# Designing Broadband Pulsed Dynamic Nuclear Polarization Sequences in Static
Solids

Nino Wili<EMAIL_ADDRESS>Department of Chemistry and Applied
Biosciences, Laboratory of Physical Chemistry, ETH Zurich, Vladimir-Prelog-Weg
2, 8093 Zurich, Switzerland  Anders Bodholt Nielsen  Interdisciplinary
Nanoscience Center (iNANO) and Department of Chemistry, Aarhus University,
Gustav Wieds Vej 14, DK-8000 Aarhus C, Denmark  Laura Alicia Völker  Lukas
Schreder  Department of Chemistry and Applied Biosciences, Laboratory of
Physical Chemistry, ETH Zurich, Vladimir-Prelog-Weg 2, 8093 Zurich,
Switzerland  Niels Chr. Nielsen  Interdisciplinary Nanoscience Center (iNANO)
and Department of Chemistry, Aarhus University, Gustav Wieds Vej 14, DK-8000
Aarhus C, Denmark  Gunnar Jeschke  Department of Chemistry and Applied
Biosciences, Laboratory of Physical Chemistry, ETH Zurich, Vladimir-Prelog-Weg
2, 8093 Zurich, Switzerland  Kong Ooi Tan<EMAIL_ADDRESS>Laboratoire
des Biomolécules, LBM, Département de Chimie, École Normale Supérieure, PSL
University, Sorbonne Université, CNRS, 75005 Paris, France

###### Abstract

Dynamic nuclear polarization (DNP) is an NMR hyperpolarization technique that
mediates polarization transfer from highly polarized unpaired electrons to
NMR-active nuclei via microwave (mw) irradiation. The ability to generate
arbitrarily shaped mw pulses using arbitrary waveform generators opens up the
opportunity to remarkably improve the robustness and versatility of DNP, in
many ways resembling the early stages of pulsed NMR. We present here novel
design principles based on single-spin vector effective Hamiltonian theory to
develop new broadband DNP pulse sequences, namely an adiabatic XiX-DNP
experiment and a broadband amplitude modulated signal enhanced (BASE)
experiment. We demonstrate that the adiabatic BASE pulse sequence may achieve
a DNP 1H enhancement factor of $\sim$ 360, a record that outperforms all
previously known pulsed DNP sequences at $\sim$ 0.35 T and 80 K in static
solids. The bandwidth of the BASE-DNP experiments is about 3 times the 1H
Larmor frequency ($\sim$50 MHz).

##  I Introduction

Dynamic nuclear polarization (DNP) is a powerful tool to increase the
sensitivity of nuclear magnetic resonance (NMR) by transferring the much
higher polarization of electron spins to nuclear spins with a theoretical
maximum enhancement factor $\varepsilon\sim 658$ Abragam and Goldman (1978);
Lilly Thankamony  _et al._ (2017) for 1H. The hyperpolarization method allows
one to study systems that suffer from poor NMR sensitivity with reduced
measurement time or cost. For instance, a DNP experiment with $\varepsilon\sim
100$ performed in one hour would have taken $\sim 1$ year without mw
irradiation for the same-quality spectrum. This opens up the possibility to
extract important structural information from small molecules, biological
samples, or inorganic materials that are otherwise inaccessible due to poor
NMR sensitivity.

Since the discovery of DNP in the fifties Overhauser (1953) tremendous
progress has been made, and there are two main DNP methods: dissolution DNP
Ardenkjaer-Larsen  _et al._ (2003) and in situ solid-state DNP NMR potentially
combined with magic-angle spinning (MAS) Becerra  _et al._ (1993). The former
category typically polarizes a static sample at low temperatures (< 2 K) and
moderate magnetic fields (3.4-10.1 T)Ardenkjær-Larsen  _et al._ (2018), where
the electron polarization approaches unity. Following that, the sample
undergoes a dissolution process prior to be transported to a high-resolution
NMR magnet or MRI system for detection in solution state. The in situ static
sample or MAS DNP NMR approach performs the hyperpolarization and NMR
detection process on solid samples in the same high-resolution magnet
typically at temperatures below 100 K. To facilitate efficient DNP processes
at these relatively higher temperatures and fields, it is necessary to use
high-power mw sources known as gyrotrons. Although these DNP methods have
successfully enabled many biological molecules and materials to be studied
with superior sensitivity Lilly Thankamony  _et al._ (2017); Jannin  _et al._
(2019), it is noted that the DNP efficiency deteriorates at higher magnetic
fields Tan  _et al._ (2019a). Despite new classes of biradicals that have been
reported to circumvent this issueBerruyer  _et al._ (2020); Cai  _et al._
(2021), the DNP performance is sample-dependent and varies with rotor sizes
(or mw penetration). Hence, it would be highly desirable to develop a general
DNP method with a consistent performance across different magnetic fields,
paramagnetic polarizing agents, and other experimental conditions. One of the
major reasons that hinders the development towards this goal is that most
contemporary DNP approaches use continuous-wave (CW) mw irradiation, where the
amplitude, phase, and frequency cannot be easily controlled. As a consequence
only four main CW-DNP mechanisms have been discovered so far, namely:
Overhauser effect (OE) Overhauser (1953), the solid effect (SE) Abragam and
Proctor (1958); C.D.Jeffries (1957), the cross effect (CE) Hwang and Hill
(1967); Kessenikh  _et al._ (1963), and thermal mixing (TM) Provotorov (1962);
Borghini (1968). In comparison, hundreds of NMR pulse sequences have been
invented to date, for purposes ranging from polarization transfer, distance
measurement, to determination of dynamics and chemical environments, etc. An
important aspect in choosing the right pulse sequence is the bandwidth, an
issue relatively less discussed in DNP. The EPR line widths of many radicals
are often broad, especially at high fields due to large g-anisotropy. The
linewidth of a typical nitroxide radical is $\sim$ 1 GHz at 9.4 T,Soetbeer
_et al._ (2018) which is orders of magnitude higher than the electron Rabi
fields conferred by the currently available mw power. The aim of this study is
to demonstrate - albeit at lower field - ways to design broadband and
efficient DNP techniques exploiting shaped pulses generated by arbitrary
waveform generators (AWG) at different mw power conditions.

Our approach to the design of broadband pulsed DNP experiments has been
inspired by previous pulsed DNP techniques, namely nuclear orientation via
spin locking (NOVEL) Henstra  _et al._ (1988); Can  _et al._ (2015); Mathies
_et al._ (2016), ramped-amplitude (RA)-NOVEL Can  _et al._ (2017a), off-
resonance NOVEL Jain, Mathies, and Griffin (2017), the integrated solid effect
(ISE) Henstra, Dirksen, and Wenckebach (1988); Can  _et al._ (2017b), the
adiabatic solid effect (ASE) Tan  _et al._ (2020), nuclear rotating frame
(NRF)-DNP Wind  _et al._ (1988), the dressed spin solid effect (DSSE) Weis
_et al._ (2000), PulsePol Schwartz  _et al._ (2018), and time-optimized pulsed
(TOP)-DNP Tan  _et al._ (2019b). Among these sequences, we would like to
emphasize that the TOP-DNP sequence is substantially different from others,
i.e., the initial truncation of the electron-nuclear dipolar couplings by the
nuclear Zeeman term can be reintroduced by mw irradiation to transfer
polarization in DNP. The methods to reintroduce these couplings are
mathematically similar to and inspired by the dipolar recoupling techniques in
MAS solid-state NMR spectroscopy, where rf irradiation interferes with the
rotational averaging of dipolar couplings. Following this, a phase-alternating
low-power X-inverse X (XiX)-DNP has been demonstrated recently using similar
average Hamiltonian and operator-based Floquet theory design principles
Redrouthu and Mathies (2022). In this work, we further introduce an
alternative design strategy based on single-spin vector effective Hamiltonian
theory Shankar  _et al._ (2017); Nielsen  _et al._ (2019), which incorporates
Fourier coefficients (exploited in Floquet theory) into average Hamiltonian
theory. This theoretical framework is applicable to any arbitrary periodic DNP
experiments, and will here be used for designing broadband DNP experiments. We
will examine these sequences by numerical simulations and experiments at 0.35
T/ 9.8 GHz/ 15 MHz on OX063 trityl radicals doped in a glycerol-water mixture
at 80 K.

##  II Theory

We will describe the overall Hamiltonian for an electron-nuclear spin system
followed by a series of transformations leading to a convergent effective
Hamiltonian. The Hamiltonian is cast as a Fourier expansion, which allows us
to identify the resonance conditions and determine the effective couplings
that govern the DNP polarization transfer. The procedure takes inspiration
from previous single-vector effective Hamiltonian approaches described
recently in relation to solid-state NMR dipolar recoupling Shankar  _et al._
(2017) and liquid-state NMR isotropic mixing Nielsen  _et al._ (2019).

While the theory outlined in the following is valid and extendable to describe
systems with multiple electrons and nuclei, we will for simplicity stick to a
two-spin system comprised of one electron spin ($S$) and one nuclear spin
($I$). The laboratory-frame Hamiltonian is given by

$\displaystyle\mathcal{H}=\omega_{S}S_{z}+\mathcal{H}_{\text{mw}}+\vec{S}\cdot\mathbf{A}\cdot\vec{I}+\omega_{I}I_{z}\quad,$
((1))

with $\omega_{S}=-\gamma_{\text{e}}B_{0}$ and
$\omega_{I}=-\gamma_{\text{n}}B_{0}$ being angular frequencies for the
electron and nuclear Zeeman interactions, respectively (for an e-1H system
$\omega_{S}>0$ and $\omega_{I}$<0). $\gamma$, $\mathbf{A}$,
$\mathcal{H}_{\text{mw}}$, and $B_{0}$ refer to the gyromagnetic ratio, the
hyperfine coupling tensor, the Hamiltonian of the mw irradiation, and the
external static magnetic field along the $z$-axis, respectively. Upon
transformation to the electron rotating frame and employing the high-field
approximation , the first-order effective Hamiltonian becomes

$\bar{\tilde{\mathcal{H}}}=\underbrace{\Omega_{S}S_{z}+\tilde{\mathcal{H}}_{\text{mw}}}_{\mathcal{H}_{\text{control}}}+\underbrace{A_{zz}S_{z}I_{z}+BS_{z}I_{x}+\omega_{I}I_{z}}_{\mathcal{H}^{\prime}}\quad,$
((2))

where $\Omega_{S}=\omega_{S}-\omega_{\text{mw}}$ is the mw offset frequency;
$A_{zz}$ and $B=\sqrt{A_{zx}^{2}+A_{zy}^{2}}$ are the secular and
pseudosecular coupling, respectively. Note that the $B$ term originates purely
from the dipolar coupling, which is averaged to zero in solution state. We
will restrict our discussions to spin systems in the regime where
$|A_{zz}|,|B|\ll|\omega_{I}|$, so that the effect of the spin diffusion
barrier can be neglected in this context, i.e. the electron-nuclear distance
falls in the range of 4-10 Å Wolfe (1973); Tan  _et al._ (2019c); Jain  _et
al._ (2021); Stern  _et al._ (2021).

We now move into the interaction frame with the control field
$\mathcal{H}_{\text{control}}$, which contains all explicit information
(amplitudes, frequencies, and phases) about mw pulses:

$\displaystyle\tilde{\mathcal{H^{\prime}}}=$ $\displaystyle
U_{\text{control}}^{\dagger}\,\mathcal{H^{\prime}}\,U_{\text{control}}$
$\displaystyle=$ $\displaystyle\sum_{\chi=x,y,z}R^{(\text{control})}_{\chi
z}(t)S_{\chi}(A_{zz}I_{z}+BI_{x})+\omega_{I}I_{z}\quad,$ ((3))

where
$U_{\text{control}}(t)=\hat{T}\exp{(-i\int_{0}^{t}\mathcal{H}_{\text{control}}(\tau)\mathrm{d}\tau)}$
and $\hat{T}$ is the Dyson time-ordering operator. The time-dependent rotation
matrix $R^{(\text{control})}_{\chi z}$ represents the interaction-frame
trajectory of the electron spin under the mw pulse sequence. Then, we can
calculate an effective time-independent Hamiltonian using standard average
Hamiltonian theory, provided $\mathcal{H}_{\text{control}}(t)$ is periodic
over a given period $\tau_{\text{m}}=2\pi/\omega_{\text{m}}$ (where
$\omega_{\text{m}}$ is the modulation frequency of the pulse sequence), i.e.,
$\mathcal{H}_{\text{control}}(t)$ =
$\mathcal{H}_{\text{control}}(t+\tau_{m})$. We note that a periodic
Hamiltonian does not necessarily need to have an identity propagator over one
cycle.

Although the control propagator ${U}_{\text{control}}$, which is equivalent to
$R_{\chi z}^{\text{(control)}}$, may not necessarily be cyclic in the normal
mw rotating frame, it is always possible to find a frame in which the new
transformed control propagator becomes cyclic. Having a cyclic propagator is
particularly useful in both characterizing and gaining useful insight into the
pulse sequence. It is also a prerequisite for the application of average
Hamiltonian theory to the interaction frame Hamiltonian. Hence, we choose a
frame with its $z$-axis aligned with the effective field
($\omega_{\text{eff}}$) Shankar  _et al._ (2017); Nielsen  _et al._ (2019),
whose magnitude and direction can be determined using quaternion algebra
Blümich and Spiess (1985); Counsell, Levitt, and Ernst (1985); Tan  _et al._
(2015). We indicate this frame in the following with a tilde on the
$\tilde{S}$-spin operators. This leads to

$\tilde{\mathcal{H^{\prime}}}=\sum_{\chi=x,y,z}R^{(\text{eff})}_{\chi
z}(t)\tilde{S}_{\chi}(A_{zz}I_{z}+BI_{x})-\omega_{\text{eff}}^{(S)}\tilde{S}_{z}+\omega_{I}I_{z}$
((4))

with

$\displaystyle R^{(\text{eff})}_{\chi z}(t)=$
$\displaystyle[R_{z}(-\omega_{\text{eff}}^{(S)}t)\cdot
R^{(\text{flip})}(\beta)\cdot R^{(\text{control})}]_{\chi z}(t)$
$\displaystyle=$ $\displaystyle\sum\limits_{k=-\infty}^{\infty}a_{\chi
z}^{(k)}e^{ik\omega_{\text{m}}t}$ ((5))

representing a three-step transformation comprised of (1) go into the control
frame; (2) flip the coordinate system by $\beta$ so that it is aligned with
$\omega_{\text{eff}}^{(S)}\tilde{S}_{z}$; (3) rotate the frame by an angle
$\omega_{\text{eff}}^{(S)}t$ around the new $z$ axis. Shankar  _et al._
(2017); Nielsen  _et al._ (2019). Moreover, Eq. (5) is cyclic, i.e.,
$[R^{(\text{eff})}_{\chi z}(t+\tau_{\text{m}})]_{\chi
z}=R^{(\text{eff})}_{\chi z}(t)$, and the elements of the overall rotation
matrix may be expressed in terms of a Fourier series as given in the right-
hand side of the equation. The term $-\omega_{\text{eff}}^{(S)}\tilde{S}_{z}$
incorporates the Coriolis term originating from step (3) above.

We transform the nuclear part ($I$) of the Hamiltonian into an interaction
frame with closest match between the mw modulation frequency
$\omega_{\text{m}}$ and $\omega_{I}$ to obtain the $I$-spin effective field

$\omega_{\text{eff}}^{(I)}=\omega_{I}-k_{I}\omega_{\text{m}}\text{ with
}k_{I}=\mathrm{round}\left(\frac{\omega_{I}}{\omega_{\text{m}}}\right),$ ((6))

which is then used to form the propagator
$U_{\text{eff}}=\exp{\left(-ik_{I}\omega_{\text{m}}tI_{z}\right)}$ that
transforms $\tilde{\mathcal{H^{\prime}}}$ (Eq. (4)) into

$\displaystyle\tilde{\tilde{\mathcal{H}}}^{\prime}$
$\displaystyle=U_{\text{eff}}^{\dagger}\,\tilde{\mathcal{H^{\prime}}}\,U_{\text{eff}}-k_{I}\omega_{\text{m}}I_{z}$
$\displaystyle=\sum\limits_{\chi=x,y,z}\sum\limits_{k=-\infty}^{\infty}a_{\chi
z}^{(k)}e^{ik\omega_{\text{m}}t}\tilde{S}_{\chi}\times$
$\displaystyle\left(A_{zz}I_{z}+\frac{B}{2}\left(e^{ik_{I}\omega_{\text{m}}t}I^{+}+e^{-ik_{I}\omega_{\text{m}}t}I^{-}\right)\right)$
$\displaystyle-\omega_{\text{eff}}^{(S)}\tilde{S}_{z}+\omega_{\text{eff}}^{(I)}I_{z}\quad.$
((7))

Next, we apply first-order average Hamiltonian theory (AHT) and recognize that
index $k$ in Eq. (7) has to be either $=\pm k_{I}$ or $=0$. All other terms
vanish upon integration over one modulation period, leading to

$\displaystyle\bar{\tilde{\tilde{\mathcal{H}}}}^{(1)}$
$\displaystyle=A_{zz}\sum\limits_{\chi=x,y,z}a_{\chi
z}^{(0)}\tilde{S}_{\chi}I_{z}+\frac{B}{2}\tilde{S}_{z}\left(a_{zz}^{(-k_{I})}I^{+}+a_{zz}^{(k_{I})}I^{-}\right)$
$\displaystyle+\frac{B}{4}\left(a_{+z}^{(-k_{I})}\tilde{S}^{+}I^{+}+a_{-z}^{(-k_{I})}\tilde{S}^{-}I^{+}+a_{+z}^{(k_{I})}\tilde{S}^{+}I^{-}+a_{-z}^{(k_{I})}\tilde{S}^{-}I^{-}\right)-\omega_{\text{eff}}^{(S)}\tilde{S}_{z}+\omega_{\text{eff}}^{(I)}I_{z}\quad,$
((8))

where we define
$a_{xz}^{(q)}\tilde{S}_{x}+a_{yz}^{(q)}\tilde{S}_{y}=\frac{1}{2}(a_{+z}^{(q)}\tilde{S}^{+}+a_{-z}^{(q)}\tilde{S}^{-})$
and $a_{\pm z}^{(q)}=a_{xz}^{(q)}\mp ia_{yz}^{(q)}$. Since the effective
Hamiltonian must be Hermitian, it is enforced that
$a_{-z}^{(q)}=\left(a_{+z}^{(-q)}\right)^{*}$.

###  II.1 Identifying resonance conditions and scaling factors

We note that the effective Hamiltonian
$\bar{\tilde{\tilde{\mathcal{H}}}}^{(1)}$ (Eq. (8)) was derived without
explicitly describing the details of the pulse sequence, i.e., it has a
general form and is applicable to _all_ periodic EPR or DNP sequences acting
on a two-spin electron-nucleus system in the regime where
$|A_{zz}|,|B|\ll|\omega_{I}|$. This is possible because the details of these
pulses sequences are implicitly encoded in the scaling factors $a_{\chi z}$ of
the corresponding resonance conditions, which will be discussed in the
following.

For polarization-transfer experiments, it is necessary to retain only either
the zero- or double-quantum (ZQ or DQ) operators and suppress all other non-
commuting operators. This is achieved by matching the effective fields
($-\omega_{\text{eff}}^{(S)}\tilde{S}_{z},\omega_{\text{eff}}^{(I)}I_{z}$) in
Eq. (8), i.e., by finding the DNP matching conditions. One can identify a
resonance condition $\omega_{\text{eff}}^{(S)}=-\omega_{\text{eff}}^{(I)}$,
where only the ZQ operators ($\tilde{S}^{\pm}I^{\mp}$) survive because the DQ
terms ($\tilde{S}^{\pm}I^{\pm}$) are truncated by the larger non-commuting
$I_{z}+\tilde{S}_{z}$ term. Note that a converse case is found in the other
resonance condition, $\omega_{\text{eff}}^{(S)}=\omega_{\text{eff}}^{(I)}$,
where only the DQ terms remain due to a similar truncation in the ZQ subspace.
All other terms can be neglected as long as they are much smaller than the
effective fields. This is a good approximation for weakly coupled protons
involved in DNP, and will be checked by numerical simulations below. While the
$A_{zz}\tilde{S}_{z}I_{z}$ term commutes with the effective fields, it shifts
both energy levels in the same direction within the respective subspace (ZQ or
DQ). Thus, the energy difference and resonance conditions remain unchanged. By
neglecting the the terms discussed above, we obtain

$\displaystyle\bar{\tilde{\tilde{\mathcal{H}}}}^{(1)}=$
$\displaystyle\frac{B}{4}\left(a_{-z}^{(\mp
k_{I})}\tilde{S}^{-}I^{\pm}+a_{+z}^{(\pm k_{I})}\tilde{S}^{+}I^{\mp}\right)$
$\displaystyle-$
$\displaystyle\omega_{\text{eff}}^{(S)}\tilde{S}_{z}+\omega_{\text{eff}}^{(I)}I_{z}\quad\text{for}\quad\omega_{\text{eff}}^{(S)}\approx\mp\omega_{\text{eff}}^{(I)}\quad,$
((9))

and the $\rho_{0}\rightarrow I_{z}$ transfer mediated by the effective
Hamiltonian can be calculated by using
$U=\exp{\left(-i\bar{\tilde{\tilde{\mathcal{H}}}}^{(1)}t\right)}$:

$\displaystyle\braket{I_{z}}(t)=\frac{\gamma_{e}}{\gamma_{n}}\braket{\rho_{0}}{\tilde{S}_{z}}Tr\\{U\tilde{S}_{z}U^{\dagger}I_{z}\\}/Tr\\{I_{z}^{2}\\}$
$\displaystyle=\pm\frac{\gamma_{e}}{\gamma_{n}}\braket{\rho_{0}}{\tilde{S}_{z}}\frac{B^{2}a_{\mp}^{2}}{4\omega_{\mp}^{2}}\sin^{2}\left(\frac{1}{2}\omega_{\mp}t\right)\quad\text{for}\quad\omega_{\text{eff}}^{(S)}\approx\mp\omega_{\text{eff}}^{(I)}$
((10))

with

$\displaystyle a_{\mp}$ $\displaystyle=\sqrt{a_{-z}^{(\mp k_{I})}a_{+z}^{(\pm
k_{I})}}$ ((11))

representing a unitless scaling factor that dictates the transfer efficiency.
The quantity

$\displaystyle\omega_{\mp}$
$\displaystyle=\sqrt{B^{2}a_{\mp}^{2}/4+(\Delta\omega_{\text{eff}}^{\mp})^{2}}$
((12))

characterizes the DNP buildup rate, and it depends on the “mismatch” of the
effective fields:

$\displaystyle\Delta\omega_{\text{eff}}^{\mp}(\Omega_{S})$
$\displaystyle=\omega_{\text{eff}}^{(S)}(\Omega_{S})\pm\omega_{\text{eff}}^{(I)}\quad,$
((13))

where the mismatch can be electron offset-dependent
$\Delta\omega_{\text{eff}}^{\mp}(\Omega_{S})$. While the buildup is faster in
case of a larger mismatch, the transfer amplitude is lower, similar to the
situation of an off-resonance pulse in a two-level system. Note that the
subscripts ’-’ and ’+’ in $a_{\mp}$ and $\omega_{\mp}$ symbolize the ZQ and
the DQ case, respectively. The prefactor
$\frac{\gamma_{e}}{\gamma_{n}}\braket{\rho_{0}}{\tilde{S}_{z}}$ in Eq. (10)
highlights that only the part of the electron density operator projected onto
the effective field will be transferred to the nucleus via the ZQ/DQ operator.

The ZQ/DQ-operator-mediated transfers may be visualized by expressing the
Hamiltonian (Eq. (9)) in terms of fictitious spin-1/2 operators Vega (1978);
Wokaun and Ernst (1977)
$I_{x}^{\pm}=\frac{1}{2}(\tilde{S}^{+}I^{\pm}+\tilde{S}^{-}I^{\mp})$,
$I_{y}^{\pm}=\frac{1}{2i}(\tilde{S}^{+}I^{\pm}-\tilde{S}^{-}I^{\mp})$, and
$I_{z}^{\pm}=\frac{1}{2}(\tilde{S}_{z}\pm I_{z})$. This results in

$\displaystyle\bar{\tilde{\tilde{\mathcal{H}}}}^{(1)}=$
$\displaystyle\frac{B}{2}\left(\operatorname{Re}(a_{-z}^{(\mp
k_{I})})I_{x}^{\mp}\mp\operatorname{Im}(a_{-z}^{(\mp
k_{I})})I_{y}^{\mp}\right)$ $\displaystyle+$
$\displaystyle\Delta\omega_{\text{eff}}^{-}I_{z}^{-}+\Delta\omega_{\text{eff}}^{+}I_{z}^{+}\quad,$
((14))

recalling the superscript signs - and + relate to ZQ and DQ operators,
respectively. Figure 1 shows the evolution of spin operators in a diabatic
(’sudden’) (Fig. 1a) or adiabatic (Fig. 1b) manner in the ZQ/DQ subspace. At
the exact resonance condition (mismatch $\Delta\omega_{\text{eff}}^{\mp}=0$),
the polarization-transfer expression simplifies to

$\braket{I_{z}}(t)=\pm\frac{\gamma_{e}}{\gamma_{n}}\braket{\rho_{0}}{\tilde{S}_{z}}\sin^{2}\left(\frac{B}{4}a_{\mp}t\right)\,\,\text{for}\,\,\omega_{\text{eff}}^{(S)}=\mp\omega_{\text{eff}}^{(I)}\quad,$
((15))

whose initial polarization buildup (small $t$) can be approximated by a Taylor
series as

$\braket{I_{z}}(t)\approx\pm\frac{\gamma_{e}}{\gamma_{n}}\braket{\rho_{0}}{\tilde{S}_{z}}\frac{B^{2}}{16}a_{\mp}^{2}t^{2}+\mathcal{O}(t^{4})\,\,\text{for}\,\,\omega_{\text{eff}}^{(S)}=\mp\omega_{\text{eff}}^{(I)}\quad.$
((16))

We define the transfer parameter

$f_{\mp}=\braket{\rho_{0}}{\tilde{S}_{z}}a_{\mp}\quad,$ ((17))

which allows us to semi-quantitatively evaluate the performance of a pulse
sequence, before performing a detailed analysis by defining a particular spin
system.

Finally, we should be aware that the choice of the effective field is not
unique, and it requires a convention. We choose
$|\omega_{\text{eff}}^{(S)}|,|\omega_{\text{eff}}^{(I)}|\leq\omega_{\text{m}}/2$,
which is an arbitrary but convenient decision. If we would allow for a larger
effective field, it would become harder to keep track of resonance conditions.
For our choice, there is one special case, when
$|\omega_{\text{eff}}^{(I)}|\approx|\omega_{\text{eff}}^{(S)}|\approx|\omega_{\text{m}}|/2$.
In this case, the ZQ and DQ resonance conditions are approximately fulfilled
at the same time, because the difference or sum of the effective field matches
the modulation frequency implying that a single scaling factor is not
sufficient to describe the spin dynamics.

Figure 1: Diagram of the effective (fictitious) spin-1/2 subspaces involved in
DNP, with the - and + signs on operators and frequencies representing ZQ and
DQ transfers, respectively. (a) A diabatic (’sudden’) sequence with a constant
mismatch. (b) Adiabatic sequence with slowly changing mismatch of the nuclear
and electron effective fields. The effective ”mismatch” field is slowly
dragged from $+z$ to $-z$, corresponding to full polarization transfer. Note
that $a_{\mp}$ is not necessarily a constant. In principle there can also be
terms proportional to $I_{y}^{\pm}$, which we ignored for the illustration.

###  II.2 Illustration of the interaction frame transformation

A central element in calculating the scaling factors $a_{\mp}$ is the
interaction frame transformations involved in Eqs. (3) \- (5). Here, we give a
concrete example of the elements of $R^{(\text{control})}$ and
$R^{(\text{eff})}$ and their relationship. We consider an XiX-DNP experiment
(parameters here not chosen to represent any good DNP sequence, but for easier
explanation) with $\nu_{1}$= 4 MHz, $t_{\text{p},1}$= 14 ns, $t_{\text{p},2}$=
28 ns, and $\Omega_{S}/2\pi$= 25 MHz (see Figure 3 for the pulse sequence
description). Figure 2(a) shows the elements of the initial interaction frame
transformation $R^{(\text{control})}(t)$ plotted over one period
$\tau_{\text{m}}$. The blue curves denote the trajectory of the normal
rotating frame operator $S_{z}$. Note that
$R^{(\text{control})}(0)=\mathbb{1}\neq
R^{(\text{control})}(\tau_{\text{m}})$, i.e. the trajectory is not cyclic with
$\tau_{\text{m}}$. This prohibits the straight-forward application of average
Hamiltonian theory and is the reason for the subsequent transformations.
Figure 2(b) shows the three-dimensional trajectory of the original $S_{z}$
operator in the initial interaction frame (the three blue components in panel
(a)). The trajectory of the first modulation period is marked in red. The end
points of the subsequent five periods are shown as black dots in panel (b).
The overall rotation from one period to the next can be described by an
effective field shown in gray. This can be understood as a constant effective
field, which can be removed by flipping the frame such that the effective
field is along $z$, and then going into an interaction frame with said
effective field. The result of this transformation, $R^{(\text{eff})}$, is
shown in Figure 2(c). The $z$-axis in this new frame points along the
effective field in Figure 2(b), and the effect of the overall rotation was
eliminated by a counter rotation, i.e. start and end points of the
trajectories are now the same. In Figure 2(c), all the coefficients are cyclic
with time $\tau_{\text{m}}$. A Fourier transform of the respective time-
dependent coefficients directly yields $a^{(k)}_{\chi z}$ in Equation (5).

Figure 2: Illustration for the interaction frames employed in this work, on
the example of XiX-DNP (details in main text). (a) Illustration of the initial
interaction frame transformation $R^{(\text{control})}(t)$. The individual
plots show the evolution of rotation matrix elements over one period
$\tau_{\text{m}}$. (b) Three-dimensional trajectory of the original $S_{z}$
operator in the initial interaction frame (the three blue components in (a)).
The trajectory of the first modulation period is marked in red. The end points
of the subsequent five periods are shown as black dots. The effective field
describes the overall rotation of the sequence. (c) The same trajectory as in
(a), but in the flipped, effective (or cyclic) frame. The start and end points
are the same.

An example script to perform these calculations in MATLAB is given in the SI.

###  II.3 Adiabatic sweeps

It is advantageous to sweep the effective fields across the resonance
conditions because it can improve the bandwidth and overall robustness of
pulse sequences. Hence, we will address the effect of adiabatic sweeps through
a resonance condition using the theoretical framework developed in this
manuscript. For simplicity, we assume (1) only one resonance condition is
swept during the experiment; (2) the changes in the scaling factors and the
electron spin component along the effective field are sufficiently slow
relative to the change in the effective fields, i.e., an adiabatic process.

To adiabatically invert the fictitious operator $I_{z}^{\mp}\rightarrow-
I_{z}^{\mp}$ in the respective ZQ/DQ subspace (Eq. (14)) , an adiabatic sweep
is implemented by varying the offset (or resonance mismatch) in the ZQ/DQ
subspace starting from large positive values
($\Delta\omega_{\text{eff}}^{\mp}\gg 0$), then slowly through zero
($\Delta\omega_{\text{eff}}^{\mp}=0$), and then continue to large negative
values ($\Delta\omega_{\text{eff}}^{\mp}\ll 0$). Note that the offset term in
the ZQ/DQ subspace is usually not equivalent to the electron offset
$\Omega_{S}$. For instance, in the adiabatic NOVEL DNP sequence, the offset in
the ZQ/DQ subspace is determined by the mismatched Rabi field
$\omega_{1S}(t)$, although the electron offset is $\Omega_{S}=0$ throughout
the sequence. Figure 1(b) shows the schematic diagram of the described
adiabatic sweep, which resulted in the spin evolution of
$I_{z}^{\mp}\rightarrow I_{z}^{\pm}$ (in a subspace), which is mathematically
equivalent to a $\tilde{S}_{z}\rightarrow I_{z}$ transfer.

To ensure that the sequence is adiabatic, one can calculate the adiabaticity
$Q_{\text{crit}}$ at the moment the resonance condition is passed Baum, Tycko,
and Pines (1985); Jeschke, Pribitzer, and Doll (2015)

$\displaystyle
Q_{\text{crit}}^{\mp}=\frac{1}{4}\frac{\left(Ba_{\mp}\right)^{2}}{\frac{\mathrm{d}}{\mathrm{d}t}\Delta\omega_{\text{eff}}^{\mp}(t)}$
((18))

which can be exploited to evaluate the polarization-transfer efficiency using
the Landau-Zener formula

$\displaystyle\braket{I_{z}}=\pm\frac{\gamma_{e}}{\gamma_{n}}\braket{\rho_{0}}{\tilde{S}_{z}}\left(1-\exp\left(-\frac{\pi}{2}Q_{\text{crit}}^{\mp}\right)\right)\qquad.$
((19))

We note that it is impractical to vary the $\omega_{\text{eff}}$ indefinitely
slow to maintain the high adiabaticity because relaxation effects will start
impeding the transfers at long mixing times, i.e., each sample/experiment has
to be individually optimized for maximum transfer. Nevertheless, the scaling
factor $a_{\mp}$ and the resonance conditions depend only on the pulse
sequence, such that good initial guesses and sequence parameter ranges can be
estimated theoretically.

##  III Materials and Methods

###  III.1 Numerical calculation of scaling factors

All numerical calculations were implemented in MATLAB (The MathWorks Inc). All
sequences presented in this work are piece-wise constant pulse sequences such
that the effective fields could be calculated by quaternion multiplication of
the individual pieces. Interaction-frame trajectories were calculated by time
slicing. Fourier coefficients were calculated with an fft of
$R^{\text{(eff)}}(t)$. The two-dimensional simulation of BASE-DNP was
implemented with the simulation package SPINACH Hogben  _et al._ (2011). A
three-spin electron-proton-proton system was used, with a $\mathbf{g}$-tensor
of [2.0046 2.0038 2.0030], e-n distances of $r_{1}$= 4.5 Å and $r_{1}$= 6.5 Å,
polar angles of $\theta_{1}$=$0^{\circ}$ and $\theta_{2}$=$90^{\circ}$ and
azimuthal angles $\phi_{1}$=$0^{\circ}$ and $\phi_{2}$=$70^{\circ}$. A simple
$T_{1}/T_{2}$ relaxation theory was used, with $T_{1,\text{e}}$= 2.5 ms,
$T_{2,\text{e}}$= 5 $\upmu$s and $T_{1,\text{n}}$= 36 s, $T_{2,\text{n}}$= 1
ms. Relaxation was implemented via the Levitt-Di Bari approachLevitt and Bari
(1992). A two-angle Lebedev gridLebedev and Laikov (1999) with 194
orientations was used.

###  III.2 Sample preparation

A 5 mM sample of OX063 trityl radical in DNP juice (glycerol-d8:D_2O:H_2O,
6:3:1 by volume) at 80 K was used for all experiments. In detail, 1.65 mg
trityl radical (MW= 1359 g mol-1, 1.2 $\upmu$mole) were dissolved in 24.3
$\upmu$L of H_2O and 72.9 $\upmu$L D_2O. Of the resulting solution, 48.6
$\upmu$L were then added to 72.9 $\upmu$L of gly-$d_{8}$. 40 $\upmu$L of the
final solution were transferred to a 3 mm OD quartz capillary and flash frozen
in liquid nitrogen before the measurements.

###  III.3 Instrumentation and EPR/NMR spectroscopy

All experimental data were acquired on a new home-built X-band spectrometer
which is based on the design described in Ref.(Doll and Jeschke (2017)).
Notable differences for the experiments described in this work were that a 1.8
GSa/s digitizer (SP Devices ADQ412) was used and that the temperature of 80 K
was achieved with a cryogen-free cryostat (Cryogenic Limited). Microwave
pulses were generated with an arbitrary waveform generator (AWG) model M8190A
(Keysight) and amplified with a 1 kW traveling wave tube (TWT) amplifier
(Applied Systems Engineering). A standard Bruker EN4118A-MD4 ENDOR resonator
was used, with an external rf tuning and matching circuit. NMR experiments
were performed using a Stelar PC-NMR spectrometer. An Arduino board was used
to count TWT gate triggers of the EPR spectrometer, each corresponding to an
$h$ increment (Fig. 3), and the Arduino board triggers the NMR acquisition
after $h$ loops.

FT EPR spectra were acquired by a chirp echo sequence with linear chirp pulses
spanning 300 MHz, a duration of 200 ns ($\pi/2$) and 100 ns ($\pi$) and an
inter-pulse delay of 2 $\upmu$s. All EPR and NMR signals were processed in
MATLAB. All experimental results presented in this work were acquired within a
single session, i.e., the sample was not moved between different DNP
experiments.

###  III.4 Pulse sequences and enhancements

Figure 3: General pulsed DNP sequence with various building blocks (NOVEL,
XiX, and BASE) repeated by $h$ loops before 1H-NMR detection with a solid
echo.

The basic structure of all DNP experiments, as addressed individually in the
following section, is shown in Figure 3. A train of 1H saturation pulses
(eleven 100∘ pulses spaced by 1 ms) was applied before the DNP element. Each
DNP block was repeated $h$ times, with a total build-up time
$T_{\text{DNP}}=h\cdot t_{\text{rep}}$. The contact time $t_{\text{contact}}$,
during which the microwaves are turned on, is generally much shorter than the
repetition time $t_{\text{rep}}$, due to a 1% duty cycle limit of the TWT. The
1H NMR signal was then read out with a solid echo sequence comprised of two
2.5 $\upmu$s $90^{\circ}$ pulses separated by a delay of $t_{\text{SE}}$= 80
$\upmu$s. A conventional eight-step phase cycle was used with
$\\{x,x,y,y,-x,-x,-y,-y\\}$ for the first pulse and detection and
$\\{y,-y,x,-x,y,-y,x,-x\\}$ for the second pulse. The proton spectrum at
thermal equilibrium was acquired using similar parameters except without
microwaves, and a delay of 180 s $\approx 5\cdot T_{1,\text{n}}$ was used in
between the 660 accumulated scans. The $T_{1,\text{n}}$= 36 s was determined
both with a saturation recovery sequence, and by the decay of polarization
after DNP (see SI).

For most cases, we report the polarization enhancement $\varepsilon_{P}$,
given by the ratio of the DNP-enhanced signal intensity divided by the signal
intensity at thermal equilibrium. These values can be different from simple mw
on/off signal enhancements recorded with the same delay, because the DNP
build-up time $T_{B}$ can be much shorter than $T_{1,\text{n}}$. For most
parameter optimizations, we used a repetition time $t_{\text{rep}}$ of 1 ms,
and a build-up time $T_{\text{DNP}}$ of 2 s. Build-up curves were acquired by
changing the value of $h$, and with variable repetition times mentioned in the
respective figures.

##  IV Results

In this section, we apply the theory and design procedures outlined earlier
for DNP experiments at X-band frequencies (9.5 GHz) using both low-power and
high-power microwaves. The former involves variants of the recently published
XiX-DNP experiments Redrouthu and Mathies (2022), while the latter involves
development of a new pulse sequence with improved performance relative to
previous NOVEL-DNP experiments Henstra  _et al._ (1988); Can  _et al._ (2015,
2017a).

###  IV.1 Low-power XiX-DNP

Figure 4:  Numerical and experimental analysis of the XiX-DNP experiment. (a)
Pulse sequence for the mw part of the DNP experiment. (b) Comparison of
$\tilde{S}_{z}\rightarrow I_{z}$ polarization transfer efficiencies calculated
using an effective Hamiltonian including all first order terms (blue) or only
the flip-flop terms (red) with a full numerical simulation (black circles). A
two-spin e-1H spin pair with a distance $r_{en}$ = 4.5 Å is used in the
numerical simulations. The green line illustrates the initial build-up in
Equation (16). (c) and (d) Resonance conditions and theoretical and
experimental enhancements for XiX-DNP with $\nu_{1}$= 4 MHz,
$t_{\text{contact}}$= 8 $\upmu$s $\tau_{\text{rep}}$= 1 ms, $T_{\text{DNP}}$=
2 s as function of the electron offset frequency. (c) The absolute value of
the effective fields $\nu_{\text{eff}}^{(S)}$ (blue) and
$\nu_{\text{eff}}^{(I)}$ (black) as a function of the mw offset, for
$t_{\text{p},1}=t_{\text{p},2}=$ 9 ns. Resonance conditions are indicated as
black circles. (d) Experimental enhancements for different combinations of
$t_{\text{p},1}$ and $t_{\text{p},2}$ (given in brackets) with fixed total
modulation period (black), and theoretical predictions based on Equation (17)
(red). Small additional peaks are due to higher-order processes involving two
protons. The calculation in (b) was done at an electron offset of 40.89 MHz.

Figure 4(a) shows the mw part of the XiX-DNP pulse sequence consisting of two
oppositely phased pulses repeated $n$ times, leading to a total contact time
of
$t_{\text{contact}}=n\cdot\tau_{\text{m}}=n\cdot(t_{\text{p},1}+t_{\text{p},2})$.
Assuming an mw field with an amplitude of $\nu_{1}$= 4 MHz (we use $\omega$
for angular frequencies and $\nu=\omega/2\pi$ for linear frequencies), and an
offset slightly above 40 MHz, $t_{p,1}$ = $t_{p,2}$ = 9 ns, this leads to the
calculated transfer profiles shown in Figure 4(b) when using fully numerical
simulations (black circles), the full first-order Hamiltonian (blue), partial
ZQ/DQ Hamiltonians (Eq. (9), red), or Taylor-expanded series (Eq. (16),
green). The offset-dependent resonance conditions become clear in Figure 4(c)
showing the electron and nuclear effective fields and the matching conditions.
In this example, $\nu_{\text{eff}}^{(I)}=\nu_{I}$ and $k_{I}=0$ for all
resonance conditions. Since low-power mw irradiation is used, the electron
effective field is mainly dominated by the electron offset. We note the
reflection at $\nu_{\text{m}}/2$, which is a consequence of our particular
choice of convention.

Figure 4(d) shows the experimental results and calculated $f_{\mp}$ (Equation
(17)) for different combinations of $t_{\text{p},1}$ and $t_{\text{p},2}$ (but
with a constant sum $t_{\text{p},1}+t_{\text{p},2}$). The bottom case with
$t_{\text{p},1}=t_{\text{p},2}$ corresponds to the sequence introduced by
Mathies et al.Redrouthu and Mathies (2022). Clearly, both the positions and
the relative intensities of the matching conditions are well predicted. The
small peaks visible in the experimental data correspond to a three-spin
electron-1H-1H transition (see SI). Interestingly, if both pulses have the
same length (the bottom trace in Figure 4(d)), the resonance condition at the
usual SE offset ($\Omega_{S}/2\pi\approx\nu_{I}$) is still fulfilled, but the
scaling factor is zero. This figure shows that the resonance conditions alone
are not enough to characterize the DNP performance and that the theoretical
scaling factors reliably predict the relative DNP enhancement.

The performance of the XiX-DNP experiment may be further improved by
adiabatically sweeping the effective fields through the resonance condition,
i.e., slowly increase $t_{\text{p},2}$ upon increasing the loop number $n$
(Figure 4(a)). The improved enhancement is demonstrated in DNP experiments
(Figure 5(a)), where the adiabatic version of XiX-DNP with the second pulse
swept from 8–10 ns (red line) clearly outperforms its diabatic counterpart
proposed by Mathies and coworkers Redrouthu and Mathies (2022) (black line).
Figure 5(b) shows the time-dependence of the effective field mismatch
$\Delta\omega^{-}_{\text{eff}}(t)$ for the diabatic and adiabatic variants.
The black lines correspond to the diabatic variant with fixed timing. Exactly
at the offset of 40.89 MHz, the effective field mismatch is exactly zero
(black solid line), and it does not change over time. Under these conditions,
the transfer is optimal. However, if the offset is 2 MHz off — a reasonable
value given that the FWHM of trityl is $\sim$5 MHz — the effective fields are
also mismatched by about 2 MHz (black dashed line). Consequently, the DNP
matching condition is not fulfilled and the DNP transfer is entirely quenched
for a system with small hyperfine couplings. On the contrary, it is evident
that the effective field of the adiabatic variant (red) crosses zero in both
cases, leading to polarization transfer for a broader distribution of electron
offsets. Finally, Figure 5(c) shows the experimental build-up curve of the two
XiX-DNP sequences, which clearly shows the advantages of implementing
adiabatic XiX-DNP.

Figure 5:  Experimental comparison of the diabatic ($t_{p,1}$=$t_{p,2}$= 9 ns,
black) and adiabatic ($t_{p,1}$= 9 ns, $t_{p,2}$= 8-10 ns, red) XiX-DNP. (a)
XiX-DNP mw offset profile with 2 s of buildup time. (b) Mismatch between
nuclear and electron effective fields for XiX (black) and its adiabatic
version (red) as a function of contact time. Solid lines show an exactly-
matched ($\Delta\nu_{\textrm{eff}}=0$) resonance condition around 40 MHz,
whereas the dashed line describes a scenario of a shifted resonance condition
by 2 MHz. All lines except the black dashed line cross the
$\Delta\nu_{\textrm{eff}}=0$ line, and hence DNP will take place. This clearly
shows the mismatch compensating feature exhibited by adiabatic sequences. (c)
Experimental 1H build-up curves with a repetition time of $t_{\mathrm{rep}}=$
1 ms. XiX-DNP: $\varepsilon_{\mathrm{max}}=$ 261, $T_{\mathrm{B}}=$ 19.0 s,
adiabatic XiX-DNP: $\varepsilon_{\mathrm{max}}=$ 327, $T_{\mathrm{B}}=$ 15.1 s

###  IV.2 High-power BASE-DNP

We will now analyze pulsed DNP sequences requiring high-power mw irradiation.
In particular, we examine NOVEL (nuclear orientation via spin locking) Henstra
_et al._ (1988); Can  _et al._ (2015); Mathies  _et al._ (2016) and its
adiabatic version, the ramped-amplitude RA-NOVEL-DNP Can  _et al._ (2017a)
(Figure 6(a)). We will then show how a simple amplitude modulation can be used
to improve its bandwidth.

For NOVEL, the spinlock strength has to match the nuclear Zeeman frequency,
$\nu_{1}\approx\nu_{I}$, while for RA-NOVEL the nutation frequency is slowly
increased from below the matching condition to above it in a linear fashion.
Although other amplitude modulation regimes were examined, no major
improvement was observed at long contact times Can  _et al._ (2017a). Figure
6(b) and (c) compare the DNP performances of the NOVEL sequences as function
of the (average) Rabi field $\nu_{1}$ and the offset $\Omega_{S}/2\pi$. The
plots shows that RA-NOVEL is more tolerant towards $\nu_{1}$ mismatch, and
hence leads to higher DNP enhancements. Additionally, the calculated mismatch
plot (Figure 6(d)) also predicts that the adiabatic sequence can moderately
improve the bandwidth for small couplings. Nevertheless, RA-NOVEL experiments
did not show an improved offset compensation, most likely due to the more
dominant mw Rabi field inhomogeneity (about 18 %) across the sample.

Figure 6:  (a) Pulse sequence for NOVEL and Ramped Amplitude (RA) NOVEL
(dashed) DNP. b) Experimental DNP enhancement as a function of the spin lock
strength (Rabi frequency) $\nu_{1}$ after 2 s of DNP. (c) DNP mw offset
profiles for both NOVEL sequences (blue) and the EPR spectrum (black) with an
arbitrary scale. (d) Calculated resonance mismatch (Eq. (13)) as a function of
the offset for RA-NOVEL. The adiabatic sequence begins with a large negative
$\Delta\nu_{\textrm{eff}}\ll 0$ (blue), which slowly increases towards
$\Delta\nu_{\textrm{eff}}\sim 0$ (gray), and ends with a large positive
$\Delta\nu_{\textrm{eff}}\gg 0$ (black). DNP occurs whenever the lines cross
$\Delta\nu_{\textrm{eff}}=0$.

Motivated by these results and also our previous works in designing broadband
ssNMR recoupling sequences Tan  _et al._ (2014), we hypothesized that a
broadband pulsed DNP sequence can be designed by combining XiX and NOVEL,
i.e., a Broadband Amplitude modulated Signal Enhanced (BASE) DNP (Figure
7(a)). Similar to NOVEL (but unlike the XiX DNP), BASE-DNP is a spin-locked
experiment. Additionally, the sequence can be made adiabatic by slowly varying
$t_{\text{p},2}$ through one of the matching conditions, which are shown as
dashed lines in the 2D plot of a theoretical prediction using Eq. (15) (Figure
7(c)). This calculation included distributions of electron offsets and Rabi
fields. The intensity and width of the resonance conditions already hint at
the robustness of them with respect to these parameters.

Figure 7(b) shows the DNP enhancement as a function of $t_{\text{p},2}$ with a
fixed value of $t_{\text{p},1}$= 20 ns. One can see that the calculated
$f_{\mp}$ (Eq. (17)) matches the observed resonance conditions and the
relative DNP performance well, despite the fact that the Rabi field
inhomogeneity and mw offsets were simply neglected in these calculations. We
have labeled the two different resonance conditions ($k_{I}=0$ and $k_{I}=1$)
for later reference (vide infra), and the sweep range of the adiabatic
variants are indicated by gray bars. The experimental enhancement as a
function of both pulse lengths is shown in Figure 7(d). Again the positions of
the resonance conditions are well predicted by the theory. There are some
differences in the width and intensity that are expected from the simplistic
two-spin model we are using. A slightly more sophisticated (though still
simplistic) numerical calculation employing Spinach Hogben  _et al._ (2011) is
shown in Figure 7(e). Overall, our theory reliably predicts the resonance
conditions for BASE-DNP. While numerical simulations of small spin systems can
include more details, such as electronic relaxation, they are still not
capturing all the complications in the complete DNP process. In this case, our
(semi-)analytical theory is very helpful in quickly identifying resonance
conditions and for choosing suitable experimental parameters.

Figure 7:  (a) Pulse sequence of BASE-DNP. (b) Experimental (black) and
calculated (red) DNP performance as a function of $t_{\text{p},2}$ with fixed
$t_{\text{p},1}$=20 ns. The ranges for the adiabatic sweeps are marked by gray
bars. (c) Calculated relative enhancement (Equation (10)) including offset
distributions (5 MHz FWHM) and $\nu_{1}$ inhomogeneity (6 MHz FWHM centered at
32 MHz). (d) Experimental BASE-DNP enhancement as a function of
$t_{\text{p},1}$ and $t_{\text{p},2}$ with $T_{\text{DNP}}$= 1 s. The observed
resonance conditions matches well with the theory (black and white dashed
lines). The solid gray line indicates the position of (b). (e) SPINACH
simulation on a spin system described in Materials and Methods. The
experiments were performed using on-resonance ($\Omega_{S}=0$) mw irradiation
and a Rabi field of $\nu_{1}\sim$ 32 MHz, which is twice of that used for
NOVEL. Other experimental details include $t_{\text{contact}}$= 8 $\upmu$s,
$\nu_{1}\approx$ 32 MHz, $t_{\text{rep}}$= 1 ms.

We will now characterize the BASE-DNP resonance conditions in more detail. The
DNP enhancement as a function of $\nu_{1}$ is shown in Figure 8(a). Is is
evident that the adiabatic BASE outperforms its diabatic counterpart. For one
of the resonance conditions, the best transfer was achieved with the highest
power available. A closer inspection at the $k_{I}=0$ and $k_{I}=1$ resonance
conditions reveals that the position of the latter is much more robust with
respect to the spin-lock field strength, in agreement with the experimental
data (Figure 8(a)). Figure 8(b) shows the BASE DNP frequency profile (constant
$B_{\textrm{0}}$ field with varying mw center frequency) for the $k_{I}=1$
case. A maximum mw power was used for the $\pi/2$ pulse for a maximum
bandwidth, but the spin-lock field was adjusted at each offset position
according to the mw resonator (see SI). Note that the mw power adjustment was
not possible for the adiabatic BASE due to the limited mw power available. The
small enhancements at larger offsets ($\frac{\Omega_{S}}{2\pi}=\pm 60$ MHz)
are due to the matched resonance conditions during the adiabatic sweep, where
the offset-dependent mismatch during the contact period is explicitly
calculated (Figure 8 (c)).

As discussed in the analysis performed for RA-NOVEL, DNP occurs when the
mismatch is zero (diabatic case) or passes zero (adiabatic case). The theory
shows that the mismatch for BASE DNP is quite offset-tolerant, as visible in
Figure 8 (c). In other words, the mismatch hardly varies by more than 1 MHz
over the $|\Omega_{s}/2\pi|<$ 30 MHz range. Hence, the theory implies that
BASE will be a broadband sequence, and indeed this was verified
experimentally.

Figure 8: Experimental BASE-DNP enhancement as a function of (a) the Rabi
field $\nu_{1}$ for the diabatic and adiabatic versions of the respective
resonance conditions and (b) offset $\Omega_{S}/2\pi$, and an EPR spectrum is
included here for reference (black). The BASE parameters were
$t_{\text{p},1}$= 20 ns, $t_{\text{contact}}$= 8 $\upmu$s,
$\tau_{\text{rep}}$= 1 ms, $T_{\text{DNP}}$= 2 s. The pulse length of the
second pulse, $t_{\text{p},2}$ was fixed in the case of (diabatic) BASE
(’$\circ$’) to 7 ns ($k_{I}=0$) and 29 ns ($k_{I}=1$), while it was swept over
4.75–9.25 ns ($k_{I}=0$) or 26–32 ns ($k_{I}=1$) for adiabatic BASE (’x’). (c)
Calculated offset-dependent mismatch for $k_{I}=1$ condition. The adiabatic
sweep begins with $t_{\text{p},2}=$ 26 ns (red), through the intermediate
stages (gray), and ends at $t_{\text{p},2}=$ 32 ns (black).

It is evident that the increased Rabi field $\nu_{1}$ employed in BASE has
resulted in a $\sim 3\times$ higher bandwidth compared to RA-NOVEL (Figure 9).

Figure 9:  (a) DNP frequency profile of RA-NOVEL and adiabatic BASE
($k_{I}=1$) DNP with 2 s build up time. (b) DNP build-up curve using a
repetition time of 5 ms. The build-up curves were fitted with exponential
functions
$\varepsilon_{\text{P}}(T_{\text{DNP}})=\varepsilon_{\text{max}}\left(1-\exp(-T_{\text{DNP}}/T_{B})\right)$.
The build-up curves were measured using different repetition times
$\tau_{\text{rep}}$ 1-20 ms, where $t_{\mathrm{rep}}=$ 5 ms yield the largest
$\varepsilon$, and $t_{\mathrm{rep}}\sim$ 2 ms resulted in the highest
$\varepsilon_{\text{max}}/\sqrt{T_{1,\text{n}}/T_{B}}$. $90^{\circ}$ flip-back
pulses were applied after each DNP contact to replenish the electron Zeeman
spin bath. Exact BASE parameters are given in Figure 8.

Moreover, we also measured the build-up curves (Figure 9(b)), and the results
are summarized in Table 1. In summary, adiabatic BASE has outperformed RA-
NOVEL with a higher $\varepsilon_{\mathrm{max}}$ of $\approx$ 361 and
$\varepsilon_{\mathrm{max}}\cdot\sqrt{{T_{\mathrm{1,n}}}/{T_{\mathrm{B}}}}\approx$
701, compared to $\varepsilon_{\mathrm{max}}$ of $\approx$ 335 and
$\varepsilon_{\mathrm{max}}\cdot\sqrt{{T_{\mathrm{1,n}}}/{T_{\mathrm{B}}}}\approx$
671 for RA-NOVEL. Although the improvement of adiabatic BASE over RA-NOVEL is
marginal ($\leq 8\%$), a larger relative gain can be envisaged when applied to
other more generic DNP polarizing agents—which usually have broader lines than
OX063, and, hence, offset compensation becomes critical.

Table 1: Enhancements $\varepsilon_{\mathrm{max}}$, build-up times $T_{\mathrm{B}}$ , and sensitivity per unit time (i.e., signal per square root of time) $\varepsilon_{\mathrm{max}}\cdot\sqrt{{T_{\mathrm{1,n}}}/{T_{\mathrm{B}}}}$ for RA-NOVEL and adiabatic BASE. All measured with an additional flip-back pulse after the DNP contact. $T_{\mathrm{1,n}}=$ 36.2 s. $T_{\mathrm{1,e}}=$ 2.5 ms. | RA-NOVEL | ad. BASE
---|---|---
|  $t_{\mathrm{rep}}=$ 2 ms
$\varepsilon_{\mathrm{max}}$ | 321 | 342
$T_{\mathrm{B}}$ / s | 8.3 | 8.7
$\varepsilon_{\mathrm{max}}\cdot\sqrt{\frac{T_{\mathrm{1,n}}}{T_{\mathrm{B}}}}$ | 671 | 701
|  $t_{\mathrm{rep}}=$ 5 ms
$\varepsilon_{\mathrm{max}}$ | 335 | 361
$T_{\mathrm{B}}$ / s | 12.1 | 12.1
$\varepsilon_{\mathrm{max}}\cdot\sqrt{\frac{T_{\mathrm{1,n}}}{T_{\mathrm{B}}}}$ | 580 | 626

##  V Conclusions and outlook

We have demonstrated a generalized theoretical treatment applicable to
periodic DNP sequences in static samples. This is realized by analyzing the
resonance conditions and determining the Fourier coefficients, which encode
the details in a sequence — Rabi fields, phases, amplitudes, and mw offsets.
We showed here an example of how theory can help design a broadband sequence
that is robust against mw offsets. With further improvement by implementing
adiabatic sweeps, we show that adiabatic BASE, to the best of our knowledge,
is the best performing pulsed DNP sequence discovered to date. This is
supported by having obtained excellent enhancement values on trityl at static
conditions, a temperature of 80 K and a field of 0.35 T — with
$\varepsilon_{\mathrm{max}}$ of $\approx$ 361 and
$\varepsilon_{\mathrm{max}}\cdot\sqrt{{T_{\mathrm{1,n}}}/{T_{\mathrm{B}}}}\approx$
701, which are higher than RA-NOVEL. While the adiabatic solid effectTan  _et
al._ (2020) can achieve similar maximal enhancements of 360, it can only reach
sensitivity enhancements of
$\varepsilon_{\mathrm{max}}\cdot\sqrt{{T_{\mathrm{1,n}}}/{T_{\mathrm{B}}}}\approx$
629 (see SI) and is inherently limited to a bandwidth equal to or lower than
the nuclear Zeeman frequency.

Moreover, our theory implies that adiabatic BASE should have field-independent
performances, provided that the mw Rabi fields are also scaled linearly with
the external magnetic fields. Although the mw power requirement for adiabatic
BASE currently cannot be fulfilled for high-field ($>5$ T) DNP NMR
applications, it could be satisfactorily fulfilled for polarizing ^13C in
diamond nitrogen-vacancy (NV) centers. For example, PulsePol, which requires
an order of magnitude more mw power than adiabatic BASE, was demonstrated to
polarize ^13C nuclei in diamond NVs at $\sim$ 0.17 T Schwartz  _et al._
(2018). In such situations, a broadband sequence that is robust against mw
power inhomogeneity and offsets could be advantageous for quantum computing
applications Scheuer and Naydenov (2020).

Lastly, we emphasize that our generalized theoretical framework is not only
applicable to DNP sequences, it is also valid for other magnetic resonance
applications. In fact, the framework developed in this paper is a direct
adaptation (with minor adjustments) from NMR theory, as it has been applied to
various recoupling and decoupling sequences. Hence, it is not too far-fetched
to envision that our theoretical framework could be applied to electron
paramagnetic resonance pulse sequences employing matching conditions Jeschke,
Rakhmatullin, and Schweiger (1998); Rizzato  _et al._ (2013), or to future
pulsed MAS-DNP experiments.

## Author contributions

NW, KOT, and GJ designed the research. NW performed all measurements with help
from LAV and LS. NW, ABN and NCN developed the theory. All simulation scripts
were written by NW, with general help and input from KOT, ABN and LAV.

## Acknowledgments

This article is dedicated to Dr. Anton Ashuiev, stranded in Ukraine at the
time of writing. Dr. Daniel Klose and René Tschaggelar are acknowledged for
building the EPR spectrometer used for this work. We thank Prof. Matthias
Ernst and Prof. Robert G. Griffin for extensive and numerous discussions.
Prof. Jan Henrik Ardankjær-Larsen generously provided OX063 for an earlier
project. KOT acknowledges the funding support from the French National
Research Agency (ANR-20-ERC9-0008) and the Respore program (petit équipement
projet n°339299). NW and GJ acknowledge funding by ETH Zürich grant ETH-48
16-1.

## References

  * Abragam and Goldman (1978) A. Abragam and M. Goldman, “Principles of dynamic nuclear polarisation,” Reports Prog. Phys. 41, 395–467 (1978).
  * Lilly Thankamony  _et al._ (2017) A. S. Lilly Thankamony, J. J. Wittmann, M. Kaushik, and B. Corzilius, “Dynamic nuclear polarization for sensitivity enhancement in modern solid-state NMR,” Prog. Nucl. Magn. Reson. Spectrosc. 102-103, 120–195 (2017).
  * Overhauser (1953) A. W. Overhauser, “Polarization of nuclei in metals,” Phys. Rev. 92, 411–415 (1953).
  * Ardenkjaer-Larsen  _et al._ (2003) J. H. Ardenkjaer-Larsen, B. Fridlund, A. Gram, G. Hansson, L. Hansson, M. H. Lerche, R. Servin, M. Thaning, and K. Golman, “Increase in signal-to-noise ratio of > 10,000 times in liquid-state NMR,” Proc. Natl. Acad. Sci. 100, 10158–10163 (2003).
  * Becerra  _et al._ (1993) L. R. Becerra, G. J. Gerfen, R. J. Temkin, D. J. Singel, and R. G. Griffin, “Dynamic nuclear polarization with a cyclotron resonance maser at 5 T,” Phys. Rev. Lett. 71, 3561–3564 (1993).
  * Ardenkjær-Larsen  _et al._ (2018) J. H. Ardenkjær-Larsen, S. Bowen, J. R. Petersen, O. Rybalko, M. S. Vinding, M. Ullisch, and N. C. Nielsen, “Cryogen-free dissolution dynamic nuclear polarization polarizer operating at 3.35 t, 6.70 t, and 10.1 t,” Magnetic Resonance in Medicine 81, 2184–2194 (2018).
  * Jannin  _et al._ (2019) S. Jannin, J. N. Dumez, P. Giraudeau, and D. Kurzbach, “Application and methodology of dissolution dynamic nuclear polarization in physical, chemical and biological contexts,” J. Magn. Reson. 305, 41–50 (2019).
  * Tan  _et al._ (2019a) K. O. Tan, S. Jawla, R. J. Temkin, and R. G. Griffin, “Pulsed dynamic nuclear polarization,” eMagRes 8, 339–352 (2019a).
  * Berruyer  _et al._ (2020) P. Berruyer, S. Björgvinsdóttir, A. Bertarello, G. Stevanato, Y. Rao, G. Karthikeyan, G. Casano, O. Ouari, M. Lelli, C. Reiter, F. Engelke, and L. Emsley, “Dynamic nuclear polarization enhancement of 200 at 21.15 t enabled by 65 kHz magic angle spinning,” The Journal of Physical Chemistry Letters 11, 8386–8391 (2020).
  * Cai  _et al._ (2021) X. Cai, A. L. Paioni, A. Adler, R. Yao, W. Zhang, D. Beriashvili, A. Safeer, A. Gurinov, A. Rockenbauer, Y. Song, M. Baldus, and Y. Liu, “Highly efficient trityl-nitroxide biradicals for biomolecular high-field dynamic nuclear polarization,” Chemistry – A European Journal 27, 12758–12762 (2021).
  * Abragam and Proctor (1958) A. Abragam and W. G. Proctor, “Une nouvelle methode de polarisation dynamique des noyaux atomiques dans les solides.” Comp. Rend. Acad. Sci. 246, 2253–2256 (1958).
  * C.D.Jeffries (1957) C.D.Jeffries, “Polarisation of Nuclei by Resonance Saturation in Paramagnetic Crystals,” Phys. Rev. 106, 164–165 (1957).
  * Hwang and Hill (1967) C. F. Hwang and D. A. Hill, “New effect in dynamic polarization,” Phys. Rev. Lett. 18, 110–112 (1967).
  * Kessenikh  _et al._ (1963) A. Kessenikh, V. Luschikov, A. Manekov, and Y. V. Taran, “Proton polarization in irradiated polyethylenes,” Sov. Physics–Solid State 5, 321–329 (1963).
  * Provotorov (1962) B. N. Provotorov, “Magnetic Resonance Saturation in Crystals,” Sov. Phys. JETP 14, 1126–1131 (1962).
  * Borghini (1968) M. Borghini, “Spin-temperature model of nuclear dynamic polarization using free radicals,” Phys. Rev. Lett. 20, 419–421 (1968).
  * Soetbeer  _et al._ (2018) J. Soetbeer, P. Gast, J. J. Walish, Y. Zhao, C. George, C. Yang, T. M. Swager, R. G. Griffin, and G. Mathies, “Conformation of bis-nitroxide polarizing agents by multi-frequency EPR spectroscopy,” Physical Chemistry Chemical Physics 20, 25506–25517 (2018).
  * Henstra  _et al._ (1988) A. Henstra, P. Dirksen, J. Schmidt, and W. T. Wenckebach, “Nuclear spin orientation via electron spin locking (NOVEL),” Journal of Magnetic Resonance (1969) 77, 389–393 (1988).
  * Can  _et al._ (2015) T. V. Can, J. J. Walish, T. M. Swager, and R. G. Griffin, “Time domain DNP with the NOVEL sequence,” J. Chem. Phys. 143, 054201 (2015).
  * Mathies  _et al._ (2016) G. Mathies, S. Jain, M. Reese, and R. G. Griffin, “Pulsed Dynamic Nuclear Polarization with Trityl Radicals,” J. Phys. Chem. Lett. 7, 111–116 (2016).
  * Can  _et al._ (2017a) T. V. Can, R. T. Weber, J. J. Walish, T. M. Swager, and R. G. Griffin, “Ramped-amplitude NOVEL,” J. Chem. Phys. 146, 154204 (2017a).
  * Jain, Mathies, and Griffin (2017) S. K. Jain, G. Mathies, and R. G. Griffin, “Off-resonance NOVEL,” J. Chem. Phys. 147, 164201 (2017).
  * Henstra, Dirksen, and Wenckebach (1988) A. Henstra, P. Dirksen, and W. T. Wenckebach, “Enhanced dynamic nuclear polarization by the integrated solid effect,” Phys. Lett. A 134, 134–136 (1988).
  * Can  _et al._ (2017b) T. V. Can, R. T. Weber, J. J. Walish, T. M. Swager, and R. G. Griffin, “Frequency-Swept Integrated Solid Effect,” Angew. Chemie - Int. Ed. 56, 6744–6748 (2017b).
  * Tan  _et al._ (2020) K. O. Tan, R. T. Weber, T. V. Can, and R. G. Griffin, “Adiabatic Solid Effect,” J. Phys. Chem. Lett. 11, 3416–3421 (2020).
  * Wind  _et al._ (1988) R. A. Wind, L. Li, H. Lock, and G. E. Maciel, “Dynamic nuclear polarization in the nuclear rotating frame,” J. Magn. Reson. 79, 577–582 (1988).
  * Weis  _et al._ (2000) V. Weis, M. Bennati, M. Rosay, and R. G. Griffin, “Solid effect in the electron spin dressed state: a new approach for dynamic nuclear polarization,” J. Chem. Phys. 113, 6795–6802 (2000).
  * Schwartz  _et al._ (2018) I. Schwartz, J. Scheuer, B. Tratzmiller, S. Müller, Q. Chen, I. Dhand, Z.-Y. Wang, C. Müller, B. Naydenov, F. Jelezko, and M. B. Plenio, “Robust optical polarization of nuclear spin baths using hamiltonian engineering of nitrogen-vacancy center quantum dynamics,” Science Advances 4, eaat8978 (2018).
  * Tan  _et al._ (2019b) K. O. Tan, C. Yang, R. T. Weber, G. Mathies, and R. G. Griffin, “Time-optimized pulsed dynamic nuclear polarization,” Science Advances 5, eaav6909 (2019b).
  * Redrouthu and Mathies (2022) V. S. Redrouthu and G. Mathies, “Efficient pulsed dynamic nuclear polarization with the x-inverse-x sequence,” Journal of the American Chemical Society 144, 1513–1516 (2022).
  * Shankar  _et al._ (2017) R. Shankar, M. Ernst, P. K. Madhu, T. Vosegaard, N. C. Nielsen, and A. B. Nielsen, “A general theoretical description of the influence of isotropic chemical shift in dipolar recoupling experiments for solid-state NMR,” The Journal of Chemical Physics 146, 134105 (2017).
  * Nielsen  _et al._ (2019) A. B. Nielsen, M. R. Hansen, J. E. Andersen, and T. Vosegaard, “Single-spin vector analysis of strongly coupled nuclei in TOCSY NMR experiments,” The Journal of Chemical Physics 151, 134117 (2019).
  * Wolfe (1973) J. P. Wolfe, “Direct Observation of a Nuclear Spin Diffusion Barrier,” Phys. Rev. Lett. 31, 907–910 (1973).
  * Tan  _et al._ (2019c) K. O. Tan, M. Mardini, C. Yang, J. H. Ardenkjær-Larsen, and R. G. Griffin, “Three-spin solid effect and the spin diffusion barrier in amorphous solids,” Sci. Adv. 5, eaax2743 (2019c).
  * Jain  _et al._ (2021) S. K. Jain, C. J. Yu, C. B. Wilson, T. Tabassum, D. E. Freedman, and S. Han, “Dynamic Nuclear Polarization with Vanadium(IV) Metal Centers,” Chem 7, 421–435 (2021).
  * Stern  _et al._ (2021) Q. Stern, S. F. Cousin, F. Mentink-Vigier, A. C. Pinon, S. J. Elliott, O. Cala, and S. Jannin, “Direct observation of hyperpolarization breaking through the spin diffusion barrier,” Sci. Adv. 7, 1–14 (2021).
  * Blümich and Spiess (1985) B. Blümich and H. W. Spiess, “Quaternions as a practical tool for the evaluation of composite rotations,” J. Magn. Reson. 61, 356–362 (1985).
  * Counsell, Levitt, and Ernst (1985) C. Counsell, M. Levitt, and R. Ernst, “Analytical theory of composite pulses,” Journal of Magnetic Resonance (1969) 63, 133–141 (1985).
  * Tan  _et al._ (2015) K. O. Tan, M. Rajeswari, P. K. Madhu, and M. Ernst, “Asynchronous symmetry-based sequences for homonuclear dipolar recoupling in solid-state nuclear magnetic resonance,” J. Chem. Phys. 142, 1–9 (2015).
  * Vega (1978) S. Vega, “Fictitious spin 1/2 operator formalism for multiple quantum NMR,” The Journal of Chemical Physics 68, 5518–5527 (1978).
  * Wokaun and Ernst (1977) A. Wokaun and R. R. Ernst, “Selective excitation and detection in multilevel spin systems: Application of single transition operators,” The Journal of Chemical Physics 67, 1752–1758 (1977).
  * Baum, Tycko, and Pines (1985) J. Baum, R. Tycko, and A. Pines, “Broadband and adiabatic inversion of a two-level system by phase-modulated pulses,” Phys. Rev. A 32, 3435–3447 (1985).
  * Jeschke, Pribitzer, and Doll (2015) G. Jeschke, S. Pribitzer, and A. Doll, “Coherence Transfer by Passage Pulses in Electron Paramagnetic Resonance Spectroscopy,” J. Phys. Chem. B 119, 13570–13582 (2015).
  * Hogben  _et al._ (2011) H. Hogben, M. Krzystyniak, G. Charnock, P. Hore, and I. Kuprov, “Spinach – a software library for simulation of spin dynamics in large spin systems,” Journal of Magnetic Resonance 208, 179–194 (2011).
  * Levitt and Bari (1992) M. H. Levitt and L. D. Bari, “Steady state in magnetic resonance pulse experiments,” Physical Review Letters 69, 3124–3127 (1992).
  * Lebedev and Laikov (1999) V. Lebedev and D. Laikov, “Quadrature formula for the sphere of 131-th algebraic order of accuracy,” Dokl. Akad. Nauk SSSR 366, 741–745 (1999).
  * Doll and Jeschke (2017) A. Doll and G. Jeschke, “Wideband frequency-swept excitation in pulsed EPR spectroscopy,” Journal of Magnetic Resonance 280, 46–62 (2017).
  * Tan  _et al._ (2014) K. O. Tan, A. B. Nielsen, B. H. Meier, and M. Ernst, “Broad-band DREAM recoupling sequence,” The Journal of Physical Chemistry Letters 5, 3366–3372 (2014).
  * Scheuer and Naydenov (2020) J. Scheuer and B. Naydenov, “Dynamic nuclear polarization (DNP) in diamond,” in  _Diamond for Quantum Applications Part 1_ (Elsevier, 2020) pp. 277–293.
  * Jeschke, Rakhmatullin, and Schweiger (1998) G. Jeschke, R. Rakhmatullin, and A. Schweiger, “Sensitivity Enhancement by Matched Microwave Pulses in One- and Two-Dimensional Electron Spin Echo Envelope Modulation Spectroscopy,” Journal of magnetic resonance 131, 261–271 (1998).
  * Rizzato  _et al._ (2013) R. Rizzato, I. Kaminker, S. Vega, and M. Bennati, “Cross-polarisation edited ENDOR,” Mol. Phys. 111, 2809–2823 (2013).