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**Magnetic Resonance**
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**Research article**
04 May 2020

**Research article** | 04 May 2020

Study of electron spectral diffusion process under DNP conditions by ELDOR spectroscopy focusing on the ^{14}N solid effect

- Department of Chemical and Biological Physics, Weizmann Institute of Science, Rehovot, Israel

- Department of Chemical and Biological Physics, Weizmann Institute of Science, Rehovot, Israel

**Correspondence**: Daniella Goldfarb (daniella.goldfarb@weizmann.ac.il) and Shimon Vega (shimon.vega@weizmann.ac.il)

**Correspondence**: Daniella Goldfarb (daniella.goldfarb@weizmann.ac.il) and Shimon Vega (shimon.vega@weizmann.ac.il)

Abstract

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Electron spectral diffusion (eSD) plays an important role in solid-state, static dynamic nuclear polarization (DNP) with polarizers that have inhomogeneously broadened EPR spectra, such as nitroxide radicals. It affects the electron spin
polarization gradient within the EPR spectrum during microwave irradiation
and thereby determines the effectiveness of the DNP process via the so-called indirect cross-effect (iCE) mechanism. The electron depolarization
profile can be measured by electron–electron double resonance (ELDOR)
experiments, and a theoretical framework for deriving eSD parameters from
ELDOR spectra and employing them to calculate DNP profiles has been
developed. The inclusion of electron depolarization arising from the
^{14}N solid effect (SE) has not yet been taken into account in this
theoretical framework and is the subject of the present work. The ^{14}N
SE depolarization was studied using W-band ELDOR of a 0.5 mM TEMPOL
solution, where eSD is negligible, taking into account the hyperfine
interaction of both ^{14}N and ^{1}H nuclei, the long microwave
irradiation applied under DNP conditions, and electron and nuclear
relaxation. The results of this analysis were then used in simulations of
ELDOR spectra of 10 and 20 mM TEMPOL solutions, where eSD is significant
using the eSD model and the SE contributions were added ad hoc employing the
^{1}H and ^{14}N frequencies and their combinations, as found from the
analysis of the 0.5 mM sample. This approach worked well for the 20 mM
solution, where a good fit for all ELDOR spectra recorded along the EPR
spectrum was obtained and the inclusion of the ^{14}N SE mechanism
improved the agreement with the experimental spectra. For the 10 mM
solution, simulations of the ELDOR spectra recorded along the *g*_{z}
position gave a lower-quality fit than for spectra recorded in the center of
the EPR spectrum. This indicates that the simple approach we used to
describe the ^{14}N SE is limited when its contribution is relatively high
as the anisotropy of its magnetic interactions was not considered
explicitly.

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How to cite.

Ramirez Cohen, M., Feintuch, A., Goldfarb, D., and Vega, S.: Study of electron spectral diffusion process under DNP conditions by ELDOR spectroscopy focusing on the ^{14}N solid effect, Magn. Reson., 1, 45–57, https://doi.org/10.5194/mr-1-45-2020, 2020.

1 Introduction

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It has been recently recognized that electron spectral diffusion (eSD) plays
a significant role in dynamic nuclear polarization (DNP) under static
conditions (Hovav
et al., 2015a; Leavesley et al., 2017). It affects the electron spin
polarization gradient within the EPR spectrum as a consequence of microwave
irradiation and thereby determines the effectiveness of the DNP process via
the so called indirect cross-effect (iCE)
mechanism (Hovav et al.,
2015a). This is particularly relevant in the case of nitroxide radicals, the
EPR spectra of which are in-homogeneously broadened in frozen solutions, at
concentrations of 20–40 mM used in DNP applications. Hovav et al. (2015a, b), Siaw et al. (2014) and Shimon et al. (2012, 2014) observed that during constant microwave (MW) irradiation there exists
an optimal radical concentration that leads to a maximum in the DNP
enhancement. At this concentration the inter-electron spin dipolar
interaction is sufficiently strong to generate a polarization gradient that
favors an efficient iCE enhancement mechanism, while at higher
concentrations the spectral diffusion saturates large parts of the EPR
spectrum and spin temperature effects can be
expected (Caracciolo et al., 2016; Kundu et al., 2018a, b). To monitor
directly the electron depolarization during MW irradiation, Hovav et al. (2015b) measured the ELDOR signals of frozen TEMPOL solutions, under
static DNP conditions, as a function of TEMPOL concentration, sample
temperature and MW irradiation time. Furthermore, they developed a model
(called the eSD model) that describes the depolarization process. This model
is based on rate equations for the electron polarizations along the EPR
spectrum, taking into account an exchange process between polarizations, in
addition to the saturation effects of the MW irradiation and the
spin-lattice relaxation. This eSD model introduces a fitting parameter
Λ^{eSD} that defines the strength of the polarization
exchange rate leading to the spectral diffusion within the EPR spectrum.
Using this eSD model, experimental ELDOR spectra could be satisfactorily
simulated and thus provide a feasible description of the eSD process.
Subsequently, it was demonstrated that once the polarization gradient within
the EPR spectrum has been determined via the eSD model simulations, the
lineshape of the associated DNP spectrum could be reproduced taking into
account the polarization differences between all electron pairs satisfying
the cross effect (CE) condition (Hovav et al.,
2015a). This approach was also implemented by Leavesley et al. (2017) when they explored the eSD process and its influence on the DNP
efficiency at a magnetic field of 7 T. They also considered the effects of
variations in the radical concentration, temperature and MW power on the
^{1}H-DNP spectra. Furthermore, Kundu et al. (2018b) used the eSD model to quantify the
dependence of the electron polarization exchange parameter Λ^{eSD} on radical concentration and temperature.

To justify the rather phenomenological eSD model, Kundu et al. (2018a, b) performed quantum-mechanics-based calculations of the spin evolution and associated EPR spectra of the electron spins in dipolarly coupled small spin systems under DNP conditions. In the case of weak dipolar coupling constants and after adding cross-relaxation (Hwang and Hill, 1967; Kessenikh et al., 1964) to the ELDOR calculations, the results were similar to those obtained using the eSD model. In the case of strong dipolar couplings a thermal mixing mechanism in the rotating frame could provide the calculated EPR spectra under MW irradiation (Abragam, 1961; de Boer, 1976; Borghini, 1968; Goldman, 1970; Provotorov, 1962; Wenckebach, 2016; Wollan, 1976). These studies also contributed to the validity of the iCE model in the weak and the strong dipolar coupling regime.

In addition to the CE mechanism, leading to the main nuclear signal
enhancements at relatively high radical concentrations, the solid effect
(SE) process also influences these enhancements. This process contributes to
the signal enhancements, but in addition causes some electron depolarization
that in turn can influence the CE enhancement
process (Hovav et al., 2015b; Leavesley et
al., 2018). When nitroxide radicals are used as DNP polarizers, these SE
depolarization effects arise from ^{1}H and ^{14}N nuclei hyperfine
interactions (Kundu et al., 2018b;
Leavesley et al., 2017). The SE-induced electron polarization depletions are
highly evident in ELDOR spectra at concentrations that are below the usual
concentration used for DNP, but their influence is observed also at
concentrations around 20 mM, which are relevant for
DNP (Harris et al.,
2011; Thankamony et al., 2017). As the Λ^{eSD} constant is determined from ELDOR lineshapes, the SE effects should be taken into
account in the eSD model to ensure the extraction of the correct value.
The purpose of this study is to account explicitly for the effects of the SE
mechanism on ELDOR lineshapes for nitroxides and to explore its influence on
the extraction of Λ^{eSD} at concentrations
relevant for static DNP.

We started this study by measuring ELDOR spectra of a 0.5 mM TEMPOL in DMSO frozen solution, in which the SE is the sole mechanism of depolarization, as
the spectral diffusion mechanism is negligible. To analyze these ELDOR
spectra we established a theoretical framework that accounts for all
^{14}N-SE and ^{1}H-SE depletions observed in these spectra. For this
low concentration, the ELDOR spectrum is identical to the ELDOR-detected NMR
(EDNMR) spectrum of nitroxide, which has already been studied and simulated
in the
past (Cox
et al., 2017; Florent et al., 2011; Jeschke and Spiess, 1998; Kaminker et
al., 2014; Nalepa et al., 2014, 2018). Yet, there is one major difference:
under EDNMR conditions, where resolution is of prime interest, the MW
irradiation period is short, in the microsecond range, and therefore
relaxation processes play a limited role during that irradiation. However,
under DNP conditions the duration of the irradiation is in the range of
milliseconds or longer and the electron and nuclear relaxation processes
influence the magnitude of the depolarization. A second, more technical,
difference is that in a full field-frequency two-dimensional (2D) EDNMR
spectrum the EPR dimension is usually obtained by stepping the magnetic
field (Florent
et al., 2011; Jeschke and Spiess, 1998; Kaminker et al., 2014; Nalepa et
al., 2014, 2018) unless chirped pulses are being
used (Wili and Jeschke, 2018), while 2D
ELDOR maps in the context of DNP are obtained by stepping the frequency. In
some earlier works the contributions from different nuclei in the EDNMR
spectra were taken into account by superimposing their individual spectra,
ignoring the contributions of combination
frequencies (Tan et al., 2019; Wang
et al., 2018). In others, the combinations were also taken into account and
reproduced in the simulated spectra (Cox et al., 2017).
The appearance of these lines depends on the experimental conditions (Cox et al., 2017). As under DNP conditions the duration
of the microwave irradiation is long we also took into account the
^{14}N–^{1}H combination lines in the ELDOR spectral simulations.

After analyzing the 0.5 mM spectrum, we proceeded to 10 and 20 mM TEMPOL
solutions, where spectral diffusion becomes significant. We measured their
ELDOR spectra and analyzed them employing the eSD
model (Hovav et al., 2015b), taking into account the SE
mechanism through an ad hoc inclusion of the ^{14}N and ^{1}H
frequencies.

2 Methods and materials

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Samples of 2–3 µL in 0.6 mm ID × 0.84 mm quartz tubes, with 0.5, 10 and
20 mM TEMPOL dissolved in a solution of DMSO ∕ H_{2}O (1:1 v/v), were
degassed by a *freeze–pump–thaw* procedure and fast-frozen with liquid nitrogen. TEMPOL and
DMSO were both purchased from Sigma Aldrich and used as is.

All measurements were carried out on our W-band (95 GHz, 3.4 T) in-house-built EPR spectrometer (Goldfarb et al., 2008; Mentink-Vigier et al., 2013) at 20 K.

Echo-detected EPR (ED-EPR) spectra were measured using the pulse sequence
*π*∕2-*τ*-*π*-*τ*-echo with *τ*=600 ns, while
increasing the magnetic field stepwise from 3370 to 3395 mT, with a 2 ms
repetition time. The pulse lengths were 100 ns for the *π*∕2 pulse and
200 ns for the *π* pulse, optimized at a detection frequency of 94.90 GHz.

Electron spin-lattice relaxation times *T*_{1e} were measured at different
positions within the EPR spectrum by saturation recovery experiments with a
long MW saturation pulse of 30 ms and echo pulses of 300 ns each, which is typical
for DNP MW power. The *T*_{1e} curves were analyzed using a superposition of
two exponential functions with time constants *t*_{1} and *t*_{2}, with the
slow (and major) component assigned to *T*_{1e}.

The ELDOR pulse sequence is shown in Fig. 1 and ELDOR spectra were
measured at different detection frequencies along the EPR spectrum. The
spectrometer was set to low power, which is typical for DNP using the detection
sequence *α*-*τ*-*α*-*τ*-echo, where *α* is a
flip angle of less than *π*∕2. While for EPR applications ELDOR is
carried out at a fixed detection frequency and the magnetic field is varied
to access different regions in the EPR spectrum, here we kept the field
constant and varied the detection frequency to access the spectrum width as
done for DNP applications. To carry out these ELDOR measurements, we
increased the bandwidth of the cavity to accommodate the full spectrum of
TEMPOL (approx. 500 MHz). The cavity resonance was tuned to 94.80 GHz. For
the 0.5 mM sample ELDOR spectra (40 in total) were recorded as a function of
the pump frequency, which was varied from 94.3 to 95.3 GHz. To obtain 2D
ELDOR data ELDOR spectra were measured at different detection frequencies in
intervals of 10 MHz from 94.55 to 94.95 GHZ, which covers most of the
EPR spectrum. The amplitude of the pump pulse, *ν*_{1}, was 0.5 MHz, as
determined by a nutation experiment at 94.8 GHz, corresponding to an
inversion pulse of 1 µs. The experimental parameters for the ELDOR
experiments are listed in Table 1.

3 Simulations

Back to toptop
In an effort to analyze the ELDOR spectra of the 0.5 nm TEMPOL solution we
rely on quantum-mechanics-based calculations considering the spin evolution
of a three-spin system consisting of an electron spin, $S=\mathrm{1}/\mathrm{2}$, coupled to a
single ^{1}H nucleus and a single ^{14}N nucleus. Simulations of these
ELDOR spectra were performed using a modified version of the computer code
developed by Kaminker et al. (2014) for a two-spin system; one electron spin and one ^{14}N
nucleus. The simulated ELDOR spectrum comprises EPR signals calculated at
fixed detection frequency positions ${\mathit{\nu}}_{\mathrm{det}}={\mathit{\omega}}_{\mathrm{det}}/\mathrm{2}\mathit{\pi}$ as
a function of the pump pulse frequency, ${\mathit{\nu}}_{\mathrm{MW}}={\mathit{\omega}}_{\mathrm{MW}}/\mathrm{2}\mathit{\pi}.$
In these calculations, we had to take into account the fact that the
duration of the MW irradiation in DNP experiments *t*_{MW} is much longer
than commonly used in EDNMR spectroscopy (millisecond vs. microsecond range,
respectively). For such long irradiation times the three-spin calculations
cannot account for the experimental spectral observations, mainly due to the
fact that the real spin system is more extended than only three spins
because of the many coupled protons present in the sample. Accordingly,
without extending the number of spins in our model we had to modify
Kaminker's procedure to reproduce the experimental observations, as
discussed next.

The three-spin system is described by the following spin Hamiltonian in the MW rotating frame, assuming the high field approximation:

$$\begin{array}{}\text{(1)}& \begin{array}{rl}{\widehat{\mathbf{H}}}_{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}& =\mathrm{\Delta}{\mathit{\omega}}_{\mathrm{e}}{\widehat{\mathbf{S}}}_{z}-{\mathit{\omega}}_{\mathrm{N}}{\widehat{\mathbf{I}}}_{z\mathrm{N}}-{\mathit{\omega}}_{\mathrm{H}}{\widehat{\mathbf{I}}}_{z\mathrm{H}}+{A}_{zz}^{\mathrm{H}}{\widehat{\mathbf{S}}}_{z}{\widehat{\mathbf{I}}}_{z\mathrm{H}}+{A}_{zz}^{\mathrm{N}}{\widehat{\mathbf{S}}}_{z}{\widehat{\mathbf{I}}}_{z\mathrm{N}}\\ & +({A}_{\mathrm{H}}^{+}{\widehat{\mathbf{I}}}_{\mathrm{H}}^{+}+{A}_{\mathrm{H}}^{-}{\widehat{\mathbf{I}}}_{\mathrm{H}}^{-}){\widehat{\mathbf{S}}}_{z}+({A}_{\mathrm{N}}^{+}{\widehat{\mathbf{I}}}_{\mathrm{N}}^{+}+{A}_{\mathrm{N}}^{-}{\widehat{\mathbf{I}}}_{\mathrm{N}}^{-}){\widehat{\mathbf{S}}}_{z}\\ & +{\widehat{\mathbf{I}}}_{\mathrm{N}}\cdot \stackrel{\mathrm{\u0303}}{Q}\cdot {\widehat{\mathbf{I}}}_{\mathrm{N}},\end{array}\end{array}$$

where

$$\begin{array}{}\text{(2)}& \mathrm{\Delta}{\mathit{\omega}}_{\mathrm{e}}{\widehat{\mathbf{S}}}_{z}=\left({\mathit{\mu}}_{B}{B}_{\mathrm{0}}{g}_{\mathrm{eff}}\right(\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi})-{\mathit{\omega}}_{\mathrm{MW}}){\widehat{\mathbf{S}}}_{z}.\end{array}$$

In Eq. (1) we neglected the dipolar interaction between the nuclei. Δ*ω*_{e} is the off-resonance electron frequency, *B*_{0} is the
strength of the external magnetic field pointing along the *z* axis of the
laboratory frame, and *g*_{eff}(*θ*, *φ*) is the effective
*g* tensor parameter for a specific orientation of the magnetic field with
respect to the principle axis system of the *g* tensor, given by the polar
angles *θ* and *φ*. The *g* tensor used for the calculation is $g=[\mathrm{2.0065},\mathrm{2.0037},\mathrm{1.9997}]$, obtained by simulating, using
Easyspin (Stoll and Schweiger, 2006), the frequency
domain EPR spectrum extracted from the echo intensity of the ELDOR spectra
with the pump pulse set far outside the EPR spectrum (see Fig. S1 in the Supplement).
The *g* values obtained from the EPR simulations and further used in the EDNMR
simulations differ from those reported by Florent et al. (2011)
($g=[\mathrm{2.00988},\mathrm{2.00614},\mathrm{2.00194}]$) as they compensate for an error of 4 mT
in the determination of *B*_{0}. These *g* values were used to determine the
selected orientations and to calculate *g*_{eff} in Eq. (2). Because the
energies and their differences depend on the product *g*_{eff}*B*_{0}, where
the error in *B*_{0} has been compensated in *g*, they are not affected by
the error in the field. The shift of 4 mT in *B*_{0} results in a shift of
the proton frequency by 0.17 MHz, which is very small compared to the EDNMR
linewidth. For ^{14}N it is even smaller and therefore the errors in the
nuclei Larmor frequencies are negligible. The Larmor frequencies of ^{1}H
and ^{14}N are *ω*_{H}=2*π**ν*_{H} and *ω*_{N}=2*π**ν*_{N}, respectively. In the EPR high field
approximation the terms that contribute to the hyperfine interaction are the
secular and pseudo-secular terms with coefficients $({A}_{zz}^{\mathrm{H}},\phantom{\rule{0.125em}{0ex}}{A}_{\mathrm{H}}^{\pm})$ for ^{1}H and $({A}_{zz}^{\mathrm{N}},\phantom{\rule{0.125em}{0ex}}{A}_{\mathrm{N}}^{\pm})$ for ^{14}N, where ${A}_{\pm}^{K}={A}_{zx}^{K}\pm i{A}_{zy}^{K}$, $K=\mathrm{H},\phantom{\rule{0.125em}{0ex}}\mathrm{N}$. In the case of ^{14}N the hyperfine tensor
contains an isotropic contribution ${a}_{\mathrm{iso}}^{\mathrm{N}}\ne \mathrm{0}$ in addition to
the anisotropic tensor elements $[{a}_{ZZ}^{K},\phantom{\rule{0.125em}{0ex}}{a}_{XX}^{K},\phantom{\rule{0.125em}{0ex}}{a}_{YY}^{K}]$,
where *X*, *Y* and *Z* are its principle axes. Assuming that the two
anisotropic hyperfine interactions are of axial symmetry (i.e., ${a}_{XX}^{K}={a}_{YY}^{K}=-\mathrm{1}/\mathrm{2}{a}_{ZZ}^{K})$ and that their major principal axes coincide
with that of the *g* tensor, the hyperfine coefficients of ${\widehat{\mathbf{H}}}_{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}$ become ${A}_{zz}^{K}\equiv {A}_{zz}^{K}\left(\mathit{\theta}\right)={a}_{\mathrm{iso}}^{K}+\frac{\mathrm{1}}{\mathrm{2}}{a}_{ZZ}^{K}(\mathrm{3}{\mathrm{cos}}^{\mathrm{2}}\mathit{\theta}-\mathrm{1})$ and ${A}_{\pm}^{K}\equiv {A}_{\pm}^{K}\left(\mathit{\theta}\right)=\frac{\mathrm{3}}{\mathrm{2}}{a}_{ZZ}^{K}\mathrm{cos}\mathit{\theta}\mathrm{sin}\mathit{\theta}$ (Schweiger and Jeschke, 2001). In the case of TEMPOL, the
isotropic ^{14}N contribution is ${a}_{\mathrm{iso}}^{\mathrm{N}}=\mathrm{44}\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\mathrm{MHz}$
and the anisotropic value is $-{a}_{ZZ}^{\mathrm{N}}=\mathrm{55}\phantom{\rule{0.125em}{0ex}}\mathrm{MHz}$. The
^{1}H hyperfine value was taken as ${a}_{ZZ}^{\mathrm{H}}=\mathrm{3},\mathrm{MHz}$. Finally, the ^{14}N nuclear quadrupole interaction
is also included in the spin Hamiltonian. Here we used the principal values
of the quadrupole tensor obtained by Florent et al. (2011),
$({Q}_{XX},\phantom{\rule{0.125em}{0ex}}{Q}_{YY},\phantom{\rule{0.125em}{0ex}}{Q}_{ZZ})=(\mathrm{0.48},\phantom{\rule{0.125em}{0ex}}\mathrm{1.29},\phantom{\rule{0.125em}{0ex}}-\mathrm{1.77})$ MHz, and again assumed
that its principal axes coincides with those of the *g* tensor.

The MW irradiation Hamiltonian in the rotating frame is defined as

$$\begin{array}{}\text{(3)}& {\widehat{\mathbf{H}}}_{\mathrm{MW}}={\mathit{\omega}}_{\mathrm{1}}{\widehat{\mathbf{S}}}_{x}.\end{array}$$

At the start of all our simulations, the Hamiltonian for each set of
(*θ*, *φ*) angles is represented in matrix form, in the 12
product states of the basis sets in the laboratory frame |*χ*_{e}〉, |*χ*_{H}〉 with ${\mathit{\chi}}_{e,\phantom{\rule{0.125em}{0ex}}H}=\mathit{\alpha},\phantom{\rule{0.125em}{0ex}}\mathit{\beta}$ and |*χ*_{N}〉 with ${\mathit{\chi}}_{\mathrm{N}}=+\mathrm{1},\phantom{\rule{0.125em}{0ex}}\mathrm{0},\phantom{\rule{0.125em}{0ex}}-\mathrm{1}$, and diagonalized according to

$$\begin{array}{}\text{(4)}& {\widehat{\mathbf{\Lambda}}}_{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}={\widehat{\mathbf{D}}}_{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}^{-\mathrm{1}}{\widehat{\mathbf{H}}}_{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}{\widehat{\mathbf{D}}}_{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}.\end{array}$$

${\widehat{\mathbf{D}}}_{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}$ is the diagonalization matrix and ${\widehat{\mathbf{\Lambda}}}_{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}$ is the diagonal matrix consisting of the eigenvalues ${E}_{i}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}$, in frequency units, corresponding to the 12 eigenstates $\left|{\mathit{\lambda}}_{i}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}\right.\u232a$ with $i=\mathrm{1},\phantom{\rule{0.125em}{0ex}}\mathrm{\dots}\phantom{\rule{0.125em}{0ex}},\mathrm{12}$. The EPR transition probabilities between levels $\left|{\mathit{\lambda}}_{i}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}\right.\u232a$ and $\left|{\mathit{\lambda}}_{j}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}\right.\u232a$ are as follows:

$$\begin{array}{}\text{(5)}& {P}_{i,\phantom{\rule{0.125em}{0ex}}j}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}=\left|\mathrm{2}\u2329{\mathit{\lambda}}_{i}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}\left|{\widehat{\mathbf{D}}}_{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}^{-\mathrm{1}}{\widehat{\mathbf{S}}}_{x}\phantom{\rule{0.125em}{0ex}}{\widehat{\mathbf{D}}}_{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}\right|{\mathit{\lambda}}_{j}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}\u232a\right|{}^{\mathrm{2}}.\end{array}$$

When $\left|{Q}_{ZZ}\right|<{\mathit{\omega}}_{\mathrm{N}}<\left|\frac{\mathrm{1}}{\mathrm{2}}{a}_{ZZ}^{\mathrm{N}}\right|,\phantom{\rule{0.125em}{0ex}}\left|{a}_{\mathrm{iso}}\right|$, the ${\mathit{\omega}}_{\mathrm{N}}{\widehat{\mathbf{I}}}_{z,\phantom{\rule{0.125em}{0ex}}\mathrm{N}}$ term in all Hamiltonians ${\widehat{\mathbf{H}}}_{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}$ has little influence on the form of the eigenstates, which are
products of the electron states |*χ*_{e}〉 with the
eigenvalues ${m}_{\mathrm{e}}=\pm \mathrm{1}/\mathrm{2}$, the hyperfine mixed proton states
approximately equivalent to |*χ*_{H}〉 with ${m}_{\mathrm{H}}\approx \pm \mathrm{1}/\mathrm{2}$ and the nitrogen states |*χ*_{N}〉,
mainly determined by the hyperfine interaction terms in ${\widehat{\mathbf{H}}}_{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}$ with ${m}_{\mathrm{N}}\approx +\mathrm{1},\phantom{\rule{0.125em}{0ex}}\mathrm{0},\phantom{\rule{0.125em}{0ex}}-\mathrm{1}$. As a result we can
easily recognize six “allowed” transitions with frequencies ${\mathit{\nu}}_{(i,\phantom{\rule{0.125em}{0ex}}j{)}_{\mathrm{a}}}(\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi})=({E}_{i}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}-{E}_{j}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}})$ that correspond to EPR transitions (*i*−*j*)_{a}, with
$\mathrm{\Delta}{m}_{\mathrm{e}}=\pm \mathrm{1}$, Δ*m*_{H}≈0 and Δ*m*_{N}≈0 and thus ${P}_{i,\phantom{\rule{0.125em}{0ex}}j}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}\approx \mathrm{1}$. We note that for
orientations along the *x*, *y* axis, the ^{14}N hyperfine interaction is close
to *ω*_{N} and therefore ${P}_{i,\phantom{\rule{0.125em}{0ex}}j}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}<\mathrm{1}$. Figure 2 presents a schematic energy level diagram of the three-spin
system for an arbitrary set of angles (*θ*, *φ*). The six allowed
transitions are indicated by red arrows. For one of these transitions the
corresponding homonuclear “single quantum” (SQ) forbidden transitions,
with $\mathrm{\Delta}{m}_{\mathrm{H}}\approx \pm \mathrm{1}$ or $\mathrm{\Delta}{m}_{\mathrm{N}}\approx \pm \mathrm{1}$, are
also indicated, in blue or green, respectively. The heteronuclear
“double-” and “zero quantum” (DQ and ZQ) forbidden transitions, with
$\mathrm{\Delta}{m}_{\mathrm{H}}\approx \pm \mathrm{1}$ and $\mathrm{\Delta}{m}_{\mathrm{N}}\approx \pm \mathrm{1}$, are shown
in purple.

Using the Orisel function in Easyspin (Stoll and
Schweiger, 2006), the values of ${E}_{i}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}$ and
${P}_{i,\phantom{\rule{0.125em}{0ex}}j}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}$ were calculated for a collection of 9609 sets
of values of (*θ*, *φ*) and from them all transition frequencies
${\mathit{\nu}}_{i,\phantom{\rule{0.125em}{0ex}}j}(\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi})$ were determined. To choose which
orientations of the spin system contribute to the allowed EPR signal at a
given *ν*_{det}, we search for those sets of angles (*θ*, *φ*) for which at least one allowed transition falls in the frequency range
${\mathit{\nu}}_{\mathrm{det}}-\mathrm{3}\phantom{\rule{0.125em}{0ex}}\mathrm{MHz}\le {\mathit{\nu}}_{(i,\phantom{\rule{0.125em}{0ex}}j{)}_{\mathrm{a}}}(\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi})\le {\mathit{\nu}}_{\mathrm{det}}+\mathrm{3}\phantom{\rule{0.125em}{0ex}}\mathrm{MHz}$. This frequency span provides a frequency bandwidth
of 6 MHz for the detection pulse, estimated as the excitation bandwidth for
a detection pulse of 300 ns length. In addition, it can account for some
*g* and hyperfine strains. This procedure generated a subset of selected
(*θ*, *φ*)_{det} pairs for each *ν*_{det}, the
size of which depends on the position of *ν*_{det} within the EPR
spectrum.

After choosing a value for *ν*_{det} we simulated the ELDOR spectra of
all crystal orientations of the subset (*θ*, *φ*)_{det}. The sum
of these spectra are compared with the measured ELDOR spectrum at that
frequency. To obtain the individual ELDOR spectra we calculated the EPR
signal at *ν*_{det} after a long MW pump pulse as a function of the
frequency of this pulse, *ν*_{MW}.

To follow the evolution of the spin system during the long MW irradiation period, prior to the EPR detection, it is sufficient to consider only the eigenstate populations ${p}_{i}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}\left(t\right)$ of all $\left|{\mathit{\lambda}}_{i}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}\right.\u232a$ for the detection subset, as described earlier (Hovav et al., 2010, 2015b). The rate equation during the MW irradiation for these populations can be presented as

$$\begin{array}{}\text{(6)}& {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{p}_{i}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}=\sum _{j=\mathrm{1},\phantom{\rule{0.125em}{0ex}}\mathrm{12}}\left\{-{R}_{ij}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}+{W}_{ij}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}\right\}{p}_{j}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}},\end{array}$$

where ${R}_{ij}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}$ denotes the elements of the 12×12 spin-lattice relaxation matrix ${\widehat{\mathbf{R}}}_{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}$, and ${W}_{ij}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}$ denotes the elements of the 12×12 MW rate matrix ${\widehat{\mathbf{W}}}_{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}$. The relaxation matrix ${\widehat{\mathbf{R}}}_{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}$ is equal
to the sum of the relaxation matrices ${\widehat{\mathbf{r}}}_{\left(ij\right)}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}$
of all transitions *{**i*−*j**}* with *E*_{j}>*E*_{i}. The non-zero matrix
elements of ${\widehat{\mathbf{r}}}_{\left(ij\right)}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}$ are derived, assuming a
linear field fluctuation causing *T*_{1e}:

$$\begin{array}{}\text{(7a)}& \begin{array}{r}{r}_{\left(ij\right),\phantom{\rule{0.125em}{0ex}}ii}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}=-{\displaystyle \frac{\mathrm{1}}{{T}_{\mathrm{1},\phantom{\rule{0.125em}{0ex}}ij}}}{\displaystyle \frac{\mathrm{1}}{(\mathrm{1}+{\mathit{\eta}}_{ij})}};\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}{r}_{\left(ij\right),\phantom{\rule{0.125em}{0ex}}ij}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}={\displaystyle \frac{\mathrm{1}}{{T}_{\mathrm{1},\phantom{\rule{0.125em}{0ex}}ij}}}{\displaystyle \frac{{\mathit{\eta}}_{ij}}{(\mathrm{1}+{\mathit{\eta}}_{ij})}},\\ {r}_{\left(ij\right),\phantom{\rule{0.125em}{0ex}}ji}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}={\displaystyle \frac{\mathrm{1}}{{T}_{\mathrm{1},\phantom{\rule{0.125em}{0ex}}ij}}}{\displaystyle \frac{\mathrm{1}}{(\mathrm{1}+{\mathit{\eta}}_{ij})}};\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}{r}_{\left(ij\right),\phantom{\rule{0.125em}{0ex}}jj}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}=-{\displaystyle \frac{\mathrm{1}}{{T}_{\mathrm{1},\phantom{\rule{0.125em}{0ex}}ij}}}{\displaystyle \frac{{\mathit{\eta}}_{ij}}{(\mathrm{1}+{\mathit{\eta}}_{ij})}}\end{array},\end{array}$$

and

$$\begin{array}{}\text{(7b)}& {\displaystyle \frac{\mathrm{1}}{{T}_{\mathrm{1},\phantom{\rule{0.125em}{0ex}}ij}}}={\displaystyle \frac{{\left|\mathrm{2}\u2329{\mathit{\lambda}}_{i}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}\left|{\widehat{\mathbf{S}}}_{x}\right|{\mathit{\lambda}}_{j}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}\u232a\right|}^{\mathrm{2}}}{{T}_{\mathrm{1}\mathrm{e}}}},\end{array}$$

with ${\mathit{\eta}}_{ij}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}={p}_{i}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi};\phantom{\rule{0.125em}{0ex}}\mathrm{eq}}/{p}_{j}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi};\phantom{\rule{0.125em}{0ex}}\mathrm{eq}}$ being the ratio between the thermal equilibrium populations defined in the laboratory frame, and

$$\begin{array}{}\text{(7c)}& {\widehat{\mathbf{R}}}_{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}=\sum _{\mathit{\{}i-j\mathit{\}}}{\widehat{\mathbf{r}}}_{\left(ij\right)}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}.\end{array}$$

The elements of ${\widehat{\mathbf{W}}}_{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}$ are equal to the sum of the
${\widehat{\mathbf{w}}}_{\left(ij\right)}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}$ matrices with non-zero elements that
express the effective irradiation strength on each transition
(*i*−*j*) (Hovav et al., 2010):

$$\begin{array}{}\text{(8a)}& \begin{array}{rl}& {w}_{\left(ij\right),\phantom{\rule{0.125em}{0ex}}ij}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}={w}_{\left(ij\right),\phantom{\rule{0.125em}{0ex}}ji}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}=-{w}_{\left(ij\right),\phantom{\rule{0.125em}{0ex}}ii}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}=-{w}_{\left(ij\right),\phantom{\rule{0.125em}{0ex}}jj}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}=\\ & {\displaystyle \frac{{\mathit{\omega}}_{\mathrm{1}}^{\mathrm{2}}{\left|\mathrm{2}\u2329{\mathit{\lambda}}_{i}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}\left|{\widehat{\mathbf{S}}}_{x}\right|{\mathit{\lambda}}_{j}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}\u232a\right|}^{\mathrm{2}}{T}_{\mathrm{2}\mathrm{mw}}}{\mathrm{1}+\mathrm{4}{\mathit{\pi}}^{\mathrm{2}}{\left\{{\mathit{\nu}}_{ij}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}-{\mathit{\nu}}_{\mathrm{MW}}\right\}}^{\mathrm{2}}{T}_{\mathrm{2}\mathrm{mw}}^{\mathrm{2}}}}\end{array}\end{array}$$

and

$$\begin{array}{}\text{(8b)}& {\widehat{\mathbf{W}}}_{ij}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}=\sum _{(i-j)}{\widehat{\mathbf{w}}}_{\left(ij\right)}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}.\end{array}$$

Here *ω*_{1} is the MW amplitude (see Eq. 3). A transverse relaxation
time *T*_{2mw}, which determines the off-resonance efficiency of the
irradiation, is introduced and for simplicity is assumed to be the same for
all transitions. Note that *T*_{2mw} is not the measured phase memory time,
*T*_{M}. After entering the values of *T*_{1e}, *ω*_{1} and an
irradiation time, it is possible to solve Eq. (6) and to use the populations
at the end of the irradiation to evaluate the EPR signals.

Setting the detection frequency at one of the allowed transition frequencies
and irradiating with a pump frequency that matches one of its associated
forbidden transitions (i.e, they share a common energy level) result in a
depletion of the EPR signal. The calculations show that the depletion can be
very significant for pump pulses on the order of tens of microseconds but
disappears for irradiation periods of the order of tens of milliseconds.
Thus using Eq. (6) works well for calculating EDNMR spectra for short pump
pulses (Kaminker
et al., 2014; Ramirez Cohen et al., 2017). However, for extended periods of
MW irradiation, longer than *T*_{1e} as is applied in DNP, the simulated
ELDOR signals are very weak at the forbidden transition frequencies. The
reason for this is that for MW irradiations longer than *T*_{1e}, the SE
spin evolution of an electron-nuclear spin pair brings the electronic
polarization back to its equilibrium value. This is, however, in contrast to
the experimental results where rather intense lines were observed even for
long irradiation. The reason for this discrepancy is that in reality the
electron spins are interacting with several equivalent coupled nuclei, which
transfer their polarization to the bulk via nuclear spin diffusion. This is
particularly true when many protons are present. Accordingly, reproducing
the experimental results, while still employing our simplified three-spin
system model, requires modification of the simulation procedure as described
next.

In order to obtain from a three-spin calculation the observed EPR signal
depletions even after long irradiation periods, we modified the form of the
MW rate matrix. Realizing that an irradiation of one of the forbidden
transitions, (*i*−*k*)_{f} or (*k*−*j*)_{f}, causes a depletion of the
population difference of an allowed transition, (*i*−*j*)_{a}, we removed the
four matrix elements of ${\widehat{\mathbf{w}}}_{(ik{)}_{\mathrm{f}}}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}$ and
${\widehat{\mathbf{w}}}_{(kj{)}_{\mathrm{f}}}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}$ from the ${\widehat{\mathbf{W}}}_{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}$ matrix. This is equivalent to removing the irradiation on the
forbidden transitions, which in turn cause the change in population
difference of the allowed transition, ${P}_{i,j}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}$. To
re-introduce the effect of the forbidden transitions on ${P}_{i,\phantom{\rule{0.125em}{0ex}}j}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}$ of the allowed transitions, we added them as an artificial
irradiation on the allowed one by adding them to the four non-zero matrix
elements of ${\widehat{\mathbf{w}}}_{(ij{)}_{\mathrm{a}}}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}:{\left\{{\widehat{\mathbf{w}}}_{(ik{)}_{\mathrm{f}}}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}+{\widehat{\mathbf{w}}}_{(kj{)}_{\mathrm{f}}}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}\right\}}_{(ij{)}_{\mathrm{a}}}$. In this way we ensure a depletion of
the population difference of (*i*−*j*)_{a}, without the relaxation mechanism
canceling it. While realizing that the depletion of polarization due to the irradiation of the forbidden transitions can be reduced by the allowed transition relaxation dictated by the value of *T*_{i, ij}, we introduce SE fitting parameters to
adjust their values during irradiation: one for each of the different
forbidden proton(${a}_{\mathrm{H}}^{\mathrm{SE}}$), nitrogen (${a}_{\mathrm{N}}^{\mathrm{SE}}$), combined
proton–nitrogen (${a}_{\mathrm{HN}}^{\mathrm{SE}}$) and even DQ nitrogen (${a}_{\mathrm{DQ}-\mathrm{N}}^{\mathrm{SE}}$) transitions. In this way an irradiation on (*i*−*k*)_{f}
reproduced the experimentally observed signal depletions, still taking into
account the effective MW irradiation strengths, ${\mathit{\omega}}_{\mathrm{1}}\times \left.\u2329{\mathit{\lambda}}_{i}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}\right|{\widehat{\mathbf{S}}}_{x}\left|{\mathit{\lambda}}_{k}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}\right.\u232a$, and its original off resonance efficiency.
Performing this procedure for all forbidden transitions, the modified
${\widehat{\mathbf{W}}}_{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}$ matrix contains only elements corresponding
to the allowed transitions (*i*−*j*)_{a}:

$$\begin{array}{}\text{(9)}& \begin{array}{rl}{\widehat{\mathbf{W}}}_{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}& =\sum _{\begin{array}{c}\mathrm{6}\phantom{\rule{0.25em}{0ex}}\text{allowed}\\ (i-j{)}_{\mathrm{a}}\end{array}}{\widehat{\mathbf{W}}}_{(ij{)}_{\mathrm{a}}}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}};\\ {\widehat{\mathbf{W}}}_{(ij{)}_{\mathrm{a}}}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}& ={\widehat{\mathbf{w}}}_{ij}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}+{a}_{\mathrm{N}}^{\mathrm{SE}}\sum _{(ik{)}_{\mathrm{N}};(kj{)}_{\mathrm{N}}}{\left\{{\widehat{\mathbf{w}}}_{(ik{)}_{\mathrm{N}}}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}+{\widehat{\mathbf{w}}}_{(kj{)}_{\mathrm{N}}}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}\right\}}_{(ij{)}_{\mathrm{a}}}\\ & +{a}_{\mathrm{DQ}-\mathrm{N}}^{\mathrm{SE}}\sum _{(ik{)}_{\mathrm{DQ}-\mathrm{N}};\phantom{\rule{0.125em}{0ex}}(kj{)}_{\mathrm{DQ}-\mathrm{N}}}{\left\{{\widehat{\mathbf{w}}}_{(ik{)}_{\mathrm{DQ}-\mathrm{N}}}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}+{\widehat{\mathbf{w}}}_{(kj{)}_{\mathrm{DQ}-\mathrm{N}}}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}\right\}}_{(ij{)}_{\mathrm{a}}}\\ & +{a}_{\mathrm{H}}^{\mathrm{SE}}\sum _{(ik{)}_{\mathrm{H}};(kj{)}_{\mathrm{H}}}\left\{{\left\{{\widehat{\mathbf{w}}}_{(ik{)}_{\mathrm{H}}}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}+{\widehat{\mathbf{w}}}_{(kj{)}_{\mathrm{H}}}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}\right\}}_{(ij{)}_{\mathrm{a}}}\right.\\ & \left.+{a}_{\mathrm{HN}}^{\mathrm{SE}}\sum _{(kl{)}_{\mathrm{HN}};\phantom{\rule{0.125em}{0ex}}(lk{)}_{\mathrm{HN}}}{\left\{{\widehat{\mathbf{w}}}_{(kl{)}_{\mathrm{HN}}}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}+{\widehat{\mathbf{w}}}_{(lk{)}_{\mathrm{HN}}}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}\right\}}_{(ij{)}_{\mathrm{a}}}\right\}.\end{array}\end{array}$$

Here the sums over *k* and *l* of $(ik{)}_{K},\phantom{\rule{0.125em}{0ex}}(kj{)}_{K},\phantom{\rule{0.125em}{0ex}}(kl{)}_{K{K}^{\prime}},\phantom{\rule{0.125em}{0ex}}(lk{)}_{K{K}^{\prime}}$ are restricted to the homonuclear and
heteronuclear forbidden transitions only. After this modification it
becomes possible to write for each allowed transition (*i*−*j*)_{a} a 2×2
rate equation for the populations ${p}_{i}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}\left(i\right)$ and
${p}_{j}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}\left(t\right)$ with a rate matrix $(-{\widehat{\mathbf{r}}}_{(ij{)}_{\mathrm{a}}}+{\widehat{\mathbf{W}}}_{(ij{)}_{\mathrm{a}}})$.

The actual relaxation pathways in the spin system are influenced by all the
elements of ${\widehat{\mathbf{R}}}_{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}$ and as a result, an irradiation
on one allowed transition can have a small effect on the populations of
another allowed
transition (Kaminker
et al., 2014). Our modification caused this effect to vanish in the
simulations. To reintroduce it we added to each ${\widehat{\mathbf{W}}}_{(ij{)}_{\mathrm{a}}}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}$ the MW rate matrices of the other transitions
${\widehat{\mathbf{W}}}_{(kl{)}_{\mathrm{a}}}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}$, while introducing an
additional small fitting parameter *a*_{a−a}:

$$\begin{array}{}\text{(10)}& {\widehat{\mathbf{W}}}_{(ij{)}_{\mathrm{a}}}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}={\widehat{\mathbf{W}}}_{(ij{)}_{\mathrm{a}}}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}+{a}_{a-a}\sum _{\begin{array}{c}(kl{)}_{\mathrm{a}}\\ k,\phantom{\rule{0.125em}{0ex}}l\ne i,\phantom{\rule{0.125em}{0ex}}j\end{array}}{\left\{{\widehat{\mathbf{W}}}_{(kl{)}_{\mathrm{a}}}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}\right\}}_{(ij{)}_{\mathrm{a}}}.\end{array}$$

Choosing values for all fitting parameters and inserting values for *T*_{1e} and *T*_{2mw}, the populations of the allowed transitions corresponding
to (*θ*,*φ*)_{det} at the end of a MW pump period *t*_{MW} at frequency *ν*_{MW} can now be obtained using Eq. (6). The EPR signal
E_{det}(*ν*_{det}, *t*_{MW}) at *ν*_{det} can then be
calculated by taking the Hamiltonian diagonalization into account and by
solving Eq. (6) with the modified MW rate matrices for each set of angles
(*φ*,*θ*). Adding all $({p}_{{i}_{\mathrm{a}}}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}-{p}_{{j}_{\mathrm{a}}}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}})\left({t}_{\mathrm{MW}}\right)$ values belonging to (*θ*,*φ*)_{det} and normalizing their sum *S*_{det}(*ν*_{MW}, *t*_{MW}) to
the sum ${S}_{\mathrm{det}}^{\mathrm{ref}}\left({t}_{\mathrm{MW}}\right)$ of all $({p}_{{i}_{\mathrm{a}}}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}}-{p}_{{j}_{\mathrm{a}}}^{\mathit{\theta},\phantom{\rule{0.125em}{0ex}}\mathit{\phi}})\left({t}_{\mathrm{MW}}\right)$ belonging to (*θ*,*φ*)_{det}, obtained by again solving Eq. (6) but this time for a *ν*_{MW} value far removed from the frequency range of all allowed and
forbidden transitions, gave the following:

$$\begin{array}{}\text{(11)}& {E}_{\mathrm{det}}({\mathit{\nu}}_{\mathrm{MW}},\phantom{\rule{0.125em}{0ex}}{t}_{\mathrm{MW}})={S}_{\mathrm{det}}({\mathit{\nu}}_{\mathrm{MW}},\phantom{\rule{0.125em}{0ex}}{t}_{\mathrm{MW}})/{S}_{\mathrm{det}}^{\mathrm{ref}}\left({t}_{\mathrm{MW}}\right).\end{array}$$

Plotting *E*_{det}(*ν*_{MW}, *t*_{MW}) as a function of *ν*_{MW}, after line smoothing over 5 MHz, results in a ELDOR spectrum at *ν*_{det}. (see Fig. 2).

To simulate the ELDOR spectra of the 10 and 20 mM samples we used the eSD
model (Hovav et al., 2015b). This computational model
divides the EPR spectrum into frequency bins and calculates the electron
polarizations *P*_{b}(*t*_{MW}) of each bin at frequency *ν*_{b}. It
consists of a set of coupled rate equations for these polarizations with
rate constants describing the effects of spin-lattice relaxation, eSD
polarization exchange and MW irradiation. To take the SE into account the MW
rate constants of each *P*_{b}(*t*_{MW}) are extended by effective SE
terms (Hovav et al., 2015b;
Kundu et al., 2018b; Wang et al., 2018):

$$\begin{array}{}\text{(12)}& \begin{array}{rl}{w}_{\mathrm{MW}}^{\mathrm{b}}& ={\displaystyle \frac{{\mathit{\omega}}_{\mathrm{1}}^{\mathrm{2}}{T}_{\mathrm{2}\mathrm{mw}}}{\mathrm{1}+\mathrm{4}{\mathit{\pi}}^{\mathrm{2}}({\mathit{\nu}}_{\mathrm{b}}-{\mathit{\nu}}_{\mathrm{MW}}{)}^{\mathrm{2}}{T}_{\mathrm{2}\mathrm{mw}}^{\mathrm{2}}}}\\ & +\sum _{K=\mathrm{H},\phantom{\rule{0.125em}{0ex}}\mathrm{N},\phantom{\rule{0.125em}{0ex}}\mathrm{H}-\mathrm{N}}{\displaystyle \frac{({A}_{K}^{\mathrm{SE}}{\mathit{\omega}}_{\mathrm{1}}{)}^{\mathrm{2}}{T}_{\mathrm{2}\mathrm{mw}}}{\mathrm{1}+\mathrm{4}{\mathit{\pi}}^{\mathrm{2}}({\mathit{\nu}}_{\mathrm{b}}\pm {\mathit{\nu}}_{K}-{\mathit{\nu}}_{\mathrm{MW}}){T}_{\mathrm{2}\mathrm{mw}}^{\mathrm{2}}}}\end{array}.\end{array}$$

Here *ν*_{K} are the ^{1}H and ^{14}N nuclear frequencies and
${A}_{\mathrm{H}}^{\mathrm{SE}}$, ${A}_{\mathrm{N}}^{\mathrm{SE}}$ and ${A}_{\mathrm{H}-\mathrm{N}}^{\mathrm{SE}}$ are fitting parameters used
to scale the MW power on the forbidden transition and they just affect the
SE peak intensities of the ELDOR peaks and not their positions. The eSD
exchange rate constants between the polarizations in bin b and bin b^{′} are
defined by the exchange rate coefficients

$$\begin{array}{}\text{(13)}& {r}_{\mathrm{b},{\mathrm{b}}^{\prime}}^{\mathrm{eSD}}={\displaystyle \frac{{\mathrm{\Lambda}}^{\mathrm{eSD}}}{\mathrm{4}{\mathit{\pi}}^{\mathrm{2}}({\mathit{\nu}}_{\mathrm{b}}-{\mathit{\nu}}_{{\mathrm{b}}^{\prime}}{)}^{\mathrm{2}}}},\end{array}$$

where the parameter Λ^{eSD} determines the timescale of the
spectral diffusion process. After solving the polarization rate equations
for an irradiation frequency *ν*_{MW} the polarization *P*_{det}(*ν*_{MW}) at the detection frequency *ν*_{det} is obtained and divided
by its Boltzman equilibrium value ${P}_{\mathrm{det}}^{\mathrm{eq}}$ to obtain the ELDOR
signal

$$\begin{array}{}\text{(14)}& E({\mathit{\nu}}_{\mathrm{MW}},{\mathit{\nu}}_{\mathrm{det}},\phantom{\rule{0.125em}{0ex}}{t}_{\mathrm{MW}})={\displaystyle \frac{{P}_{\mathrm{det}}\left({\mathit{\nu}}_{\mathrm{MW}}\right)}{{P}_{\mathrm{det}}^{\mathrm{eq}}}}.\end{array}$$

4 Results and discussion

Back to toptop
Experimental ELDOR spectra of the 0.5 mM TEMPOL were obtained by recording
EPR echo intensities as a function of *ν*_{MW} for fixed *ν*_{det} and *t*_{MW} values, using the experimental parameters summarized in the Sect. 2. The results $\mathrm{E}({\mathit{\nu}}_{\mathrm{MW}};\phantom{\rule{0.125em}{0ex}}{\mathit{\nu}}_{\mathrm{det}},\phantom{\rule{0.125em}{0ex}}{t}_{\mathrm{MW}})$ were analyzed using the procedure described in Sect. 3.
From the many ELDOR spectra measured in this way, we show in Fig. 3 (black
traces) only three, each one with a different detection frequency *ν*_{det} within the EPR spectrum. The dips in the ELDOR spectra, also
referred to as EDNMR spectra, appear at the frequencies of the allowed and
forbidden transitions, dictated by the ^{1}H and ^{14}N Larmor
frequencies *ν*_{H} and *ν*_{N} and their hyperfine interactions
$({A}_{zz}^{\mathrm{H}},\phantom{\rule{0.125em}{0ex}}{A}_{\mathrm{H}}^{\pm})$ for ^{1}H and $({A}_{zz}^{\mathrm{N}},\phantom{\rule{0.125em}{0ex}}{A}_{\mathrm{N}}^{\pm})$ along with the quadrupole interaction for ^{14}N (Aliabadi
et al., 2015; Cox et al., 2013, 2017; Kaminker et al., 2014; Nalepa et al.,
2014; Ramirez Cohen et al., 2017; Rapatskiy et al., 2012). At W-band
frequencies (∼95 GHz) the ^{1}H frequencies are around 144 MHz and the ^{14}N frequencies are in the range ∼20–70 MHz, as reported earlier in EDNMR
experiments (Florent
et al., 2011; Kaminker et al., 2014; Nalepa et al., 2014; Wili and Jeschke,
2018). Thus we expect in addition to the homonuclear forbidden transition
signals additional signals around −144, 0 and +144 MHz each with a
possible spread of −70–+70 MHz, due to the heteronuclear forbidden
transitions.

Figure 3b shows the ELDOR spectrum for *ν*_{det}=94.55 GHz,
where this frequency falls in the *g*_{z} region of the EPR spectrum (Fig. 3a), which is characterized by its “single-crystal-like” features. As a
result the ^{14}N signals are only slightly powder broadened and well
resolved (Florent
et al., 2011; Kaminker et al., 2014). At this detection frequency the
contributions to the echo signal originate only from the two low-frequency
allowed transitions (red in the $\mathrm{\Delta}{\mathit{\nu}}_{\mathrm{det}}=-\mathrm{250}$ MHz
stick diagram), split by the ^{1}H hyperfine interaction, of the
crystallites belonging to the “single crystal”. The MW excitation is not
selective enough to resolve the protons splitting. In Table S1 in the Supplement the
frequency assignments of the lines in the ELDOR spectra are correlated to
the (*i*−*j*)_{a} and (*i*−*j*)_{f} transitions in Fig. 2, together with the
color coding in the stick spectrum shown in Fig. 3b. The assignments of the
other four allowed transitions are also tabulated, together with their
^{1}H and ^{14}N homonuclear forbidden transitions and the
^{1}H–^{14}N heteronuclear forbidden transitions. In the ELDOR spectra
the two ^{1}H transitions (in blue) and the four ^{14}N transitions (in
green) are clearly present. The ^{1}H–^{14}N transitions (in purple) are
also detected. The additional spectral features must originate from the four
non-directly detected allowed transitions with their forbidden transitions.
Stick spectra of these allowed transitions and their ^{1}H forbidden
transitions are also added in Fig. 3b, and it is interesting to see that
some of these lines appear in the experimental ELDOR
spectrum (marked by arrows in Fig. 3b). The appearance of signals
corresponding to the non-directly excited allowed transition has been
reported
earlier (Kaminker
et al., 2014) and was attributed to the combination of off-resonance and
relaxation effects. In Fig. 3c the experimental ELDOR spectrum at *ν*_{det}=94.8 GHz (*g*_{y}) is plotted and a schematic stick
spectrum is added on the top. All possible allowed transitions contribute to
this spectrum and the spectral features are broadened and even hard to
distinguish. The stick spectrum represents only one typical contribution to
the observed powder spectrum. The same is true for the spectrum in Fig. 3d
at *ν*_{det}=94.9 GHz (*g*_{x}).

To simulate the experimental ELDOR spectra we needed to measure the
*T*_{1e} values. These were measured at several frequency positions within
the EPR spectrum: 20.8 ms at *ν*_{det}=94.6 GHz,
13.8 ms at *ν*_{det}=94.8 GHz and 15.8 ms at
*ν*_{det}=94.9 GHz, with the highest value obtained for the *g*_{z} region. In the
simulations we used the average value of *T*_{1e}=16.7 ms.

The best-fit simulated spectra that resemble the three experimental ELDOR
spectra in Fig. 3 are shown in red. To achieve these spectra we used the
following parameters: *T*_{2mw}=100 µs, *t*_{MW}=100 ms and the SE fitting parameters ${a}_{\mathrm{H}}^{\mathrm{SE}}={\mathrm{10}}^{\mathrm{3}}$,
${a}_{\mathrm{N}}^{\mathrm{SE}}=\mathrm{0.5}$, ${a}_{\mathrm{H}-\mathrm{N}}^{\mathrm{SE}}={\mathrm{10}}^{\mathrm{3}}$ and ${a}_{a-a}^{\mathrm{SE}}=\mathrm{0.5}\times {\mathrm{10}}^{-\mathrm{3}}$. These parameters were determined via manual fitting of the
intensities of the different lines in the spectrum in Fig. 3b. The same
parameters were used for the simulated spectra in Fig. 3c and d. The fact
that the SE parameter of the ^{1}H forbidden transitions is large seems
to be connected with the many protons involved in the SE process in the
sample. In addition to the abovementioned forbidden transitions, we also added
^{14}N double quantum effect in the simulations by introducing a SE
parameter of ${a}_{\mathrm{DQ}-\mathrm{N}}^{\mathrm{SE}}=\mathrm{5}$. Comparing the simulated and experimental
spectra we observe all expected forbidden transitions and some lines
originating from the non-observed allowed transitions and their forbidden
transitions. The double quantum lines expected around Δ*ν*=200 MHz are not clearly resolved. The calculated spectra in Fig. 3c and d resemble
the experimental spectra, although the relative intensities of the lines do
not agree so well.

A contour plot of the experimental 2D-ELDOR spectrum of the 0.5 mM sample is
shown in Fig. 4a. The positions of the lines corresponding to the allowed
transitions appear at the intense central diagonal of the spectrum. The
signals associated with the *{**e*−^{14}N*}*
forbidden transitions are close to the central diagonal and clearly reveal
the anisotropic character of the hyperfine interaction. Namely, the
strongest shifts of the line positions, with respect to the allowed line
positions, are about 40 MHz in the *g*_{z} region of the EPR spectrum and
reduce to 20 MHz in the *g*_{x, y} regime. The signals associated with the
*{**e*−^{1}H*}* forbidden transitions are the
intense lines parallel to the diagonal and are surrounded by the signals
coming from the $\mathit{\{}e-{}^{\mathrm{1}}\mathrm{H}--{}^{\mathrm{14}}\mathrm{N}\mathit{\}}$ forbidden
transitions. Figure 4b shows the simulated 2D-ELDOR contour plot, which
reproduces most of the features observed in the experimental contours. Some
discrepancies can be observed in the intensities of the forbidden transition
lines which can be attributed to the simplifications of the model.

The 2D ELDOR spectrum for a 10 mM TEMPOL solution, presented in Fig. 5,
displays the main features of the ^{1}H SE solid effect lines,
which run parallel to the diagonal. ^{14}N and combination lines are
detectable but they are not as nicely resolved as in the 0.5 mM sample. In
addition, broad features that correspond to the depolarization of the
electron spins owing to the eSD process are evident. To consider both SE and
eSD effects we simulated the ELDOR spectra using the eSD model, including
the influence of ^{14}N and ^{1}H SE by incorporating the SE features as
described in Sect. 3 and Eq. (12). We also measured *T*_{1e}
along the EPR spectrum and the results are given in Fig. 6. *T*_{1e}
displays anisotropic behavior; namely it depends on the position within
the EPR spectrum with the largest variations observed in the *g*_{z} region
(similar to our earlier observation for the 0.5 mM solution). Similar
*T*_{1e} variations was also reported by Weber et al. (2017). To
include the experimental *T*_{1e} values in the simulations, we
assigned to each group of five consecutive bins, each one with a width of 2 MHz, the value of *T*_{1e} measured at the position in the EPR spectrum
that correspond to those bins. Examples of experimental and simulated ELDOR
spectra for three positions of the detection frequency in the EPR spectrum
are shown in Fig. 7.

Initially the spectra were simulated using the eSD model considering only
the ^{1}H SE effect (blue traces in Fig. 7), and the best fit gave an eSD
parameter of Λ^{eSD}=60 µs^{−3}. A better fit was obtained
when taking into account ^{14}N SE, including the ^{14}N–^{1}H
combinations (green traces). This addition broadened the ELDOR lines,
resulting in a better match with the experimental result, with the same
Λ^{eSD} value. Nevertheless, when *ν*_{det} reached the
*g*_{z} region of the EPR spectrum (Fig. 7a, $\mathrm{\Delta}{\mathit{\upsilon}}_{\mathrm{det}}=-\mathrm{100}$ MHz), the fit was not as good as in *g*_{x} (Fig. 7b, Δ*υ*_{det}=0 MHz) and *g*_{y} (Fig. 7c,
Δ*υ*_{det}=100 MHz). This implies that Λ^{eSD} might be anisotropic, which is unexpected. At this point we
attribute this “apparent” anisotropy to the oversimplified ad hoc
inclusion of the SE mechanism into the eSD model which does not fully
account for the anisotropy of the ^{14}N hyperfine interaction.

To examine the degree of the influence of the ^{14}N SE on the electron
depolarization at higher radical concentrations, where the ELDOR spectrum is
shaped primarily by the eSD process, we also tested the 20 mM sample and
used the eSD model to simulate the ELDOR lineshape recorded with *ν*_{det} set to the center of the EPR spectrum, as shown in Fig. 8.
Because of the high electron spin concentration, the eSD causes a large degree of depolarization of the EPR spectrum, which translates to extensive broadening
of the ELDOR spectrum.

Figure 8 shows in red the experimental ELDOR spectrum, where although the
lineshape of this spectrum is determined by the eSD process, we can still
see small signals coming from the ^{14}N SE. Simulation including both the
^{1}H and ^{14}N SE with Λ^{eSD}=400 µs^{−3} gave a
good agreement with the experimental spectrum. In contrast, setting Λ^{eSD}=400 µs^{−3} and taking into account only the
contributions of the ^{1}H SE did not result in a good fit. This shows
that even at relatively high radical concentrations, the effect of the
depolarization due to the ^{14}N SE can still be significant and if not
included can introduce inaccuracies in the eSD parameters and thus also in
the DNP spectra, derived from the depolarized EPR lineshapes that are
constructed using these parameters. Earlier measurements showed that for a 20 mM
TEMPOL concentration, ELDOR spectra measured at the *g*_{y} and *g*_{z}
position gave the same quality fit with the same Λ^{eSD}, implying
that at this concentration the relative contribution of the ^{14}N SE
mechanism is small and can be accounted for by the simple model presented in
this work.

5 Conclusions

Back to toptop
In this work we use ELDOR measurements to determine the contributions of the
^{14}N SE to the depolarization gradient within the EPR spectrum of
TEMPOL during long MW irradiation, as commonly used in DNP measurements. For
a low concentration (0.5 mM) TEMPOL sample, where the SE dominates and eSD
is negligible, we have successfully reproduced all the SE-related
depolarization signals, including those involving combinations of
^{1}H–^{14}N associated forbidden EPR transitions and those arising from
off-resonance effects. Subsequently, we used the eSD
model (Hovav et al., 2015c) to simulate
ELDOR spectra of 10 and 20 mM TEMPOL samples with ad hoc addition of
electron depolarization due to the ^{14}N SE based on the frequencies
determined from the 0.5 mM sample. We observed that simulations including
the ^{14}N SE improved the fit with experimental ELDOR spectra for the 10 mM sample. However, we noticed that at the *g*_{z} region of the EPR spectrum
the fit was not as good, indicating that the model is does not account
sufficiently well for the large ^{14}N SE contributions in this
region. For the 20 mM concentration the model works well and the ^{14}N SE
effect is still significant and can affect the best fitted value of Λ^{eSD}. We conclude that including ^{14}N SE in the eSD model is
essential for obtaining reliable fitting at high radical concentrations.

Data availability

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Data availability.

The data in the figures are available at https://doi.org/10.5281/zenodo.3757682 (Ramirez-Cohen et al., 2020.). The software used for simulations is available upon request.

Supplement

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Supplement.

The supplement related to this article is available online at: https://doi.org/10.5194/mr-1-45-2020-supplement.

Author contributions

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Author contributions.

MRC did the measurements and part of the simulations. SV and AF developed the theory and the simulations. DG and SV conceived and supervised the work. The paper was written by all authors.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

This work was made possible in part by the historic generosity of the Harold Perlman Family (Daniella Goldfarb). Daniella Goldfarb holds the Erich Klieger Professorial Chair in Chemical Physics.

Financial support

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Financial support.

This research has been supported by the US–Israel Binational Science Foundation (grant no. 2014149).

Review statement

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Review statement.

This paper was edited by Konstantin Ivanov and reviewed by two anonymous referees.

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Short summary

DNP is a method used to enhance NMR signals by transferring polarization from radicals to nuclear spins using microwave irradiation. The EPR spectrum of the radical during the irradiation is determined by a process called spectral diffusion and affects the DNP efficiency. This process can be measured by electron–electron double resonance. We explored experimentally and theoretically the contribution of the hyperfine coupling of the ^{14}N nuclei in the nitroxide radical to the spectral diffusion.

DNP is a method used to enhance NMR signals by transferring polarization from radicals to...

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