Double nutation cross-polarization between heteronuclear spins in solids †

We study transfer of magnetization from one nuclear spin species to another in solid-state nuclear magnetic resonance by cross-polarization (CP) employing radiofrequency irradiation that causes simultaneous nutations around a pair of orthogonal axes. Under such DOuble NUTation (DONUT), polarization transfer proceeds in an unexplored arena of what we refer to as the nutation frame, which represents the interaction frame with respect to the Hamiltonian that drives nutation. The eﬀect of DONUT is to develop either the zero-quantum or double-quantum secular component of the heteronuclear dipolar interaction, causing flip-flop or flop-flop exchange of the spin states. We demonstrate DONUT CP in polycrystalline adamantane, glycine, and histidine, also examining folding of the CP spectrum under magic-angle spinning as well as the buildup behavior of the magnetization in comparison with the conventional CP scheme. In addition, we put forth a concept of spin relaxation in the nutation frame, which is a straightforward extension of the well-known concept of spin relaxation in the rotating frame.


Introduction
Cross-polarization (CP) is a technique of solid-state nuclear magnetic resonance (NMR) spectroscopy to enhance the sensitivity [1][2][3] by transferring the magnetization from one nuclear spin species, I, to another, S, through simultaneous applications of radio-frequency (rf) irradiation in such a way that the Hartmann-Hahn condition is fulfilled. 4Usually, CP takes place in a doubly rotating frame, where the coordinate systems for I and S are rotated separately around the static magnetic field at their respective Larmor frequencies.][7][8] In this work, we report on CP proceeding in a different arena realized through simultaneous nutations of the source spins I around two separate magnetic fields B 1 and B 2 at frequencies o 1I = Àg I B 1 and o 2I = Àg I B 2 , where g I is the gyromagnetic ratio of spin I.Such DOuble NUTation (DONUT) is driven by a propagator U I (t) of the form U I (t) = exp(Àio 1I tB 1 ÁI) exp(Àio 2I tB 2 ÁI).The idea of multiple nutation was first put forth by Khaneja and Nielsen, who studied homonuclear dipolar recoupling with the Triple Oscillating Field techniqUe (TOFU 12 ), and then extended by Straasø et al. to the Four-Oscillating FielD (FOLD) technique. 13Later on, DONUT was applied to heteronuclear decoupling by Takeda et al. 14 Here, we restrict ourselves to DONUT with B 1 = (Ào 1I /g I ,0,0) and B 2 = (0,0,Ào 2I /g I ), so that the propagator U I (t) is represented in the form U I (t) = exp(Àio 1I tI x ) exp(Àio 2I tI z ). ( The rf Hamiltonian H I rf (t) that results in the propagator U I (t) in eqn ( 1) is straightforwardly obtained with H I rf (t) = i : U I U I À1 14,15 : DONUT, as well as nutation around triple-and quadrupleoscillating fields, 12,13 requires smooth modulations of the rf pulses, which can be implemented with modern NMR spectrometers.
In DONUT CP, it is in the interaction frame with respect to H 1 = o 1I I x (eqn (3)) that the source spins I contact with the target spins S. To represent this interaction frame, we coin a term, the nutation frame.Then, the essence of DONUT CP can be stated conveniently as polarization transfer that takes place between the source spin I in the nutation frame and the target spin S in the rotating frame.Transformation from the rotating frame to the nutation frame is analogous to a well-known transformation from the laboratory frame to the rotating frame, in which a static magnetic field vanishes, whereas a resonant rf field in the former becomes stationary in the latter, as described in Fig. 1(a).In DONUT, another time-dependent field B 2 is applied, such that, in the rotating frame, B 2 follows a circular trajectory around the stationary field B 1 at the nutation frequency o 1I .By performing a transformation from the rotating frame into the nutation frame, B 1 vanishes and B 2 becomes stationary (Fig. 1(b)).

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][18][19][20][21][22][23][24][25][26][27][28][29] They improve robustness against the mismatch of the Hartmann-Hahn condition and/or the efficiency of spin-locking.Conversely, one noteworthy feature of DONUT CP is the way that the spin interactions are manipulated in the nutation frame; they acquire time dependence with mixed frequencies o 1I AE o 2I .This is analogous to a frequency mixer, an electrical device used to literally mix a pair of alternating current signals with different frequencies, say, o p and o q , into another pair of signals with frequencies o p AE o q .The mixing leads to a different condition for polarization transfer from the conventional Hartmann-Hahn condition.
In the following, we describe the principle of DONUT CP, deriving the modified Hartmann-Hahn condition both in static and magic-angle spinning (MAS) cases.Then, we discuss the practical aspects of implementing DONUT using rf irradiation with modulation, with a note on the sign of the phase of the rf signals set in the transmitter of the NMR spectrometer and that of the rf field that develops inside the coil and is felt by the nuclear spins.We demonstrate 1 H- 13 C DONUT CP in adamantane and glycine, and 1 H- 15 N DONUT CP in histidine.

Theory
We consider a dipolar-coupled heteronuclear spin system, I and S, under DONUT CP, taking account of the dipolar interaction H IS , the interaction H S rf corresponding to monotonic rf irradiation applied to the target spin S with an intensity o 1S along, say, the x axis, and the interaction H I rf = H 1 + H 2 (t) of the source spin I with the rf field that drives DONUT.The net Hamiltonian H rot (t) in the rotating frame is represented as Here, H 1 and H 2 (t) are given in eqn ( 3) and ( 4), and H S rf and H IS are where d IS is the dipolar coupling constant, given, with the gyromagnetic ratios g I and g S and the internuclear distance r IS , by and o r is the sample spinning frequency.Under MAS, the spinning axis is tilted by cos À1 ð1= ffiffi ffi 3 p Þ from the static field B 0 , so that the coefficient c k in eqn ( 7) is represented as where b and g are the Euler angles of the internuclear vector with respect to the reference frame fixed in the spinning sample container.The time evolution of the system is governed by a propagator U(t), which is written as with H nut (t) is the Hamiltonian in the nutation frame, i.e., in the interaction frame with respect to H 1 , given by .This work follows the convention in which the Larmor precession is clockwise for positive g, and nutation is counterclockwise for both positive and negative g. [9][10][11] This journal is © the Owner Societies 2023 with By tilting the rotating frame for spin S such that the effective field points in the z direction, we rewrite H nut (t) as Another transformation of the dipolar interaction H IS nut (t) to the interaction frame with respect to o 2I I z + o 1S S z leads to which can be rewritten as where D and S are defined to be the difference and the sum of the frequencies o 1I and o 2I : Now, the Hartmann-Hahn conditions can be obtained by invoking the time independence of the dipolar Hamiltonian H ˜IS (eqn ( 23)).In contrast to the conventional CP and a number of its variants, it is the mixed frequencies D and S that determine the Hartmann-Hahn conditions.

Static case
Let us first consider the case of static samples, setting o r = 0. Since o 1I and o 2I are positive, so is S. It follows that o 1S = S 4 0 leads to a time-independent, zero-quantum component which is quite similar to the secular contribution found in the conventional CP schemes, except that the coefficient of the flip-flop term, d IS /8, is halved in the case of DONUT CP compared to that in the conventional CP, d IS /4.To obtain the Hartmann-Hahn conditions that involve D, we separately deal with the following three cases.
Case (i): The matching conditions and the corresponding secular terms responsible for polarization transfer are summarized in Table 1.

MAS case
In the case of MAS, the time dependence induced by the sample spinning can compensate for the energy mismatch.Thus, the Hartmann-Hahn conditions are found, irrespective of the mutual size relation between o 1I and o 2I .The zero-quantum secular term follows from and the double-quantum term from where n = AE1, AE2.Now, the nutation frequency o 1S for the target spins that fulfils the Hartmann-Hahn condition can be negative, as is also the case for the conventional CP. 30 The effect of negative o 1S will be discussed below.Let us recall the two conditions for CP to take place: 3 (a) the Hartmann-Hahn condition is fulfilled, (b) the magnetization of the source spins is locked along the effective field in the relevant frame of reference.So far, we have considered (a) alone.In the conventional CP, to fulfill (b), a p/2 pulse is applied to flip the magnetization of the source spin into the xy-plane before the pulse for the Hartmann-Hahn contact is applied, and the phase of the former has to be shifted by p/2 with respect to that of the latter.Interestingly, in DONUT CP, the initial p/2 pulse is unnecessary, provided that the z axis is chosen to be the axis of the second nutation, since, in the nutation frame, the only existing field points in the z direction, which is parallel to the initial magnetization of the source spin.
It should be noted that, if the second nutation axis is not parallel to the z axis, the initial p/2 pulse is required.

Experimental
To let spins I evolve under H I rf (t) given in eqn (2), the rf irradiation is applied with an amplitude o a (t) given by with a phase f(t) and a frequency offset Do(t) To obtain the profile of such additional phase modulation that incorporates the frequency modulation (eqn (34)), we rewrite H I rf (t) as We also rewrite U I (t), originally given in eqn (1), as where (41) In the interaction frame with respect to H I a (t), the profile of the phase modulation is given by while that of the amplitude modulation is unchanged and is given simply by o a (t).Examples of the waveforms of the amplitude o a (t) and the phase f(t) À c(t) are given in the ESI.† The relationship between the sign of the phase of the rf signal programmed by the user of an NMR spectrometer and the sign of the phase of the rf field that is actually produced has been discussed extensively.According to the instructive papers by Levitt, 10,11 when a pulse sequence involves phase modulation, it is necessary to set the sign of the phase depending on both the sign of the gyromagnetic ratio and the way that the carrier wave at the Larmor frequency o 0 is generated in the transmitter of the spectrometer, through mixing of a pair of synthesized rf signals at an intermediate frequency o IF and a local frequency o LO .For spins with a positive gyromagnetic ratio, the sign of the phase needs to be reversed when the carrier signal is generated through up-conversion, namely o 0 = o IF + o LO , whereas the sign has to be kept unaltered in the case of down-conversion, i.e., o 0 = |o IF + o LO |.It is recommended that the user of the spectrometer is aware of whether upconversion or down-conversion is employed in the transmitter being used, so as to make the right choice of the sign of the phase.
In the present work, we implemented DONUT of the1 H spins (positive gyromagnetic ratio) on home-built NMR spectrometers [31][32][33]  To examine the trajectory of the 1 H magnetization in the Bloch sphere, we numerically solved the Bloch equations, 34 under the rf irradiation corresponding to the Hamiltonian given in eqn (41), with a normalized initial magnetization to be (0,0,1).In Fig. 2(b)-(f), also shown are the calculated three-dimensional trajectories of the magnetization and its projection onto the xy plane, which were found to explain the observed dependence of the transverse magnetization on the duration of the DONUT pulse, and therefore convinced us that the pulse sequence of DONUT was indeed working as we intended.mixed frequencies |o 1I AE o 2I |, namely, ca.2pÁ30 kHz and 2pÁ70 kHz, is expected to develop.Indeed, when the 13 C nutation frequency o 1S was closer to these values, the 13 C magnetization was enhanced, as demonstrated in Fig. 3(a).For (o 1I /2p, o 2I /2p) = (50 kHz, 20 kHz), the 13 C magnetization was negatively enhanced for o 1S B D of around 2pÁ30 kHz.This is consistent with the theory discussed above, which derived the double-quantum nature of the secular component of the heteronuclear dipolar interaction in the nutation frame for the source spins and in the rotating frame for the target spins for the case of o 1I 4 o 2I .

Results and discussion
In Fig. 3(b), the Hartmann-Hahn matching profiles, obtained with DONUT CP under MAS at 9 kHz, are shown.Now, the observed DONUT-CP spectrum was such that the individual peak in the static case (Fig. 3 4 is assigned as the n = À2 sideband of the DONUT-CP spectrum, which ought to be at around À2pÁ16 kHz, but folded back to the positive position at |D À 2o r | of ca.+2pÁ16 kHz.In addition, the zero-quantum exchange, instead of the double-quantum exchange, was observed for the folded-back peak.It is interesting to make a comparison with the well-known peak folding in the conventional CP under MAS, where the double-quantum exchange alone takes place for the folded-back peak. 302 Magnetization buildup Fig. 5(a) shows a 15 N spectrum of a polycrystalline sample of 15 N-labeled histidine in the form of a t tautomer (Fig. 5(b)).35 The three peaks are assigned as the a, e 2 , and d 1 sites, respectively, as indicated in the figure.Fig. 6 compares the buildup behaviors of the 15 N magnetizations in histidine under the conventional CP and the DONUT CP schemes under MAS at 20 kHz.In the former, the source 1 H spins were irradiated with continuous-wave (cw) rf with an intensity o 1I /2p of 35 kHz, while in the latter, DONUT with o 1I /2p = 80 kHz and o 2I /2p = 45 kHz was applied. Tothe target 15 N spins, cw rf irradiation was applied with intensities of 17 kHz and 16 kHz, which were experimentally found to be optimal in the respective measurements.
For all three nitrogen sites in histidine, the initial buildup was slower in DONUT CP than in the conventional CP.Indeed, the slopes of the buildup curve at the beginning of the contact time were less steep by factors of 2.5, 2.1, and 2.0 for the a, e 2 , and d 1 sites for DONUT CP compared to the conventional CP.
For the e 2 nitrogen, which has a covalent bond to the hydrogen atom and is therefore close to the latter, transient oscillation was observed.The period of oscillation in DONUT CP was found to be longer than that in the conventional CP by a factor of ca.1.6.The slower buildup and the longer period of the transient oscillation are ascribed to the down-scaling of the heteronuclear dipolar interaction in the nutation frame by the factor of 2.
Interestingly, for the a and e 2 sites, as the contact time was increased up to several hundreds of microseconds, the buildup curves for the conventional CP began to decay, while those for DONUT CP continued to grow.Eventually, for all three sites, DONUT CP led to larger 15 N magnetization when increasing the contact time further, up to several milliseconds.Since the way that the rf irradiation was applied to the 15 N spins was the same for both CP schemes, it was the relaxation of the 1 H spins that led to the difference.
When a spin species with a relatively low gyromagnetic ratio, like 15 N, is the target, it is often not feasible or desirable to apply too much rf power, such that one has to bear a relatively low nutation frequency o 1S .Then, in the conventional CP, one has to reduce the intensity of the rf field applied to the source spins as well to fulfill the Hartmann-Hahn condition.Even  This journal is © the Owner Societies 2023 though this is not the case if one employs ultrafast MAS, access to such a state-of-art facility is still limited.In addition, ultrafast MAS is not compatible with samples with a relatively large volume, which is often desirable to gain sensitivity.Even though the concept of time-averaged precession frequency (TAPF) can be a solution in CP to the low-gyromagnetic-ratio spin species, the 1 H relaxation can be so fast that it is difficult to optimize the performance of magnetization transfer. 36,37onversely, in DONUT CP, various combinations of o 1I and o 2I can be chosen such that the effect of relaxation is not so serious.
The overall profile of the CP buildup curve is determined by the balance between the two competing processes, namely transfer of polarization and relaxation.As discussed above, the polarization transfer was indeed slower in DONUT CP than in the conventional CP, because of the smaller magnitude of the secular heteronuclear dipolar interaction in the nutation frame.In DONUT CP, the arena in which spins I undergo relaxation is the nutation frame, in contrast to the rotating frame in the conventional CP schemes.Here, we introduce a symbol T 1n to represent the time constant of spin-lattice relaxation of spins I in the nutation frame, in analogy with T 1r that represents spin-lattice relaxation in the rotating frame.
The subscript n (nu) appropriately reminds one of the nutation frame, just like the subscript r (rho) does of the rotating frame.
The dynamics of spin-lattice relaxation in the nutation frame require development of the method to experimentally determine T 1n , collection of data in a number of systems under various experimental conditions, and an extension of the relaxation theory that is well-established in the case of the rotating frame 3 by incorporating the relevant spin interactions.The expression for the effective homonuclear dipolar interaction under DONUT, already given in ref. 12 in the case of simultaneous nutations around the x and the y axes with a common rate o 1I = o 2I = C, includes quite a few terms, some of which can interfere with the time dependence caused by sample spinning.As a result, T 1n as a function of o 1I , o 2I , and o r would show a somewhat complex profile.Although systematic studies on these subjects are outside the scope of the current work, what the results of DONUT CP demonstrated in this work imply is that, in some cases, spinlattice relaxation in the nutation frame can be slower than that in the rotating frame.When this is indeed the case, the slower buildup rate due to the reduced heteronuclear dipolar interaction can be more than compensated for by the benefit of the longer T 1n .

Summary
The nutation frame is the interaction frame with respect to the Hamiltonian responsible for driving nutation.DONUT CP causes transfer of nuclear spin polarization in solids between the source spins in the nutation frame and the target spins in the rotating frame.The Hartmann-Hahn condition D = o 1S for o 1I 4 o 2I leads to double-quantum exchange, and the other Hartmann-Hahn conditions to zero-quantum exchange.Under MAS, the peak of the DONUT-CP spectrum at the negative frequency folds back to the positive region, accompanying sign inversion in the intensity of the magnetization.In DONUT CP, the magnitude of the secular part of the dipolar interaction is half that in the conventional CP, resulting in a slower buildup rate.Nevertheless, there exist cases in which the maximum attainable enhancement exceeds that in the conventional CP, when spin relaxation in the nutation frame is slow enough.

Fig. 1
Fig.1(a) (left) A schematic drawing of a static magnetic field B 0 and a magnetic field B 1 rotating about B 0 at the Larmor frequency o 0 = ÀgB 0 , where g is the gyromagnetic ratio of the spins of interest.In a reference frame rotating around B 0 at o 0 , B 0 vanishes and B 1 is stationary (right).(b) In addition to the rotating field B 1 , a time-dependent component, B 2 , is applied (left) such that, in the rotating frame, it draws a circular trajectory rotating at an angular frequency o 1 = |g|B 1 (middle).Another transformation through rotation around B 1 at o 1 leads to a frame, which we call the nutation frame, in which B 1 vanishes while B 2 now becomes stationary (right).This work follows the convention in which the Larmor precession is clockwise for positive g, and nutation is counterclockwise for both positive and negative g.[9][10][11] ) which carries the double-quantum flop-flop term.Thus, the polarization of the target spin S is expected to be enhanced negatively.The double-quantum exchange in stationary samples is one interesting feature of DONUT CP, in contrast to the conventional CP where it is possible only in rotating samples.30Case (ii):o 1I = o 2I D = 0,and no o 1S 4 0 interfering with the e AEiDt term is found.Case (iii): o 1I o o 2I D o 0, and o 1S = ÀD 4 0 leads to the zero-quantum secular term, so that positive enhancement of the magnetization of the target spins S is expected.
HI b ðtÞ ¼ U a À1 ðtÞH I b ðtÞU a ðtÞ ¼ o a ðtÞ I x cos fðtÞ À cðtÞ ½ þ I y sin fðtÞ À cðtÞ ½ È É ; in nominal 9.4 T and 7 T magnets, setting the intermediate frequency o IF at 180 MHz and the local frequency o LO at 220 MHz (120 MHz) + d, mixing them in the transmitter to yield signals at o IF AE o LO , and passing them through a 400 MHz (300 MHz) bandpass filter, thereby choosing o IF + o LO = 400 MHz (300 MHz) + d.Here, d, determined by the actual field strength, was on the order of several tens of kHz in this work.Thus, the signal was up-converted.Taking advantage of employing the open-architecture home-built spectrometer, we unambiguously chose to reverse the sign of the phase given in the pulse program.To check if the sequence worked as intended, we carried outThis journal is © the Owner Societies 2023

4. 1
Fig.3(a) shows13 C magnetizations enhanced by DONUT CP obtained for a stationary powder sample of adamantane as a function of the amplitude o 1S of the rf field applied to the target13 C spins.Following Meier,30 we henceforth refer to the o 1S dependence of the S magnetization as the CP spectrum.The CP spectrum depicted with blue circles in the figure was obtained with DONUT applied to the source 1 H spins with nutation frequencies (o 1I /2p, o 2I /2p) of (20 kHz, 50 kHz), whereas the CP spectrum with the swapped combination, i.e., (o 1I /2p, o 2I /2p) = (50 kHz, 20 kHz), is shown with red circles.For both of these cases, time dependence with the

Fig. 2
Fig. 2 Trajectories of the 1 H magnetization of liquid water under double nutation applied with frequencies (o 1 /2p, o 2 /2p) of (a) (20 kHz, 0 kHz), (b) (20 kHz, 10 kHz), (c) (20 kHz, 20 kHz), (d) (20 kHz, 30 kHz), (e) (20 kHz, 40 kHz), and (f) (20 kHz, 50 kHz) for incremented time intervals up to 50 ms.The position of each circle represents the in-phase and the quadrature parts of the first sampling point of the measured FID.The magnitude of the complex data point was normalized to that obtained with the conventional p/2 pulse.Solid lines represent the trajectories of the magnetization (inset) and their projection onto the (x,y) plane obtained by numerically solving the Bloch equation with the initial magnetization being along the z axis and under application of the double-nutation by rf irradiation with amplitude o a (t) and phase f(t) À c(t) given in eqn (32) and (43).

Fig. 3
Fig. 3 1 H-13 C Hartmann-Hahn matching profiles of DONUT CP measured for adamantane in a magnetic field of 7 T under (a) the static condition and (b) MAS at 9 kHz.Red and blue circles represent the o 1S dependence obtained with (o 1I /2p, o 2I /2p) of (50 kHz, 20 kHz) and (20 kHz, 50 kHz), respectively.The contact time was 7 ms.
(a)) split into sidebands with a separation corresponding to the spinning speed, just like in the case of the conventional CP under MAS.The feature of negative enhancement for the case of o 1I 4 o 2I and o 1S = D is retained.Fig. 4 shows a 13 C DONUT-CP spectrum of the methylene carbon obtained in polycrystalline glycine under MAS at 23 kHz with DONUT applied at the 1 H spins with (o 1I /2p, o 2I /2p) of (90 kHz, 60 kHz).With o 1I 4 o 2I , D/2p = 30 kHz, and o r /2p = 23 kHz, the expected double-quantum exchange with negative enhancement of the 13 C magnetization was observed at around o 1S /2p of 76 kHz (n = 2), 53 kHz (n = 1), and 7 kHz (n = À 1).The positive peak of the DONUT-CP spectrum in Fig.

Fig. 5
Fig. 5 (a) 15 N spectrum of polycrystalline 15 N-labeled histidine, obtained with DONUT CP in 9.4 T under MAS at 20 kHz with a contact time of 3 ms.(b) The structure of a neutral t tautomer of histidine.

Table 1
The Hartmann-Hahn (H-H) conditions for stationary samples and the secular terms of the dipolar interaction under DONUT CP