the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Improved NMR transfer of magnetization from protons to half-integer spin quadrupolar nuclei at moderate and high magic-angle spinning frequencies

### Jennifer S. Gómez

### Andrew G. M. Rankin

### Julien Trébosc

### Frédérique Pourpoint

### Yu Tsutsumi

### Hiroki Nagashima

### Olivier Lafon

### Jean-Paul Amoureux

Half-integer spin quadrupolar nuclei are the only
magnetic isotopes for the majority of the chemical elements. Therefore, the
transfer of polarization from protons to these isotopes under magic-angle
spinning (MAS) can provide precious insights into the interatomic
proximities in hydrogen-containing solids, including organic, hybrid,
nanostructured and biological solids. This transfer has recently been
combined with dynamic nuclear polarization (DNP) in order to enhance the NMR
signal of half-integer quadrupolar isotopes. However, the cross-polarization
transfer lacks robustness in the case of quadrupolar nuclei, and we have
recently introduced as an alternative technique a *D*-RINEPT (through-space
refocused insensitive nuclei enhancement by polarization transfer) scheme
combining a heteronuclear dipolar recoupling built from adiabatic pulses
and a continuous-wave decoupling. This technique has been demonstrated at
9.4 T with moderate MAS frequencies, *ν*_{R}≈10–15 kHz, in
order to transfer the DNP-enhanced ^{1}H polarization to quadrupolar
nuclei. Nevertheless, polarization transfers from protons to quadrupolar
nuclei are also required at higher MAS frequencies in order to improve the
^{1}H resolution. We investigate here how this transfer can be achieved at
*ν*_{R}≈20 and 60 kHz. We demonstrate that the *D*-RINEPT
sequence using adiabatic pulses still produces efficient and robust
transfers but requires large radio-frequency (rf) fields, which may not be compatible with
the specifications of most MAS probes. As an alternative, we introduce
robust and efficient variants of the *D*-RINEPT and PRESTO (phase-shifted
recoupling effects a smooth transfer of order) sequences using
symmetry-based recoupling schemes built from single and composite *π* pulses. Their performances are compared using the average Hamiltonian
theory and experiments at *B*_{0}=18.8 T on *γ*-alumina and
isopropylamine-templated microporous aluminophosphate (AlPO_{4}-14),
featuring low and significant ^{1}H–^{1}H dipolar interactions,
respectively. These experiments demonstrate that the ^{1}H magnetization
can be efficiently transferred to ^{27}Al nuclei using *D*-RINEPT with
$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(270_{0}90_{180}) recoupling and using PRESTO with $\mathrm{R}{\mathrm{22}}_{\mathrm{2}}^{\mathrm{7}}$(180_{0}) or $\mathrm{R}{\mathrm{16}}_{\mathrm{7}}^{\mathrm{6}}$(270_{0}90_{180}) schemes at
*ν*_{R}=20 or 62.5 kHz, respectively. The *D*-RINEPT and PRESTO
recoupling schemes complement each other since the latter is affected by
dipolar truncation, whereas the former is not.

We also analyze the losses during these recoupling schemes, and we show how
these magnetization transfers can be used at *ν*_{R}=62.5 kHz to
acquire in 72 min 2D HETCOR (heteronuclear correlation) spectra between ^{1}H and quadrupolar nuclei,
with a non-uniform sampling (NUS).

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Quadrupolar nuclei with a nuclear spin quantum number $S=\mathrm{3}/\mathrm{2}$, $\mathrm{5}/\mathrm{2}$, $\mathrm{7}/\mathrm{2}$ or
$\mathrm{9}/\mathrm{2}$ are the only NMR-active isotopes for over 60 % of the chemical
elements of the first six periods of the periodic table, including six of
the eight most abundant elements by mass in the Earth's crust: O, Al, Ca,
Na, Mg and K (Ashbrook and Sneddon, 2014). A wide
range of materials, including organic compounds, biological macromolecules,
and nanostructured or hybrid materials, contain half-integer spin
quadrupolar nuclei and protons. Proximities between these isotopes have
notably been probed in solid-state NMR experiments by transferring the
polarization of protons to half-integer quadrupolar nuclei through dipolar
couplings under magic-angle spinning (MAS) conditions
(Rocha
et al., 1991; Hwang et al., 2004; Peng et al., 2007; Vogt et al., 2013; Chen
et al., 2019). More recently, this polarization transfer has been combined
under MAS with dynamic nuclear polarization (DNP) in order to enhance the
NMR signals of half-integer spin quadrupolar nuclei
(Vitzthum
et al., 2012; Perras et al., 2015a; Nagashima et al., 2020). This approach
has notably allowed for the detection of insensitive quadrupolar nuclei with low
natural abundance, such as ^{17}O or ^{43}Ca, or low gyromagnetic ratio,
*γ*, such as ^{47,49}Ti, ^{67}Zn or ^{95}Mo, near surfaces of
materials (Perras
et al., 2015a, 2016, 2017; Blanc et al., 2013; Hope et al., 2017; Lee et
al., 2017; Nagashima et al., 2020, 2021; Li et al., 2018).

This transfer has originally been achieved using cross-polarization under
MAS (CPMAS) (Harris and Nesbitt, 1988). Nevertheless, this
technique lacks robustness for quadrupolar nuclei since the spin-locking of
the central transition (CT) between energy levels $\pm \mathrm{1}/\mathrm{2}$ is sensitive
to the strength of the quadrupole interaction, the offset, the chemical
shift anisotropy (CSA) and the radio-frequency (rf) field inhomogeneity
(Vega,
1992; Amoureux and Pruski, 2002; Tricot et al., 2011). Furthermore, CPMAS
experiments require a careful adjustment of the rf field applied to the
quadrupolar isotope in order to fulfill the Hartmann–Hahn conditions,
$(S+\mathrm{1}/\mathrm{2}){\mathit{\nu}}_{\mathrm{1}\mathrm{S}}+\mathit{\epsilon}{\mathit{\nu}}_{\mathrm{1}\mathrm{H}}=n{\mathit{\nu}}_{\mathrm{R}}$,
where *ν*_{1S} and *ν*_{1H} denote the amplitudes of the
rf fields applied to the *S* quadrupolar isotope and to the protons,
respectively; $\mathit{\epsilon}=\pm \mathrm{1}$, $n=\pm \mathrm{1}$, or ±2; and
*ν*_{R} denotes the MAS frequency, while avoiding the rotary resonance
recoupling (*R*^{3}) ${\mathit{\nu}}_{\mathrm{1}\mathrm{S}}=p{\mathit{\nu}}_{\mathrm{R}}/(S+\mathrm{1}/\mathrm{2})$
with *p*=0, 1, 2 and 3 (Amoureux and
Pruski, 2002; Ashbrook and Wimperis, 2009). Moreover, the magnetization of
the quadrupolar nuclei cannot be spin-locked for some crystallite
orientations, which leads to line-shape distortions
(Barrie, 1993; Hayashi and
Hayamizu, 1993; Ding and Mcdowell, 1995).

These issues have been circumvented by use of the PRESTO (phase-shifted
recoupling effects a smooth transfer of order) scheme
(Perras et al., 2015a, b; Zhao et al., 2004) and, more recently, the through-space refocused
INEPT (denoted RINEPT hereafter) (Nagashima et
al., 2020; Giovine et al., 2019). These schemes benefit from higher
robustness than CPMAS since they do not employ a spin-lock on the
quadrupolar channel but instead a limited number (two or three) of CT
selective pulses. In these sequences, the dipolar interactions between
protons and quadrupolar nucleus are reintroduced by applying on the ^{1}H
channel symmetry-based recoupling sequences, such as $\mathrm{R}{\mathrm{18}}_{\mathrm{2}}^{\mathrm{5}}$ for
PRESTO or $\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$ for RINEPT (Zhao
et al., 2001; Brinkmann and Kentgens, 2006a). In the case of recoupling
sequences built from single square *π* pulses, the RINEPT sequence using
$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$ (denoted RINEPT-$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$) is more efficient than PRESTO
at *ν*_{R}≥60 kHz because of its higher robustness to rf field
inhomogeneity and ^{1}H offset and CSA. At *ν*_{R}<20 kHz,
the PRESTO technique is more efficient since the efficiency of
RINEPT-$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$ is reduced by the increased losses due to
^{1}H–^{1}H interactions during the $\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$ recoupling and the
windows used to rotor synchronize the $\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$ blocks, whereas the
PRESTO sequence is devoid of these windows
(Giovine et al., 2019).

Recently, we have introduced a novel variant of the RINEPT sequence
by employing the $\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$ recoupling built (i) from tanh/tan (tt)
adiabatic inversion pulses, (ii) continuous-wave (CW) irradiations during
the windows, and (iii) composite $\mathit{\pi}/\mathrm{2}$ and *π* pulses on the ^{1}H
channel, in order to limit the losses due to ^{1}H–^{1}H interactions
and improve the transfer efficiency at moderate MAS frequencies
(Nagashima et al., 2020, 2021).
This novel RINEPT variant, denoted RINEPT-CWc-$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}\left(\mathrm{tt}\right)$, is
more efficient than PRESTO and CPMAS at *ν*_{R}≈12.5 kHz, and
it has been combined with DNP to detect the NMR signals of quadrupolar nuclei
with small dipolar coupling with protons, including the low-*γ*
isotopes, such as ^{47,49}Ti, ^{67}Zn or ^{95}Mo, and unprotonated
^{17}O nuclei. Furthermore, for quadrupolar nuclei subject to large
dipolar interactions, such as ^{17}O nuclei of OH groups, we have shown
that a RINEPT-CWc-$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}\left(\mathrm{tt}\right)$ version with only two pulses on
the quadrupolar channel is more efficient that its PRESTO counterpart
(Nagashima et al., 2021).

However, several NMR experiments require the transfer of ^{1}H
magnetization to quadrupolar nuclei at *ν*_{R}>12.5 kHz.
In particular, MAS frequencies of *ν*_{R}≥20 kHz are needed to
avoid the overlap between the center bands and the spinning sidebands of
satellite transitions (STs) in ^{27}Al NMR spectra at 18.8 T. In addition,
magnetization transfers at *ν*_{R}≥60 kHz are advantageous to
acquire through-space heteronuclear correlation (*D*-HETCOR) 2D spectra
between protons and quadrupolar nuclei endowed with high resolution along
the ^{1}H dimension since fast MAS averages out the ^{1}H–^{1}H
dipolar couplings.

Concurrently, we have demonstrated that the efficiency of PRESTO transfers
using the $\mathrm{R}{\mathrm{16}}_{\mathrm{7}}^{\mathrm{6}}$ recoupling can be improved at *ν*_{R}=62.5 kHz using (270_{0}90_{180}) composite *π* pulses as a basic
inversion element, where the standard notation for the pulses is used: *ξ*_{ϕ} denotes a rectangular, resonant rf pulse with flip angle *ξ*
and phase *ϕ* in degrees
(Giovine et al., 2019). More
recently, $\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$ and $\mathrm{R}{\mathrm{12}}_{\mathrm{3}}^{\mathrm{5}}$ recoupling schemes built from
(90_{−45}90_{45}90_{−45}) composite *π* pulses have been proposed,
but they have not yet been incorporated into RINEPT transfers
(Perras et al., 2019). Globally, no systematic study
of the $\mathrm{R}{N}_{n}^{\mathit{\nu}}$ recouplings built from composite *π* pulses has been carried out.

In the present article, we investigate the use of RINEPT-CWc using an
adiabatic recoupling scheme at the higher MAS frequencies of *ν*_{R}=20 and 62.5 kHz. We demonstrate using numerical simulations of spin
dynamics and experiments on *γ*-alumina and isopropylamine-templated
microporous aluminophosphate (AlPO_{4}-14) (hereafter AlPO_{4}-14) that
the rf requirement of this technique increases with the ^{1}H–^{1}H
dipolar interactions. In practice, this rf requirement is not compatible
with the specifications of most MAS probes at *ν*_{R}≥20 kHz,
even for moderate ^{1}H–^{1}H dipolar interactions. As an alternative,
we introduce variants of the PRESTO and RINEPT sequences by selecting with
average Hamiltonian (AH) theory recoupling schemes built from single
rectangular or composite *π* pulses. Finally, using experiments on
*γ*-alumina and AlPO_{4}-14, which feature small and moderate
^{1}H–^{1}H dipolar interactions, respectively, we identify the most
robust and efficient PRESTO and RINEPT transfers at *B*_{0}=18.8 T with
*ν*_{R}=20 and 62.5 kHz.

## 2.1 PRESTO

### 2.1.1 Single-quantum heteronuclear dipolar recoupling

A $\mathrm{R}{N}_{n}^{\mathit{\nu}}$ sequence, where *N* is an even positive integer
and *n* and *ν* are integers, consists of $N/\mathrm{2}$ pairs of elements ${\mathcal{R}}_{\mathit{\varphi}}{\mathcal{R}}_{-\mathit{\varphi}}^{\prime}$, with $\mathit{\varphi}=\mathit{\pi}\mathit{\nu}/N$ radians an overall phase
shift. ℛ_{ϕ} is an inversion pulse with a duration of $n{T}_{\mathrm{R}}/N$,
where ${T}_{\mathrm{R}}=\mathrm{1}/{\mathit{\nu}}_{\mathrm{R}}$ is the rotor period, and ${\mathcal{R}}_{-\mathit{\varphi}}^{\prime}$ is an
inversion pulse derived from ℛ by changing the sign of all phases. ℛ and ℛ^{′}
are identical when they are amplitude modulated; i.e., all phase shifts are
multiples of *π*. The rf field requirement of $\mathrm{R}{N}_{n}^{\mathit{\nu}}$
is equal to

where ${\mathit{\xi}}^{\mathrm{tot}}=\sum _{i=\mathrm{1}}^{P}{\mathit{\xi}}^{i}$ is the sum of the flip
angles of the *P* individual pulses of the ℛ element.

In the PRESTO sequence (Fig. 1a), symmetry-based *γ*-encoded $\mathrm{R}{N}_{n}^{\mathit{\nu}}$ schemes applied to the ^{1}H channel reintroduce
the |m|=2 space components and the single-quantum (SQ)
terms of the heteronuclear dipolar couplings between the protons and the
quadrupolar nuclei, as well as the ^{1}H CSA, while they suppress the
contributions of ^{1}H isotropic chemical shifts, the heteronuclear
*J* couplings with protons, and the ^{1}H–^{1}H dipolar couplings to the
first-order AH (Zhao et al., 2004). The heteronuclear
dipolar interaction is characterized by a space rank *l* and a spin rank
*λ*. A *γ*-encoded |m|=2 SQ heteronuclear
dipolar recoupling must selectively reintroduce the two components
$\mathit{\{}l,m,\mathit{\lambda},\mathit{\mu}\mathit{\}}=\mathit{\{}\mathrm{2},\mathrm{2},\mathrm{1},\mathit{\mu}\mathit{\}}$ and $\mathit{\{}\mathrm{2},-\mathrm{2},\mathrm{1},-\mathit{\mu}\mathit{\}}$ of the heteronuclear dipolar coupling and ^{1}H CSA with
$\mathit{\mu}=\pm \mathrm{1}$, while all other components must be suppressed.

During these recoupling schemes, the contribution of the dipolar coupling
between *I*=^{1}H and *S* nuclei to the first-order Hamiltonian is equal to
(Zhao et al., 2004)

where ${I}^{\pm}={I}_{x}\pm i{I}_{y}$ symbols represent the shift operators, and the
magnitude and phase of the recoupled ** I**–

**dipolar coupling are given by**

*S*and

respectively, where *b*_{IS} is the dipolar coupling constant in rad/s, and
*κ* is the scaling factor of the recoupled heteronuclear dipolar
interaction, which depends on the $\mathrm{R}{N}_{n}^{\mathit{\nu}}$ symmetry and
the ℛ element. The Euler angles $\left\{\mathrm{0},{\mathit{\beta}}_{PR}^{D,IS},{\mathit{\gamma}}_{PR}^{D,IS}\right\}$ relate the ** I**–

** vector to the MAS rotor frame, and**

*S**t*

^{0}refers to the starting time of the recoupling. The norm of ${\stackrel{\mathrm{\u203e}}{H}}_{D,IS}^{\left(\mathrm{1}\right)}$ does not depend on the ${\mathit{\gamma}}_{PR}^{D,IS}$ angle, since these recoupling schemes are

*γ*encoded (Pileio et al., 2007; Martineau et al., 2012). The Hamiltonian of Eq. (2) does not commute among different spin pairs; hence, the PRESTO sequence is affected by dipolar truncation; i.e., the transfer to distant nuclei is attenuated by the stronger couplings with nearby spins (Bayro et al., 2009).

As mentioned above, the SQ heteronuclear dipolar recoupling schemes also
reintroduce the ^{1}H CSA with the same scaling factor *κ* but
without commuting with the recoupled ^{1}H–*S* dipolar interactions.
Therefore, in the case of large ^{1}H CSA, for instance at high magnetic
fields, this interaction can interfere with the ^{1}H–*S* dipolar couplings,
especially with the small ones. These interferences can be limited by the
use of the PRESTO-III variant, depicted in Fig. 1a, c
(Zhao et al., 2004), in which three CT-selective
pulses are applied to the *S* channel. Indeed, the CT-selective *π* pulses
partly refocus the ^{1}H CSA, which limits these interferences.

### 2.1.2 Selection of the recoupling sequence

On the basis of the AH and spin dynamics simulations, the
$\mathrm{R}{\mathrm{18}}_{\mathrm{1}}^{\mathrm{7}}$ and $\mathrm{R}{\mathrm{18}}_{\mathrm{2}}^{\mathrm{5}}$
schemes built from single rectangular *π* pulses were selected for
heteronuclear dipolar recoupling at moderate MAS frequencies, *ν*_{R}
≈10 kHz (Zhao et al., 2001), while, more recently,
sequences based on symmetries $\mathrm{R}{\mathrm{12}}_{\mathrm{5}}^{\mathrm{4}}$,
$\mathrm{R}{\mathrm{14}}_{\mathrm{6}}^{\mathrm{5}}$, $\mathrm{R}{\mathrm{16}}_{\mathrm{7}}^{\mathrm{6}}$, $\mathrm{R}{\mathrm{14}}_{\mathrm{8}}^{\mathrm{5}}$, $\mathrm{R}{\mathrm{18}}_{\mathrm{8}}^{\mathrm{7}}$,
$\mathrm{R}{\mathrm{16}}_{\mathrm{9}}^{\mathrm{6}}$, $\mathrm{R}{\mathrm{20}}_{\mathrm{9}}^{\mathrm{8}}$ and
$\mathrm{R}{\mathrm{18}}_{\mathrm{10}}^{\mathrm{7}}$ using (270_{0}90_{180}) as inversion
element were chosen for the measurement of ^{1}H CSA at fast MAS
frequencies, *ν*_{R}≈60–70 kHz
(Pandey et al., 2015). We also
transferred the proton polarization to ^{27}Al nuclei at *ν*_{R}=62.5 kHz using PRESTO with $\mathrm{R}{\mathrm{16}}_{\mathrm{3}}^{\mathrm{2}}$ recoupling
built from a single rectangular *π* pulse
(Giovine et al., 2019).

We screened here the $\mathrm{R}{N}_{n}^{\mathit{\nu}}$ schemes built from single
rectangular and composite *π* pulses to achieve *γ*-encoded |m|=2 heteronuclear SQ dipolar recoupling at *ν*_{R}=20
or 62.5 kHz. Dipolar recoupling at *ν*_{R}≥60 kHz is useful to
correlate the signals of quadrupolar nuclei with high-resolution ^{1}H
spectra without using homonuclear dipolar decoupling. We tested the three
following composite *π* pulses: (1) (270_{0}90_{180}), which is
offset compensated and amplitude modulated and has been employed in several
$\mathrm{R}{N}_{n}^{\mathit{\nu}}$ sequences (Giovine
et al., 2019; Carravetta et al., 2000; Levitt, 2002; Pandey et al., 2015);
(2) (90_{0}240_{90}90_{0}), which compensates both rf inhomogeneity
and offset (Freeman et al.,
1980; Duong et al., 2019); and (3) (90_{−45}90_{45}90_{−45}), which
has homonuclear decoupling properties (Madhu et al.,
2001). Adiabatic pulses cannot be employed for SQ heteronuclear dipolar
recoupling since they yield vanishing scaling factors for the rotational
components with *μ*≠0
(Nagashima et al., 2018).

A total of 109 $\mathrm{R}{N}_{n}^{\mathit{\nu}}$ symmetries with $\mathrm{2}\le N\le \mathrm{30}$, $\mathrm{2}\le n\le \mathrm{7}$ and $\mathrm{1}\le \mathit{\nu}\le \mathrm{11}$ were found which
recouple the $\mathit{\{}\mathrm{2},\pm \mathrm{2},\mathrm{1},\pm \mathrm{1}\mathit{\}}$ or
$\mathit{\{}\mathrm{2},\mp \mathrm{2},\mathrm{1},\pm \mathrm{1}\mathit{\}}$ rotational components
of the ^{1}H–*S* dipolar coupling and ^{1}H CSA. We selected the $\mathrm{R}{N}_{n}^{\mathit{\nu}}$ recouplings based on those symmetries with rf field
limited to *ν*_{1}≤120 and 190 kHz for *ν*_{R}=20 and
62.5 kHz, respectively. We only considered the $\mathrm{R}{N}_{n}^{\mathit{\nu}}$
symmetries with $\mathrm{45}\le \mathit{\varphi}\le \mathrm{135}$^{∘} since sequences with
*ϕ* close to 90^{∘} are better compensated for rf field errors
and inhomogeneities (Brinkmann and Kentgens, 2006b). The scaling
factor, *κ*, of the recoupled ^{1}H–*S* dipolar interaction was
calculated using the “C and R symmetries” Mathematica package
(Carravetta et al., 2000;
Brinkmann and Levitt, 2001; Brinkmann et al., 2000; Brinkmann and Edén,
2004).

These $\mathrm{R}{N}_{n}^{\mathit{\nu}}$ symmetries eliminate the contribution of
^{1}H–^{1}H dipolar interactions to the first-order Hamiltonian but not
their contribution to the second order. The cross terms between
^{1}H–^{1}H interactions in the second-order Hamiltonian can be written
(Brinkmann and Edén, 2004):

where the sum is taken over all second-order cross terms *{*1,2*}* between the $\mathit{\{}{l}_{\mathrm{1}},{m}_{\mathrm{1}},{\mathit{\lambda}}_{\mathrm{1}},{\mathit{\mu}}_{\mathrm{1}}\mathit{\}}$ and $\mathit{\{}{l}_{\mathrm{2}},{m}_{\mathrm{2}},{\mathit{\lambda}}_{\mathrm{2}},{\mathit{\mu}}_{\mathrm{2}}\mathit{\}}$ rotational components of
DD_{1} and DD_{2} ^{1}H–^{1}H dipolar interactions, respectively.
${\mathit{\kappa}}_{\mathit{\{}\mathrm{1},\mathrm{2}\mathit{\}}}^{{\mathrm{DD}}_{\mathrm{1}}\times {\mathrm{DD}}_{\mathrm{2}}}$ is the scaling factor of this
cross term; ${\left[{A}_{{l}_{i}{m}_{i}}^{{\mathrm{DD}}_{i}}\right]}^{R}$ and ${T}_{{\mathit{\lambda}}_{i}{\mathit{\mu}}_{i}}^{{\mathrm{DD}}_{i}}$ denote the component *m*_{i} of the
*l*_{i}th rank spatial irreducible spherical tensor ${A}^{{\mathrm{DD}}_{i}}$ in
the MAS rotor-fixed frame and the component *μ*_{i} of the *λ*_{i}th rank spin irreducible spherical tensor operator
${T}^{{\mathrm{DD}}_{i}}$. Equation (5) indicates that the amplitude of the second-order
Hamiltonian decreases at higher MAS frequencies. The magnitude of the
cross terms between ^{1}H–^{1}H interactions was estimated by
calculating the Euclidean norm
(Hu
et al., 2009; Gansmüller et al., 2013):

For each basic element ℛ, we selected the $\mathrm{R}{N}_{n}^{\mathit{\nu}}$
schemes with the highest ratio $\mathit{\kappa}/{\u2225{\mathit{\kappa}}_{\mathit{\{}\mathrm{1},\mathrm{2}\mathit{\}}}^{{\mathrm{DD}}_{\mathrm{1}}\times {\mathrm{DD}}_{\mathrm{2}}}\u2225}_{\mathrm{2}}$ in order to minimize the
interference of ^{1}H–^{1}H dipolar interactions with the ^{1}H–*S*
dipolar recoupling. Besides ^{1}H–^{1}H dipolar interactions, other
cross terms involving ^{1}H CSA and offset can also interfere with the
^{1}H–*S* dipolar recoupling. These cross terms can be expressed by Eq. (5), in
which DD_{1} and DD_{2} indexes are substituted by other interactions,
such as ^{1}H CSA or isotropic chemical shift (*δ*_{iso}). For the
selected symmetries, we estimated the magnitude of the cross terms between
^{1}H CSA or offset by calculating the Euclidean norms ${\u2225{\mathit{\kappa}}_{\mathit{\{}\mathrm{1},\mathrm{2}\mathit{\}}}^{\mathrm{CSA}\times \mathrm{CSA}}\u2225}_{\mathrm{2}}$ and ${\u2225{\mathit{\kappa}}_{\mathit{\{}\mathrm{1},\mathrm{2}\mathit{\}}}^{\mathit{\delta}\mathrm{iso}\times \mathit{\delta}\mathrm{iso}}\u2225}_{\mathrm{2}}$ given by Eq. (6).

The corresponding selected $\mathrm{R}{N}_{n}^{\mathit{\nu}}$ sequences are listed
in Tables S1 and S2 in the Supplement for *ν*_{R}=20 and 62.5 kHz, respectively.

For *ν*_{R}=20 kHz, according to the AH, the $\mathrm{R}{N}_{n}^{\mathit{\nu}}$ sequence with the highest robustness to
^{1}H–^{1}H dipolar interactions is
$\mathrm{R}{\mathrm{22}}_{\mathrm{2}}^{\mathrm{7}}({\mathrm{180}}_{\mathrm{0}}$). However, this recoupling is
slightly less robust to ^{1}H CSA and offset than
$\mathrm{R}{\mathrm{18}}_{\mathrm{2}}^{\mathrm{5}}({\mathrm{180}}_{\mathrm{0}}$), which has already been
reported. For this MAS frequency, the $\mathrm{R}{N}_{n}^{\mathit{\nu}}$ schemes
using the chosen composite pulses either required rf fields greater than 120 kHz, e.g., *ν*_{1}=130 and 173 kHz for the
R26${}_{\mathrm{3}}^{\mathrm{7}}$ schemes built from
(90_{−45}90_{45}90_{−45}) and (270_{0}90_{180}) pulses, or did not
suppress efficiently the second-order cross terms between ^{1}H–^{1}H
interactions because of small rf field (*ν*_{1}≤62.5 kHz).

For *ν*_{R}=62.5 kHz, the $\mathrm{R}{N}_{n}^{\mathit{\nu}}$ sequences
using composite *π* pulses recouple the ^{1}H–*S* dipolar interaction with
a higher scaling factor than those built from single *π* pulses.
According to AH, the (90_{0}240_{90}90_{0}) basic element leads to the
highest robustness to ^{1}H–^{1}H interferences. Even if the amplitude
of the cross terms is inversely proportional to the MAS frequency (Eq. 5),
the amplitude of these terms is lower at *ν*_{R}=20 than 62.5 kHz.
The (270_{0}90_{180}) element is less robust to ^{1}H–^{1}H
interferences but benefits from a high robustness to offset. The selected
$\mathrm{R}{N}_{n}^{\mathit{\nu}}$ symmetries for this element include
$\mathrm{R}{\mathrm{14}}_{\mathrm{6}}^{\mathrm{5}}$ and $\mathrm{R}{\mathrm{16}}_{\mathrm{7}}^{\mathrm{6}}$,
which have already been employed for the measurement of ^{1}H CSA and the
transfer of ^{1}H polarization to half-integer quadrupolar nuclei at *ν*_{R}≥60 kHz (Giovine
et al., 2019; Pandey et al., 2015). The scaling factors *κ* of the
^{1}H–*S* dipolar interaction of the $\mathrm{R}{N}_{n}^{\mathit{\nu}}$ schemes
built from single *π* pulses are small with $\mathrm{45}\le \mathit{\varphi}\le \mathrm{135}$^{∘}; hence, we also selected in Table S3 those with an
extended *ϕ* range of 20–160^{∘}. These recoupling schemes are
less robust to offset than the $\mathrm{R}{N}_{n}^{\mathit{\nu}}$ schemes built
from (270_{0}90_{180}) element.

## 2.2 *D*-RINEPT

### 2.2.1 Zero-quantum heteronuclear dipolar recoupling

In the *D*-RINEPT sequence, the ^{1}H–*S* dipolar interactions are reintroduced
under MAS by applying non-*γ*-encoded two-spin order dipolar
recoupling to the ^{1}H channel. These schemes reintroduce the
|m|=2 space components and the zero-quantum (0Q) terms of
the ^{1}H–*S* dipolar interaction and ^{1}H CSA; i.e., the rotational components
$\mathit{\{}l,m,\mathit{\lambda},\mathit{\mu}\mathit{\}}=\mathit{\{}\mathrm{2},\pm \mathrm{2},\mathrm{1},\mathrm{0}\mathit{\}}$, while they suppress the contributions of
^{1}H isotropic chemical shifts, the heteronuclear *J* couplings with
protons, and the ^{1}H–^{1}H dipolar couplings to the first-order AH
(Brinkmann and Kentgens, 2006a, b). The
contribution of the ^{1}H–*S* dipolar coupling to this Hamiltonian is equal
to
(Giovine
et al., 2019; Brinkmann and Kentgens, 2006a; Lu et al., 2012)

where

The norm of ${\stackrel{\mathrm{\u203e}}{H}}_{D,IS}^{\left(\mathrm{1}\right)}$ depends on the *φ* phase, given by
Eq. (4), and hence on the ${\mathit{\gamma}}_{PR}^{D,IS}$ angle. Therefore, these
two-spin order dipolar recoupling schemes are non-*γ*-encoded. The
Hamiltonian of Eq. (7) commutes among different spin pairs; hence, these
recoupling schemes are not affected by dipolar truncation. Similarly, the
recoupled ^{1}H CSA contribution to the first-order Hamiltonian is
proportional to *I*_{z} and hence also commutes with the recoupled
^{1}H–*S* dipolar interactions and does not interfere with the heteronuclear
dipolar recoupling.

### 2.2.2 Selection of the recoupling sequence

Different $\mathrm{R}{N}_{n}^{\mathit{\nu}}$ sequences have been proposed to
achieve non-*γ*-encoded |m|=2 two-spin order
dipolar recoupling, including (i) symmetries $\mathrm{R}(\mathrm{4}n{)}_{n}^{\mathrm{2}n-\mathrm{1}}=\mathrm{R}{\mathrm{12}}_{\mathrm{3}}^{\mathrm{5}}$, $\mathrm{R}{\mathrm{16}}_{\mathrm{4}}^{\mathrm{7}}$,
$\mathrm{R}{\mathrm{20}}_{\mathrm{5}}^{\mathrm{9}}$, $\mathrm{R}{\mathrm{24}}_{\mathrm{6}}^{\mathrm{11}}$,
$\mathrm{R}{\mathrm{28}}_{\mathrm{7}}^{\mathrm{13}}$ and $\mathrm{R}{\mathrm{32}}_{\mathrm{8}}^{\mathrm{15}}$
for *n*=3, 4, 5, 6, 7 and 8 using single *π* pulses as basic element,
which have been employed to measure ^{1}H–^{17}O dipolar couplings at
*ν*_{R}=50 kHz (Brinkmann and Kentgens, 2006b); (ii) $\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$ recoupling built from a single *π* pulse, which
corresponds to the ${\left[\mathrm{R}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}\mathrm{R}{\mathrm{4}}_{\mathrm{1}}^{-\mathrm{2}}\right]}_{\mathrm{0}}{\left[\mathrm{R}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}\mathrm{R}{\mathrm{4}}_{\mathrm{1}}^{-\mathrm{2}}\right]}_{\mathrm{120}}{\left[\mathrm{R}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}\mathrm{R}{\mathrm{4}}_{\mathrm{1}}^{-\mathrm{2}}\right]}_{\mathrm{240}}$ sequence and has been employed in the RINEPT scheme
(Nagashima et al.,
2021; Giovine et al., 2019); (iii) $\mathrm{R}{\mathrm{12}}_{\mathrm{3}}^{\mathrm{5}}$ and
$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$ schemes using a (90_{−45}90_{45}90_{−45})
composite *π* pulse as a basic element, which have been incorporated into
*D*-HMQC (heteronuclear multiple quantum coherence) at *ν*_{R}=36 kHz (Perras et al.,
2019), and (iv) $\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$ schemes built from a (tt) adiabatic
pulse, which have been used in the RINEPT sequence
(Nagashima et al., 2021, 2020).
During the (tt) pulse, the instantaneous rf amplitude is equal to

where ${\mathit{\omega}}_{\mathrm{1},max}$ is the peak amplitude of the rf field, *t* refers to
the time since the start of the pulse, which lasts ${T}_{\mathrm{R}}/\mathrm{4}$ when
incorporated into the $\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$ recoupling scheme. The
parameter *ξ* determines the rise and fall times of the pulse. Hence, in
the frequency-modulated (FM) frame (Garwood and
DelaBarre, 2001), the frequency offset is

where $\mathrm{\Delta}{\mathit{\nu}}_{\mathrm{0},max}$ is the peak amplitude of the carrier
frequency modulation, and *θ* determines the frequency sweep rate in
the center of the pulse. Here, we employed *ξ*=10 and $\mathit{\theta}=\mathrm{87}{}^{\circ}=\mathrm{atan}\left(\mathrm{20}\right)$
(Kervern
et al., 2007; Nagashima et al., 2018, 2020).

We screened here the $\mathrm{R}{N}_{n}^{\mathit{\nu}}$ schemes built from
(180_{0}), (270_{0}90_{180}), (90_{0}240_{90}90_{0}) and
(90_{−45}90_{45}90_{−45}) inversion elements. A total of 58 $\mathrm{R}{N}_{n}^{\mathit{\nu}}$ symmetries with $\mathrm{2}\le N\le \mathrm{30}$, $\mathrm{2}\le n\le \mathrm{7}$ and $\mathrm{1}\le \mathit{\nu}\le \mathrm{11}$ were found which recouple the
$\mathit{\{}\mathrm{2},\pm \mathrm{2},\mathrm{1},\mathrm{0}\mathit{\}}$ rotational components of
the ^{1}H–*S* dipolar coupling and ^{1}H CSA. We only considered the $\mathrm{R}{N}_{n}^{\mathit{\nu}}$ symmetries with $\mathrm{60}\le \mathit{\varphi}\le \mathrm{120}$^{∘} since the currently employed non-*γ*-encoded |m|=2 two-spin order heteronuclear dipolar recoupling schemes have
$\mathrm{75}\le \mathit{\varphi}\le \mathrm{90}$^{∘}.

We calculated the scaling factor of the recoupled ^{1}H–*S* dipolar
interaction and the Euclidean norm ${\u2225{\mathit{\kappa}}_{\mathit{\{}\mathrm{1},\mathrm{2}\mathit{\}}}^{{\mathrm{DD}}_{\mathrm{1}}\times {\mathrm{DD}}_{\mathrm{2}}}\u2225}_{\mathrm{2}}$ of the cross terms between
^{1}H–^{1}H interactions using the “C and R symmetries” Mathematica
package. For each basic element ℛ, we selected the $\mathrm{R}{N}_{n}^{\mathit{\nu}}$ schemes with the highest ratios $\mathit{\kappa}/{\u2225{\mathit{\kappa}}_{\mathit{\{}\mathrm{1},\mathrm{2}\mathit{\}}}^{{\mathrm{DD}}_{\mathrm{1}}\times {\mathrm{DD}}_{\mathrm{2}}}\u2225}_{\mathrm{2}}$. The selected $\mathrm{R}{N}_{n}^{\mathit{\nu}}$ sequences are listed in Table S4, along with the
parameters of the $\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$ schemes built from the different
basic elements ℛ for the sake comparison. For these sequences, we calculated
the Euclidean norms, ${\u2225{\mathit{\kappa}}_{\mathit{\{}\mathrm{1},\mathrm{2}\mathit{\}}}^{\mathrm{CSA}\times \mathrm{CSA}}\u2225}_{\mathrm{2}}$ and ${\u2225{\mathit{\kappa}}_{\mathit{\{}\mathrm{1},\mathrm{2}\mathit{\}}}^{\mathit{\delta}\mathrm{iso}\times \mathit{\delta}\mathrm{iso}}\u2225}_{\mathrm{2}}$, in order to estimate the magnitudes of the cross terms
between ^{1}H CSA and offset.

According to the AH, the (90_{0}240_{90}90_{0}) composite *π* pulse
yields the highest robustness to ^{1}H–^{1}H dipolar interactions.
However, the rf field requirement of the $\mathrm{R}{N}_{n}^{\mathit{\nu}}$
sequences built from this composite pulse, ${\mathit{\nu}}_{\mathrm{1}}=\mathrm{1.16}N{\mathit{\nu}}_{\mathrm{R}}/n$, is not compatible at *ν*_{R}=62.5 kHz with most 1.3 mm
MAS probes (e.g., *ν*_{1}=291 kHz for $\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$).
Furthermore, the highest robustness to ^{1}H CSA and offset are achieved
using the (270_{0}90_{180}) composite *π* pulse. The
$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$ schemes benefit from the highest robustness to
^{1}H CSA because of the three-step multiple-quantum super-cycle
(Brinkmann and Edén, 2004; Brinkmann
and Kentgens, 2006a). Contrary to the $\mathrm{R}{N}_{n}^{\mathit{\nu}}$
with |m|=2 SQ heteronuclear dipolar
recouplings, the rf field of the $\mathrm{R}{N}_{n}^{\mathit{\nu}}$
with |m|=2 two-spin order schemes is always
higher than 2*ν*_{R} since these symmetries with 2*n*>*N*, such
as $\mathrm{R}{\mathrm{12}}_{\mathrm{9}}^{\mathrm{5}}$, have smaller *κ* scaling factors
for the basic elements employed here.

In the case of the adiabatic $\mathrm{R}{N}_{n}^{\mathit{\nu}}$ (tt) sequences,
the determination of the scaling factors of the first- and second-order
terms of the effective Hamiltonian is more cumbersome since they depend on
the ${\mathit{\nu}}_{\mathrm{1},max}$, $\mathrm{\Delta}{\mathit{\nu}}_{\mathrm{0},max}$, *ξ* and *θ*
parameters (Nagashima et al.,
2018). For example, the scaling factor of the
$\mathrm{R}{\mathrm{12}}_{\mathrm{3}}^{\mathrm{5}}$ and $\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$ schemes is *κ*=0.31 for ${\mathit{\nu}}_{\mathrm{1},max}/\mathrm{\Delta}{\mathit{\nu}}_{\mathrm{0},max}=\mathrm{0.685}$, *ξ*=10 and *θ*=87^{∘}, and this value monotonously
decreases for increasing ${\mathit{\nu}}_{\mathrm{1},max}/\mathrm{\Delta}{\mathit{\nu}}_{\mathrm{0},max}$ ratios.

### 2.2.3 *D*-RINEPT-CWc sequence

The *D*-RINEPT-CWc sequence is displayed in Fig. 1b and c. The ^{1}H–*S* dipolar
couplings are reintroduced by applying the $\mathrm{R}{N}_{n}^{\mathit{\nu}}$
schemes listed in Table S4 during the defocusing and refocusing delays *τ*, which are identical in this article, even if distinct delays can improve
the transfer efficiency (Nagashima et al.,
2021). As the two-spin order recoupling schemes are non-*γ*-encoded,
they must be rotor synchronized. We used here a delay of *T*_{R} between two
successive $\mathrm{R}{N}_{n}^{\mathit{\nu}}$ blocks. In the *D*-RINEPT-CWc sequence,
a CW irradiation is applied during these delays in order to limit the losses
due to ^{1}H–^{1}H dipolar interactions (Nagashima et
al., 2021). The nutation during this CW irradiation is eliminated by
employing CW irradiations with opposite phases. Furthermore, the robustness
to ^{1}H rf field inhomogeneity is improved by replacing the first *π*
and second $\mathit{\pi}/\mathrm{2}$ pulses by composite (90_{0}180_{90}90_{0}) and
(90_{90}90_{0}) pulses, respectively, with the CW irradiation being applied
between the individual pulses (Freeman et al.,
1980; Levitt and Freeman, 1979).

## 3.1 Simulation parameters

All simulations were performed using version 4.1.1 of the SIMPSON
package (Bak et al., 2000). The powder
average was performed using 462 $\mathit{\{}{\mathit{\alpha}}_{\mathrm{MR}},{\mathit{\beta}}_{\mathrm{MR}},{\mathit{\gamma}}_{\mathrm{MR}}\mathit{\}}$ Euler angles relating the
molecular and rotor frames. This set of angles was obtained by considering
66 *{**α*_{MR},*β*_{MR}*}* pairs and
7 *γ*_{MR} angles. The *{**α*_{MR},*β*_{MR}*}* values were selected according to the REPULSION
algorithm (Bak and Nielsen, 1997), while the *γ*_{MR} angles were regularly stepped from 0 to 360^{∘}.

To accelerate the simulations, we used a ^{1}H → ^{15}N RINEPT
transfer instead of the ^{1}H → ^{27}Al one, because the computing
time is proportional to the cube of the size of the density matrix.
Furthermore, in RINEPT experiments, only CT-selective pulses are applied to
the quadrupolar nuclei; hence, the contribution of the STs to the signal
can be disregarded. The ^{1}H → ^{15}N RINEPT transfer was simulated
for a ^{15}N^{1}H_{4} spin system. A similar approach has already been
applied for the simulation of the RINEPT transfer from protons to
quadrupolar nuclei (Nagashima et al.,
2021; Giovine et al., 2019). This ^{15}N^{1}H_{4} spin system
comprises a tetrahedron of four protons with a ^{15}N nucleus on one of
its symmetry axes. The dipolar coupling constants between protons are all
equal to $\left|{b}_{\mathrm{HH}}\right|/\left(\mathrm{2}\mathit{\pi}\right)=\mathrm{1}$, 7 or 15 kHz. The dipolar
coupling between the ^{15}N nucleus and its closest ^{1}H neighbor is
$\left|{b}_{\mathrm{HN}}\right|/\left(\mathrm{2}\mathit{\pi}\right)=\mathrm{2575}$ Hz, corresponding to a
^{1}H–^{27}Al distance of 2.3 Å, typical of the distance between the
protons of hydroxyl groups and the Al atoms of the first surface layer of
hydrated *γ*-alumina (Lee et al., 2014). All protons
were subject to a CSA of 6 kHz, i.e., 7.5 ppm at 18.8 T, with a null asymmetry
parameter (Liang et al., 2018). We simulated the
^{1}H → ^{15}N RINEPT-CWc sequences by incorporating either
$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(tt) or $\mathrm{R}{\mathrm{12}}_{\mathrm{3}}^{\mathrm{5}}$(tt)
recoupling schemes. We used a static magnetic field of 18.8 T, for which the
^{1}H and ^{15}N Larmor frequencies were equal to 800 and 81 MHz,
respectively, and MAS frequencies of *ν*_{R}=20 or 62.5 kHz. The
defocusing and refocusing periods were both equal to their optimal values
*τ*=650 or 640 µs at *ν*_{R}=20 or 62.5 kHz,
respectively. The rf field nutation frequency on the ^{1}H channel was
equal to 200 kHz during the $\mathit{\pi}/\mathrm{2}$ and *π* pulses that do not belong to
the recoupling sequence, as well as the CW irradiation, whereas the pulses
applied to *S* =^{15}N nuclei were considered ideal Dirac pulses. For
the (tt) adiabatic pulses, the simulations were performed with ${\mathit{\nu}}_{\mathrm{1},max}/{\mathit{\nu}}_{\mathrm{R}}$ and $\mathrm{\Delta}{\mathit{\nu}}_{\mathrm{0},max}/{\mathit{\nu}}_{\mathrm{R}}$ ratios
ranging from 0.5 to 10 and from 10 to 200, respectively. All other pulses
were applied on resonance. The density matrix before the first pulse was
equal to ${I}_{\mathrm{1}z}+{I}_{\mathrm{2}z}+{I}_{\mathrm{3}z}+{I}_{\mathrm{4}z}$. We normalized the
transfer efficiency of the ^{1}H → ^{15}N RINEPT sequences to the
maximal signal for a ^{1}H → ^{15}N through-bond RINEPT sequence
made of ideal Dirac pulses in the case of a ^{15}N–^{1}H spin system
with a *J*-coupling constant of 150 Hz.

## 3.2 Optimal adiabatic recoupling

The transfer efficiency of RINEPT using $\mathrm{R}{N}_{n}^{\mathit{\nu}}$ schemes
built from adiabatic (tt) pulses depends on ${\mathit{\nu}}_{\mathrm{1},max}$ and $\mathrm{\Delta}{\mathit{\nu}}_{\mathrm{0},max}$ parameters. For a similar ^{15}N^{1}H_{4} spin
system with $\left|{b}_{\mathrm{HN}}\right|/\left(\mathrm{2}\mathit{\pi}\right)=\mathrm{2.575}$ and
$\left|{b}_{\mathrm{HH}}\right|/\left(\mathrm{2}\mathit{\pi}\right)=\mathrm{7}$ kHz, spinning at *ν*_{R}=12.5 kHz, we showed using numerical simulations that a maximal transfer
efficiency was achieved provided that ${\mathit{\nu}}_{\mathrm{1},max}=\mathrm{0.07}\mathrm{\Delta}{\mathit{\nu}}_{\mathrm{0},max}$ and ${\mathit{\nu}}_{\mathrm{1},max}/{\mathit{\nu}}_{\mathrm{R}}\ge \mathrm{8}$
(Nagashima et al., 2021). In practice, we used ${\mathit{\nu}}_{\mathrm{1},max}=\mathrm{11}{\mathit{\nu}}_{\mathrm{R}}=\mathrm{137}$ kHz and $\mathrm{\Delta}{\mathit{\nu}}_{\mathrm{0},max}=\mathrm{160}{\mathit{\nu}}_{\mathrm{R}}=\mathrm{2}$ MHz.

Similar simulations were performed here for *ν*_{R}=20 or 62.5 kHz.
As seen in Fig. 2a–c, at a given MAS frequency, higher ^{1}H–^{1}H
dipolar couplings require higher rf field and broader carrier frequency
sweep so that the (tt) pulses remain adiabatic in spite of the modulation of
the ^{1}H–^{1}H dipolar couplings by MAS (Nagashima et al., 2021; Kervern et al., 2007). For
$\left|{b}_{\mathrm{HH}}\right|/\left(\mathrm{2}\mathit{\pi}\right)=\mathrm{7}$ kHz, the minimal ${\mathit{\nu}}_{\mathrm{1},max}/{\mathit{\nu}}_{\mathrm{R}}$ ratio decreases for higher MAS frequencies (compare
Fig. 2b and d) since the contribution of the modulation of ^{1}H–^{1}H
dipolar couplings by MAS to the first adiabaticity factor is proportional to
(${\mathit{\nu}}_{\mathrm{1},max}{)}^{\mathrm{2}}/{\mathit{\nu}}_{\mathrm{R}}$; hence, ${\mathit{\nu}}_{\mathrm{1},max}$ values
proportional to $\sqrt{{\mathit{\nu}}_{\mathrm{R}}}$, i.e., ${\mathit{\nu}}_{\mathrm{1},max}/{\mathit{\nu}}_{\mathrm{R}}$ ratio
inversely proportional to $\sqrt{{\mathit{\nu}}_{\mathrm{R}}}$, are sufficient to maintain the
adiabaticity of the pulses (Kervern et al., 2007).
Nevertheless, Fig. 2d indicates that the $\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(tt)
recoupling requires ${\mathit{\nu}}_{\mathrm{1},max}\ge \mathrm{313}$ kHz for *ν*_{R}=62.5 kHz, which is hardly compatible with the specifications of most 1.3 mm
MAS probes. Similar transfer efficiencies were simulated for the RINEPT
sequence with $\mathrm{R}{\mathrm{12}}_{\mathrm{3}}^{\mathrm{5}}$(tt) recoupling scheme (not
shown).

## 4.1 Samples and experimental conditions

L-[U-^{15}N]-histidine ⋅ HCl (hereafter referred to as
“histidine”) and isotopically unmodified *γ*-alumina were purchased
from Merck, and AlPO_{4}-14 was prepared as described previously
(Antonijevic et al., 2006).

All ^{1}H → *S* RINEPT-CWc and PRESTO-III experiments were
performed at *B*_{0}=18.8 T on Bruker BioSpin Avance NEO spectrometers
equipped with double-resonance ^{1}H/X probes.

^{1}H → ^{15}N RINEPT-CWc-$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(tt) experiments on
histidine were performed with 1.3 and 0.7 mm MAS probes spinning at *ν*_{R}=40 or 62.5 kHz, with defocusing and refocusing delays equal to
*τ* = 375 and 384 µs, respectively. The rf field of the ^{1}H
$\mathit{\pi}/\mathrm{2}$ and *π* pulses, which do not belong to the recoupling scheme,
was equal to 200 kHz, that of the continuous-wave irradiation to 100 kHz,
and that of the ^{15}N pulses to 62 kHz. ^{1}H decoupling with an
rf field of 16 kHz was applied during the acquisition. The pulses on the
^{1}H channel were applied on resonance, whereas those on the ^{15}N
channel were applied at the isotropic chemical shift of the ^{15}NH^{τ} signal (172 ppm). These 1D spectra resulted from averaging eight transients
with a relaxation delay of 3 s. The ^{15}N isotropic chemical shifts were
referenced to an aqueous saturated solution of NH_{4}NO_{3} using
[^{15}N]-glycine as a secondary reference.

^{a} AlO_{6} signal normalized to that with ^{1}H → ^{27}Al RINEPT-CWc-$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(tt). ^{b} The relative error bars were determined from the S $/$ N, for the AlO_{6} signal intensity, and they are equal to ±0.03. ^{c} FWHM (full width at half maximum) of the robustness to offset. ^{d} FWHM of the robustness to rf field. ^{e} Only a lower bound of rf field could be determined due to probe rf specifications (Fig. 4).

^{a} Intensities of AlO_{6}, AlO_{5} and AlO_{4} resonances
normalized to their intensities with ^{1}H → ^{27}Al
RINEPT-CWc-$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(tt). The relative errors for the signal intensities are ^{b} ± 0.02, ^{c} ±0.03 and ^{d} ±0.01, for AlO_{6}, AlO_{5} and AlO_{4}, respectively. ^{e} FWHM of the robustness to rf field was not measured for RINEPT-$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(tt) or RINEPT-$\mathrm{R}{\mathrm{12}}_{\mathrm{3}}^{\mathrm{5}}$(tt) (Fig. S2).

^{1}H → ^{27}Al RINEPT-CWc and PRESTO-III experiments on *γ*-alumina and AlPO_{4}-14 were performed with a 1.3 mm MAS probe spinning
at *ν*_{R}=20 (to test the $\mathrm{R}{N}_{n}^{\mathit{\nu}}$ schemes
with large rf field requirement) or 62.5 kHz. The tested recoupling schemes
are listed in Tables 1 and 2 for *ν*_{R}=20 kHz and Tables 3 and 4 for *ν*_{R}=62.5 kHz. The rf field of the ^{1}H $\mathit{\pi}/\mathrm{2}$ and *π* pulses, which do not belong to the recoupling scheme, was equal to 208 kHz, that of the continuous-wave irradiation to 147 kHz, and the ^{27}Al
CT-selective one for $\mathit{\pi}/\mathrm{2}$ and *π* pulses to 10 kHz. The defocusing
and refocusing delays *τ* are given in Tables 1 to 4. The pulses on the
^{1}H channel were applied on resonance, whereas those on ^{27}Al
channel were applied (i) on resonance with AlO_{6} signal of *γ*-alumina in Figs. 4 and 7, Tables 1 and 3, and in Figs. 5 and 8 when
the offset is null; (ii) on resonance with AlO_{4} signal of AlPO_{4}-14
in Figs. S2 and S4, Tables 2 and 4 as well as in Figs. S3 and S5 when the
offset is null; and (iii) in the middle of the AlO_{4} and AlO_{6} peaks
for the 1D spectra shown in Figs. 3 and 6. These differences in offset
explain some changes in the relative efficiencies of the recoupling between
the figures. These 1D spectra resulted from averaging 64 transients with a
relaxation delay of 1 s. The ^{27}Al isotropic chemical shifts were
referenced at 0 ppm to 1 mol L^{−1} [Al(H_{2}O)_{6}]^{3+}
solution.

^{a} AlO_{6} signal normalized to that with ^{1}H → ^{27}Al RINEPT-CWc-$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(tt). ^{b} The relative error on AlO_{6} signal intensity is ±0.08. ^{c} FWHM of the robustness to rf field was not measured for
RINEPT-$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(tt) or RINEPT-$\mathrm{R}{\mathrm{12}}_{\mathrm{3}}^{\mathrm{5}}$(tt) (Fig. 7).

^{a} Intensities of AlO_{6}, AlO_{5} and AlO_{4} resonances normalized to their intensities with ^{1}H → ^{27}Al
RINEPT-CWc-$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(tt). The relative errors for the signal intensities are ^{b} ±0.04, ^{c} ±0.06 and ^{d} ±0.02, for AlO_{6}, AlO_{5} and AlO_{4}, respectively. ^{e} FWHM of the robustness to rf field was not measured for RINEPT-$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(tt) or
RINEPT-$\mathrm{R}{\mathrm{12}}_{\mathrm{3}}^{\mathrm{5}}$(tt) (Fig. S4).

We also measured the decay of the transverse proton magnetization of
AlPO_{4}-14 during a spin echo sequence, in which the refocusing *π* pulse was identical to that used in the defocusing part of the RINEPT-CWc
sequences (Fig. 1b). This decay was measured at *ν*_{R}=20 and 62.5 kHz either without any recoupling or by applying a $\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$
recoupling built from (180_{0}), (270_{0}90_{180}) and (tt) pulses
during the delays of the spin echo sequence. The rf fields during the
recoupling two blocks were equal to their optimal values given in Tables 2
and 4.

We also acquired several 2D ^{1}H → ^{27}Al *D*-HETCOR spectra of
AlPO_{4}-14 using RINEPT-CWc-$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$ with (180_{0}),
(270_{0}90_{180}) and (tt) pulses as well as
PRESTO-$\mathrm{R}{\mathrm{16}}_{\mathrm{7}}^{\mathrm{6}}$(270_{0}90_{180}).
These 2D spectra were acquired using a non-uniform sampling (NUS) with an
exponentially biased sampling retaining 25 % of the points with respect to
uniform sampling. The 2D spectra resulted from eight transients for each of the
500 *t*_{1} increments with a recycle delay of 1 s, i.e., an acquisition time
of 72 min.

## 4.2 Optimal adiabatic recoupling

Figure 2e and f show the efficiency of the ^{1}H → ^{15}N
RINEPT-$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(tt) transfer for histidine as a function of
the ${\mathit{\nu}}_{\mathrm{1},max}/{\mathit{\nu}}_{\mathrm{R}}$ and $\mathrm{\Delta}{\mathit{\nu}}_{\mathrm{0},max}/{\mathit{\nu}}_{\mathrm{R}}$
ratios for *ν*_{R} = 40 or 62.5 kHz, respectively. These experimental
data indicate that at higher MAS frequencies, an efficient adiabatic
recoupling can be achieved for lower ${\mathit{\nu}}_{\mathrm{1},max}/{\mathit{\nu}}_{\mathrm{R}}$ and
$\mathrm{\Delta}{\mathit{\nu}}_{\mathrm{0},max}/{\mathit{\nu}}_{\mathrm{R}}$ ratios. This result agrees with the
numerical simulations of Fig. 2b and d.

## 4.3 PRESTO and RINEPT performances for *ν*_{R}=20 kHz

### 4.3.1 *γ*-alumina

The 1D spectra of *γ*-alumina acquired using ^{1}H → ^{27}Al
RINEPT and PRESTO sequences, shown in Fig. 3, exhibit two resonances at 70
and 10 ppm, assigned to tetra- (AlO_{4}) and hexa-coordinated (AlO_{6})
resonances, respectively (Morris and Ellis, 1989). The
signal of penta-coordinated (AlO_{5}) sites, which are mainly
located in the first surface layer, is barely detected because of the lack
of sensitivity of conventional solid-state NMR spectroscopy
(Lee et al., 2014). The most intense peak, AlO_{6}, was
used to compare the transfer efficiencies of RINEPT and PRESTO sequences
with different recoupling schemes. Table 1 lists the measured performances
of ^{1}H → ^{27}Al RINEPT-CWc and PRESTO transfers using various
recoupling for *γ*-alumina at *ν*_{R}=20 kHz. We notably
compared the PRESTO sequences using
$\mathrm{R}{\mathrm{22}}_{\mathrm{2}}^{\mathrm{7}}$(180_{0}) and
$\mathrm{R}{\mathrm{18}}_{\mathrm{2}}^{\mathrm{5}}$(180_{0}) recoupling with the
RINEPT-CWc scheme using $\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$ and
$\mathrm{R}{\mathrm{12}}_{\mathrm{3}}^{\mathrm{5}}$ with single (180_{0}),
composite (270_{0}90_{180}), and (90_{−45}90_{45}90_{−45}) or (tt)
adiabatic pulses. A low transfer efficiency was obtained for
RINEPT-CWc-$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(90_{0}240_{90}90_{0})
because of its low scaling factor, *κ* = 0.131; hence, its
performances are not reported in Table 1. We also tested the recoupling
schemes based on the symmetry $\mathrm{SC}{\mathrm{2}}_{\mathrm{1}}^{\mathrm{0}}$,
corresponding to the ${\left[\mathrm{C}{\mathrm{2}}_{\mathrm{1}}^{\mathrm{0}}\right]}_{\mathrm{0}}{\left[\mathrm{C}{\mathrm{2}}_{\mathrm{1}}^{\mathrm{0}}\right]}_{\mathrm{120}}{\left[\mathrm{C}{\mathrm{2}}_{\mathrm{1}}^{\mathrm{0}}\right]}_{\mathrm{240}}$ sequence with a basic element *C* = (90_{45}90_{135}90_{45}90_{225}90_{315}90_{225}) or
$\mathrm{C}{\mathrm{6}}_{\mathrm{3}}^{\mathrm{0}}$ built from *C*^{′} = (90_{30}90_{120}90_{30}90_{240}90_{330}90_{240}). These basic
elements, which derive from (90_{−45}90_{45}90_{−45}), have recently
been proposed (Perras et al., 2019). As seen in
Table 1 and Fig. 3a, the sequences yielding the highest transfer efficiencies
are by decreasing order RINEPT-CWc with
$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(tt) or
$\mathrm{R}{\mathrm{12}}_{\mathrm{3}}^{\mathrm{5}}$(tt) > PRESTO-$\mathrm{R}{\mathrm{22}}_{\mathrm{2}}^{\mathrm{7}}$(180_{0}) > RINEPT-CWc-$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(270_{0}90_{180}) ≈ PRESTO-$\mathrm{R}{\mathrm{18}}_{\mathrm{2}}^{\mathrm{5}}$(180_{0}) > RINEPT-CWc-$\mathrm{R}{\mathrm{12}}_{\mathrm{3}}^{\mathrm{5}}$(270_{0}90_{180}).
Figures 4 and 5 display the signal intensity of these sequences as
a function of the rf field amplitude and offset, respectively.

The highest transfer efficiencies are obtained with the RINEPT-CWc sequence incorporating a (tt) adiabatic pulse. This recoupling also leads to the highest robustness to offset and rf inhomogeneity, and $\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(tt) and $\mathrm{R}{\mathrm{12}}_{\mathrm{3}}^{\mathrm{5}}$(tt) yield identical transfer efficiency and robustness. Hence, the three-step multiple-quantum super-cycle of the $\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$ symmetry does not improve the robustness in the case of a (tt) basic element. However, these recoupling schemes require maximum rf fields of ${\mathit{\nu}}_{\mathrm{1},max}\ge \mathrm{8}{\mathit{\nu}}_{\mathrm{R}}=$ 160 kHz, which may exceed the rf power specifications of most 3.2 mm MAS probes.

The PRESTO sequences using $\mathrm{R}{\mathrm{22}}_{\mathrm{2}}^{\mathrm{7}}$(180_{0}) and
$\mathrm{R}{\mathrm{18}}_{\mathrm{2}}^{\mathrm{5}}$(180_{0}) recoupling also result
in good transfer efficiencies but lower than
RINEPT-CWc-$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(tt). However, they use
rf fields of ${\mathit{\nu}}_{\mathrm{1}}/{\mathit{\nu}}_{\mathrm{R}}=$ 5.5 and 4.5, which are compatible
with the specifications of 3.2 mm MAS probes. The higher transfer efficiency
of $\mathrm{R}{\mathrm{22}}_{\mathrm{2}}^{\mathrm{7}}$(180_{0}) with respect to
$\mathrm{R}{\mathrm{18}}_{\mathrm{2}}^{\mathrm{5}}$(180_{0}) stems from its weaker
second-order cross terms between ^{1}H–^{1}H interactions (Table S1).

The efficiency of the RINEPT-CWc-$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(270_{0}90_{180})
sequence, with *ν*_{1}=4*ν*_{R}, is comparable to that of
PRESTO-$\mathrm{R}{\mathrm{18}}_{\mathrm{2}}^{\mathrm{5}}$(180_{0}) but with a
higher robustness to offset and rf inhomogeneity. We can notice that
amplitude modulated recoupling schemes, for which the phase shifts are equal
to 180^{∘}, such as
$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(270_{0}90_{180}) and
$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(180_{0}), exhibit a high
robustness to rf field maladjustments (Fig. 5) (Carravetta
et al., 2000). The use of (270_{0}90_{180}) composite pulses with
$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$ symmetry, instead of single *π* pulses, improves its transfer efficiency as well as its robustness to
offset and rf field inhomogeneity.

In summary, for *ν*_{R}=20 kHz in *γ*-alumina, the
RINEPT-CWc-$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(270_{0}90_{180})
sequence achieves efficient and robust transfers of magnetization from
protons to ^{27}Al nuclei using a moderate rf field of *ν*_{1}=4*ν*_{R}. For ^{1}H spectra with a width smaller than 20 kHz and MAS probes with a good rf homogeneity,
PRESTO-$\mathrm{R}{\mathrm{22}}_{\mathrm{2}}^{\mathrm{7}}$(180_{0}) can result in
slightly higher transfer efficiencies.

### 4.3.2 AlPO_{4}-14

Figure 6a shows the ^{1}H → ^{27}Al RINEPT and PRESTO 1D spectra of
AlPO_{4}-14 recorded with *ν*_{R}=20 kHz. They exhibit three
^{27}Al resonances at 43, 21 and −2 ppm assigned to AlO_{4}, AlO_{5}
and AlO_{6} sites, respectively (Ashbrook et
al., 2008) The AlO_{5} and AlO_{6} sites are directly bonded to OH
groups. The ^{1}H MAS spectrum is shown in Fig. S1. According to the
literature, the ^{27}AlO_{4} signal subsumes the resonances of two
AlO_{4} sites with quadrupolar coupling constants *C*_{Q} = 1.7 and 4.1 MHz, whereas those of AlO_{5} and AlO_{6} sites are equal to 5.6 and 2.6 MHz, respectively
(Fernandez et
al., 1996; Antonijevic et al., 2006). The ^{1}H–^{1}H dipolar couplings
within the isopropylamine template molecule are larger than in *γ*-alumina. We used the most intense peak, AlO_{4}, to compare the ^{1}H → ^{27}Al transfer efficiencies of RINEPT-CWc and PRESTO sequences
with different recoupling schemes, and the results are given in Table 2. The
six sequences yielding the highest transfer efficiencies are the same as for
*γ*-alumina and their relative efficiencies are comparable for the
AlO_{4} peak of AlPO_{4}-14 and the AlO_{6} signal of *γ*-alumina.

Nevertheless, the rf requirement of the $\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(tt) and
$\mathrm{R}{\mathrm{12}}_{\mathrm{3}}^{\mathrm{5}}$(tt) schemes is higher for
AlPO_{4}-14 than for *γ*-alumina because of the larger
^{1}H–^{1}H dipolar couplings, in agreement with the numerical
simulations of Fig. 2a–c. This rf requirement prevents the use of these
adiabatic recoupling schemes at *ν*_{R}=20 kHz with most 3.2 mm MAS
probes. That of the other sequences and their robustness to offset and
rf field homogeneity are similar for both samples (Table 2 and Figs. S2 and
S3).

In the case of AlPO_{4}-14, PRESTO yields a higher efficiency than RINEPT
for AlO_{5} and AlO_{6}, contrary to the AlO_{4} resonance, since
(i) these Al sites are directly bonded to OH groups and (ii) $\mathrm{R}{\mathrm{22}}_{\mathrm{2}}^{\mathrm{7}}$(180_{0}) and
$\mathrm{R}{\mathrm{18}}_{\mathrm{2}}^{\mathrm{5}}$(180_{0}) schemes are subject to
dipolar truncation (Sect. 2.1.1), which prevents to transfer the ^{1}H
magnetization of these OH groups to ^{27}AlO_{4} nuclei.

Hence, at *ν*_{R}=20 kHz, for both AlPO_{4}-14 and *γ*-alumina, the RINEPT-CWc-$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(270_{0}90_{180})
and PRESTO-$\mathrm{R}{\mathrm{22}}_{\mathrm{2}}^{\mathrm{7}}$(180_{0}) sequences
are the best choices to transfer the ^{1}H magnetization to ^{27}Al
nuclei.

## 4.4 PRESTO and RINEPT performances for *ν*_{R}=62.5 kHz

Similar comparisons of the performances of the various RINEPT-CWc and PRESTO
sequences were performed for *γ*-alumina and AlPO_{4}-14 at *ν*_{R}=62.5 kHz.

### 4.4.1 *γ*-alumina

The corresponding data for *γ*-alumina are given in Table 3. The
sequences yielding the highest transfer efficiencies are by decreasing
order: RINEPT-CWc with $\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(tt) or
$\mathrm{R}{\mathrm{12}}_{\mathrm{3}}^{\mathrm{5}}$(tt) > RINEPT-CWc-$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(270_{0}90_{180}) ≈ PRESTO-$\mathrm{R}{\mathrm{16}}_{\mathrm{7}}^{\mathrm{6}}$(270_{0}90_{180}) > PRESTO-$\mathrm{R}{\mathrm{14}}_{\mathrm{6}}^{\mathrm{5}}$(270_{0}90_{180}) > RINEPT-CWc-$\mathrm{R}{\mathrm{12}}_{\mathrm{3}}^{\mathrm{5}}$(270_{0}90_{180}).

The nominal rf requirements of the RINEPT sequences using adiabatic or
(270_{0}90_{180}) composite *π* pulses correspond to ${\mathit{\nu}}_{\mathrm{1},max}\approx \mathrm{5}{\mathit{\nu}}_{\mathrm{R}}$ (313 kHz: Fig. 2d) or 4*ν*_{R} (250 kHz), which
exceed the specifications of our 1.3 mm MAS probe, and the sequences were
tested only up to ${\mathit{\nu}}_{\mathrm{1},max}=\mathrm{208}$ kHz (Fig. 7). This suboptimal rf
field may limit the transfer efficiencies of these sequences.

The PRESTO-$\mathrm{R}{\mathrm{16}}_{\mathrm{7}}^{\mathrm{6}}$(270_{0}90_{180})
and PRESTO-$\mathrm{R}{\mathrm{14}}_{\mathrm{6}}^{\mathrm{5}}$(270_{0}90_{180})
sequences yield transfer efficiencies comparable to those of
RINEPT-CWc-$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(270_{0}90_{180})
but with a significantly lower rf field, *ν*_{1}≈137 kHz ≈2.3*ν*_{R}. Furthermore, the robustness to offset of these
PRESTO sequences is comparable to that of
RINEPT-CWc-$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(270_{0}90_{180})
(Fig. 8). PRESTO-$\mathrm{R}{\mathrm{22}}_{\mathrm{4}}^{\mathrm{3}}$(180_{0}) and
PRESTO-$\mathrm{R}{\mathrm{16}}_{\mathrm{3}}^{\mathrm{2}}$(180_{0}) sequences with a small phase shift of
2*ϕ*≤52^{∘} are less efficient, because they are sensitive
to rf inhomogeneity.

### 4.4.2 AlPO_{4}-14

In the case of AlPO_{4}-14, the relative transfer efficiencies for
^{27}AlO_{4} species follow a similar order as for *γ*-alumina,
except that the transfer efficiencies of
PRESTO-$\mathrm{R}{\mathrm{16}}_{\mathrm{7}}^{\mathrm{6}}$(270_{0}90_{180}) and
PRESTO-$\mathrm{R}{\mathrm{14}}_{\mathrm{6}}^{\mathrm{5}}$(270_{0}90_{180}) are
significantly lower than that of
RINEPT-CWc-$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(270_{0}90_{180})
(Table 4). This decreased efficiency of the PRESTO schemes for AlO_{4}
stems notably from the dipolar truncation, which prevents the transfer of
magnetization from the OH groups bonded to AlO_{5} and AlO_{6} sites to
AlO_{4}, since these ^{27}AlO_{4} nuclei are significantly more
distant from protons (see Table S5). Furthermore, the amplitude-modulated
$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(270_{0}90_{180}) recoupling
benefits from a higher robustness to rf field inhomogeneity than the PRESTO
schemes (Fig. S4). Conversely, the robustness to offset of these three
sequences are comparable (Fig. S5), whereas the rf requirements of
$\mathrm{R}{\mathrm{16}}_{\mathrm{7}}^{\mathrm{6}}$(270_{0}90_{180}) and
$\mathrm{R}{\mathrm{14}}_{\mathrm{6}}^{\mathrm{5}}$(270_{0}90_{180}) are much
lower than that of
$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(270_{0}90_{180}).

In summary, at *ν*_{R}=62.5 kHz, for both *γ*-alumina and
isopropylamine-templated AlPO_{4}-14,
PRESTO-$\mathrm{R}{\mathrm{16}}_{\mathrm{7}}^{\mathrm{6}}$(270_{0}90_{180}) and
RINEPT-CWc-$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(270_{0}90_{180})
are the best methods to transfer the polarization of protons to quadrupolar
nuclei. However, the first sequence requires a much lower rf field than the
second does.

## 4.5 Decay of transverse ^{1}H magnetization during recoupling

We also measured the decay of the ^{1}H transverse magnetization during a
spin echo experiment, in which the refocusing *π* pulse was the composite
one employed in the defocusing part of the RINEPT-CW sequence shown in Fig. 1b. We performed these experiments on AlPO_{4}-14 since the
^{1}H–^{1}H dipolar interactions are larger in this sample than in
*γ*-alumina. This decay was measured either in the absence of any
recoupling or under a $\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$
recoupling built from (180_{0}), (270_{0}90_{180}) or (tt) inversion
element. The three ^{1}H signals featured a mono-exponential decay with a
time constant ${T}_{\mathrm{2}}^{\prime}$ reported in Table 5.

At *ν*_{R}=20 kHz, the ${T}_{\mathrm{2}}^{\prime}$ constants are significantly shorter under $\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(180_{0}) and
$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(270_{0}90_{180})
than without recoupling. This faster decay can stem from the reintroduction
of ^{1}H–^{1}H dipolar interactions in the second- and higher-order
terms of the AH by the recoupling as well as the effect of pulse transients
(Wittmann et al., 2016). Conversely, the ${T}_{\mathrm{2}}^{\prime}$
constants under $\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(tt)
are much longer than without recoupling, showing that the adiabatic pulses
using large rf field efficiently decouple the ^{1}H–^{1}H dipolar
interactions, whereas the continuous variation of the phase and amplitude
during these pulses minimizes the transients.

At *ν*_{R}=62.5 kHz, the ${T}_{\mathrm{2}}^{\prime}$ constants without recoupling are
lengthened with respect to those at *ν*_{R}=20 kHz since faster MAS better averages the ^{1}H–^{1}H dipolar interactions
(Mao et al., 2009). Conversely, the ${T}_{\mathrm{2}}^{\prime}$ constants
under $\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(270_{0}90_{180})
recoupling are shorter at *ν*_{R}=62.5 than at 20 kHz.
This counterintuitive reduction may stem from the shorter pulse lengths at
*ν*_{R}=62.5 kHz, which result in a larger number of transients.
Indeed, the recoupling time, *τ*, only depends on the sample; hence,
the number of transients is proportional to *ν*_{R}, because the
recoupling sequences are rotor synchronized. Moreover, it also increases
with the use of composite pulses and as a result there are 6.25 times more
transients at *ν*_{R}=62.5 kHz and (270_{0},90_{180}) pulses
than at 20 kHz MAS and (180_{0}) pulses. Additionally, the rf power
increases with the spinning speed and the use of composite pulses, and then
also the amplitude of the transients. For the same reason, the ${T}_{\mathrm{2}}^{\prime}$
constants under $\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(180_{0}) are only
slightly longer at high MAS frequency. The ${T}_{\mathrm{2}}^{\prime}$ constants under
$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(tt) recoupling are much shorter
at *ν*_{R}=62.5 than at 20 kHz, because the adiabaticity criterion is not fulfilled at *ν*_{R}=62.5 kHz; hence, the elimination of
^{1}H–^{1}H dipolar interactions is less effective (Figs. 2f and S4).

## 4.6 2D ^{1}H–^{27}Al *D*-HETCOR of AlPO_{4}-14

Figure 9 demonstrates the possibility to acquire 2D ^{1}H–^{27}Al
*D*-HETCOR spectra using
RINEPT-CWc-$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(270_{0},90_{180})
transfer at *ν*_{R}=62.5 kHz. This spectrum was recorded using a
NUS scheme retaining 25 % of the *t*_{1} points, which would be acquired
using uniform sampling. In this spectrum, the CH proton only correlates with
the AlO_{4} site since it is too distant from AlO_{5} and AlO_{6}
sites (see Table S5). The other two ^{1}H signals correlate with the three
Al environments. Similar 2D spectra (not shown) were acquired using
RINEPT-CWc transfer based on $\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(180_{0}) and
$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(tt) recoupling as well as
PRESTO-$\mathrm{R}{\mathrm{16}}_{\mathrm{7}}^{\mathrm{6}}$(270_{0}90_{180}).
Their skyline projections are shown in Figs. S6 and S7.

In this work, we have introduced novel symmetry-based heteronuclear dipolar
recoupling schemes, which can be incorporated into the RINEPT and PRESTO
sequences to transfer the magnetization from protons to half-integer
quadrupolar nuclei at *ν*_{R}=20 or 62.5 kHz. These new recouplings
have been compared to the existing ones. We have shown that the
RINEPT-CWc-$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(tt) sequence with
adiabatic pulses, which produces efficient and robust transfers at *ν*_{R}≈10–15 kHz (Nagashima et
al., 2020), requires rf fields incompatible with the specifications of most
MAS probes for *ν*_{R}≥20 kHz. Conversely, the introduced
RINEPT-CWc-$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(270_{0}90_{180})
and PRESTO-$\mathrm{R}{\mathrm{22}}_{\mathrm{2}}^{\mathrm{7}}$(180_{0}) techniques
with rf fields of ca. 4*ν*_{R} and 5.5*ν*_{R}, respectively, are
the methods of choice at *ν*_{R}=20 kHz to transfer the
magnetization from protons to quadrupolar nuclei. At *ν*_{R}=62.5 kHz, the
RINEPT-CWc-$\mathrm{SR}{\mathrm{4}}_{\mathrm{1}}^{\mathrm{2}}$(270_{0}90_{180})
and PRESTO-$\mathrm{R}{\mathrm{16}}_{\mathrm{7}}^{\mathrm{6}}$(270_{0}90_{180})
sequences with rf requirements of ca. 4*ν*_{R} and 2.3*ν*_{R},
respectively, result in the most robust and efficient transfers. At both MAS
frequencies, the RINEPT and PRESTO techniques complement each other since
the latter is dipolar truncated, whereas the former is not. As a result, the
RINEPT sequences must be chosen to observe simultaneously protonated and
unprotonated sites, whereas the PRESTO schemes can be employed for the
selective observation of quadrupolar nuclei in proximity to protons. These
techniques are expected to be useful for transferring the DNP-enhanced
magnetization of protons to quadrupolar nuclei in indirect MAS DNP
experiments at *ν*_{R}≥20 kHz, notably used at high magnetic
fields (Nagashima
et al., 2021, 2020; Rankin et al., 2019; Berruyer et al., 2020). We also
show that they can be used to correlate the NMR signals of protons and
quadrupolar nuclei at high MAS frequencies.

The raw data are available on the Zenodo site at https://doi.org/10.5281/zenodo.4896852 (Gómez et al., 2021).

The supplement related to this article is available online at: https://doi.org/10.5194/mr-2-447-2021-supplement.

JSG, AGMR and JT carried out the NMR
experiments on *γ*-alumina and AlPO_{4}-14. YT performed the spin
dynamics simulations and carried out the NMR experiments on
L-histidine ⋅ HCl. OL derived average Hamiltonian theory for the
investigated recoupling sequences. HN and FP contributed to the analysis and interpretation of the results. OL and JPA wrote the article. All the authors contributed to the editing of the article.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Geoffrey Bodenhausen Festschrift”. It is not associated with a conference.

This article is dedicated to Francis Taulelle, our friend, who passed away very recently. The Chevreul Institute (FR 2638), Ministère de l'Enseignement Supérieur, de la Recherche et de l'Innovation, Hauts-de-France Region, and FEDER are acknowledged for supporting and funding partially this work. Olivier Lafon acknowledges financial support from Institut Universitaire de France (IUF) and contract ANR-18-CE08-0015-01 (ThinGlass). Frédérique Pourpoint acknowledges financial support from I-site contract OPE-2019-0043 (5400-MOFFIN).

This research has been supported by the IR-RMN-THC FR-3050 CNRS, the European Union's Horizon 2020 research and innovation program (EUSMI, grant no. 731019), Institut Universitaire de France (IUF), Agence Nationale de la Recherche (contract ANR-18-CE08-0015-01, ThinGlass), and I-site UNLE (contract OPE-2019-0043, 5400-MOFFIN).

This paper was edited by Fabien Ferrage and reviewed by one anonymous referee.

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