Article pubs.acs.org/JPCA
Improved Phase-Modulated Homonuclear Dipolar Decoupling for Solid-State NMR Spectroscopy from Symmetry Considerations Meghan E. Halse and Lyndon Emsley* Centre de RMN à Très Hauts Champs, Institut de Sciences Analytiques (CNRS/ENS-Lyon/UCB Lyon 1), Université de Lyon, 5 rue de la Doua, 69100 Villeurbanne, France S Supporting Information *
ABSTRACT: We explore the effects of symmetry on the performance of phase-modulated homonuclear dipolar decoupling in 1H solid-state NMR. We demonstrate that the symmetry of the DUMBO family of decoupling sequences is the result of two well-defined symmetry expansions. The first is an antipalindromic expansion that arises from the symmetrization step that was built into the original architecture of the DUMBO sequence. The second is a mirror-pair expansion that inverts the sign of the phase modulation in the second half of the pulse sequence relative to the first. The combination of these two symmetry expansions generates a sequence of four Lee−Goldburg-type rotations in the rotating frame. The axes of rotation, oriented at the magic angle, are separated in the transverse plane by 2α, where α is chosen to minimize the sensitivity of the sequence to instrument imperfections such as rf inhomogeneity. The efficiency of the DUMBO symmetry for decoupling is demonstrated experimentally, and the effect of the α-phase-shift parameter is investigated. A new decoupling sequence (LG4) that combines the DUMBO symmetry with α = 55° is introduced and is shown to produce very efficient decoupling as well as a nearly 2-fold increase in coherence lifetimes when compared to standard PMLG/FSLG decoupling.
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INTRODUCTION Homonuclear dipolar decoupling plays a key role in a wide range of contemporary high-resolution solid-state NMR spectroscopy applications.1−4 The importance of homonuclear decoupling in solid-state NMR arises from fact that in powdered solids dense networks of strongly dipolar-coupled 1 H spins give rise to broad, featureless spectra. Thus, some form of dipolar decoupling is required to obtain high-resolution information, such as chemical shifts or structural restraints, from 1H NMR spectra. In natural abundance solids, magicangle spinning alone is typically not sufficient even at the highest rates available today. Thus, homonuclear dipolar decoupling by radio frequency (rf) pulses is usually required. Since the seminal contributions of Lee and Goldburg5 and Waugh and co-workers,6−8 there has been considerable development in the field of homonuclear dipolar decoupling.9−17 Modern homonuclear decoupling schemes achieve efficient decoupling through a combination of rf-pulse sequences and magic-angle spinning (MAS), the so-called CRAMPS approach.18 Over the past decade, two groups of decoupling sequences have emerged as the gold standard for achieving high-resolution 1H NMR spectra in the solid state. The first group includes the phase-modulated16,19−22 (PMLG) and frequency-switched15,23 (FSLG) Lee−Goldburg sequences. They were developed using a theoretical framework based on the original Lee−Goldburg experiment5 in which the spin system is subjected to a rotation about the magic angle in the rotating frame under the influence of an off-resonance rf © 2013 American Chemical Society
irradiation. The second group, the DUMBO sequences, were developed using an optimization approach that sought to produce the most efficient line-narrowing performance in situ relative to simulation17 or experiment.24−26 Both of these families of experiments have been successfully applied in a range of solid-state NMR applications, yielding important advancements in fields such as NMR crystallography.27−33 Although both the PMLG and DUMBO sequences use constant-amplitude phase-modulated rf irradiation to achieve 1 H line narrowing, the theoretical relationship between these approaches was previously poorly understood. Primarily, this was due to the absence of a clear theoretical explanation for the excellent decoupling performance of the DUMBO methods. Recently, however, it has been shown that a Legendre polynomial basis can be used to describe the phase modulation of both the PMLG and DUMBO families of sequences.34 A direct comparison of these decoupling methods is made possible by this common basis and reveals that the dominant contribution to the phase modulation in all of the DUMBO sequences is, in fact, a series of linear-phase ramps akin to those used in Lee−Goldburg decoupling. Thus, the DUMBO sequences can be described in terms of a series of Lee− Goldburg-type rotations, analogous to those used in PMLG/ FSLG experiments. The key difference is that in a DUMBO Received: April 18, 2013 Revised: June 10, 2013 Published: June 10, 2013 5280
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sequence there are four such rotations compared to only two for standard PMLG/FSLG. In addition, the DUMBO sequences include a supplementary low-amplitude phase modulation that acts to improve in situ line-narrowing performance beyond that achieved by the series of LG rotations alone. In this Article, we expand on this work by exploring the symmetry properties of the DUMBO sequences revealed by the use of the Legendre polynomial basis. In particular, we formally define the symmetry relationship between the successive Lee− Goldburg-type rotations that make up the core of a DUMBO sequence and investigate through simulation and experiment the effect of symmetry and the phase shift parameter, α, on decoupling performance. Finally, we introduce a new decoupling sequence, LG4, that is made up of a series of four Lee−Goldburg-type rotations combined using DUMBO symmetry with α = 55°, and we show that this sequence yields optimal 1H line widths and coherence lifetimes, particularly in the case of significant rf inhomogeneity.
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THEORY Common Theory of Phase-Modulated Homonuclear Decoupling. The basic theory behind phase-modulated homonuclear dipolar decoupling has been described extensively in the literature.16,35−40 Herein, we will briefly overview the theoretical aspects of these sequences that are relevant to the present discussion of the symmetry properties of phasemodulated homonuclear decoupling. As discussed above, in theory the source of decoupling efficiency is the same for both families of decoupling sequences.34 Both PMLG/FSLG and DUMBO can be decomposed into a series of rotations about axes oriented at an angle θLG with respect to the z axis of a rocking, rotating frame that is in resonance with the effective frequency offset of the rf irradiation. This angle, θLG, is typically at or close to the magic angle of 54.7°. We will refer to these as Lee−Goldburg (LG) rotations, after the original homonuclear decoupling sequence proposed by Lee and Goldburg in 1965.5 Figure 1a demonstrates how off-resonance irradiation can be used to generate an LG rotation. Consider a rotating frame that rotates at an angular frequency of ω = ω0 + ΔωLG, where ω0 is the Larmor frequency of the nucleus of interest and ΔωLG is the frequency offset of the off-resonance rf irradiation. In this rotating frame, the nuclei will feel the combined effects of the rf-field amplitude, ω1, oriented in the transverse plane and an offset component, ΔωLG, oriented along the negative z axis. These two components combine to yield an effective field, ωeff, oriented at an angle θLG with respect to the z axis (eq 1). θLG = tan−1(ω1/ΔωLG) ω1
Figure 1. (a) Vector representation of the effective field, ωeff, generated by an off-resonance rf irradiation with ω = ω0 + ΔωLG, where ω0 is the Larmor frequency of the nuclei and the rotating frame is resonant with the rf field. (b) Transverse projection of the four LG rotation axes located in each of the four quadrants in the transverse plane of the rotating frame labeled A, B, C, and D, respectively. Each projection is oriented at an angle of ±α with respect to the ±y axis. (c) Vector diagrams of two LG rotation axes (A in red and B in blue) oriented at θLG with respect to the z axis. The rotating frame is defined such that (i) it rotates in resonance with the effective off-resonance rf irradiation in each case (ω = ω0 ± ΔωLG) and (ii) the rotation axes are oriented at ±α with respect to the ±y axis of the transverse plane.
The off-resonance irradiation used to generate an LG rotation can be implemented by explicitly setting the transmitter frequency to ω = ω0 ± ΔωLG, as in an FSLG sequence. Alternatively, as demonstrated by Vega and coworkers,16 it is equivalent to use a nominally on-resonance rf irradiation coupled with a phase modulation, ϕ(t) = ± ΔωLGt, where as few as five phase increments can be used to define the linear-phase ramp. This is the PMLG sequence. In this Article, we will focus on the PMLG implementation so that a straightforward comparison can be made between the PMLG phase modulation and that of the DUMBO family of sequences. Nevertheless, because they are formally identical all of the arguments in this Article can be equally applied to the FSLG sequence. For a single LG rotation or a pair of complementary positive and negative rotations about the same axis, the orientation of the LG rotation axis in the transverse plane is arbitrary. In this case, the LG rotation can be fully described by the two angles θLG and γLG along with the sense (positive or negative) of the rotation. However, DUMBO sequences combine rotations about multiple LG axes, so the relative orientation of these axes in the transverse plane takes on a significant importance. Thus, we need two additional parameters to describe the LG rotation: (i) the quadrant of the transverse plane (A, B, C, and D as defined in Figure 1b) in which the transverse projection of the
(1)
If /ΔωLG = 2 , then the effective field is oriented at the magic angle θm = 54.7°. The angle of rotation about the effective field, γLG, is dependent on the duration of the irradiation, τLG, and the amplitude of the effective field according to eq 2. 1/2
γLG = ωeff τLG = (ω1/sin θLG)τLG
(2)
For an ideal LG rotation, θLG is set to the magic angle and the duration is set to τLG = (2/3)1/2(2π/ω1) so that the rotation angle, γLG, is equal to 2π. Shifting the phase of the rf field, ω1, by π and reversing the sign of the resonance offset, ΔωLG, will generate an inverse rotation around the same axis. 5281
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positive LG rotation axis lies and (ii) α, the orientation of the LG rotation axis relative to the ±y axis in the transverse plane of the rotating frame. These final two parameters could be combined into a single phase-offset parameter; however, it will be more convenient for the symmetry discussion to maintain a distinction between the quadrant of the rotation axis and α, the relative orientation of the axis within the quadrant. Using the above nomenclature, we can describe any PMLG/ FSLG or DUMBO sequence as a series of LG rotations with fixed values of θLG, γLG, and α. An ideal PMLG/FSLG sequence is the combination of a forward LG rotation with γLG = +2π and θLG = θm = 54.7° and an inverse rotation (γLG = −2π) about the same axis. If we define the positive rotation axis to be in quadrant A, then this sequence can be written as a series of two ideal LG rotations, AA̅ , where the overbar denotes the negative sense of the second rotation. The orientation of the rotation axis, and hence the value of α, is arbitrary in this case. Similarly, the supercycled PMLG sequence of Leskes et al.21 with p = 2 can be represented as a series of four ideal LG rotations, AA̅ BB̅ with α = 90°. In the following sections, we will demonstrate that DUMBO sequences can be written as a series of four LG rotations of the form AA̅ B̅ B and that the value of α plays a key role in the overall performance of the sequence. Symmetry in DUMBO Decoupling. Symmetry has been used extensively in the design of homonuclear decoupling sequences to improve decoupling performance, as, for example, in the development of the BR13 and BLEW14 sequences. For the following discussion, we will use the nomenclature for symmetry expansions presented by Levitt.41 To generate a DUMBO sequence, we first built a PMLG sequence by applying an antipalindromic expansion to a single LG rotation. The resultant sequence is antipalindromic, meaning that if you reverse the order of the elements then you obtain the negative of the original sequence. This process, commonly referred to as symmetrization in the NMR literature, has frequently been used in the design of homonuclear decoupling sequences to eliminate even-order average Hamiltonian correction terms.7,9 The antipalindromic expansion can be written down formally as shown in eq 3, where S is the original pulse sequence and S′ is the new symmetry-expanded pulse sequence. The symmetry operations are defined as follows: E is the identity operator, Φ(ϕ) is a phase-shift operator that shifts the phase of every element in the rf pulse sequence by ϕ, Orev is an order-reversal operator that inverts the order of the elements in the pulse sequence, and σxy is an xy-reflection operator that changes the sign of all frequency offsets in the sequence.41 S′ = {E,Φ(π )Orev σxy}S
Figure 2. (a) Phase-modulation waveform for a series of four ideal LG rotations of the form AA̅ B̅ B. The two pairs of complementary rotations, AA̅ and B̅ B, are phase shifted relative to each other by 2α, where α = 55° in this case. (b) Phase-modulation waveform generated by applying an antipalidromic expansion followed by a mirror-pair expansion to the A rotation highlighted in red. This A rotation corresponds to the first quarter of the eDUMBO-122 waveform.24
pulse sequence. Thus, this new sequence can be written as AA̅ without the need to specify the value of the other parameters separately for each element of the sequence. As was discussed in the previous section, AA̅ is a basic PMLG/FSLG sequence. The first half of Figure 2a (0 ≤ t ≤ 2τLG) is representative of a phase-modulated AA̅ decoupling sequence. The second step in the construction of a DUMBO-type sequence is the application of a mirror-pair expansion. This symmetry expansion is most commonly used in NMR for the construction of R-type recoupling sequences.42 Mirror-pair symmetry is generated by two successive reflections of the pulse sequence, about the xy plane and about the xz plane.41 The result is a π rotation of the effective field around the x axis. For pure phase-modulated decoupling sequences (with nominally on-resonance irradiation), this is achieved by simply inverting the sign of each phase in the pulse sequence. For a frequencymodulated sequence (i.e., off-resonance rf irradiation as in FSLG), the sign of the frequency offset must also be inverted. The mirror-pair expansion can be written formally as in eq 4, where σxz is an xz-reflection operator and σxy is an xy-reflection operator.41
(3)
The result is a sequence for which the rf propagator has palindromic symmetry; that is, the propagator is invariant to time reversal. Consider a positive rotation, γLG, about an axis oriented at θLG with respect to the z axis and α with respect to the y axis in a rotating frame that is defined such that the positive rotation axis lies in quadrant A. Such a rotation is the result of the phase modulation highlighted in red in Figure 2a and is illustrated by the vector diagram labeled A in Figure 1c. The antipalindromic expansion pairs this forward rotation, A, with the identical inverse rotation, A̅ (Figure 1c). Note that the antipalindromic expansion does not change the magnitude of γLG, θLG, or α; it simply reverses the sense of rotation in the second half of the
S ′′ = {E, σxyσxz}S′
(4)
Consider the positive A rotation axis depicted in red in Figure 1c. If this axis is rotated by π about the x axis, then we obtain the negative B rotation axis, B̅ . Therefore the mirror-pair of an A rotation is a B̅ rotation. Similarly, the mirror-pair of an A̅ rotation is a B rotation. Accordingly, the result of applying a mirror-pair expansion to the AA̅ sequence (PMLG) is a sequence of four rotations of the form AA̅ B̅ B. The phase modulation for an AA̅ B̅ B sequence, using ideal LG rotations and α = 55°, is shown in Figure 2a. This is the basic symmetry 5282
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as Floquet theory.43 Nevertheless, the first-order average Hamiltonian theory results presented in this Article remain broadly relevant, even in the case where there exists additional non-negligible contributions resulting from the interaction between the rf and MAS. As will be seen below, these approximations appear valid given the excellent agreement between experiment and prediction. Average Hamiltonian theory (AHT) is a well-established technique for the analysis of solid-state NMR experiments.44 A comprehensive discussion of the use of AHT for calculating dipolar and chemical-shift scaling factors for homonuclear dipolar decoupling sequences has previously been given by Salager et al.45 and for the convenience of the reader, we repeat here the definition of the dipolar and chemical-shift scaling factors presented by them. For a system of nonrotating, dipolar-coupled spin-1/2 nuclei, the internal Hamiltonian of the system, Hs, can be written as the sum of the chemical-shift and dipolar terms, as shown in eq 5, where ωCS and ωD are the chemical-shift and dipolar coupling frequencies, respectively.
of all DUMBO sequences. It is important to note that in contrast to the PMLG/FSLG sequences a DUMBO sequence combines both A and B rotations; therefore, the relative orientation of these axes in the transverse plane is no longer arbitrary. Thus, the phase parameter, α, will have a significant impact on the properties and performance of the sequence. Example: eDUMBO-122. The phase waveform presented in Figure 2a generates a series of ideal LG rotations. However, as discussed in the Introduction, a full DUMBO waveform contains not only a series of linear phase ramps but also additional low-amplitude phase modulation. In the Legendre polynomial parametrization scheme introduced in ref 34, this corresponds to contributions from Legendre polynomials, Pn, with n > 1. Consequently, we refer to this component as the high-order DUMBO phase modulation. This high-order modulation acts to improve the in situ line-narrowing performance and fulfills the same symmetry relationships as the series of LG rotations. Therefore, the pair of symmetry expansions described in the previous section can be used to construct the entire DUMBO sequence. As an example, consider the waveform in Figure 2b. The portion of the waveform highlighted in red is the first quarter of an eDUMBO-122 phase-modulation waveform constructed from a Legendre polynomial basis using the coefficients in ref 34. This basic unit of phase modulation generates a rotation that we will call A. To construct the full sequence, we apply an antipalindromic expansion and then a mirror-pair expansion to this A rotation. The resultant four-step waveform, of the form AA̅ B̅ B, is shown in Figure 2b. The eDUMBO-122 waveform, as first published in ref 24, is related to the waveform in Figure 2b by a reversal of the order of the four rotations, BB̅ A̅ A, followed by the cyclic permutation of the A rotation from the end of the waveform to the beginning, ABB̅ A̅ . In the Results and Discussion section, we will show that the order reversal and cyclic permutation operations do not significantly impact the properties of the sequence. It is the underlying DUMBO symmetry, generated by the antipalindromic and mirror-pair expansions, that is the root of the decoupling efficiency for these sequences. Scaling Factors and Radio Frequency Inhomogeneity. One of the key factors limiting homonuclear dipolar decoupling performance is rf inhomogeneity. Therefore, designing sequences that are insensitive to variations in rf amplitude is an important step toward improving the efficiency of homonuclear dipolar decoupling. The effect of rf inhomogeneity on the efficacy of a decoupling sequence can be characterized through an analysis of the effect of changing the rf irradiation amplitude on the sequence of dipolar and chemical-shift scaling factors. In the following analysis of the effect of rf pulse sequence symmetry on the sensitivity of the chemical-shift and dipolar scaling factors to rf inhomogeneity, we neglect the effects of magic-angle spinning, assuming that if the resonance conditions between the rf irradiation and the spinning are carefully avoided, then the quasi-static approximation is appropriate for the spinning rate used herein (ωr = 2π*12.5 kHz).25 That is, we assume that the effects of the rf irradiation and the magic-angle spinning can be treated separately, allowing us to investigate the relative performance of the rf sequences using calculations of scaling factors that do not explicitly include MAS. At higher spinning rates or close to resonance conditions, the interaction between MAS and rf irradiation will become increasingly important and can be investigated using other approaches, such
Hs =
∑ ωiCST10i + ∑ ∑ ωijDT20ij i
i
(5)
j
