuclei can get transferred to the nuclei of another type by means of electrons pairs satisfying the hn-CE matching condition. The same four-spin mechanism may also potentially play a role in a previously observed cross-relaxation between nuclei of different types such as 1 H and 13  Another possibility for the creation of the frequency shift between the electrons arises if the model is extended to include many strongly dipolar coupled electron spins. As Fig. 1C illustrates, in this scenario the electron levels split into bands, corresponding to a projection of the total electron angular momentum | ⟩, while the energy difference between these bands is ~the central frequency of the ESR line. The characteristic width of a band arises primarily due to dipolar couplings between electrons but could also have contributions due to differences in -values and hyperfine interactions. A pair of 1 H and 19 F nuclei coupled to such an electronic system produces splitting in these bands as shown in Fig. 1C

Quantum mechanical model of heteronuclear DNP
We where ̂m w describes the microwave irradiation energy, ̂e n describes the semi-secular hyperfine (HF) interaction energy, and ̂0 contains the electron and nuclear Zeeman interaction energies, the secular HF interaction energy and the energy of the mutual electron dipolar coupling. Specifically, the ̂m w and ̂e n have the form: where ( ) , ( ) are the semi-secular hyperfine interactions of 1 H and 19 F nuclei respectively, and ℎ. . stands for Hermitian conjugates of the operators. The summation over " " in eqn (2) and (3) is carried out over all electrons in the system, while summation over " " and " " in eqn (3) is carried out over all nuclei coupled to the electron spin " ". The Hamiltonian term ̂0 can be conveniently split into two parts, where ̅ 0 represents the Zeeman part , and ̂0 are the remaining terms: Here = − denotes the offset of the microwave frequency from the centre of the electron resonance line . In addition, Δ ( ) = − are the electron frequency shifts due to -anisotropy, ( ) are the strengths of secular hyperfine interactions of 1 H nuclei, and ′ is the mutual dipolar electron spin interaction strengths. For simplicity, we neglect the 19 F hyperfine secular interaction as this species is remote from the electrons. We also neglect the dipolar interactions between the considered nuclear spins as they are too weak compared to other interactions. Eqn (6) describes the broadening of ESR line. There, the first term is responsible for inhomogeneous broadening due to -anisotropy and hyperfine couplings, whereas the second term describes homogeneous broadening due to electron spin dipolar couplings.
In this system there are several possible mechanisms for polarisation transfer between electrons and nuclear spins. In the simplest case polarisation transfer can be mediated by SE DNP between an electron and its coupled 1 H nuclei ( ) as well as between the same electron and Since the ESR linewidth of BDPA radicals is much smaller than the Larmor frequencies of both 1 H and 19 F nuclei, the difference of electron Larmor frequencies 1 , 2 does not match any of those nuclear frequencies as needed for conventional CE 5,8 On the other hand, the ESR linewidth is wide enough to fulfil the condition for the hn-CE, i.e.
In this case, the polarisation transfer is mediated by an energy conserving four-spin flip-flop that involves two electrons, one 1 H and one 19 F nucleus. There, the difference of electron polarisations between electron spins is transferred to the nuclear spin pair. However, such four-spin flip-flops can also be brought about by the heteronuclear thermal mixing (hn-TM) which was qualitatively described above. The properties of the electron spin system, such as the strength of interactions and the timescales of relaxation processes, determine which of the two mechanisms would dominate. The strength of the two terms in ̂0 becomes of particular importance. As pointed out earlier, the first term in ̂0 describes the inhomogeneous broadening, whereas the second term describes the mutual electron spin dipolar interactions. Qualitatively, in the CE mechanism, the inhomogeneous broadening (i.e. the first term) is large enough to treat the evolution under the dipolar interactions (i.e. the second term) as a perturbation. In contrast, under TM mechanism, the evolution of the spin system is dominated by the mutual electron dipolar interactions, whereas the first term acts only as a perturbation.
The Electronic Supplementary Information (ESI) provides the details of a procedure for averaging eqn (1) in order to obtain an effective Hamiltonian representing the macroscopic electron-nuclear dynamics arising due to the different possible DNP mechanisms. The two necessary key steps of this procedure are the transformation of eqn (1) to a frame rotating with zero-quantum spin transitions and the adiabatic elimination of oscillating non-secular terms.
The calculations of effective Hamiltonians for SE, hn-CE and hn-TM were carried out, and a summary of those calculations for the hn-TM pathway is presented below.
The hn-TM mechanism can be described by a minimal model Hamiltonian ̂h nTM that involves two effective electrons 1 and 2 and two unlike nuclear spins and , which represent the where is the effective electron dipolar interaction strength and , are the effective pseudo-secular hyperfine interactions of the 1 H and 19 F nuclei respectively. In addition, ± (Δ) and ± are the shorthand notations for: where normalization factor represents the average frequency gap between the multiple electron lines that arise from the secular hyperfine interaction of 1 H nuclei close to the electron.
As seen from eqn (8) The dispersion parameter plays an important role of breaking the symmetry in the Hamiltonian by shifting one electron spin frequency with respect to the other, as was previously discussed for regular TM mechanism 18 . When the dispersion parameter = 0, the effective spin system Hamiltonian shown in eqns (7)(8)(9) is symmetric with respect to swapping 1 and 2 operators, making the two electron spins indistinguishable. As a result, when MW is applied to such a system, both 1 and 2 get saturated with the same effective MW strength as seen from eqn (8), so that the polarization difference required for the DNP cannot be created.
Furthermore, when = 0 , parameters ± in eqn (9) have the same magnitude, with ̂h nTM en producing no difference in nuclear polarizations. Such highly symmetric scenario with = 0 however, can only be realized in an ideal single crystal without any heterogeneities. In any real sample, there are many heterogeneities, meaning that electron spin frequencies at each site are different leading to ≠ 0. In this case, the symmetry of the effective Hamiltonian with respect to swapping 1 and 2 is broken as seen in eqns (7)(8)(9). The departure from the highly symmetric case, described by the dispersion parameter , is thus a very important characteristic of the system needed for producing the DNP.
The effective Hamiltonians for hn-TM and other DNP mechanisms can be used to derive the master equations for the density matrix in the Lindblad form, as shown in the "Density Matrix Simulations" section of the ESI. In these equations, the relaxation is taken into account using a standard approach with single-spin Markovian jump operators. The simulations based on these master equations will be compared to experimental data in the "Results" section. The calculations use various relaxation and other common parameters shown in Table 1. In addition, the ESR lineshape (Δ) in these calculations is taken as a Gaussian, described by a full width at half height (FWHH) parameter .

Sample preparation
The and finally, the sample was thawed and the normal pressure is restored by letting the nitrogen gas into the system.

Equipment
All experiments were performed using a NMR-DNP system described previously 23   Previously, DNP spectra of 1 H in m-xylene-2,2-d6 doped with BDPA at 2.5 T magnetic field 9 , were shown to exhibit a similar small feature explained by a rather inefficient thermal mixing mechanism, which could be the case for our measurements as well. However, the 19 F-DNP spectrum also shows negative and positive enhancements at the frequencies of ( − 0 )/2 ≈ ∓20 MHz respectively. Such features have signs opposite to what is expected from a regular TM mechanism, and therefore cannot be explained by it. However, as the spectrum reveals, the feature corresponding to ( − 0 )/2 = ( + )/2 = +277 MHz is not present in the spectrum, therefore features at ( − 0 )/2 = ∓20 MHz cannot be attributed to double-SE.

DNP enhancements
In contrast, four-spin models of heteronuclear DNP (hn-DNP), such as hn-CE developed by Shimon et al. 19 and hn-TM explained in the "Theoretical Background" section, agree better with the observed DNP spectrum. Both models rely on a strong mixing of electron and nuclear levels for 1 H and 19 F when electron frequencies differ by | 1 − 2 | ≈ − . In both hn-CE and hn-TM scenarios, MW irradiation applied to an allowed ESR transition, followed by four-spin flip-flops, creates non-equilibrium populations of | ⟩ and | ⟩ states, thereby producing enhancements of opposite signs for 1 H and 19 F nuclei.
As pointed out in the "Theoretical Background" section the dominance of hn-TM or hn-CE mechanism is determined by the properties of the electron spin system, such as its homogeneous and inhomogeneous broadening and relaxation times. In order to help resolve the two mechanisms in our study, ESR spectra of BDPA at two concentrations were recorded using a high-field ESR spectrometer, as described in ESI. As can be seen in Fig. S1   and it shows a very good qualitative agreement with the experimental data in Fig. 2A.
To support further our assignment of SE and hn-DNP peaks, we explore the dependence of hn-DNP on the concentration of BDPA. The build-up times and enhancements extracted from the build-up curves of the samples with various BDPA concentrations are summarized in Table 2.
The transition probability needed for the double-SE mechanism is smaller compared to regular SE, for that reason, the build-up time for the double-SE is expected to be noticeably longer than for regular SE 9,19 . As Table 2 shows, the build-up times for hn-DNP and SE-DNP are indeed different, yet have the same order of magnitude, which provides additional evidence that spectral lines assigned as hn-DNP cannot arise due to double-SE. The build-up times for SE and hn-DNP become shorter and enhancements increase with the concentration of the polarizing agent, and yet for hn-DNP the dependence is somewhat stronger. Since the SE mechanism involves excitation of only one electron, this concentration dependence qualitatively agrees with the fact that the hn-DNP process involves more than one unpaired electron spin. The build-up times of thermal signals were measured without MW irradiation.
They also become shorter as the BDPA concentration increases, meaning that 19 F nuclear relaxation times are not dominated by the intrinsic processes, independent on the unpaired electrons. As we show later, in fact, the observed thermal signal build-ups rates depend significantly on the polarization transfer between 1 H and 19 F.  Fig. 3C. There, the concentration dependence of the DNP spectral shape is noticeable, but not very pronounced.
Furthermore, the simulations show an increase of the enhancement up to the concentration of 8 mM, followed by a decrease for concentrations higher than that, in stark contrast with the experimental data. The simulations in Fig. 3B and 3C therefore provide support for hn-TM as a possible mechanism.
The DNP-enhancement of 19 F signals due to SE and hn-DNP were also measured at a 100 mW microwave power level for a sample with [BDPA]=30 mM, as shown in Table 2  represented by hn-TM model. Overall, these simulations provide additional evidence that the observed hn-DNP arises due to hn-TM mechanism.

H-19 F cross-relaxation
As pointed out in the "Theoretical Background" section, the strong mixing of electron and nuclear states in a four-spin system can produce cross-relaxation between the 1 H and 19 In both cases, the magnetization builds up with the same time-constant ( + ) and the ratio of steady-state polarizations is given by: In another experiment, which sequence is shown in Fig. S2A of the ESI, the recovery of 19 F signals was followed after turning off the MW irradiation and saturation of 19 F. In other words, the system evolution without MW irradiation starts with the "hot" (high spin temperature) 19 F nuclei and "cold" (low spin temperature) 1 H nuclei. The recovery of 19 F and 1 H signals for a sample with [BDPA]=40 mM is shown in Fig. S2B of the ESI. There, 1 H signals decay to thermal equilibrium with the rate close to the intrinsic , while the 19 F signals quickly recover from zero to some value, which is higher than thermal equilibrium polarization , , and then gradually decay to , . Such behaviour is much more prominent in the sample containing Further evidence for a cross-relaxation between 19 F and 1 H can be observed in the DNP spectra of solid effect. Fig. S3  Gaussians, however, produced very large variance in the fitted parameters, and therefore is not shown.

Heteronuclear-DNP
In this work we observed the hn-DNP effect arising in the system containing 1 H and 19  The hn-CE proposed earlier 19 and hn-TM explained in the "Theoretical Background" section are related to one another as regular CE and TM. The transfer of energy between the electron and nuclear spins in both hn-CE and hn-TM is based on the four-spin mechanism described previously 19 , however the polarization gradient or the difference of polarization between the neighbouring electrons is created in a different manner. In CE, the polarization gradient is produced by irradiating individual electron spin in a pair, whereas in TM it is produced by the microwave irradiation of a system of many coupled electron spins.
At the same time, it is also important to point out, that while the existence of electron dipolar energy temperature needed for the thermal mixing has been confirmed experimentally by measuring saturated ESR lineshapes in systems such as paramagnetic impurities in single crystals as reviewed in ref. 38 , to the best of our knowledge, for stable organic radicals used in modern DNP applications it was done only indirectly by comparing the spin temperatures of polarizable nuclei in the system 12,13,15,39 . ELDOR measurements, probing the shape of the ESR line after the saturation would help to provide a more quantitative view of the observed hn-DNP effect 40 , however that is beyond the scope of this work.

H-19 F cross-relaxation.
The mixing of electron and nuclear states in a four-spin model required for both hn-CE and hn-TM predicts the presence of a cross-relaxation between nuclei of different types in the absence of MW irradiation. Such cross-relaxation was observed via a recovery of saturated 19 41 . In a similar manner, the electron relaxation may also involve terms ̂̂±̂∓ leading to a flip-flop of two nuclei, one 1 H and one 19 F, which produce the cross-relaxation between 1 H and 19 F, and affect the rate . Since the probability of the "forbidden" transitions involving a flip of two nuclei depends on the hyperfine and Larmor frequencies in the higher order of perturbation theory 19,30 , such an effect is expected to be significantly weaker. In other words, the increase in rate due to increase in BDPA concentration would be greater than the induced increase in . This in turn, disagrees with experimental data shown in Table 3  The rate of cross-relaxation in TEMPO compared to BDPA is faster, which cannot be explained well at our current level of understanding. In a sample with a broad ESR line the probability of finding a "good" pair of electrons satisfying the matching condition | 1 − 2 | ≈ | − | should be smaller, yielding smaller observed cross-relaxation rate , in contrast with our observations. However, the rate of polarization transfer depends on the hyperfine couplings, due to the factor shown in eqn (9). As pointed out earlier, the unpaired electron spin density in BDPA is delocalized over many atoms, leading to small hyperfine couplings with matrix 19 F and 1 H nuclei. In contrast, in the TEMPO radical, the spin density is partitioned almost equally between the nitrogen and oxygen atoms of the NO fragment, which may lead to bigger couplings with nearby nuclei 43 . Therefore, while the likelihood of finding a suitable pair of electrons is smaller for TEMPO, the polarization transfer rates in TEMPO could be greater due to larger hyperfine couplings and may lead to a larger observed nuclear cross-relaxation rate.
Interestingly, a rather noticeable cross-relaxation is present even in the degassed sample without any polarizing agent. Two possible explanations for this can be offered. First, previous experiments in LiF crystals have revealed that energy reservoirs formed by the dipolar couplings of Li and F nuclei acquire the same temperature 44 , and therefore saturation of one type of nuclei can be transferred to the other. Alternatively, the crosstalk may be caused by some other non-coherent cross-relaxation process and may be specific to the molecular structure of the matrix molecules. Recently it was found that a nuclear Overhauser effect-like cross-relaxation between 1 H and 13 C exists under conditions of solid state magic angle spinning DNP (magnetic field ~9.4T , temperature ~100 K) 45,46 . Such a mechanism requires a difference in the rates, and effectively spectral densities, associated with double-quantum and zeroquantum relaxation. One potential process that might cause this at the temperature of 1.4 K could be the tunnelling of hindered methyl groups in toluene molecules 47 . However, a detailed investigation and thereby distinguishing between the two potential mechanisms goes beyond the scope of this work.

Conclusions.
This work provides the evidence of hn-DNP effects in a system with polarizable 19