heteronuclear decoupling interference during symmetry ... · heteronuclear decoupling interference...

Heteronuclear decoupling interference duringsymmetry-based homonuclear recoupling in solid-state NMR

Ildefonso Marin-Montesinos a, Darren H. Brouwer a, Giancarlo Antonioli a,Wai Cheu Lai a, Andreas Brinkmann b, Malcolm H. Levitt a,*

a School of Chemistry, University of Southampton, Southampton SO17 1BJ, UKb Institute for Molecules and Materials, Radboud University Nijmegen, 6525 ED Nijmegen, The Netherlands

Received 2 June 2005; revised 13 July 2005Available online 19 September 2005

Abstract

We examine the influence of continuous-wave heteronuclear decoupling on symmetry-based double-quantum homonuclear dipo-lar recoupling, using experimental measurements, numerical simulations, and average Hamiltonian theory. There are two distinctregimes in which the heteronuclear interference effects are minimized. The first regime utilizes a moderate homonuclear recouplingfield and a strong heteronuclear decoupling field; the second regime utilizes a strong homonuclear recoupling field and a weak orabsent heteronuclear decoupling field. The second regime is experimentally accessible at moderate or high magic-angle-spinning fre-quencies and is particularly relevant for many realistic applications of solid-state NMR recoupling experiments to organic or bio-logical materials.� 2005 Elsevier Inc. All rights reserved.

Keywords: Symmetry-based recoupling; Decoupling; Double-quantum NMR; Magic-angle spinning; Average Hamiltonian theory

1. Introduction

Recoupling methods reintroduce specific nuclear spininteractions that are normally averaged out by magic-angle-spinning (MAS), so that both the high spectralresolution afforded by MAS and the structural informa-tion provided by these nuclear spin interactions are pre-served. The symmetry theory for rotor-synchronizedpulse sequences [1] allows the design of pulse sequencesthat selectively recouple desired classes of nuclear spininteraction under MAS conditions. Pulse sequencesbased on the two classes of symmetries, CN m

n and RN mn,

have been employed for a wide variety of experiments[1–42]. In particular, some of the most important andsuccessful applications of symmetry-based sequenceshave involved double-quantum (DQ) homonuclear

dipolar recoupling. These applications include two-di-mensional DQ correlation spectroscopy [4,9,10,14,22,23,33], high-order multiple-quantum excitation in solids[13,15,36,41,42], and the estimation of internuclear dis-tances [26,39], torsional angles [3,5,31], and motional or-der parameters [38].

For organic and biological samples, the most com-mon case concerns recoupling of 13C nuclei introducedby isotopic labelling (similar conditions apply to therecoupling of 15N or 31P nuclei). In organic solids, thehomonuclear interactions between 13C nuclei are usuallyaccompanied by strong heteronuclear dipolar interac-tions with the surrounding 1H nuclei. These heteronucle-ar interactions generally interfere with the homonuclear13C–13C recoupling. The heteronuclear interference isnormally suppressed by applying a strong resonant RFfield on the 1H channel while the symmetry-based recou-pling sequence is applied to the 13C channel. Most earlywork in this area was conducted at low spinning

1090-7807/$ - see front matter � 2005 Elsevier Inc. All rights reserved.

doi:10.1016/j.jmr.2005.07.020

* Corresponding author.E-mail address: [email protected] (M.H. Levitt).

www.elsevier.com/locate/jmr

Journal of Magnetic Resonance 177 (2005) 307–317

frequencies (around 6 kHz or less), where good hetero-nuclear decoupling during homonuclear recoupling isachieved by maintaining a ratio of 2.5 or more betweenthe 1H and 13C nutation frequencies [8–10]. This ap-proach is generally not feasible at moderate to high spin-ning frequencies (around 10 kHz or more), since therequired RF field on the 1H channel would be unaccept-ably high.

One approach to reduce heteronuclear interference athigher MAS frequencies is to use sequences with symme-tries that require lower ratios of 13C nutation frequencyto MAS frequency. For example, the SPC5 [16] and thesupercycled C1454 sequences [18] require 13C nutationfrequencies that are 5 and 3.5 times the MAS frequency,respectively. Kristiansen et al. [35] showed that the sym-metry class may be chosen to allow efficient 13C dipolarrecoupling at 30 kHz spinning frequency in the presenceof high power 1H decoupling. The disadvantage of thisapproach is that the required symmetries have relativelysmall scaling factors for the homonuclear dipolar inter-actions and are often less robust with respect to the ef-fects of RF inhomogeneity, isotropic chemical shiftdifferences, and anisotropic chemical shift interactions.

Ishii [43] showed that some pulse sequences, such asfpRFDR (finite-pulse radio-frequency driven recou-pling) function well at very high spinning frequencies(around 30 kHz) in the complete absence of a 1H decou-pling field. To some extent, this is expected, since heter-onuclear dipolar interactions transform in the same wayas the 13C chemical shift anisotropy (CSA) under rota-tions of the 13C spins by the RF field and by the samplerotation, providing that a 1H RF field is absent. Sincethe 13C CSA is compensated by most homonuclearrecoupling sequences, it is not surprising that the heter-onuclear interactions are suppressed without furtherintervention, under suitable conditions. However, inthe case of fpRFDR, this ‘‘no-decoupling regime’’ isonly reached for very fast sample rotation frequenciesin the vicinity of 30 kHz. Many solid-state NMR appli-cations require relatively large sample volumes, forwhich the maximum available spinning frequencies arecurrently limited to around 15 kHz.

Hughes et al. [44] demonstrated that the 13C DQ fil-tering efficiency for [13C2]glycine obtained with thePOST-C712 sequence [7] without 1H decoupling at12 kHz spinning frequency was comparable to that ob-tained with high power 1H decoupling at 7 kHz spinningfrequency. This suggests that the no-decoupling regimeappears at conveniently low spinning frequencies forthe symmetry-based pulse sequences. This makes sense,since the symmetry-based design of these sequences en-sures their high compensation for CSA, and by exten-sion, for heteronuclear interactions in the absence ofheteronuclear decoupling.

Fig. 1 shows that this effect also occurs for sequencesbelonging to the RN m

n symmetry class. The spectra shown

in Fig. 1 demonstrate 13C DQ filtering efficiencies great-er than 35% in three 13C2-labelled organic compounds ata MAS frequency of 12 kHz, using a pulse sequence withsymmetry R2092, without using a heteronuclear decou-pling field during the homonuclear recoupling sequence.

In this paper, we present an experimental and theo-retical investigation of heteronuclear interference during13C homonuclear dipolar recoupling, for the case ofdouble-quantum homonuclear recoupling pulsesequences of the symmetry class RN m

n. We explore the ef-fects of the spinning frequency, the nutation frequencieson both the 13C and 1H channels, and the symmetrynumbers. For simplicity, we concentrate on RN m

n

sequences applied in the presence of unmodulated (con-tinuous-wave, CW) 1H irradiation, applied on-reso-nance. We show that the interference caused byheteronuclear interactions may be simulated fairly accu-rately by numerical simulations and that the general fea-tures of heteronuclear decoupling during homonuclearrecoupling sequences may be predicted by first-orderaverage Hamiltonian theory. Heteronuclear interference

20 10 0 -10 -20kHz

20 10 0 -10 -20kHz

1313

C13C13

H H

O

O N15 H3

+

C13

H C13

O

O NH3

+

HH

H

20 10 0 -10 -20kHz

20 10 0 -10 -20kHz

20 10 0 -10 -20kHz

20 10 0 -10 -20kHz

(NH )4 2+C C

H

H

COO

OOC

CH

CO2-

CO2-

C

C

C

A

B

C

Fig. 1. Comparison of 13C CP MAS NMR spectra (left column) withdouble-quantum filtered spectra (right column) for (A) 10%-labelleddiammonium-[2,3-13C2]fumarate, (B) [15N,13C2]glycine, and (C)[2,3-13C2]alanine. The double-quantum-filtered spectra were obtainedat a spinning frequency of xr/2p = 12 kHz using the R2092 sequencewithout 1H decoupling during the recoupling period.

308 I. Marin-Montesinos et al. / Journal of Magnetic Resonance 177 (2005) 307–317

with the CN mn class of pulse sequences is considered brief-

ly in the Supporting information.

2. Pulse sequences

2.1. Symmetry-based recoupling

This article concerns DQ 13C homonuclear dipolarrecoupling with symmetry-based RN m

n sequences, assketched in Fig. 2. A pulse sequence with the symmetryRN m

n may be constructed as follows:

RN mn ¼ ½R/R0

�/�N=2

; ð1Þ

where the basicR element is a pulse or series of pulses thatrotates the spins by an odd number of p rotations aboutthe x-axis, andR 0 is derived fromR by changing the signsof all phases. The subscript / denotes the overall RFphase shifts of the R elements and is given ideally by

/ ¼ pm=N ð2Þwhile the superscript N/2 denotes the number of repeti-tions of the bracketed elements making up a completecycle (in practice, it is usually sufficient to complete aneven number of R elements). In the pulse sequences ex-plored in this article, the basic R element is always givenby the composite p pulse

R0 ¼ R00 ¼ 900270180 ð3Þ

specifying flip angles and phases in degrees. This basicelement provides particularly robust performance withrespect to chemical shift interactions.

The 13C nutation frequency xSnut is set so that one

RN mn sequence occupies exactly n rotor periods. In the

case of the R element in Eq. (3), the 13C nutation fre-quency is set to exactly N/n times the spinning frequencyxr. For example, a R1252 sequence based on the elementin Eq. (3) may be written explicitly as

½9075:0270255:090285:0270105:0�6; ð4Þwhere the subscripts are the phases of the pulses (in de-grees) while the superscript denotes six repetitions of the

bracketed elements. One complete cycle spans two rotorperiods and requires an RF field strength correspondingto a nutation frequency which is exactly six times thespinning frequency.

2.2. Symmetry series

A symmetry analysis of the sequence R1252, using thetreatment given in [1,27,42] shows that the only homonu-clear first-order average Hamiltonian terms H ð1Þ

‘mkl whichare symmetry-allowed have quantum numbers of theform {‘, m, k, l} = {2, ± 1, 2, ± 2} (homonuclear dou-ble-quantum terms) or {‘, m, k, l} = {0, 0, 0, 0} (homo-nuclear J-couplings). Here {‘, m} define the tensorialrank and component index for spatial rotations, while{k, l} define the tensorial rank and component indexfor spin rotations, as discussed in [1]. As mentionedabove, this pulse sequence requires a ratio of 13C nuta-tion frequency to spinning frequency of xS

nut=xr ¼ N=n,assuming that the basic element in Eq. (3) is used.

The sequence R1252 is the first member of a seriesfR1252;R14

62;R16

72;R18

82; . . .g whose elements share the

same first-order selection rules, but which have different13C nutation frequencies xS

nut ¼ Nxr=n. It is thereforepossible to explore the role of 13C nutation frequencyat fixed spinning frequency by running along the seriesRN ð�1þN=2Þ

n with NP 12. We use this property extensive-ly in the following discussion.

2.3. Double-quantum filtering

The RN mn recoupling sequences are incorporated into

double-quantum filtered (DQF) experiments as shownin Fig. 2. After 1H fi 13C ramped cross polarization[45], a 90� pulse is applied with a phase shift of �90�to generate longitudinal 13C magnetization. A recou-pling sequence of duration s excites DQ coherenceswhich are reconverted into longitudinal magnetizationby a second recoupling sequence of the same durations. An on-resonance continuous-wave 1H decouplingfield with nutation frequency xI

nut is applied simulta-neously with the 13C recoupling sequence. After a shortdelay, a final 90� pulse is applied on the 13C channel and1H SPINAL-64 decoupling [46] is applied during theacquisition period. The shaded elements are given afour-step phase cycle to select signals passing throughDQ coherences.

In the experiments discussed below, the recouplingefficiency is studied as a function of the followingparameters: the magic-angle-spinning frequency xr, the1H decoupling nutation frequency xI

nut during thehomonuclear recoupling sequence, the 13C recouplingnutation frequency xS

nut, and the symmetry number N

of the RN ð�1þN=2Þn recoupling sequence. The symmetry

number n is held fixed at n = 2. The synchronizationrequirements of the RN m

n sequences requires the

R

90 270

-

90 270

x

-y y

DQcoherence

CW decouplingCP

1H

13C

SPINAL-64

rn/N

RNn RNn

R

Fig. 2. Pulse sequence diagram for 13C double-quantum homonucleardipolar recoupling with a symmetry-based RN m

n recoupling sequence.

I. Marin-Montesinos et al. / Journal of Magnetic Resonance 177 (2005) 307–317 309

relationship xSnut ¼ Nxr=n. There are therefore three

independent parameters to consider.

3. Experimental

3.1. Measurement of double-quantum-filtering efficiency

For all of the results presented below, the DQ efficien-cies were estimated according to the following proce-dure. For a given RN m

n sequence and spinning frequencyxr, the number of R elements giving the maximumDQF signal was determined. This number was held fixedwhile the efficiency was optimized by small adjustmentsto the 13C RF power and R element phase shift / (docu-mented in Supplementary Material). Once optimized, aseries of 13C DQ filtered spectra were obtained (with fouror eight acquisitions each), with the strength of the 1Hdecoupling during 13C homonuclear recoupling incre-mented from 0 to 120 kHz in increments of 3 kHz. TheDQ filtering efficiencies were calculated by comparingthe integrals of the DQ filtered spectra to a cross polari-zation spectrum obtained under identical conditions.

A range ofMAS frequencies between 6 and20 kHzanda series of RN m

n sequences with n = 2,N = 12, 14, 16. . . andm = �1 + N/2 were studied. All experiments were per-formed in a field of 9.4 T using well-filled 3.2 mm zirconiarotors on a Varian InfinityPlus console. The 1H nutationfrequency during the SPINAL-64 decoupling sequenceapplied during signal acquisition was fixed at 100 kHz.The full details of the experimental parameters used areprovided in the Supplementary Material.

3.2. Samples

These investigations were carried out three represen-tative organic compounds: diammonium-[2,3-13C2]fumarate diluted to 10% in natural abundance diammo-nium fumarate (DAF), [15N,13C2]glycine with 99% 15Nand 13C labelling (Gly), and [2,3-13C2]alanine with 99%13C labelling (Ala). These compounds broadly representtypical classes of 13C spin systems involving CH, CH2,and CH3 groups, respectively. Their molecularstructures are shown in Fig. 1.

Note that the relatively low labelling level of the DAFsample leads to prominent natural abundance peaks fromisolated 13C nuclei (Fig. 1A, left column). These signalsdisappear after double-quantumfiltration (right column).

4. Results

4.1. Diammonium-[2,3-13C2]fumarate (DAF)

Fig. 3 presents the effect of the 1H decoupling nuta-tion frequency on the 13C DQ filtering efficiency for

DAF at spinning frequencies of 6 and 12 kHz. Atxr/2p = 6 kHz (Fig. 3A), the sequence RN m

n exhibitsthe familiar behaviour in which the best DQ filtering effi-ciency is achieved with a large 1H decoupling nutationfrequency of xI

nut=2p ¼ 120 kHz, which is approximate-ly three times the 13C nutation frequency used for therecoupling, xS

nut=2p ¼ 36 kHz. As the symmetry numberN is increased (corresponding to an increase in 13C nuta-tion frequency since xS

nut ¼ Nxr=n) the DQ filtering effi-ciency increases at zero decoupling field, and a dip in theDQ filtering efficiency becomes evident in the vicinity ofHartmann–Hahn match, xI

nut ¼ xSnut. This minimum

shifts to higher xInut as the N value of the sequence

increases. For the sequence with the largest N ðR40192 Þ,and therefore the largest 13C nutation frequency, theDQ filtering efficiency without 1H decoupling is 37%and is comparable to the 40% recoupling efficiencyobtained with the lowest N and maximum 1Hdecoupling power ðR1252Þ.

For xr/2p = 12 kHz (Fig. 3B), similar trends in theeffects of 1H decoupling on the 13C DQ filtering efficien-cy are seen. At this increased spinning frequency, peri-odic local minima in the plots appear, separated bythe spinning frequency of 12 kHz. A respectable DQ fil-tering efficiency of 48% was achieved at xr/2p = 12 kHzusing the R2092 sequence without 1H decoupling.

To display the interaction of 1H and 13C nutation fre-quencies for the entire series of RN m

n sequences, two-di-mensional contour plots were constructed. Contour

0

20

40

60

0 30 60 90 1201H nutation frequency /KHz

DQ

F e

ffic

ien

cy/%

0

20

40

60

0 30 60 90 120 0 30 60 90 120

R1252 R167

2 R2092

0 30 60 90 1200 30 60 90 1200 30 60 90 1201H nutation frequency /kHz

0

20

40

60

DQ

F e

ffic

ien

cy/%

R24112 R3215

2 R40192

R1252 R167

2 R2092

A

B

Fig. 3. Plots of 13C double-quantum recoupling efficiency vs. 1Hdecoupling nutation frequency for DAF with indicated RN m

n sequencesat (A) xr/2p = 6 kHz and (B) xr/2p = 12 kHz. The experimental andsimulated data are plotted as solid and dashed lines, respectively. Someof the experimental data sets are incomplete, due to power-handlinglimitations of the RF probe.


plots of experimental DQ filtering efficiencies for DAFat spinning frequencies of 6 and 12 kHz are presentedin Figs. 4A and B, respectively. The traces shown inFig. 3 are horizontal slices through these contouredsurfaces.

These contour plots clearly show two regimes for effi-cient DQ homonuclear dipolar recoupling in the pres-ence of heteronuclear dipolar interactions, separatedby a region in which the DQ filtering is poor. The firstregime, indicated by the circled number 1 in the plotof Fig. 4A, utilizes strong 1H decoupling to removethe 1H–13C heteronuclear dipolar interactions whilethe 13C nutation frequency and the spinning frequencyare kept moderate, to avoid Hartmann–Hahn matching.The second regime, indicated by the circled number 2 inthe plot of Fig. 4A, uses strong 13C irradiation to re-move the 1H–13C heteronuclear dipolar interactions. Inthis regime, good homonuclear recoupling may beachieved without 1H decoupling, as reported for severalsymmetries of the CN m

n type by Hughes et al. [44].The second regime was explored further at higher

MAS frequencies. Fig. 5 displays DQ filtering efficiencyas a function of 1H decoupling nutation frequency forDAF using the R1252 sequence at MAS frequencies from6 to 20 kHz. As the spinning frequency increases, theDQ filtering efficiency increases under no-decouplingconditions. The periodic dips in the DQ filtering efficien-cy as a function of 1H decoupling nutation frequency be-come increasingly evident as the spinning frequency is

increased. The increase of DQ filtering efficiencies withincreasing spinning frequency, under no-decouplingconditions, is shown more clearly in Fig. 6.

4.2. [15N,13C2]Glycine (Gly)

The effects of 1H decoupling during 13C recouplingwere also studied for [15N,13C2]glycine for a series ofRN m

n sequences and MAS frequencies of 6 and 12 kHz(Fig. 4). The appearance of these plots is qualitativelysimilar to DAF, but the region of poor recoupling ismuch broader. As discussed below, this is due to the

A

B

Fig. 4. Contour plots of the experimental 13C double-quantum filtering efficiency at a spinning frequency of (A) 6 kHz and (B) 12kHz for DAF, Gly,and Ala. The 1H decoupling nutation frequency xI

nut=2p is plotted along the horizontal axis while the 13C recoupling nutation frequency xSnut=2p is

plotted along the vertical axis. The corresponding symmetry number N is indicated on the right-hand vertical axis. The black regions are inaccessible,due to technical limitations on the probe performance.

14 kHz

18 kHz

12 kHz6 kHz

16 kHz

DQ

F e

ffic

ien

cy /

%

20 kHz

0

20

40

60

30 60 90 120 0 30 60 90 120 0 30 60 90 1201H nutation frequency / kHz

0

20

40

60

0

Fig. 5. Double-quantum filtering efficiencies for DAF using therecoupling sequence R1252 as a function of 1H decoupling nutationfrequency, at the indicated MAS frequencies (xr/2p). The experimentaland simulated data are plotted as solid and dashed lines, respectively.


strong 1H–1H homonuclear interactions in the CH2

group of glycine. Nonetheless, it was still possible toachieve a DQ filtering efficiency of 36% for glycine usingthe R2092 sequence with no 1H decoupling at 12 kHzspinning frequency and 18% using the R1252 sequencewith no 1H decoupling at 20 kHz.

4.3. [2,3-13C2]Alanine (Ala)

The effects of 1H decoupling during 13C recouplingwere studied for [2,3-13C2]alanine for a series of RN m

n

sequences and MAS frequencies of 6 and 12 kHz(Fig. 4). These data are similar to the DAF data, prob-ably since the 1H–1H homonuclear interactions are rela-tively weak due to the rapid rotation of the CH3 group.It was possible to achieve a DQ filtering efficiency of48% for alanine using the R2092 sequence with no 1Hdecoupling at 12 kHz spinning frequency and 38% usingthe R1252 sequence with no 1H decoupling at 20 kHz.

5. Simulations

Simulations of the effects of 1H decoupling on 13Chomonuclear recoupling were performed using theSIMPSON platform [47]. The simulations consideredthe 13C nuclei and their directly-attached 1H nuclei ina single molecule of DAF, Gly, or Ala, ignoring the pro-tons attached to nitrogens, the 15N nucleus (in the caseof Gly) and all intermolecular interactions. Representa-tive SIMPSON input files, containing full details of thesimulated spin systems are provided in Supplementary

information. For each simulation, the signal intensitieswere averaged over 100 crystallite orientations. The re-sults of these simulations accompany the experimentaldata in Figs. 3,5, and 6. The full set of simulations aredisplayed as contour plots in Fig. 7 to facilitate compar-ison with the experimental data displayed in Fig. 4.

The simulated efficiencies were typically 20–40%higher than those observed experimentally. This discrep-ancy may be attributed to long-range spin–spin interac-tions, instrumental imperfections (such as phasetransients and RF inhomogeneity) and incoherent relax-ation. Despite the discrepancy in the overall scale, all themain experimental trends are reproduced qualitativelyin the simulations, including the existence of the two re-gimes, the differences between Gly and the other com-pounds, and the periodic structures in the plots of DQfiltering efficiency vs. 1H decoupling nutation frequencyobserved at high spinning frequency.

We have verified by simulation that the large homo-nuclear coupling between the two Ha protons is respon-sible for the different behaviour of glycine, compared toDAF and Ala. If the 1H–1H coupling is removed, thesimulated glycine plot is similar to that of DAF (notshown).

6. Theory

Hughes et al. [44] addressed the problem of heteronu-clear interference with homonuclear recoupling sequenc-es by calculating the magnitude of second-order averageHamiltonian terms, in the absence of a decoupling field.In the discussion below, we show that first-order averageHamiltonian theory provides a sufficient framework forunderstanding many of the essential features of hetero-nuclear interference, and allows one to explore the fullspace of decoupler field variation.

The theory of symmetry-based recoupling in hetero-nuclear systems has been given in [27]. A slightly modi-fied formulation is required for the case of CWirradiation of the 1H nuclei. In the following discussionthe 13C nuclei are denoted as S, while the protons aredenoted I.

The spin Hamiltonian during the RN mn sequence may

be written

HðtÞ ¼ HSðtÞ þ HISðtÞ þ HIðtÞ; ð5Þwhere the various terms involve the interactions of thespins with each other, with the molecular environmentand with the applied magnetic fields:

HSðtÞ ¼ HCSS ðtÞ þ HSSðtÞ þ HRF

S ðtÞ;HIðtÞ ¼ HCS

I ðtÞ þ HIIðtÞ þ HRFI ðtÞ;

ð6Þ

where HCSS ðtÞ and HCS

I ðtÞ represent chemical shift inter-actions, HSS (t), HIS (t) and HII (t) represent spin–spincouplings, and HRF

S ðtÞ and HRFI ðtÞ represent interactions

DQ

F e

ffic

ien

cy / %

120108968472604836

6 8 12 14 16 18 2010

70

60

50

40

30

20

10

0

13C nutation frequency / kHz

r /2π)/kHz(

Fig. 6. Double-quantum filtering efficiencies for DAF using therecoupling sequence R1252, without proton decoupling. The spinningfrequency xr/2p is varied along the horizontal axis. The corresponding13C recoupling nutation frequency is given by xS

nut ¼ 6xr for all points.The experimental and simulated data are plotted as solid and dashedlines, respectively.


with the resonant RF fields. The spin Hamiltonian termsare time-dependent due to the mechanical sample rota-tion and the modulation of the applied RF fields. Inthe case of the S-spins, the applied RF field is phase-modulated according to the RN m

n scheme

HRFS ðtÞ ¼ xS

nutðSx cos/SRFðtÞ þ Sy cos/

SRFðtÞÞ; ð7Þ

where /SRFðtÞis the time-modulated phase of the S-spin

RF field. For the sequences considered here, the ampli-tude of the S-spin RF field, expressed as a nutation fre-quency xS

nut ¼ 12jcSBS

RFj, is kept constant.In the case of the I-spins, we consider only CW irra-

diation with a constant nutation frequency

HRFI ¼ xI

nutIx. ð8ÞThe effect of the RF fields is conveniently described byintroducing two RF propagation operators, which obeythe equations:

d

dtURF

S ðt; t0Þ ¼ �iHRFS ðtÞURF

S ðt; t0Þ;

d

dtURF

I ðt; t0Þ ¼ �iHRFI ðtÞURF

I ðt; t0Þ;ð9Þ

where t0 is the time point at the start of the recouplingsequence. For the I-spins, this propagator has the simpleform of a rotation around the x-axis of the rotating ref-erence frame

URFI ðt; t0Þ ¼ RI

xðxInutðt � t0ÞÞ; ð10Þ

where

RIxðbÞ ¼ expf�ibIxg. ð11Þ

For the S-spins, the propagator under the modulatedRF field is much more complicated. It may be written ingeneral as

USRFðt; t0Þ ¼ RS

z ðaSRFðtÞÞRSy ðb

SRFðtÞÞRS

z ðcSRFðtÞÞ

¼ expf�iaSRFðtÞSzg expf�ibSRFðtÞSyg

� expf�icSRFðtÞSzg; ð12Þ

where faSRF; bSRF; c

SRFg are the three Euler angles describ-

ing the rotation induced by the modulated RF field.It proves convenient to express the I-spin rotation in

an analogous form to the S-spin rotation. If an addition-al �p/2 rotation of the I-spins is introduced (equivalentto a transformation into a tilted reference frame), wemay write

UIRFðt; t0ÞRI

yð�p=2Þ ¼ RIzðaIRFÞRI

yðbIRFÞRI

zðcIRFðtÞÞ; ð13Þ

where the I-spin Euler angles are given by:

aIRF ¼ 0;

bIRF ¼ � p

2;

cIRFðtÞ ¼ �xInutðt � t0Þ.

ð14Þ

The RN mn sequence is designed to impose a periodic

time-translation symmetry on the S-spin RF Euler an-gles. As described in [27], this time-translation symmetryhas the form:

bSRFðt þ qsRÞ ¼ bS

RFðtÞ þ qp;

cSRFðt þ qsRÞ ¼ cSRFðtÞ �2pmSN

q;ð15Þ

A

B

Fig. 7. Contour plots of the simulated 13C double-quantum filtering efficiency at a spinning frequency of (A) 6 kHz and (B) 12 kHz for DAF, Gly,and Ala. The 1H decoupling nutation frequency xI

nut=2p is plotted along the horizontal axis while the 13C recoupling nutation frequency xSnut=2p is

plotted along the vertical axis. The corresponding symmetry number N is indicated on the right-hand vertical axis.


where sR = nsr/N is the duration of one R element and q

is an integer. From now on, the winding number of theRN m

n sequence is written as mS, for reasons which willshortly become clear.

From Eq. (14), the I-spin RF Euler angles obey theequation:

bIRFðt þ qsRÞ ¼ bS

RFðtÞ;cIRFðt þ qsRÞ ¼ cIRFðtÞ � xI

nutqsR.ð16Þ

Now suppose that the I-spin nutation frequency isgiven by

xInut ¼

mIxr

n; ð17Þ

where mI is an integer. In this case, the symmetry rela-tionship Eq. (16) may be written:

bIRFðt þ qsRÞ ¼ bS

RFðtÞ;

cIRFðt þ qsRÞ ¼ cIRFðtÞ �2pmIN

q;ð18Þ

which has a similar form to Eq. (15) but is of the CN mn

class of symmetries [1]. The combination of the time-translation symmetries in Eqs. (15) and (18) therefore de-fine a dual CRN mI ;mS

n sequence on the IS spin system [27].The symmetry theory for dual CRN mI ;mS

n sequences, asdescribed in [27], may therefore be applied directly to theproblem of RN m

n recoupling in the presence of CW heter-onuclear decoupling, providing that the decouplingnutation frequency is such that Eq. (17) is fulfilled foran integer winding number mI, and that the additionalp/2 rotation given in Eq. (13) is taken into account. Inprinciple, RF fields which do not conform to Eq. (17)may be analyzed by splitting the I-spin RF field terminto two parts: a large part which conforms to Eq.(17), and a smaller perturbation term which is incorpo-rated into the average Hamiltonian theory. In this sec-tion, we avoid this complication by assuming that Eq.(17) is exactly fulfilled for an integer winding number mI.

The CRN mI ;mSn symmetry of the recoupling pulse se-

quence in the presence of CW proton decoupling leadsdirectly to the following selection rule for the first-orderaverage Hamiltonian terms [27]:

�H ð1Þ‘;m;kI ;lI ;kS ;lS

¼ 0 if mn� lImI � lSmS 6¼N2ZkS ; ð19Þ

where a general spin interaction is defined by the quan-tum numbers {‘, m, kI, lI, kS, lS}, representing the rankand component index under spatial rotations, the rankand component index under rotations of the I-spins,and the rank and component index under rotations ofthe S-spins, respectively. The symbol ZkS representsany integer with the same parity as the S-spin rank kS.If kS is odd, then ZkS takes values ±1,±3, . . . ±5, whileif kS is even, then ZkS takes values 0,±2, . . . ±4.

As an example, consider a R1462 sequence applied inthe presence of a CW heteronuclear decoupling with a

strength corresponding to xInut ¼ 15

2xr. This is very close

to the Hartmann–Hahn match condition, which isexpected to be damaging for the homonuclear recou-pling. From Eq. (17), this I-spin RF field correspondsto a I-spin winding number mI = 15. The selection ruleEq. (19) shows that a first-order average Hamiltonianterm vanishes unless the equality2m� 15lI � 6lS ¼ 7ZkS is satisfied for an integer ZkS

with the same parity as kS. The double-quantum homo-nuclear recoupling properties of the sequence R1462 fol-low from the fact that the terms {‘, m, kI, lI, kS,lS} = {2, ±1, 0, 0, 2, «2} are symmetry-allowed since2(±1) � 15(0) � 6(«2) = 7(±2). At the same time, theheteronuclear dipole–dipole coupling terms {‘, m, kI,lI, kS, lS} = {2, ±1, 1, ±1, «1} are symmetry-allowedsince 2(±1) � 15(±1) � 6(«1) = 7(«1). It follows thatseveral heteronuclear terms are symmetry-allowed atthe I-spin nutation frequency xI

nut ¼ 152xr, and that this

condition should therefore be avoided when applyingCW proton decoupling during R1462 recoupling.

The selection rule Eq. (19) allows logging of the sym-metry-allowed first-order average Hamiltonian terms asa function of the symmetry numbers {N,n,mI,mS}, provid-ing that the I-spin nutation frequency is an integer mul-tiple of xr/n. In addition, the magnitude of thesymmetry-allowed heteronuclear dipolar coupling termsmay be assessed, through the scaling factors jIS

2;m;1;lI ;1;lS,

which may be calculated using the method described in[27]. In general, there are several symmetry-allowed het-eronuclear terms for each combination of symmetrynumbers {N,n,mI,mS}. In this case, a qualitative heteronu-clear interference factor may be assessed by taking theroot sum square of the symmetry-allowed scaling factors

R ¼ RðN ; n; mI ; mSÞ

¼X

fm;lI ;lSgsymm allowed

j2;m;1;lI ;1;lS ðCRNmI ;mSn Þ

�� 22664

3775

1=2

; ð20Þ

where the sum is taken over all symmetry-allowed com-binations of quantum numbers {m,lI,lS} deriving froma heteronuclear dipole–dipole interaction.

An example of the Mathematica code used for thesecalculations is given in the Supplementary information.

Although Eq. (18) defines CN mn symmetry for the I-

spins (in the tilted frame defined by Eq. (13)), the CWirradiation does not have the form of a series ofphase-shifted elements, normally used to constructsequences with this symmetry. This illustrates an impor-tant point: the mathematical symmetry properties of theinteraction frame Hamiltonian are more general thanthe usual construction principles for these symmetries.In other words, the usual construction principles ofCN m

n and RN mn sequences are sufficient, but not necessary,

to generate the CN mn and RN m

n selection rules. In the


present case, CN mn symmetry is generated in a tilted

frame by an unmodulated RF field.Fig. 8 plots the heteronuclear interference factor

RðN ; n; mI ; mSÞ for a parameter space corresponding tothe simulations of DQ filtered efficiency shown inFig. 7A. As in those simulations, the condition n = 2is kept fixed, while N and mS are varied simultaneouslyalong the vertical axis, according to mS ¼ 1

2N � 1, so as

to maintain the homonuclear double-quantum recou-pling condition. To assist comparison with Fig. 7A,the top horizontal axis has been labelled according tothe I-spin nutation frequency corresponding to the plot-ted values of mI, at the same spinning frequencyxr/2p = 6 kHz used in the simulations. The right-handvertical axis has been labelled with the S-spin nutationfrequency corresponding to the plotted values of N

and mS, under the same conditions.There is a striking qualitative resemblance between

the plots in Figs. 7A and 8. Conditions providing a largevalue of R correspond very well to regions of poor DQfiltering performance. The two regimes under whichgood DQ filtering performance is observed correspondto regions free from points displaying large values ofR. The broad, roughly diagonal, band of very poorDQ filtering efficiency in Fig. 7A corresponds closelyto a similar band of finite R values in Fig. 8.

A closer examination of the two plots does, however,reveal some discrepancies. The symmetry analysis pre-dicts that a small change in I-spin nutation frequencyby one-half of the spinning frequency should radicallychange the degree of heteronuclear interference.Although simulations do agree with this prediction in

some regions of the plot, they do not confirm the pre-dicted sensitivity to the I-spin nutation frequency inother regions—especially near the broad, near-diagonal,band of very poor DQ filtering efficiency, where the DQfiltering efficiency appears to be insensitive to smallchanges in xI

nut. In addition, the first-order averageHamiltonian analysis cannot explain the qualitativelydifferent behaviour of glycine, which is due to the stronghomonuclear interactions in that compound.

Brinkmann and Eden [49] showed that the magnitudeof second-order average Hamiltonian terms is often in-creased in the vicinity of conditions which lead to finitefirst-order terms. A region of parameter space which hasa high density of finite first-order terms is thereforelikely to provide considerable second-order terms, evenif the first-order terms disappear under particular condi-tions. This effect is probably sufficient to explain thequalitative appearance of the simulated plots, includingthe expanded region of poorer performance for glycine.The experimental plots of double-quantum-filtering effi-ciency will be further broadened by effects such as RFinhomogeneity.

We have also calculated heteronuclear interferencefactors for homonuclear recoupling sequences of thetype CN 1

2, where 7 6 N 6 20 (see Supplementary Infor-mation). The first-order average Hamiltonian theorypredicts a very similar behaviour to the correspondingset of R-sequences.

7. Discussion

The main conclusions of this study are that there aretwo experimental regimes under which heteronuclearinterference with RN m

n sequences is minimized. In the firstregime, the S-spin RF field is moderated, while the I-spinRF field is as strong as possible. This regime is usuallyonly suitable for low spinning frequencies. In the secondregime, the S-spin RF field is as strong as possible, whilethe I-spin RF field is weak or absent. This regime allowsgood recoupling performance at moderate to high spin-ning frequencies. The second regime is a better match tothe typical experimental conditions for most current sol-id-state NMR experiments, and is expected to becomepopular. As an example, we have successfully performeddouble-quantum experiments on 13C2-labelled rhodopsinsamples at a spinning frequency of 10 kHz, using a R2092sequence without proton decoupling [48].

Although the two regimes are observed for a varietyof samples, the details of the behaviour are sample-de-pendent. As usual CH2 groups present the most severechallenges for 13C recoupling experiments, due to thestrong heteronuclear 1H–13C and homonuclear 1H–1Hcouplings. For substances like glycine, the no-decou-pling regime is only accessible when the strongest S-spinRF fields are used.

Fig. 8. Heteronuclear recoupling factor R, as defined in Eq. (20),plotted as a function of the symmetry numbers mI (horizontal axis) andN (vertical axis), for the set of sequences CRN mI ;mS

n , with n = 2 andmS = �1 + N/2. The chosen parameter space corresponds to thesimulation conditions in Fig. 7A, with which this plot should becompared. The corresponding nutation frequencies at a spinningfrequency of 6 kHz are indicated at the top and right edges of the plot.


We have also performed preliminary investigations of13C–13C recoupling over longer distances, where super-cycles are necessary to stabilize the long-term perfor-mance [40]. The effects of heteronuclear interferenceare similar to those observed at short range.

In summary, we have confirmed the conclusions ofHughes et al. [44], showing that symmetry-based recou-pling sequences produce good homonuclear recouplingwithout heteronuclear decoupling, even at low or moder-ate spinning frequencies. We have extended the study ofHughes et al. [44] to a range of RN m

n sequences, exploringa wide range of nutation frequencies on both spin species.A first-order average Hamiltonian analysis describes thesimulated and experimental spin dynamics rather well. Amore quantitative understanding of these phenomenamay require simulations of larger numbers of spins,and the inclusion of effects such as RF inhomogeneity.

Acknowledgments

This research was supported by the EPSRC (UK).D.H.B. was supported by the Natural Sciences andEngineering Research Council of Canada. We thankO.G. Johannessen for instrumental support and A. Se-bald for providing the 13C-labelled diammonium fuma-rate sample.

Appendix A. Supplementary data

Supplementary data associated with this article canbe found, in the online version, at doi:10.1016/j.jmr.2005.07.020. Full-colour versions of the plotsmay be viewed in the on-line version.

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