Hiroki Nagashima

Aany Sofia

Lilly Thankamony

Julien Trébosc

Lionel Montagne

Gwendal Kerven

Jean-Paul Amoureux
email: jean-paul.amoureux@univ-lille.fr

Olivier Lafon
email: olivier.lafon@univ-lille.fr

Sofia Lilly Aany

Thankamony

Observation of proximities between spin-1/2 and quadrupolar nuclei in solids: Improved robustness to chemical shielding using adiabatic symmetry-based recoupling

Keywords: quadrupolar isotope, adiabatic pulse, heteronuclear dipolar recoupling, 207 Pb-11 B correlation

come

recently, we have reported the possibility to observe dipolar couplings between 77 Se and 71 Ga nuclei in gallium selenide crystals and glasses using NMR [START_REF] Nagashima | 71 Ga-77 Se connectivities and proximities in gallium selenide crystal and glass probed by solidstate NMR[END_REF].

For the studies which are listed above, NMR spectra were acquired under Magic-Angle Spinning (MAS) in order to improve their resolution and their signal-to-noise ratio. Under MAS, the dipolar couplings are averaged out and they have to be reintroduced by recoupling schemes. This can be done using cross-polarization (CP) [START_REF] Vega | MAS NMR spin locking of half-integer quadrupolar nuclei[END_REF]. However, CP transfers under MAS (CPMAS) suffer from several limitations when applied to quadrupolar nuclei [START_REF] Vega | CP/MAS of quadrupolar S = 3/2 nuclei[END_REF][START_REF] Amoureux | Theoretical and experimental assessment of single-and multiple-quantum cross-polarization in solid state NMR[END_REF]. First, they exhibit a low efficiency since the magnetization of all the crystallites cannot be spin-locked simultaneously for quadrupolar nuclei [START_REF] Vega | MAS NMR spin locking of half-integer quadrupolar nuclei[END_REF]. Second, for half-integer spin quadrupolar nuclei, CPMAS is often sensitive to resonance offset and Chemical Shift Anisotropy (CSA) of this nucleus since the selective spin-lock of its Central-Transition (CT) requires low radiofrequency (rf) field [START_REF] Amoureux | Theoretical and experimental assessment of single-and multiple-quantum cross-polarization in solid state NMR[END_REF]. Furthermore, CPMAS experiments involving halfinteger spin quadrupolar nuclei require a previous optimization of the spin-lock of this nucleus since it is inefficient at the Rotary Resonance Recoupling (R 3 ) conditions, i.e. when the nutation frequency of the CT is a multiple of the MAS frequency, R [START_REF] Ashbrook | Spin-locking of half-integer quadrupolar nuclei in nuclear magnetic resonance of solids: second-order quadrupolar and resonance offset effects[END_REF]. Finally, the CPMAS technique may not be efficient for all sites since its efficiency largely depends on the quadrupolar coupling constant, C Q [START_REF] Tricot | Structural characterisation of phosphate materials: new insights into the spatial proximities between phosphorus and quadrupolar nuclei using the D-HMQC MAS NMR technique[END_REF].

These shortcomings can be alleviated by the use of techniques that reintroduce the dipolar couplings by only irradiating the spin-1/2 isotopes, thus limiting the interference of the quadrupole interaction with the recoupling sequence. For such case, the heteronuclear dipolar couplings have been reintroduced using techniques, such as REDOR (Rotational-Echo DOuble Resonance) [START_REF] Gan | 13 C/14N heteronuclear multiple-quantum correlation with rotary resonance and REDOR dipolar recoupling[END_REF], SFAM (Simultaneous Frequency and Amplitude Modulation) [START_REF] Hu | Comparison of several hetero-nuclear dipolar recoupling NMR methods to be used in MAS HMQC/HSQC[END_REF][START_REF] Lafon | Indirect Detection via Spin-1/2 Nuclei in Solid State NMR Spectroscopy: Application to the Observation of Proximities between Protons and Quadrupolar Nuclei[END_REF][START_REF] Lu | Observation of proximities between spin-1/2 and quadrupolar nuclei: which heteronuclear dipolar recoupling method is preferable?[END_REF], R 3 with rf-field amplitude 1 = R [START_REF] Trebosc | Through-space R3-HETCOR experiments between spin-1/2 and half-integer quadrupolar nuclei in solid-state NMR[END_REF], and symmetrybased schemes, such as R [START_REF] Nagashima | Independent through-space hetero-nuclear correlation between spin-1/2 and quadrupolar nuclei in solids[END_REF]. All these recoupling techniques reintroduce the space component 16 5

3

| | of the heteronuclear dipolar interaction. The REDOR and SFAM schemes are = 1 whereas R 3 and R are [START_REF] Martineau | High-resolution through-space correlations between spin-1/2 and half-integer quadrupolar nuclei using the MQ-D-R-INEPT NMR experiment[END_REF]. 16 5 3 In order to measure the strength of the dipolar coupling between spin-1/2 and quadrupolar nuclei, and hence their distances, these heteronuclear recoupling schemes have been employed in dipolar dephasing experiments, such as REDOR [START_REF] Fyfe | Dipolar dephasing between quadrupolar and spin-nuclei. REDOR and TEDOR NMR experiments on VPI-5[END_REF], its frequency-selective variant [START_REF] Trebosc | Simultaneous frequency-selective solid-state NMR analysis of internuclear distances and through-bond connectivities in the presence of quadrupolar nuclei[END_REF][START_REF] Pourpoint | Quantitative Analysis of the Proximities of OH Ligands and Vanadium Sites in a Polyoxovanadate Cluster Using Frequency-Selective 1 H-51 V Solid-State NMR Spectroscopy[END_REF], TRAPDOR (TRAnsfer of Populations in DOuble Resonance) [START_REF] Van Eck | A novel application of nuclear spin-echo double-resonance to aluminophosphates and aluminosilicates[END_REF][START_REF] Grey | The detection of weak heteronuclear coupling between spin 1 and spin 1/2 nuclei in MAS NMR; 14 N/ 13 C/ 1 H triple resonance experiments[END_REF][START_REF] Grey | Rotational echo 14 N/ 13 C/ 1 H triple resonance solid state nuclear magnetic resonance: A probe of 13 C-14 N internuclear distances[END_REF][START_REF] Grey | Determination of the Quadrupole Coupling Constant of the Invisible Aluminum Spins in Zeolite HY with 1 H-27 Al TRAPDOR NMR[END_REF], REAPDOR (Rotational-Echo Adiabatic-Passage DOuble-Resonance) [START_REF] Gullion | Measurement of dipolar interactions between spin-1/2 and quadrupolar nuclei by rotational-echo, adiabatic-passage, double-resonance NMR[END_REF], SPIDER (Saturation-Pulse Induced Dipolar Exchange with Recoupling) [START_REF] Schmidt-Rohr | Selective observation of nitrogen-bonded carbons in solid-state NMR by saturation-pulse induced dipolar exchange with recoupling[END_REF], RESPDOR (Rotational-Echo Saturation-Pulse DOuble-Resonance) [START_REF] Gan | Measuring amide nitrogen quadrupolar coupling by high-resolution 14 N/ 13 C NMR correlation under magic-angle spinning[END_REF][START_REF] Chen | Measurement of hetero-nuclear distances using a symmetry-based pulse sequence in solid-state NMR[END_REF][START_REF] Chen | Distance measurement between a spin-1/2 and a half-integer quadrupolar nuclei by solid-state NMR using exact analytical expressions[END_REF][START_REF] Lu | Detailed analysis of the S-RESPDOR solidstate NMR method for inter-nuclear distance measurement between spin-1/2 and quadrupolar nuclei[END_REF], and PM-RESPDOR its phase-modulated version [START_REF] Nimerovsky | Efficient rotational echo double resonance recoupling of a spin-1/2 and a quadrupolar spin at high spinning rates and weak irradiation fields[END_REF][START_REF] Nimerovsky | Insights into the spin dynamics of a large anisotropy spin subjected to long-pulse irradiation under a modified REDOR experiment[END_REF][START_REF] Nimerovsky | Phase-modulated LA-REDOR: a robust, accurate and efficient solid-state NMR technique for distance measurements between a spin-1/2 and a quadrupole spin[END_REF][START_REF] Nimerovsky | Analysis of large-anisotropy-spin recoupling pulses for distance measurement under magic-angle spinning NMR shows the superiority and robustness of a phase modulated saturation pulse[END_REF].

These single-channel heteronuclear recoupling techniques have also been used in D-HETCOR (Dipolar-mediated HETeronuclear CORrelation) sequences in order to identify unambiguously the proximities between distinct sites occupied by spin-1/2 and quadrupolar isotopes. These D-HETCOR sequences include the dipolar-mediated D-RINEPT (Refocused Insensitive Nuclei Enhanced by Polarization Transfer) [START_REF] Martineau | High-resolution through-space correlations between spin-1/2 and half-integer quadrupolar nuclei using the MQ-D-R-INEPT NMR experiment[END_REF][START_REF] Fyfe | Dipolar dephasing between quadrupolar and spin-nuclei. REDOR and TEDOR NMR experiments on VPI-5[END_REF][START_REF] Trebosc | Through-space R 3 -HETCOR experiments between spin-1/2 and half-integer quadrupolar nuclei in solid-state NMR[END_REF] and D-HMQC (Heteronuclear Multiple-Quantum Correlation) sequences [START_REF] Trebosc | Through-space R 3 -HETCOR experiments between spin-1/2 and half-integer quadrupolar nuclei in solid-state NMR[END_REF][START_REF] Gan | 13 C/ 14 N heteronuclear multiple-quantum correlation with rotary resonance and REDOR dipolar recoupling[END_REF][START_REF] Gan | Proton-detected 14 N MAS NMR using homonuclear decoupled rotary resonance[END_REF]. These two methods complement each other since the former employs direct detection, i.e. the excited and detected isotopes differ, whereas the latter uses indirect detection, i.e. the detected isotope is the same as the excited one. The relative sensitivities of the direct and indirect detections depend on the gyromagnetic ratios, the longitudinal relaxation times, T 1 , and the spectral widths of the correlated nuclei [START_REF] Ishii | Sensitivity Enhancement in Solid State 15 N NMR by Indirect Detection with High-Speed Magic Angle Spinning[END_REF]. Recently, we have also introduced the D-HUQC (Dipolar-mediated Heteronuclear Universal-Quantum Correlation) sequence [START_REF] Nagashima | Independent through-space hetero-nuclear correlation between spin-1/2 and quadrupolar nuclei in solids[END_REF]. This indirectly detected D-HETCOR experiment employing recoupling schemes improves the robustness to MAS fluctuations, hence reducing the t 1 -noise, when the detected spin-1/2 nuclei are subject to large CSA.

We focus herein on the single-channel heteronuclear recoupling, which generally exhibits higher robustness to rf field inhomogeneity and resonance offset than the ones. In practice, given the inhomogeneity of the rf-field in the sample space [START_REF] Gupta | RF inhomogeneity and how it controls CPMAS[END_REF][START_REF] Purusottam | Effects of inherent rf field inhomogeneity on heteronuclear decoupling in solid-state NMR[END_REF][START_REF] Nagashima | Imaging the spatial distribution of radiofrequency field, sample and temperature in MAS NMR rotor[END_REF][START_REF] Odedra | Imaging of the B 1 distribution and background signal in a MAS NMR probehead using inhomogeneous B 0 and B 1 fields[END_REF][START_REF] Tošner | Radiofrequency fields in MAS solid state NMR probes[END_REF], the recoupling often results in higher sensitivity than the ones. So far, no detailed comparison of the efficiency and the robustness of the two most employed heteronuclear recoupling techniques, REDOR and SFAM, has been reported in the literature.

It has been shown that SFAM recoupling features high robustness to resonance offset and spatial rfinhomogeneity. Such robustness stems from the modulation of the carrier frequency. However, the sinusoidal time-modulations of rf-field amplitude and phase are not designed to achieve adiabatic inversions. It has been shown that broadband inversions, which are very tolerant to the rfinhomogeneity, can be achieved in solids using frequency-swept adiabatic pulses [START_REF] Heise | REDOR with adiabatic dephasing pulses[END_REF][START_REF] Leppert | REDOR: an assessment of the efficacy of dipolar recoupling with adiabatic inversion pulses[END_REF][START_REF] Hardy | Fast-MAS total through-bond correlation spectroscopy using adiabatic pulses[END_REF][START_REF] Leppert | Broadband RFDR with adiabatic inversion pulses[END_REF][START_REF] Riedel | Homonuclear chemical shift correlation in rotating solids via RNnu n symmetry-based adiabatic RF pulse schemes[END_REF][START_REF] Riedel | TEDOR with adiabatic inversion pulses: Resonance assignments of 13 C/ 15 N labelled RNAs[END_REF][83][START_REF] Riedel | Tailoring broadband inversion pulses for MAS solid state NMR[END_REF][START_REF] Riedel | Broadband homonuclear double-quantum NMR/filtering via zero-quantum dipolar recoupling in rotating solids[END_REF][START_REF] Kervern | Fast adiabatic pulses for solid-state NMR paramagnetic systems[END_REF][START_REF] Deschamps | Superadiabaticity in magnetic resonance[END_REF][START_REF] Siegel | Sensitivity enhancement of NMR spectra of half-integer spin quadrupolar nuclei in solids using hyperbolic secant pulses[END_REF][START_REF] Nakashima | Sensitivity enhancement of solid-state NMR spectra of half-integer spin quadrupolar nuclei: Double-or single-frequency sweeps? Insights from the hyperbolic secant experiment[END_REF][START_REF] Siegel | Signal enhancement of NMR spectra of half-integer quadrupolar nuclei in solids using hyperbolic secant pulses[END_REF][START_REF] Pell | Broadband inversion for MAS NMR with single-sideband-selective adiabatic pulses[END_REF][START_REF] Sanders | Broadband MAS NMR spectroscopy in the low-power limit[END_REF]. These pulses have notably been used as basic element for symmetry-based C and R recoupling [START_REF] Heise | REDOR with adiabatic dephasing pulses[END_REF][START_REF] Leppert | REDOR: an assessment of the efficacy of dipolar recoupling with adiabatic inversion pulses[END_REF][START_REF] Hardy | Fast-MAS total through-bond correlation spectroscopy using adiabatic pulses[END_REF][START_REF] Leppert | Broadband RFDR with adiabatic inversion pulses[END_REF][START_REF] Riedel | Homonuclear chemical shift correlation in rotating solids via RNnu n symmetry-based adiabatic RF pulse schemes[END_REF][START_REF] Riedel | TEDOR with adiabatic inversion pulses: Resonance assignments of 13 C/ 15 N labelled RNAs[END_REF][83][START_REF] Riedel | Tailoring broadband inversion pulses for MAS solid state NMR[END_REF][START_REF] Riedel | Broadband homonuclear double-quantum NMR/filtering via zero-quantum dipolar recoupling in rotating solids[END_REF], and notably the REDOR xy4 scheme [START_REF] Heise | REDOR with adiabatic dephasing pulses[END_REF][START_REF] Leppert | REDOR: an assessment of the efficacy of dipolar recoupling with adiabatic inversion pulses[END_REF][START_REF] Riedel | TEDOR with adiabatic inversion pulses: Resonance assignments of 13 C/ 15 N labelled RNAs[END_REF], which is based on R symmetry [START_REF] Levitt | Encyclopedia of Nuclear Magnetic Resonance[END_REF]. However, to the 4 1 2 best of our knowledge, this adiabatic REDOR scheme has never been employed to reintroduce dipolar couplings between spin-1/2 and quadrupolar nuclei.

In the present article, we introduce a heteronuclear dipolar recoupling sequence based on R symmetry using an adiabatic pulse as basic element. In the case of D-HMQC experiment with 2 1 1 indirectly detected spin-1/2 nucleus, we compare its performances to those of already reported encoded heteronuclear dipolar recoupling schemes: REDOR, SFAM and R with adiabatic inversion 4 1 2 pulses. The performances are compared using spin dynamics numerical simulations and experiments between indirectly detected 31 P nuclei and directly detected 27 Al ones in microporous crystalline aluminophosphate VPI-5. The robustness of D-HMQC sequence using an adiabatic R symmetry-2 1 1 based recoupling is used to record the first 207 Pb-{ 11 B} D-HETCOR 2D spectra on a lead borate Pb 4 O(BO 3 ) 2 crystal used for non-linear optics [START_REF] Yu | A new congruent-melting oxyborate, Pb4O(BO3)2 with optimally aligned BO3 triangles adopting layered-type arrangement[END_REF].

II. Pulse sequences and theory

We consider hereafter that the recoupling scheme is always applied to the spin I = 1/2.

II-1. REDOR

In the REDOR scheme, two are applied per rotor period, R = R (see Fig. 1b). The lengths of the are denoted p in the following and their phases are usually set to the xy-4 (xyxy) or xy-8 (xyxy yxyx) schemes [START_REF] Gullion | New, compensated Carr-Purcell sequences[END_REF]. The first sequence, REDOR xy4 , corresponds to R symmetry, whereas 4 1 2 the second one, REDOR xy8 , corresponds to the phase inversion super-cycle R R [START_REF] Levitt | Encyclopedia of Nuclear Magnetic Resonance[END_REF].

4 1 2 4 1 2
The symmetry relationships between the mechanical and the rf rotations allows generating a firstorder average Hamiltonian containing the desired rotational components of the spin interactions, while other components are suppressed. A rotational component is defined by four quantum numbers: the space rank, l, the spin rank, and the indexes m and running from -l to +l and -to + , respectively, in integer steps.

The REDOR sequence reintroduces the components with and = 0 of the heteronuclear | | = 1 dipolar interaction and the CSA of the spin-I nucleus (CSA I ), whereas the spin-I resonance offset, and the heteronuclear J-couplings are suppressed in the first-order average Hamiltonian. This sequence does not suppress the homonuclear dipolar interaction between the spin-I nuclei. The contribution of I-S dipolar interaction to the REDOR first-order average Hamiltonian is equal to [START_REF] Jaroniec | Dipolar Recoupling: Heteronuclear[END_REF] .

(1) (1) =

The magnitude, D,IS , of the recoupled I-S coupling in Eq.1 is given by (2) = sin( 2)cos( ) with a scaling factor equal to: the internuclear I-S vector to the MAS rotor-fixed frame, f = p R denotes the pulse fraction, 0 denotes the initial rotor position, R = R , and t 0 refers to the starting time of the symmetry-based scheme. According to Eq.3, for Dirac pulses, we have f = 0 and = , whereas for windowless rf 2 irradiation f = 1 and = . 1/(2 2 ) As seen in Eqs. 1 to 4, the norm of depends on the angle and the REDOR scheme is hence

(3) = 2 cos ( 2 )
(1)
The recoupled Hamiltonian of Eq.1 commutes among different spin-pairs and hence, REDOR is not affected by dipolar truncation. This facilitates the observation of long inter-nuclear distances. Furthermore, the CSA I term recoupled by such scheme is proportional to I z and thus commutes with the I-S dipolar interaction of Eq.1. Hence, the CSA I does not interfere with the evolution under I-S dipolar interaction during these recoupling schemes.

II-2. SFAM

We consider in the following the SFAM scheme with a frequency of the amplitude and phase modulations equal to the MAS frequency [START_REF] Fu | Recoupling of heteronuclear dipolar interactions in solid state magic-angle spinning NMR by simultaneous frequency and amplitude modulation[END_REF]. For such scheme, the instantaneous rf-amplitude is equal to:

(5)

( ) = sin [ ( 0 )]
where is the rf peak amplitude. The instantaneous rf-phase in radians is

(6) ( ) = sin [ ( 0 )]
where is the peak amplitude of the carrier pulsation modulation. In the frequency-modulated (FM) frame [START_REF] Garwood | The Return of the Frequency Sweep: Designing Adiabatic Pulses for Contemporary NMR[END_REF], rotating around the z-axis by an angle of with respect to the usual rotating frame, ( ) the deviation of the instantaneous carrier pulsation from its average value is equal to:

(7) ( ) = cos [ ( 0 )]
In this FM frame, the effective rf-field has a magnitude of ), the tilt angle of the effective field from the z-axis of the FM frame, is defined by and .

(

) cos [ )] = ( ) ( ) sin [ )] = ( ) ( ) 10
SFAM achieves an efficient and robust recoupling when , which corresponds to the ( ) ( )

'
. ( 12
) 1 = 2 0 + 0 cos [ )]cos [ ( 0 )] C 1 is a function of the parameter = / . ( 13
)
1 0 For = =1), C 1 = 1 and the scaling factor of SFAM is identical to that of the windowless REDOR scheme.

Apart from the scaling factor, REDOR and SFAM schemes have identical properties, i.e. they (i) achieve recoupling of the space component of the heteronuclear dipolar | | = 1 interaction and CSA I , (ii) eliminate the spin-I resonance offset and the heteronuclear J-couplings, (iii) do not suffer from dipolar truncation and interference of the CSA I , and (iv) do not eliminate the I-I dipolar interactions in the first-order average Hamiltonian [START_REF] Lafon | Indirect Detection via Spin-1/2 Nuclei in Solid State NMR Spectroscopy: Application to the Observation of Proximities between Protons and Quadrupolar Nuclei[END_REF].

In ref. [START_REF] Lu | Observation of proximities between spin-1/2 and quadrupolar nuclei: which heteronuclear dipolar recoupling method is preferable?[END_REF], we have calculated the trajectories of I-spin magnetization during the basic element of SFAM. These trajectories indicate that efficient SFAM recoupling can be achieved when the I-spin magnetization is inverted twice per rotor period. In that case, SFAM recoupling corresponds to R 2 1 1 symmetry.

II-3. Adiabatic symmetry-based recoupling

The adiabatic recoupling schemes investigated in this article are based on the R symmetry, i.e. As basic element, we used the tanh/tan (tt) adiabatic inversion pulse [START_REF] Hwang | Fast broadband inversion by adiabatic pulses[END_REF] with the R (tt) and R 4 1 (tt) recoupling schemes. This tanh/tan pulse is well suited for the purpose of fast broadband inversion and has already been employed as SHAP (Short High-power Adiabatic Pulses) to achieve broadband inversion in solids [START_REF] Leppert | REDOR: an assessment of the efficacy of dipolar recoupling with adiabatic inversion pulses[END_REF][START_REF] Hardy | Fast-MAS total through-bond correlation spectroscopy using adiabatic pulses[END_REF][START_REF] Leppert | Broadband RFDR with adiabatic inversion pulses[END_REF][START_REF] Riedel | Homonuclear chemical shift correlation in rotating solids via RNnu n symmetry-based adiabatic RF pulse schemes[END_REF][START_REF] Riedel | TEDOR with adiabatic inversion pulses: Resonance assignments of 13 C/ 15 N labelled RNAs[END_REF][83][START_REF] Riedel | Tailoring broadband inversion pulses for MAS solid state NMR[END_REF][START_REF] Riedel | Broadband homonuclear double-quantum NMR/filtering via zero-quantum dipolar recoupling in rotating solids[END_REF][START_REF] Kervern | Fast adiabatic pulses for solid-state NMR paramagnetic systems[END_REF], notably in REDOR schemes [START_REF] Leppert | REDOR: an assessment of the efficacy of dipolar recoupling with adiabatic inversion pulses[END_REF][START_REF] Riedel | TEDOR with adiabatic inversion pulses: Resonance assignments of 13 C/ 15 N labelled RNAs[END_REF][START_REF] Kervern | Fast adiabatic pulses for solid-state NMR paramagnetic systems[END_REF]. During a tanh/tan pulse, the instantaneous rf-amplitude is equal to:

(14) ( ) = tanh [ ] 0 < /2 tanh [ (1 )] /2 <
where refers to the time since the start of the pulse, which lasts p = R /2, and determines = 0 the rise and fall time of the pulse. The instantaneous rf-phase in radians is .

(15) ( ) = tan( ) ln cos 1 2 ' In the FM frame, the frequency offset is .

(

) ( ) = tan( ) tan 1 2 ' 16
The tanh/tan pulse produces a linear frequency sweep in the centre of the pulse [START_REF] Kervern | Fast adiabatic pulses for solid-state NMR paramagnetic systems[END_REF]:

. ( 17
) d d ' = /2 = tan( )
For given and p values, Eq.17 shows that the parameter determines the frequency sweep rate in the centre of the pulse. Here, we employed = 10 and = 1.52 rad = 87° = 20), as in refs. [START_REF] Leppert | REDOR: an assessment of the efficacy of dipolar recoupling with adiabatic inversion pulses[END_REF][START_REF] Kervern | Fast adiabatic pulses for solid-state NMR paramagnetic systems[END_REF]. We discuss below the choice of these parameters. Just as SFAM, the effective rf-field during the tanh/tan pulse has a magnitude given by Eq.8 and a tilt angle ) defined by Eq.10. It has been shown that the tanh/tan pulse can achieve an efficient inversion of the magnetization of the irradiated nuclei, whenever this magnetization is locked along the effective field in any of the superadiabatic frames [START_REF] Deschamps | Superadiabaticity in magnetic resonance[END_REF]. A criterion to judge the efficiency of the locking is the super-adiabaticity factor,

S Q = max n (Q n ) ( 18
)
where Q n is the n th adiabaticity factor, which characterizes the efficiency of the magnetization locking in the n th adiabatic frame. An efficient magnetization inversion is achieved when S Q >> 1. An analytical expression of the Q 1 factor can be derived for a tanh/tan pulse. It can be shown that in the absence of any anisotropic interaction and for on-resonance irradiation ( = 0), i.e. the carrier frequency in the centre of the pulse is equal to the resonance frequency, the Q 1 factor of tanh/tan pulses in R (tt) and

2 1 1 R (tt) schemes is equal to [86] 4 1 2 (19) 1 = 0) = ( ) 2 ( ) tanh 2 ( ) = ( ) tanh 2 ( )
Therefore, a large ratio between rf-peak amplitude and spinning speed, / R , favors adiabatic inversions, whereas a too large frequency sweep, , can prevent an efficient magnetization inversion.

For a nucleus with no CSA I but a resonance offset, with respect to the average carrier frequency, the adiabaticity of tanh/tan pulse is generally the weakest at the time, t , when the resonance condition is achieved, . At this time, the effective field, sum of the offset field, I z , and the effective

( ) =
rf-field, is orthogonal to B 0 and its magnitude is equal to provided is significantly smaller than . Under this condition, the Q 1 factor is equal to

Q 1 ( ) = Q 1 ( = 0)/{ } (20) 1 + tan( ) 0 2
Hence, the and parameters control the robustness to offset of tanh/tan pulses. Furthermore, when the I nuclei are subject to large anisotropic interactions, their resonance frequency is modulated by MAS. When such modulation is significant with respect to the effective field, it can prevent an efficient spin locking and magnetization inversion. Therefore, for a tanh/tan pulse applied on-resonance for I nuclei, it has been shown that when the frequency modulation produced by an axially symmetric CSA I tensor dominates the offset frequency sweep [START_REF] Kervern | Fast adiabatic pulses for solid-state NMR paramagnetic systems[END_REF]:

(21) 1 = ( ) 2 aniso tanh 2 ( )
where denotes the anisotropic chemical deshielding constant of the I spin and 0I is its Larmor aniso frequency.

However, it has been shown that the Q 1 factor is not sufficient to judge the adiabaticity of a process and that an efficient inversion using a tanh/tan pulse can be achieved with Q 1 close to 1, provided there exists an adiabatic frame for which Q n >> 1. These factors were calculated with 1 n 16 for the tanh/tan pulse used in R (tt) and R (tt) schemes according to the procedure described in ref.

[87] 2 1 1 4 1 2 (see Fig. 2). The frequency modulation produced by the CSA I , as well as the frequency offset, were taken into account. For all conditions tested in Fig. 2, the Q n values exhibit a maximum for 1 < n < 16, which corresponds to the S Q factor given in Eq.18, and which is one to two orders of magnitude larger than Q 1 .

The n value corresponding to the maximum of Q n is roughly proportional to / R [START_REF] Deschamps | Superadiabaticity in magnetic resonance[END_REF]. For all the conditions tested in Fig. 2,S Q >> 1 and the tanh/tan pulse hence achieves an efficient magnetization inversion.

We did similar calculations for SFAM recoupling and also obtained S Q >> 1. Therefore, the SFAM scheme can be considered as a R symmetry-based recoupling using half-cosine and half-sine shapes 2 1 1 for rf-field amplitude and frequency sweeps, respectively. Nevertheless, the S Q factors for the tanh/tan pulse are significantly greater than those of SFAM. This result is expected since tanh/tan pulse is a better adiabatic pulse than the cos/sin one. The comparison of Figs. 2a andb shows how the S Q factors of both recoupling schemes are strongly reduced in the case of offset irradiation. Nevertheless, such decrease can be compensated by the use of higher rf-field, i.e. larger value, especially with R (tt) and R 2 1 1 (tt) schemes (Fig. 2c). The comparison of Figs. 2c andd indicates that the S Q factor decreases for 4 1 2 increasing CSA I , when the frequency modulation produced by CSA I dominates the offset frequency sweep, in agreement with Eq.21. The comparison of Figs. 2c ande shows that the S Q factor decreases for increasing MAS frequency, the other parameters being unchanged. Such trend agrees with Eqs. 19 and 21.

Assuming Q 1 >> 1, i.e. the magnetization of the I nuclei is locked along the effective field in the first adiabatic frame, the scaling factor of the rotational component corresponding to quantum numbers {l, m, recoupled by R (tt) and R (tt) schemes can be calculated as [100]

2 1 1 4 1 2 (22) = 1 ( ) 0 [ )]exp ) + d
where denotes a reduced Wigner element, is the angle between B 0 and the rotor axis.

( )

Here, the rotor was spun at the magic angle and . In R (tt) and R (tt) schemes, = arctan( 2)

2 1 1 4 1 2
the effective field and the spin magnetization are inverted during half a rotor period. Hence, the average nutation frequency, which corresponds to the average time-variation of , is R . Typically ) tanh/tan pulses use >> R , and hence, for , the complex exponential of oscillates 0 ( ) much faster than the reduced Wigner element in Eq. 22, and hence the integral of this rapidly oscillating function cancels out. Therefore, the rotational components with for R (tt) and R (tt) 0 2 with = 0.05, = 10 and = atan (20) = 1.52 rad = 87°. Under those conditions, the scaling factor of these schemes is identical to that of windowless REDOR and SFAM with = 1). The = scaling factor is plotted as function of and parameters in Fig. 3a, b and c, respectively. As seen in Fig. 3a, monotonically decreases for increasing value. However, as seen in Eq.19,

Q 1 factor is | | proportional to
and hence a small values may prevent adiabatic inversion. Similarly | | monotonically decreases inversely to parameter (see Fig. 3c). However, Q 1 factor is proportional to tan 2 (see Eqs.19 and 21 and Fig. 3e) and small values are not compatible with adiabatic inversion. In the case of the parameter, the Q 1 factor is maximal when approaches (see Eq.19 and Fig. 3d). Nevertheless, monotonically decreases over the interval [0, and cancels for = (Fig. 3b). | | Therefore, the optimal and values result from a compromise between the magnitude of the scaling factor and the adiabaticity factor, i.e. the efficiency of the inversion. Fig. 3f shows the variation of

Q 1 Q 1 = 0) and s Q s Q = 0
) ratios as function of the offset As expected, those ratios decrease for increasing offset. Furthermore, there is a reasonably good agreement between those ratios, especially for 75 kHz, which indicates that the dependence of s Q factor with offset is similar to that of Q 1 . Therefore, the robustness to offset of tanh/tan pulse mainly depends on the and parameters (see Eq. 20). a pair of coupled IS spins, the in-phase S-spin magnetization (S x ) evolves into antiphase one (2S y I z ) during the first period, i.e. the defocusing period. The first /2 I-pulse with phase y converts it into heteronuclear MQ coherences (2S y I x ), which evolve under the isotropic I shift during the indirect evolution time, t 1 . The -pulse applied on S-spin in the middle of the t 1 period refocuses the evolution of the S-spin magnetization under isotropic shift during the and t 1 delays as well as its evolution under I-S J-coupling. At the end of t 1 , the second /2 I-pulse converts back the MQ coherences into anti-phase S-spin magnetization and these coherences evolve during the second period, i.e. the refocusing period, into observable transverse S-spin magnetization. When both ±1-quantum I-spin coherences are selected during the t 1 period, the NMR signal of this experiment is proportional to [START_REF] Lu | Observation of proximities between spin-1/2 and quadrupolar nuclei: which heteronuclear dipolar recoupling method is preferable?[END_REF] (23)

II-4. D-HMQC sequence

( ) 1 2 1 2 4 1/4 ( ) 1/4 ( )
where J ±1/4 (x) denotes the Bessel functions of the first kind and ±1/4-order. In the absence of losses, the shortest value producing maximal S intensity is opt = ).

III. Simulations and experimental section III-1. Numerical simulations

All numerical simulations of spin dynamics were performed with the SIMPSON software (version 4.1.1) [START_REF] Bak | SIMPSON: a general simulation program for solid-state NMR spectroscopy[END_REF]. The powder average was calculated using 168 { MR , MR } pairs and 13 MR angles. The 168 { MR , MR } Euler angles, which relate the molecular and rotor frames, were selected according to the REPULSION algorithm [START_REF] Bak | REPULSION, a novel approach to efficient powder averaging in solid-state NMR[END_REF], while the MR angle was equally stepped from 0 to 360°. The spin system was an isolated 13 C-31 P spin pair. Simulations were performed for two spin-1/2 isotopes in order to limit the size of the density matrix and accelerate the simulations, since the duration of these simulations is proportional to the third power of the dimensionality of the spin space [START_REF] Veshtort | SPINEVOLUTION: a powerful tool for the simulation of solid and liquid state NMR experiments[END_REF], i.e. a simulation for an isolated 13 C-31 P spin pair is about 27-fold faster than for a 27 Al-31 P pair. Furthermore, as the sequence applied to the observed nucleus is simply an echo, the recoupling scheme applied to the 31 P channel can be analyzed by replacing the 27 Al nucleus by a 13 C one without loss of generality. The 13 C-31 P dipolar coupling constant was b IS = Hz, which corresponds to the typical 27 Al-31 P dipolar couplings for Al-O-P bonds found in aluminophosphate materials. The CSA 31P values are given in the figure captions and CSA 13C = 0. The CSA 31P asymmetry parameter was null and the orientation of the CSA 31P principal axis system with respect to the 13 C-31 P vector was described by the Euler angles (0, 30°, 0).

For all simulations, the static magnetic field was fixed at B 0 = 9.4 T ( 0,13C = 100.6 MHz, 0,31P = 161.9 MHz). The MAS frequency was R = 10 or 20 kHz. We simulated the D-HMQC sequence with 13 C detection and 31 P indirect detection (denoted 13 C-{ 31 P} hereafter) using the sequence displayed in Fig. 1a incorporating five heteronuclear dipolar recoupling: REDOR xy4 , REDOR xy8 , SFAM, R (tt) 2 1 1 and R (tt). These schemes, which were always applied to the 31 P channel, were compared using 4 1 2 identical peak values for the rf-field strength. For the simulations of 1D 13 C-{ 31 P} D-HMQC sequence, we selected only the +1-quantum 31 P coherence during t 1 . The transfer efficiency is defined with respect to the 13 C signal detected after a spin echo. Except during the recoupling period, the peak rf amplitudes were equal to 100 kHz during the pulses applied to 13 C and 31 P isotopes. The pulses on both channels were always applied on resonance, except in Fig. 6, where the offset between the carrier frequency on 31 P channel and the resonance frequency is varied. For the tanh/tan pulses, we used = 10 and = atan (20) = 87°, in order to achieve a good compromise between a large Q 1 factor and a large one (see | | section II-3 and Fig. 3). The other parameters, such as R , and are indicated in the figure captions.

III.2. NMR D-HMQC experiments

One 1D MAS and the 2D MQMAS 11 B spectra of Pb 4 O(BO 3 ) 2 , displayed in Figs. S1 andS2, were acquired on a narrow-bore 18.8 T Bruker spectrometer equipped with Avance III console. One 1D MAS 207 Pb spectrum of Pb 4 O(BO 3 ) 2 , displayed in Figs. S3 andS4, was acquired on a wide-bore 2.35 T Bruker spectrometer equipped with Avance II console. All other experimental results were performed on a wide-bore 9.4 T Bruker spectrometer equipped with Avance II console. The rotors were fully filled with the samples. The heteronuclear dipolar recoupling schemes were always applied on the spin-1/2 channel since the number of pulses applied to the quadrupolar nucleus must be minimized. For the tanh/tan pulses, we used = 10 and = 87° = atan [START_REF] Greiser | 29 Si{ 27 Al} TRAPDOR MAS NMR to distinguish Q n (mAl) sites in aluminosilicates. Test case: Faujasite-type zeolites[END_REF]. A preliminary Hyper-Secant (HS) pulse, lasting HS , was used to enhance the CT polarization of the detected quadrupolar isotope by manipulating the population of satellite transitions [START_REF] Siegel | Signal enhancement of NMR spectra of half-integer quadrupolar nuclei in solids using hyperbolic secant pulses[END_REF]. The pulse program of D-HMQC sequence is similar to that given in the supporting information of reference [START_REF] Lu | Observation of proximities between spin-1/2 and quadrupolar nuclei: which heteronuclear dipolar recoupling method is preferable?[END_REF]. The program to generate the R (tt) recoupling is given in 2 1 1 the supporting information of this article. 27

Al-{ 31 P} D-HMQC 1D experiments

These experiments were recorded using 3.2 mm triple resonance HXY MAS probes. The 31 P and 27 Al isotropic chemical shifts were referenced to 85% H 3 PO 4 and a 1 mol.L -1 solution of Al(NO 3 ) 3 , respectively. 27 Al-{ 31 P} 1D spectra were recorded on a crystalline aluminophosphate material, VPI-5, at R = 10 (Figs.4, 5 and 6c1,2) or 20 kHz (Fig. 6c3). The 1D experiments were performed using the sequence of Fig. 1a with t 1 = 0 using REDOR xy4 , REDOR xy8 , SFAM or R (tt) recoupling schemes. 2 1 1 Just as in simulations, these schemes were compared using identical peak values for the rf field strength. The rf field strength of and pulses on 31 P and 27 Al channels were 81 and 9.3 kHz, respectively. The pulses on 27 Al channel achieve CT selective excitation. The optimal HS enhancement was obtained for HS = 3 ms, an rf field of 11 kHz, and a frequency sweep from 180 to 160 kHz. The other parameters, such as R , and are indicated in the figure captions. For these 1D experiments, only the +1-quantum 31 P coherence was selected during t 1 , and 16 transients were acquired with a recycle delay of RD = 1 s. The intensities have been normalized with respect to that of the 27 Al signal created by a soft CT-selective /2 pulse following the same HS pulse as the one used in 27 Al-{ 31 P} D-HMQC experiment. In Figs. 4 to 6, we plotted the intensity of the most intense 1D signal of 27 Al-{ 31 P} D-HMQC spectra of VPI-5, which corresponds to the tetrahedrally coordinated 27 Al 2,3 sites [START_REF] Engelhardt | Assignment of the 27 Al and 31 P NMR spectra of the aluminophosphate molecular sieve VPI-5[END_REF]. These sites are dipolarcoupled to three inequivalent 31 Pb 4 O(BO 3 ) 2 crystalline lead borate material was synthesized as described previously [START_REF] Yu | A new congruent-melting oxyborate, Pb4O(BO3)2 with optimally aligned BO3 triangles adopting layered-type arrangement[END_REF]. Experiments on Pb 4 O(BO 3 ) 2 were acquired at 2.35 T using double-resonance HX MAS probe with rotor of 4 mm external diameter, as well as 9.4 and 18.8 T using triple resonance HXY MAS probes with rotors of 4 and 3.2 mm external diameter, respectively.

The 1D single-pulse 11 B MAS spectra at 9.4 and 18.8 T of Fig. S1 were acquired at R = 10 and 20 kHz, respectively and were simultaneously simulated using the DMfit software [START_REF] Massiot | Modelling one-and two-dimensional solid-state NMR spectra[END_REF]. The best-fit parameters are reported in Table S2. The 2D 11 B 3QMAS NMR spectrum displayed in Fig. S2 was acquired at 18.8 T with R = 20 kHz.

The 1D 207 Pb MAS spectra displayed in Figs.S3 and S4 were acquired at 2.35 T with R = 11.5 kHz using a full echo acquisition as well as at 9.4 T with R = 12.5 and 14 kHz using Carr-Purcell Meiboom-Gill (CPMG) sequence consisting of a train of rotor-synchronized pulses. Such technique enhances the sensitivity by the acquisition of a train of rotor-synchronized echoes since for that isotope, the NMR signal is broadened by a large CSA and the decay of the top of the echo signal for increasing spin-echo delay is often much slower than that of the Free-Induction Decay. The 11 B and 207 Pb isotropic chemical shifts were referenced to NaBH 4 and Pb(NO 3 ) 3 , respectively.

The 2D 11 B-{ 207 Pb} D-HMQC experiments were acquired at 9.4 T using the sequence of Fig. 1a incorporating R (tt) scheme with R = 12.5 kHz, RD = 35 s and = 5 ms. The rf field strength of 2 1 1 and pulses on 207 Pb and 11 B nuclei were 46 and 17 kHz, respectively. The pulses on 11 B channel achieve CT selective excitation. The optimal HS enhancement was obtained with HS = 2 ms, an rf field strength of 17 kHz and a frequency sweep from 115 to 125 kHz. The and the frequencies were equal to 40 kHz and 1 MHz, respectively, which corresponds to = 0.04. We acquired a 11 B-{ 207 Pb} 2D spectrum with rotor-synchronized t 1 increment, t 1 = R = 80 (Fig. 7a), which results in an indirect spectral width of SW 1 = R = 12.5 kHz. Under that condition, all spinning sidebands produced by 207 Pb CSA are folded back onto a single band, which enhances the sensitivity. The 2D spectrum of Fig. 7a results from averaging 128 transients for each of 20 t 1 increments. We also acquired another 2D spectrum with t 1 = 5 (Fig. 7b), which results in SW 1 = 200 kHz. Such approach allows the indirect detection of the full MAS powder pattern produced by 207 Pb CSA [START_REF] Rossini | Rapid acquisition of wideline MAS solid-state NMR spectra with fast MAS, proton detection, and dipolar HMQC pulse sequences[END_REF], at the expense of a lower sensitivity since the total integrated intensity for each species is distributed among 20 distinct bands in Fig. 7b, instead of a single one in Fig. 7a. As the R (tt) scheme is the delay between the end of the 2 1 1 defocusing period and the start of the refocusing one was always equal to the least upper bound of t 1 + , which was equal to an even integer number of rotor periods. The 2D spectrum of Fig. 7b results from averaging 16 transients for each of 45 t 1 increments.

IV.

Simulations and experimental results

IV-1. 1D on-resonance build-up curves and efficiencies

Fig. 4a shows the simulated build-up curves with no CSA 31P using R (tt) recoupling with = 2 1 1 50 kHz and values ranging from 0 to 3 MHz. For = 0, the tanh/tan pulse does not achieve adiabatic inversion and the efficiency is very low. For = 0.5 and 1 MHz, corresponding to Q 1 = 10.3 and 5.2, respectively, the build-up curves agree well with the expression of Eq. 23. For 2 MHz, the maximal transfer efficiency slightly decreases since Q 1 2.6 and the tanh/tan pulse does not achieve perfect adiabatic inversion. We can also notice that increasing , i.e. decreasing leads to shorter opt time in agreement with Fig. 3a. The opt values determined from the simulated build-up curves agree well with those predicted using Eqs. 12 and 23.

Fig. 4b shows the R (tt) efficiency versus / R and / R . The simulations were performed 2 1 1 at R = 10 kHz. However, as this recoupling is rotor-synchronized, such plot does not depend on the employed MAS frequency. This plot shows that maximal transfer efficiency is achieved when 50 150 R . For instance, efficient transfer with moderate rf field requirement corresponds to = 100 R and = 5 R = 0.05). Outside of this region, the transfer efficiency decreases for decreasing / R ratio and increasing / R one, which result in lower Q 1 factors, and hence prevents an efficient adiabatic inversion (Eq. [START_REF] Cadars | New Insights into the Molecular Structures, Compositions, and Cation Distributions in Synthetic and Natural Montmorillonite Clays[END_REF].

Fig. 4c compares the build-up curves with CSA 31P = 10 kHz using REDOR xy4 , REDOR specifications lead to identical build-up curves, whereas REDOR schemes benefit from slightly faster transfers. The lower maximum transfer efficiencies of REDOR schemes result from their lower robustness to CSA I (see section IV-3).

Fig. 4d displays the experimental build-up curves of the 27 Al 2,3 sites in VPI-5 in 1D 27 Al-{ 31 P} D-HMQC experiments using REDOR xy4 , REDOR xy8 , SFAM and R (tt) recoupling. As in the 2 1 1 simulations, REDOR xy4 and REDOR xy8 schemes, which benefit from a larger scaling factor, lead to slightly faster transfers, i.e. shorter opt value, than SFAM and R (tt). Nevertheless, for all recouplings 2 1 1 the maximal transfer efficiency and the opt times are lower than in simulations (Fig. 4c) since: (i) the Al 2,3 -(OP) 4 spin system is not an isolated spin-pair like in simulations, and (ii) the recoupled 27 Al-31 P dipolar interactions are smaller than the 1/ rate of transverse losses, which are not refocused by a ' 2 pulse [START_REF] Wang | Signal enhancement of J-HMQC experiments in solid-state NMR involving half-integer quadrupolar nuclei[END_REF].

IV-2. 1D robustness to rf inhomogeneity

There exists an inherent distribution of rf-field amplitudes in the sample coil of solid-state NMR probes, mostly along the axis of solenoid coil. Typically this amplitude is maximal at the center of the coil and drops by about 50% near the edges [START_REF] Gupta | RF inhomogeneity and how it controls CPMAS[END_REF][START_REF] Purusottam | Effects of inherent rf field inhomogeneity on heteronuclear decoupling in solid-state NMR[END_REF][START_REF] Nagashima | Imaging the spatial distribution of radiofrequency field, sample and temperature in MAS NMR rotor[END_REF][START_REF] Tošner | Radiofrequency fields in MAS solid state NMR probes[END_REF]. The robustness to rf inhomogeneity of REDOR xy4 , REDOR xy8 , SFAM, R (tt) and R (tt) schemes has been compared using simulations 13 C-{ 31 P} D-HMQC experiments on one isolated 13 C-31 P spin pair with CSA 31P = 0 (Fig. 5a) and 10 kHz (Fig. 5b). The length of the in REDOR recoupling was kept constant at 10 s since we want to compare the robustness to rf inhomogeneity and the pulse length is identical in all sample regions. These simulations indicate that SFAM, R (tt) and R (tt) recoupling schemes exhibit 2 1 1 4 1 2 similar robustness to rf inhomogeneity and are more robust than REDOR schemes. We can also notice that REDOR xy4 is more robust than REDOR xy8 . The comparison of Figs. 5a andb shows that CSA 31P decreases this robustness, and that such effect is more pronounced for REDOR schemes.

2 1 1 4 1 2 of 1D
The robustness to rf inhomogeneity of REDOR xy4 , REDOR xy8 , SFAM and R (tt) schemes has also 2 1 1 been compared using 1D 27 Al-{ 31 P} D-HMQC experiments on VPI-5 (Fig. 5c). Experimentally, R2 1 1 (tt) and SFAM schemes are more robust than REDOR xy4 , which is itself more robust than REDOR xy8 . Such results are consistent with the simulations of Fig. 5a. (tt) recoupling schemes using simulations of 1D 13 C-{ 31 P} D-HMQC experiments on one isolated 13 C-31 P spin pair, the 31 P nucleus being irradiated on resonance. For the three investigated conditions, R2 1 1 (tt) scheme is always the most robust recoupling and REDOR xy8 the least robust one. Furthermore, the comparison of Figs. 6a1-3 shows that the robustness to CSA I of R (tt) and R (tt) schemes 2 1 1 4 1 2 increases for increasing ( ) 2 / R ratio, which is equal to 250, 640 and 320 kHz for Figs. 6a1, a2 and a3, respectively, in agreement with Eq. 21, whereas the robustness to CSA I of REDOR schemes mainly depends on the rf field strength. Such result indicates that the robustness to CSA I of R (tt) scheme is 2 1 1 determined by the adiabaticity of the inversion process.

The robustness to 31 P offset of the investigated recouplings has been compared using similar simulations with CSA 31P = 10 kHz (see Figs. 6b1-3). These simulations indicate that recouplings based on tanh/tan pulses are the most robust ones to offset, whereas REDOR schemes are the least robust ones. The comparison of Figs.6b1-3 indicates that the robustness to 31 P offset for R (tt) scheme mainly 2 1 1 depends on the frequency provided the tanh/tan pulse achieves adiabatic inversion, i.e. the adiabatic factor, s Q, remains high. Such result agrees with Eq. 20. In Fig. 6b3, the transfer efficiency for R (tt) recoupling decreases significantly for particular offsets, whereas these oscillations are much 2 1 1 smaller in Fig. 6b2. Such difference may stem from the very different s Q factors, which are equal to 326 and 17 for the conditions in Figs. 6b2 andb3, respectively. The robustness to 31 P offset was also investigated using 1D 27 Al-{ 31 P} D-HMQC experiments on VPI-5 (Fig. 6c1-3). The experimental results agree well with the simulated ones shown in Figs. 6b1-3, which confirm that the R (tt) scheme is the 2 1 1 most robust recoupling to offset.

The robustness to MAS frequency fluctuations of the various studied recoupling schemes was also tested experimentally using 1D 27 Al-{ 31 1H . Nevertheless, 207 Pb solid-state NMR experiments are often challenging owing to the broad range of isotropic chemical shifts, the large CSA and the long longitudinal relaxation times [START_REF] Dybowski | Solid state 207 Pb NMR spectroscopy[END_REF]. Therefore, the indirect detection of 207 Pb nuclei via fast-relaxing half-integer quadrupolar nuclei could be advantageous. Nevertheless, to the best of our knowledge, no through-space HETCOR experiment between 207 Pb and quadrupolar nuclei has been reported so far. One challenge for such experiments is the reintroduction of heteronuclear dipolar couplings between 207 Pb and the quadrupolar nuclei since both isotopes are subject to large anisotropic interactions.

The possibility to acquire and correlate through-space 11 B and 207 Pb nuclei was tested on Pb 4 O(BO 3 ) 2 . This crystalline lead borate contains two crystallographically inequivalent triangular BO 3 sites and four Pb polyhedra (one PbO 3 , one PbO 4 and two inequivalent PbO 5 ones) [START_REF] Yu | A new congruent-melting oxyborate, Pb4O(BO3)2 with optimally aligned BO3 triangles adopting layered-type arrangement[END_REF]. All Pb atoms are distant by less than 3.75 Å to B atoms (see Table S1). To the best of our knowledge, NMR spectra of such material have never been acquired so far. The signals of the two 11 B sites overlap at 9.4 and 18.8 T (Fig. S1a,b). However, these resonances are well separated on the 2D MQMAS spectrum (Fig. S2). The simultaneous simulation of 11 B spectra at 9.4 and 18.8 T yields the NMR parameters reported in Table S2. The two sites exhibit close C Q values but distinct iso shifts and Q parameters. The 1D 207 Pb MAS spectra exhibit a broad powder pattern, which is spread over ca. 2500 ppm, i.e. 210 kHz at 9.4 T (Fig. S3). The acquisition of 1D 207 Pb spectra at different MAS frequencies permitted us to identify two resonances, the frequency of which does not depend on the MAS frequency, at 125 and ca. 375 ppm (Fig. S4). However, the later resonance is about twice broader than the former one, and may result from the overlap of distinct 207 Pb signals. It has reported that the iso values of 207 Pb nuclei decrease for increasing coordination number [START_REF] Fayon | Empirical Correlations between 207Pb NMR Chemical Shifts and Structure in Solids[END_REF] and hence, the 207 Pb signal at 125 ppm can be assigned to PbO 5 sites, whereas the overlapping peaks at 375 ppm could be those of PbO 3 and PbO 4 sites.

Here, we took advantage of the robustness of R (tt) recoupling to offset and CSA in order to 2 1 1 reintroduce the dipolar couplings between 11 dimension, the two 11 B sites are not resolved at such B 0 field. The indirect dimension ranges from 250 to 400 ppm. Such interval includes the double broad 207 Pb resonance at ca. 375 ppm, but not the other one at 125 ppm, which is folded back at 275 ppm (Fig. S4). Furthermore, Fig. 7b demonstrates the possibility to indirectly detect the broad 207 Pb full sideband comb using 2D 11 B-{ 207 Pb} D-HMQC spectrum with non-rotor-synchronized t 1 period [START_REF] Engelhardt | Assignment of the 27 Al and 31 P NMR spectra of the aluminophosphate molecular sieve VPI-5[END_REF].

V. Conclusion

We have introduced in this article R (tt) recoupling. This scheme depends on four parameters, 2 1

1

,

, and We have derived an analytical expression for the contributions of the heteronuclear max 1 max 0 dipolar interaction to the first-order Average Hamiltonian during this recoupling scheme. We have shown that the parameters = 10 and = 87° result in a good compromise between an efficient adiabatic inversion and the magnitude of the recoupled dipolar interaction. The rf peak amplitude must be max 1

chosen as large as possible, within the probe specifications, in order to improve the robustness to CSA, whereas the maximum frequency sweep, , must be smaller than / R -150 in order to achieve recoupling is more robust to offset, CSA and rf-inhomogeneity than REDOR and exhibits higher robustness to offset and CSA than SFAM. Its scaling factor is comparable to that of SFAM and about 25% smaller than that of conventional REDOR owing to the absence of window. Such comparison also indicates that the xy8 super-cycle does not improve the performance of the REDOR sequence, when it is applied on the indirectly detected isotope. The main limitation of R (tt) scheme is its significant rf 2 1 1 field requirement: 4 R , which may prevent its use at high MAS frequency. We also took Institute (FR 2638), Ministère de l'Enseignement Supérieur et de la Recherche, Région Hauts-de-France and FEDER are acknowledged for supporting and funding partially this work. Financial support from the IR-RMN-THC FR-3050 CNRS for conducting the research is gratefully acknowledged. Authors also thank contract CEFIPRA n°85208-E, PRC CNRS-NSFC and ANR-17-ERC2-0022 (EOS). This project has also received funding from the European Union's Horizon 2020 research and innovation program under grant agreement n°731019 (EUSMI).

- --------------------------------------------------------------------------------------------------------------- s Q = 0) (black disks) versus offset Q 1 was calculated according to Eq. 20, whereas s Q was calculated according to the procedure described in ref. [START_REF] Deschamps | Superadiabaticity in magnetic resonance[END_REF]. Except the offset, the other parameters are identical to those of Figs. 2a andb: { , R , , 0 aniso } = {50, 10, 1000, 10} kHz. - -------------------------------------------------------------------------------------------------------------------------

0

  In Eqs. 1 to 4, b IS denotes the dipolar coupling constant in rad.s -1 , the Euler angles relate {0, , }

  around the y-axis of the FM frame with an apparent pulsation

4 1 2 1 1

 11 that of REDOR xy4 , as well as on the R one. Compared to REDOR xy4 scheme and R symmetry, contributions of the heteronuclear dipolar coupling, the isotropic 2 chemical shift of I-spin, its CSA I and the I-I dipolar interactions to the first-order average Hamiltonian. In particular, it allows the terms of the heteronuclear dipolar coupling and the CSA I with and| | = 2| | , as well as of the isotropic chemical shift of the I-spin with , whereas these terms are =

Fig.1a shows

  Fig.1ashows the D-HMQC sequence with heteronuclear dipolar recoupling applied to the indirectly detected spin I during two periods, Here, we have employed the five following |m| = 1 heteronuclear dipolar recoupling schemes: REDOR xy4 , REDOR xy8 , SFAM, R (tt) and R (tt). For 21 1

  11 B-{ 207 Pb} D-HMQC 2D experiments

IV- 3 .

 3 Figs.6a1-3 compare the robustness to CSA 31P of REDOR xy4 , REDOR xy8 , SFAM, R (tt) and R 2 1 1

  P} D-HMQC experiments on VPI-5. These experimental results [not shown] indicate that all the investigated recoupling techniques exhibit identical robustness to MAS frequency fluctuations. Such result is consistent with the fact that they are all IV-4. 2D 11 B-{ 207 Pb} D-HMQC experiments 207 Pb isotope has a spin quantum number I = 1/2, a natural abundance of 22.1% and a gyromagnetic ratio 207Pb =

  B and 207 Pb nuclei under MAS conditions. The 2D 11 B-{ 207 Pb} D-HMQC spectra of Pb 4 O(BO 3 ) 2 acquired at B 0 = 9.4 T with R (tt) and R , t 1 ) = (12.5 kHz, kHz, 5 are shown in Figs.7a and b, respectively. The two large cross-peaks in Fig.7ademonstrate the possibility to reintroduce 11 B-207 Pb dipolar couplings under MAS conditions using the R (tt) scheme and to probe 11 B-207 Pb proximities using the D-HMQC sequence. In the direct 21 1

  frequency sweep determines the robustness to offset of R (tt).

1

  Numerical simulations and 1D27 Al-{ 31 P} D-HMQC experiments on VPI-5 show that R

max 1 advantage 1 1

 11 of the high robustness of R (tt) recoupling to CSA to acquire the first 2D 11 B-{ 207 Pb} D-2 HMQC spectra.

- Fig. 1 . 1 (

 11 (a) S-{I}D-HMQC pulse sequence, in which the heteronuclear dipolar recoupling is applied to the indirectly detected spin-1/2 nucleus, I. Pulse sequences of (b) REDOR xy4 , (c) SFAM, and (d) R2 1 tt) dipolar recoupling.

Fig. 2 .

 2 Fig.2. Q n factors for SFAM (blue disks) as well as R (tt) and R (tt) (red disks) for { , R } 2 1 1

Fig. 3 .=

 3 Fig.3. (a-c) Plot of the scaling factor = -C 1 of R (tt) and R (tt), as function of (a) with 2 1 1

Fig. 4 . 1 1 1 1

 411 Fig.4. (a-c) Simulated 13 C-{ 31 P} on-resonance D-HMQC experiment at R = 10 kHz for one isolated spin pair. (a,c) Build-up curves with = 50 kHz: (a) CSA 31P = 0 using R (tt), (c) CSA 31P = 10 kHz 2 1 1 using REDOR xy4 , REDOR xy8 , SFAM, R (tt) and R (tt) with = 50 kHz for SFAM or 1 MHz 2 1 1

Fig. 5 . 1 1= 1 . 1 1

 5111 Fig.5. (a,b) Simulated 1D 13 C-{ 31 P} D-HMQC on-resonance signal at R = 10 kHz versus for one isolated 13 C-31 P spin pair with CSA 31P of (a) 0 and (b) 10 kHz using REDOR xy4 , REDOR xy8 , SFAM ( = 50 kHz), R (tt) and R (tt) ( = 1 MHz). The recoupling times were fixed to their 2 1 1

Fig. 6 . 1 ( 1 2 1 1

 6111 Fig.6. (a,b) Simulated 1D 13 C-{ 31 P} D-HMQC efficiency using REDOR xy4 , REDOR xy8 , SFAM, R2 1 1 (tt) and R (tt) recoupling schemes for one isolated 13 C-31 P spin pair versus (a) CSA 31P with on-4 1 2 resonance 31 P irradiation or (b) 31 P offset with CSA 31P = 10 kHz. The refocusing times were fixed to their optimal values deduced from Fig.4c. (c) 1D signal of Al 2,3 site in VPI-5 as function of 31 P offset in

Fig. 7 . 1 1

 71 Fig.7. Experimental 11 B-{ 207 Pb} D-HMQC 2D spectra of Pb 4 O(BO 3 ) 2 at B 0 = 9.4 T using R (tt) 2 1 1 recoupling with t 1 , R ) = (a) (80 12.5 kHz) and (b) (5 10 kHz). Other experimental parameters are given in the section III-2 of the article. Note the different scales for 207 Pb axis.

  C 1 /(2 2), just as SFAM. Nevertheless, in the case of recoupling based on tanh/tan element, and hence C 1 and parameters depend not only on but also on and parameters.

	cos [	)]
	Combining Eqs. 10, 12, 14 and 16, we calculated	2 2	for R	2 1 1	(tt) and R 4 1 (tt) schemes 2
						1 1	4 1

2

schemes have a scaling factor equal to zero. Hence, these symmetry-based schemes using adiabatic pulses recouple the same rotational components as REDOR. The Hamiltonian and D,IS angular

(1)

frequency of these schemes are identical to those of REDOR and SFAM (see Eqs. 1 and 2), whereas =

  P sites resonating at and ppm, with CSA 31P = 26.3, 33.7 and 34.6 ppm, i.e. 4.26, 5.46 and 5.60 kHz at B 0 = 9.4 T, respectively.

Acknowledgments. We thank Jean-Luc Paugam for the synthesis of Pb 4 O(BO 3 ) 2 . We thank the reviewers for their constructive comments, which improve the quality of the manuscript. Chevreul