Comparison of double-quantum NMR normalization schemes to measure homonuclear dipole-dipole interactions Kay Saalwächter Citation: The Journal of Chemical Physics 141, 064201 (2014); doi: 10.1063/1.4890996 View online: http://dx.doi.org/10.1063/1.4890996 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in DQ-DRENAR: A new NMR technique to measure site-resolved magnetic dipole-dipole interactions in multispin1/2 systems: Theory and validation on crystalline phosphates J. Chem. Phys. 138, 164201 (2013); 10.1063/1.4801634 Zero-quantum frequency-selective recoupling of homonuclear dipole-dipole interactions in solid state nuclear magnetic resonance J. Chem. Phys. 131, 045101 (2009); 10.1063/1.3176874 Frequency-selective homonuclear dipolar recoupling in solid state NMR J. Chem. Phys. 124, 194303 (2006); 10.1063/1.2192516 Through-space contributions to two-dimensional double-quantum J correlation NMR spectra of magic-anglespinning solids J. Chem. Phys. 122, 194313 (2005); 10.1063/1.1898219 Double-quantum homonuclear correlation magic angle sample spinning nuclear magnetic resonance spectroscopy of dipolar-coupled quadrupolar nuclei J. Chem. Phys. 120, 2835 (2004); 10.1063/1.1638741

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THE JOURNAL OF CHEMICAL PHYSICS 141, 064201 (2014)

Comparison of double-quantum NMR normalization schemes to measure homonuclear dipole-dipole interactions Kay Saalwächtera) Institut für Physik – NMR, Martin-Luther-Universität Halle-Wittenberg, Betty-Heimann-Str. 7, D-06120 Halle, Germany

(Received 9 May 2014; accepted 10 July 2014; published online 8 August 2014) A recent implementation of a double-quantum (DQ) recoupling solid-state NMR experiment, dubbed DQ-DRENAR, provides a quantitative measure of homonuclear dipole-dipole coupling constants in multispin-1/2 systems. It was claimed to be more robust than another, previously known experiment relying on the recording of point-by-point normalized DQ build-up curves. Focusing on the POSTC7 and BaBa-xy16 DQ pulse sequences, I here present an in-depth comparison of both approaches based upon spin-dynamics simulations, stressing that they are based upon very similar principles and that they are largely equivalent when no imperfections are present. With imperfections, it is found that DQ-DRENAR/POST-C7 does not fully compensate for additional signal dephasing related to chemical shifts (CS) and their anisotropy (CSA), which over-compensates the intrinsic CS(A)-related efficiency loss of the DQ Hamiltonian and leads to an apparent cancellation effect. The simulations further show that the CS(A)-related dephasing in DQ-DRENAR can be removed by another phase cycle step or an improved super-cycled wideband version. Only the latter, or the normalized DQ build-up, are unaffected by CS(A)-related signal loss and yield clean pure dipolar-coupling information subject to unavoidable, pulse sequence specific performance reduction related to higher-order corrections of the dipolar DQ Hamiltonian. The intrinsically super-cycled BaBa-xy16 is shown to exhibit virtually no CS(A) related imperfection terms, but its dipolar performance is somewhat more challenged by CS(A) effects than POST-C7, which can however be compensated when applied at very fast MAS (>50 kHz). Practically, DQ-DRENAR uses a clever phase cycle separation to achieve a significantly shorter experimental time, which can also be beneficially employed in normalized DQ build-up experiments. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4890996] I. INTRODUCTION

The continuing development and optimization of solidstate NMR experiments aimed at a determination of homoand heteronuclear dipole-dipole coupling constants between nuclear spins is of high relevance for studies of molecular structure and dynamics in condensed matter.1–8 In this context, methods that provide an internal compensation for signal loss during the pulse sequence are of particular relevance to obtain truly accurate results. The classic REDOR experiment solves this problem quantitatively and elegantly for the case of heteronuclear couplings via the separate recording of a reference intensity.9 Extending this concept to homonuclear coupling determinations is less straightforward, and different approaches have been published, as reviewed recently.8 In two recent publications,10, 11 a magic-angle (MAS) spinning NMR experiment termed “DQ-DRENAR” for measuring homonuclear dipole-dipole coupling constants in multispin-1/2 systems was presented, and claimed to be more robust with regards to imperfections than previous approaches. It is based upon any pulse sequence with a doublequantum (DQ) Hamiltonian H¯ DQ , specifically (but not limited to) the well known POST-C7,12 and its dephasing effect on z magnetization, yielding the S = {C, C} signal function, a) Electronic mail: [email protected]. URL: www.

physik.uni-halle.de/nmr. 0021-9606/2014/141(6)/064201/12/$30.00

following the notation in the previous paper. Imperfectionrelated signal decay is normalized by a suitable, very similar reference experiment, in which the second half of the DQ sequence is 90◦ phase-shifted with respect to the first half (C ). This phase shift leads to a sign change of the DQ Hamiltonian, providing −H¯ DQ during the second half of the sequence. This means that a {C, C } sequence provides a multi-spin dipolar echo function S0 in which no coherent dephasing occurs, but which is, to a good approximation, subject to the same imperfection- or dynamics-related signal decay (loosely: “relaxation”). The analyzed signal is Snorm = 1 −

S {C, C} , =1− S0 {C, C  }

(1)

which is an intensity build-up function from which the dipolar coupling constant can be extracted by a suitable fit. Since the length of the C element is two rotor periods (2τ R ), increasing pulse sequence durations are realized by performing experiments at different MAS frequencies and/or by repeating two combined blocks, {C, C}n and {C, C }n . An earlier approach to normalizing for relaxation is the recording of a DQ build-up curve which relies on a four-step DQ filter selecting DQ coherences (strictly, all 4N + 2 coherence orders) between two blocks of a DQ pulse sequence, and dividing it by the sum of the DQ curve and a reference signal, the latter being obtained by a simple switch in the receiver

141, 064201-1

© 2014 AIP Publishing LLC

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Kay Saalwächter

J. Chem. Phys. 141, 064201 (2014)

phase cycle. Time incrementation is here realized by lengthening (repeating) the DQ excitation and reconversion periods separately, i.e., {Cm , Cm }, which for m = 1 corresponds to the above approach. Details can be found in the original publications for static13 and MAS14, 15 implementations, and in more recent reviews.5, 8 With regards to this method, it was claimed11 that the 0→±2 and ±2→0 two-step coherence order selection theme realized by a four-step phase cycle can suffer limited efficiencies related to chemical-shift anisotropy (CSA) and other imperfection effects. In order to provide a balanced picture, I here address the differences and commonalities of the two approaches. I provide an in-depth comparison of the way the signal functions are obtained in the two experiments, and pinpoint where differences arise, realizing that without imperfections in a purely dipolar spin system both experiments in fact yield the very same information. We will find differences in the way both approaches deal with CSA-related additional signal dephasing and efficiency loss in the actual DQ excitation by way of spin dynamics simulations and a few experiments, focusing on 1 H and 31 P spin systems in organic polymer samples and in organic phosphates, respectively.

II. IMPLEMENTATIONS AND THEORETICAL CONSIDERATIONS

DQ spectroscopy relies on a pulse sequence featuring a DQ avarage Hamiltonian, (j k) (j k) (j k) H¯ DQ = ωeff,22 Ij + Ik+ + ωeff,2−2 Ij − Ij − (j k) (j k) (j k) (j k) = ωeff,22 Tˆ2,2 + ωeff,2−2 Tˆ2,−2 ,

FIG. 1. Implementation schemes for DQ spectroscopy with (a) single-cycle excitation-reconversion blocks which are repeated jointly, as suggested for DQ-DRENAR, and (b) separately incremented excitation and reconversion periods, as usually done to record DQ build-up curves. The table shows the different phase cycling options, with the different individual (separated) and phase-cycle summed signal functions.

that τ DQ is the length of separate DQ excitation and reconversion periods. A relevant difference of the implementation of the two approaches (which does not affect the final signal) is the way in which experiments taken with different phase shifts ϕ are combined. In DQ-DRENAR, they are measured separately, and its final signal function, Eq. (1) in new notation, reads

(2)

which can be designed for static16 or magic-angle spinning (MAS) conditions.12, 15, 17, 18 In the present work, I focus on the POST-C7 pulse sequence,12 which has the same H¯ DQ (j k) (j k) as its uncompensated variant C717 with |ωeff,22 | = |ωeff,2−2 | (j k) = ωeff = 343 D [1 + sin(π/14)]1/2 sin 2β, as well as on 260 j k 15 BaBa-xy16, which is the compensated variant of BaBa18 (j k) (j k) jk 3 with ωeff,22 = ωeff,2−2 = ωeff = π √ D sin 2β sin γ . In all 2 jk cases, Djk is the distance-dependent dipole-dipole coupling constant between the nuclei j and k in units of rad/s, which is the quantity of interest.

A. Pulse sequence implementations

Before turning to explicit time-evolution calculations, I introduce a suitable pulse sequence notation that allows us to clearly describe and compare the two approaches under discussion. Figs. 1(a) and 1(b) show block diagrams, where each rectangle is associated with a DQ pulse sequence block with a cycle time τ c characterized by a certain set of phases ϕ i of its constituent pulses. The second block features a variable overall phase shift ϕ, and the versions in (a) and (b) represent the stroboscopic and separate blockwise time incrementation schemes of DQ-DRENAR11 and normalized DQ spectroscopy,5, 8 respectively. The final overall length of the pulse sequence is 2τ DQ , in keeping with the usual convention

n =1− Snorm

n S0,0 n S0,90

,

(3)

where each of the two partial signal functions are either the result of a single scan, or comprise a CYCLOPS on the final read-out pulse. For simplicity, they can be treated as singlescan intensities. In normalized DQ spectroscopy, a 4N + 2 coherence order filter is used to record the so-called DQ intensity, which in its relevant time range is dominated by 2Q coherences.13 This corresponds to summing up 4 phase-shifted versions along with receiver phase inversion (180◦ ) when ϕ is 0◦ or 180◦ , m m m m m SDQ = S0,90 − S0,180 + S0,270 − S0,0 .

(4)

The reference experiment employs a 4N coherence filter; it is thus dominated by modulated longitudinal magnetization (LM), and to a lesser degree by 0Q and 4Q coherences. In a pulse sequence with a pure DQ Hamiltonian it provides the perfect intensity complement: m m m m m Sref = S0,90 + S0,180 + S0,270 + S0,0 .

(5)

The two measured functions are summed up to obtain the sum multiple-quantum reference intensity, S MQ = SDQ + Sref , which represents a fully refocused reference signal subject only to relaxation. Point-by-point normalization gives

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064201-3

Kay Saalwächter

J. Chem. Phys. 141, 064201 (2014)

the normalized DQ build-up curve SnDQ = SDQ /S MQ , m SnDQ =

m m m m S0,90 − S0,180 + S0,270 − S0,0  m m m m 2S0,270 + S0,180 − S0,180 + S0,0

m 2S0,90 +   m m + S0,0 S0,180 1 = . 1− m m 2 S0,90 + S0,270

m − S00



(6)

The terms between brackets [. . . ] in the denominator in the first line may appear superfluous as they cancel to zero, however, they have an important contribution to the actual experimental signal, as these 4 scans do contribute added noise that deteriorates the final signal. This is an unwanted consequence that is built into any coherence selection phase cycle. Yet Eq. (6) demonstrates that the problem can be circumvented for the SnDQ signal by simply skipping the selection phase cycle, and taking the four signals separately. Thus, assembling the final signal from separately measured partial signals offers a signal-to-noise advantage of a factor of 2 (a 4-fold time advantage!), since the bracketed terms as well as part of the fraction that reduces to 1 (second line) represent the added noise of 4 scans each if selection phase cycles are used. Now comparing Eqs. (3) and (6), I note the following differences. First, a trivial factor of 2 is identified that can be removed by re-definition. Second, the incrementation schemes (n vs. m) differ. This will be addressed below, but I note already here that the relevant imperfection effects are already apparent for the short-time behavior, where the approaches are identical if n = m = 1. From the time being, I will only address this case and skip the index. Third, the nDQ approach combines each S0,ϕ signal function with its 180◦ phase shifted variant, S0, ϕ+180 . To first order, one can argue that the two are identical if only dipole-dipole couplings and no imperfections are considered, as the pure DQ Hamiltonian excites only even-quantum orders and is thus invariant under a 180◦ phase shift.16, 19 Thus, with the mentioned restrictions, the signal functions of the two sequences in the short-time limit are mathematically equivalent. However, this additional and reasonable phase cycling step has a subtle consequence. The addition of two scans with inverted the second half of the sequence suppresses the odd quantum orders that have possibly been excited by the first half of the sequence, and these can and do arise from imperfections, as demonstrated in detail below. For comparisons, the following notations for the two experiments, and also new permutations of the implementations, are introduced: n S0,0 n = 1− n DQ-DRENAR, (7) Snorm S0,90 i m Snorm

= 1−

m m S0,180 + S0,0 m m S0,90 + S0,270

m Snorm = 1−

i n Snorm

= 1−

m ≡ 2SnDQ

m S0,0 m S0,90

n n S0,90 + S0,270

(8)

(9)

,

n n S0,180 + S0,0

nDQ,

.

(10)

Equations (9) and (10) define potentially improved DQDRENAR versions which test the difference between blockwise and stroboscopic time incrementation, and the potential benefit of including the additional phase-inversion step into the original blockwise implementation, respectively. In actual experiments, only the incrementation scheme represents a relevant difference in the pulse sequences; the different signal functions S0, ϕ should be acquired in four separate experiments for each DQ evolution time to realize the mentioned significant time advantage. B. Time evolution calculations

The time evolution of z magnetization during N rotor cycles of length τ R = 2π /ωR in an MAS experiment with rotor frequency ωR under such a pulse sequence, denoted as C, for a spin pair is calculated as (j k)

NτR H¯ DQ

(Ij z + Ikz ) −−−−−→ (Ij z + Ikz ) cos φ  +Oˆ DQ sin φ  ,

(11)

where φ  = ωeff N τR is a pulse-sequence duration dependent phase factor, and Oˆ DQ denotes a two-spin operator associated with DQ coherence that is “90◦ phase-shifted” with respect to H¯ DQ (meaning that both obey a cyclic commutation relationship with Ijz + Ikz ). For BaBa, it reads Oˆ DQ = i(Ij + Ik+ − Ij − Ik− ), while for (POST-)C7 it takes a more complicated form11, 12 due to its γ -encoded feature. Equation (11) describes the action of a single {C} or multiple {C}m blocks of the DQ pulse sequence. For another evolution step under the pulse sequence C, completing the S0, 0 = {C, C} experiment, we obtain (j k)

(Ij z + Ikz ) cos φ + Oˆ DQ sin φ (j k)

NτR H¯ DQ

−−−−−→(Ij z + Ikz )(cos2 φ − sin2 φ) +Oˆ DQ 2 sin φ cos φ.

(12)

This calculation follows the spirit of simple product operator theory,20 realizing that (Ijz + Ikz ), Oˆ DQ and the spin (j k) part of H¯ DQ form a cyclically commuting operator subset of Liouville space. The benefit of this calculation is that it is now seen that the sin 2 φ term, associated with DQ coherence after the first half of the pulse sequence, is in fact contributing equally and (anti)symmetrically to the overall dephasing of z magnetization. Simply, dephasing (cos terms) and coherence creation (sin terms) are intimately related in coherent spin evolution and a distinction between them is only semantic. Note that of course, simple addition theorems, e.g., (cos 2 φ − sin 2 φ) = cos 2φ, demonstrate that Eq. (12) is identical to Eq. (11) when φ = φ  /2. The DQ-DRENAR method11 is now based upon normalization (point-by-point division) by S0, 90 signal, for which it can be argued that a 90◦ phase-shifted Hamiltonian of the second half of the sequence has an inverted DQ Hamiltonian, i.e., −H¯ DQ . This means that to zeroth (and in fact probably to higher) order its average Hamiltonian is zero,

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Kay Saalwächter

J. Chem. Phys. 141, 064201 (2014)

corresponding to an echo. Calculating this echo explicitly and neglecting imperfections reveals that DQ-DRENAR approach is in this case identical to normalized DQ build-up curve analysis.5, 8 Starting again from Eq. (11) with φ  replaced by φ, we obtain (j k)

−NτR H¯ DQ

−−−−−→ (Ij z + Ikz )t=0 (cos2 φ + sin2 φ) +Oˆ DQ (sin φ cos φ − cos φ sin φ ).  

(13)

absolute-scale signals: Snorm = 2sin2 φ ≈ 1 − exp{−2φ 2 } 2 2 ≈ 2φ 2  ≈ k Deff τDQ , (16) where in the multi-spin case Deff = ( pairs Dj2k )1/2 . The prefactor is calculated as a powder average of ωeff from Eq. (2). It is thus pulse-sequence specific and reads kC7 = 0.86/15 ≈ 0.05733 for (POST-)C711 and kBaBa = 12/(5π 2 ) ≈ 0.24317 for BaBa(-xy16).15

0 C. Time evolution due to imperfections

Of course, the DQ term vanishes at the end of this sequence as it has been reconverted to z magnetization, while the cos 2 φ + sin 2 φ = 1 factor represents echo formation (full rephasing). However, its is again stressed that the sin 2 φ term modulating z magnetization arises from the DQ coherence present after the first part of the sequence. The powder-averaged . . .  ratio of the prefactors of z magnetization in Eqs. (11) and (14) is

 S0,0 cos2 φ − sin2 φ = , (14) S0,90 cos2 φ + sin2 φ and using cos 2 φ = 1 − sin 2 φ leads to Snorm = 1 −

S0,0 = 2sin2 φ = 1 − cos φ  , S0,90

(15)

where φ  = 2φ. In this way, it is again proven that by its definition, the analyzed signal function of DQ-DRENAR corresponds to the build-up of DQ coherence, while its intensity complement 1 − Snorm corresponds to the dephasing of z magnetization. There is no further formal distinction between the two. It is noted that a dephasing of z magnetization due to anything else than the creation of DQ coherence (e.g., imperfection-related dephasing) is certainly unwanted and would bias the result as long is it is not certain that the normalization can account for it. Summarizing, we have seen how DQ coherence creation and z magnetization dephasing is reflected in the different signal functions, and how this can be used for normalization. Although these insights have been obtained on the basis of simple two-spin calculations, it is stressed that they hold equally well for the action of multiple couplings in a multi-spin system:13 the DQ filtering phase cycle selects all 4N + 2 coherence orders,16, 19 while the reference experiment by its definition is a 4N coherence order filter. Neglecting relaxation, both add up to the full signal with no dephasing due to higher-order coherences—all are rephased completely. This can be demonstrated by spin-counting experiments,16, 19 adapted to the present line of arguments concerning SDQ and Sref in Ref. 13. Since the spin dynamics under the individual transients in DQ-DRENAR is the same as in the DQ-filtered experiment, the equivalence is proven also for multiple spins. In multispin systems, as well as for spin pairs, it is recommendable and has been common practice5, 10, 11, 13, 15 to use a second-moment approximation for a simple analysis of the

Imperfections of course invalidate the equalities S0, 0 = S0,180 and S0,90 = S0,270 , which are the only relevant difference between Eqs. (3) and (6), i.e., between DQ-DRENAR and nDQ build-up analysis. However, the addition of the inverted variants can remove possibly relevant remaining CS(equivalently offset-), CSA-, and radio-frequency-related imperfection terms to first order. An analytical assessment of the POST-C7 pulse sequence12 has demonstrated that the corresponding imperfection Hamiltonians follow ∼Ix/y , with prefactors that depend on the order up to which imperfections are considered. For a qualitative demonstration of their effect (neglecting higher-order terms arising from commutators between H¯ 0 and Ix/y ), I calculate the single-spin evolution under such a term, φimp Ix

Iz −−−−−→ Iz cos φimp − Iy sin φimp ±φimp Ix

−−−−−→ Iz (cos2 φimp ∓ sin2 φimp ) −Iy (cos φimp sin φimp ± sin φimp cos φimp ), (17) where φ imp = ωeff, imp Nτ R . The second line describes the second half of the S0,0 and S0,180 experiments (a 180◦ phase shift leads to an inverted imperfection Hamiltonian). For the S0,90 and S0,270 experiments, the crucial change is that the imperfection Hamiltonian is rotated by ±90◦ , respectively. The second step thus reads ±φimp Iy

Iz cos φimp − Iy sin φimp −−−−→ Iz cos2 φimp − Iy sin φimp ±Ix cos φimp sin φimp . (18) The relevant terms in Eqs. (17) and (18) are the modulation factors of Iz , which describe the imperfection-related signal contributions to S0,ϕ . Equivalently to Eq. (3), it is found for DQ-DRENAR

 cos2 φimp − sin2 φimp sin2 φimp  imp , (19) Snorm = 1 − = cos2 φimp cos2 φimp  where as for nDQ analysis, we have

 2 cos2 φimp − sin2 φimp + sin2 φimp imp SnDQ = 1 − = 0. 2 cos2 φimp (20)

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Kay Saalwächter

III. RESULTS AND DISCUSSION

Spin dynamics calculations, explicitly implementing the POST-C712 and BaBa-xy1615 pulse sequences, were performed using a home-written code21 as well as the popular SIMPSON package.22 Both approaches afforded identical results. Powder averaging was performed over 100 REPULSION {α, β} angle pairs23 and 21 linear increments in the rotor angle γ . The shown data represent spin-pair calculations with a coupling constant D/2π of 400 Hz (yet testing other values and ensuring the invariance of the results and conclusions), and optional CSAs and offsets (chemical-shift separations) which are typical for 31 P in some phosphates11 at an equivalent 1 H Larmor frequency of 500 MHz. The data thus represent typical “most challenging’ situations for dipolar recoupling pulse sequences.12 All simulations represent the signal of only one of the two simulated spins, which corresponds to the experimental situation that the coupled partners can be distinguished by their chemical shift. The cycle times of POST-C7 and BaBa-xy16 are 2 (4 for the supercycled wideband version) and 8 τ R , respectively, which represents a constraint to the analysis of buildup curves. It should thus be stressed that both sequences can be incremented in shorter time intervals. POST-C7 can be incremented in single C elements of 2τ R /7 length, while BaBaxy16 can be incremented in single τ R steps. Full imperfection compensation is of course only achieved after the full cycles are complete, yet this is practically not always relevant at short times, as addressed below. The implementation of small steps in blockwise time incremetation is straightforward (one block is always terminated by a potentially incomplete cycle), while for the stroboscopic implementation fixed-phase and phase-shifted full cycles are alternated, and the whole train is then terminated by two potentially incomplete cycles. For POST-C7 it is necessary to apply an additional phase shift of i×(720/7)◦ for the ith single C element of the incomplete phase-shifted cycle, while for BaBa the phase-shifted cycles are always applied backwards in time with respect to the preceding cycle irrespective if they are complete or not.15 It is important to note that irrespective of the time incrementation scheme (blockwise or stroboscopic), the phaseshifted cycles of BaBa-xy16 are always applied backwards in time for reasons of imperfection compensation.15 This realizes an intrinsic 90◦ phase shift and a time inversion, and means that the S0;0 and S0;180 experiments do not produce de-

phased but rather refocused signal functions. The denotation of the two is thus interchanged.

A. Simulation results: POST-C7

Simulated Snorm build-up curves are shown in Fig. 2 for all four possible POST-C7 implementations of the build-up experiment as described by Eqs. (7)–(10). Data for a rather large CSA of 20 kHz are compared to the ideal spin-pair solution (solid line), and considerable deviations are apparent. The two experiments that do not include the phase-inverted signal part (open symbols, including DQ-DRENAR) show much increased intensity and even a divergence, while the experiments with the inversion step underestimate the build-up. The original blockwise-incremented nDQ experiment is the only one that does not show irregular intensity values smaller or larger than unity. Intensities lower than ideal are a priori expected, considering that the efficiency of DQ coherence creation should be reduced by imperfections. This already suggests that without phase inversion, dephasing due to other contributions (corresponding to an apparently stronger buildup) does play a role.

(a)

id. CSA 20 kHz i m S norm (nDQ)

2.0

m

S

1.5

norm n

i

S

norm

n

Snorm

Thus, DQ-DRENAR should suffer from additional, imperfection-related signal dephasing, even though it was claimed to be free from them.10, 11 Ultimately, the absence of the imperfection term in nDQ analysis is a simple consequence of the fact that the imperfections create odd-order coherence contributions to the final signal, which are filtered out by the DQ selection phase cycle. It appears recommendable to include this phase cycling step into DQ-DRENAR. Below, I will compare it with a wideband implementation that includes the inverted variants in doubled cycles.11

J. Chem. Phys. 141, 064201 (2014)

S

1.0

norm

(DQ-DRENAR)

0.5 0.0 -0.5 0

1

2

3

4

5

6

5

6

DQ evolution time / ms

(b)

id. CSA 20 kHz spin 2 only i m S norm (nDQ)

2.0

m

S

1.5

norm n

i

S

norm

n

S

Snorm

064201-5

1.0

norm

(DQ-DRENAR)

0.5 0.0 -0.5 0

1

2

3

4

DQ evolution time / ms FIG. 2. Normalized build-up curves for POST-C7 implementations of the four different experiments described by Eqs. (7)–(10) at 14 kHz MAS for a spin pair with D/2π = 400 Hz coupling and optional CSA of ωaniso /2π = 20 kHz (2/3 of the full tensor width) on (a) both nuclei and (b) only on the non-detected nucleus.

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064201-6

(a)

Kay Saalwächter

J. Chem. Phys. 141, 064201 (2014)

2.5

2.0

MAS rate / kHz 10 14 18 22

S β

1.6 Dapp/D0

Dapp/D0

2.0

i

S

1.8

1.5

1.4 1.2 1.0

1.0

0.8

-20

-15

-10

-5

0

5

10

15

20

-20 -15 -10

-5

5

10

15

20

π

/ kHz

2.5 β

0.6

1.5

1.0 0

5

10

15

20

25

30

35

40

2

ωaniso /(2π ωR) / kHz

FIG. 3. Relative apparent dipole-dipole coupling constants from Snorm (without phase inversion, corresponding to DQ-DRENAR) simulated for POSTC7 at different MAS rates, analogous to the data in Fig. 11(a) of Ref. 11 (a) as a function of CSA magnitude and (b) as a function of rescaled CSA 2 ωaniso /(2π ωR ).

The latter point is proven by the data in Fig. 2(b), where only spin 2 carries a CSA. I remind that in all simulations, spin 1 is the detected one. No irregular intensities are observed any more, with all curves lying initially well below the ideal result. Thus, the irregular surplus intensities are directly related to the CSA of the detected spin. In order to study the CSA effect on Snorm , I turn to investigating only the initial rise of the build-up curves. This is most easily done by simulating single-point intensities for τ DQ taking its minimum value, i.e., 2τ R for POST-C7. The second-moment approximation, Eq. (16), demonstrates that the single-point intensity is proportional to the square of the 2 . Fig. 3 and all apparent dipole-dipole coupling constant, Dapp related plots below show the ratio Dapp /D0 calculated from the square-root of the simulated single-point intensity, using the correct prefactor in Eq. (16). This corresponds to simulating the initial part of the build-up curve and fitting the initial rise. The analyzed quantity does not depend on the time incrementation scheme, so we only need to distinguish between Snorm and i Snorm , the latter including the phase-inversion step. Dapp /D0 data analogous to those of Fig. 3(a) for DQDRENAR (Snorm ), varying the MAS rate and the CSA, was published before in Ref. 11; this ratio represents a correction factor quantifying adverse CSA effects. Fig. 3(b) shows a rescaled master-curve representation of the same data and

2

MAS rate / kHz 10 14 18 22

0.4

2

Dapp/D0

1/2

2.0

( (Dtot/2π) - (Ddip/2π) )

(b)

0

ω

CSA (ωaniso/2π) / kHz

0.2

D

0.0 0

5

10

15

ω

πω

20

25

30

FIG. 4. (a) Relative apparent dipole-dipole coupling constants determined from the initial rise of the i Snorm and Snorm signal functions for POST-C7 at 14 kHz MAS as a function of CSA magnitude and (b) apparent CSA contribution taken from the difference of the respective apparent coupling con2 stants as a function of rescaled CSA ωaniso /(2π ωR ).

clarifies its origin. It is thus proven that the correction fac2 /ωR , which is exactly the dependence pretor scales as ωaniso dicted for the leading CS(A)-related imperfection term ∼Ix/y in POST-C7,12 an assessment of which was presented in Sec. II C. Fig. 4(a) shows another set of simulations of Dapp /D0 at fixed MAS rate, now including the i Snorm signal function, for different CS(A) situations. While Snorm (DQ-DRENAR) always exhibits similar high correction factors above 1, i Snorm is seen to be much more robust, with corrections smaller than unity and never much below 0.8, and some dependence on the relative tensor orientation or the presence of only single CSA tensors or additional isotropic shifts. It must be stressed that performance factors below unity are in fact physical, as they describe the expected reduction of the efficiency of the dipolar DQ Hamiltonian due to higher-order terms arising from CS(A).12 This salient point was not mentioned in Ref. 11 when reporting and discussing the data in Fig. 3(a). A series expansion of the theoretical imperfection-related dephasing, Eq. (19), suggests that 2 2

2   2 imp ∝ ωCS(A) /ωR τDQ , (21) Snorm = φimp which provides an additive contribution to the apparent second moment taken from the intial rise. Thus, it can be

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064201-7

Kay Saalwächter

J. Chem. Phys. 141, 064201 (2014)

obtained from the difference of the initial rise of the two different experiments,

(a)

imp

Snorm = Snorm −iSnorm

n S0,0;180,0

. n S0,180;90,270

(22)

(23)

The experiment was designed to minimize the CS(A)-related correction factor, and does not follow a strict logic where every other block is phase-inverted. I test other, symmetric alternatives, n wbSnorm = 1−

n S0,180;0,180 n S0,180;90,270

,

m

norm m

wb S

(24)

Snorm

i

norm

n

wb S

1.0

(nDQ)

norm n

wb S

norm

(DQ-DRENAR)

0.5 0.0 -0.5 0

1

2

3

4

5

6

DQ evolution time / ms

(b) id. CSA 20 kHz m m (S 0,90 + S 0,270)/2 (ΣMQ)

1.00 0.75

m

(S

m

0,180;90,270

+S

)/2

0,180;270,90

m

S

S0

0.50

0,180;90,270 n

(S S

0.25

n

0,180;90,270

n

0,180;90,270

+S

)/2

0,180;270,90

(wbDQ-DR.)

0.00 -0.25 0

1

2

3

4

5

6

DQ evolution time / ms

(c) 2.0 i

1.8

Snorm Snorm regular cycle wb orig. wb improved

1.6

Dapp/D0

where Dtot and Ddip are associated with the apparent couplings provided by the Snorm and i Snorm experiments, respectively. The latter, by virtue of its implicit DQ filtration phase cycle does not depend on imperfection-related single-quantum contributions. The correctness of these considerations is proven by the data in Fig. 4(b), where the apparent CSA contribution obtained from (Snorm − i Snorm )1/2 is plotted vs. the master vari2 /ωR . We observe an almost universal, near-linear able ωaniso relation, the slope of which is related to the unknown prefactor in the imperfection Hamiltonian. A crucial consistency test is provided by the simulations in which spin 1 has no CSA (asterisks) and in which D0 has been set to zero (lines). Expectedly, we observe a zero value (no CSA-related dephasing any more) for the former, and no change in the result for the latter in all other simulations, demonstrating the difference Snorm − i Snorm to arise from purely CSA-related single-quantum dephasing and not from dipolar coupling. In this way, we confirm that POST-C7 is a second-order CSA recoupling pulse sequence (of course with a small prefactor), in agreement with the assessment of Hohwy et al.,12 and show that this is not compensated for in DQ-DRENAR. The interesting option arising from the analysis of the difference between Snorm and i Snorm is that the magnitude of the CSA can be experimentally estimated from the near-linear correlation in Fig. 4(b). It must, however, be mentioned that such a procedure is only approximate, since the CSA asymmetry parameter η also plays a role. Further simulations (data not shown) revealed that η generally increases the CSA influence, simply because the span (full width) of the tensor increases for a given fixed ωaniso (i.e., the anisotropy parameter). However, just using the actual span instead does not collapse data for variable η in a plot as in Fig. 4(b). An approximate collapse can be achieved by rescaling ωaniso with the square root of the ratio of the spans for finite η and η = 0. Thus, if the given linear correlation is used, the resulting estimate for ωaniso is somewhat larger than the true anisotropy parameter if η is zero. In Ref. 11, a wideband implementation was introduced, which is based upon an alternation with 180◦ phase-shifted pulse sequence blocks. In our notation, the signal function reads orig

i

wb S

1.5

2  2 2 2 ∝ Dtot − Ddip ∝ ωCS(A) /ωR ,

wbSnorm = 1 −

id. CSA 20 kHz i m S norm (nDQ)

2.0

1.4 1.2 1.0 0.8 -20 -15 -10 -5 0 5 10 CSA (ωaniso/2π) / kHz

15

20

FIG. 5. (a) Normalized build-up and (b) reference decay curves for the wideband implementations of the different signal functions for POST-C7 at 14 kHz MAS with optional CSA, as well as (c) relative apparent dipole-dipole coupling constants determined from the initial rise in the same experiment as a function of CSA magnitude.

n wbi Snorm = 1−

n n + S0,180;180,0 S0,180;0,180 n n S0,180;90,270 + S0,180;270,90

,

(25)

which can be constructed from a set of four phase-shifted n experiments. Two corresponding blockS0,180;0+ϕ,180+ϕ wise rather than stroboscopically time-incremented variations can likewise be defined. Figs. 5(a) and 5(b) show simulated build-up and reference decay curves, respectively for these four experiments for the case of large CSA. We notice that all build-up curves are not qualitatively different from the ones

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Kay Saalwächter

(a)

1.1 1.0

Dapp/D0

shown in Fig. 2, where the CSA-related dephasing effect is not present. This shows that the wideband versions largely suppress the imperfection term even without the additional inclusion of phase inversion in the signal functions (wbSnorm vs. wbi Snorm ). However, the original wideband version, Eq. (23), does not have this property, as demonstrated by the CSA dependence of Dapp /D0 in Fig. 5(c) (open circles). This is not surprising, considering the one missing phase inversion in n . However, the surplus CSAthe dephased intensity S0,0,180,0 dephasing is now adjusted so as to better counterbalance the intrinsic CSA-related efficiency loss of the DQ Hamiltonian that can never be avoided. The data show that the reduction effect of the latter is in never much larger than the surplus contribution, which suggests that the new, improved wideband experiments are the ones of choice. However, it is practically not necessary to use them, as their results (Fig. 5(a)) do not differ from the non-supercycled original i Snorm variants. The reference decay curves in Fig. 5(b), show that the CSA-related overall intensity decay at longer pulse sequence times is minimized in the wideband versions. For the stroboscopic Sn implementation, it is even completely absent. While this may represent an advantage in special cases, I emphasize an argument in favor of somewhat more quickly decaying reference intensities: it allows for an easier separation of signal components that do not belong to the dipolar-coupled reservoir, and if present, they have to be subtracted before normalization. Uncoupled fractions (isolated or isotropically mobile spins) are common in inhomogeneous materials, and conveniently identified as more slowly relaxing tails of S0 or S MQ. 5, 13 Lengthening the decay of the dipolar-coupled part renders the separation more difficult. The Dapp /D0 data in Fig. 4(a) have demonstrated that the CSA-related performance reduction features a non-trivial dependence on the tensor orientations, which is why I refrain from developing a correction procedure. In the given data range, Dapp never underestimates the true coupling by more than 20%, which is a quite reasonable accuracy. Effects of isotropic shifts (frequency offests) are less complex, and are addressed in Fig. 6. Part (a) shows apparent couplings for a ±20 kHz range of offsets at different MAS rates and situations where only one spin has an offset or the offsets have inverted signs, the latter corresponding to the situation where the spin system is irradiated between the two resonances. Again, all data can be brought to near coincidence by representing the Dapp /D0 values as a function of a scaled master variable, which in this case was found to be x = i ωi /ωR (using frequency units). This is obviously a qualitatively different functional dependence as compared to the adverse CSA dephasing, which is not surprising, as the discussed higherorder effect describes the efficiency reduction of the average dipolar DQ Hamiltonian of the sequence. Notably, offsets of different sign cancel each other, which is an advantageous property of POST-C7, and means that the on-resonance condition should be located between the two spin pair resonances of interest to minimize offset problems. A shifted parabola, D/D0 = 1.03 − 0.0245(x − 1)2 , is found to describe the data well, and can be used for correction purposes. CSA of course leads to similar effects, and from a comparison of the data in

J. Chem. Phys. 141, 064201 (2014)

0.9

i

Snorm Snorm 9 kHz MAS 22 kHz MAS 14 kHz MAS 1 spin only Δω2 = −Δω1

0.8 0.7

-20 -15 -10 -5 0 5 10 offset Δω1,2/2π / kHz

(b)

15

20

1.1 1.0

Dapp/D0

064201-8

0.9 0.8

9 kHz MAS 22 kHz MAS 14 kHz MAS 1 spin only Δω2 = −Δω1

0.7

y = 1.03−0.0245(x+1)

-3

-2

-1

2

0

1

2

3

ΣΔωi /ωR

FIG. 6. Relative apparent dipole-dipole coupling constants determined from the initial rise of the i Snorm and Snorm signal functions for POST-C7 at different MAS frequencies and offset situations (a) as a function of spectral offset and (b) as a function of rescaled summed offset ωi /ωR along with a shifted parabolic polynomial fit.

Figs. 4(a) and 6(a) we take that its effect on Dapp /D0 is roughly a factor of 2–3 weaker.

B. Simulation results: BaBa-xy16

Fig. 7 presents build-up and reference decay curve simulations analogous to those of Figs. 2(a) and 5(b), now for the case of BaBa-xy16 and large CSA. It is seen that apart from some differences at longer τ DQ , all experimental implementations feature a very good overall performance and no significant reductions in long-time DQ excitation efficiency, in contrast to POST-C7. Importantly, no effects of direct CSArelated dephasing are seen. The implementations with strobon ) are seen preserve the overscopic time incrementation (Snorm all shape of the build-up curve somewhat better. To be fair, it should be stressed that the BaBa-xy16 simulations are performed for twice the MAS frequency (28 kHz), at and above which the sequence has its specific strengths.15 Such fast spinning is not feasible for POST-C7 with its strong rf requirements, but always helps to reduce CS(A)-related deficiencies. Nevertheless, it must be noted that the initial build-up is more strongly reduced as compared to POST-C7 even in the given case. Notably, even without the phase inversion step, none of the implementations demonstrates significant problems. While the reference intensities, Fig. 7(b), suffer somewhat for incomplete (odd multiples of 4τ R ) cycles, the intensity loss is compensated for upon normalization, see Fig. 7(a). The good performance of the full 8τ R cycle is obviously due to its very good intrinsic compensation,

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064201-9

(a)

Kay Saalwächter

J. Chem. Phys. 141, 064201 (2014)

1.25

1.0

0.75

Dapp/D0

Snorm

1.00

id. CSA 20 kHz i m S norm (nDQ)

0.50

0.8

β

m

S

0.25

i

norm norm

n

S

0.00 0

1

2

3

0.6

n

S

norm

(DQ-DRENAR)

4

5

-20 -15 -10

6

0

ω

DQ evolution time / ms

(b)

-5

5

10

15

20

π

(b) 1.0

1.00

Dapp/D0

S0

0.75

id. CSA 20 kHz m m (S 0;0 + S 0;180)/2 (ΣMQ)

0.50

0.8 D/2π = 400 Hz + 10 kHz CSA

m

S

0.25

0;180 n

(S

0;0

n

S

0.00 0

1

0;180

0.6

n

+S

)/2

0;180

(DQ-DRENAR)

2

3

4

5

6

DQ evolution time / ms FIG. 7. (a) Normalized build-up and (b) reference decay curves for the BaBa-xy16 implementations of the different signal functions at 28 kHz MAS and 125 kHz rf frequency with optional CSA. The points are from simulations with 8τ R increments, and the small connected points are from 4τ R increments.

comparable to the wideband versions based upon POST-C7 discussed above. CSA-related imperfections are addressed in Fig. 8(a), which shows single-point Dapp /D0 data for the i Snorm implementations (the Snorm intensities being virtually identical). The overall reduction effect is considerably stronger than for POST-C7, and stresses that BaBa-xy16 should be applied at even faster MAS rates in excess of 50 kHz if the CSA is large and quantitative D values are required. The scaled data for 56 kHz spinning demonstrate that the CSA-related performance reduction depends on the ratio (ωCSA /ωR )2 , similar to POST-C7. Also, the additional effects of isotropic shifts are seen to be larger than CSA effects at comparable frequencies. Again, relative tensor orientations play a significant role, precluding simple correction strategies. For completeness, Fig. 8(b) shows a dependence of Dapp /D0 as a function of the rf nutation frequency, reminding that the sequence features four 90◦ pulses per rotor period that should be as short as possible. The observed performance reduction is rather weak at rf values that are realistic for a 2.5 mm MAS probe (125 kHz), and is further seen to not have a large impact if CSA is present in addition. I again attempt to quantify the offset-related performance reduction by plotting data for Dapp /D0 simulated for differ-

0

50

100

150 rf / kHz

200

250

FIG. 8. Relative apparent dipole-dipole coupling constants determined from the initial rise of the build-up functions for BaBa-xy16 (τ DQ = 8τ R ) at 28 kHz MAS (a) as a function of CSA magnitude at 125 kHz nutation frequency and (b) at variable nutation frequency (=inverse pulse length). The x axis for the simulation for 58 kHz (250 kHz rf) in (a) was scaled by the MAS frequency ratio.

ent offset situations (Fig. 9(a)) as a function of a master variable. In the present case, I found that instead of the offset sum (as was the case for POST-C7), the data are best mastered (Fig. 9(b)) by x = i ωi2 /ωR2 . Thus, BaBa-xy16 does not benefit as much from placing the spectral offset between two resonances of interest. Further, slightly different (linear) dependencies on x are found if the offset of both spins have the same or opposite signs. Obviously, the BaBa-xy16 DQ average Hamiltonian as reduced by imperfections features a rather complex shift and CSA dependence, yet the two trend lines indicated in the legend of Fig. 9(b) can be used to correct a measured Dapp value for offset effects. A comparison with Fig. 8(a) shows that a rough correction for a CSA that has a magnitude equal to an offset should be cum grano salis about half as large. C. Experimental results: BaBa-xy16 and weak 1 H–1 H couplings

In order to exemplarily demonstrate the near-equivalence of the two approaches to DQ build-up analysis, the possibility to acquire data for partial pulse sequence cycles, and asses potential differences arising for the case of a weak offset, a few 1 H NMR experiments were performed for the case of

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064201-10

(a)

Kay Saalwächter

J. Chem. Phys. 141, 064201 (2014)

1.0 0.9

Dapp/D0

0.8 0.7

18 kHz MAS 56 kHz MAS 28 kHz MAS 1 spin only Δω2 = −Δω1

0.6 0.5 0.4 0.3

(b)

Δω2 = −Δω1/2

-20 -15 -10 -5 0 5 10 offset Δω1,2/2π / kHz

20

18 kHz MAS 56 kHz MAS 28 kHz MAS 1 spin only Δω2 = −Δω1

1.0 0.9 0.8

Dapp/D0

15

Δω2 = −Δω1/2

0.7 0.6 0.5 0.4 0.3 0.0

y = 1−2.33x y = 1−1.7x 0.1

0.2 2

0.3

2

ΣΔωi / ωR

FIG. 9. Relative apparent dipole-dipole coupling constants determined from the initial rise of the build-up functions for for BaBa-xy16 (τ DQ = 8τ R ) at different MAS frequencies and offset situations (a) as a function of spectral offset and (b) as a function of rescaled squared-summed offset ωi2 /ωR2 along with linear fits.

BaBa-xy16 as applied to a natural rubber sample (for details see Ref. 15). I used a Bruker Avance III spectrometer operating at 400 1 H frequency, and a Bruker 4 mm double-resonance MAS probe. Fig. 10 shows experimental 1 H NMR data taken on-resonance for the aliphatic (CH2 ) signal of a natural rubber (polyisoprene) sample, see also Ref. 15. Homogeneous rubbers exhibit well-defined apparent residual dipolar couplings (Dapp ) among the protons of its monomer units as a result of anisotropic fast-limit chain fluctuations,5, 13 making it a feasible model sample with a well-defined Dapp /2π ≈ 260 Hz. Off-resonance effects can be gauged for the olefinic CH resonance, data for which is shown in Fig. 10(b). Note that this resonance corresponds to an isolated, more weakly coupled proton. The fact that its build-up curve is governed by the very same apparent Dres is a consequence of the dipolar truncation effect, as explained previously.8, 24 BaBa-xy16 is used with single 1τ R increments, however, the chemical-shift compensation is not perfect for incomplete cycles, in particular not for odd-numbered 1τ R increments. This enables us to gauge the differences among the different experimental realizations. The data in Fig. 10 demonstrate convincingly the near-equivalence of all the tested signal function permutations for 2τ R increments. The data also include SnDQ and S MQ signal functions constructed in the usual way, relying on DQ and reference filtration phase cycles. These merely suffer from a higher noise contribution for reasons explained above, but are otherwise fully equivalent

FIG. 10. 1 H reference decay (black) and dipolar build-up (red) functions measured for a natural rubber sample using BaBa-xy16 at 10 kHz MAS with 3 μs 90◦ pulses at 400 MHz Larmor frequency for (a) the methylene and (b) the olefinic CH resonance, the former being on resonance. Single rotor period (1τ R ) increments are used up to an excitation time of 1.6 ms. The dashed lines represent a fit based upon a second-moment approximation of the build-up curves, Eq. (16), which is identical for the two resonances. The arrows indicate full 8τ R increments.

and within the experimental accuracy expectedly identical to m m m and (S0;0 + S0;180 )/2 signals, respectively. the iSnorm It is seen that the intensity oscillations observed for the odd-τ R increments of the decaying refocused reference intensity functions are partially compensated in the relevant buildup functions by the normalization. Only when the phaseinverted signal functions are left out, as is the case for the original DQ-DRENAR, we see deviations from the expected build-up, and is straightforwardly explained by the discussed effect of additional chemical-shift related dephasing. Yet overall, it is clear that for even-τ R increments, all experimental permutations yield practically the same results, in agreement with the expectations from our simulations.

IV. SUMMARY AND CONCLUSIONS

I have presented an in-depth comparison of DQDRENAR and normalized DQ (nDQ) build-up analysis based upon the POST-C7 and BaBa-xy16 pulse sequences, mainly using spin dynamics simulations. A physically expected reduction of the average dipolar DQ Hamiltonian due to chemical shifts and their anisotropy (CSA), leading to a delayed intensity build-up and thus reduced apparent dipoledipole coupling (DDC) constants, was confirmed. It is stressed that this effect cannot be avoided in any implementation of the given pulse sequences. For large CS(A) and moderate MAS, the effect was found to be less serious for POST-C7.

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064201-11

Kay Saalwächter

J. Chem. Phys. 141, 064201 (2014)

TABLE I. Applications and recommendations for the different experiments and pulse sequences. For a definition of the experiments, see Eqs. (7)–(10) and Fig. 1. POST-C7

BaBa-xy16

n Snorm (DQ-DRENAR)

n only useful in comparison with iSnorm to gauge CS(A) effects

not recommended; large CS(A) effects for odd-τ R increments

iS n norm

moderate MAS, large CS(A); inferior long-time performance m than iSnorm

moderate MAS, low couplings and CS(A) or fast MAS, large CS(A)

m Snorm

m only useful in comparison with iSnorm to gauge CS(A) effects

not recommended; large CS(A) effects for odd-τ R increments

best for moderate MAS, large CS(A)

moderate MAS, low couplings and CS(A) or fast MAS, large CS(A)

iS m norm

(nDQ)

The original implementation of the DQ-DRENAR experiment in combination with POST-C7 was shown to provide increased apparent DDCs, which is a priori unphysical, but can be explained by additional direct CSA-related signal dephasing (taking “advantage” of POST-C7 being a higherorder CSA recoupling sequence, of course with a very small prefactor). Such a dephasing is usually compensated for in a DQ selection phase cycle, as it leads to the creation of singlequantum coherences. The performance of DQ-DRENAR can be improved by adding a phase inversion step to the phase cycle, which removes the unwanted CSA dephasing effect and renders it largely equivalent to nDQ analysis. The remaining difference to nDQ analysis is the stroboscopic rather than blockwise time incrementation. This has virtually no influence on the relevant initial rise of the build-up curves, but affects the build-up behavior towards longer times. From the observations herein, which are limited to moderate evolution times not too far beyond the first maximum of the build-up curves, it was found that POST-C7 is best combined with nDQ analysis (blockwise time incrementation). For BaBa-xy16, the full cycle of which does not recouple the CSA, rather minor differences are found for the different experiments. For both experiments and both pulse sequences, empirical functions have been established that can be used to correct for the adverse effect of spectral offsets. The necessary corrections differ qualitatively for the two pulse sequences: the efficiency loss for POST-C7 depends on the sum of the offsets of the two participating nuclei, while for BaBa-xy16 it depends on the sum of the squared offsets, with a further difference as to whether the resonance offset is located between the nuclei or outside of their range. Adverse CSA effects are generally weaker for POST-C7, but dependencies on relative orientation and the asymmetry parameter preclude quantitative corrections in both cases. Apart from its less serious CS(A)-related performance reduction at moderate MAS, POST-C7 has the additional advantage of a rather fine time step resolution of 2τ R /7. However, its high required rf power (7 times the spinning frequency) poses serious limitations at faster MAS, and it is also rather susceptible to set-up imperfections. BaBa-xy16 in turn is very easy to set-up and rather forgiving to rf mis-settings, but given large CS(A) and large DDCs, it is best applied at rather fast (>30 kHz) MAS. A very advantageous principle behind DQ-DRENAR, which is based upon the acquisition of signal functions with a defined and fixed phase shift between consecutive DQ se-

quence blocks, can be extended to nDQ build-up analysis. Separate experiments for the four required fixed phase shifts that are usually part of a DQ (or reference) selection phase cycle allow for a more efficient post-processing, where the signals can be suitably combined to form the DQ build-up and sum decay functions, which are then used to construct the nDQ build-up. In this way, a factor of two in signal-to-noise, thus fourfold gain in experimental time, can be realized. A second advantage of this “taking apart” of the phase cycle is that the signal functions with and without the phase inversion step can be constructed. The difference between these is a direct measure of direct CS(A)-related signal dephasing, and can be used to obtain an estimate of the CS(A) and of its adverse effect on the apparent DDC. The above conclusions and recommendations are summarized in Table I. A part of these were backed up by BaBaxy16 experiments on a rubber sample with weak residual 1 H–1 H dipole-dipole couplings, but it is stressed that our simulations, as well as the POST-C7 and BaBa-xy16 applications reported earlier11, 15 have demonstrated that the experimental concepts are readily applicable to low-γ nuclei pairs such as 13 C-13 C or 31 P-31 P, possibly using heteronuclear decoupling from 1 H if necessary. Future experimental work on nuclei and samples with appreciable CSA should of course be done to further validate the suggested most suitable approaches to DDC determination by intensity-normalized DQ NMR. ACKNOWLEDGMENTS

An intense exchange of ideas and stimulating discussions with Jinjun Ren and Hellmut Eckert are gratefully acknowledged. I further thank Jinjun Ren for initially providing his data analogous to Fig. 3(a) and the related SIMPSON simulation routines. 1 J. M. Griffiths and R. G. Griffin, Anal. Chim. Acta 283, 1081–1101 (1993). 2 S.

Dusold and A. Sebald, Ann. Rep. NMR Spectrosc. 41, 185–264 (2000). P. Brown and H. W. Spiess, Chem. Rev. 101, 4125–4155 (2001). 4 M. H. Levitt, in Symmetry-Based Pulse Sequences in Magic-angle Spinning Solid-state NMR, Encyclopedia of Nuclear Magnetic Resonance Vol. 9, edited by D. M. Grant and R. K. Harris (John Wiley and Sons, Chichester, 2002), pp. 165–196. 5 K. Saalwächter, Progr. NMR Spectrosc. 51, 1–35 (2007). 6 S. P. Brown, Progr. Nucl. Magn. Reson. 50, 199–251 (2007). 7 D. Reichert and K. Saalwächter, “Dipolar coupling: Molecular-level mobility,” NMR Crystallography, Encyclopedia of Magnetic Resonance, edited by R. K. Harris, R. E. Wasylishen, and M. J. Duer (John Wiley and Sons, Chichester, 2009), pp. 177–193. 8 K. Saalwächter, ChemPhysChem 14, 3000–3014 (2013). 9 T. Gullion and J. Schaefer, J. Magn. Reson. 81, 196–200 (1989). 10 J. Ren and H. Eckert, Angew. Chem., Int. Ed. 51, 12888–12891 (2012). 3 S.

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064201-12 11 J.

Kay Saalwächter

Ren and H. Eckert, J. Chem. Phys. 138, 164201 (2013). Hohwy, H. J. Jakobsen, M. Edén, M. H. Levitt, and N. C. Nielsen, J. Chem. Phys. 108, 2686–2694 (1998). 13 K. Saalwächter, P. Ziegler, O. Spyckerelle, B. Haidar, A. Vidal, and J.-U. Sommer, J. Chem. Phys. 119, 3468–3482 (2003). 14 R. Graf, A. Heuer, and H. W. Spiess, Phys. Rev. Lett. 80, 5738–5741 (1998). 15 K. Saalwächter, F. Lange, K. Matyjaszewski, C.-F. Huang, and R. Graf, J. Magn. Reson. 212, 204–215 (2011). 16 J. Baum and A. Pines, J. Am. Chem. Soc. 108, 7447–7454 (1986). 17 Y. K. Lee, N. D. Kurur, M. Helmle, O. G. Johannessen, N. C. Nielsen, and M. H. Levitt, Chem. Phys. Lett. 242, 304–309 (1995). 12 M.

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Feike, D. E. Demco, R. Graf, J. Gottwald, S. Hafner, and H. W. Spiess, J. Magn. Reson. A 122, 214–221 (1996). 19 J. Baum, M. Munowitz, A. N. Garroway, and A. Pines, J. Chem. Phys. 83, 2015–2025 (1985). 20 O. W. Sørensen, G. W. Eich, M. H. Levitt, G. Bodenhausen, and R. R. Ernst, Prog. Nucl. Magn. Reson. Spectrosc. 16, 163–192 (1983). 21 K. Saalwächter and I. Fischbach, J. Magn. Reson. 157, 17–30 (2002). 22 M. Bak, J. T. Rasmussen, and N. C. Nielsen, J. Magn. Reson. 147, 296–330 (2000). 23 M. Bak and N. C. Nielsen, J. Magn. Reson. 125, 132–139 (1997). 24 M. J. Bayro, M. Huber, R. Ramachandran, T. C. Davenport, B. H. Meier, M. Ernst, and R. G. Griffin, J. Chem. Phys. 130, 114506 (2009).

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