Méthodes désynchronisées ajustables

Les techniques de découplage homonucléaire désynchronisées, telles que PMLG ou DUMBO, sont, jusqu’à présent, les méthodes, qui ont permis d’obtenir la meilleure résolution spectrale. Cependant, elles ne sont pas parfaites, et ce pour deux raisons :

i. elles ont été conçues ou optimisées sans fenêtre d’acquisition. Par consé- quent, l’insertion de fenêtres d’acquisition dans les expériences 1D diminue l’efficacité du découplage et la résolution spectrale ;

ii. elles ont été développées dans les conditions statiques ou pour des fréquences MAS modérées [199,200,207]. De ce fait, elles ne tirent pas forcément profit de la rotation MAS, qui élimine partiellement l’interaction dipolaire.

Une première possibilité pour surmonter ces limitations consiste à optimiser expérimentalement les séquences DUMBO à νr = 65 kHz à partir d’expériences

1D 1H comportant des fenêtres. Cette stratégie a conduit à l’introduction des séquences e-DUMBO-PLUS [208].

En parallèle, nous avons proposé une autre stratégie. Elle consiste à employer une séquence de découplage désynchronisée, baptisée TIMES, ajustable suivant la fréquence MAS et dont l’élément de base comporte des fenêtres pour l’acquisition. Dans cette séquence, l’utilisation d’une rampe de phase de pente variable permet d’optimiser l’inclinaison, θ, du champ effectif dû aux déplacements chimiques par rapport à l’axe z. En jouant sur la valeur de θ et sur la longueur des fenêtres, la méthode TIMES peut être équivalente à l’enregistrement de spectres MAS sans champ rf (θ = 0◦), en présence de découplages FSLG ou PMLG (θ = 54, 7◦), ou encore MSHOT [194] (θ = 90◦). Nous avons montré expérimentalement qu’à νr = 10, 65 ou 80 kHz, la séquence TIMES conduit à des spectres 1H mieux

résolus que PMLG, MSHOT ou le simple MAS.

Cette meilleure résolution de TIMES s’explique par la possibilité d’ajuster l’orientation du champ effectif en fonction de la fréquence MAS. En effet, la ré- solution spectrale pour les expériences de découplage homonucléaire résulte d’un compromis entre la réduction des couplages dipolaires et celle des déplacements chimiques. Il y a donc différents régimes pour les expériences de découplage homo- nucléaire. Pour des fréquences MAS faibles, la rotation de l’échantillon n’est pas suffisamment rapide pour éliminer les couplages dipolaires. La réduction des cou- plages dipolaires repose donc uniquement sur le champ rf et il faut choisir θ proche de 90◦. En revanche, à haute vitesse (νr ≥ 30kHz), le MAS moyenne en grande

partie les couplages dipolaires1H–1H. Le champ rf n’a donc pas besoin d’éliminer complètement l’interaction dipolaire. Il est préférable pour obtenir une résolu- tion spectrale optimale, d’utiliser un champ rf effectif incliné d’environ 20◦ à 40◦ par rapport à B0. Ce champ élimine moins les couplages dipolaires qu’un champ orthogonal à B0, mais il réduit également moins fortement les déplacements chi- miques isotropes. Ce champ proche de B0 permet d’obtenir, à haute-vitesse, une meilleure résolution spectrale qu’un champ orthogonal à B0. Ce régime de dé- couplage homonucléaire haute-vitesse est comparable à celui employé pour les séquences CNnN/2 [34] et e-DUMBO-PLUS [208].

4.3

Perspectives

Un point important est aujourd’hui de mieux comprendre le fonctionnement des découplages homonucléaires CNnN/2 et TIMES. Pour les séquences CNnN/2

dépourvues de fenêtre, la théorie de l’hamiltonien moyen prenant en compte les termes, d’ordres zéro et un, est suffisante [209], tandis que pour les méthodes

CNnN/2 avec fenêtres et TIMES, il est nécessaire de recourir à la théorie de Flo-

quet [210]. Ces outils théoriques peuvent permettre de comparer l’efficacité des différentes méthodes de découplage en terme de résolution et de prévoir les champs rf optimaux pour chaque technique.

Il est aussi utile d’évaluer l’effet des transitoires pour les séquences CNnN/2

et TIMES. Ceci passe par la détection des transitoires lors du découplage homo- nucléaire ainsi que la réalisation de simulations numériques [211]. Nous pouvons prévoir que l’effet des transitoires doit être particulièrement marqué à haute vi- tesse, où les durées d’impulsion sont courtes. Pour diminuer ces transitoires, une solution consiste, en théorie, à lisser les impulsions, comme nous l’avons proposé pour les séquences CNnN/2. Cette méthode nécessite, cependant, de disposer de

spectromètres RMN dont l’électronique soit suffisamment rapide.

De façon pratique, il est nécessaire de comparer les performances des diffé- rentes séquences de découplage homonucléaire utilisables pour νr ≥ 10 kHz. Ces

séquences comprennent les méthodes CNnN/2, RNnN/2, PMLG et DUMBO. Cette

comparaison doit être effectuée pour différentes applications, telles que l’accrois- sement de la résolution des spectres1_{H ou l’augmentation de l’efficacité des trans-}

ferts via les couplages J hétéronucléaires. Dans tous les cas, les performances des découplage ne doivent pas uniquement être évaluées en terme de résolution spec- trale. Il est important d’inclure des critères tels que la puissance rf nécessaire, la robustesse de la séquence et sa facilité de mise en œuvre.

Signalons, enfin, que l’intérêt du découplage homonucléaire ne se limite pas, en principe, à la RMN du 1_{H. Cette stratégie pourrait aussi être appliquée pour}

les noyaux 19_{F, qui subissent souvent des couplages dipolaires homonucléaires}

élevés. Cependant, la conception du découplage homonucléaire 19F représente un défi pour la méthodologie RMN autrement plus difficile que le découplage 1_H– 1_{H en raison de l’amplitude élevée de l’écrantage électronique. Seules quelques}

tentatives ont été réalisées dans ce domaine à ce jour [212].

Proton-proton homonuclear dipolar decoupling in solid-state NMR using rotor-synchronized z-rotation pulse sequences

Olivier Lafon,1,a兲Qiang Wang,1,2,3Bingwen Hu,1Julien Trébosc,1Feng Deng,2and

Jean-Paul Amoureux1

1_{Unité de Catalyse et de Chimie du Solide (UCCS), UMR CNRS 8181, École Nationale Supérieure de} Chimie de Lille, Université de Lille 1, Bât. C7, B.P. 90108, 59652 Villeneuve d’Ascq Cedex, France 2_{Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071,} People’s Republic of China

3_{Graduate School of the Chinese Academy of Sciences, Beijing 100049, People’s Republic of China} 共Received 2 October 2008; accepted 20 November 2008; published online 6 January 2009兲 We present a theoretical analysis of rotor-synchronized homonuclear dipolar decoupling schemes

that cause a z-rotation of the spins. These pulse sequences applicable at high spinning rates共␯r

ⱖ30 kHz兲 yield high-resolution proton NMR spectra that are free of artifacts, such as zero lines and image peaks. We show that the scaled isotropic chemical-shift positions of proton lines can be calculated from the zero-order average Hamiltonian and that the scaling factor does not depend on

offset. The effects of different adjustable parameters共rf field, spinning rate, pulse shape, offset兲 on

the decoupling performance are analyzed by numerical simulations of proton spectra and by1_H

solid-state NMR experiments on NaH2PO4and glycine. © 2009 American Institute of Physics.

关DOI:10.1063/1.3046479兴

I. INTRODUCTION

The high gyromagnetic ratio of proton as well as its 99.985% isotopic natural abundance renders this nucleus the

ideal candidate for NMR detection.1,2If 1_{H NMR is rou-}

tinely used in solution state,1H signal detection and1H –1H

NMR constraints have only been exploited recently in the

solid state, for characterizing materials,3–7 small

molecules,6,8and biological macromolecular systems.9–11

Up to now, the main limitation of proton solid-state NMR remains its small chemical-shift range and the spectral resolution of each resonance. Actually the extensive network

of dipole-dipole共DD兲 couplings among the protons leads to

broad spectral lines, thus obscuring the chemical-shift

information.12Moreover, as the homonuclear DD couplings

are homogeneous interactions,13the individual1_{H linewidths}

only decrease as the inverse magic-angle spinning共MAS兲

frequency.14–16As a consequence, rotation of the sample at

the magic angle alone does not generally achieve very high

resolution, even when spinning rates as high as␯r= 60 kHz

are employed.17

The resolution can be further enhanced by combining

MAS with radio-frequency共rf兲 irradiation schemes that ma-

nipulate the spin part of the1_{H –}1_{H dipolar interaction. Such}

combined rotation and multiple-pulse spectroscopy

共CRAMPS兲 was originally designed for the quasistatic limit, in which low spinning frequencies ensure an approximately

static sample on the time scale of the rf pulse sequence.18

Recently these schemes were modified to be applicable at

spinning speeds in the range of 10–25 kHz.6A remaining

challenge is the development of homonuclear decoupling se-

quences at high MAS frequencies共␯rⱖ30 kHz兲. High MAS

rates can lead to substantial benefits in terms of resolution

and sensitivity not only for1H but also for nuclei, such as

13_{C and} 31_{P, which experience large chemical-shift aniso-}

tropy共CSA兲 at high magnetic fields. In practice, correlation

NMR experiments of these nuclei with1H are crucial for

analyzing structural properties. Furthermore, fast spinning speed decreases the irreversible losses and increases the

spectral width in the indirect dimension of rotor-

synchronized two-dimensional共2D兲 experiments.

Very recently, the phase-modulated Lee–Goldburg19,20

共PMLG兲 and decoupling using mind-boggling

optimization21–23共DUMBO兲 experiments have been shown

to perform well at high MAS rates up to␯r= 65 kHz.

24

How- ever, PMLG and DUMBO schemes were derived, respec-

tively, from quasistatic Lee–Goldburg25and BLEW12共Ref.

26兲 experiments, and these decoupling schemes may not be

optimal at high spinning frequencies. In particular, as the sample is spun faster, the resolution is not improved and

higher rf field strength are required.24,27 Furthermore, the

scaling factors of the chemical shifts under PMLG and

DUMBO irradiations are lower than 1/冑3, which limits the

achievable resolution.22,24 At last, as these sequences are

nonsynchronized with the period of the sample rotation, sev-

eral rotor-radio-frequency共RRF兲 lines appear in the NMR

spectra and can cause line broadening.

To develop homonuclear decoupling at high MAS rates, an alternative approach consists in applying rf pulse se- quences that are rotor synchronized. In this way, pulse se- quences can be designed from scratch by explicitly taking into account the sample rotation. Symmetry arguments can be used to help in the construction of these pulse

sequences.28,29Pulse sequences with C4₁0_{and RN}

n

N/2_symme-

tries have been shown to be efficient in yielding highly re-

solved 1_{H spectra.}30–34

However, for these pulse schemes, a兲_{Electronic mail: olivier.lafon@ensc-lille.fr.}

THE JOURNAL OF CHEMICAL PHYSICS 130, 014504共2009兲

0021-9606/2009/130共1兲/014504/13/$25.00 130, 014504-1 © 2009 American Institute of Physics

the effective field resulting from isotropic chemical-shift

points away from the B0_{axis共also called z axis兲. This creates}

undesirable zero and image peaks.

In this publication we will explore the properties of re-

cently introduced broad-banded z-rotation CN_nN/2_decoupling

schemes.35,36As already shown in the case of PMLG, an

effective z-rotation sequence leads to favorable properties of

homonuclear decoupling:共i兲 cleaner 1H spectra with lesser

artifacts, such as zero and image peaks,共ii兲 improved reso-

lution, and共iii兲 possible use of conventional phase-cycling

schemes in multidimensional experiments.27,37–39 These

CN_nN/2_{pulse sequences have been implemented in the indi-}

rect dimension of 2D experiments35 as well as in a one-

dimensional _{共1D兲 fashion after insertion of acquisition}

windows.36Moreover, they have been shown to yield highly

resolved1_{H double-quantum spectra.}40

The basicC element of CNn

N/2_{sequences can be either a}

composite 0° rectangular pulse or its smoothly amplitude-

modulated共SAM兲 version. The SAM sequence allows reduc-

ing rf pulse transients, which can affect the decoupling

performances.37,41 This is especially important for homo-

nuclear decoupling sequences at high MAS frequencies, since they can involve short rf pulses, not much longer than the duration of pulse transients.

This article is organized as follows. Section II introduces

zero-order average Hamiltonian theory共AHT兲12,42of CNn

N/2 schemes and derives analytical expressions for the scaling factors of isotropic chemical shifts. Section III presents nu- merical simulations that demonstrate the validity of the the- oretical predictions and analyzes the effects of different pa-

rameters共rf field, spinning rates, pulse shape, offset兲 on the

decoupling performance. In Sec. IV, results of 2D1H CNn

N/2

experiments for NaH2PO4 and glycine are presented and

compared with the theoretical predictions and numerical simulations.

II. ZERO-ORDER AHT

A. Spin interactions and average Hamiltonian In order to describe the evolution of a proton spin system that rotates at the magic angle and is subjected to rf irradia- tion close to resonance, a convenient representation of its spin Hamiltonian must first be introduced. The Hamiltonian

H⌳of a spin interaction⌳ is characterized by its space rank

l and its spin rank_{␭. Each spin interaction may therefore be}

regarded as a superposition of共2l+1兲⫻共2␭+1兲 components,

with component indexes m and␮ running, respectively, from

−l to +l and −␭ to ␭ in integer steps,

H⌳=兺 m=−l +l 兺 ␮=−␭ +␭ Hlm⌳␭␮. 共1兲

The various nuclear spin interactions are distinguished by the

values of l and␭. For instance, the pair 兵l,␭其 is equal to

兵2, 2其 for homonuclear DD interaction, 兵0, 1其 for isotropic

chemical shifts, and_{兵2, 1其 for CSA. A summary of the spin}

interaction components, in the case of exact MAS, is given

in Ref.43.

When the rf irradiation is synchronized with the sample

rotation, as in CNn

N/2 _{sequences, the dynamics of nuclear}

spins can be described in the framework of AHT.12,42,43This

leads to a time-independent average Hamiltonian 共AH兲,

which may be approximated using the Magnus expansion. In this work, the theoretical analysis is restricted to the zero-

order term of AH. As in Eq.共1兲, the zero-order AH in the

interaction frame,H共0兲, can be expressed as a sum of many

rotational components,Hlm⌳,共0兲␭␮,29,43,44

H共0兲= 兺

⌳lm␭␮Hlm␭␮

⌳,共0兲_{. 共2兲}

B. Symmetry classes and zero-order selection rules Setting up periodic symmetry relationships between the mechanical and the rf rotations facilitates the design of homonuclear decoupling sequences, since these symmetry arguments allow generating a zero-order AH containing de-

sired rotational components in Eq._共2兲while other compo-

nents are suppressed. There are two major classes of

symmetry-based decoupling sequences, denoted CN_n␯共Refs.

43and45–47兲 and RNn␯.

29,43,48

They are both composed of N

elements兵⑀0⑀1¯⑀N−1其, each of which has the same duration

␶E= n␶r/N, where␶r= 1/␯ris a period of the magic sample

rotation. In the case of CNn␯sequences, the elements are de-

rived from a basic elementC by an incremental phase shift,

⑀q=C2␲q␯/N. 共3兲

In the case of RN_n␯sequences, the phases alternate between

two values,

⑀q=

再

R␲␯/N 共q even兲

R−␲␯/N⬘ 共q odd兲,

冎

共4兲

where the prime indicates a change in the sign of all phases

internal to the basic element_{R. For CNn}␯symmetry, the basic

C element must be a rf cycle, meaning that in the absence of

other spin interactions, the rf pulses of_{C induce a rotation of}

the nuclear spins through an integer multiple of 360°共includ-

ing 0°兲. In other words, the rf propagator of the completed C

element is proportional to the identity operator. The RNn␯

schemes are built from the basic inversion elementR, which

rotates the nuclear spins through an odd multiple of 180° about the rotating-frame x-axis.

The symmetry properties of CNn␯ and RNn␯ pulse se-

quences lead to the following selection rules on the zero-

order AH:43,46

Hlm⌳,共0兲␭␮共CNn␯兲 = 0 if mn −␮␯ ⫽ NZ, 共5兲

where Z is any integer, including zero, and Hlm⌳,共0兲␭␮共RNn␯兲 = 0 if mn −␮␯ ⫽

NZ_␭

2 , 共6兲

where Z_␭is an integer with the same parity as␭. If H_lm⌳,共0兲_␭␮is

zero, the term共l,m,␭,␮兲 is said to be symmetry forbidden in

the zero-order AH, otherwise the term共l,m,␭,␮兲 is symme-

try allowed.

014504-2 Lafon et al. J. Chem. Phys. 130, 014504共2009兲

C. Selection of isotropic chemical shifts

High-resolution isotropic shift spectra may be obtained in solids if all DD couplings and CSA terms are suppressed, while isotropic chemical shifts are preserved. In the absence of rf irradiation, the DD coupling and CSA terms in the zero-order AH are zero due to sample spinning. In contrast, the rf irradiation can interfere with MAS in a destructive manner and may reintroduce the undesired interactions at

zero order. CNn␯and RNn␯sequences offer the advantage that

the zero-order selection rules given by Eq.共5兲and共6兲pre-

vent these interferences, whatever the spinning and rf nuta- tion frequencies. Nevertheless, the symmetry numbers N, n,

and␯ must be carefully chosen.

The DD couplings and CSA terms have a space rank

l = 2, while isotropic chemical shifts have l = 0. Therefore, the

suitable CNn␯and RNn␯sequences must have symmetry num-

bers N, n, and ␯ such that some terms of the form

共l,m,␭,␮兲=共0,0,1,␮兲 with ␮=−1,0, +1 are symmetry al-

lowed, while all terms of the form共2,m,␭,␮兲, with m⫽0,

are symmetry forbidden. Sequences satisfying these condi- tions produce a zero-order AH containing only isotropic chemical shifts and hetero- and homonuclear J-couplings.

A list of CN_n␯and RN_n␯symmetries suitable for high-

resolution1H spectroscopy in the solid state was given in

Ref.33. The RN_n␯sequences belong to the series RN_nN/2_and

thus only involve phase shifts of 180° between subsequent

elements. The suitable CNn␯ sequences include not only

CNn

N/2_{series, with phase shifts of 180°, but also CN}

n

0 se-

quences, with no phase shift, and CNn␯sequences共e.g., C913

and C10₁3_{兲, with phase shifts lower than 180°.}

D. Effective z-rotation sequences

As explained in Sec. I, it is highly favorable that isotro- pic chemical-shift interaction leads to effective rotation of

the spins around the B0_{axis. For the different suitable CN}

n

␯

and RNn␯pulse sequences, it is thus important to determine

the effective field resulting from isotropic chemical shift. In first approximation, this effective field can be calculated from the terms of isotropic chemical shift in the zero-order

AH. For instance, all RNn

N/2 _{sequences select terms}

共l,m,␭,␮兲=共0,0,1, ⫾1兲 of isotropic chemical shift, thus

generating an effective field along the x-axis.33,34Conversely

CN_n␯ sequences with ␯⫽0 only allow terms 共l,m,␭,␮兲

=共0,0,1,0兲 and the effective field points in the z-direction.

At last, the zero-order AH during CN_n0_{sequences contains}

terms共l,m,␭,␮兲=共0,0,1,0兲 and 共0,0,1, ⫾1兲. In this case,

the zero-order selection rules do not restrain the direction of the effective field.

In this article, we focus on CN_nN/2_{symmetries关see Fig.}

1共a兲兴, which cause a z-rotation of the spins. Furthermore,

they only involve phase shifts of 180° between subsequent cycles, and hence can be implemented as amplitude-

modulated sequences, if the basic elementC corresponds it-

self to an amplitude modulation. In this article, we employ as

basic C element either a composite 0° rectangular pulse,

␣0␣180, or one of its SAM version, a half-cosine pulse, de-

noted half-cos in the text. Here the standard notation for rf

pulses is used:␰_␾indicates a rectangular, resonant rf pulse

with flip angle␰ and phase␾, and the angles are written in

degrees. To unify the notations of this article, the CNn

N/2_se- quences built from either basic elements are denoted

CNn N/2_共␣

0␣180兲 and CNn

N/2_{共half-cos兲 共instead of SAM兲. These}

two types of homonuclear decoupling sequences are exem-

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