TOP-PASS: A processing algorithm to reduce 2D PASS acquisition time

Michael C. Davis, Kimberly M. Shookman, Jacob D. Sillaman, Philip J. Grandinetti∗

Department of Chemistry, The Ohio State University, 120 W. 18th Avenue, Columbus, Ohio 43210-1173

Abstract

A slow speed MAS spectrum contains a pattern of spinning sideband resonances separated by integermultiples of the rotor frequency and centered about an isotropic frequency. The 2D signal acquired in atwo-dimensional Phase Adjusted Spinning Sideband (PASS) experiment correlates this slow speed MASspectrum, obtained in the direct dimension, to an indirect dimension spectrum containing the same patternof spinning sideband resonances centered about a frequencyof zero. An affine transformation is used toconvert the acquired 2D PASS signal into a 2D signal that correlates a spectrum of pure isotropic frequenciesto a spectrum of spinning sideband resonances with no isotropic frequency contributions. The conventionalaffine transform applied to 2D PASS consists of an active shear ofthe signal parallel to the indirect timedomain coordinate followed by a passive scaling of the indirect time domain coordinate. Here we show thatan alternative affine transform, previously employed in the Two-dimensional One Pulse (TOP) experiment,can be employed to create the same 2D signal correlation withan enhanced spectral width in the anisotropic(spinning sideband) dimension. This enhancement can provide a significant reduction in the minimumexperiment time required for a 2D PASS experiment, particularly for spectra where the individual spinningsideband patterns are dispersed over a wider spectral rangethan the isotropic resonance frequencies. TheTOP processing consists of an active shear of the signal parallel to the direct time domain, followed by anactive shear of the signal parallel to the new indirect time domain coordinate followed by a passive scaling ofthe new direct time domain coordinate. A theoretical description of the affine transformation in the contextof 2D PASS is given along with illustrative examples of29Si in Clinoenstatite and13C in L-Histidine.

1. Introduction

Frequency anisotropy in magnetic resonancespectroscopy is a rich source of detail concerningstructure and dynamics at the macroscopic leveldown to the molecular level. At the macroscopiclevel these anisotropies can occur as a result of in-homogeneities in the external magnetic field, vari-ations in magnetic susceptibilities, or through the

∗Corresponding authorURL: http://www.grandinetti.org (Philip J.

Grandinetti)

intentional use of magnetic field gradients as inmagnetic resonance imaging. At the molecularlevel frequency anisotropy arises through magneticdipolar couplings amongst nuclei and through in-teractions of the nuclear multipole moments withsurrounding electrons. While the manifestation ofthese molecular level anisotropies in solution stateNMR is primarily through relaxation, its effectsare seen directly in solid-state NMR spectra as thepowder pattern lineshape.

Early in the history of NMR it was realizedthat inhomogeneous anisotropic broadenings can

Preprint submitted to Elsevier November 16, 2012

be removed through sample rotation[1–3]. In so-lution state NMR, sample rotation is a standardapproach to average away broadenings from mag-netic field inhomogeneities[1], while magic-anglesample spinning (MAS)[2, 3] has become a pop-ular and routine method in solid-state NMR foreliminating second-rank anisotropic broadenings,particularly when combined with the sensitivity en-hancement of cross-polarization (CP/MAS)[4, 5].With sample rotation, the inhomogeneous line-shape breaks up into a set of spinning sidebandscentered about a centerband lineshape and spacedat integer multiples of the spinning frequency. Asthe spinning frequency is increased the intensitiesof the spinning sidebands are reduced and trans-ferred into the centerband. In the limit of infinitespinning speed only the centerband frequency re-mains.

A few years after the introduction of CP/MAS,Herzfeld and Berger[6] developed an approach foranalyzing the sideband intensities in a slow speedMAS spectrum to obtain the same details about theanisotropic spin interactions as found in the staticpowder pattern lineshape. With the introduction oftwo-dimensional NMR spectroscopy[7], a numberof approaches have been designed to correlate ahigh or infinite speed MAS spectrum with eitherthe static[8–13], slow speed MAS spectrum[14–18] or only spinning sideband intensities[19, 14–18]. One of the more popular approaches is the el-egant Phase Adjusted Spinning Sidebands (PASS)experiment of Dixon[19]. While 2D PASS[16] ro-bustly correlates isotropic frequencies to spinningsideband patterns, it has a drawback that experi-ment times can become lengthy when spectra con-tain sites having a large number of spinning side-bands. In such situations, the sampling rate inthe indirect dimension is increased or the spinningspeed is increased. The latter approach, however,could result in a loss of sideband information fromother sites with smaller anisotropies. Here, we de-

scribe an alternative approach for processing 2DPASS signals, based on the TOP experiment[20–22], which increases the spectral width in theanisotropic (spinning sideband) dimension withoutincreasing the sampling rate in the indirect (ro-tor pitch) dimension (or even changing the spin-ning speed). We refer to this combination of TOPprocessing applied to the 2D PASS signal as theTOP-PASS experiment. This new approach canprovide a significant reduction in the minimum ex-periment time required for a 2D PASS experiment,particularly for samples with large anisotropies aswell as those simply having long relaxation times.

2. Experimental

NMR experiments on Clinoenstatite, MgSiO3,were performed on a Bruker Avance operating ata field strength of 9.4 Tesla corresponding to anoperating frequency of 79.576 MHz for29Si witha 4 mm Bruker MAS probe spinning at 1000± 2Hz using29Si rf field strength of 94 kHz. NMR ex-periments onL-Histidine were performed on a hy-brid Tecmag Apollo-Chemagnetics CMX II NMRspectrometer operating at a field strength of 9.4Tesla, corresponding to an13C NMR frequency of100.605 MHz and a1H NMR frequency of 400.068MHz, and using a 4 mm Chemagnetics double res-onance MAS probe spinning at 1500± 2 Hz with a1H rf field strength of 83 kHz for initial excitationand TPPM decoupling, a1H-13C cross-polarizationcontact rf field strength of 104.17 kHz and contacttime of 1.1 ms. An exponential apodization of 50Hz in the direct dimension and 100 Hz in the in-direct dimension was applied to all 2D datasets.L-Histidine monochloride monohydrate was ob-tained from Sigma Aldrich and used without fur-ther purification. The acquisition time for a single2D PASS cross-section in the indirect dimensionfor Clinoenstatite andL-Histidine was 9 hours and1.5 hours, respectively.

2

3. Results and Discussion

3.1. Bloch Decay MAS

The first-order contribution of the nuclearshielding to the NMR frequency for a single crys-tallite as a function of rotor angle and phase can beexpanded in a Fourier series[23] as

Ω(1)σ (θR, φR) = 0(θR, α, β)

+∑

m,0

m(θR, α, β)eim(φR+γ),

(1)

whereθR is the rotor angle,φR is the rotor phase,α, β, andγ are the Euler angles between the ro-tor coordinate frame and the crystallite coordinateframe. In the simple Bloch decay MAS experi-ment the signal phase as function of time,t, whereφR(t) = ωRt + χR, is

Φ(t) =

t∫

0

Ω(1)σ (θR, φR(t′)) dt′

= W0 t +∑

m,0

Wmeim(χR+γ)[

eimωRt − 1]

,

(2)

whereχR is the initial rotor phase, and we define

W0 = 0(θM, α, β), (3)

and

Wm =m(θM , α, β)

imωR. (4)

HereθM = cos−1(1/√

3). One can show[23] thatthe Bloch decay MAS signal is given by

sB(α, β, t, χR) = se(t) eiW0t

×∑

N1,N2

AN1A∗N2e−iN1ωRtei(N2−N1)(χR+γ),

(5)

where

AN =12π

2π∫

0

exp

i∑

m,0

WmeimΘ

eiNΘdΘ, (6)

and se(t) represents the envelope function due to

the relaxation. A partial averaging of the Bloch de-cay signal over the angleγ eliminates the depen-dence on the initial rotor phase,χR, yielding

〈sB(α, β, t)〉γ = se(t) eiW0t

∑

N

|AN |2e−iNωR t. (7)

The Bloch decay MAS signal from all crystallites,SB(t), is then given by

SB(t) =∫ 2π

0dα∫ π

0sinβ dβ 〈sB(α, β, t)〉γ

= se(t) eiW0t

∑

N

INe−iNωRt,(8)

where

IN =

∫ 2π

0dα∫ π

0sinβ dβ |AN |2. (9)

Notice that W0 during MAS is only dependenton isotropic frequency contributions, whereas thespinning sideband intensities,IN , are only depen-dent on anisotropic frequency contributions.

3.1.1. TOP

One can rewrite Eq. (8) in the form

S(t1, t2) = se(t2)eiW0t2

∑

N

INe−iNωR t1, (10)

and visualize the 1D Bloch decay MAS signal froma rotating sample as a signal filling a 2Dt1–t2 co-ordinate system, as illustrated in Fig. 1. Note that

SB(t1 + ntR, t2) = SB(t1, t2), (11)

wheretR = 2π/ωR andn is an integer. The signifi-cance of this coordinate system is that the isotropicfrequency contributions are removed along thet1dimension, and the anisotropic frequency contri-butions are removed along thet2 dimension. Thiscoordinate system is particularly useful for under-standing a number of important solid-state NMRexperiments for manipulating rotary echoes and

3

Figure 1: Solid circles and arrows represent the sampling tra-jectory of the 1D Bloch decay MAS signal (blue circles) andits complex conjugate signal (brown circles) in thet1–t2 co-ordinate system. The slope of the sampling trajectory in thet1-t2 coordinate system is 1. Identical signal run parallel, sep-arated bytR in t1 and t2, respectively. Only anisotropic fre-quency contributions are present along thet1 dimension, andonly isotropic frequency contributions are present along thet2dimension.

their associated spectral spinning sidebands. Prob-ably the simplest of these is the Two-dimensionalOne Pulse (TOP) processing approach [20–22],which samples the 2D signal of Eq. (10) usingidentical 1D Bloch decay signals running parallelto each other and separated bytR in both t1 andt2,as shown in Fig. 1.

In the TOP approach, one first generates apseudo-2D signal,Stop(ǫ, t), in a ǫ–t coordinatesystem from the 1D Bloch decay (or rotor synchro-nized Hahn echo) signal,SB(t). This is illustratedin Fig. 2A, where a sampling ofStop(ǫ, t) for posi-tive and negative values ofǫ is generated using therelationship

Stop(ǫn = n2π/ωR, t) = SB(t), (12)

for a range of positive and negative integern val-ues. The 2D signal constructed in this manner hasno decay from relaxation along theǫ dimension.This 2D signal produces a pure absorption mode

[ []*= ]0 0

Figure 2: Scheme for generating the pseudo-2D signal forTOP processing from the 1D Bloch decay signal in a rotat-ing sample. (A) Identical Bloch decay signal are laid downin a coordinate system at integer multiples of the rotor pe-riod along theǫ dimension. (B) The complex conjugate of the1D Bloch decay signal is used to generate a Bloch decay sig-nal going backward in time. (C) Identical complex conjugateBloch decay signals are placed in opposite quadrant in thet–ǫcoordinate system. (D) The TOP signal before application ofthe affine transformation.

4

2D spectrum since it is generated for positive andnegative values ofǫ, as illustrated in Fig. 2A. Ifdesired, a full sampling of the TOP signal in allfour quadrants can be generated using the complexconjugate of the Bloch decay signal,S∗B(t), as il-lustrated in Fig. 2B and 2C. In the generated 2Dsignal, Stop(ǫ, t), acquired as a function ofǫ andt2, the anisotropic frequency contributions are refo-cused into a echo along the lineǫ + t1 = 0, and theisotropic frequency contribution is refocused alongt = 0, as shown in Fig. 2D.

After the 2D signal,Stop(ǫ, t), is created, anaffine transformation, as illustrated in Fig. 3, is ap-plied to separate the isotropic and anisotropic fre-quency contributions into orthogonal dimensions.For the TOP signal this is performed as a doubleshear, starting with a shear parallel to thet coordi-nate with a shear ratio ofκt = −1, followed by ashear parallel to thet′2 coordinate with a shear ratioof κt′2 = 1, according to

t2

t1

=

1 1

0 1

︸ ︷︷ ︸

Kt′2

1 0

−1 1

︸ ︷︷ ︸

Kt

ǫ

t

=

0 1

−1 1

ǫ

t

.

(13)

The application of this affine transformation toStop(ǫ, t) yields the TOP[22] signal in Eq. (10)whose Fourier transform yields a 2D spectrumcorrelating isotropic and anisotropic frequencies.Prior to the final 2D Fourier transformation, amatched-filter apodization can be applied alongthe t2 (isotropic) dimension to improve sensitiv-ity. Any apodization along thet1 (anisotropic, side-band) dimension, however, is applied entirely forcosmetic reasons, creating a purely artificial linebroadening of the sideband resonances. Most im-portantly, note that the application of this affinetransformation to the original digital sampling

Figure 3: TOP transformation ofS(ǫ, t) into a 2D signal thatcorrelates isotropic and anisotropic frequencies. An affinetransformation,At, consisting of a shear parallel tot, createsa 2D signal with anisotropic frequencies refocused along thet′2 axis and the isotropic frequences refocused alongt + ǫ = 0.The green dashed line in the top figure represents the passiveaffine transformation of the 2D coordinate system to create atime coordinate,t′2, along which the 2D signal is unaffectedby the anisotropic frequency contributions. AfterAt′2

, a shearparallel tot′2, a 2D signal with anisotropic frequencies refo-cused along thet2 axis and isotropic frequencies is refocusedalongt1 is obtained. The red dashed line in the middle figurerepresents the passive affine transformation of the 2D coordi-nate system to create a time coordinate,t1, along which the 2Dsignal is unaffected by the isotropic frequency contributions.5

rates,∆ǫ and∆t, yields transformed sampling ratesof ∆t2 = ∆ǫ = tR and∆t1 = ∆t, respectively, afterthe double shear transformation.

In Fig. 4 is an example of the application of theTOP processing applied to a 2D signal,Stop(ǫ, t),constructed from the29Si NMR Hahn echo sig-nal of polycrystalline Clinoenstatite. Clinoenstatitehas two inequivalent tetrahedrally coordinated (i.e.,Q(2)) silicon sites, resolved at -85.2 ppm and -82.8ppm in the isotropic dimension, and exhibit similarnuclear shielding anisotropies, as expected[24, 25],in the sideband dimension.

A limitation of TOP processing applied to a 1DBloch decay (or rotor synchronized Hahn echo)MAS signal is that the spectral width inω2, theisotropic frequency dimension, is limited to inte-ger divisors of the rotor frequency. This cannot beeasily corrected by increasing the rotor frequencysince it also reduces the information content in thesideband intensities. Nonetheless, TOP process-ing applied to a 1D Bloch decay has a numberof added advantages, particularly when applied tohalf-integer quadrupolar nuclei[22] where its mainstrength lies in the rapid interpretation of MAS sig-nals. The experimental simplicity of TOP process-ing applied to a 1D Bloch decay makes it a com-pelling method, and it is surprising that it has notbeen more widely utilitized.

3.1.2. 2D PASS

Generally, the NMR signal in a rotating sam-ple can be manipulated into a number of desirableforms by applying a series ofπ pulses between theinitial excitation pulse and the start of signal acqui-sition. In the PASS experiment [19, 16], a time co-ordinate is defined where the initial excitation pulseis applied att = −T and signal acquisition beginsat t = 0. Between the initial excitation pulse andsignal acquisition areQ π-pulses, applied at times−T + τ1,−T + τ2, . . . ,−T + τQ. The signal phaseat t = 0, a duration ofτQ+1 = T after the initial

excitation pulse, is given by

ΦQ(t = 0) = (−1)QQ∑

q=0

(−1)q−T+τq+1∫

−T+τq

Ω(t′)dt′,

= W0

T − 2(−1)Q

Q∑

q=1

(−1)qτq

+∑

m,0

Wmeim(χR+γ)

×

1−2(−1)Q

Q∑

q=1

(−1)qeimθq e−imθT −(−1)Qe−imθT

,

(14)

whereτ0 = 0, θT = ωRT , andθq = ωRτq.In the PASS experiment [19, 26] the timings of

theQ π pulses are manipulated so the signal phaseat t = 0 has a form

Φpass(ǫ, t = 0) =

W0 [ 0 ] +∑

m,0

Wmeim(χR+γ)[

1− e−imωRǫ]

,(15)

where ǫ is a function of theQ π pulse timings.Evolving forward fromt = 0 the PASS signal phasethen becomes similar to the Bloch decay MAS sig-nal

Φpass(ǫ, t) =

W0 t +∑

m,0

Wmeim(χR+γ)[

eimωRt − e−imωRǫ]

,(16)

with the important exception thatǫ is non-zero andcan be varied independent oft. The PASS ap-proach for obtaining the signal phase of Eq. (15)comes from equating Eqs. (14) and (15) to obtainthe PASS equations:

θT − 2(−1)QQ∑

q=1

(−1)qθq = 0, (17)

and

e−imΘ =

2(−1)QQ∑

q=1

(−1)qeimθq e−imθT + (−1)Qe−imθT ,

(18)

6

0 -40 -804080

-78

-80

-82

-84

-86

-88

0 -40 -804080

Figure 4: Application of TOP processing to a29Si NMR rotor synchronized Hahn echo signal of Clinoenstatite. Isotropic frequen-cies are referenced to TMS. Contour levels are plotted from 5%-100% in increments of 5% of the maximum intensity.

whereΘ = ωRǫ. Levitt and coworkers[16] sug-gested a fiveπ pulse (Q = 5) 2D PASS experiment,of constant durationT , with θT = 2π, and obtainedthe equations

25∑

q=1

(−1)qθq + 2π = 0, (19)

and

−25∑

q=1

(−1)qeimθq −1 = e−imΘ, for m = 1, 2. (20)

These equations can be solved numerically for theπ pulse timings shown in Fig. 5 and are tabulatedelsewhere[16].

The phase of Eq. (16) leads to a PASSsignal[23], when averaged over the crystallite an-gles, of the form

Spass(ǫ, t) = eiW0t∑

N

I(N)e−iNωR(t+ǫ). (21)

The 2D PASS signal, just like the Bloch decayMAS signal, can be used to fill the 2D signal in aǫ–t coordinate system, as shown at the top of Fig. 6.

In the case of PASS, a single shear parallel to theǫ coordinate with a shear ratio ofκǫ = −1 followed

0.0

0.2

0.4

0.6

0.8

1.0

Figure 5: Pulse sequence and timings for the fiveπ pulse con-stant time 2D PASS experiment of Antzutkin et al.[16]. Here,ωR is the rotor spinning frequency.

7

Figure 6: Affine transformation,Aǫ , for transforming the 2DPASS signal,S(ǫ, t), into a 2D signal that correlates isotropicand anisotropic frequencies, consists of a shear parallel toǫ, giving a 2D signal with anisotropic frequencies refocusedalong thet2 axis and isotropic frequencies refocused alongt1.The green dashed line in the top figure represents the passiveaffine transformation of the 2D coordinate system to create atime coordinate,t2, along which the 2D signal is unaffected bythe isotropic frequency contributions.

by a scaling ofς(ǫ∗) = −1, as outlined in Fig. 6, andgiven by

t1

t2

=

−1 0

0 1

︸ ︷︷ ︸

Sǫ∗

1 −1

0 1

︸ ︷︷ ︸

Kǫ

ǫ

t

=

−1 1

0 1

ǫ

t

,

(22)

has been the conventional approach for obtaining a2D spectrum correlating isotropic and anisotropicfrequencies. The original acquisition digital sam-pling rates,∆ǫ and∆t, after the single shear be-come∆t1 = ∆ǫ = tR and∆t2 = ∆t, respectively.In Fig. 7 is an example of the application of thesingle shear to a 2D PASS29Si signal from Cli-noenstatite as a function of the number of stepstaken in varyingǫ from zero to one full rotor pe-riod. Only after 16 steps does the spectral widthin the anisotropic (spinning sideband) dimensionhave sufficient width. The central idea of thiswork is that the application of TOP processing,that is, a double shear, can allow one to obtain a2D spectrum with sufficient spectral width in boththe isotropic and anisotropic dimension with sig-nificantly fewer steps in theǫ dimension. This issimply because the application of the TOP trans-formation to the original digital sampling rates,∆ǫ and ∆t, yields transformed sampling rates of∆t2 = ∆ǫ = tR and∆t1 = ∆t, respectively. Thus,a 29Si 2D PASS experiment on Clinoenstatite onlyneeds one step inǫ when using the double shear(TOP) processing approach to resolve the twoQ(2)

sites in both dimensions, compared to the 16 stepsneeded with single shear (conventional) process-ing approach. This is illustrated in Fig. 8, wherethe TOP processed 2D PASS signals of Clinoen-statite are presented as a function of the numberof steps taken in varyingǫ from zero to one fullrotor period. Close examination of the Clinoen-statite 2D PASS spectrum reveals a third additional

8

-40

-80

-120

-160

0 -40 -8040800 -4040000

1 Step 2 Steps 4 Steps 8 Steps 16 Steps

Figure 7: Single shear processing of 2D PASS signal of Clinoenstatite. Isotropic frequencies are referenced to TMS. Contour levels are plotted from 5%-100% in incrementsof 5% of the maximum intensity.

9

2 Steps

4 Steps

8 Steps

16 Steps

0 -40 -804080

0

-40

-80

-120

-160

-40

-80

-120

-80

-80

-80

1 Step

Figure 8: Double shear (TOP) processing of 2D PASS signal of Clinoenstatite. Isotropic frequencies are referenced to TMS.Contour levels are plotted from 5%-100% in increments of 5% of the maximum intensity.

10

resonance appearing at -63.7 ppm in the isotropicdimension due to a minor impurity of crystallineForsterite. A proper acquisition of this third siteonly require 4 steps in the indirect dimension ofTOP-PASS, whereas, the conventional 2D PASSexperiment would still require 16 steps in the in-direct dimension. An additional advantage of theTOP transformation over the single shear transfor-mation is that the spectral width required in theindirect dimension is easily obtained from a highspeed MAS spectrum.

When the spectral width in the isotropic dimen-sion is significantly smaller than the spectral widthof the anisotropic dimension then the TOP process-ing can greatly reduce the number of steps requiredin the ǫ dimension. In contrast, the time savingsafforded by TOP processing is less significant asthe spectral width of the isotropic dimension ap-proaches that of the anisotropic dimension. Tohighlight this point, we compare the13C 2D PASSNMR spectra ofL-Histidine[27] processed with theconventional single shear approach, and shown inFig. 9, with the13C TOP-PASS spectra, processedwith a double shear, and shown in Fig. 10. Inthis example, the conventionally processed PASSsignal required 16 steps when varyingǫ to havesufficient spectral width to prevent aliasing of thecarbonyl sideband resonances, whereas the TOP-PASS spectrum required approximately the samenumber of steps to obtain sufficient spectral widthfor the full isotropic spectrum.

Finally, it is worth noting that the reason the sin-gle shear processing approach fails when appliedto the pseudo-2D signal constructed from a Blochdecay MAS signal is because the original digitalsampling rates,∆ǫ and∆t, after the single shearbecome∆t1 = ∆ǫ = tR and∆t2 = ∆t, respectively,and such a sampling rate alongt1, the sideband-only coordinate, simply cannot resolve any spin-ning sidebands inω1.

4. Summary

We have shown that the simple application ofthe TOP processing approach (i.e., a double shearaffine transformation) to 2D PASS signal can sig-nificantly reduce the number of measurementsneeded in the indirect dimension. When applyingthe TOP transformation to the 2D PASS signal it isthe spectral width in the “infinite speed” isotropicdimension that determines the dwell time neededin the indirect dimension. This can lead to signifi-cant time savings for systems where the span of theanisotropic lineshapes exceeds the minimum spec-tral width required for the isotropic dimension. Thesame approach can be identically applied to sig-nals from the 2D PASS experiment designed for thesecond-order broadened central transition of half-integer quadrupolar[28, 17].

Acknowledgments

This material is based upon work supported inpart by the National Science Foundation. Anyopinions, findings and conclusions or recommen-dations expressed in this material are those of theauthor(s) and do not necessarily reflect the viewsof the National Science Foundation (NSF).

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0 -40 -804080

0

100

200

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0 -4040000

Figure 9: Single shear processing of 2D PASS signal fromL-Histidine. Isotropic frequencies are referenced to TMS. Contour levels are plotted from 5%-100% in incrementsof 5% of the maximum intensity.

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