# Room Temperature DNP of Diamond Powder Using Frequency Modulation

Daphna Shimon  Institute of Chemistry, The Hebrew University of Jerusalem,
Edmond J. Safra, Givat Ram, Jerusalem, Israel<EMAIL_ADDRESS>Kelly A. Cantwell  Department of Physics and Astronomy, Dartmouth College,
Hanover, NH 03755, U.S.A.  Linta Joseph  Department of Physics and Astronomy,
Dartmouth College, Hanover, NH 03755, U.S.A.  Chandrasekhar Ramanathan
Department of Physics and Astronomy, Dartmouth College, Hanover, NH 03755,
U.S.A.

###### Abstract

Dynamic nuclear polarization (DNP) is a method of enhancing NMR signals via
the transfer of polarization from electron spins to nuclear spins using on-
resonance microwave (MW) irradiation. In most cases, monochromatic continuous-
wave (MCW) MW irradiation is used. Recently, several groups have shown that
the use of frequency modulation of the MW irradiation can result in an
additional increase in DNP enhancement above that obtained with MCW. The
effect of frequency modulation on the solid effect (SE) and the cross effect
(CE) has previously been studied using the stable organic radical 4-hydroxy
TEMPO (TEMPOL) at temperatures under 20 K. Here, in addition to the SE and CE,
we discuss the effect of frequency modulation on the Overhauser effect (OE)
and the truncated CE (tCE) in the room-temperature 13C-DNP of diamond powders.
We recently showed that diamond powders can exhibit multiple DNP mechanisms
simultaneously due to the heterogeneity of P1 (substitutional nitrogen)
environments within diamond crystallites. We explore the enhancement obtained
via the two most important parameters for frequency modulation 1) Modulation
frequency, $f_{m}$ (how fast the modulation frequency is varied) and 2)
Modulation amplitude, $\Delta\omega$ (the magnitude of the change in microwave
frequency) influence the enhancement obtained via each mechanism. Frequency
modulation during DNP not only allows us to improve DNP enhancement, but also
gives us a way to control which DNP mechanism is most active. By choosing the
appropriate modulation parameters, we can selectively enhance some mechanisms
while simultaneously suppressing others.

##  1 Introduction

Dynamic nuclear polarization (DNP) enhances NMR signals by transferring spin
polarization from electrons to nuclei by on-resonance microwave (MW)
irradiation of the electron spins 1, 2, 3. In most cases, monochromatic
continuous-wave (MCW) MW irradiation is used and the single frequency of the
MW irradiation is chosen to maximize the DNP enhancement 1, 2, 3. A DNP
spectrum (the DNP enhancement measured as a function of MW frequency) is
typically recorded to determine the optimal MW frequency for the MCW. The
shape of the DNP spectrum depends on the nature of the electron spin, the
nucleus that is being enhanced, and the DNP mechanism involved 4, 5, 6, 7.

Recently, several groups have demonstrated that employing MW irradiation
schemes that are more complex than simple MCW irradiation can result in an
additional increase in the DNP enhancement (above that obtained with MCW). 8,
9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 One method that works well at
high magnetic fields ($>$ 3.3 T) and for the case of relatively limited MW
power is frequency modulation, or trains of broadband chirp pulses. Under
static (non-spinning) conditions, this irradiation can improve the DNP
performance several fold compared to MCW irradiation 8, 9, 10, 11, 12, 13, 14,
15, 16, 17, 18, 19, 20. The increase in the enhancement is usually attributed
to the fact that a larger number of electrons can be excited with broadband MW
irradiation 13, 19. However, the size of the additional enhancement depends on
many experimental factors, such as the modulation frequency $f_{m}$ (i.e. the
rate of change of the MW frequency), the modulation amplitude $\Delta\omega$
(the bandwidth of the frequency change), the nuclei involved, the radical type
and other physical parameters 13, 19, 20. Several groups have shown that for a
given sample, the improvement of the DNP enhancement is maximal (optimal) at
certain combinations of $\Delta\omega$ and $f_{m}$, with other values often
resulting in sub-optimal DNP enhancements 13, 20.

The effect of frequency modulation on the solid effect (SE) and the cross
effect (CE) has previously been studied using the stable organic radical
TEMPOL at temperatures under 20 K 21. We showed that the maximum of the DNP
spectrum for the SE and the CE depends both on the modulation amplitude and on
electron spectral diffusion (eSD), which spreads the electron depolarization
throughout the EPR line. We also showed that modulation results in a
transition from SE to CE enhancement in those model samples.21 To come to this
conclusion, it was necessary to update the original DNP model for large spin
systems (known as the indirect CE model) by Hovav et al. to include the
effects of frequency modulation 13.

Here, in addition to the SE and CE, we discuss the effect of frequency
modulation on the Overhauser effect (OE) and the truncated CE (tCE) in room-
temperature 13C-DNP experiments on diamond powders. We recently showed that
diamond powders can exhibit multiple DNP mechanisms simultaneously due to the
heterogeneity of P1 environments within diamond crystallites, while also
enabling a greater than 100-fold enhancement of the 13C signal 22.

The P1 center is a substitutional nitrogen defect (spin-1/2) in diamond that
has been used to hyperpolarize nuclei at both low and high magnetic fields.
Previously, P1 centers have been shown to produce a significant increase in
the nuclear spin polarization during DNP at cryogenic temperatures 23, 24, 25,
26, 27, 28. Bretschneider et al. also reported a room temperature 13C DNP
enhancement of 130 at 9.4 T under 8 kHz MAS conditions with 10 W of microwave
power at 263 GHz 27.

Since all four mechanisms are observed in the same diamond powder sample, we
are able to examine and compare how the modulation frequency and modulation
amplitude influence the DNP enhancement obtained via each of these mechanisms.
Frequency modulation during DNP allows us to improve DNP enhancement. More
importantly, it gives us a way to control which DNP mechanism is most active,
by selectively enhancing some mechanisms and suppressing other mechanisms,
depending on the modulation parameters.

##  2 Results and Discussion

We performed frequency modulated DNP experiments on a diamond powder sample at
3.34 T. Details about the sample and experimental methods can be found in the
Methods Section at the end of the paper. This powder sample exhibits four DNP
mechanisms, the SE, CE, OE and tCE, which we identified in a previous
publication by fitting a non-modulated DNP spectrum 22.

In Figure 1, we compare the DNP spectrum acquired with a constant modulation
amplitude ($\Delta\omega=50$ MHz), at various modulation frequencies, $f_{m}$,
ranging from 200 Hz to 200 kHz. Each spectrum is acquired by changing the
central irradiation frequency while applying the same modulation parameters
and measuring the DNP enhancement across the central frequency range of
interest for the diamond powder sample. The figure shows that the shape of the
DNP spectrum measured at large modulation frequencies looks identical to that
without modulation. As the modulation frequency is lowered, we see that the
sharper features of the DNP spectrum begin to broaden and blur, and that this
broadening increases as the $f_{m}$ is decreased. The enhancement increased by
around 20% at $f_{m}=$3 kHz and then decreased back to its original value as
$f_{m}$ is decreased further.

Figure 1: 13C-DNP spectra, showing normalized DNP enhancement as a function of
the MW irradiation frequency at different modulation frequencies, $f_{m}$, as
indicated in the figure. The same modulation amplitude, $\Delta\omega=50$ MHz,
and build-up time (120 s) were used in all experiments.

Next, we kept the modulation frequency constant (at 200 Hz) and observed how
the modulation amplitude affected the DNP spectrum. In Figure 2 we compare the
DNP spectra acquired with $\Delta\omega=50$ MHz (also shown in Figure 1) and
$\Delta\omega=150$ MHz. We also fit these DNP spectra using the same model we
used in our previous publication on diamond DNP 22, which is described in the
Methods section. Increasing $\Delta\omega$ results in a large change to the
DNP spectrum. Fitting the data allows us to identify two key effects: (i)
blurring of the features due to the modulation, and (ii) changes in the
relative intensities of each DNP mechanism due to the modulation. We see that
with a larger $\Delta\omega$, the contribution of the SE decreases, while the
contribution of the CE and tCE increases, which match our previous results 21.

Figure 2: Fit of experimental constant-frequency DNP spectrum (black circles)
of the diamond powder sample using a sum (green line) of the OE (red line), SE
(magenta line), the truncated CE (blue line), and the CE (black line) for
(left) $\Delta\omega=50$ MHz and (right) $\Delta\omega=50$ MHz. The DNP build-
up time was 120 s.

In order to better understand these observations we must try and separate the
various DNP mechanisms. This is not an easy task, as the mechanisms overlap in
many cases. However, we have identified specific frequencies that mark the
maximum DNP enhancement from each mechanism, and contain the least
‘contamination’ from other mechanisms. These frequencies were chosen based on
the fit of the non-modulated DNP spectrum 22. The frequencies are:
$\omega$(OE) = 93.814 GHz, $\omega$(SE) = 93.777 GHz, $\omega$(tCE) = 93.725
GHz, and $\omega$(CE) = 93.702 GHz. Note, however, that the overlap between
mechanisms grows larger as the frequency modulation amplitude is increased.

When thinking about how frequency modulation affects different DNP mechanisms
it is helpful to consider two different cases: 1) small spin systems, where
there are only one or two electron spin packets; and 2) large spin systems,
where there are many electron spin packets. Small spin systems featuring the
SE and the CE under modulation have previously been described,13 and here we
expand the theory to include the OE and tCE in small spin systems. In large
spin systems, the addition of frequency modulation will increase the number of
electron spins participating in the DNP process, because, instead of
irradiating a single spin-packet, we are able to irradiate many more electron
spins packets 13, 29, 21. By irradiating more electrons, we see an increase in
the DNP enhancement, an important reason why frequency modulation is employed
13, 29, 21. This is, in principle, true for all DNP mechanisms. However, when
considering large spin systems, we must also consider the effect of electron-
electron spectral diffusion (eSD) on the DNP process. eSD can spread electron
depolarization throughout the EPR line, and thus also affects the DNP
enhancement, by affecting the amount of electron polarization available for
DNP 21. Additionally, because each DNP mechanism behaves differently under
frequency modulation, each mechanism must be considered separately, and there
is often a balance between involving more electrons to increase the
enhancement and saturating additional electron or electron-nuclear transitions
that decrease the enhancement.

###  2.1 Solid Effect and Cross Effect

####  2.1.1 Experimental Results

We begin by exploring the effect of changing the modulation amplitude and
frequency on the DNP enhancement at the SE and the CE frequencies. Figures
3(a) and (b) show the non-modulated DNP spectrum and the locations of the
center frequencies (solid blue line) of the MW irradiation
($\omega_{\text{MW}}$) used for the SE and CE respectively. The blue dashed
lines in these figures mark the frequency range that corresponds to a
modulation amplitude of $\Delta\omega=\omega_{C}$, the 13C Larmor frequency,
on either side of $\omega_{\text{MW}}$.

Figure 3(c) and (d) show the effect of changing the modulation amplitude and
frequency at the SE and CE frequencies respectively. The colors represent the
ratio of the DNP enhancement with frequency modulation to that measured
without frequency modulation. The black dashed lines in panels (c) and (d)
correspond to the two T1e values (110 $\mu$s and 1.3 ms) measured at room
temperature at 2.5 GHz 22. The shorter timescale (right line) likely
represents reorganization of the electron polarization as a result of eSD, or
the recovery time of clusters of P1 centers or other paramagnetic defects. The
long timescale (left line) is likely to be the recovery time of isolated P1
centers. We did not adjust these values despite the fact that the DNP was
performed at a higher field. The dashed white line marks position where the
modulation amplitude is equal to $\omega_{C}$.

The key features that we observe are: (i) The SE is very sensitive to the
combination of low modulation frequencies fm and large modulation amplitudes
$\Delta\omega$; (ii) Frequency modulation results only in a very small
improvement of the SE enhancement (from $\sim 26$ to $\sim 27$), and in most
cases results in a decrease in the enhancement; and (iii) The CE shows a range
of optimal conditions where there is an increase of about $\sim 20\%$ to the
enhancement (from $\sim 40$ to $\sim 51$).

Figure 3: Non-modulated experimental DNP spectra showing the position of the
central irradiation frequency (black symbols), $\omega_{\text{MW}}$, for the
a) SE (solid black line) and b) CE (solid magenta line) mechanisms. The dashed
lines represent the edges of the MW irradiation when
$\Delta\omega=\omega_{C}$. DNP enhancement at the c) SE and d) CE frequencies
as a function of the modulation frequency, $f_{m}$, and the modulation
amplitude, $\Delta\omega$. The dashed white line marks the position where
$\Delta\omega=\omega_{C}$. The black dashed lines represent the two timescales
of T1e values of a bi-exponential fit of an inversion-recovery experiment
using pulsed EPR measured at 2.5 GHz, previously published.22

####  2.1.2 Small spin system simulations

The effect of frequency modulation on the SE- and CE-DNP enhancement in small
spin systems was originally described by Hovav et al.13. Figure 4 shows the
effect of $\Delta\omega$ for the SE and the CE, at a fixed modulation
frequency $f_{m}=50$ kHz, for the diamond sample under study and the key
features are summarized in Table 1.

In our simulations, we simplify the diamond sample, and consider a single
electron or a pair of electrons hyperfine coupled to a 13C nucleus. The
addition of a 14N nucleus will just result in three EPR manifolds, each of
which will behave independently, as we previously described 22. For an e–C
system, the Hamiltonian of the spin system in the MW rotating frame is

$H_{0}=\Delta\omega_{e}S_{z}-\omega_{C}I_{z}^{C}+A_{x}^{C}S_{z}I_{x}^{C}+A_{y}^{C}S_{z}I_{y}^{C}$

where $S$ and $I^{C}$ are the electron and carbon spin operators,
respectively. $\Delta\omega_{e}=\omega_{e}-\omega_{\text{MW}}$ is the electron
off-resonance (compared to the rotating frequency $\omega_{\text{MW}}$),
$\omega_{C}$ is the 13C Larmor frequency. $A_{x}^{C}$ and $A_{y}^{C}$ are the
pseudo-secular terms of the dipolar hyperfine interaction. Note that no e-C
secular hyperfine, $A_{z}^{n}=$0 MHz, was added to the simulation. Here,
$A_{x}^{C}=A_{y}^{C}$. For an e2–e1–C system the Hamiltonian is

$H_{0}=\Delta\omega_{e1}S_{z1}+\Delta\omega_{e2}S_{z2}+D_{ee}S_{z1}S_{z2}-\omega_{C}I_{z}^{C}+A_{x}^{C}S_{z1}I_{x}^{C}+A_{y}^{C}S_{z1}I_{y}^{C}$

where $D_{ee}$ is the electron-electron dipolar interaction.

|  Enhancement conditions |  Enhancement when $\Delta\omega<\omega_{C}$ |  Enhancement when $\Delta\omega\geq\omega_{C}$
---|---|---|---
SE
e–C system |  Positive or negative enhancement when irradiating at $\omega_{e}\pm\omega_{C}$, respectively

(see Figure 4(a) red) |  unchanged

(see Figure 4(b) red) |  sharp decline when $\omega_{e}-\Delta\omega<\omega_{\text{MW}}<\omega_{e}+\Delta\omega$ and a decrease of enhancement outside this range due to a reduction in the irradiation efficiency

(see Figure 4(c-d) red)
CE
ea-eb-C
system

$\omega_{e}^{a}<\omega_{e}^{b}$ $\omega_{e}^{b}-\omega_{e}^{a}\approx\omega_{C}$ |  Electron polarization difference results in nuclear polarization enhancement according to $P_{C}=\frac{P_{b}-P_{a}}{1-P_{b}P_{a}}$

(see Figure 4(a) blue) |  slight increase due to the ability to irradiate on both sides of the EPR line that is split by the electron-electron dipolar coupling

(see Figure 4(b) blue) |  sharp decline as a result of partial/full saturation of both CE-electrons and a drop in the CE enhancement even when only a single electron is irradiated due to a reduction in the irradiation efficiency

(see Figure 4(c-d) blue)
Table 1: Description of the enhancement for the SE and the CE under different
modulation amplitude conditions. $\omega_{e}$ and $\omega_{C}$ are the
electron and 13C Larmor frequencies.

In the original work by Hovav et al.13, they showed that it is best if the
modulation rate (the inverse of the frequency of modulation, $1/f_{m}$) is set
to be smaller than the electron relaxation time, T1e so that the electron spin
does not lose its polarization between irradiation periods.13 This is true for
both the SE and the CE.

The modulation amplitude, $\Delta\omega$, depends on the sample under study,
and specifically on which nucleus we would like to enhance. For the SE, the MW
irradiation $\omega_{\text{MW}}$ must be placed on the double quantum (DQ:
$\omega_{e}-\omega_{C}$) or zero quantum (ZQ: $\omega_{e}+\omega_{C}$)
transition in order to achieve positive or negative DNP enhancement,
respectively (Figure 4a red). At modulation amplitudes of
$\Delta\omega<\omega_{C}$ the DNP enhancement is unchanged (Figure 4b red).
However, once $\Delta\omega\geq\omega_{C}$, there is a sharp decline in the
DNP enhancement of the SE in the range of
$\omega_{e}-\Delta\omega<\omega_{\text{MW}}<\omega_{e}+\Delta\omega$ (Figure
4c-d red). This is a result of the irradiation saturating the electron single
quantum (SQ) electron transition (at $\omega_{e}$) in addition to the e–C DQ
or ZQ transition. When this occurs, there is a decrease in the electron
polarization that can be transferred to the nucleus. In addition, the SE
enhancement outside of the frequency range given above drops due to a
reduction in the irradiation efficiency at each MW frequency, with the
increase in the modulation amplitude $\Delta\omega$.

For the CE, we must have two electrons at the CE condition (two electrons
separated by the nuclear Larmor frequency $\omega_{C}$). The polarization that
is transferred to the nucleus is equal to the polarization difference between
the two electrons. When irradiating on the SQ transition of the low frequency
electron, positive enhancement is achieved, and on the SQ transition of the
high frequency electron negative enhancement is achieved (Figure 4a blue). At
low modulation amplitudes of $\Delta\omega<\omega_{C}$ there is an increase in
the CE enhancement due to the ability to irradiate on both sides of the EPR
line that is split by the electron-electron dipolar coupling (Figure 4b blue).
However, once $\Delta\omega\geq\omega_{C}$, there is a sharp decline in the
DNP enhancement of the CE as a result of the irradiation saturating both CE-
electrons, resulting in no polarization difference available to be transferred
to the nucleus (Figure 4c-d blue). In addition, as $\Delta\omega$ is
increased, there is a drop in the CE enhancement even when only a single
electron is irradiated due to a reduction in the irradiation efficiency. The
effective irradiation intensity depends on the strength of the state mixing,
and is of the order of $2^{-1/2}\omega_{1}$ 13 where $\omega_{1}$ corresponds
to the strength of applied MW field. Therefore, in order to observe the effect
of decreasing MW irradiation efficiency under the CE conditions, Hovav et al.
performed the simulations at low MW field strengths of 30 kHz 13. The CE
simulations shown in Figure 4 were performed at $\omega_{1}/2\pi=50$ kHz.

Figure 4: Simulated DNP spectra for an e–C system exhibiting the SE (red) and
an e–e–C system exhibiting the CE (blue). Nuclear polarization (as a function
of MW irradiation frequency, referenced to $\omega_{ref}=\omega_{e}=94$ GHz.
All polarizations are normalized to the electron polarization at thermal
equilibrium. a) no modulation, b) $\Delta\omega=0.5\omega_{C}$, c)
$\Delta\omega=\omega_{C}$, and d) $\Delta\omega=3\omega_{C}$. The dashed
orange lines mark the position of the e–C DQ and ZQ transitions. The
simulation parameters: For SE (e–C): $\omega_{e}=$94 GHz, $\omega_{C}=$36 MHz,
$A_{z}^{n}=$0 MHz, $A_{x}^{n}=A_{y}^{n}=$0.5 MHz, $\omega_{1}=$0.5 MHz,
T${}_{1e}=$1.3 ms, T${}_{1C}=$10 s, T${}_{2e}=$10 $\mu$s, $T_{2C}=$100 $\mu$s,
$t_{\text{MW}}=$100 s and $f_{m}=$50 kHz. No cross relaxation was added to the
system. For CE (e–e–C): $\omega_{e1}=$94 GHz, $\omega_{e2}=$94.035876 GHz
$\omega_{C}=$36 MHz, $D_{ee}=3$ MHz, $A_{z1}^{C}=$0 MHz,
$A_{x1}^{C}=A_{y1}^{C}=$0.5 MHz, $A_{z2}^{C}=$0 MHz, $A_{x2}^{C}=A_{y2}^{C}=$0
MHz, $\omega_{1}=$0.05 MHz, T${}_{1e1}=$T${}_{1e2}=$1.3 ms, T${}_{1C}=$10 s,
T${}_{2e1}=$T${}_{2e2}=$10 $\mu$s, $T_{2C}=$100 $\mu$s, $t_{\text{MW}}=$100 s,
$f_{m}=$50 kHz and T=$273$ K. No cross relaxation was added to the system.

####  2.1.3 Large spin system simulations

The small spin simulation results do not align closely with the experimental
data. In order to describe a real sample, we must next consider how modulation
affects the SE and the CE in large spin systems, when there are many electrons
that are possible DNP sources. As noted earlier it is essential to consider
the effects of eSD on the DNP process. In Figure 5 we consider the cases of
(i) negligible eSD ($\Lambda=1\mu$s-3 – circles); (ii) intermediate eSD
($\Lambda=25\mu$s-3 – crosses); and (iii) large eSD ($\Lambda=200\mu$s-3 –
diamonds). In order to show the effect of the modulation amplitude without
considering the addition of more electron spin packets as DNP sources, we plot
the normalized DNP enhancement from a single electron spin packet in Figure 5.
This gives an indication to how a broad EPR line affects the DNP achieved from
a single electron spin packet within the wider EPR line.

Negligible electron spectral diffusion
For the SE (red circles), at low modulation amplitudes (below the optimum
$\Delta\omega$), when irradiating at a given MW frequency
$\omega_{\text{MW}}$, we are irradiating on the DQ and ZQ transitions of
electrons that have a SQ transition at $\omega_{\text{MW}}\pm\omega_{C}$.
However, we are also partially saturating the SQ transition of the electrons
at $\omega_{\text{MW}}$. Partial saturation of these electrons results in a
low DNP enhancement. As we increase the modulation amplitude $\Delta\omega$,
the saturation of these SQ transitions becomes less effective, and we observe
an increase in DNP enhancement. When we reach $\Delta\omega\approx\omega_{C}$,
where we begin saturating both the DQ/ZQ transition and the SQ transition, and
this causes a sharp decrease in the DNP enhancement. This results in an
optimum $\Delta\omega\approx\omega_{C}$ (or slightly larger). The CE (blue
circles) behaves in a similar manner to the SE. Also here, the optimum
modulation amplitude is $\Delta\omega\approx\omega_{C}$.

Intermediate electron spectral diffusion
When eSD is non-negligible, the electron depolarization spreads throughout the
EPR line even when $\Delta\omega<\omega_{C}$, and saturates additional
electron transitions that we are not directly irradiating.21 As a result, we
observe a lower optimum $\Delta\omega$ frequency, for the SE and the CE.21 In
the case of the CE, there is an increase because the eSD spreads the
polarization throughout the line, so that even when we do not directly
irradiate a CE electron we still observe its saturation.

Large electron spectral diffusion
At very large values of eSD, modulation does not improve the DNP contribution
of a single electron to the SE or CE, and just results in a decrease in the
enhancement.

Figure 5: The SE (red) and CE (blue) enhancement of a single electron bin as a
function of the $\Delta\omega$ for various electron spectral diffusion values
($\Lambda^{eSD}$ -shown in the figure). The simulation parameters: The
simulations were done using a model Gaussian EPR line with a full width at
half max of 40 MHz, centered around $\omega_{e}=94$ GHz. $\omega_{C}=36$ MHz,
T=$273$ K, T${}_{\text{MW}}=100s$, $\omega_{1}=0.5$ MHz,
$\bar{A^{\pm}}=0.5MHz$, T${}_{1e}=1.3$ ms, T${}_{1C}=10$ s and
T${}_{2e}=10\mu$s.

Going back to Figure 3, we are now in the position where we can discuss the
effect of modulation on the SE and the CE. We see that SE is very sensitive to
the combination of low modulation frequencies fm and large modulation
amplitudes $\Delta\omega$. In this regime, the modulation rate ($1/f_{m}$) is
likely on the order of, or larger than T1e making the effective MW power very
low, and making it difficult to saturate the DNP transitions. Thus we observe
a decrease in the enhancement. This is true in particular at large
$\Delta\omega$ values, where the time spent irradiating each MW frequency is
very short. It makes sense that the SE would be more sensitive than the CE, as
the effective irradiation strength for the SE is already much weaker than for
the CE.3

In our sample, we show that the SE enhancement only slightly improves with the
application of modulation, and in most cases does not improve at all (Figure
3). This suggests that adding many electron spin packets to the DNP process is
able to make up for the decrease in the SE enhancement from a single electron
spin packet, such that the effects are balanced. Because we are observing the
SE from the central EPR line, which is quite narrow (width at base of
$\approx$ 20 MHz), increasing the modulation amplitude to values that are
larger than the width of the central EPR line does not improve the DNP, but
rather decreases it, especially at small fm values.

Moreover, the fact that we do not see a large increase in the SE enhancement
in Figure 3 also suggests that our sample has non-negligible eSD, and perhaps
even quite strong eSD. Otherwise, we would expect to find optimal modulation
parameters that would result in an increase in the SE enhancement. This was
observed in the TEMPOL sample recently. 21.

In the CE case, we do find optimal modulation parameters that increase the
enhancement. Therefore, we can conclude that the addition of more electron
spin-packets into the DNP process is strong enough to overcome the fact that
each spin packet is contributing less to the overall enhancement. This was
also observed in the TEMPOL sample recently. 21. Increasing the modulation
amplitude, and eSD likely effect the CE more than the SE because the CE is
resultant from a broader EPR line. When the line is broader, exciting the full
EPR line is difficult at the low powers we are using. Increasing the
modulation amplitude and eSD can help make up for this. We must also remember
that at large modulation amplitudes many of the DNP mechanisms overlap, making
it difficult to draw hard conclusions about their behaviour under frequency
modulation (see Figure 2).

Overall, the results from the SE and the CE suggest that by carefully
selecting modulation parameters it is possible to change the relative ratios
of the various DNP mechanisms, in systems where many mechanisms are active.

###  2.2 Overhauser Effect and Truncated Cross Effect

####  2.2.1 Experimental Results

Next, we proceeded to explore the effect of frequency modulation at the OE and
the tCE frequencies. Once again we plot the change in the enhancement compared
to the non-modulated case as a function of the modulation amplitude and
frequency.

Figure 6: Non-modulated experimental DNP spectra showing the position of the
central irradiation frequency(black symbols), $\omega_{\text{MW}}$, for the a)
OE (solid red line) and b) tCE (solid blue line) mechanisms. The dashed blue
lines represent the edges of the MW irradiation when
$\Delta\omega=\omega_{C}$. DNP enhancement at the c) OE and d) tCE frequencies
as a function of the modulation frequency, $f_{m}$, and the modulation
amplitude, $\Delta\omega$. The dashed white line marks the position where
$\Delta\omega=\omega_{C}$. The black dashed lines represent the two timescales
of T1e values of a bi-exponential fit of an inversion-recovery experiment
using pulsed EPR measured at 2.5 GHz, previously published.22

Both mechanisms (but in particular the OE) are very sensitive to the
combination of low modulation frequencies fm and large modulation amplitudes
$\Delta\omega$ when enhancement is seen to drop sharply. Frequency modulation
results in a slight increase for the OE (from -25.0 to -27.2) and a larger
increase for the tCE (32.7 to 36.3) for large modulation frequencies and
amplitudes.

####  2.2.2 Small spin system simulations

In order to better explain these observations, we expand the work of Hovav et
al.13 and describe the effect of frequency modulation on the OE and the tCE in
small spin systems. The results are described in Figure 7 and summarized in
Table 2.

Overhauser Effect
The simulated DNP spectrum with and without frequency modulation are shown in
Figure 7, for several $\Delta\omega$ values – (a) no frequency modulation, (b)
modulation with $\Delta\omega<\omega_{C}$, (c) $\Delta\omega=\omega_{C}$, and
(d) $\Delta\omega>\omega_{C}$. These spectra were calculated for an e–C spin
system, and show a prominent negative OE-DNP mechanism when irradiating
directly on the electron SQ transition. The e–C DQ and ZQ transitions are
marked in the figure with dashed orange lines. At irradiation strengths of
$\omega_{1}/(2\pi)=500$ kHz, the enhancement does not change as a function of
the modulation amplitude (not shown), because of the large effective
irradiation on the electron SQ transition. At lower irradiation strengths such
as $\omega_{1}/(2\pi)=50$ kHz the enhancement decreases as the modulation
amplitude increases (solid magenta lines), due to a reduction in the
irradiation efficiency at each MW frequency, with the increase in the
modulation amplitude $\Delta\omega$. In Figure 7(a) a very small SE-DNP is
seen when irradiating on the e–C DQ/ZQ transitions. It is small due to the
weak effective irradiation, and is not visible in the other panels because the
modulation further weakens it.

Next, we consider the effect of the frequency of the modulation on the DNP
enhancement of the OE. As can be seen from Figure 7(e), for the OE case, as in
the SE and CE case, in order to effectively saturate the electron transitions
it is best if $f_{m}$T${}_{1e}>1$ where T1e is marked by the dashed green
lines. Here, as in the Hovav et al. paper, 13 we will not discuss the case
where $1/(f_{m})<$T2e.

Truncated Cross Effect
The simulated DNP spectrum with and without frequency modulation are shown in
Figure 7, for several $\Delta\omega$ values. These spectra were calculated for
an e1-e2-n spin system, and show tCE-DNP when irradiating on e1. The values of
$\Delta\omega$ are given in the figure caption. They represent a) no frequency
modulation, b) modulation with $\Delta\omega<\omega_{C}$, c)
$\Delta\omega=\omega_{C}$, and d) $\Delta\omega>\omega_{C}$ conditions. The
e–C DQ and ZQ transitions are marked in the figure with dashed orange lines.
At irradiation strengths of $\omega_{1}/(2\pi)=500$ kHz, the enhancement does
not change as a function of the modulation amplitude (not shown), because of
the large effective irradiation on the electron SQ transition. At lower
irradiation strengths such as $\omega_{1}/(2\pi)=50$ kHz, the enhancement
decreases as the modulation amplitude increases (solid magenta lines), due to
a reduction in the irradiation efficiency at each MW frequency, with the
increase in the modulation amplitude $\Delta\omega$.

Finally, we consider the effect of the frequency of the modulation on the DNP
enhancement of the tCE. As can be seen from Figure 7e, for the tCE case, as in
the SE and CE case, in order to effectively saturate the electron transitions
we need $f_{m}$T${}_{1e1}>1$, where T1e1 is the relaxation time of the of the
slower relaxing electron. T1e2 must be at least 100 times faster than T1e1 for
the tCE to be active (for slower T1e2 values, we are in the range of CE). In
addition, working at much larger modulation rates should be advantageous for
the tCE. We will not discuss the case where $1/(f_{m})<$T2e.

|  Enhancement conditions |  Enhancement when $\Delta\omega<\omega_{C}$ |  Enhancement when $\Delta\omega\geq\omega_{C}$
---|---|---|---
OE
e–C system |  Positive or negative enhancement when irradiating at $\omega(e)$, depending on the relative ZQ/DQ cross relaxation rates

(see Figure 7(a) black) |  slight decrease due to a reduction in the irradiation efficiency

(see Figure 7(b) black) |  continued slight decrease due to a continued reduction in the irradiation efficiency

(see Figure 7(c-d) black)
tCE
ea–eb–C
system

$\omega_{e}^{a}<\omega_{e}^{b}$ and $\omega_{e}^{b}-\omega_{e}^{a}\approx\omega_{C}$ |  Electron polarization difference results in nuclear polarization enhancement according to $P_{C}=\frac{P_{b}-P_{a}}{1-P_{b}P_{a}}$

(see Figure 7(a) magenta) |  slight decrease due to a reduction in the irradiation efficiency

(see Figure 7(b) magenta) |  continued slight decrease due to a reduction in the irradiation efficiency

(see Figure 7(c-d) magenta)
Table 2: Description of the enhancement for the OE and the tCE under different
modulation amplitude conditions Figure 7: Simulated DNP spectra for an e–C
system exhibiting the OE (black) and an e–e–C system exhibiting the tCE
(magenta). Nuclear polarization (as a function of MW irradiation frequency,
referenced to $\omega_{ref}=\omega_{e}=94$ GHz. All polarizations are
normalized to the electron polarization at thermal equilibrium. a) no
modulation, b) $\Delta\omega=0.5\omega_{C}$, c) $\Delta\omega=\omega_{C}$, and
d) $\Delta\omega=3\omega_{C}$. The dashed orange lines mark the position of
the e–C DQ and ZQ transitions. e) Polarization as a function of $1/f_{m}$ for
$\Delta\omega=\omega_{C}$. The green dashed lines represent 1/T1e and 1/T1n.
The simulation parameters: For OE (e–C): $\omega_{e}=$94 GHz, $\omega_{C}=$36
MHz, $A_{z}^{n}=$0 MHz, $A_{x}^{C}=A_{y}^{C}=$0.5 MHz, $\omega_{1}=$0.05 MHz,
T${}_{1e}=$1.3 ms, T${}_{1C}=$10 s, T${}_{2e}=$10 $\mu$s and $T_{2C}=$100
$\mu$s. $t_{\text{MW}}=$100 s, $f_{m}=$50 kHz and T=$273$ K. Cross-relaxation
of T${}_{1eZQ}=1000$T ${}_{1e}=$1.3 s and T${}_{1eDQ}=10$T${}_{1e}=$13 ms were
used. For tCE (e–e–C): $\omega_{e1}=$94 GHz, $\omega_{e2}=$94.035876 GHz
$\omega_{C}=$36 MHz, $D_{ee}=3$ MHz, $A_{z1}^{C}=$0 MHz,
$A_{x1}^{C}=A_{y1}^{C}=$0.5 MHz, $A_{z}2^{C}=$0 MHz, $A_{x}2^{C}=A_{y}2^{C}=$0
MHz, $\omega_{1}=$0.05 MHz, T${}_{1e1}=$1.3 ms, T${}_{1e2}=$0.013 ms,
T${}_{1C}=$10 s, T${}_{2e1}=$T${}_{2e2}=$10 $\mu$s and $T_{2C}=$100 $\mu$s.
$t_{\text{MW}}=$100 s, $f_{m}=$50 kHz and T=$273$ K. No cross relaxation was
added to the system.

####  2.2.3 Large spin system simulations

In Figure 8 we plot the effect of the modulation amplitude on the DNP
enhancement from a single electron packet within a wider EPR line. We again
consider the cases of (i) negligible eSD ($\Lambda=1\mu$s-3 – circles); (ii)
intermediate eSD ($\Lambda=25\mu$s-3 – crosses); and (iii) large eSD
($\Lambda=200\mu$s-3 – diamonds).

Negligible electron spectral diffusion
For the OE (black circles) the enhancement is not sensitive to the modulation
amplitude. This is because the effective irradiation strength on the SQ
electron transition is large even under large modulation amplitudes,
$\Delta\omega$. Moreover, because we are irradiating on the SQ electron
transition, the irradiation of the DQ/ZQ transitions does not affect the DNP
enhancement, exactly as described above for small spin systems.

For the tCE (magenta circles), the enhancement is mediated via the
polarization difference between the fast-relaxing pool of electrons (which is
always fully polarized) and the slowly-relaxing pool of electrons. At low
modulation amplitudes, the polarization difference is quite large because we
are able to saturate the slow relaxing electrons. As the modulation amplitude
increases, we see a decrease in the enhancement because the efficiency of the
saturation of the slow electrons decreases as we spend less time irradiating
at each frequency. However, at even larger modulation amplitudes we again see
an increase in the enhancement. This comes from a second increase in the
saturation of the slow electrons, as the modulation amplitudes become large
enough to cover the whole width of the EPR line. When this happens, even
irradiation that is not directly on the electron bin we are observing still
results in saturation of that bin.

Intermediate electron spectral diffusion
When the eSD is non-negligible, the electron depolarization spreads throughout
the EPR line, and saturates additional electron transitions that we are not
directly irradiating even when $\Delta\omega<\omega_{C}$.21 As a result, in
the case of the tCE, there is an increase because the eSD spreads the
polarization throughout the line, so that even when we do not directly
irradiate on the slow relaxing electron pool, we still observe its saturation
(see Figure 8 – magenta). The OE remains insensitive (all OE curves in Figure
8 in black overlap)

Large electron spectral diffusion
At very large values of eSD we still observe a slight increase in the tCE
enhancement, until the slow relaxing EPR line is fully saturated (see Figure 8
– magenta diamonds).

Figure 8: The OE (black) and tCE (magneta) enhancement of a single electron
bin as a function of the $\Delta\omega$ for various electron spectral
diffusion values (shown in the figure). The simulation parameters: the
simulations were done using a model Gaussian EPR line with a full width at
half max of 40 MHz, centered around $\omega_{e}=94$ GHz. $\omega_{n}=36$ MHz,
T$=273$ K, T${}_{\text{MW}}=100s$, $\omega_{1}=0.5$ MHz,
$\bar{A^{\pm}}=0.5$MHz, T${}_{1e}=1.3$ ms, T${}_{1n}=10$ s, T${}_{2e}=10\mu$s
and $\Lambda^{eSD}$ as written in the figure

Going back to Figure 6, we are now in the position where we can discuss the
effect of modulation on the OE and the tCE. Looking at Figure 6a-b more
closely, we notice that the OE is very sensitive to the combination of low
modulation frequencies fm and large modulation amplitudes $\Delta\omega$,
similar to the SE. For the OE, the effective irradiation strength is much
stronger because it relies on SQ irradiation. However, in this regime, the
modulation rate ($1/f_{m}$) is likely of the order of, or larger than T1e
making it difficult to saturate the DNP transitions, and thus resulting in a
decrease in the enhancement. This is true in particular at large
$\Delta\omega$ values, where the time spent irradiating each MW frequency is
very short.

For both the OE and the tCE, we get a slight increase in the enhancement (of
the order of 10 %) at large modulation amplitudes and large modulation
frequencies. For the OE, the enhancement is a result of the narrow central EPR
line, and therefore we would expect that increasing the modulation amplitude
beyond the width of the central EPR line does not improve the DNP, but rather
does not change it, or even decreases it, especially at small fm values. Note,
also, that the DNP mechanisms overlap at large modulation amplitudes many of,
making it difficult to draw hard conclusions about their behaviour under
frequency modulation (see Figure 2).

For the tCE, the simulations of a single bin suggest that we should see an
increase in the tCE enhancement. The simulations show that the increase in the
tCE enhancement becomes smaller as the eSD becomes larger. This suggests that
the slow electron pool in our system has a relatively large eSD connecting the
electrons that make up the two outer EPR lines. This matches with our
conclusions from the SE and CE enhancement described above. Thus, it seems
like we are in the regime of large eSD, combined with a decrease in the
effective irradiation power due to the modulation that decreases the tCE. We
must also remember that the tCE overlaps with the CE and SE at large
modulation amplitudes, making it difficult to separate the mechanisms.

###  2.3 Conclusions

Microwave frequency-modulation has recently been shown to improve the nuclear
spin enhancements obtained during DNP. While the effects of frequency-
modulation on the SE and CE mechanisms have been studied previously, we have
extended this analysis to the Overhauser effect (OE) and the truncated CE
(tCE). We compare theoretical simulations of both small and large spin systems
with experimental results from the the room-temperature 13C-DNP of diamond
powders. We recently showed that DNP via P1 defects in diamond can exhibit all
of these mechanisms simultaneously due to the heterogeneity of P1
(substitutional nitrogen) environments within diamond crystallites 22 \- a
property which we utilize here in this study.

We have shown that the overall resultant DNP enhancement observed in a sample
during frequency modulation is a non-trivial combination of five factors,
which we identified in this work: (1) The larger the modulation amplitude, the
more electron spins are involved in the enhancement process; (2) The
enhancement from a single electron bin as a function of modulation amplitude
varies for different DNP mechanisms; (3) The frequency of the modulation also
affects the efficiency of the DNP; (4) Which nucleus we are enhancing changes
how the modulation affects each DNP mechanism; and (5) The shape of the EPR
line and especially its width compared to $\omega_{n}$ changes how the
modulation affects each DNP mechanism. By carefully picking modulation
parameters, it is possible to suppress some mechanisms and enhance others,
effectively changing how DNP proceeds in a very complex sample.

##  3 Materials and Methods

###  3.1 Experiments

####  3.1.1 Sample preparation:

Diamond powder used in these experiments were donated by Element6. The type Ib
diamond is made by HPHT synthesis (High Pressure, High Temperature). The
powder was 15-25 $\mu$m in size, and with a P1 concentration of 110-130 ppm.

####  3.1.2 DNP spectrometer:

The experiments were performed on a homebuilt DNP spectrometer, at a field of
3.4 T, corresponding to an electron Larmor frequency of 94 GHz, a 1H Larmor
frequency of 142 MHz and a 13C Larmor frequency of 36 MHz. All experiments
were performed at room temperature. The homebuilt DNP spectrometer and the MW
bridge were described in detail recently.22

The modulation amplitude reported in this work refers to half the amplitude of
modulation, $\Delta\omega$, such that the full bandwidth of irradiation is:
$\Delta\omega_{m}=\omega_{MW}\pm\Delta\omega$. To achieve frequency
modulation, a ramp-up sawtooth modulation was applied to the VCO, around a
central value of 2.5 V. In order to convert from mV to MHz, a calibration was
performed, using a spectrum analyzer to record the output of the voltage
controlled oscillator (VCO) as a function of the voltage applied.

####  3.1.3 13C NMR and DNP experiments:

The following settings were used for all the experiments, unless otherwise
noted:

The DNP enhanced NMR signal was recorded using 20 or 24 scans, with a
90-acquire pulse sequence, using a $\pi$/2 pulse of 10 $\mu$s in length. Eight
step phase cycling was used in all cases (no difference was found between 20
scans and 24 scans despite the phase cycle being incomplete). The thermal NMR
signal was recorded using 100 scans. All experiments began with a train of 100
30 ms saturation pulses separated by 20 $\mu$s. We did not record the DNP
enhancement at steady state. The length of the MW irradiation or the thermal
recovery time after saturation was set to 120 s, as a compromise between the
length of the experiment and the signal intensity.

Modulation grids were recorded by systematically changing the modulation
amplitude and modulation frequency, while keeping all other experimental
parameters unchanged. The 13C relaxation time T1n and the enhancement buildup
time Tbu measurements were described in a previous publication.22

####  3.1.4 Data processing:

Data processing was performed in MATLAB using custom scripts. 3 points of
left-shift were used, in all cases. The data was then baseline corrected,
phase corrected and 200 Hz exponential line broadening was applied. We report
the integrated intensity of the 13C resonance. For DNP enhancement
calculations we divide the integrated intensity of the MW-on signal with the
MW-off signal. No enhancement is equal to 1.

###  3.2 Simulations

####  3.2.1 Small spin systems

Quantum mechanical simulations of a e–C and e2–e1–C spin systems which include
relaxation effects and frequency modulation were performed according to the
methods described by Hovav et al.13 In order to simulate the OE, we added
unequal cross-relaxation terms on the ZQ and DQ transitions, T1ZQ and T1DQ,
respectively. The addition of the cross relaxation to the relaxation
superoperator was achieved by assuming fluctuations of S+I- \+ S-I+ for the ZQ
interaction and S+I+ \+ S-I- for the DQ interaction, as described by Hovav et
al.13 S± and I± are the raising and lowering operators for the electron and
the nucleus, respectively. In order to simulate the tCE, we repeated the CE
simulations, but with unequal T1e values for the two electrons.

The simulations include: The electron and nuclear Larmor frequencies,
$\Delta\omega_{e}$ and $\omega_{C}$, respectively. Secular and pseudo secular
hyperfine interactions of Az, and A${}^{x}=$Ay, respectively. The electron and
nuclear spin-lattice and spin-spin relaxation times, T1e, T1C, T2e and, T2C,
respectively. Electron-nuclear cross-relaxation on the DQ and ZE transition,
T${}_{\text{1ZQ}}$ and T${}_{\text{1DQ}}$, respectively. MW irradiation at a
frequency of $\omega_{\text{MW}}$, for a time $t_{\text{MW}}$, with a step
size of $\Delta$t and of a strength $\omega_{1}$. Sawtooth modulation (ramp
up) is employed, with a modulation amplitude of $\pm\Delta\omega$ surrounding
the central irradiation frequency $\omega_{\text{MW}}$, a modulation frequency
of $f_{m}$. No electron spectral diffusion is included in these simulations.

####  3.2.2 Large spin systems

In order to simulate a large spin system, we rely on a previously published
model by Hovav et al. 13 which we updated to include modulation effects. 21
The model describes the changes in the polarization of the electrons in the
sample using rate equations of their polarization. The polarization of a
single hyperfine-coupled nucleus is also included, to mimic the average of all
nuclei in the sample. The EPR line is split into bins of index j, with a
frequency $\omega_{j}$, an amplitude $f_{j}$, and of width
$\omega_{\text{bin}}$. The effect of frequency modulation with an amplitude
$\Delta\omega$ surrounding a central frequency $\omega_{\text{excite}}$ is
also taken into consideration. The model assumes modulation with a frequency
that is larger then $1/T_{1e}$, but does not consider an explicit $f_{m}$
value. The polarization of each bin,
$P_{e}(\omega_{j};\omega_{\text{excite}};\Delta\omega)$ is then described by
coupled rate equations. We assume that the frequency modulation affects N
bins, according to $N=(\Delta\omega)/\omega_{\text{bin}}$, and we scale the
rates for the single quantum (SQ) and double/zero quantum (DQ/ZQ) irradiation
according to the inverse of the amount of time the MW irradiation spends in
each bin, $\omega_{\text{bin}}/\Delta\omega$. For every bin, we include SQ
irradiation, DQ/ZQ irradiation, $1/T_{1e}$ relaxation, $1/T_{1n}$. We also
include electron spectral diffusion (eSD) which spreads polarization between
every pair of bins j and j’. The spectral diffusion constant is described by
$\lambda^{eSD}$, with units of $\mu s^{-3}$, and is a phenomenological
parameter. All the rates are included in a single rate matrix $R$.

The rate equations for the electron polarization across the EPR line (which
has the length of the number of bins N) and the polarization of a single
nucleus are solved by calculating:

$\displaystyle\begin{bmatrix}\overrightarrow{P_{e}(\Delta\omega;t_{\text{MW}})}\\\
P_{n}(\Delta\omega;t_{\text{MW}})\end{bmatrix}=R\begin{bmatrix}\overrightarrow{P_{e}(\Delta\omega;0)}\\\
P_{e}(\Delta\omega;0)\end{bmatrix}$ (1)

where the electron polarization across the EPR line,
$\overrightarrow{P_{e}(\Delta\omega;t_{\text{MW}})}$, is represented by a
vector of length N, which matches the number of bins in the EPR line, N. The
electron polarization and the shape of the EPR line are then used to simulate
the DNP spectra for the various DNP mechanisms. The nuclear polarization is
not used further, but is necessary to reproduce the effect of the hyperfine
interaction on the electron polarization during the MW irradiation. The
details of the SE and CE under frequency modulation were published
previously.13 This model in conjunction with frequency modulation was used to
analyze experimental data for model samples with nitroxide radicals 21. In
this work we expand the model to include the OE and the tCE.

##### The SE:

$S_{SE}(\omega_{\text{MW}})=\sum_{\omega_{\text{MW}}}\Bigg{(}\sum_{\Delta\omega_{\text{mod}}=-\Delta\omega/2}^{\Delta\omega/2}\big{(}g(\omega_{\text{MW}}-\omega_{n})P_{e}(\omega_{\text{MW}}-\omega_{n}+\Delta\omega_{\text{mod}},\omega_{\text{MW}})\\\
-g(\omega_{\text{MW}}+\omega_{n})P_{e}(\omega_{\text{MW}}+\omega_{n}+\Delta\omega_{\text{mod}},\omega_{\text{MW}})\big{)}\Bigg{)}$
(2)

Where $\omega_{\text{MW}}$ is the central MW irradiation frequency,
$\Delta\omega_{\text{mod}}$ is position of the irradiation around the central
irradiation frequency, and goes from $-\Delta\omega/2$ to $\Delta\omega/2$ in
steps that match the bin size, $\omega_{n}$ is the nuclear Larmor frequency,
$P_{e}(\omega_{x},\omega_{y})$ is the electron polarization at a position
$\omega_{x}$, when irradiating at a frequency $\omega_{y}$.13

##### The CE:

$\sum_{\omega_{\text{MW}}}\left(g(\omega_{\text{MW}})g(\omega_{\text{MW}}-\omega_{n})\frac{\left(P_{e}(\omega_{\text{MW}}-\omega_{n},\omega_{\text{MW}})-P_{e}(\omega_{\text{MW}},\omega_{\text{MW}})\right)}{\left(1-P_{e}(\omega_{\text{MW}}-\omega_{n},\omega_{\text{MW}})P_{e}(\omega_{\text{MW}},\omega_{\text{MW}})\right)}\right)$

##### The OE:

$S_{OE}(\omega_{\text{MW}})=\sum_{\omega_{\text{MW}}}\left(\sum_{\Delta\omega_{\text{mod}}=-\Delta\omega/2}^{\Delta\omega/2}g(\omega_{\text{MW}}+\Delta\omega_{\text{mod}})(P_{e}^{0}-P_{e}(\omega_{\text{MW}}+\Delta\omega_{\text{mod}},\omega_{\text{MW}}))\right)$

Where $P_{e}^{0}$ is the electron polarization at thermal equilibrium (i.e.,
fully polarized). Here, we assume that the polarization that is transferred to
the nucleus matches the amount of polarization that the electron loses during
the DNP, in order to simulate negative enhancement.

##### The tCE:

$\sum_{\omega_{\text{MW}}}\left(g(\omega_{\text{MW}})g(\omega_{\text{MW}}-\omega_{n})\frac{\left(P_{e}(\omega_{\text{MW}}-\omega_{n},\omega_{\text{MW}})-P_{e}^{0}\right)}{\left(1-P_{e}(\omega_{\text{MW}}-\omega_{n},\omega_{\text{MW}})P_{e}^{0}\right)}\right)$

Here, we assume that there are two electron pools, one at
$\omega_{\text{MW}}-\omega_{n}$ that is partially saturated, and another at
$\omega_{\text{MW}}$ that is fully polarized. In order to achieve that, we
always keep the polarization of the second electron equal to the thermal
electron polarization $P_{e}^{0}$. As a result we assume that there are two
identical EPR lines, one for a slow relaxing electron pool, and one for a fast
relaxing electron pool that cannot be saturated.

###  3.3 EasySpin Simulation of the EPR line

The EPR line was simulated via EasySpin, as described in our previous
publication. 22

###  3.4 Fitting the Modulation DNP Spectra

The DNP spectra with modulation were fitted using a crude method of
convoluting the EPR line with delta functions to form the basic shapes for the
SE, OE (same shape used for tCE) and CE DNP mechanisms, and changing their
amplitude to achieve the best agreement with the experimental spectrum. This
method was previously described in Banerjee et al. 30, for the SE and the CE,
and for the OE and tCE in our previous publication on diamond DNP.22

We added the effect of ramp-up (sawtooth) frequency modulation by convoluting
the shapes for the SE, CE, OE and tCE with square functions. The width of
these square functions matches the width of $\Delta\omega$ used in the
experiment.

##  4 Acknowledgements

The authors thank Element 6 for the donation of the powder samples used in
these experiments. This work was partially supported by funding from the
National Science Foundation under cooperative agreement under grants
OIA-1921199.

## References

  * Maly et al. 2008 Maly, T.; Debelouchina, G. T.; Bajaj, V. S.; Hu, K. N.; Joo, C. G.; Mak-Jurkauskas, M. L.; Sirigiri, J. R.; Van Der Wel, P. C.; Herzfeld, J.; Temkin, R. J.; Griffin, R. G. Dynamic nuclear polarization at high magnetic fields. _Journal of Chemical Physics_ 2008, _128_ , 1–19
  * Lilly Thankamony et al. 2017 Lilly Thankamony, A. S. et al. Dynamic nuclear polarization for sensitivity enhancement in modern solid-state NMR. _Progress in Nuclear Magnetic Resonance Spectroscopy_ 2017, _102-103_ , 120–195
  * Kundu et al. 2019 Kundu, K.; Mentink-Vigier, F.; Feintuch, A.; Vega, S. DNP mechanisms. _eMagRes_ 2019, _8_ , 295–338
  * Hovav et al. 2015 Hovav, Y.; Shimon, D.; Kaminker, I.; Feintuch, A.; Goldfarb, D.; Vega, S.; Material, E. S.; Chemistry, P.; Physics, C.; Societies, O.; Po, E.; Polarization, D. N. Effects of the electron polarization on uynamic Nuclear Polarization in Solids. _Physical Chemistry Chemical Physics_ 2015, _17_ , 6053–6065
  * Kaminker et al. 2016 Kaminker, I.; Shimon, D.; Hovav, Y.; Feintuch, A.; Vega, S. Heteronuclear DNP of protons and deuterons with TEMPOL. _Physical Chemistry Chemical Physics_ 2016, _18_ , 11017–11041
  * Shimon et al. 2012 Shimon, D.; Hovav, Y.; Feintuch, A.; Goldfarb, D.; Vega, S. Dynamic nuclear polarization in the solid state: A transition between the cross effect and the solid effect. _Physical Chemistry Chemical Physics_ 2012, _14_ , 5729–5743
  * Leavesley et al. 2017 Leavesley, A.; Shimon, D.; Siaw, T. A.; Feintuch, A.; Goldfarb, D.; Vega, S.; Kaminker, I.; Han, S. Effect of electron spectral diffusion on static dynamic nuclear polarization at 7 Tesla. _Physical Chemistry Chemical Physics_ 2017, _19_ , 3596–3605
  * Kisselev 1995 Kisselev, Y. F. The modulation effect on the dynamic polarization of nuclear spins. 1995, _356_ , 99–101
  * Thurber et al. 2010 Thurber, K. R.; Yau, W. M.; Tycko, R. Low-temperature dynamic nuclear polarization at 9.4 T with a 30 mW microwave source. _Journal of Magnetic Resonance_ 2010, _204_ , 303–313
  * Equbal et al. 2019 Equbal, A.; Tagami, K.; Han, S. Pulse-Shaped Dynamic Nuclear Polarization under Magic-Angle Spinning. _Journal of Physical Chemistry Letters_ 2019, _10_ , 7781–7788
  * Judge et al. 2020 Judge, P. T.; Sesti, E. L.; Alaniva, N.; Saliba, E. P.; Price, L. E.; Gao, C.; Halbritter, T.; Th, S.; Kyei, G. B.; Barnes, A. B. Characterization of frequency-chirped dynamic nuclear polarization in rotating solids. _Journal of Magnetic Resonance_ 2020, _313_ , 106702
  * Cassidy et al. 2013 Cassidy, M. C.; Chan, H. R.; Ross, B. D.; Bhattacharya, P. K.; Marcus, C. M. In vivo magnetic resonance imaging of hyperpolarized silicon particles. _Nature Nanotechnology_ 2013, _8_ , 363–368
  * Hovav et al. 2014 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
  * Bornet et al. 2014 Bornet, A.; Milani, J.; Vuichoud, B.; Perez, A. J.; Bodenhausen, G.; Jannin, S. Microwave frequency modulation to enhance Dissolution Dynamic Nuclear Polarization. 2014, _602_ , 63–67
  * Idehara et al. 2015 Idehara, T.; Khutoryan, E. M.; Tatematsu, Y. High-Speed Frequency Modulation of a 460-GHz Gyrotron for Enhancement of 700-MHz DNP-NMR Spectroscopy. 2015, _36_ , 819–829
  * Yoon et al. 2016 Yoon, D.; Soundararajan, M.; Cuanillon, P.; Braunmueller, F.; Alberti, S.; Ansermet, J.-p. Dynamic nuclear polarization by frequency modulation of a tunable gyrotron of 260 GHz. _Journal of Magnetic Resonance_ 2016, _262_ , 62–67
  * Lipsø et al. 2017 Lipsø, K. W.; Bowen, S.; Rybalko, O.; Ardenkjær-Larsen, J. H. Large dose hyperpolarized water with dissolution-DNP at high magnetic field. _Journal of Magnetic Resonance_ 2017, _274_ , 65–72
  * Ajoy et al. 2018 Ajoy, A.; Liu, K.; Lv, X.; Nazaryan, R.; Wang, G.; Druga, E.; Reimer, J. A.; Suter, D.; Ramanathan, C.; Meriles, C. A.; Pines, A. Enhanced dynamic nuclear polarization via swept microwave frequency combs. _Proceedings of the National Academy of Sciences of the United States of America_ 2018, _115_ , 1–11
  * Kaminker and Han 2018 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 Chemical physical chemistry letters_ 2018, _9_ , 3110–3115
  * Guy et al. 2017 Guy, M. L.; Van Schooten, K. J.; Zhu, L.; Ramanathan, C. Chemisorption of Water on the Surface of Silicon Microparticles Measured by Dynamic Nuclear Polarization Enhanced NMR. _Journal of Physical Chemistry C_ 2017, _121_ , 2748–2754
  * Shimon and Kaminker 2022 Shimon, D.; Kaminker, I. A transition from solid effect to indirect cross effect with broadband microwave irradiation. _Physical Chemistry Chemical Physics_ 2022, _24_ , 7311–7322
  * Shimon et al. 2022 Shimon, D.; Cantwell, K. A.; Joseph, L.; Williams, E. Q.; Peng, Z.; Takahashi, S.; Ramanathan, C. Large Room Temperature Bulk DNP of 13C via P1 Centers in Diamond. 2022; https://arxiv.org/abs/2207.11961
  * Lock et al. 1992 Lock, H.; Maciel, G. E.; Johnson, C. E. Natural-abundance 13C dynamic nuclear polarization experiments on chemical vapor deposited diamond film. _Journal of materials research_ 1992, _7_ , 2791–2797
  * Reynhardt and High 1998 Reynhardt, E. C.; High, G. L. Dynamic nuclear polarization of diamond. I. Solid state and thermal mixing effects. _The Journal of chemical physics_ 1998, _109_ , 4090–4099
  * Reynhardt and High 2001 Reynhardt, E.; High, G. Nuclear magnetic resonance studies of diamond. _Progress in Nuclear Magnetic Resonance Spectroscopy_ 2001, _1_ , 37–81
  * Casabianca et al. 2011 Casabianca, L. B.; Shames, A. I.; Panich, A. M.; Shenderova, O.; Frydman, L. Factors affecting DNP NMR in polycrystalline diamond samples. _The Journal of Physical Chemistry C_ 2011, _115_ , 19041–19048
  * Bretschneider et al. 2016 Bretschneider, C. O.; Akbey, U.; Aussenac, F.; Olsen, G. L.; Feintuch, A.; Oschkinat, H.; Frydman, L. On The Potential of Dynamic Nuclear Polarization Enhanced Diamonds in Solid‐State and Dissolution 13C NMR Spectroscopy. _ChemPhysChem_ 2016, _17_ , 2691–2701
  * Kwiatkowski et al. 2018 Kwiatkowski, G.; Jähnig, F.; Steinhauser, J.; Wespi, P.; Ernst, M.; Kozerke, S. Direct hyperpolarization of micro-and nanodiamonds for bioimaging applications–considerations on particle size, functionalization and polarization loss. _Journal of Magnetic Resonance_ 2018, _286_ , 42–51
  * Shimon and Kaminker 2021 Shimon, D.; Kaminker, I. Electron spin effects in static DNP with broadband excitation. _eMagRes_ 2021, _9_
  * Banerjee et al. 2013 Banerjee, D.; Shimon, D.; Feintuch, A.; Vega, S.; Goldfarb, D. The interplay between the solid effect and the cross effect mechanisms in solid state 13C DNP at 95 GHz using trityl radicals. _Journal of Magnetic Resonance_ 2013, _230_ , 212–219