Cross-Effect Dynamic Nuclear Polarization Explained: Polarization

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Perspective

Cross-Effect Dynamic Nuclear Polarization Explained: Polarization, Depolarization and Oversaturation Asif Equbal, Alisa Leavesley, Sheetal K. Jain, and Songi Han J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.8b02834 • Publication Date (Web): 15 Jan 2019 Downloaded from http://pubs.acs.org on January 17, 2019

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The Journal of Physical Chemistry Letters

Invited Perspective Cross-Effect Dynamic Nuclear Polarization Explained: Polarization, Depolarization and Oversaturation Asif Equbal ,† Alisa Leavesley,† Sheetal Kumar Jain ,† and Songi Han∗,†,‡ †Department of Chemistry and Biochemistry, University of California, Santa Barbara, Santa Barbara, California 93106, United States. ‡Department of Chemical Engineering, University of California, Santa Barbara, Santa Barbara, California 93106, United States. E-mail: [email protected]

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Abstract The scope of this perspective is to analytically describe NMR hyper-polarization by the three-spin cross effect (CE) dynamic nuclear polarization (DNP) using an effective Hamiltonian concept. We apply, for the first time, the bimodal operator-based Floquet theory in the Zeeman-interaction frame for two and three coupled spins to derive the known interaction Hamiltonian for CE-DNP. With a unified understanding of CEDNP, and supported by empirical observation of the state of electron spin polarization under the given experimental conditions, we explain diverse manifestations of CE from oversaturation, enhanced hyper-polarization by broad-band saturation to nuclear spin depolarization under magic-angle spinning.

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Introduction. Nuclear magnetic resonance spectroscopy (NMR) is intrinsically a low sensitivity technique due to weak Zeeman interactions resulting from a low gyromagnetic ratio (γ) that leads to a relatively low nuclear spin polarization, even at the highest magneticfield available today. Transferring spin polarization from a sensitive (high γ) spin to an insensitive (low γ) spin is a popular approach in NMR spectroscopy to augment its sensitivity. Dynamic nuclear polarization (DNP) is a technique that enables the transfer of polarization from electron spins to nuclear spins (γe >> γn ), and has become one of the most exciting add-ons to solid-state NMR (ssNMR). 1–8 In principle, DNP enhancement is equal to the ratio γe /γn , but experimentally, signal amplification corresponding to this limit is not achieved, while the polarization transfer efficiency varies dramatically with the sample system and condition. Of the various DNP mechanisms in insulating solids, cross effect (CE) DNP is more efficient at higher magnetic-field compared to the solid effect (SE). This is attributed to the lower power requirement of CE that relies on allowed spin transitions. A major research focus in DNP has been the optimization of the radicals used as polarizing agents, and the experimental conditions (temperature, glassing-solvent) to achieve greater CE efficiency. 9–11 In this perspective article, we present a generalized and extensible theoretical framework for describing the polarization transfer according to the CE-DNP mechanism 12–16 from first principles that eases the hurdle of simulating experimental systems and predicting new CEDNP modalities. We describe the effective Hamiltonian operational in CE-DNP, and show that the evolution of the density operator under the effective Hamiltonian can result in nuclear spin polarization enhancement or depletion, depending on the prevailing density operator and steady-state polarization of the electron spins. With a unified understanding of the CE-DNP theory, and supported by experimental measurement of the state of electron spin polarization under the given experimental conditions, we describe three different nuances of CE-DNP: (i) Oversaturation in CE-DNP resulting in a decreased DNP enhancement above a threshold microwave (µw) power, (ii) Improved CE-DNP performance by adiabatic chirppulse train saturation of electron spins, and (iii) Nuclear spin depolarization causing a loss

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in NMR signal under magic-angle spinning (MAS) in the absence of µw irradiation. All three phenomena are governed by the same CE Hamiltonian for polarization transfer in an e−e−n three-spin system that, however, manifest themselves drastically differently depending on the polarization difference between the donor electron spin (∆Pe ) and the acceptor nuclear spin (Pn ) ensemble in frequency and time. In the magnetic resonance literature, the optimum DNP condition and its mechanism have been depicted mostly using a degeneracy-of-state or product-state picture, which is cumbersome beyond the two-spin case. 14,15,17 Using an operator-based formalism, we show an alternative, simpler, derivation of the effective three-spin coupling Hamiltonian leading to CE, where the polarization differences between the three involved spins are the main quantities modulated by the effective Hamiltonian. The framework is very similar to the ones used for elucidating ssNMR experiments under MAS, which requires calculating the effective (or time-averaged) Hamiltonian using the Average Hamiltonian Theory (AHT) 18,19 or Floquet theory 20–23 in the appropriate frame. The density operator formalism presented here is an effective, straightforward and versatile tool to model and rationalize currently available experimental observations of CE-DNP effects, and has the benefit of being extensible to describe multi-frequency and time-modulated DNP experiments in the future. Theory for Polarization Transfer Mechanism. We first pedagogically present a theoretical framework to describe the polarization transfer mechanism using two- and threespin model systems. In general, any transfer of magnetization or polarization between spins is possible in the presence of coupling interaction between the spins, either direct or indirect. Quantum mechanically, the spin-spin dipolar coupling Hamiltonian can be constituted as a sum of zero-quantum (ZQ), single-quantum (SQ) and double-quantum (DQ) operators (Eq. 1). In the (rotating) frame of Zeeman interactions, these terms are modulated by frequencies that depend on the Zeeman interaction strengths of the individual spins , I and S (here, Larmor frequencies ω0I and ω0S ). In the high B0 field limit, the zz or “ZQ-diagonal” term remains unmodulated, while the

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ZQ-off-diagonal term is modulated by the difference of the two Zeeman frequencies, ω0I −ω0S . The SQ terms are modulated by either ω0I or ω0S , depending on which spin (I or S) has coherence, and the DQ-off-diagonal terms are modulated by the sum of the two Zeeman frequencies, ω0I + ω0S . ˆ dipolar = H

A ωIS IS | {zz }z

ZQ, diagonal (zz)

B C D + ωIS (I + S − + I − S + ) + ωIS (I + Sz + Iz S + ) + ωIS (I − Sz + Iz S − ) | {z } | {z } ZQ, of f −diagonal

SQ, of f −diagonal

E F + ωIS (I + S + ) + ωIS (I − S − ) | {z }

(1)

DQ, of f −diagonal

In general, any term in the Hamiltonian with a large net non-zero frequency modulation will be averaged out to first-order under the secular approximation. In the case of a heteroγ coupled two-spin system, without external perturbation, only the diagonal (zz) part of the ZQ dipolar term survives upon time-averaging, while all other terms have effectively no contribution, owing to the secular approximation valid at high magnetic-field. In a homo-γ coupled two-spin system, where the Larmor frequencies of the two spins are close in magnitude, all ZQ terms (diagonal and off-diagonal components) remain effective. The modulation frequencies of the various coupling terms ZQ/SQ/DQ of the Hamiltonian (Eq. 1) in the Zeeman frame is depicted for each dipolar term in Scheme 1, including the complex conjugate (c.c.) terms for I − S. This ensures that the net Hamiltonian is Hermitian. It is important to emphasize here that for any coherent transfer of polarization to occur, there must exist an effective off-diagonal, ZQ (flip-flop) or DQ (flip-flip), operator in the Zeeman basis. In the case of a hetero-γ spin system, however, only the diagonal, zz part of the dipolar term survives the time-averaging, requiring RF irradiation for the generation of off-diagonal ZQ or DQ terms. One approach to achieve polarization transfer is to transform the zz dipolar term (Eq. 1) to effectively act as an off-diagonal ZQ or DQ operator upon RF irradiation. Irradiation on both spins is a must for the Iz Sz coupling term to create coherence −i(ω

I +ω

S )t

1I x 1S x e on both spins (ZQ or DQ), specifically, Iz Sz −−−−−−−−−−→ I˜z S˜z + I˜+ S˜− + I˜+ S˜+ + c.c..

rotating frame

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The Hartman-Hahn cross-polarization (CP) between hetero-γ spins under MAS is the best known example in this category, and the effect of broad-band irradiation on both spins and the resultant Hamiltonian well-known. 24 Another approach to achieve polarization transfer is to transform the SQ dipolar operator to generate the coherent ZQ or DQ terms. As the SQ dipolar operators consist of pseudo-secular dipolar terms, they require irradiation on e−iω1S Sx t

only one spin for coherence generation (i.e. I ± Sz −−−−−−−−→ I˜± S˜z + I˜± S˜− + I˜± S˜+ ); hence rotating frame

irradiation on that single spin suffices to transfer polarization between the two spins. 25 The pseudo-secular SQ term plays an important role in EPR experiments and DNP. In fact, µw induced SE-DNP falls into this category. See SI, section-A1 for more details. Passive coupling can also generate polarization transfer between two uncoupled spins with the aid of a third spin that is coupled to both of these uncoupled spins. 26,27 Following the general principle described here, irradiation on both spins (donor as well as acceptor of polarization) is needed to achieve polarization transfer between hetero-γ spins, unless there is a strong pseudo-secular SQ interaction (e.g. hyperfine coupling) between the hetero-γ that can be exploited. In the case of DNP, irradiation on both electron and nuclear spins with the required power cannot be easily performed with the currently available hardware. However, if the pseudo-secular (SQ) hyperfine interaction between the electron and nuclear spins is strong, polarization transfer from an electron spin to a nuclear spin by irradiation on just the electron spin is effective. An advantage of this mechanism is that the polarization to nuclear spin is transferred along the direction of the B0 field, and that the transfer rate is limited by T1 rather than T1ρ of the nuclear spin. The derivation of the effective Hamiltonian and mechanism of polarization transfer from S (electron) to I (nuclear) spins, as operational in CE-DNP upon µw irradiation of only the S spins, is the core subject to be presented here, followed by its implementation to rationalize and simulate experimental observation of oversaturation, 28 broad-band µw saturation, 29,30 and depolarization 31–33 , described above. Polarization Transfer in Two-Spin Case. We start with a two-spin case (denoted

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as 1 and 2), where two spins are homo-γ dipolar coupled. In the context of DNP, this is to describe the polarization exchange and transfer between electron spins with similar frequencies (with differences smaller than their coupling strength) in the presence or absence of µw irradiation. This applies when many electron spins with a broad frequency-distribution are coupled, or under MAS conditions when electron spin frequencies level-cross at some rotor-position. We start with this description because the framework established for the two spin- 12 homo-γ (dipolar) coupled case forms the theoretical basis for describing many spin cases, including the three-spin case of two electron spins and one nuclear spin. In the absence of external-perturbation, the Hamiltonian is the sum of Zeeman and coupling interactions: ˆ = ω01 S1z + ω02 S2z + H ˆ dipolar . In the rotating frame of the two Zeeman interactions, H transformed using the propagator U = e−i(ω01 S1z +ω02 S2z )t , the Hamiltonian is time-modulated and takes the form ˆ˜ = U † HU ˆ − ω01 S1z − ω02 S2z H A B C = ω12 S1z S2z + ω12 (S1+ S2− ei(ω01 −ω02 )t + S1− S2+ e−i(ω01 −ω02 )t ) + ω12 (S1z S2+ eiω02 t + S1+ S2z eiω01 t ) D E + + i(ω01 +ω02 )t F − − −i(ω01 +ω02 )t + ω12 (S1z S2− e−iω02 t + S1− S2z e−iω01 t ) + ω12 S1 S2 e + ω12 S1 S2 e .

(2)

This Hamiltonian is equivalent to the one presented in Eq. 1 (for two S spins), but in the rotating frame of the two Zeeman interactions. The first-order effective (time-averaged) ¯ˆ˜ 1 R τ ˆ˜ Hdt, can then be obtained by integrating the Hamiltonian over a Hamiltonian, H= τ 0 time period characterized by the frequency of each term. 18 The SQ and DQ terms are always modulated by large frequencies, leading to zero average in the first-order Hamiltonian, and therefore can be ignored here. Only the ZQ term can have a non-zero average, depending on ˆ˜ = ω A S S . the value of ω01 − ω02 . When ω01 6= ω02 , the time-averaged Hamiltonian is H ef f 12 1z 2z This zz term only causes broadening of the individual spin’s signal, but cannot change the ˆ˜ = (ω A S S + ω B (S + S − + state of the spin system. When ω01 = ω02 , the average is H 1 2 12 1z 2z 12 S1− S2+ )). Here, only the off-diagonal ZQ (flip-flop) (Eq. 1) part of this Hamiltonian survives

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in the rotating frame of the two Zeeman interactions, which in turn can lead to polarization transfer, as will be demonstrated next. We first define operators: Sx23 = 12 (S1+ S2− + S1− S2+ ), Sy23 =

1 (S1+ S2− − S1− S2+ ), 2i

Sz14 = 21 (S1z + S2z ) and Sz23 = 12 (S1z − S2z ), and consider the initial

density operator to be

ρ0 = p1 S1z + p2 S2z = ΣpSz14 + ∆pSz23 ,

(3)

with Σp and ∆p being the sum and difference of p1 and p2 , respectively. The effective Hamiltonian, that evolves this initial state-of-the-system, is ¯ˆ˜ B + − − + B 23 H ef f = ω12 (S1 S2 + S1 S2 ) = 2ω12 Sx .

(4)

In other words, the initial density operator represent spin states in which spins 1 and 2 are prepared as z magnetization, with polarizations of magnitude p1 and p2 , respectively. Using a simple calculation, it can be seen that Sx23 , Sy23 , and Sz23 act as “fictitious” spin1 2

operators 34 that satisfy the commutation relation [Sj23 , Sk23 ] = iεjkl Sl23 (where j, k, l are

the elements of the set {x, y, z} in all possible permutations and ε denotes the Levi-Civita symbol). Only the second part of the initial density operator (Eq. 3), the off-diagonal ZQ term Sz23 , can evolve under the effective Hamiltonian (Eq. 4). The first part of the initial density operator, Sz14 , remains unaffected as it commutes with the effective Hamiltonian i.e. ˆ˜ , S 14 ] =[S 23 , S 14 ]=0. The modulation of the individual spin polarization under the [H ef f z x z effective Hamiltonian can be seen by calculating the trace, hSiz (t)i, of the Siz operator, here shown for hS1z (t)i and hS2z (t)i.

hS1z (t)i =

Σp ∆p Σp ∆p B B + cos(ω12 t) & hS2z (t)i = − cos(ω12 t) 2 2 2 2

(5)

Equation.5 shows that the polarizations of coupled spins 1 and 2 oscillate around a constant value (average of their polarization, Σp/2) with an amplitude proportional to the half of the

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difference of the polarization (∆p/2) and rate, ω12 /2π, proportional to the the magnitude of the effective coupling Hamiltonian (Eq. 4). This shows that net polarization oscillation/transfer is observed if spins 1 and 2 have unequal polarization, which can be achieved by selective saturation and/or with thermal polarization of a spin system with a large g (or chemical shift) anisotropy. Figure 1a maps the polarization transferred from S1 to S2 spins as a function of the dipolar evolution time and the polarization difference, p1 − p2 . Simply relying on Eqns.4-5, we show that the polarization on spins S1 and S2 oscillate with B , while the maximum polarfrequency corresponding to the dipolar coupling strength, ω12

ization transferred depends linearly on the polarization difference between the spins, ∆p. This is illustrated in the traces extracted from the contours of Fig.1a, shown in Fig.1b-c (the script for this calculation can be found in the SI). The exchange of polarization between two homo-γ spins (with chemical shift difference smaller than their coupling strength) is an important phenomenon that leads, e.g., to large spectral diffusion among dipolar coupled electron spins, and is in turn critical in the context of nuclear spin polarization enhancement and depolarization under MAS, which will be discussed later. The equivalent mechanism in nuclear spins is important for diffusion of nuclear polarization from core to bulk. 14,17 Polarization Transfer in Three-Spin Case. Next, we consider the three-spin case. There are three subcategories: (i) all three are hetero-γ spins (I, N, S), (ii) all three are homo-γ spins (S1 , S2 , S3 ), (iii) and two are homo-γ spins (S) and one hetero-γ spin (I). The first case is a simple extension of second-order CP, e.g. in 1 H-13 C-15 N PAIN-CP. 26 The second case is similar to the two-spin case discussed above. However, unlike in the hetero-γ spin case, the three homo-γ spin case cannot be considered as a mere sum of two homo-γ spins. This is because of the non-commutativity of the homo-γ coupling Hamiltonian, i.e [HS1 S2 , HS1 S3 ] 6=0, where HSi Sj is the dipolar coupling Hamiltonian between homogeneous-spins i and j. This leads to dipolar truncation effects, a phenomenon elaborated extensively in the ssNMR literature. 35 The third case, in the scenario where the two S spins are the electron spins and I the nuclear spin, is the most important one for DNP, and will be discussed here.

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The rotating frame dipolar Hamiltonian is time-modulated by the frequencies ω0i , ω03 , (ω0i ± ω03 ), and (ω01 ± ω02 ), where ω0i and ω03 refer to the Larmor frequencies of I and Si (i = 1, 2) spins, respectively (see SI, section-A2). The dipolar Hamiltonian terms modulated by the large frequencies (∼ electron spin Larmor frequency) will be ignored, as they do not contribute to the effective S to I polarization transfer. We only include the ISi - SQ and S1 S2 - ZQ coupling terms that are modulated by lower frequencies (typically sub GHz). These terms will play an important role in CE-DNP transfer, as will be explained later. The truncated rotating frame Hamiltonian defined by the propagator, U = e−i(ωo1 S1z +ωo2 S2z +ωo3 I3z )t becomes, ˜ˆ A B H =ω12 S1z S2z + ω12 (S1+ S2− ei(ω01 −ω02 )t + S1− S2+ e−i(ω01 −ω02 )t ) X A C + iω03 t D − −iω03 t + (ωi3 I3z + ωi3 I3 e + ωi3 I3 e )Siz .

(6)

i=1,2

Note that the flip-flop (off-diagonal) part of the homo-γ S1 -S2 coupling is modulated by a frequency equal to the difference between the two electron spin’s precession frequencies (as shown earlier in Eq. 2), ∆12 = (ω01 − ω02 ), and the pseudo-secular part of the heteronuclear ISi coupling (also known as the hyperfine term in electron-nuclear coupling) modulated by the I spin’s Larmor frequency, ω03 . If ∆12 and ω03 are very large compared to the strength of the corresponding coupling strengths, ω12 and ωi3 (i=1 or 2), the frequency-modulated terms can be ignored to first-order in the Hamiltonian owing to secular approximation. However, the cross-terms (higher-order Hamiltonian) of these modulated terms can cause polarization transfer among two coupled S to I spins, which can be obtained using perturbation theory. The Hamiltonian in Eq. 6 is time-dependent, modulated periodically with characteristic frequencies ∆12 and ω03 , but does not commute with itself at different times. In pursuit of getting physical insights using such a time-dependent, non-commutating Hamiltonian, an effective time-independent Hamiltonian needs to be computed. This is done using a special operator-based perturbation analysis of the AHT or the Floquet theory, commonly

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employed in ssNMR under MAS. Both of these theories exploit the periodicity of the timedependent Hamiltonian to solve for higher-order perturbation corrections. Although AHT and Floquet theory approaches are equivalent for the here discussed case, we choose the Floquet-theory as it has an advantage over AHT in solving time-dependent Hamiltonian multiple incommensurate frequencies. The Floquet formalism is well documented in the ssNMR literature, 22,23,36,37 and therefore we emphasize on the standard analytic equations to find the effective Hamiltonian for DNP (see SI section-A2 for details). Using Ladder, F∆n and Fωk03 , and Number, 37 N∆ and Nω03 , operators the bimodal (two frequencies, here ∆12 and ω03 ) Floquet Hamiltonian of Eq. 6 can be represented as:

eF = H

∞ X

∞ X

n=−∞

k=−∞

e (n,k) F n F k + ∆12 N∆ + ω03 Nω03 H ∆ ω03

(7)

where n and k are the Fourier indices representing the frequency multiplicity. The indices n and k in the above Hamiltonian can be truncated to [-1, 1], as the Hamiltonian (in Eq. 6) simply has ±1 multiples of the two characteristic frequencies, ∆12 and ω03 . The relevant Floquet Hamiltonian terms are then:

e (0,0) = H

X

A A e (1,0) = ω B S + S − , H e (−1,0) = ω B S − S + S1z S2z , H ωi3 I3z Siz + ω12 } | 12 {z1 2} | {z 12 1 2} | {z i=1,2 S1 S2 , (ZQ) S1 S2 , (ZQ) {z } S1 S2 , (ZZ) | Si I3 , (ZZ)

e (0,1) = H

X

C + e (0,−1) = ωi3 I3 Siz , H

i=1,2

|

X

D − ωi3 I1 Siz

(8)

i=1,2

{z

Si I3 , (SQ)

}

|

{z

Si I3 , (SQ)

}

We apply the van Vleck perturbation theory 21,23,36 to get the effective Hamiltonian in the rotating frame. Such a treatment is valid when the condition n∆12 +kω03 6= 0 is satisfied (i.e. avoiding resonance or accidental degeneracy conditions). The effective Floquet Hamiltonian

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with up to second-order of the van Vleck expansion 21,23,36 is given by:

e ef f = H

X X [H e (n0 −n,k0 −k) , H e (n,k) ] e (n0 ,k0 ) − 1 H 2 n ,k n,k n∆12 + kω03 n0 ,k0 0 0 {z } | {z } | X

e1 H ef f

(9)

e2 H ef f

where, n0 and k0 are the Fourier indices, and will sum over only those values where the resonance condition n0 ∆12 + k0 ω03 = 0 is satisfied, in both first and second-order. The first order term of the effective Hamiltonian is zero in our case. If the ω03 and ∆12 frequencies are significantly larger than the coupling strengths, the main contribution to the effective Hamiltonian comes from the second-order term, which is simply the commutator between different first-order terms. The summation of Fourier indices in the second order physically means that the net modulation frequency of the commutator of two terms is the sum of frequencies of the individual terms. Of all the possible hetero cross-terms, [S1z S2z , Siz I3z ], [S1z S2z , Siz I3± ], [S1± S2∓ , Siz I3z ], and [S1± S2∓ , Siz I± ], only the last cross-term, i.e the commue (±1,0) ) and the Si I3 - SQ (H e (0,∓1) ) coupling term can lead tator between the S1 S2 - ZQ (H to a three-spin polarization transfer operator, as will be further discussed. Adding all the possible resonance conditions, (n0 , k0 )=(±1 , ∓1), the second order effective Hamiltonian operational for polarization transfer between three spins, referred henceforth as the CE Hamiltonian, then becomes: e (1,0) , H e (0,−1) ] [H e (−1,0) , H e (0,1) ] [H e (0,1) , H e (−1,0) ] [H e (0,−1) , H e (1,0) ] e 2 = −1 [ [H H + + + ] ef f 2 ∆12 −∆12 ω03 −ω03 C ω B (ω C − ω23 ) + − − (10) = 12 13 S1 S2 I3 + c.c. ω03 As mentioned, this second-order cross-term is resonant and will be dominant, only if the resonance condition, n0 ∆12 + k0 ω03 = 0, is satisfied, i.e. ∆12 = ω03 since n0 = −k0 . This entails that the Larmor frequency difference between the two coupled S spins must equal the Larmor frequency of the third spin. In order to understand how this effective Hamiltonian

12


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causes oscillations in the polarization, let us consider the following initial density operator

ρ0 =p1 S1z + p2 S2z + p3 I3z =(∆ps − p3 )

0.5(S1z − S2z ) + I3z (S1z + S2z ) 0.5(S1z − S2z ) − I3z + (∆ps + p3 ) + Σps , 2 2 2 (11)

where ∆ps and Σps are the difference (p1 − p2 ) and sum (p1 + p2 ) of polarization of the two S spins, respectively. The initial density operator can be written as a sum of three terms with coefficients representing (a) difference of ∆ps and p3 , (b) sum of ∆ps and p3 and (c) Σps . Using a simple calculation, it can be seen that only the first part of the density operator in Eq. 11 can evolve under the CE Hamiltonian (Eq. 10). This is because the latter two operators, [(S1z − S2z ) + I3z ] and [S1z + S2z ], commute with the effective Hamiltonian operator (S1+ S2− I3− + c.c.), and therefore do not evolve under CE Hamiltonian. Just as the effective two spin flip-flop operator (Eq. 4) modulates ∆p between two homogeneous-spins, the three-spin flip-flop-flip operator (Eq. 10) modulates ∆ps − p3 . Hence this three-spin effective Hamiltonian yields a perfect archetype for CE-DNP e CE = ωe1 e2 (ωe1 n − ωe2 n ) (S + S − I − ) + c.c. H 1 2 3 ef f ω0n

(12)

According to the effective CE Hamiltonian, the rate of the polarization transfer is directly proportional to the strength of the e − e coupling, ωe1 e2 , and the difference of the two e − n (hyperfine) couplings, ωe1 n - ωe2 n , while inversely proportional to the nuclear Larmor frequency, ω0n (see SI section-B2 for further discussions). The CE theory has been presented in literature using different approaches. A common approach is the diagonalization 13 of the electron spin Hamiltonian using a unitary transformation, as recently described by Thankamony et al., 8 showing that the hyperfine-coupling term under a specific unitary transformation yields a Hamiltonian similar to Eq. 12. Note that this Hamiltonian only gives us information about the rate of CE transfer, but not the amount of transfer, which 13



is proportional to the term ∆ps − p3 . Therefore, large CE enhancement of nuclei ideally requires a large polarization difference, ∆ps , between the two electron spins that fulfill the CE condition compared to the nuclear spin polarization, pI . Without µw perturbation, ∆ps − pI is small since the constants, pi (i = 1, 2, 3), are simply proportional to the Boltzmann polarization. Therefore in practice, efficient saturation of part of the electron paramagnetic resonance (EPR) spectrum is required for CE-driven DNP to generate a large electron spin polarization difference that can lead to large nuclear spin polarization enhancement. To illustrate this subtlety of the CE Hamiltonian, Fig. 2 maps the polarizations, hSiz (t)i, of the three-spin system as a function of time for two different cases: (a) ∆ps > pI and (b) ∆ps < pI . It can be clearly seen that for a large initial ∆ps in S-spins (blue and green), a large polarization is transferred to the I-spin (red) (Fig. 2a). The oscillations of all involved spins have the same modulation rate, as the underlying Hamiltonian is the same. It should be noted that these oscillations or transfer rates are independent of the initial density operator of the system, and instead depend on the strength of the effective CE Hamiltonian. In the limit where the polarization difference between the two S spins is smaller than the polarization of the I spin, ∆ps < pI , a reverse polarization transfer would occur, i.e. the I spin will lose its polarization to the two S spins. This can lead to depolarization, as depicted in Fig. 2b. The smaller the value of p1 − p2 compared to half of p3 , the more will the I-spin polarization be depleted to the S-spins. Experimentally, these oscillations are difficult to capture owing to (i) nuclear spin diffusion, (ii) powder averaging, and (iii) relaxation effects, which are not included in the simulations presented here. Also, continuous µw saturation of the electron spins further adds to the multifaced complexity of electron spin dynamics. Simulations of changes in polarization in experimentally more realistic scenarios, involving powder orientation, relaxation effects and continuous µw saturation, are demonstrated using the SpinEvolution package in the SI (section-B2). The CE mechanism is the most efficient mechanism in the limit where the nuclear Larmor frequency ω0n is larger than the strength of the S1 S2 coupling. Otherwise, the equalization

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of the electron spin polarization (or the reduction in ∆p ) can not be circumvented owing to the first-order S1 S2 - ZQ Hamiltonian itself, as discussed in the homonuclear two-spin case, diminishing the CE-DNP efficiency (see SI, section-B2). For this reason, CE requires an inhomogeneously broadened EPR line, so that a polarization difference between electron spin populations can be created and maintained during the course of polarization transfer, yielding NMR signal enhancement by CE-DNP. In the limit of thermal mixing of a strongly coupled S spin bath (high concentration) or high µw power, the CE-DNP efficiency decreases, as will be demonstrated experimentally in the next section. The various polarization transfer mechanisms are summarized in Table. 1. The CEDNP mechanism falls into the three-spin case, where the effective Hamiltonian of the form S1+ S2− I ± originates from the cross-term or commutator between the S1 S2 - ZQ (flip-flop) Q and Si I - SQ (pseudo-secular) coupling terms. In general, a Hamiltonian of the form i Ii± P evolves a density matrix term of the form i ±pi Iiz . An important difference between an effective CE and SE-DNP Hamiltonian (SI, section-A1) is that the effective SE Hamiltonian is generated only in the presence of off-resonant µw irradiation, whereas the CE Hamiltonian is operational with or without µw irradiation; the CE Hamiltonian is “intrinsically” operational by the dynamics of the given spin system that satisfies the required resonance condition. e (±1,0) ) and the Si I3 - SQ (H e (0,∓1) ) coupling the commutator between the S1 S2 - ZQ (H term Table 1: Summary: Polarization transfer between coupled spins Spin-spin coupling operator Iz Sz I ± Sz + c.c. S1+ S2− + c.c. S1+ S2− + I ± Siz + c.c.

External perturbation B1 Irradiation B1 Irradiation No irradiation No irradiation

on both I and S spins on only S spin (if ω0S1 ≈ ω0S2 ) (if ω0I ≈ ω0S1 − ω0S2 )

Example Cross polarization Solid effect DNP Spin diffusion Cross effect DNP

Experimental. The µw-amplitude (ν1µw ) currently available for high magnetic-field DNP is tiny (sub-MHz) compared to the strengths of the g-anisotropy (∼0.8 GHz for ni-

15



troxide radicals at 7 T), hyperfine-coupling (a few MHz), and e − e coupling, i.e. including dipolar (∼24 MHz for 1.3 nm) and exchange (∼40 MHz for AMUPOL) 38 interactions. Hence, the electron to nuclear spin polarization transfer rate in CE-DNP depends mainly on the e − e and e − n coupling strengths and ω0n . 15 The role of µw irradiation is essentially to saturate (or depolarize) a portion of the EPR line (inhomogeneously broadened), where the degree of its saturation (electron-depolarization) is dependent on µw power, as well as intrinsic spin dynamics. 39 Specifically, the extent of electron spin saturation depends on electron spectral-diffusion (eSD), which equilibrates the polarization via e − e dipolar coupled spin network and cross-relaxation, besides on T1 e. The eSD mechanism broadens the width of the hole burnt in the EPR line by µw irradiation, and a new quasi steady-state electron spin polarization is reached that is different from the Boltzmann polarization under the Zeeman Hamiltonian. The cumulative polarization difference (∆Pe ) generated by µw saturation between the electron spins (satisfying the CE conditions) across the EPR line, and the evolution of this quasi steady-state under the CE Hamiltonian, leads to enhanced polarization of the coupled nuclear spins. The total enhancement is the weighted sum of enhancements from different electron spin pairs satisfying CE conditions. Different crystallite orientations lead to different CE enhancements as the essential interactions are anisotropic in nature. Polarization of the bulk-nuclei is achieved via the spin-diffusion mechanism, similar to those operating in coupled electron spins. 14 Using this mechanistic understanding of CE-DNP, we will next illustrate the experimental nuances of select CE-DNP features. The experiments shown here were acquired with a home-built, quasi-optics based, dual NMR/EPR spectrometer at 7 Tesla. 40–43 Details and parameters of the experimental setup are given in the SI (section-C1). Oversaturation: The 1 H DNP enhancement () recorded as a function of µw frequency (f µw ) for 40 mM 4-amino Tempo (4AT) in DNP-juice at 4 K and 7 T field is shown in Fig. 3a. Here, is defined as the ratio of NMR signal obtained with and without µw irradiation. Two f µw conditions displaying maximal , 193.67 GHz and 194.1 GHz, typical for CE-DNP

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can be seen. 44 This frequency profile is obtained with the maximum available µw power, ν1µw =120 mW. The positive condition (f µw =193.7 GHz) for as a function of ν1µw is shown in Fig. 3b, whose shape a priori is surprising. Intuitively, is expected to increase with increased ν1µw , but here reaches an optimum value at 25 mW and subsequently decreases with increasing ν1µw . For instance, at ν1µw =120 mW is less than half of that obtained at ν1µw =25 mW. This phenomenon is a consequence of “oversaturation” of the EPR line, and can be rationalized using the theoretical framework presented here. As discussed earlier, the CE enhancement is proportional to ∆Pe between CE electron spin pairs across the EPR line. The quasi steady-state electron spin polarization profile, and hence ∆Pe , induced by µw can be experimentally accessed by pump-probe ELDOR experiments. 45,46 In Fig. 3c, experimental ELDOR profiles acquired at the same conditions for the four different ν1µw (marked with arrows in Fig. 3b) are shown, with fixed saturation pulse duration of 100 ms (∼10T1e ). The ELDOR profiles show the phenomenological exchange of polarization between electron spins across the EPR line with those at the probe frequency, i.e. 193.7 GHz in this experiment. The ELDOR profiles clearly reveal that the extent and width of the electron spin saturation increases with increasing ν1µw . The µw power of ν1µw =1.7 mW (orange line) is too low to measurably saturate the EPR line, and correspondingly is small. In contrast, at maximum ν1µw =120 mW, the EPR line is dramatically oversaturated that leads to a net lower ∆Pe , and therefore lower . Maximum is achieved at an intermediate ν1µw (here ∼25 mW), where ∆Pe is maximal for this system. The ELDOR experiments clearly demonstrate that maximizing the CE efficiency can be essentially achieved by maximizing ∆Pe . The saturation factor that utilizes the eSD mechanism is determined by a combined effect of electron spin relaxation rates and the e − e coupling strengths or the flip-flop transition rate. Therefore, the extent of saturation via eSD is also governed by electron spin concentrations and temperatures, as these parameters markedly influence the e − e coupling strength and/or the electron spin relaxation rates. Higher concentrations and lower temperatures,

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generally, lead to a larger saturation via eSD. To demonstrate how temperature influences the saturation factor, and in turn , we measure the normalized enhancement vs. the µw frequency, as well as power. Figure 4a shows normalized vs. f µw at four different temperatures (4K (red), 10K (green), 25K (blue), and 80K (orange)) of a sample containing 40 mM 4AT in DNP-juice. The normalized frequency profiles at different temperatures appear similar, with some small changes occurring only in f µw values where positive and negative maximum enhancements are observed. This is attributed to changes in the relative contribution of SE and CE to the overall DNP effect with temperature. At higher temperature, the SE contribution decreases, which reduces the difference between f µw corresponding to positive and negative maxima – but we will not focus on this effect. 47 Here, we monitor how temperature influences the optimal ν1µw in CE-DNP. This is shown in Fig. 4b. At lower temperatures, where saturation via eSD is very effective (due to long T1e ), optimum is reached at much lower ν1µw . 39,46 Increase in ν1µw beyond optimum attenuates via oversaturation. Raising temperature decreases the time-frame over which eSD occurs (shorter T1e ), so that larger ν1µw is needed to generate the maximum ∆Pe and , as validated at 10 K and 80 K. This implies that for a fixed high ν1µw , might be higher at elevated temperatures. In the literature, the cause of decreasing with increased ν1µw has sometimes been ascribed to sample-heating. 48 However, relying on the analytical theory of CE-DNP, and the shape of the experimental ELDOR profile, we clearly demonstrate that such effects result from oversaturation of the EPR spectrum as first debuted by Siaw et al. 28 Chirp-DNP: In general, DNP enhancement is achieved using CW µw irradiation, but recent hardware developments have shown that enhancements can be dramatically increased when using frequency-modulated pulse-trains 49 (e.g. Chirp) instead of CW irradiation. The benefit of broad-band pulsed µw irradiation is more notable in the regime of low spectraldiffusion, i.e. at higher temperatures and/or lower electron spin concentrations, where ∆Pe (Chirp) > ∆Pe (CW) . The use of linear Chirp-pulses, in which the µw frequency varies linearly in time, can lead to broad-band electron spin saturation (i.e larger ∆Pe ⇒ larger ), whose

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effect is amplified in systems where small eSD is the limiting factor. A major advancement in this area has been made by Kaminker et al. 30 , who reported on the implementation of arbitrary phase-shaped pulses for DNP at 7 T, further enhancing the work of Hovav et al. on Chirp-DNP at 3.34 T. 50 . Kaminker et al. and Hovav et al. have demonstrated that in the limit of faster frequency modulation rate (δf µw /δt) than the relaxation rate, the DNP efficiency of Chirp irradiation increases with increased Chirp bandwidth, δf µw (or the frequency-modulation amplitude of Chirp), until a threshold is reached where the oversaturation condition is reached. It is straight-forward to understand that the optimal δf µw must be smaller than the ω0n , otherwise it will cause a decrease in net ∆Pe between the CE electron pair and hence the DNP enhancement. Indeed, using a 20 mM 4AT radical system, we demonstrate that Chirp-DNP can lead to ¿two-fold greater enhancement compared to conventional CW DNP (Fig. 5a). This is due to larger ∆Pe induced at steady-state, as confirmed by ELDOR (see SI section-C2). The Chirp irradiation scheme was implemented using an arbitrary waveform generation (AWG) setup in the Han Lab, as debuted by Kaminker et al. 42 Notably, Chirp-DNP not only increases the magnitude of , but also its robustness with respect to the µw irradiation (carrier) frequency (see Fig. 5a). Nuclear spin Depolarization: The effective Hamiltonian used above holds true for static-samples, but not exactly for magic-angle spinning (MAS) experiments, as under MAS the energies of spin are time-modulated due to anisotropic (g and Dipolar) interactions. Modulation of ESR frequencies over the course of the MAS rotor period leads to multiple different energy-level crossings (diabatic transitions) and avoided-crossings (adiabatic transitions) of the coupled e−e−n spins, as detailed by Thurber et al. 15 . MAS therefore renders the spin-B1 (µw induced electron transition/saturation) and spin-spin (e − e homogeneousmixing and e−e−n CE-mixing) interactions time-dependent, also called rotor-events. 16 The transition caused at the rotor-events depends on the magnitude of the perturbations, i.e. the off-diagonal terms (e.g. ν1µw for electron spin saturation, ωee for e − e mixing, and

ωee ωen ω0n

for

CE transitions) and the rate of the energy change at the crossing, i.e. the diagonal terms

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(mainly dependent on g-anisotropy and MAS rate). An analytic solution to the transition dynamics for a time-dependent Hamiltonian was first proposed by Landau 51 and Zener 52 for a two energy-level system. According to the Landau-Zener (LZ) model, the adiabatic transition probability,

P(adiabatic or avoided crossing) = 1 − exp(−πE 2 /A).

(13)

The P at the crossing is inversely proportional to the energy-crossing-rate (A), and directly proportional to (scales up) the magnitude of perturbation (E) that mixes the two states of the system. P=1 means that the transition is purely adiabatic and that the state-of-thesystem completely change from one to another. This would happen when the perturbationfactor (E 2 ) is much larger than the rate of energy change (A) at the crossing. P=0 (no perturbation) would essentially mean that the two states do not mix at the energy-crossing condition. Intermediate values of P would basically employ partial mixing of the states-ofthe-system at the crossing. Thurber et al. 15 first used the LZ concept to explain the CE mechanism under MAS. Using this theory, they also uncovered the phenomenon of nuclear depolarization - that leads to a decrease in nuclear spin polarization under MAS, in the absence of µw irradiation. 31,32 Thurber et al. 31 and Mentink-Vigier et al. 32 showed that under MAS electron spin pairs (partially) exchange their polarization through dipolar/exchange coupling (similar to Eq. 4) at some specific rotor-positions that depend on the orientation of the individual g-tensors. If the transition probability (P) is small due to weak ωee , ∆Pe will be reduced after crossing. Under the condition of long T1e compared to the rotor-period, the electron spins do not relax fast enough to reach their Boltzmann equilibrium. Subsequently, the rotor-positions satisfying the CE-resonance condition can cause a reverse transfer of polarization, i.e. from the nuclear spins to electron spins if ∆Pe < Pn . This depletion of the nuclear polarization is termed nuclear spin depolarization, where this effect increases with increased spinning-rate

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before reaching a plateau. We show in Fig. 5b a numerically simulated 1 H depolarization profile map, displaying increased depolarization with increased MAS spinning rate and increased T1e . This results from the effect of small ωee coupling (leading to non-adiabatic mixing, Eq. 4) and an operative CE Hamiltonian (Eq. 12) under the condition of long T1e , where the steady-state polarization difference between the electron spins drops below that of the nuclear Boltzmann polarization. Clearly, a larger ωee leading to an adiabatic e − e crossing would be a way to reduce the depletion of nuclear spin polarization (see SI, Fig. S6 for more details). Recently, Lund et al. experimentally elucidated that for similar radical systems and comparable spin concentrations, nuclear depolarization is dominantly modulated by T1e , with longer T1e leading to greater nuclear depolarization. Further, it was shown that the depolarization effect can be significantly attenuated by purposefully shortening T1e (without affecting T1n ) by adding paramagnetic-relaxation-agents, e.g. Gdchelate complexes. 33 Using a mixture of narrow and broad radicals with distinct isotropic g-tensors will also circumvent this conundrum. To this end, mixed radicals, which exhibit a truncated-CE 53 would be the most favourable radical-system to minimize depolarization under MAS, as it has been shown recently that one of the two electron spins always remain fully polarized owing to its fast electron spin T1e relaxation property (this is demonstrated in SI, section-B2, in greater detail). In this paper, we have described the polarization, depolarization, and oversaturation features of CE-DNP, aided by the theory of an effective Hamiltonian and experimentally supported by ELDOR data. We recapitulated the knowledge in the literature that the CE mechanism relies on the intrinsic dynamics of the spin system, while the role of lowpower µw irradiation is merely to generate a large electron spin polarization differential by selectively saturating one of the electron spin population. However, if the EPR line is oversaturated under conditions of high µw power and/or broad Chirp-bandwidth, large spectral-diffusion supervenes with increased e − e coupling (high concentration) and/or slow relaxation (low temperature) conditions, decreasing the net polarization difference. This in

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turn can decrease the CE-DNP enhancement, as highlighted in this article. Here, we also discussed the advantage of frequency-modulated shaped-pulses (linear-Chirp) that increases the net ∆Pe by recruiting more electron spins into the operation, which is beneficial especially under conditions of low spectral diffusion. Nuclear spin depolarization is another salient feature of the CE mechanism, which leads to a reverse polarization transfer i.e. from nuclear spins to electron spins under conditions of MAS-induced non-adiabatic polarization mixing (at the energy-crossings) between electron spins with long T1e . All examples presented here share the same underlying mechanism, but are effective at different steady-state electron spin polarizations, as probed and verified by pump-probe ELDOR experiments under DNP conditions. Using the same perspective, it can be envisaged that mixed bi-radicals with different relaxation rates (a slow and a fast relaxing electron spin combination) will lead to larger ∆Pe , as only one of the two electron spins can be selectively saturated, and therefore lead to higher CE enhancement, especially under MAS conditions. We have analytically derived the CE Hamiltonian using an elegant, simple and effective operator-based Floquet theoretical framework that, in this form, has not been derived and presented before in the literature. The main advantage of using Floquet Theory is for solving multi-modal time dependent Hamiltonian, e.g. as applied to DNP under multifrequency application or time-optimized DNP experiments– that the AHT cannot solve. With the advent of high power µw amplifier technology, 54 coherent EPR manipulation that rely on time-modulated µw pulse schemes will become increasingly viable for highly efficient DNP enhancement and electron-nuclear decoupling under DNP conditions 55,56 . Given this trajectory, the real benefit of the operator-based Floquet framework will become apparent in time-dependent DNP experiments, which represent exciting future developments.

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Acknowledgement This work was supported by the National Science Foundation (CHE #1505038 to SH), the National Institute of Health (NIBIB #1R21EB022731 and #R21GM103477 to SH) and Binational Science Foundation (Grant #2014149 to SH). The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.

Supporting Information The supporting information on experimental and numerical results and methods, matlab code, and Spin-Evolution code is available free of charge on the ACS publication website at DOI: XXX.

Authors Biographies Asif Equbal received his BS-MS degree from IISER Mohali, India in 2013. He then pursued his Ph.D. (2013 to 2016) from iNANO, Aarhus University, Denmark under the supervision of Prof. Niels Christian Nielsen on development of heteronuclear spin decoupling pulse scheme under MAS. He is currently working as a post doctoral fellow in the lab of Prof. Songi Han, at University of California Santa Barbara, on development of dynamic nuclear polarization enhancement technique. Alisa Leavesley completed her Ph.D. (2013 to 2018) at the University of California, Santa Barbara under the supervision of Prof. Songi Han. Her Ph.D. work focused on the development of solid-state source-based high field EPR and dynamic nuclear polarization (DNP) instrumentation and the study of radical and electron-spin clustering on DNP mechanisms and efficiency. She is currently working at Thomas Keating LTD. to design custom quasi optical based EPR and DNP instruments.

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Sheetal Kumar Jain received his Ph.D. from Aarhus university under the supervision of Prof. Niels Christian Nielsen in 2014. His Ph.D. was focused on development of crosspolarization methods for solid-state NMR spectroscopy and applications. In 2014, he joined the group of Prof. Robert G. Griffin at Massachusetts Institute of Technology (MIT), USA, where he worked on developing methods for pulsed DNP. Currently, Sheetal is a postdoc in the Han group at University of California Santa Barbara. His current research is centred around DNP/NMR methods and applications. Songi Han received her Doctoral Degree in Natural Sciences (Dr.rer.nat) from Aachen University of Technology (RWTH), Germany, in 2001. She pursued her postdoctoral studies at the University of California Berkeley sponsored by the Feodor Lynen Fellowship of the Alexander von Humboldt Foundation. Dr. Han joined the faculty at University of California Santa Barbara (UCSB) in 2004, received tenure in 2010 and was promoted to full professor in 2012. She is currently a Professor in the Department of Chemistry and Biochemistry and the Department of Chemical Engineering at UCSB. She is a recipient of the 2008 Packard Fellowship for Science and Engineering, the 2010 Dreyfus-Teacher Scholar Award, the 2011 NIH Innovator Award, the 2015 Bessel Prize of the Alexander von Humboldt Foundation, and the 2018 recipient of the Biophysical Society Innovator Award. Her research group focuses on mechanistic studies of DNP, as well as broadening the application scope of DNP and EPR.

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(29) Hovav, Y.; Feintuch, A.; Vega, S.; Goldfarb, D. Dynamic nuclear polarization using frequency modulation at 3.34 T. Journal of Magnetic Resonance 2014, 238, 94–105. (30) Kaminker, I.; Han, S. Amplification of Dynamic Nuclear Polarization at 200 GHz by Arbitrary Pulse Shaping of the Electron Spin Saturation Profile. The Journal of Physical Chemistry Letters 2018, 9, 3110–3115, PMID: 29775537. (31) Thurber, K. R.; Tycko, R. Perturbation of nuclear spin polarizations in solid state NMR of nitroxide-doped samples by magic-angle spinning without microwaves. The Journal of chemical physics 2014, 140, 184201–184211. (32) 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, 21824– 21836. (33) Lund, A.; Equbal, A.; Han, S. Tuning nuclear depolarization under MAS by electron T 1e. Physical Chemistry Chemical Physics 2018, 20, 23976–23987. (34) Vega, S. Fictitious spin 1/2 operator formalism for multiple quantum NMR. The Journal of Chemical Physics 1978, 68, 5518–5527. (35) Bayro, M. J.; Huber, M.; Ramachandran, R.; Davenport, T. C.; Meier, B. H.; Ernst, M.; Griffin, R. G. Dipolar truncation in magic-angle spinning NMR recoupling experiments. The Journal of Chemical Physics 2009, 130, 114506. (36) Scholz, I.; Meier, B. H.; Ernst, M. Operator-based triple-mode Floquet theory in solidstate NMR. The Journal of chemical physics 2007, 127, 204504. (37) Boender, G.; Vega, S.; De Groot, H. A physical interpretation of the Floquet description of magic angle spinning nuclear magnetic resonance spectroscopy. Molecular Physics 1998, 95, 921–934. 28


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Scheme 1: Modulation of various dipolar terms (alphabets) in the Zeeman rotating frame of individual spins constituting the I-S dipolar Hamiltonian shown in Eq. 1.

Figure 1: (a)Polarization oscillations between two homonuclear coupled spins (5 kHz) without any external perturbation, as a function of coupling evolution time and the polarization difference between the two spins. The contour maps the polarization on spin 2. (b) trace along time-axis to show polarization of spin 2 (blue) and spin 1 (red) vs. time. (c) trace along y-axis to show polarization transferred from 1 to 2 as a function of initial polarization difference.

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Figure 2: Polarization oscillations between e1 − e2 − H spins satisfying the CE condition leading to (a) nuclear enhancement for ∆Pe > Pn and (b) depolarization ∆Pe < Pn . The Larmor frequency of 1 H was set to 300 MHz, same as the difference of electron-spins Larmor frequencies. e1 − e2 , e1 − H and e2 − H couplings were to 20, 12.5 and 0 MHz, respectively.

Figure 3: (a) Normalized DNP enhancement vs. µw frequency for 120 mW power, 60 s build up time, at 7 T and 4 K, using 40 mM 4AT in DNP-juice. (b) DNP enhancement vs. µw power for fixed µw frequency, 193.7 GHz. (b) ELDOR saturation profile for four different µw powers. The probe frequency is set to 193.7 GHz, which is the optimum µw frequency for DNP.

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Figure 4: (a) Normalized DNP enhancement vs. µw frequency for 120 mW power at 7 T using 40 mM 4AT acquired at four temperatures. (b) DNP enhancement vs. µw power for the four temperatures at their optimum µw frequencies.

Figure 5: (a) DNP frequency profile using 40 mM 4AT at 7 T and 4 K acquired using CW (red) and frequency modulated (black) irradiation. (b) Numerically simulated 1 H depolarization profile as function spin rate and T1e . The simulation was done for 3 spins (eeH) system with electron g-tensors of Nitroxide based radical, e − e anisotropic coupling of 37 MHz, and using a 7 T field conditions. 34


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Cross-Effect Dynamic Nuclear Polarization Explained: Polarization

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