- Email: [email protected]
Heteronuclear decoupling in the NMR of solids
Progress in Nuclear Magnetic Resonance Spectroscopy 46 (2005) 197–222 www.elsevier.com/locate/pnmrs
Heteronuclear decoupling in the NMR of solids Paul Hodgkinson* Department of Chemistry, University of Durham, South Road, Durham DH1 3LE, UK Received 23 February 2005 Available online 18 July 2005
Contents 1. 2.
3.
4.
5.
6.
7.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Decoupling in solution-state NMR and the contrast to solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Average Hamiltonian theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. The ‘direct’ approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 H decoupling in experimental practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Continuous wave decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Two pulse phase modulation and related sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Other common sequences: SPINAL-64 and XiX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Understanding decoupling performance in solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Rare spin linewidths in solid-state NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1. Refocusable and non-refocusable linewidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2. Lineshapes under decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Factors determining decoupling performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1. Off-resonance behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2. Interaction with sample spinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3. Interaction with 1H dipolar couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4. Interference with exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5. RF inhomogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Characterisation and development of decoupling methods for solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Characterisation of decoupling performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Strategies for development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1. Symmetry-based methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2. Numerical simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3. ‘Direct optimisation’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diversions in decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. 1H decoupling in 19F-containing systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Decoupling of other nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1. Spin-1/2 nuclei: 31P, 19F and 15N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2. Half-integer spin quadrupolar nuclei: 27Al . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3. Integer spin nuclei: 2H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
197 199 199 200 203 203 204 205 206 206 207 208 209 209 210 212 214 214 214 214 216 216 218 218 218 218 220 220 220 220 220 221 221
1. Introduction * Tel.: C44 191 334 2019; fax: C44 191 384 4737. E-mail address: [email protected]
0079-6565/$ - see front matter q 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.pnmrs.2005.04.002
Effective decoupling of heteronuclear spin interactions is critical in many Nuclear Magnetic Resonance applications. This is particularly true for rare spin NMR of organic
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Nomenclature D
Heff
effective Hamiltonian over a given period (of the decoupling sequence) I, S decoupled and observed spins, respectively (i.e. decoupling RF is always applied to I spins) T2 time constant for spin–spin (T2) relaxation T20 time constant for decay of S spin magnetisation under spin-echo refocusing nr sample rotation frequency nrf nutation frequency of I spins subject to onresonance RF tc period of decoupling RF sequence tr period of sample rotation tp, Df pulse duration and phase difference respectively for TPPM and related sequences
offset of I spin Larmor frequency from (decoupler) transmitter frequency X figure of merit for decoupling band-width CM Continuous Modulation decoupling (Ref. [31]) CSA Chemical Shift Anisotropy CW Continuous Wave HORROR Homonuclear Rotary Resonance (Ref. [58]) MAS Magic Angle (sample) Spinning RF Radio Frequency (irradiation) TPPM Two Pulse Phase Modulation decoupling (Ref. [22]) SPINAL Small Phase Incremental Alternation decoupling (Ref. [27]) XiX x-inverse-x decoupling (Ref. [37])
molecules. As illustrated in Fig. 1, the resonances of 13C spectra of solid powders in the absence of decoupling, Fig. 1(a), are strongly broadened by (dipolar) coupling to surrounding 1H nuclei. The through-space nature of dipolar coupling means that even non-protonated carbons (such as the carbonyl resonances around 180 ppm) are poorly resolved. Decoupling of the protons has a dramatic effect on the spectral sensitivity and resolution, Fig. 1(b). Considerable progress has been made in heteronuclear decoupling in solution-state NMR since the first use of simple single-frequency (continuous wave) irradiation to decouple heteronuclear interactions [1]. ‘Broad-band decoupling sequences’ allow the complete 1H spectrum to be decoupled with a minimum of RF power (typically less
than a Watt). In contrast, considerably higher power (typically 10 s, even 100 s of Watts) continuous wave (CW) decoupling is widely applied in solid-state NMR. Over the past decade, however, it has become clear that CW decoupling is increasingly ineffective as static magnetic fields and sample spinning rates have increased, prompting renewed interest in the problem of 1H decoupling for dipolar-coupled systems. The current article sets out the background to these developments: outlining how the nature of decoupling differs in dipolar and J coupled systems, what techniques have been developed for high-performance decoupling, and the challenges that remain, both theoretical and experimental. The focus is on 1H decoupling in rare spin NMR, where
(a) x 10
180
160
140
120
100
80
100
80
60
40
20
0
60
40
20
0
(b)
* 180
160
140
120
ppm Fig. 1. 13C CP/MAS spectra (spin rate, nrZ5.1 kHz, 13C Larmor frequency, 75.40 MHz) of a glycine/alanine mixture obtained (a) without 1H decoupling, and (b) using continuous wave 1H decoupling (1H nutation rate, nrf, 60 kHz). The asterisk marks spinning sidebands from the carbonyl resonances.
P. Hodgkinson / Progress in Nuclear Magnetic Resonance Spectroscopy 46 (2005) 197–222
decoupling is of most direct benefit, although the decoupling of other nuclei is briefly surveyed in Section 6.2. Although superficially related, the issues involved homonuclear decoupling (for high resolution spectroscopy of abundant spin systems) are signficantly different and are not discussed here. Decoupling in liquid crystals provides some interesting points of comparison, but Ref. [2] contains a more complete and directed account of 1H decoupling in 13 C liquid crystal NMR.
2. Decoupling in solution-state NMR and the contrast to solids This section describes the theory and development of decoupling for solution-state NMR in some detail, to provide the necessary background for decoupling in the solid state. Complete descriptions and reviews of broadband decoupling for solution-state NMR can be found in the literature [3–5]. The spin Hamiltonian for a heteronuclear system can be written schematically: HðtÞ Z HS C HIS C HI C HII C Hrf;I ðtÞ
(1)
where the terms represent the spin Hamiltonians for the (observed) dilute S spins, the coupling between I and S, I spin chemical shifts, homonuclear I spin interactions, and RF irradiation applied to the I spins, respectively. The condition that jHrf,IjOHISCHI[HII is generally easily achieved in solution-state NMR with RF nutation rates of a few kHz (compared to 1 J CH w200 Hz, 2 J HH w20 Hz). This separation of scales allows us to consider the time-averaged Hamiltonian over the time-scale of the I irradiation. In hand-waving terms, the effect of continuous RF irradiation is to continuously invert I spin magnetisation, and the time average of heteronuclear coupling terms (which are of the form JIzSz at high field) will be hHIS iZJSzh Iz iZ0 (hHI i is also zero). Under these circumstances, the I and S spin systems are now ‘decoupled’: hH iZHSChHII i. Note that decoupling is the ‘separation’ of I and S spin systems; it does not imply that all terms involving the decoupled spins can be ignored [6]. For instance, heteronuclear multiple-quantum coherences, such as IxSx, will continue to evolve under the (averaged) Hamiltonian h HII i. This Hamiltonian is generally complex (homonuclear interactions, for example, will only be scaled, not eliminated by the I spin irradiation). Fortunately, such coherences are currently rarely encountered in solid-state NMR, and so it is normally reasonable to simplify the Hamiltonian under I spin decoupling to hH iZHS. Note that in contrast to other solution-state decoupling techniques, which generally ignore II interactions, the DIPSI decoupling sequences [7] are specifically designed to create a well-defined averaged Hamiltonian, hHII i.
199
The S spin Hamiltonian, HS, for dilute spins commutes with the other terms of the Hamiltonian, and so is not directly relevant to the decoupling problem (other than being the ultimate goal of effective decoupling). This separation does not strictly apply if the S spin system is not dilute. In this case [HSS, HIS]s0 and it is necessary to consider the entire spin Hamiltonian when considering decoupling. These effects are only likely to be significant for the already fraught case of 1Hdecoupling in 19F NMR, which is considered in Section 6.1. Since, we can generally neglect HS (isotropic shift of S) and HII (negligibly small homonuclear couplings) 1H decoupling in solution-state NMR can thus be described by the Hamiltonian HðtÞ Z JIz Sz C DIz C Hrf;I ðtÞ
(2)
where JIzSz is the J coupling between I and S, and D is the offset of the I spin resonance from the decoupler frequency. 2.1. Average Hamiltonian theory The classic approach to describing the evolution of a spin system under a time-dependent perturbation is Average Hamiltonian Theory (AHT) [8,9]. We separate the Hamiltonian of Eq. (2) into a time-independent term, H0 and a larger, time-dependent perturbation from the RF irradiation, Hrf(t). The propagator for the evolution of the density matrix from the time origin to time t, U(t), is then factored into two components: ðt UðtÞ Z T exp Ki H0 C Hrf ðt 0 Þdt 0 Z U1 ðtÞU~ 0 ðtÞ (3) 0
where ðt 0 0 U1 ðtÞ Z T exp Ki Hrf ðt Þdt
and
0
ðt U~ 0 ðtÞ Z T exp Ki H~ 0 ðt 0 Þdt 0
(4)
0
H~ 0 is the Hamiltonian transformed into an interaction (or ‘toggling’) frame defined by the time-dependent perturbation: H~ 0 ðtÞZ U1† ðtÞH0 U1 ðtÞ. We assume that the perturbation is periodic, i.e. Hrf (tCtc)ZHrf(t) where tc is the period. The spectrum, within a spectral width of 1/tc, is then defined by the propagator over one period, Uðtc ÞZ U1 ðtc ÞU~ 0 ðtc Þ. So called ‘cycling sidebands’ can arise from transitions outside this spectral width if tc is long (e.g. the decoupling field is very weak), but these will be negligibly small for a well-adjusted decoupling sequence. Sidebands from inadequate decoupling can be observed for liquid-crystalline samples, but the low quality of solid-state spectra under conditions of poor decoupling obscures these weak transitions for more typical dipolarcoupled systems. Despite this simplification, the evaluation of U(tc) is only reasonably straightforward if, in addition, the RF
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irradiation is also cyclic in the sense that the toggling frame returns to its starting point over the period of the perturbation, i.e. U1(tc)Z1. In this case, Uðtc ÞZ U~ 0 ðtc Þ. Finally, we try to describe U~ 0 ðtc Þ in terms of an ‘average Hamiltonian’ over the perturbation, H 0 , with U~ 0 ðtc ÞZ expðKiH 0 tc Þ. This Hamiltonian can be determined from the Magnus expansion ð1Þ H 0 Z H ð0Þ 0 C H 0 C/ whose first term is given by ð 1 tc ~ H ð0Þ H ðtÞ dt 0 Z 0 tc 0 0
(5)
(6)
Applying this analysis to decoupling with continuous (CW) RF irradiation is straightforward. We start with the Hamiltonian of Eq. (2) written in terms of the I spin sub-space: HGðtÞ Z ðDGJ=2ÞIz C nrf Ix
(7)
where HC is the Hamiltonian for the Sa sub-space (and HK corresponds to S in the b state). If we assume that DGJ/2/ nrf (i.e. the decoupling field is large), than the transformation to the toggling frame due to the RF irradiation (here applied along x) is simply described by Iz/Iz cos 2pnrftKIy sin 2pnrft. The average Hamiltonian over the cycle of the RF is simply ð tcZ1=nrf H ð0Þ ZðDGJ=2Þ Iz cos 2pnrf t KIy sin 2pnrf t dt Z0 0 0
(8) which vanishes, as expected. The drawbacks of the AHT treatment become apparent as we move away from this trivial case. If D is no longer small in comparison to nrf, i.e. we consider off-resonance irradiation, then the transformation into the interaction frame is nontrivial. Moreover, if the coupling itself cannot be neglected (a realistic situation when we consider decoupling in solids), pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2rf C ðDGJ=2Þ2 , then the effective nutation rate, neff rf Z now depends on the state of S. In addition, the normal requirement for cyclic RF irradiation is a major constraint; many of the most effective decoupling sequences are not cyclic and so cannot be readily analysed within this framework.
If necessary, an ‘effective Hamiltonian’ over the rotor cycle can be determined, HeffZKi ln U(tc)/tc. This is distinct from the interaction-frame ‘average Hamiltonian’ of Eq. (5). As the S spin state is a good quantum number, the propagator (or effective Hamiltonian) is block diagonal ! UC 0 Uðtc Þ Z (9) 0 UK where UG are the propagators over a period within the a and b sub-spaces of S. The S spin signal at multiples of the cycle time is given by Sðntc Þ Z trðSCUðtc Þn Sx Uðtc Þ†n Þ
(10)
† n Þ Z 1=2 tr½ðU_UC
(11)
Perfect decoupling would correspond to no evolution of the S spin signal (apart from the evolution due to HS, which † can be factored out, as explained above), i.e. UKUC Z 1 or, equivalently, UCZUK. This leads to the important conclusion that ideal decoupling requires that the propagators describing the evolution of the I spin system be identical for both the a and b states of S. In other words, if the I spin evolution is insensitive to the state of the S spin, then the I and S spins must be decoupled. Note that no assumptions are made about the nature of the I spin evolution. The RF is required to be periodic, but not necessarily cyclic; it does not matter where the evolution finishes, as long as both a and b states finish at the same point. We can describe the effect of an arbitrary propagator, U, on a single spin-1/2 as a rotation through an angle f about an axis n (specified by a unit vector). Expressing Eq. (11) in terms of the distinct rotations corresponding to the a and b states of S, (fC, nC) and (fK, nK), gives the signal [10] 1 fC K fK Sðntc Þ Z ð1 C nC$nKÞcos n 2 2 1 fC C fK C ð1 K nC$nKÞcos n ð12Þ 2 2 This cosine modulated signal corresponds to a sum of four frequencies: GUC and GUK with fC C fK f K fK and UK Z C 2tc 2tc
2.2. The ‘direct’ approach
UC Z
Although the Average Hamiltonian approach is extremely powerful when the goal is a well-defined ‘effective Hamiltonian’. It is less suited to problems, such as heteronuclear decoupling, in which the goal is a ‘zero’ Hamiltonian, since the performance will be determined by the behaviour of complex high order terms. Waugh introduced a simpler and more direct approach to describing decoupling [10], which is described below. As before, we neglect cycling sidebands and focus on the evolution over the cycle of the RF irradiation, U(tc).
with intensities given by the corresponding coefficients of Eq. (12). The UC terms can generally be neglected as they will have minimal intensity (nCznK), Fig. 2(a). The quality of the decoupling is determined by the frequency difference of the transitions at GUK (relative to the S spin Larmor frequency), in comparison to the original coupling, J:
2UK fC K fK
Z (14) l Z
J Jtc
(13)
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201
J (a)
λJ
−nνc
0
nνc
0.8
(b)
(c)
y n+
0.6
n−
λ 0.4 c n+
nc−
x
0.2
0 −1
−0.5
0
∆ / vrf
0.5
1
Fig. 2. Decoupling in a J-coupled CH system: (a) schematic decoupled spectrum showing scaling of the original doublet (dotted lines) to a doublet of separation lJ (where l should be sufficiently small that the residual splitting is unresolved), plus a pair of weak ‘forbidden’ transitions and, in the case of multi-pulse decoupling, potential ‘cycling sidebands’ centred at multiples of the cycle frequency of the sequence, nc. (b) Scaling factor, l, as a function of transmitter offset for continuous wave decoupling (dotted line) and an idealised decoupling sequence (solid line) which is broadband over a range DwGnrf/2. (c) Plots of the x and y components of the magnetisation vectors corresponding to the two spin states of the 13C over the course of a supercycled decoupling sequence (adapted from Ref. [90]).
In the limit of perfect decoupling, fKZfC and so lZ0, i.e. the transitions are coincident. A final simplification can be made if we take into account the relative magnitudes of the different interactions involved. Returning to Eq. (7), the net rotation angles, f will be functions of the effective offset DGJ/2 (in addition to parameters describing the RF). Hence the decoupling scaling factor is lðDÞ Z
fðD C J=2Þ K fðD K J=2Þ 1 vf Z Jtc tc vD
(15)
in the limit of J being much smaller than the range of D (A 10 ppm 1H chemical shift range at 500 MHz for example, corresponds to a D range of 5 kHz, which is considerably larger than the couplings involved.) This is invariably the case for 1H decoupling in solution, reducing the problem to one of finding sequences that give low values of l over an appropriate range of D values. This analysis for an S spin coupled to a single I can be readily extended to multiple I spins, although the ‘quality parameter’ is more difficult to define, especially if II interactions are significant [11]. Armed with this exact picture of decoupling, we can rationalise the behaviour of different decoupling sequences.
The scaling factor for continuous wave decoupling (familiar from off-resonance decoupling) is given by D ffi lCW Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2rf C D2
(16)
As can be seen in Fig. 2(c), this is only acceptably small (say less than 1%) for D/nrf. Excessive RF power would be needed to satisfy this criterion for even modest chemical shift ranges. The decoupling bandwidth may be quantified by the figure of merit [5] XZ
DF nrf
(17)
where nrf is the decoupling amplitude (expressed as a nutation frequency) and DF is the range of transmitter offsets over which the height of the decoupled signal (after applying defined processing) remains over 80% of its maximum value (i.e. on-resonance). CW decoupling, for example, has a X value of w0.0075, while effective decoupling schemes have X values around unity or greater, i.e. nrf is no longer required to be much larger than the range of 1H NMR frequencies. A tempting, but flawed, way used to increase the bandwidth of the the decoupling irradiation, is to introduce
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‘additional frequencies’. Noise decoupling, for example, involves randomly switching the phase of the irradiation [12]. Fourier transformation of the resulting time-domain signal would reveal a spread of frequencies around the RF carrier frequency, with a ‘band-width’ determined by the frequency of the phase modulation. Unfortunately, such Fourier arguments are only applicable if the response of the spin system to the RF is linear. While this is a reasonable approximation for small nutation angles, it is definitely not applicable to decoupling, and noise decoupling, while an improvement on straight CW irradiation, is not particularly efficient in practice. Noise decoupling was rapidly displaced by much more effective techniques based on composite pulses that could be derived using the exact decoupling theory. The classic design routes mostly involve ‘cycles’, that is RF irradiation that returns magnetisation to its starting point, since the requirement of forming a closed loop is relatively unrestrictive. The widely used WALTZ-16 sequence [13], for example, uses a basic inversion + composite pulse of RZ 90+x 180Kx 270+x (abbreviated in terms of multiples of 908 to 123, hence the choice of acronym). This cycle is moderately robust with respect to transmitter offset, and, in common with other sequences with only 1808 phase shifts, is insensitive to imperfections in phase setting. The combination of inversion pulses, RRR R creates a reasonably broadband decoupling sequence (WALTZ-4). As illustrated in Fig. 2(c), the decoupling performance of this basic sequence can be significantly improved by ‘supercycling’. The figure plots the x and y components of the magnetisation vectors corresponding to the two spin states of the 13C, nC and nK. At the end of the basic cycle, the magnetisation vectors are close to the origin, but there is a non-negligible difference between nC and nK, corresponding to a finite scaling factor, l. However, by permuting 908 pulse elements, the errors in the trajectories largely cancel over the course of the supercycle, i.e. nCwnK. In the specific case of WALTZ decoupling, the resulting four step supercycled sequence WALTZ-16 gives excellent broadband decoupling with a X value of 1.8. If even larger decoupling of bandwidths are required, e.g. for nuclei such as 31P or 19F which have a much larger spread of NMR frequencies than 1H, then specially optimised sequences such as GARP-1 [14] (XZ4.8) may be used. These trade a greater bandwidth factor, X, against somewhat higher scaling factors, l. An alternative strategy uses ‘adiabatic inversion’ pulses. These use a careful modulation of the RF amplitude and/or phase to ‘sweep’ magnetisation between z and Kz (and vice versa). If all the spins are perfectly inverted, perfect decoupling is achieved. The strength of adiabatic techniques is their built-in robustness with respect to many experimental imperfections, e.g. modest RF inhomogeneities are irrelevant in an amplitude modulated sequence. Sequences such as WURST [15] achieve extremely high bandwidths in proportion to the RF fields used, e.g. XZ16.7 (although
the X parameter is not entirely appropriate for adiabatic sequences since its value varies with nrf). Full treatments of decoupling sequences for solutionstate NMR can be found elsewhere [4,5], but it should be clear from the overview above that the theory of solutionstate decoupling is fully established and 1H decoupling can be regarded as essentially a solved problem. Although there is no ‘perfect’ decoupling sequence, the different trade-offs between bandwidth, residual splittings and artifact levels (e.g. from cycling sidebands), are well understood. In turning to decoupling in solids and liquid crystals, it is important to consider why decoupling in these phases is such a different problem. The most significant difference is the presence of strong homonuclear interactions, due to dipolar couplings that are not averaged out by molecular motion, unlike in the solution state. The dipolar coupling between the protons within a CH2 unit, for example, is of the order of 20 kHz, and the overall width of the proton spectrum in typical organic solids is approximately 100 kHz. As a result, much larger RF field strengths are required to manipulate the proton magnetisation. Although nutation rates of, say, 50 kHz, may be sufficient in undemanding applications, considerably higher nutation rates are desirable in many cases, e.g. at high magnetic fields. This puts considerable strain on the RF handling capabilities of probes, and both decoupling power levels and RF duty cycles must be chosen carefully to avoid overloading the probe (‘arcing’). Unlike solution-state NMR, where the 1H is often kept on throughout an experiment (generally boosting the rare-spin polarisation through the nuclear Overhauser effect), decoupling in solid samples is invariably ‘gated’, i.e. decoupling is only applied when strictly necessary, e.g. during acquisition. The strong homonuclear couplings also have a profound effect on the ‘spin dynamics’. In reducing Eqs. (1) to (2), we assumed that the I spin homonuclear interactions due to the J couplings could be neglected. This is clearly not the case with dipolar Hamiltonians, and treatments of decoupling in systems with significant dipolar couplings need to consider an InS spin system, where n is potentially infinite due to the presence of dipolar couplings between any pair of I spins. The many-spin nature of the problem remains a major obstacle to fundamental understanding of decoupling in dipolar-coupled systems. Less obviously, the role played by the transmitter offset is less well defined. When decoupling JCH, we could assume that this coupling was significantly smaller than the other terms involving the I spins. As a result, rather than considering the evolution under the effective offsets DCJ/2 and DKJ/2, it was sufficient to consider the behaviour around D, i.e. as a function of transmitter offset (Eq. (15)). Hence decoupling in solution-state NMR is generally tested by following the linewidth of a single 13C resonance as the proton transmitter is stepped through resonance. The effective bandwidth, DF, of the decoupling sequence is immediately obvious. Unfortunately,
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the heteronuclear dipolar coupling, d, cannot generally be neglected in comparison to D. Indeed, the interaction of offresonance decoupling and large couplings can result in differential line-broadenings and small apparent frequency shifts [16]. These effects are particularly striking in the presence of 19F, and are explored in Section 6.1. Although changing the transmitter offset does strongly influence decoupling performance (Fig. 10), the fact that performance is a function of the accummulated phases at both DCd/2 and DKd/2 makes the significance of the offset dependence much less clear cut. This is illustrated schematically in Fig. 3 for two functions fJ and fd which nominally correspond to decoupling sequences suitable for decoupling small (J) vs. large (d) interactions, respectively. Optimal functions for solution-state decoupling require that the gradient of the net rotation, f, be small, (Eq. (15)). However, f(D 00 Cd)Kf(D 00 Kd) may be large if d is comparable to the transmitter offset, resulting in poor decoupling. In contrast, the variation in fd is limited over a given ‘bandwidth’, so Df does not become unacceptably large, even if local gradients may be large compared to fJ. Hence an offset dependence which is suitable for decoupling J may not be appropriate for decoupling dipolar interactions, and vice versa. In short, we have moved from the case where jHrf,IjO HIOHIS[HII to a situation where all the interactions are of a similar order of magnitude, jHrf,IjOHICHIS[HII. To further complicate matters, the spin Hamiltonian is typically time dependent due to magic-angle spinning (MAS), which is necessary in solid-state NMR of powdered materials to remove the orientation dependent component of the chemical shift (the Chemical Shift Anisotropy, CSA). The resulting time-dependence greatly complicates theoretical analysis.
3. 1H decoupling in experimental practice 3.1. Continuous wave decoupling
203
on-resonance continuous wave irradiation is an effective decoupling strategy for typical organic solids. Fig. 4 shows how the linewidth (and so sensitivity) improves as the decoupler power, expressed as the approximate 1H nutation rate nrf, is increased. At these modest spinning speeds, the spectrum with very weak decoupling is worse than the spectrum obtained without decoupling. The linewidth then decreases rapidly with increased decoupling power until only modest increases in sensitivity are achieved as the RF amplitude is increased towards the limit for the probe. Performance degrades rapidly more than a few kHz away from the 1H resonance (Fig. 10), but optimisation of the transmitter offset is straightforward and is reasonably consistent between samples. CW decoupling under these conditions is effective because of, rather than in spite of, the width of the proton spectrum. In hand-waving terms, the multiple homonuclear dipolar interactions between the protons make the spin system ‘homogeneous’, i.e. continuous irradiation does not burn a hole in the proton spectrum, but rather the entire spin system is affected. Provided the RF field strength (expressed as a nutation rate) is at least comparable to the 1 H bandwidth, the protons respond collectively to the decoupling field. Interactions linear in I, including the IS couplings, will be scaled by P1(cos q)Zcos q, (where q is the nutation axis), while bilinear terms, i.e. the proton–proton couplings, are scaled by P2 ðcos qÞZ 1=2ð3 cos2 qK 1Þ. If the proton transmitter is approximately on-resonance (qZp/2), then the heteronuclear couplings are averaged to zero, while the homonuclear couplings are scaled by K1/2. The continuing presence of homonuclear couplings is actually a benefit. As discussed below, the continuing presence of homonuclear coupling is actually a benefit, and the decoupling performance is degraded if proton ‘spin-diffusion’ is suppressed.
At relatively modest magnetic fields, corresponding to I spin Larmor frequencies of, say, less than 400 MHz, simple φ φd ∆
φJ ∆" −d
∆"+ d
∆'− J / 2
∆' +J / 2
Fig. 3. Schematic illustration of the different role of transmitter offset, D, in sequences suitable for decoupling J(fJ) vs. dipolar (fd) interactions. While the gradient of fJ (dotted line) remains small, the differences in net rotation, f(D 00 Cd)Kf(D 00 Kd), may be large. In contrast, the maximum difference is limited for function fd.
0 kHz
28 kHz
36 kHz
48 kHz
60 kHz
Fig. 4. Effect of changing 1H decoupler power on the 13C CP/MAS spectrum of a glycine/alanine mixture (nrZ5.1 kHz, 13C Larmor frequency of 75.40 MHz).
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3.2. Two pulse phase modulation and related sequences At first thought, we would not expect the sample spinning rate to influence rare spin resolution. MAS is required to deal with the S spin chemical shift anisotropy, but the CSA behaves inhomogeneously under spinning [17] and so manifests itself as spinning sidebands of nominally zero width (in the absence of other line-broadening factors). As shown in Fig. 5(a), however, 13C linewidths under MAS increase significantly at spin rates above ca. 10 kHz. This effect can be straightforwardly explained by a weakening of the homonuclear coupling network by the magic-angle spinning [18,19]. The strong dipolar couplings between the protons play, in fact, a useful role in decoupling, leading to a partial ‘self-decoupling’ effect. Indeed, if the rate of mixing of the proton states were sufficiently fast (in comparison to the heteronuclear couplings), then decoupling would not be required at all. Faster MAS reduces the rate of this ‘spin diffusion’, weakening its helpful effects on decoupling. But because the network of homonuclear couplings behaves ‘homogeneously’ under MAS (in contrast to the CSA), modest MAS spin rates have little effect on the proton spectrum. As the spinning speed increases, however, resolved spinning sidebands are observed in the 1H spectrum. This co-incides with the degradation of decoupling performance. In principle, sufficiently fast MAS would average out the dipolar couplings, leaving only the relatively small J coupling to be removed by decoupling. However, the ‘undecoupled’ 13C linewidth only decreases relatively
(b)
(a)
weakly (Dnf1/nr) with spin rate [20] and fast magic-angle spinning on its own is unable to eliminate the effect of dipolar couplings, even for plastic crystals such as adamantane [21]. The degradation of rare spin linewidths with increasing nr becomes problematic as static magnetic field strengths, B0, increase. Since the CSA scales with B0, proportionately faster spin rates are required to prevent excessive complication by spinning sidebands or rotational resonance effects in isotopically enriched samples. In addition, 1H–X correlation spectra also become increasingly useful at high MAS rates due to the improvement in 1H resolution. Hence we require effective 1H decoupling at high spin rates. Minor degradations could be offset by increasing the power of the CW decoupling, but, as previously observed in solutionstate NMR, this is not an effective long-term strategy. A major step forward was made with the observation that simple modulations of the decoupler phase could dramatically improve the performance of the decoupling relative to simple CW irradiation [18,22]. In Two Pulse Phase Modulation (TPPM) [22], the repeating unit of the decoupling sequence consists of two pulses of the same nominal nutation angle, tp, and phase difference Df. Experimentally, optimal values typically lie around tpz1708 and Dfz158. The parameters are correlated and dependent on the sample and experimental conditions, such as spin rate (Fig. 14), although in practice, Df is often fixed at 158 and only tp (which would require calibration in any case) is optimised. The effects of TPPM decoupling can be seen in Fig. 5(c). Note that the increase in peak intensity
(d)
(c)
νr / kHz
k / Hz
νr / kHz
k / Hz
30
350
30
350
25
420
25
420
20
500
20
500
15
700
15
700
1100
10 1
0
−1
ν(13C) / kHz
1
1100
10 0
−1
ν(13C) / kHz
1
0
−1
ν(13C) / kHz
1
0
−1
ν(13C) / kHz
Fig. 5. 13C spectra of polycrystalline 2-13C-alanine as a function of sample spin rate, nr, using (a) CW and (c) TPPM 1H decoupling with nrfZ100 kHz (1H Larmor frequency of 600 MHz, DfZ158). (b) and (d) are corresponding spectra simulated using experimental values of 1H spin diffusion rate, k (Section 4.2.3 for details of the model used). Figure adapted from Ref. [19].
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with MAS rate seen here is not always observed [36], and this probably reflects the sensitivity of TPPM performance to experimental parameters (Fig. 14). Following the success of TPPM, a number of variants have been proposed. A combined phase and frequency modulated decoupling, FMPM [23], provides some insight into the mechanism of decoupling using TPPM, while AMTPPM [24] added an amplitude modulation. SPARC [25] essentially uses the ‘super-cycling’ established from solution-state composite-pulse decoupling with TPPM as a basic element. Although SPARC-16 performed better than TPPM for some [25] (but not all [26]) liquid-crystalline systems, the application of super-cycling without a welldefined basic element may lead to worse rather than better performance [11]. Its use in solid-state decoupling has not been reported, and in any case, SPINAL-64 (discussed below) has proved a more effective solution for liquidcrystal and solid-state decoupling. Relatively slow phase changes (1–2 ms) have also been used to improve decoupling performance in liquid crystals [27] (phase steps of p/2) and solids in which the 1H coupling network is not particularly extensive [18] (phase step of p). These schemes clearly ‘refocus’ some component of the rare-spin linewidth, although the similarities with schemes such as TPPM (much faster phase switching) and XiX (strongly synchronised to spinnning rate) are only partial, and they have not been widely applied. Rather than switch the RF phase between discrete values, other TPPM variants use a continuous modulation of the phase, e.g. f(t)Za cos(2pnct) where 2a is the total range of the modulation (equivalent to Df in TPPM) and nc is the frequency of modulation (equivalent to the frequency of TPPM modulation, 1/2tp). With echoes of noise decoupling, cosine phase modulation, CPM [28] and nPPM [29], use more than one modulation frequency and ‘supercycling’ to try to increase the ‘band-width’ of decoupling. Recently, single-frequency Continuous Modulation, CM [30], has been analysed in some detail (see Fig. 17 and associated text for further discussion). Modifications to the CM approach have included imposing an overall Gaussian envelope to one or more periods of the CM modulation with the goal of improving the robustness of the decoupling [31]. 3.3. Other common sequences: SPINAL-64 and XiX In the absence of a full theory of heteronuclear decoupling in the presence of strong homonuclear interactions, the rationalisations behind these various sequences are frequently questionable, and their practical performances have yet to show significant and consistent improvements over TPPM. It is unlikely, however, that a sequence as simple as TPPM is the last word in such a non-trivial problem. Several other sequences have been proposed and are discussed in Section 5.2, but a couple of sequences, developed in two different extremes have found successful applications beyond model compounds, and merit more attention.
205
The SPINAL sequences [26], and most notably SPINAL64, were originally developed for decoupling in liquidcrystal 13C NMR. The basic element of the SPINAL sequences is Q Z Pð108ÞPð108ÞPð158ÞPð158ÞPð208ÞPð208ÞPð158ÞPð158Þ (18) where P(m) denotes a pulse with nutation angle of w1658 and of phase m, and a bar indicates that the output phases are all negated, e.g. Pð108Þ corresponds to an output phase of K108. This element is then ‘super-cycled’ to generate a family of sequences, such as SPINAL-64 QQQ SPINAL-64 : QQ QQ Q
(19)
although, as remarked earlier, the significance of ‘super cycling’ in this context is unclear, especially as the basic element is acyclic (magnetisation does not return to its starting point after applying Q). As for TPPM, tp needs to be optimised for best performance, but good performance can be obtained without modifying the phases used, making the experimental application of SPINAL straightforward. Although TPPM was used as the starting point, the very different characteristics of SPINAL and TPPM suggest they should be considered as distinct sequences. Given its origins, it is unsurprising that SPINAL-64 is effective in decoupling static liquid-crystal [32] and ‘soft solid’ samples at low spinning rates [33]. Its better performance relative to TPPM under the very different conditions of high speed MAS of rigid solids is surprising, but frequently observed [34,35]. Although the long period might be expected to make it sensitive to interference with spinning (Section 4.2.2)—even the base element, Q, is four times longer than TPPM—its performance is almost independent of spinning speed [36]. The other sequence of note is XiX decoupling [37]. Like TPPM, this consists of a pair of pulses of alternating phase. In this case, however, the phase difference is fixed at 1808 (e.g. x and Kx, hence its name, x-inverse-x), and the duration tp is defined with respect to the rotation period and is not a nutation angle. Indeed, a better comparison may be the early use of phase alternation to improve decoupling performance in solution-state NMR [38]. XiX is discussed further in Section 4.2.2, but, in summary, the performance of XiX is strongly dependent on the ratio tp/tr (where tr is the rotor period). Optimisation about initial values of tp/ trz1.85 is generally sufficient in experimental practice to find a satisfactory optimum. Since the XiX period lengthens as the spinning speed is reduced, it is not surprising that its performance tends towards that of CW decoupling in the limit of slow spinning (Fig. 13); TPPM and SPINAL are invariably better choices. The performance of XiX steadily improves with spinning speed, however, and generally compares favourably with TPPM at spinning rates above w30 kHz.
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4. Understanding decoupling performance in solids 4.1. Rare spin linewidths in solid-state NMR The goal of decoupling is to narrow rare spin linewidths and so to improve both sensitivity and resolution. Before examining the characteristics of decoupling in dipolar-coupled solids, it is useful to review the factors that determine rare-spin linewidths and their relation to decoupling. Rare spin linewidths have been discussed extensively in the literature [39–42] from the viewpoint of individual contributions to linewidth, which can be grouped into three independent components to linewidths. Inhomogeneous linewidth of an individual resonance results from variations of the fundamental resonance frequency. These can arise from the nature of the sample itself, e.g. amorphous or poorly crystalline samples, or reflect variations of the local magnetic field. Good sample preparation is often critical in biomolecular NMR since larger molecules are often difficult to obtain as high quality crystalline powders. As shown in Fig. 6 for different samples of the decapeptide antamanide, the inhomogeneous linewidth (and hence the spectral resolution) can be highly sensitive to the sample preparation. Clearly, optimising decoupling will have little measurable impact on the quality of spectra from samples (a) and (b). Local magnetic field variations most obviously arise from inadequate shimming. Although the shimming characteristics of a normal MAS probe are poor in comparison to typical solution-state probes, shimming is only a limiting factor for materials with very narrow intrinsic linewidths, e.g. plastic crystals, such as adamantane. Variations in the bulk susceptibility are a more subtle source of inhomogeneous broadening. The presence of interfaces, such as grain boundaries, voids, etc. modify the bulk susceptibility,
c, and hence the net magnetic field, leading to linebroadening. If the susceptibilities of the bulk materials are isotropic, then these variations will be eliminated by magic angle spinning. However, MAS is only partially successful if the c of the sample is anisotropic [39,41]. The effects of anisotropy of the bulk magnetic susceptibility (ABMS), Dc, are reduced, but not eliminated, by magic-angle spinning, and is often the limiting factor in rare spin linewidths; contrast the broad lines observed for samples high in ordered aromatic rings (and so high ABMS), such as hexamethylbenzene [39], with the sharp lines commonly observed in samples with very small values of Dc, such as typical steroids. The division into ‘inhomogeneous’ and ‘homogeneous’ contributions to linewidths is unproblematic in solutionstate NMR. For solids, however, it is important to divide the homogeneous linewidth into ‘incoherent’ contributions resulting from random processes, and so which cannot be reversed, and ‘coherent’ contributions from the nuclear spin Hamiltonian and which can, in principle, be manipulated, and even reversed. Linewidth due to relaxation and exchange process clearly fall into the former category, and can be termed ‘incoherent homogeneous’ linewidth. The relaxation linewidth, 1/pT2, represents the ultimate limit on resolution. It is relatively straightforward to describe relaxation in solution-state NMR; we can consider an individual spin, or small set of spins, loosely coupled to its surroundings, and the decay of xy magnetisation can be described in terms of fluctuating random fields, resulting in a well-defined time constant for decay, T2. Unfortunately, little is known about T2 relaxation in typical molecular solids, either experimentally or theoretically. Extrapolation of classical relaxation theory to the limit of long correlation times (slow motions) is highly suspect,
13C
15N
(a)
(b)
(c)
70
60
50
40
30
20
10
140
130
120
110
100 ppm
Fig. 6. 1H-decoupled 13C and 15N spectra (1H Larmor frequency of 60 MHz) of three samples of the cyclic decapeptide antamanide prepared (a) as a lyophilized powder, (b) by fast solvent evaporation and (c) by slow evaporation (over several days). Figure adapted from Ref. [91]; see reference for further details.
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between ‘coherent’ and ‘incoherent’ contributions is unclear. The question is of great importance, however, as it is only the ‘incoherent’ linewidth that sets the fundamental limit on resolution in solids.
since ‘random field’ models assume that fluctuations of the Hamiltonian occur on a timescale that is fast compared to the evolution of the NMR signal. This model cannot be applied to solids where spins are generally strongly coupled together. The decrease in xy magnetisation is not primarily due to decoherence due to random external fluctuations, but is essentially a ‘dephasing’ process driven by the spin Hamiltonian. Initial xy magnetisation spreads into multiple quantum coherences between large number of spins and the system quickly reaches a ‘quasi-equilibrium’ state whose density matrix, seq, commutes with the system Hamiltonian, Hsys [43,44]. This quasi-equilibrium state slowly evolves to thermal equilibrium via T1 relaxation. Such states have been strikingly demonstrated in cross-polarisation experiments under MAS [45]. Although from the viewpoint of an individual spin, such dipolar evolution can be satisfactorily modelled as a random process, the initial evolution is purely coherent and can be reversed by appropriate manipulation of the Hamiltonian, e.g. in the context of cross-polarisation [46]. Since the evolution of coupled 1H spins is a coherent process, the ‘T2’ values used to characterise the 1H signal decay are purely phenomenological. Similarly, in the absence of perfect 1H decoupling, we also expect the 13C linewidth to be partly coherent in origin. This is confirmed by the often strong dependence of the 13C linewidth on factors, such as MAS rotation rate or 1H decoupling, that clearly modulate Hsys rather than relaxation processes (see below). Given the difficulty of modelling extended spin networks, and of distinguishing experimentally between contributions to homogeneous linewidth, the balance
4.1.1. Refocusable and non-refocusable linewidth A simple spin-echo is sufficient in solution-state NMR to distinguish inhomogeneous and homogeneous contributions to linewidth; the former are refocused, while the latter are not. The decay of xy magnetisation during a spin-echo train thus provides the time constant for homogeneous decay, T2. The situation is less clear in solids, as we cannot be sure how the coherent evolution responds to spin-echoes. If we can neglect homonuclear couplings between the I spins, then the ‘effective Hamiltonian’ during the decoupling will contain only heteronuclear terms. These will be refocused by a spin echo on S, i.e. the only decay will be due to relaxation of S spin magnetisation. Under these circumstances (which can be observed even in solids for isolated spin pairs), it is more efficient to refocus the dipolar evolution with a spin echo rather than resort to decoupling. If, however, the I spins are strongly interacting, then a spin-echo on the S spins is insufficient to reverse the evolution of the coherences, and any refocussing is likely to be partial. This can be seen in Fig. 7(b), which shows the intensity of the 13C signal after a tKpKt spin-echo, with different decoupling schemes applied during the spin-echo. As expected, the decays are much slower than would be calculated directly from the natural linewidth (for example, the linewidth of 46.4 Hz obtained for optimised CM would 21.37
(a)
21.34
19.72 TPPM CW
9.44
207
SPINAL64
TDOP-CM
48.2 Hz
46.4 Hz
51.3 Hz
81.2 Hz
(b) 1.0 0.8
T2' =47.0 ms
a.u.
0.6 0.4
T2' =28.3 ms
T2' =24.6 ms
SPINAL64 TPPM CW
0.2 0.0
T2' =6.5 ms 10
TDOP-CM
20
30
40
2τ / ms
Fig. 7. (a) Ca resonance of polycrystalline 2-13C-glycine for different decoupling sequences (nrfZ80 kHz, nrZ10 kHz, nominal 1H Larmor frequency of 500 MHz). The figures give the linewidths and peaks heights (arbitrary units). (b) Intensity of 13C signal after a tKpKt spin-echo as a function of delay period, using the different decoupling schemes during 2t. Adapted from Ref. [92].
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correspond to a ‘T2’ of 6.9 ms). However, the strong dependence of the decay of the spin echo on decoupling sequence indicates that it cannot be identified with true 13C T2 relaxation. In addition to true relaxation, the decay must also reflect coherent evolution (due indirectly to the 1H coupling network) that is not refocused by the spin echo. The resulting time constants have been denoted T20 to distinguish them from true relaxation time constants [92], and it is important to note that large variations in T20 values can be observed for decoupling sequences that perform identically (as quantified by peak intensity or line width), e.g. SPINAL vs. TDOP-CM. The latter sequence (Transverse Dephasing Optimised CM) refers to CM-decoupling specifically optimised for T20 , i.e. the parameters of the CM sequence have been optimised in order to maximise the spin echo signal after a fixed period. Although such optimisation does not improve the onedimensional spectrum, the long T20 is of great significance for experiments involving spin echoes, since the resolution (and sensitivity) of such experiments will be determined by the achievable T20 and not by the linewidth in the onedimensional spectrum. This is particularly significant for amorphous and disordered samples, as illustrated in Fig. 8 for
13
C-enriched cellulose. The linewidths in this sample are dominated by inhomogeneous contributions and there is little resolution improvement in the simple CP/MAS spectrum when an optimised decoupling is applied, Fig. 8(b). In contrast, the refocused INADEQUATE spectrum involves relatively long spin-echo periods during which doublequantum coherence is created and reconverted through J-couplings. Hence, the overall sensitivity of the experiment is strongly dependent on T20 , as can be seen in the intensity of the traces (c) and (d). The TDOP-CM decoupled spectrum provides detailed information on the connectivity between sites and correlations between inhomogeneous broadenings on connected sites (via the peak shapes). 4.1.2. Lineshapes under decoupling It is important to note that line shape, as well as its width, can be strongly dependent on the decoupling method (provided the linewidth is not dominated by inhomogeneous contributions). This is illustrated in Fig. 9 for the Ca resonance of alanine. Note in particular that the TPPM and XiX full-widths-at-half-maximum are very similar (33 and 31 Hz, respectively), even though the height of the XiX-decoupled signal is significantly greater. The
CW decoupling
TDOP-CM decoupling (b)
(a)
carbon chemical shift
carbon chemical shift
(d)
(c)
double quantum frequency / ppm
x3
x3
(f)
(e) 140
140
150
150
160
160
170
170
180
180
190
190 120
110
100 90 80 70 60 single quantum frequency / ppm
110
100 90 80 70 60 single quantum frequency / ppm
Fig. 8. 13C CP/MAS spectra of 10% 13C-cellulose under (a) CW decoupling and (b) TDOP-CM decoupling (nrfZ100 kHz, nrZ12.5 kHz, nominal 1H Larmor frequency of 500 MHz), (e) and (f) show corresponding refocused INADEQUATE spectra, while (c) and (d) are traces through the 2D spectra at a double quantum frequency of 137 ppm. Adapted from Ref. [92]; see reference for further details.
P. Hodgkinson / Progress in Nuclear Magnetic Resonance Spectroscopy 46 (2005) 197–222
1.5
209
(a) CW
TPPM
XiX
peak height
1
0.5
0
x10 (b) 1 kHz Fig. 9. Ca resonance of polycrystalline 13C-labelled alanine under different (optimised) decoupling schemes, using nrfZ150 kHz and a spin rate of 30.03 kHz (nominal 1H Larmor frequency of 600 MHz). Adapted from Ref. [37].
baseline expansions show that the TPPM-decoupled signal has a significantly wider foot, indicating that the difference in peak heights is largely a consequence of the much tighter lineshape for the XiX-decoupled signal. 4.2. Factors determining decoupling performance The difficulty of establishing a simple model for decoupling performance in solids has already been alluded to. This section attempts to unravel some of the factors that determine decoupling performance, focussing on CH and CH2 groups, since quaternary and CH3 carbons are relatively ‘easy’ to decouple. For instance, if we consider samples that are not subject to significant inhomogeneous broadenings, typical linewidths for CH, CH2 and CH3 under optimised decoupling are w20 Hz for CH3, w35 Hz for CH and w50 Hz for CH2 (due to the presence of strong homonuclear coupling). The behaviour of methyl groups is distinct from the other protonated carbons [36], which is presumably a consequence of the fast internal rotation in CH3, which scales the 13CK1H couplings by a factor of K1/3 and homonuclear 1H couplings by K1/2. 4.2.1. Off-resonance behaviour As discussed in Section 2.2, the performance of a decoupling sequence for solution-state NMR can be simply characterised in terms of its off-resonance behaviour. Observing the height of the NMR signal as a function of transmitter offset provides a direct indication of the range, DF, over which the decoupling is effective. Similar considerations apply to solid-state decoupling. As illustrated in Fig. 10 for sample used in Figs. 1 and 4, the decoupling performance degrades as the 1H transmitter frequency is moved away from the centre of the 1H spectrum. Note the very different responses of the CH and
−6
0 transmitter offset / kHz
6
Fig. 10. CW 1H-decoupled 13C NMR spectra of (a) the CH2 resonance of glycine and the (b) CH resonance of alanine acquired under the same conditions (nrZ5.1 kHz, nrfZ60 kHz, 1H Larmor frequency of 300 MHz) as a function of 1H transmitter offset (steps of 600 Hz).
CH2 resonances; as expected, the CH2 peak is much more sensitive to offset variations, reflecting the strong dipolar coupling between the protons. Although the optimum offset is essentially the same for different 13C sites under these conditions, it is clearly much more difficult to define the ‘decoupling bandwidth’ compared to solution-state decoupling. In mobile systems where the 1H sites are resolved, such as liquid crystals, however, it may be difficult to find a common offset that efficiently decouples all sites (e.g. aromatic vs. alkyl carbons). Note that a somewhat different behaviour is often observed in static samples. Rather than increasing monotonically with increasing offset, the linewidth has a clear maximum when the tilt axis of offresonance irradiation is at the magic angle [47]. This provides clear evidence of the ‘self-decoupling’ effect of fast spin-diffusion, which is suppressed by magic-angle (or Lee–Goldburg [48]) decoupling. In spinning samples, the resonance offset can be timedependent due to the presence of 1H CSA, resulting in a broadening effect distinct from that of an overall resonance offset [49,50]. The interaction between the time-dependent CSA, dipolar coupling and RF is rather subtle, and the derivation is only presented in outline. Following previous work [49], the spin Hamiltonian for a simple IS system under CW decoupling is given by:
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H Z uIS ðUIS Þ$2Iz Sz C uI ðUI Þ C urf Ix
(20)
where uIS and uI are the IS dipolar coupling and I spin shift, respectively (u indicating angular frequency). These interactions are dependent on orientation through the sets of Euler angles UIS and UI; urf is the RF nutation frequency. If we assume that urf is significantly larger than the other interactions, perturbation theory can be used to second order to derive the frequencies of the two strong transitions corresponding to the S spin resonance (Fig. 2): uG zG
uIS ðUIS ÞuI ðUI Þ urf
(a)
2
1
0
−1
−2
1
0
−1
−2
0
−1
−2
(b)
(21)
If 2pnr/urf, then the averaging due to spinning and RF irradiation can be treated independently and the centre-band frequencies for spinning samples are determined by the time average over a rotor cycle: uIS ðUIS ÞuI ðUI Þ huGi zG (22) urf This non-zero splitting corresponds to a cross-term between the heteronuclear coupling and the shielding of the I spin. Since the associated tensors are both of rank 2, this cross-term contains components of rank 0, 2 and 4. Under normal decoupling conditions, the resulting isotropic term is negligibly small and the rank 2 term is eliminated, like other rank 2 interactions, by magic-angle spinning. However, the rank 4 terms survive the averaging. At finite values of urf, this term separates the otherwise degenerate transitions, and the S spin spectrum from an individual crystallite appears as a doublet. The size of the splitting depends on the crystallite orientation (as well as the relative orientations of the tensor interactions) and so the spectrum of a powdered sample appears as a broad double-peaked lineshape. These effects can be clearly seen in the spectra of isolated spin pairs, as illustrated in Fig. 11 for a 15NK1H pair. The spectrum in the absence of decoupling, (c), is a simple doublet due to the 1 J NH coupling, but the CW-decoupled spectra, (a) and (b), show broad lineshapes due to the interaction between the 1H CSA, 15NK1H dipolar coupling and decoupling. As expected, these effects increase at higher magnetic fields, (b), due to the proportionately larger CSA. This interaction between the CSA and dipolar interaction can be ‘refocused’ by modifying the decoupling RF. For instance, the sequence (2p)x(2p)Kx. completely suppresses this line-broadening, but is an ineffective decoupling sequence otherwise. Similarly, it can be shown from average Hamiltonian theory that TPPM with a nutation angle of 1808 strongly reduces, without completely refocusing, this broadening mechanism. Note that this broadening is refocused in spin-echo experiments and does not contribute (to first order) to the T20 decays discussed in Section 4.1.1.
2
(c)
2
1
ν( N) / kHz 15
Fig. 11. 1H CW decoupled 15N NMR spectra of polycrystalline d9trimethyl-15N-ammonium chloride at proton resonance frequencies of (a) 600 MHz and (b) 300 MHz at an MAS spinning rate of 30 kHz and decoupling field strength, nrf, of 100 kHz. (c) Is the corresponding spectrum at 600 MHz acquired without decoupling (the splitting is due to the 1JNH coupling of ca. 100 Hz). Figure adapted from Ref. [49].
These effects have been observed experimentally on model systems [49] and in cases where the effective CSAs are particularly large, e.g. when decoupling 31P rather than 1 H [51], or when augmented by strong couplings to a third spin species [33,52] (The latter case is discussed in more detail in Section 6.1.) The importance of these effects in more typical organic systems is less clear. The difficulty of decoupling tends to correlate with the dipolar environment, e.g. CH2 vs. CH rather than paralleling trends in 1H CSAs, for example. It is also notable that in the systems where this mechanism is clearly at work, different decoupling conditions are optimal for sites strongly perturbed by large effective CSAs (e.g. sites close to 19F) and unperturbed sites. In the former case, the optimal TPPM/SPINAL angles are z1808, as predicted from simple two-spin models, while the optimal angles for TPPM/SPINAL for ‘normal’ sites is consistently, and significantly, less than 1808 [52]. Although good decoupling sequences obviously need to cope with the effects of 1H CSA, there are clearly other factors at work. 4.2.2. Interaction with sample spinning We have assumed so far that the nutation rate associated with the decoupling irradiation, nrf, significantly exceeds the
P. Hodgkinson / Progress in Nuclear Magnetic Resonance Spectroscopy 46 (2005) 197–222
spinning rate, nr. In this ‘quasi-static’ limit, the system Hamiltonian is assumed to be approximately constant over the decoupling cycle, and so we can sequentially determine the effects of the decoupling and sample spinning. This separation of time-scales is more difficult to maintain as the spin rate is increased. In particular, ‘rotary resonance’ conditions can occur at certain ratios of nrf and nr. The rotating frame Hamiltonian for a spinning sample under CW decoupling can be written as a Fourier series [53,54]: HðtÞ Z nrf Ix C
2 X
Hm e2pimnr t
(23)
mZK2
44 kHz
The index runs from K2 to 2 since the interactions (chemical shift and dipolar) are rank 2 tensors, and it is unsurprising that interference effects are observed when pZnrf/nr is equal to 1 or 2. These ‘rotary resonance’ conditions [55] have the effect of ‘recoupling’ the heteronuclear dipolar interaction, which is then no longer refocused over the rotor cycle, greatly reducing the decoupling efficiency [56]. These effects can be clearly seen in Fig. 12; the width of the 19F resonance is strongly broadened at nrfZnr and 2nr, with weaker resonances observed at pZ3 and 4. Note that the observed guest molecules are highly mobile in these systems, and such resonances are rarely so clearly observed in more conventional solids. Especially in more strongly coupled systems, a recoupling phenomenon may also be observed at pZ0.5.
11 kHz
22 kHz
33 kHz
100 kHz
νrf −210
−220
−230
δF / ppm Fig. 12. CW 1H-decoupled 19F spectra of the inclusion compound formed between 1-fluorotetradecane and urea as a function of 1H nutation rate. Strong interferences are observed when the (approximate) nrf matches a multiple of the spin rate, nrZ12 kHz. The increase in linewidth at the highest decoupling fields is the consequence of significant ‘Bloch–Seigert’ shifts at this field (1H resonant at 200.13 MHz), cf. Section 6.1. Figure adapted from Ref. [93].
211
This so-called HORROR condition [57] recouples the homonuclear dipolar interaction and has a less direct effect on the decoupling efficiency. These resonant effects should not be confused with the general trends in decoupling performance as a function of spinning speed discussed in Section 3.2, which result from the weakening of the 1H dipolar network due to MAS. Resonance conditions can also be expected for more complex irradiation patterns than CW. Moreover, as there is an additional frequency associated with the cycle rate, nc, we may expect more complex interactions. This is certainly the case for homonuclear decoupling, where rational values of nc/nr are associated with major drops in performance [9]. The analogy is not perfect, however. Homonuclear decoupling sequences are designed to be cyclic, and have a well-defined effective Hamiltonian. In contrast, solid-state decoupling sequences (with the obvious exception of CW decoupling) are generally acyclic, and the effective Hamiltonians are much less well defined (Even for TPPM, for example, the effective Hamiltonian is barely tractable for all but special cases.) This may explain why the interaction between RF cycle rate and sample spin rate varies so widely. For example, the interaction is extremely strong for XiX, as seen in Fig. 13. Indeed, effective XiX decoupling involves finding values of nc that avoid these strong resonance conditions. On the other hand, the performance of SPINAL-64 barely varies with spinning speed, even though its very long cycle length appears to make it vulnerable to resonances between nr and nc [36]. While most effort has been made to find high power decoupling sequences that continue to function effectively at high spinning speeds, there have also been investigations of decoupling with low RF powers, i.e. nrf/nr [20]. Here, magic angle spinning is used as the primary means of averaging the dipolar interactions, assisted by relatively weak RF irradiation. But as maximum sample rotation rates fall well short of maximum RF nutation rates (50–70 kHz and 150–200 kHz, respectively, at time of writing), it is unsurprising that low power CW-decoupling is consistently less effective than conventional high-power decoupling. This approach may be valuable, however, for experiments that involve simultaneous 1H decoupling and RF irradiation on an X nucleus (e.g. for recoupling). Effective recoupling often requires X nutation rates that are multiples of nr (e.g. 7nr in the case of C7 and derivatives [58]), while 1H decoupling fields generally need to be a factor of three higher still [59] to avoid perturbing effects associated with matched X and 1H nutation rates. Such large nutation rates require large RF fields (especially for lossy samples), which is risky for both the probe and for temperature-sensitive samples. It would, therefore, be extremely useful to find sequences that are effective at nutation rates much lower than, or even comparable to, the spinning rate. The potential advantages of low-power, or even no, decoupling in X nucleus recoupling sequences at higher spin rates has recently been demonstrated experimentally [60].
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τp /τr
50
100
150
7 5 6
4 3
5
peak height
2 4 1 0
3
0
0.1
0.2
τp / ms
0.3
0.4
0.5
2
1 CW 0 0
1
2
τp / ms
3
4
5
Fig. 13. Performance of XiX decoupling as a function of XiX pulse width, tp. The plots show the peak height of the Ca resonance of alanine relative to CW decoupling at the same nrf (150 kHz), at an MAS frequency of 30.03 kHz and nominal 1H Larmor frequency of 600 MHz. The inset is an expansion for pulse widths up to 500 ms (15 rotor periods, tr). Points within the grey circles correspond to optimal values of tp/tr. Figure adapted from Ref. [37].
4.2.3. Interaction with 1H dipolar couplings The weakness of scalar couplings relative to the RF nutation rate, nrf, and chemical shift offsets means that homonuclear 1H, 1H interactions can be neglected in theories of decoupling for solution-state NMR, allowing the problem to be expressed in terms of a single I spin. This is clearly not appropriate for decoupling in typical proton-containing solids. Some progress can be made analytically for static samples. For instance, models of the coupled proton system using memory function approaches predict an inverse dependence of the S spin linewidth with decoupling power, which is confirmed experimentally in static solids with a continuous ‘network’ of proton–proton couplings [61,62]. In contrast, exact modelling of InS systems predicts that the S spin linewidth remains constant for on-resonance irradiation, but that low power decoupling in multi-spin systems creates significant cycling sidebands [63]. Increasing the decoupler power reduces the intensity of these signals (which are typically broad and unresolved), transferring their increasing intensity to the centreband without significantly modifying its width. The behaviour of typical solids under MAS falls between these limits [36,63]. Note that these ‘on-resonance’ effects are quite distinct from the normal ‘off-resonance’ behaviour illustrated in Fig. 2, which involves the broadening of the centreband into resolved transitions as the transmitter frequency is moved away from exact resonance. The 13 C1 H2 spin system is an obvious starting point for exact simulations of decoupling which attempt to include
homonuclear coupling. Unfortunately, numerical simulations of a CH2 system under MAS do not reproduce the homogeneous character of the 1H coupling network. A simple H2 dipolar-coupled pair, for instance, behaves ‘inhomogeneously’, with the dipolar interaction refocusing under spinning, producing sharp spinning sidebands [17]. Similarly, simulations of an isolated CH2 system under MAS are unrepresentative of the behaviour of methylene groups in typical organic solids. In principle, these problems can be overcome by adding a sufficiently large number of neighbouring protons. However, exact analysis rapidly becomes intractable, especially for spinning samples, as the number of spins increases. Even if we restrict our simulation to, say, 10 dipolar coupled spin-1/2 nuclei, the Hilbert space is 1024!1024, and diagonalising such large matrices becomes prohibitively slow. Even the selection of 10 spins out of a continuous network is problematic. Largescale simulations (O15 spin-1/2 nuclei) have been applied to geometries in which spatial periodicity allows the Hamiltonian to be block-diagonalised [64], but these techniques have yet to be applied to the more involved problem of decoupling under arbitrary RF. The difficulties of numerical simulation of decoupling are compounded if we move beyond the ‘quasi-static’ approximation and incorporate sample spinning. The time dependence means that density matrix propagators must be determined by small time-step integration. If the rotor period, tr, can be set to a multiple of the RF cycle time, tc, then it is generally only necessary to integrate the propagator over a single rotation cycle (see Ref. [54] for
P. Hodgkinson / Progress in Nuclear Magnetic Resonance Spectroscopy 46 (2005) 197–222
^ iH^ C R, ^ contains an additional term R^ the Liouvillian, LZ intended to mimic the effect of spin diffusion amongst the protons. The exact form of this term is not critical, but the significant feature is that including a dissipative term causes the 1H magnetisation to decay, e.g. Iz(t)ZIz(0)exp(Kkt), where k is a spin diffusion or relaxation rate. Although this is an incoherent model (random spin exchange due to ‘spin diffusion’ or relaxation) of a coherent ‘dephasing’ process, the approach is reasonable; the experimental decays of proton magnetisation generally fit to decaying exponentials. These models cannot themselves predict appropriate values of the ‘spin diffusion’ rate, k, as a function of spin rate. However, since the decay of 1H magnetisation under CW decoupling corresponds to T1r relaxation, measurements of T1r provide a straightforward means of accessing k. As expected, k decreases approximately linearly with spin rate, and using the experimental values of k in the ‘CHC bath’ models gives good agreement between experimental and simulated spectra [19,66] (Fig. 5). Fig. 14 shows a comparison between model and experiment for a 13C, 1H and a 15N, 1H system which have significantly different proton spin-diffusion rates.
a full discussion). As we have seen, however, many decoupling sequences are not rotor synchronised; XiX, for example, explicitly avoids synchronisation conditions. Unless the time-scales for the RF and rotation permit sequential averaging, we must either resort to bimodal Floquet treatments [53,65], which are analytically elegant but are poorly adapted to numerical work, or to small timestep integration over the complete NMR signal. Since decoupling performance is dependent on the behaviour at relatively long (10 s of ms) timescales, we would need to compute the NMR signal out to long times (brute force methods) or use extremely large matrices (in Floquet approaches). Since performing exact simulations may be impractically slow and is unlikely to provide detailed physical insights, most workers have chosen to model the effect of the coupling network empirically (or ignore it completely). One particularly simple model which is able to reproduce the essential features of CW decoupling as a function of spinning speed involves a Liouville space treatment of a 13 C, 1H spin system [19]. In addition to the Hamiltonian ^ which describes the coherent evolution, superoperator, H, (a) 10
213
10
0.9
0.9
0.8
9
9 0.7
τp / µs
0.7
8
8
7
7
0.6
0.5
0.5
0.4
6
6
5
5
0.3
0.3
0.2
0.1
0.1
4
0
40
80
120
160
4
0
40
∆φ / °
80
120
160
∆φ / °
(b) 10
10
9
9
8
8
7
7
6
6
5
5
0.9
τp / µs
0.7
0.5
4
0.3
0.1
0
40
80
∆φ / °
120
160
4
0
40
80
120
160
∆φ / °
Fig. 14. Plots of decoupling performance as a function of TPPM parameters (pulse length, tp, and phase difference, Df) as measured by the intensity of (a) the Ca resonance of 2-13C-alanine at a 1H resonance frequency of 300 MHz and nrZ28 kHz, nrfZ100 kHz, and (b) the 15N resonance of d9-trimethyl-15Nammonium chloride. High values correspond to good decoupling performance. The plots on the right are simulated line intensities using the model of an X, 1H spin system coupled to dissipative bath with a 1H spin diffusion rate of (a) kZ500 Hz and (b) kZ0 Hz. Figure adapted from Ref. [66]. Further simulation details can be found in this reference.
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The plots show that a reasonable qualitative agreement is obtained between the model (which also includes the interaction between 1H CSA, dipolar coupling and RF) and experiment over a wide range of TPPM parameters. The detailed agreement is less striking: the experimental plots are significantly different for the two systems, while, as may be expected, the overall appearance of the two simulated plots is very similar, with the increased spin diffusion tending to ‘smooth out’ the calculated surface. The application of such models to more complex decoupling sequences is more problematic. How can we measure the decay rates of 1H magnetisation (k) if there is no well-defined spin-lock axis, as is the case in CW decoupling? In the absence of experimental values, k must be treated as an empirical parameter. Morever, the effective strength of the homonuclear coupling network is a key factor in the efficiency of a decoupling sequence. Given that these small-spin models cannot themselves predict k, they are not expected to be particularly effective in comparing the efficiencies of very different decoupling sequences. In other words, while small-spin models are useful for retrodictive rationalisation, they will not be particularly reliable for predictive purposes. 4.2.4. Interference with exchange Any process that occurs on the same time-scale can potentially interact with decoupling. The interaction between sample spinning and decoupling has already been discussed in Section 4.2.2, but exchange processes are another possible source of interference. As expected, the line-broadening resulting from such interference is most severe when the rates associated with the decoupling, e.g. nrf matches the rate of the exchange. The stochastic nature of the process means that the effects are observed over a larger range compared to the resonancelike behaviour of MAS and decoupling interference (Fig. 12). If the complexity of the decoupling sequence is increased, the interference effects are likely to occur over a wider range of time-scales, although such effects have to date only been clearly documented for homonuclear decoupling [67]. Altering the exchange rate by changing the sample temperature is the simplest way to diagnose (and alleviate) such line-broadenings. On the other hand, these effects can be used to probe dynamics on the time scale of RF nutation rates, although 1H T1r measurements will often provide the same information in a more direct fashion [68]. The interference can be modelled quantitatively in terms of simple relaxation models of the effect of exchange [69], or, more satisfactorily, using a full Liouville space treatment [67,70]. 4.2.5. RF inhomogeneity A drawback of multi-pulse techniques compared to simple CW decoupling is potentially an increased sensitivity to variations in the B1 field across the sample. Nutation experiments show a variation in RF nutation rates
across a full sample of about G10% relative to the mean [30]. Restricting the sample volume in order to reduce the impact of RF inhomogeneities is common practice in multipulse techniques for abundant spins, but is undesirable for rare-spin NMR. Recent work described in Section 5.1 suggests that the effectiveness of existing sequences such as TPPM (and variants such as CM) and XiX, is due, in part, to their relative robustness with respect to variations in nrf. The distribution of nutation rates due to RF inhomogeneity can be incorporated without difficulty into numerical simulations, and sequences such as SDROOPY [71] (described below), which are optimised to be insensitive to the exact nrf, have performed well in practice. An alternative to trying to model every aspect of spectrometer performance, including RF inhomogeneities, is to search for and optimise decoupling directly on the spectrometer, as described in Section 5.2.3.
5. Characterisation and development of decoupling methods for solids Rigorous approaches to developing decoupling sequences, whether through analysis of the spin Hamiltonian or through numerical simulation, are extremely challenging, principally because of the need to suppress interactions by about three orders of magnitude. This can be contrasted with the successful development of ‘recoupling’ sequences designed to re-introduce interactions that would otherwise be averaged by magic-angle spinning. In this case, imperfections of a few percent are quite acceptable. This section reviews some of the different strategies that have been used to develop and characterise decoupling methods in the absence of a comprehensive model for 1H decoupling in solids. 5.1. Characterisation of decoupling performance As discussed in Section 4.2.1, decoupling sequences for solution-state NMR state can be fully characterised simply by measuring the intensity of a single X resonance as a function of 1H offset. In the presence of strong dipolar couplings, however, the decoupling behaviour of carbons having different numbers of bonded protons varies substantially (Fig. 10). The CH case is obviously the more straightforward. It can be modelled satisfactorily in terms of the damped CH spin system discussed above, and the performance of the decoupling sequence determined from the height of a simulated NMR signal. Alternatively, as discussed in Section 2.2, we can determine the divergence between the 1 H magnetisation vectors for the a and b states of the 13C over a cycle of the decoupling. This quantity, nC$nK in terms of Eq. (12), will be zero for perfect decoupling. The CH 2 system is considerably more involved. Although, being the hardest system to decouple, it should
P. Hodgkinson / Progress in Nuclear Magnetic Resonance Spectroscopy 46 (2005) 197–222
Cα
C4
TPPM
1.5
0.5
SPINAL-64
relative nutation angle
1.0
1.5 1.0 0.5
DROOPY-1
1.5 1.0
1.5 1.0 0.5
DROOPY-2
1.5 1.0 0.5
SDROOPY-2
8.5 8.0 7.5 7.0 6.5 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0
SDROOPY-1
0.5
1.5 1.0 0.5 −3
−1
1
3 5 −3 −1 transmitter offset / kHz
1
3
5
Fig. 15. Intensity of a representative aromatic (C4) and aliphatic (Ca) peak of the 13C spectrum of 5CB (4-pentyl-4 0 -cyanobiphenyl) as a function of transmitter offset and RF power (expressed as the overall nutation angle relative to the correctly calibrated angle for the sequence) for different decoupling schemes: TPPM (DfZ158), SPINAL-64, DROOPY-1, DROOPY-2 and ‘supercycled’ versions of the latter, SDROOPY-1 and SDROOPY-2. Adapted from Ref. [71].
logically receive the most attention, relatively few studies to date have attempted to tackle this problem directly. The methodology of Eq. (12) has been adapted to InS systems [11] and the resulting composite pulse-based sequences, such as COMARO-2, perform satisfactorily for 2H in liquidcrystal samples [72], but do not transfer well to spinning solids [73]. De Pa¨epe et al. [71] minimise the contribution of heteronuclear IiSj terms in the effective Hamiltonian over a decoupling cycle, plus a fairly ad hoc modification of the CH2 spin system Hamiltonian to model the effect of
215
additional homonuclear couplings. The sequences were also optimised for robustness with respect to offset and RF inhomogeneity. Fig. 15 compares the performance of two solutions from this optimisation, DROOPY-1 and DROOPY-2, together with ‘supercycled’ versions, with TPPM and SPINAL-64 for a liquid-crystalline sample. The optimised performance of SPINAL-64, as shown by its higher maximum peak intensities, is clearly superior to TPPM, although the performance is quite sensitive to the parameter settings, especially for the Ca methylene carbon. The lack of correlation between offset sensitivity and peak decoupling performance is in marked contrast to the behaviour of solution-state decoupling sequences. The unpredictable nature of ‘supercycling’ applied to acyclic sequences is also noteworthy. SDROOPY-1 strongly outperforms its parent, DROOPY-1, and is competitive with SPINAL-64, while SDROOPY-2 is significantly less effective in decoupling the aliphatic site than DROOPY-2. Given the complex nature of the optimisation and the uncertain correspondence between predicted and measured performance, it is difficult to judge the effectiveness of the model and metric in isolation. It is interesting to note that the strong performance of SPINAL-64 and SDROOPY-1 is maintained even under conditions of fast sample spinning [36], despite their relatively long cycle lengths, tc. It has recently been suggested that proton nutation spectra provide insight into the mode of action of different decoupling sequences [30]. Nutation spectra might not be expected to be particularly informative for cyclic decoupling sequences, since we would expect the magnetisation to remain locked if sampled once per decoupling cycle. However, decoupling sequences for solids are frequently acyclic, and so the evolution of the 1H magnetisation may provide greater insight than the intensity of a single peak in the X spectrum. Fig. 16 shows 1H nutation spectra for two different proton environments. In Fig. 16(a), the sample is an isotropic liquid, and the finite width of the CW nutation spectrum provides a direct indication of the RF inhomogeneity (Section 4.2.5). The nutation spectra for TPPM and Continuous Modulation (CM) in particular are much ‘tighter’, implying a greater tolerance to RF inhomogeneity. It is also noteworthy that the nutation spectra are peaked at the cycle frequency, nc, rather than the RF nutation rate, nrf (This behaviour is reproduced in simulations of simple spin1/2 systems and can be rationalised analytically.) The interpretation of corresponding nutation spectra in solid samples, Fig. 16(b), is less straightforward. For instance, a broad nutation spectrum may correspond to an undesirable sensitivity to RF inhomogeneity, or it may be the result of ‘recoupling’ the homonuclear dipolar interactions, leading to rapid dephasing of the 1H magnetisation, which is generally desirable. Although sequences involving modest phase shifts can be approximated in terms of an overall nutation, the significance of nutation spectra for highly acyclic sequences, such as SPINAL-64, is less obvious.
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(a)
(b)
×2
CW
CW
×4
TPPM
×4
TPPM CM
×4
SPINAL64
CM
XiX
×4 −100
−50 1
0
50
100
H nutation frequency / kHz −100
0 1
100
H nutation frequency / kHz
Fig. 16. 1H nutation spectra as a function of 1H decoupling sequence: (a) spectra from residual protons in a D2O sample using nrfz97 kHz with Continuous Wave, TPPM and Continuous Modulation (CM) decoupling, (b) spectra from 2-13C-glycine with nrZ20 kHz, nrfz100 kHz and nominal 1H Larmor frequency of 500 MHz, using CW, TPPM, CM, SPINAL-64 and XiX decoupling. Figure adapted from Ref. [30]. See reference for further experimental details.
Despite clearly being informative, it is difficult to use the spectra of Fig. 16(b) in a predictive manner. The small angle phase variations of sequences such as TPPM and CM make them amenable to analysis [22], with the Continuous Modulation form requiring less approximation [30] in comparison to its discretely jumped counterpart. To first order, there is a well-defined spinlock axis associated with the RF irradiation, with an effective nutation rate, neff, and angle relative to the z axis, feff. The effect of the 1H homonuclear couplings on 13C decoupling can then be probed in a relatively direct fashion. Fig. 17 shows the height of the CH2 resonance of glycine as a function of the parameters of CM decoupling. Like TPPM, the performance is strongly dependent on decoupling parameters; indeed Fig. 17 is comparable to the results from the CH resonance of alanine, Fig. 14 (with the vertical axis reversed). As might be expected, it is impossible to find satisfactory decoupling under rotary resonance conditions, i.e. when neff matches a multiple of the spin rate (point c is a weak local maximum). On the other hand, decoupling is effective when the HORROR condition is matched, point a, i.e. recoupling of the 1H homonuclear interactions is associated, in this case at least, with effective decoupling. This is supported by the lack of correlation between decoupling performance and an effective tilt axis at the magic angle (feffZGqm), which would be associated with weakened homonuclear interactions. It is interesting to note the presence of additional ‘resonance’-like conditions in the plots of Fig. 17, which are also present, but less clearly resolved, in Fig. 14. Fig. 17(c) shows an expansion of Fig. 17(a) for the region containing the local optima a–d, but acquired under
spin-echo conditions, i.e. it plots the intensity of the 13C signal after a tKpKt evolution period (with 2tZ20 ms) as a function of the decoupling parameters during 2t. As discussed in Section 4.1.1, this removes the refocusable component, which corresponds to, but is not necessarily synonymous with, the inhomogeneous linewidth. This significantly accentuates differences in decoupling performance. The optimal decoupling performance is obtained for point a (on the HORROR condition) rather than point b; although these points have indistinguishable directlymeasured performances, they have clearly distinct T20 values of 58 and 52 ms, respectively. In these cases at least, good ‘spin-echo’ performance also implies good ‘conventional’ decoupling performance, i.e. decoupling optimised under spin-echo conditions can be applied throughout an experiment. 5.2. Strategies for development Given the difficulty of defining even models and metrics for analysing decoupling in strongly coupled systems, such as CH2, it is unsurprising perhaps that serendipity has often been a feature in the discovery of many decoupling sequences. There are, however, some less ad hoc approaches that have provided useful insights. 5.2.1. Symmetry-based methods Levitt et al. have proposed a framework for the design of rotor-synchronised pulse sequences under MAS. One family of such sequences is denoted CNnn where C is an element corresponding to an rf cycle (i.e. returning irradiated spins to their initial state). The C element is repeated N times with
P. Hodgkinson / Progress in Nuclear Magnetic Resonance Spectroscopy 46 (2005) 197–222
(a)
100
φeff = θm
1.1
80
H
1.0
νc / νrf
217
a
b d
60
R1
0.9 R2 c
40
φeff = π/2
0.8
20 0.7 φeff = −θm
0 0
0.2 10
0.4 20
νc / νrf
(b)
0.6
a
30
0.8 40
(rad)
(degree)
100
(c) 1.0
1.0
0.95
0.95
0.90
0.90
80
60
40
20 0.85
0.85 0
0.1
0.225
0.35
a / rad
0.1
0.225
0.35
a / rad
Fig. 17. (a) Experimental 13C peak height for the CM-decoupled CP-MAS spectrum of 2-13C-glycine as a function of the CM parameters, a (total phase excursion) and the cycle frequency, nc, relative to the nutation rate, using nrfZ115 kHz and nrZ20 kHz, at a nominal 1H Larmor frequency of 500 MHz. Dotted lines mark significant values for the tilt of the effective lock axis, feff. Solid lines mark the rotary resonance (neffZpnr, pZ1,2) and ‘HORROR’ (neffZnr/2) conditions between the effective nutation rate and the spin rate. (b) is an expansion of the region containing the local optima marked a-d, while (c) shows the results from the same region, but obtained under spin-echo conditions. Adapted from Ref. [30].
a phase shift of 2pn/N and a total duration of ntr (where tr is the rotor period). Note that this implies a fixed ratio between the RF nutation and spinning rates, nrf/nrZN/n. Each term of the nuclear spin Hamiltonian can be written as the product of a spatial component of rank l and a spin component of rank l (with respect to the rotation produced by the RF sequence). In the same way as phase cycles are designed to suppress undesired coherence pathways, we can choose values of N, n and n such that terms of the Hamiltonian with different values of l and l are suppressed in the average Hamiltoninan for the sequence. With this basic selection in place, we can choose elements C that are robust with respect to experimental parameters, such as RF inhomogeneities. This provides a systematic route to the design of RF pulse sequences. It was originally applied to ‘recoupling’ problems such as the original C7 dipolar recoupling
sequence [58], but heteronuclear decoupling sequences can be designed within this framework [74]. As discussed in Section 2.1, however, first order AHT is not adequate to describe the suppression of interactions by three orders of magnitude, and extensions of the symmetry principles become unwieldy at higher orders. Moreover, the synchronisation conditions, e.g. that the RF nutation rate be six times the spin rate (C12K1 2 ) make these sequences less flexible compared to simple techniques such as TPPM. The experimental performance of C12K1 2 and related sequences, such as R2412 , is comparable to TPPM [75]. The combination of symmetry-based sequences with adiabatic C elements (consisting of a pair of adiabatic inversion pulses) has also been evaluated [76]. Although such sequences are expected to exhibit greater tolerance to, say, RF inhomogeneities, improved performance relative to TPPM has yet to be demonstrated.
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5.2.2. Numerical simulation Although less elegant than analytical approaches, numerical simulation is able (given the correct model system) to cope with arbitrarily complex problems. As discussed in Section 4.2.3, however, a truly predictive model for heteronuclear decoupling in solids has yet to be devised; such an absolute predictive model would need to be able to calculate a ‘quality parameter’ (say peak intensity) as a function of any experimental conditions. In optimisation problems, however, we are only really interested in relative performances within a restricted family of sequences (Is the decoupling better with parameter set F1 or set F2?). Provided the model is capable of predicting such trends, then we may be able to work with fairly simplistic models. The model proposed by De Pae¨pe et al. [71], for example, is relatively unsophisticated, but, given suitable optimisation criteria, it is sufficient to devise useful and robust decoupling sequences. The phase-modulated decoupling sequence is expressed as a Fourier series, which allows a wide range of modulations to be included. The mean decoupling performance for a given sequence is then evaluated over an appropriate range of transmitter offsets and RF powers (to model RF inhomogeneity), ensuring that the solution will be experimentally robust. The resulting pulse sequences are clearly superior to TPPM in terms of decoupling performance and robustness, although edged out in peak performance by SPINAL-64 on the liquid crystal test system. Numerical simulations have yet to be used predictively for decoupling under MAS. Simulations have been used to rationalise the performance of XiX decoupling [37], for example, but it is noteworthy that the two spin model was only able to reproduce some of the important features, e.g. unfavourable ‘resonance’ conditions between the decoupling period, tc, and the rotor period, tr. Subtle, but important details, such as optimal values of tc/tr, could not be derived from small spin models. 5.2.3. ‘Direct optimisation’ An alternative to spin system modelling is to determine and optimise the decoupling performance on a sample directly [77]. As before, we start with a family of decoupling sequences, but now apply the decoupling sequence from a given set of parameters directly on a sample and measure, say, the peak intensity in the 13C spectrum (The linewidth is a poorer choice of parameter since a decoupling sequence may be effective at concentrating intensity without significantly changing the full-width-at-half maximum (cf. Fig. 9). Many experimental features, such as RF inhomogeneity, are automatically accounted for by this direct experimental approach; good solutions must necessarily be robust with respect to the experimental effects of RF inhomogeneity. Others, such as insensitivity to transmitter offset would need to be included specifically (e.g. by averaging the decoupling performance over a range of offsets). As, however,
insensitivity to 1H offset and ‘broadbandedness’ are not entirely equivalent for solids, it may be more useful to use model systems with a range of 13C sites and compute an average performance. The results to date [77] have involved optimisation of using a single resonance. It is unsurprising, therefore, that a different decoupling solution was found to that obtained by the numerical simulation discussed above. Starting from the same general Fourier series, the optimal solutions consistently reduced to a single frequency modulation of the phase, i.e. f(t)Za cos (2pnct). The overall performance of this Continuous Modulation (CM) is comparable to TPPM, although local high-performance solutions are often found by this iterative optimisation. These parameters are then transferrable to other samples run under the same conditions. Perhaps the most interesting feature of this work is emergence of a single frequency phase modulation as an optimal solution, under the conditions used. The family of functions used does impose limits on the solution space—it is unlikely, for example, that a long acyclic sequence such as SPINAL-64 could have emerged as a solution—but direct optimisation may continue to provide useful new starting points in the future.
6. Diversions in decoupling The previous sections have described the principles, complexities and current solutions to the problem of 1H decoupling in solids. This final section considers some other practical issues that arise in the application of decoupling in solid-state NMR, including the decoupling of nuclei other than 1H. 6.1. 1H decoupling in
19
F-containing systems
19
F is an attractive nucleus for NMR due to its 100% natural abundance and high magnetogyric ratio. However, the similarity of its Larmor frequency to 1H (just 6% lower) significantly complicates double-resonance experiments involving both 1H and 19F. As well as requiring appropriate hardware to ensure effective isolation of the 1H and 19F channels, 1H irradiation will result in a shift in the 19F resonance frequencies of Df z
n2rf 2Dn
(24)
where nrf is the 1H nutation rate and Dn is the difference in H and 19F Larmor frequencies. This shift is described, perhaps inappropriately, as a Bloch–Seigert shift [78]. Care is required when referencing, for example, to ensure that reference and sample spectra are acquired under the same decoupling conditions to ensure that the shifts are consistent.
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These effects are illustrated in Fig. 18. It can be seen that H decoupling significantly improves the linewidth compared to the undecoupled spectrum (top), but causes a measurable frequency shift which increases with increasing RF power. The Bloch–Seigert shifts are fairly modest at these Larmor frequencies. When the shifts are more significant, 19F linewidths may increase at higher decoupling powers due to B1 inhomogeneities creating different Bloch–Seigert shifts across the sample and hence an overall line-broadening (Fig. 12). It is also worth noting that in many systems, 19F linewidths are strongly determined by 19 F homonuclear coupling, and 1H decoupling is not a limiting factor for 19F resolution. Especially at higher magnetic fields, fast magic-angle spinning may render the complications of 1H decoupling in 19F NMR unnecessary. Effects due to 19F can also be observed in protondecoupled spectra of rare spins. It had been widely noted, for example, that the resolution of 13C sites close to a 19F nucleus was often poor [79,80]. This is illustrated in Fig. 19 for the 13C NMR spectra of the liquid crystal I35. The resonances from carbon sites close to the fluorine should be simple doublets due to 13C–19F coupling (dipolar plus scalar). Instead, the doublet components are strongly, and 1
asymmetrically, distorted (C1–3), or even completely unresolved (C4) in the CW-decoupled spectrum. (The same effects are present, but less strikingly, in simple 13C, 1 H systems, with the asymmetric broadening resulting in a peak maximum that changes with CW decoupler offset [16].) Changing the phase modulation of the decoupling sequence has a dramatic effect, and the SPINAL-64decoupled spectrum shows good resolution for all 13C sites. These broadening effects were traced to the presence of strong 19F–1H couplings (the dipolar coupling between an adjacent 19F and 1H on an aromatic ring, for instance, is dz5 kHz). The dipolar coupling modifies the effective offset (or ‘local field’) of the 1H by Gd. This cannot be removed by optimisation of the 1H transmitter frequency, and requires an appropriately robust decoupling sequence. These decoupling-dependent line-broadenings are most evident in liquid crystal and soft solid samples, but analagous effects are observed in MAS spectra of conventional solids. Other line-broadening mechanisms complicate, however, the task of obtaining well-resolved 13 C signals from sites close to 19F. The most significant of these follows from the relatively short lifetime of 19F coherences. A short 19F ‘T2’ causes the expected doublet for a 13C scalar coupled to 19F to broaden or coalesce, F
I35
9 CH2CH2
CH3CH2CH2
280 Hz
219
8
2 3
10 4
7 1 6
505 Hz
CH2CH2CH2CH2CH3
5
(a) 100 kHz
280 Hz
(b) 80 kHz 310 Hz
C9
55 kHz 795 Hz
30 kHz 15 kHz
1240 Hz −50
−55
−60
−65
−70 δ F / ppm
Fig. 18. CW 1H-decoupled 19F NMR spectra of a CF3-containing system as a function of 1H nutation rate, nrf (figures to left) at a Larmor frequency of 282 MHz for 19F (300 MHz for 1H) and a spin rate of 10 kHz. The chemical shift scale is indirectly referenced to CFCl3 via C6F6 at K116.4 ppm. Figures by the peaks are full widths at half-maximum. The dashed line shows the 19F spectrum in the absence of decoupling. Note that TPPM did not lead to further line narrowing under these conditions.
C8
C6
(c) C4
C7
C10
C2
C5
C3
C1
200
150
50
0
ppm
Fig. 19. 13C NMR spectra of the liquid crystal I35 using (a) CW, (b) TPPM and (c) SPINAL-64 decoupling, (nrfZ50 kHz). Adapted from Ref. [33], which contains further experimental details.
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degrading resolution [52]. Similar effects can be expected from other systems in which a third spin species has an appreciable coupling to the nucleus being decoupled. Simultaneous high-power decoupling of both 1H and 19F eliminate such line-broadenings. Although technically demanding, the use of such double decoupling to simplify rare-spin spectra has been reported for both organic materials [79,80] and organometallics, e.g. 119Sn NMR [81]. Note that 1H and 19F nutation rates are often similar (and both high) as no particular harm is done by HartmannHahn contact between the 1H and 19F spin baths. 6.2. Decoupling of other nuclei There are relatively few reports of decoupling of heteronuclei other than 1H and 19F in solids. This is presumably because homonuclear couplings between other nuclei are less significant, and decoupling of dilute spins is not generally necessary since the heteronuclear couplings may be well resolved and potentially informative. In addition, if the heteronuclear couplings are not required, they can be refocused by spin-echoes, which is not possible in the prescence of an extended coupling network (a simple spin echo will not refocus decoherence into multiple quantum states). 6.2.1. Spin-1/2 nuclei: 31P, 19F and 15N The much larger CSAs of nuclei other than 1H are expected to complicate any attempts at decoupling. In Ref. [51], for example, CW 31P decoupled 1H spectra show marked effects of the CSA/dipolar/RF interference effects discussed in Section 4.2.1. More effective decoupling methods, such as TPPM, reduce, but could not consistently eliminate, the line-splittings and broadenings. Continuous wave decoupling is also expected to be relatively inefficient for 19F decoupling, and TPPM decoupling has been shown to give substantial resolution enhancements over CW decoupling in materials such as chiolite, Na5Al3F14 [82]. In inverse-detection experiments, the rare-spin evolves (under heteronuclear decoupling) during an indirect dimension, and the NMR signal is acquired, after polarisation transfer, on the abundant high-g spin, typically 1H. By definition, the observed 1H spins will be coupled to the rare spin and hence broadened by 1 J XH . These broadenings can lead to a detectable sensitivity decrease, particularly if the 1 H linewidths have been reduced by 2H isotopic substitution. The larger chemical shift range of the X spins means that a broad-band decoupling sequence is required to decouple the X nuclei while the proton signal is acquired, justifying the use of 15N GARP decoupling in inverse-detection experiments on 2H-substituted microcrystalline protein samples [83]. 6.2.2. Half-integer spin quadrupolar nuclei: 27Al 27 Al decoupling (in combination with simultaneous 1 H decoupling) has been used to substantially improve
the resolution of 31P spectra of aluminophosphates, while 1 H decoupling in isolation had no observable effects on linewidths [84]. However, the interaction of RF irradiation, sample spinning and substantial quadrupole couplings can be extremely complex. For example, RF irradiation is often sufficient to cause recoupling of the dipolar interactions via the TRAPDOR effect [85], which is clearly counterproductive for decoupling. Experimentally, relatively low power 27Al decoupling (nutation rates of 10–15 kHz) was found to be optimal; further increases of the decoupling power led to steady increases in 31P linewidth. This effect was not reproduced in simulations of a simple 27Al, 31P spin system. 6.2.3. Integer spin nuclei: 2H Continuous wave decoupling is not very effective for decoupling 2H in static samples due to the large width of the 2 H spectrum (in turn due to the quadrupolar coupling). Irradiation of the double quantum coherence (which is unaffected to first order by the quadrupole coupling) allows 2 H to be decoupled [86,87], but uses RF power inefficiently. Composite-pulse based decoupling provides a more efficient decoupling strategy [72]. MAS significantly changes the problem since the quadrupole coupling is modulated by the spinning. Recently 2H decoupling has been used to improve the resolution of 13C spectra of biomolecules in which the protons have been diluted with 2H, by suppressing the 1JCD coupling (ca. 30 Hz) [88].
7. Conclusions With the exception of systems with high degrees of mobility, or particularly large susceptibility anisotropies, the linewidths of rare spins in organic solids are often limited by the quality of heteronuclear 1H decoupling. Conventional continuous wave decoupling becomes increasingly inefficient as the 1H spectrum is resolved, e.g. through faster magic angle spinning and/or higher static magnetic fields. However, significant progress has been made in recent years in finding practical solutions, such as TPPM, SPINAL-64, XiX, etc. that allow the resolution of 13 C and 15N spectra to be maintained at higher fields and spin rates. At the same time, it has become clear that the resolution and sensitivity in many solid-state NMR experiments is determined by the decay of the rare spin signal during a spin-echo due to coherent evolution. Although most effects of imperfect decoupling are refocused in a spin-echo, the time constant for the decay is strongly dependent on how efficiently the rare-spin is decoupled from the dephasing of the 1H magnetisation. Lengthening this ‘T20 ’ decay by optimising the 1H decoupling has a major impact on the viability of key experiments, such as natural abundance 13C double quantum experiments [89]. It also raises fundamental questions about the final limits of sensitivity in solid-state
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NMR; what if the true relaxation-limited linewidths were extremely small? Despite significant improvements in observed decoupling performance and our understanding of the factors behind these developments, a complete picture of 1H heteronuclear decoupling remains elusive. In many ways, heteronuclear decoupling is the ultimate challenge in ‘spin control’ for solid-state NMR; the suppression of interactions by at least three orders of magnitude, while accounting for the complex inter-relations between hetero- and homo-nuclear coupling networks, sample spinning, shift variations, etc. is a formidable intellectual challenge. Although it is unlikely that a single picture or decoupling sequence can reconcile such diverse factors, it is reasonable to suppose that the analytical and numerical tools developed to tackle this challenge will find wide application beyond the immediate problem of heteronuclear decoupling for solids.
Acknowledgements The support of authors in providing the original artwork for many of the figures reproduced in this article is gratefully acknowledged. In particular, I thank Matthias Ernst (ETH, Zu¨rich), Lyndon Emsley and Gae¨l De Pae¨pe (ENS, Lyon) for many fruitful discussions over heteronuclear decoupling of solids over the years. Thanks are also due to Dr D.C. Apperley (Durham) for his assistance in providing Figs. 1, 4, 10 and 18.
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Heteronuclear decoupling in the NMR of solids Paul Hodgkinson* Department of Chemistry, University of Durham, South Road, Durham DH1 3LE, UK Received 23 February 2005 Available online 18 July 2005
Contents 1. 2.
3.
4.
5.
6.
7.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Decoupling in solution-state NMR and the contrast to solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Average Hamiltonian theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. The ‘direct’ approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 H decoupling in experimental practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Continuous wave decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Two pulse phase modulation and related sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Other common sequences: SPINAL-64 and XiX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Understanding decoupling performance in solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Rare spin linewidths in solid-state NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1. Refocusable and non-refocusable linewidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2. Lineshapes under decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Factors determining decoupling performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1. Off-resonance behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2. Interaction with sample spinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3. Interaction with 1H dipolar couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4. Interference with exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5. RF inhomogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Characterisation and development of decoupling methods for solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Characterisation of decoupling performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Strategies for development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1. Symmetry-based methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2. Numerical simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3. ‘Direct optimisation’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diversions in decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. 1H decoupling in 19F-containing systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Decoupling of other nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1. Spin-1/2 nuclei: 31P, 19F and 15N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2. Half-integer spin quadrupolar nuclei: 27Al . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3. Integer spin nuclei: 2H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
197 199 199 200 203 203 204 205 206 206 207 208 209 209 210 212 214 214 214 214 216 216 218 218 218 218 220 220 220 220 220 221 221
1. Introduction * Tel.: C44 191 334 2019; fax: C44 191 384 4737. E-mail address: [email protected]
0079-6565/$ - see front matter q 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.pnmrs.2005.04.002
Effective decoupling of heteronuclear spin interactions is critical in many Nuclear Magnetic Resonance applications. This is particularly true for rare spin NMR of organic
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Nomenclature D
Heff
effective Hamiltonian over a given period (of the decoupling sequence) I, S decoupled and observed spins, respectively (i.e. decoupling RF is always applied to I spins) T2 time constant for spin–spin (T2) relaxation T20 time constant for decay of S spin magnetisation under spin-echo refocusing nr sample rotation frequency nrf nutation frequency of I spins subject to onresonance RF tc period of decoupling RF sequence tr period of sample rotation tp, Df pulse duration and phase difference respectively for TPPM and related sequences
offset of I spin Larmor frequency from (decoupler) transmitter frequency X figure of merit for decoupling band-width CM Continuous Modulation decoupling (Ref. [31]) CSA Chemical Shift Anisotropy CW Continuous Wave HORROR Homonuclear Rotary Resonance (Ref. [58]) MAS Magic Angle (sample) Spinning RF Radio Frequency (irradiation) TPPM Two Pulse Phase Modulation decoupling (Ref. [22]) SPINAL Small Phase Incremental Alternation decoupling (Ref. [27]) XiX x-inverse-x decoupling (Ref. [37])
molecules. As illustrated in Fig. 1, the resonances of 13C spectra of solid powders in the absence of decoupling, Fig. 1(a), are strongly broadened by (dipolar) coupling to surrounding 1H nuclei. The through-space nature of dipolar coupling means that even non-protonated carbons (such as the carbonyl resonances around 180 ppm) are poorly resolved. Decoupling of the protons has a dramatic effect on the spectral sensitivity and resolution, Fig. 1(b). Considerable progress has been made in heteronuclear decoupling in solution-state NMR since the first use of simple single-frequency (continuous wave) irradiation to decouple heteronuclear interactions [1]. ‘Broad-band decoupling sequences’ allow the complete 1H spectrum to be decoupled with a minimum of RF power (typically less
than a Watt). In contrast, considerably higher power (typically 10 s, even 100 s of Watts) continuous wave (CW) decoupling is widely applied in solid-state NMR. Over the past decade, however, it has become clear that CW decoupling is increasingly ineffective as static magnetic fields and sample spinning rates have increased, prompting renewed interest in the problem of 1H decoupling for dipolar-coupled systems. The current article sets out the background to these developments: outlining how the nature of decoupling differs in dipolar and J coupled systems, what techniques have been developed for high-performance decoupling, and the challenges that remain, both theoretical and experimental. The focus is on 1H decoupling in rare spin NMR, where
(a) x 10
180
160
140
120
100
80
100
80
60
40
20
0
60
40
20
0
(b)
* 180
160
140
120
ppm Fig. 1. 13C CP/MAS spectra (spin rate, nrZ5.1 kHz, 13C Larmor frequency, 75.40 MHz) of a glycine/alanine mixture obtained (a) without 1H decoupling, and (b) using continuous wave 1H decoupling (1H nutation rate, nrf, 60 kHz). The asterisk marks spinning sidebands from the carbonyl resonances.
P. Hodgkinson / Progress in Nuclear Magnetic Resonance Spectroscopy 46 (2005) 197–222
decoupling is of most direct benefit, although the decoupling of other nuclei is briefly surveyed in Section 6.2. Although superficially related, the issues involved homonuclear decoupling (for high resolution spectroscopy of abundant spin systems) are signficantly different and are not discussed here. Decoupling in liquid crystals provides some interesting points of comparison, but Ref. [2] contains a more complete and directed account of 1H decoupling in 13 C liquid crystal NMR.
2. Decoupling in solution-state NMR and the contrast to solids This section describes the theory and development of decoupling for solution-state NMR in some detail, to provide the necessary background for decoupling in the solid state. Complete descriptions and reviews of broadband decoupling for solution-state NMR can be found in the literature [3–5]. The spin Hamiltonian for a heteronuclear system can be written schematically: HðtÞ Z HS C HIS C HI C HII C Hrf;I ðtÞ
(1)
where the terms represent the spin Hamiltonians for the (observed) dilute S spins, the coupling between I and S, I spin chemical shifts, homonuclear I spin interactions, and RF irradiation applied to the I spins, respectively. The condition that jHrf,IjOHISCHI[HII is generally easily achieved in solution-state NMR with RF nutation rates of a few kHz (compared to 1 J CH w200 Hz, 2 J HH w20 Hz). This separation of scales allows us to consider the time-averaged Hamiltonian over the time-scale of the I irradiation. In hand-waving terms, the effect of continuous RF irradiation is to continuously invert I spin magnetisation, and the time average of heteronuclear coupling terms (which are of the form JIzSz at high field) will be hHIS iZJSzh Iz iZ0 (hHI i is also zero). Under these circumstances, the I and S spin systems are now ‘decoupled’: hH iZHSChHII i. Note that decoupling is the ‘separation’ of I and S spin systems; it does not imply that all terms involving the decoupled spins can be ignored [6]. For instance, heteronuclear multiple-quantum coherences, such as IxSx, will continue to evolve under the (averaged) Hamiltonian h HII i. This Hamiltonian is generally complex (homonuclear interactions, for example, will only be scaled, not eliminated by the I spin irradiation). Fortunately, such coherences are currently rarely encountered in solid-state NMR, and so it is normally reasonable to simplify the Hamiltonian under I spin decoupling to hH iZHS. Note that in contrast to other solution-state decoupling techniques, which generally ignore II interactions, the DIPSI decoupling sequences [7] are specifically designed to create a well-defined averaged Hamiltonian, hHII i.
199
The S spin Hamiltonian, HS, for dilute spins commutes with the other terms of the Hamiltonian, and so is not directly relevant to the decoupling problem (other than being the ultimate goal of effective decoupling). This separation does not strictly apply if the S spin system is not dilute. In this case [HSS, HIS]s0 and it is necessary to consider the entire spin Hamiltonian when considering decoupling. These effects are only likely to be significant for the already fraught case of 1Hdecoupling in 19F NMR, which is considered in Section 6.1. Since, we can generally neglect HS (isotropic shift of S) and HII (negligibly small homonuclear couplings) 1H decoupling in solution-state NMR can thus be described by the Hamiltonian HðtÞ Z JIz Sz C DIz C Hrf;I ðtÞ
(2)
where JIzSz is the J coupling between I and S, and D is the offset of the I spin resonance from the decoupler frequency. 2.1. Average Hamiltonian theory The classic approach to describing the evolution of a spin system under a time-dependent perturbation is Average Hamiltonian Theory (AHT) [8,9]. We separate the Hamiltonian of Eq. (2) into a time-independent term, H0 and a larger, time-dependent perturbation from the RF irradiation, Hrf(t). The propagator for the evolution of the density matrix from the time origin to time t, U(t), is then factored into two components: ðt UðtÞ Z T exp Ki H0 C Hrf ðt 0 Þdt 0 Z U1 ðtÞU~ 0 ðtÞ (3) 0
where ðt 0 0 U1 ðtÞ Z T exp Ki Hrf ðt Þdt
and
0
ðt U~ 0 ðtÞ Z T exp Ki H~ 0 ðt 0 Þdt 0
(4)
0
H~ 0 is the Hamiltonian transformed into an interaction (or ‘toggling’) frame defined by the time-dependent perturbation: H~ 0 ðtÞZ U1† ðtÞH0 U1 ðtÞ. We assume that the perturbation is periodic, i.e. Hrf (tCtc)ZHrf(t) where tc is the period. The spectrum, within a spectral width of 1/tc, is then defined by the propagator over one period, Uðtc ÞZ U1 ðtc ÞU~ 0 ðtc Þ. So called ‘cycling sidebands’ can arise from transitions outside this spectral width if tc is long (e.g. the decoupling field is very weak), but these will be negligibly small for a well-adjusted decoupling sequence. Sidebands from inadequate decoupling can be observed for liquid-crystalline samples, but the low quality of solid-state spectra under conditions of poor decoupling obscures these weak transitions for more typical dipolarcoupled systems. Despite this simplification, the evaluation of U(tc) is only reasonably straightforward if, in addition, the RF
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irradiation is also cyclic in the sense that the toggling frame returns to its starting point over the period of the perturbation, i.e. U1(tc)Z1. In this case, Uðtc ÞZ U~ 0 ðtc Þ. Finally, we try to describe U~ 0 ðtc Þ in terms of an ‘average Hamiltonian’ over the perturbation, H 0 , with U~ 0 ðtc ÞZ expðKiH 0 tc Þ. This Hamiltonian can be determined from the Magnus expansion ð1Þ H 0 Z H ð0Þ 0 C H 0 C/ whose first term is given by ð 1 tc ~ H ð0Þ H ðtÞ dt 0 Z 0 tc 0 0
(5)
(6)
Applying this analysis to decoupling with continuous (CW) RF irradiation is straightforward. We start with the Hamiltonian of Eq. (2) written in terms of the I spin sub-space: HGðtÞ Z ðDGJ=2ÞIz C nrf Ix
(7)
where HC is the Hamiltonian for the Sa sub-space (and HK corresponds to S in the b state). If we assume that DGJ/2/ nrf (i.e. the decoupling field is large), than the transformation to the toggling frame due to the RF irradiation (here applied along x) is simply described by Iz/Iz cos 2pnrftKIy sin 2pnrft. The average Hamiltonian over the cycle of the RF is simply ð tcZ1=nrf H ð0Þ ZðDGJ=2Þ Iz cos 2pnrf t KIy sin 2pnrf t dt Z0 0 0
(8) which vanishes, as expected. The drawbacks of the AHT treatment become apparent as we move away from this trivial case. If D is no longer small in comparison to nrf, i.e. we consider off-resonance irradiation, then the transformation into the interaction frame is nontrivial. Moreover, if the coupling itself cannot be neglected (a realistic situation when we consider decoupling in solids), pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2rf C ðDGJ=2Þ2 , then the effective nutation rate, neff rf Z now depends on the state of S. In addition, the normal requirement for cyclic RF irradiation is a major constraint; many of the most effective decoupling sequences are not cyclic and so cannot be readily analysed within this framework.
If necessary, an ‘effective Hamiltonian’ over the rotor cycle can be determined, HeffZKi ln U(tc)/tc. This is distinct from the interaction-frame ‘average Hamiltonian’ of Eq. (5). As the S spin state is a good quantum number, the propagator (or effective Hamiltonian) is block diagonal ! UC 0 Uðtc Þ Z (9) 0 UK where UG are the propagators over a period within the a and b sub-spaces of S. The S spin signal at multiples of the cycle time is given by Sðntc Þ Z trðSCUðtc Þn Sx Uðtc Þ†n Þ
(10)
† n Þ Z 1=2 tr½ðU_UC
(11)
Perfect decoupling would correspond to no evolution of the S spin signal (apart from the evolution due to HS, which † can be factored out, as explained above), i.e. UKUC Z 1 or, equivalently, UCZUK. This leads to the important conclusion that ideal decoupling requires that the propagators describing the evolution of the I spin system be identical for both the a and b states of S. In other words, if the I spin evolution is insensitive to the state of the S spin, then the I and S spins must be decoupled. Note that no assumptions are made about the nature of the I spin evolution. The RF is required to be periodic, but not necessarily cyclic; it does not matter where the evolution finishes, as long as both a and b states finish at the same point. We can describe the effect of an arbitrary propagator, U, on a single spin-1/2 as a rotation through an angle f about an axis n (specified by a unit vector). Expressing Eq. (11) in terms of the distinct rotations corresponding to the a and b states of S, (fC, nC) and (fK, nK), gives the signal [10] 1 fC K fK Sðntc Þ Z ð1 C nC$nKÞcos n 2 2 1 fC C fK C ð1 K nC$nKÞcos n ð12Þ 2 2 This cosine modulated signal corresponds to a sum of four frequencies: GUC and GUK with fC C fK f K fK and UK Z C 2tc 2tc
2.2. The ‘direct’ approach
UC Z
Although the Average Hamiltonian approach is extremely powerful when the goal is a well-defined ‘effective Hamiltonian’. It is less suited to problems, such as heteronuclear decoupling, in which the goal is a ‘zero’ Hamiltonian, since the performance will be determined by the behaviour of complex high order terms. Waugh introduced a simpler and more direct approach to describing decoupling [10], which is described below. As before, we neglect cycling sidebands and focus on the evolution over the cycle of the RF irradiation, U(tc).
with intensities given by the corresponding coefficients of Eq. (12). The UC terms can generally be neglected as they will have minimal intensity (nCznK), Fig. 2(a). The quality of the decoupling is determined by the frequency difference of the transitions at GUK (relative to the S spin Larmor frequency), in comparison to the original coupling, J:
2UK fC K fK
Z (14) l Z
J Jtc
(13)
P. Hodgkinson / Progress in Nuclear Magnetic Resonance Spectroscopy 46 (2005) 197–222
201
J (a)
λJ
−nνc
0
nνc
0.8
(b)
(c)
y n+
0.6
n−
λ 0.4 c n+
nc−
x
0.2
0 −1
−0.5
0
∆ / vrf
0.5
1
Fig. 2. Decoupling in a J-coupled CH system: (a) schematic decoupled spectrum showing scaling of the original doublet (dotted lines) to a doublet of separation lJ (where l should be sufficiently small that the residual splitting is unresolved), plus a pair of weak ‘forbidden’ transitions and, in the case of multi-pulse decoupling, potential ‘cycling sidebands’ centred at multiples of the cycle frequency of the sequence, nc. (b) Scaling factor, l, as a function of transmitter offset for continuous wave decoupling (dotted line) and an idealised decoupling sequence (solid line) which is broadband over a range DwGnrf/2. (c) Plots of the x and y components of the magnetisation vectors corresponding to the two spin states of the 13C over the course of a supercycled decoupling sequence (adapted from Ref. [90]).
In the limit of perfect decoupling, fKZfC and so lZ0, i.e. the transitions are coincident. A final simplification can be made if we take into account the relative magnitudes of the different interactions involved. Returning to Eq. (7), the net rotation angles, f will be functions of the effective offset DGJ/2 (in addition to parameters describing the RF). Hence the decoupling scaling factor is lðDÞ Z
fðD C J=2Þ K fðD K J=2Þ 1 vf Z Jtc tc vD
(15)
in the limit of J being much smaller than the range of D (A 10 ppm 1H chemical shift range at 500 MHz for example, corresponds to a D range of 5 kHz, which is considerably larger than the couplings involved.) This is invariably the case for 1H decoupling in solution, reducing the problem to one of finding sequences that give low values of l over an appropriate range of D values. This analysis for an S spin coupled to a single I can be readily extended to multiple I spins, although the ‘quality parameter’ is more difficult to define, especially if II interactions are significant [11]. Armed with this exact picture of decoupling, we can rationalise the behaviour of different decoupling sequences.
The scaling factor for continuous wave decoupling (familiar from off-resonance decoupling) is given by D ffi lCW Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2rf C D2
(16)
As can be seen in Fig. 2(c), this is only acceptably small (say less than 1%) for D/nrf. Excessive RF power would be needed to satisfy this criterion for even modest chemical shift ranges. The decoupling bandwidth may be quantified by the figure of merit [5] XZ
DF nrf
(17)
where nrf is the decoupling amplitude (expressed as a nutation frequency) and DF is the range of transmitter offsets over which the height of the decoupled signal (after applying defined processing) remains over 80% of its maximum value (i.e. on-resonance). CW decoupling, for example, has a X value of w0.0075, while effective decoupling schemes have X values around unity or greater, i.e. nrf is no longer required to be much larger than the range of 1H NMR frequencies. A tempting, but flawed, way used to increase the bandwidth of the the decoupling irradiation, is to introduce
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‘additional frequencies’. Noise decoupling, for example, involves randomly switching the phase of the irradiation [12]. Fourier transformation of the resulting time-domain signal would reveal a spread of frequencies around the RF carrier frequency, with a ‘band-width’ determined by the frequency of the phase modulation. Unfortunately, such Fourier arguments are only applicable if the response of the spin system to the RF is linear. While this is a reasonable approximation for small nutation angles, it is definitely not applicable to decoupling, and noise decoupling, while an improvement on straight CW irradiation, is not particularly efficient in practice. Noise decoupling was rapidly displaced by much more effective techniques based on composite pulses that could be derived using the exact decoupling theory. The classic design routes mostly involve ‘cycles’, that is RF irradiation that returns magnetisation to its starting point, since the requirement of forming a closed loop is relatively unrestrictive. The widely used WALTZ-16 sequence [13], for example, uses a basic inversion + composite pulse of RZ 90+x 180Kx 270+x (abbreviated in terms of multiples of 908 to 123, hence the choice of acronym). This cycle is moderately robust with respect to transmitter offset, and, in common with other sequences with only 1808 phase shifts, is insensitive to imperfections in phase setting. The combination of inversion pulses, RRR R creates a reasonably broadband decoupling sequence (WALTZ-4). As illustrated in Fig. 2(c), the decoupling performance of this basic sequence can be significantly improved by ‘supercycling’. The figure plots the x and y components of the magnetisation vectors corresponding to the two spin states of the 13C, nC and nK. At the end of the basic cycle, the magnetisation vectors are close to the origin, but there is a non-negligible difference between nC and nK, corresponding to a finite scaling factor, l. However, by permuting 908 pulse elements, the errors in the trajectories largely cancel over the course of the supercycle, i.e. nCwnK. In the specific case of WALTZ decoupling, the resulting four step supercycled sequence WALTZ-16 gives excellent broadband decoupling with a X value of 1.8. If even larger decoupling of bandwidths are required, e.g. for nuclei such as 31P or 19F which have a much larger spread of NMR frequencies than 1H, then specially optimised sequences such as GARP-1 [14] (XZ4.8) may be used. These trade a greater bandwidth factor, X, against somewhat higher scaling factors, l. An alternative strategy uses ‘adiabatic inversion’ pulses. These use a careful modulation of the RF amplitude and/or phase to ‘sweep’ magnetisation between z and Kz (and vice versa). If all the spins are perfectly inverted, perfect decoupling is achieved. The strength of adiabatic techniques is their built-in robustness with respect to many experimental imperfections, e.g. modest RF inhomogeneities are irrelevant in an amplitude modulated sequence. Sequences such as WURST [15] achieve extremely high bandwidths in proportion to the RF fields used, e.g. XZ16.7 (although
the X parameter is not entirely appropriate for adiabatic sequences since its value varies with nrf). Full treatments of decoupling sequences for solutionstate NMR can be found elsewhere [4,5], but it should be clear from the overview above that the theory of solutionstate decoupling is fully established and 1H decoupling can be regarded as essentially a solved problem. Although there is no ‘perfect’ decoupling sequence, the different trade-offs between bandwidth, residual splittings and artifact levels (e.g. from cycling sidebands), are well understood. In turning to decoupling in solids and liquid crystals, it is important to consider why decoupling in these phases is such a different problem. The most significant difference is the presence of strong homonuclear interactions, due to dipolar couplings that are not averaged out by molecular motion, unlike in the solution state. The dipolar coupling between the protons within a CH2 unit, for example, is of the order of 20 kHz, and the overall width of the proton spectrum in typical organic solids is approximately 100 kHz. As a result, much larger RF field strengths are required to manipulate the proton magnetisation. Although nutation rates of, say, 50 kHz, may be sufficient in undemanding applications, considerably higher nutation rates are desirable in many cases, e.g. at high magnetic fields. This puts considerable strain on the RF handling capabilities of probes, and both decoupling power levels and RF duty cycles must be chosen carefully to avoid overloading the probe (‘arcing’). Unlike solution-state NMR, where the 1H is often kept on throughout an experiment (generally boosting the rare-spin polarisation through the nuclear Overhauser effect), decoupling in solid samples is invariably ‘gated’, i.e. decoupling is only applied when strictly necessary, e.g. during acquisition. The strong homonuclear couplings also have a profound effect on the ‘spin dynamics’. In reducing Eqs. (1) to (2), we assumed that the I spin homonuclear interactions due to the J couplings could be neglected. This is clearly not the case with dipolar Hamiltonians, and treatments of decoupling in systems with significant dipolar couplings need to consider an InS spin system, where n is potentially infinite due to the presence of dipolar couplings between any pair of I spins. The many-spin nature of the problem remains a major obstacle to fundamental understanding of decoupling in dipolar-coupled systems. Less obviously, the role played by the transmitter offset is less well defined. When decoupling JCH, we could assume that this coupling was significantly smaller than the other terms involving the I spins. As a result, rather than considering the evolution under the effective offsets DCJ/2 and DKJ/2, it was sufficient to consider the behaviour around D, i.e. as a function of transmitter offset (Eq. (15)). Hence decoupling in solution-state NMR is generally tested by following the linewidth of a single 13C resonance as the proton transmitter is stepped through resonance. The effective bandwidth, DF, of the decoupling sequence is immediately obvious. Unfortunately,
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the heteronuclear dipolar coupling, d, cannot generally be neglected in comparison to D. Indeed, the interaction of offresonance decoupling and large couplings can result in differential line-broadenings and small apparent frequency shifts [16]. These effects are particularly striking in the presence of 19F, and are explored in Section 6.1. Although changing the transmitter offset does strongly influence decoupling performance (Fig. 10), the fact that performance is a function of the accummulated phases at both DCd/2 and DKd/2 makes the significance of the offset dependence much less clear cut. This is illustrated schematically in Fig. 3 for two functions fJ and fd which nominally correspond to decoupling sequences suitable for decoupling small (J) vs. large (d) interactions, respectively. Optimal functions for solution-state decoupling require that the gradient of the net rotation, f, be small, (Eq. (15)). However, f(D 00 Cd)Kf(D 00 Kd) may be large if d is comparable to the transmitter offset, resulting in poor decoupling. In contrast, the variation in fd is limited over a given ‘bandwidth’, so Df does not become unacceptably large, even if local gradients may be large compared to fJ. Hence an offset dependence which is suitable for decoupling J may not be appropriate for decoupling dipolar interactions, and vice versa. In short, we have moved from the case where jHrf,IjO HIOHIS[HII to a situation where all the interactions are of a similar order of magnitude, jHrf,IjOHICHIS[HII. To further complicate matters, the spin Hamiltonian is typically time dependent due to magic-angle spinning (MAS), which is necessary in solid-state NMR of powdered materials to remove the orientation dependent component of the chemical shift (the Chemical Shift Anisotropy, CSA). The resulting time-dependence greatly complicates theoretical analysis.
3. 1H decoupling in experimental practice 3.1. Continuous wave decoupling
203
on-resonance continuous wave irradiation is an effective decoupling strategy for typical organic solids. Fig. 4 shows how the linewidth (and so sensitivity) improves as the decoupler power, expressed as the approximate 1H nutation rate nrf, is increased. At these modest spinning speeds, the spectrum with very weak decoupling is worse than the spectrum obtained without decoupling. The linewidth then decreases rapidly with increased decoupling power until only modest increases in sensitivity are achieved as the RF amplitude is increased towards the limit for the probe. Performance degrades rapidly more than a few kHz away from the 1H resonance (Fig. 10), but optimisation of the transmitter offset is straightforward and is reasonably consistent between samples. CW decoupling under these conditions is effective because of, rather than in spite of, the width of the proton spectrum. In hand-waving terms, the multiple homonuclear dipolar interactions between the protons make the spin system ‘homogeneous’, i.e. continuous irradiation does not burn a hole in the proton spectrum, but rather the entire spin system is affected. Provided the RF field strength (expressed as a nutation rate) is at least comparable to the 1 H bandwidth, the protons respond collectively to the decoupling field. Interactions linear in I, including the IS couplings, will be scaled by P1(cos q)Zcos q, (where q is the nutation axis), while bilinear terms, i.e. the proton–proton couplings, are scaled by P2 ðcos qÞZ 1=2ð3 cos2 qK 1Þ. If the proton transmitter is approximately on-resonance (qZp/2), then the heteronuclear couplings are averaged to zero, while the homonuclear couplings are scaled by K1/2. The continuing presence of homonuclear couplings is actually a benefit. As discussed below, the continuing presence of homonuclear coupling is actually a benefit, and the decoupling performance is degraded if proton ‘spin-diffusion’ is suppressed.
At relatively modest magnetic fields, corresponding to I spin Larmor frequencies of, say, less than 400 MHz, simple φ φd ∆
φJ ∆" −d
∆"+ d
∆'− J / 2
∆' +J / 2
Fig. 3. Schematic illustration of the different role of transmitter offset, D, in sequences suitable for decoupling J(fJ) vs. dipolar (fd) interactions. While the gradient of fJ (dotted line) remains small, the differences in net rotation, f(D 00 Cd)Kf(D 00 Kd), may be large. In contrast, the maximum difference is limited for function fd.
0 kHz
28 kHz
36 kHz
48 kHz
60 kHz
Fig. 4. Effect of changing 1H decoupler power on the 13C CP/MAS spectrum of a glycine/alanine mixture (nrZ5.1 kHz, 13C Larmor frequency of 75.40 MHz).
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3.2. Two pulse phase modulation and related sequences At first thought, we would not expect the sample spinning rate to influence rare spin resolution. MAS is required to deal with the S spin chemical shift anisotropy, but the CSA behaves inhomogeneously under spinning [17] and so manifests itself as spinning sidebands of nominally zero width (in the absence of other line-broadening factors). As shown in Fig. 5(a), however, 13C linewidths under MAS increase significantly at spin rates above ca. 10 kHz. This effect can be straightforwardly explained by a weakening of the homonuclear coupling network by the magic-angle spinning [18,19]. The strong dipolar couplings between the protons play, in fact, a useful role in decoupling, leading to a partial ‘self-decoupling’ effect. Indeed, if the rate of mixing of the proton states were sufficiently fast (in comparison to the heteronuclear couplings), then decoupling would not be required at all. Faster MAS reduces the rate of this ‘spin diffusion’, weakening its helpful effects on decoupling. But because the network of homonuclear couplings behaves ‘homogeneously’ under MAS (in contrast to the CSA), modest MAS spin rates have little effect on the proton spectrum. As the spinning speed increases, however, resolved spinning sidebands are observed in the 1H spectrum. This co-incides with the degradation of decoupling performance. In principle, sufficiently fast MAS would average out the dipolar couplings, leaving only the relatively small J coupling to be removed by decoupling. However, the ‘undecoupled’ 13C linewidth only decreases relatively
(b)
(a)
weakly (Dnf1/nr) with spin rate [20] and fast magic-angle spinning on its own is unable to eliminate the effect of dipolar couplings, even for plastic crystals such as adamantane [21]. The degradation of rare spin linewidths with increasing nr becomes problematic as static magnetic field strengths, B0, increase. Since the CSA scales with B0, proportionately faster spin rates are required to prevent excessive complication by spinning sidebands or rotational resonance effects in isotopically enriched samples. In addition, 1H–X correlation spectra also become increasingly useful at high MAS rates due to the improvement in 1H resolution. Hence we require effective 1H decoupling at high spin rates. Minor degradations could be offset by increasing the power of the CW decoupling, but, as previously observed in solutionstate NMR, this is not an effective long-term strategy. A major step forward was made with the observation that simple modulations of the decoupler phase could dramatically improve the performance of the decoupling relative to simple CW irradiation [18,22]. In Two Pulse Phase Modulation (TPPM) [22], the repeating unit of the decoupling sequence consists of two pulses of the same nominal nutation angle, tp, and phase difference Df. Experimentally, optimal values typically lie around tpz1708 and Dfz158. The parameters are correlated and dependent on the sample and experimental conditions, such as spin rate (Fig. 14), although in practice, Df is often fixed at 158 and only tp (which would require calibration in any case) is optimised. The effects of TPPM decoupling can be seen in Fig. 5(c). Note that the increase in peak intensity
(d)
(c)
νr / kHz
k / Hz
νr / kHz
k / Hz
30
350
30
350
25
420
25
420
20
500
20
500
15
700
15
700
1100
10 1
0
−1
ν(13C) / kHz
1
1100
10 0
−1
ν(13C) / kHz
1
0
−1
ν(13C) / kHz
1
0
−1
ν(13C) / kHz
Fig. 5. 13C spectra of polycrystalline 2-13C-alanine as a function of sample spin rate, nr, using (a) CW and (c) TPPM 1H decoupling with nrfZ100 kHz (1H Larmor frequency of 600 MHz, DfZ158). (b) and (d) are corresponding spectra simulated using experimental values of 1H spin diffusion rate, k (Section 4.2.3 for details of the model used). Figure adapted from Ref. [19].
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with MAS rate seen here is not always observed [36], and this probably reflects the sensitivity of TPPM performance to experimental parameters (Fig. 14). Following the success of TPPM, a number of variants have been proposed. A combined phase and frequency modulated decoupling, FMPM [23], provides some insight into the mechanism of decoupling using TPPM, while AMTPPM [24] added an amplitude modulation. SPARC [25] essentially uses the ‘super-cycling’ established from solution-state composite-pulse decoupling with TPPM as a basic element. Although SPARC-16 performed better than TPPM for some [25] (but not all [26]) liquid-crystalline systems, the application of super-cycling without a welldefined basic element may lead to worse rather than better performance [11]. Its use in solid-state decoupling has not been reported, and in any case, SPINAL-64 (discussed below) has proved a more effective solution for liquidcrystal and solid-state decoupling. Relatively slow phase changes (1–2 ms) have also been used to improve decoupling performance in liquid crystals [27] (phase steps of p/2) and solids in which the 1H coupling network is not particularly extensive [18] (phase step of p). These schemes clearly ‘refocus’ some component of the rare-spin linewidth, although the similarities with schemes such as TPPM (much faster phase switching) and XiX (strongly synchronised to spinnning rate) are only partial, and they have not been widely applied. Rather than switch the RF phase between discrete values, other TPPM variants use a continuous modulation of the phase, e.g. f(t)Za cos(2pnct) where 2a is the total range of the modulation (equivalent to Df in TPPM) and nc is the frequency of modulation (equivalent to the frequency of TPPM modulation, 1/2tp). With echoes of noise decoupling, cosine phase modulation, CPM [28] and nPPM [29], use more than one modulation frequency and ‘supercycling’ to try to increase the ‘band-width’ of decoupling. Recently, single-frequency Continuous Modulation, CM [30], has been analysed in some detail (see Fig. 17 and associated text for further discussion). Modifications to the CM approach have included imposing an overall Gaussian envelope to one or more periods of the CM modulation with the goal of improving the robustness of the decoupling [31]. 3.3. Other common sequences: SPINAL-64 and XiX In the absence of a full theory of heteronuclear decoupling in the presence of strong homonuclear interactions, the rationalisations behind these various sequences are frequently questionable, and their practical performances have yet to show significant and consistent improvements over TPPM. It is unlikely, however, that a sequence as simple as TPPM is the last word in such a non-trivial problem. Several other sequences have been proposed and are discussed in Section 5.2, but a couple of sequences, developed in two different extremes have found successful applications beyond model compounds, and merit more attention.
205
The SPINAL sequences [26], and most notably SPINAL64, were originally developed for decoupling in liquidcrystal 13C NMR. The basic element of the SPINAL sequences is Q Z Pð108ÞPð108ÞPð158ÞPð158ÞPð208ÞPð208ÞPð158ÞPð158Þ (18) where P(m) denotes a pulse with nutation angle of w1658 and of phase m, and a bar indicates that the output phases are all negated, e.g. Pð108Þ corresponds to an output phase of K108. This element is then ‘super-cycled’ to generate a family of sequences, such as SPINAL-64 QQQ SPINAL-64 : QQ QQ Q
(19)
although, as remarked earlier, the significance of ‘super cycling’ in this context is unclear, especially as the basic element is acyclic (magnetisation does not return to its starting point after applying Q). As for TPPM, tp needs to be optimised for best performance, but good performance can be obtained without modifying the phases used, making the experimental application of SPINAL straightforward. Although TPPM was used as the starting point, the very different characteristics of SPINAL and TPPM suggest they should be considered as distinct sequences. Given its origins, it is unsurprising that SPINAL-64 is effective in decoupling static liquid-crystal [32] and ‘soft solid’ samples at low spinning rates [33]. Its better performance relative to TPPM under the very different conditions of high speed MAS of rigid solids is surprising, but frequently observed [34,35]. Although the long period might be expected to make it sensitive to interference with spinning (Section 4.2.2)—even the base element, Q, is four times longer than TPPM—its performance is almost independent of spinning speed [36]. The other sequence of note is XiX decoupling [37]. Like TPPM, this consists of a pair of pulses of alternating phase. In this case, however, the phase difference is fixed at 1808 (e.g. x and Kx, hence its name, x-inverse-x), and the duration tp is defined with respect to the rotation period and is not a nutation angle. Indeed, a better comparison may be the early use of phase alternation to improve decoupling performance in solution-state NMR [38]. XiX is discussed further in Section 4.2.2, but, in summary, the performance of XiX is strongly dependent on the ratio tp/tr (where tr is the rotor period). Optimisation about initial values of tp/ trz1.85 is generally sufficient in experimental practice to find a satisfactory optimum. Since the XiX period lengthens as the spinning speed is reduced, it is not surprising that its performance tends towards that of CW decoupling in the limit of slow spinning (Fig. 13); TPPM and SPINAL are invariably better choices. The performance of XiX steadily improves with spinning speed, however, and generally compares favourably with TPPM at spinning rates above w30 kHz.
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4. Understanding decoupling performance in solids 4.1. Rare spin linewidths in solid-state NMR The goal of decoupling is to narrow rare spin linewidths and so to improve both sensitivity and resolution. Before examining the characteristics of decoupling in dipolar-coupled solids, it is useful to review the factors that determine rare-spin linewidths and their relation to decoupling. Rare spin linewidths have been discussed extensively in the literature [39–42] from the viewpoint of individual contributions to linewidth, which can be grouped into three independent components to linewidths. Inhomogeneous linewidth of an individual resonance results from variations of the fundamental resonance frequency. These can arise from the nature of the sample itself, e.g. amorphous or poorly crystalline samples, or reflect variations of the local magnetic field. Good sample preparation is often critical in biomolecular NMR since larger molecules are often difficult to obtain as high quality crystalline powders. As shown in Fig. 6 for different samples of the decapeptide antamanide, the inhomogeneous linewidth (and hence the spectral resolution) can be highly sensitive to the sample preparation. Clearly, optimising decoupling will have little measurable impact on the quality of spectra from samples (a) and (b). Local magnetic field variations most obviously arise from inadequate shimming. Although the shimming characteristics of a normal MAS probe are poor in comparison to typical solution-state probes, shimming is only a limiting factor for materials with very narrow intrinsic linewidths, e.g. plastic crystals, such as adamantane. Variations in the bulk susceptibility are a more subtle source of inhomogeneous broadening. The presence of interfaces, such as grain boundaries, voids, etc. modify the bulk susceptibility,
c, and hence the net magnetic field, leading to linebroadening. If the susceptibilities of the bulk materials are isotropic, then these variations will be eliminated by magic angle spinning. However, MAS is only partially successful if the c of the sample is anisotropic [39,41]. The effects of anisotropy of the bulk magnetic susceptibility (ABMS), Dc, are reduced, but not eliminated, by magic-angle spinning, and is often the limiting factor in rare spin linewidths; contrast the broad lines observed for samples high in ordered aromatic rings (and so high ABMS), such as hexamethylbenzene [39], with the sharp lines commonly observed in samples with very small values of Dc, such as typical steroids. The division into ‘inhomogeneous’ and ‘homogeneous’ contributions to linewidths is unproblematic in solutionstate NMR. For solids, however, it is important to divide the homogeneous linewidth into ‘incoherent’ contributions resulting from random processes, and so which cannot be reversed, and ‘coherent’ contributions from the nuclear spin Hamiltonian and which can, in principle, be manipulated, and even reversed. Linewidth due to relaxation and exchange process clearly fall into the former category, and can be termed ‘incoherent homogeneous’ linewidth. The relaxation linewidth, 1/pT2, represents the ultimate limit on resolution. It is relatively straightforward to describe relaxation in solution-state NMR; we can consider an individual spin, or small set of spins, loosely coupled to its surroundings, and the decay of xy magnetisation can be described in terms of fluctuating random fields, resulting in a well-defined time constant for decay, T2. Unfortunately, little is known about T2 relaxation in typical molecular solids, either experimentally or theoretically. Extrapolation of classical relaxation theory to the limit of long correlation times (slow motions) is highly suspect,
13C
15N
(a)
(b)
(c)
70
60
50
40
30
20
10
140
130
120
110
100 ppm
Fig. 6. 1H-decoupled 13C and 15N spectra (1H Larmor frequency of 60 MHz) of three samples of the cyclic decapeptide antamanide prepared (a) as a lyophilized powder, (b) by fast solvent evaporation and (c) by slow evaporation (over several days). Figure adapted from Ref. [91]; see reference for further details.
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between ‘coherent’ and ‘incoherent’ contributions is unclear. The question is of great importance, however, as it is only the ‘incoherent’ linewidth that sets the fundamental limit on resolution in solids.
since ‘random field’ models assume that fluctuations of the Hamiltonian occur on a timescale that is fast compared to the evolution of the NMR signal. This model cannot be applied to solids where spins are generally strongly coupled together. The decrease in xy magnetisation is not primarily due to decoherence due to random external fluctuations, but is essentially a ‘dephasing’ process driven by the spin Hamiltonian. Initial xy magnetisation spreads into multiple quantum coherences between large number of spins and the system quickly reaches a ‘quasi-equilibrium’ state whose density matrix, seq, commutes with the system Hamiltonian, Hsys [43,44]. This quasi-equilibrium state slowly evolves to thermal equilibrium via T1 relaxation. Such states have been strikingly demonstrated in cross-polarisation experiments under MAS [45]. Although from the viewpoint of an individual spin, such dipolar evolution can be satisfactorily modelled as a random process, the initial evolution is purely coherent and can be reversed by appropriate manipulation of the Hamiltonian, e.g. in the context of cross-polarisation [46]. Since the evolution of coupled 1H spins is a coherent process, the ‘T2’ values used to characterise the 1H signal decay are purely phenomenological. Similarly, in the absence of perfect 1H decoupling, we also expect the 13C linewidth to be partly coherent in origin. This is confirmed by the often strong dependence of the 13C linewidth on factors, such as MAS rotation rate or 1H decoupling, that clearly modulate Hsys rather than relaxation processes (see below). Given the difficulty of modelling extended spin networks, and of distinguishing experimentally between contributions to homogeneous linewidth, the balance
4.1.1. Refocusable and non-refocusable linewidth A simple spin-echo is sufficient in solution-state NMR to distinguish inhomogeneous and homogeneous contributions to linewidth; the former are refocused, while the latter are not. The decay of xy magnetisation during a spin-echo train thus provides the time constant for homogeneous decay, T2. The situation is less clear in solids, as we cannot be sure how the coherent evolution responds to spin-echoes. If we can neglect homonuclear couplings between the I spins, then the ‘effective Hamiltonian’ during the decoupling will contain only heteronuclear terms. These will be refocused by a spin echo on S, i.e. the only decay will be due to relaxation of S spin magnetisation. Under these circumstances (which can be observed even in solids for isolated spin pairs), it is more efficient to refocus the dipolar evolution with a spin echo rather than resort to decoupling. If, however, the I spins are strongly interacting, then a spin-echo on the S spins is insufficient to reverse the evolution of the coherences, and any refocussing is likely to be partial. This can be seen in Fig. 7(b), which shows the intensity of the 13C signal after a tKpKt spin-echo, with different decoupling schemes applied during the spin-echo. As expected, the decays are much slower than would be calculated directly from the natural linewidth (for example, the linewidth of 46.4 Hz obtained for optimised CM would 21.37
(a)
21.34
19.72 TPPM CW
9.44
207
SPINAL64
TDOP-CM
48.2 Hz
46.4 Hz
51.3 Hz
81.2 Hz
(b) 1.0 0.8
T2' =47.0 ms
a.u.
0.6 0.4
T2' =28.3 ms
T2' =24.6 ms
SPINAL64 TPPM CW
0.2 0.0
T2' =6.5 ms 10
TDOP-CM
20
30
40
2τ / ms
Fig. 7. (a) Ca resonance of polycrystalline 2-13C-glycine for different decoupling sequences (nrfZ80 kHz, nrZ10 kHz, nominal 1H Larmor frequency of 500 MHz). The figures give the linewidths and peaks heights (arbitrary units). (b) Intensity of 13C signal after a tKpKt spin-echo as a function of delay period, using the different decoupling schemes during 2t. Adapted from Ref. [92].
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correspond to a ‘T2’ of 6.9 ms). However, the strong dependence of the decay of the spin echo on decoupling sequence indicates that it cannot be identified with true 13C T2 relaxation. In addition to true relaxation, the decay must also reflect coherent evolution (due indirectly to the 1H coupling network) that is not refocused by the spin echo. The resulting time constants have been denoted T20 to distinguish them from true relaxation time constants [92], and it is important to note that large variations in T20 values can be observed for decoupling sequences that perform identically (as quantified by peak intensity or line width), e.g. SPINAL vs. TDOP-CM. The latter sequence (Transverse Dephasing Optimised CM) refers to CM-decoupling specifically optimised for T20 , i.e. the parameters of the CM sequence have been optimised in order to maximise the spin echo signal after a fixed period. Although such optimisation does not improve the onedimensional spectrum, the long T20 is of great significance for experiments involving spin echoes, since the resolution (and sensitivity) of such experiments will be determined by the achievable T20 and not by the linewidth in the onedimensional spectrum. This is particularly significant for amorphous and disordered samples, as illustrated in Fig. 8 for
13
C-enriched cellulose. The linewidths in this sample are dominated by inhomogeneous contributions and there is little resolution improvement in the simple CP/MAS spectrum when an optimised decoupling is applied, Fig. 8(b). In contrast, the refocused INADEQUATE spectrum involves relatively long spin-echo periods during which doublequantum coherence is created and reconverted through J-couplings. Hence, the overall sensitivity of the experiment is strongly dependent on T20 , as can be seen in the intensity of the traces (c) and (d). The TDOP-CM decoupled spectrum provides detailed information on the connectivity between sites and correlations between inhomogeneous broadenings on connected sites (via the peak shapes). 4.1.2. Lineshapes under decoupling It is important to note that line shape, as well as its width, can be strongly dependent on the decoupling method (provided the linewidth is not dominated by inhomogeneous contributions). This is illustrated in Fig. 9 for the Ca resonance of alanine. Note in particular that the TPPM and XiX full-widths-at-half-maximum are very similar (33 and 31 Hz, respectively), even though the height of the XiX-decoupled signal is significantly greater. The
CW decoupling
TDOP-CM decoupling (b)
(a)
carbon chemical shift
carbon chemical shift
(d)
(c)
double quantum frequency / ppm
x3
x3
(f)
(e) 140
140
150
150
160
160
170
170
180
180
190
190 120
110
100 90 80 70 60 single quantum frequency / ppm
110
100 90 80 70 60 single quantum frequency / ppm
Fig. 8. 13C CP/MAS spectra of 10% 13C-cellulose under (a) CW decoupling and (b) TDOP-CM decoupling (nrfZ100 kHz, nrZ12.5 kHz, nominal 1H Larmor frequency of 500 MHz), (e) and (f) show corresponding refocused INADEQUATE spectra, while (c) and (d) are traces through the 2D spectra at a double quantum frequency of 137 ppm. Adapted from Ref. [92]; see reference for further details.
P. Hodgkinson / Progress in Nuclear Magnetic Resonance Spectroscopy 46 (2005) 197–222
1.5
209
(a) CW
TPPM
XiX
peak height
1
0.5
0
x10 (b) 1 kHz Fig. 9. Ca resonance of polycrystalline 13C-labelled alanine under different (optimised) decoupling schemes, using nrfZ150 kHz and a spin rate of 30.03 kHz (nominal 1H Larmor frequency of 600 MHz). Adapted from Ref. [37].
baseline expansions show that the TPPM-decoupled signal has a significantly wider foot, indicating that the difference in peak heights is largely a consequence of the much tighter lineshape for the XiX-decoupled signal. 4.2. Factors determining decoupling performance The difficulty of establishing a simple model for decoupling performance in solids has already been alluded to. This section attempts to unravel some of the factors that determine decoupling performance, focussing on CH and CH2 groups, since quaternary and CH3 carbons are relatively ‘easy’ to decouple. For instance, if we consider samples that are not subject to significant inhomogeneous broadenings, typical linewidths for CH, CH2 and CH3 under optimised decoupling are w20 Hz for CH3, w35 Hz for CH and w50 Hz for CH2 (due to the presence of strong homonuclear coupling). The behaviour of methyl groups is distinct from the other protonated carbons [36], which is presumably a consequence of the fast internal rotation in CH3, which scales the 13CK1H couplings by a factor of K1/3 and homonuclear 1H couplings by K1/2. 4.2.1. Off-resonance behaviour As discussed in Section 2.2, the performance of a decoupling sequence for solution-state NMR can be simply characterised in terms of its off-resonance behaviour. Observing the height of the NMR signal as a function of transmitter offset provides a direct indication of the range, DF, over which the decoupling is effective. Similar considerations apply to solid-state decoupling. As illustrated in Fig. 10 for sample used in Figs. 1 and 4, the decoupling performance degrades as the 1H transmitter frequency is moved away from the centre of the 1H spectrum. Note the very different responses of the CH and
−6
0 transmitter offset / kHz
6
Fig. 10. CW 1H-decoupled 13C NMR spectra of (a) the CH2 resonance of glycine and the (b) CH resonance of alanine acquired under the same conditions (nrZ5.1 kHz, nrfZ60 kHz, 1H Larmor frequency of 300 MHz) as a function of 1H transmitter offset (steps of 600 Hz).
CH2 resonances; as expected, the CH2 peak is much more sensitive to offset variations, reflecting the strong dipolar coupling between the protons. Although the optimum offset is essentially the same for different 13C sites under these conditions, it is clearly much more difficult to define the ‘decoupling bandwidth’ compared to solution-state decoupling. In mobile systems where the 1H sites are resolved, such as liquid crystals, however, it may be difficult to find a common offset that efficiently decouples all sites (e.g. aromatic vs. alkyl carbons). Note that a somewhat different behaviour is often observed in static samples. Rather than increasing monotonically with increasing offset, the linewidth has a clear maximum when the tilt axis of offresonance irradiation is at the magic angle [47]. This provides clear evidence of the ‘self-decoupling’ effect of fast spin-diffusion, which is suppressed by magic-angle (or Lee–Goldburg [48]) decoupling. In spinning samples, the resonance offset can be timedependent due to the presence of 1H CSA, resulting in a broadening effect distinct from that of an overall resonance offset [49,50]. The interaction between the time-dependent CSA, dipolar coupling and RF is rather subtle, and the derivation is only presented in outline. Following previous work [49], the spin Hamiltonian for a simple IS system under CW decoupling is given by:
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H Z uIS ðUIS Þ$2Iz Sz C uI ðUI Þ C urf Ix
(20)
where uIS and uI are the IS dipolar coupling and I spin shift, respectively (u indicating angular frequency). These interactions are dependent on orientation through the sets of Euler angles UIS and UI; urf is the RF nutation frequency. If we assume that urf is significantly larger than the other interactions, perturbation theory can be used to second order to derive the frequencies of the two strong transitions corresponding to the S spin resonance (Fig. 2): uG zG
uIS ðUIS ÞuI ðUI Þ urf
(a)
2
1
0
−1
−2
1
0
−1
−2
0
−1
−2
(b)
(21)
If 2pnr/urf, then the averaging due to spinning and RF irradiation can be treated independently and the centre-band frequencies for spinning samples are determined by the time average over a rotor cycle: uIS ðUIS ÞuI ðUI Þ huGi zG (22) urf This non-zero splitting corresponds to a cross-term between the heteronuclear coupling and the shielding of the I spin. Since the associated tensors are both of rank 2, this cross-term contains components of rank 0, 2 and 4. Under normal decoupling conditions, the resulting isotropic term is negligibly small and the rank 2 term is eliminated, like other rank 2 interactions, by magic-angle spinning. However, the rank 4 terms survive the averaging. At finite values of urf, this term separates the otherwise degenerate transitions, and the S spin spectrum from an individual crystallite appears as a doublet. The size of the splitting depends on the crystallite orientation (as well as the relative orientations of the tensor interactions) and so the spectrum of a powdered sample appears as a broad double-peaked lineshape. These effects can be clearly seen in the spectra of isolated spin pairs, as illustrated in Fig. 11 for a 15NK1H pair. The spectrum in the absence of decoupling, (c), is a simple doublet due to the 1 J NH coupling, but the CW-decoupled spectra, (a) and (b), show broad lineshapes due to the interaction between the 1H CSA, 15NK1H dipolar coupling and decoupling. As expected, these effects increase at higher magnetic fields, (b), due to the proportionately larger CSA. This interaction between the CSA and dipolar interaction can be ‘refocused’ by modifying the decoupling RF. For instance, the sequence (2p)x(2p)Kx. completely suppresses this line-broadening, but is an ineffective decoupling sequence otherwise. Similarly, it can be shown from average Hamiltonian theory that TPPM with a nutation angle of 1808 strongly reduces, without completely refocusing, this broadening mechanism. Note that this broadening is refocused in spin-echo experiments and does not contribute (to first order) to the T20 decays discussed in Section 4.1.1.
2
(c)
2
1
ν( N) / kHz 15
Fig. 11. 1H CW decoupled 15N NMR spectra of polycrystalline d9trimethyl-15N-ammonium chloride at proton resonance frequencies of (a) 600 MHz and (b) 300 MHz at an MAS spinning rate of 30 kHz and decoupling field strength, nrf, of 100 kHz. (c) Is the corresponding spectrum at 600 MHz acquired without decoupling (the splitting is due to the 1JNH coupling of ca. 100 Hz). Figure adapted from Ref. [49].
These effects have been observed experimentally on model systems [49] and in cases where the effective CSAs are particularly large, e.g. when decoupling 31P rather than 1 H [51], or when augmented by strong couplings to a third spin species [33,52] (The latter case is discussed in more detail in Section 6.1.) The importance of these effects in more typical organic systems is less clear. The difficulty of decoupling tends to correlate with the dipolar environment, e.g. CH2 vs. CH rather than paralleling trends in 1H CSAs, for example. It is also notable that in the systems where this mechanism is clearly at work, different decoupling conditions are optimal for sites strongly perturbed by large effective CSAs (e.g. sites close to 19F) and unperturbed sites. In the former case, the optimal TPPM/SPINAL angles are z1808, as predicted from simple two-spin models, while the optimal angles for TPPM/SPINAL for ‘normal’ sites is consistently, and significantly, less than 1808 [52]. Although good decoupling sequences obviously need to cope with the effects of 1H CSA, there are clearly other factors at work. 4.2.2. Interaction with sample spinning We have assumed so far that the nutation rate associated with the decoupling irradiation, nrf, significantly exceeds the
P. Hodgkinson / Progress in Nuclear Magnetic Resonance Spectroscopy 46 (2005) 197–222
spinning rate, nr. In this ‘quasi-static’ limit, the system Hamiltonian is assumed to be approximately constant over the decoupling cycle, and so we can sequentially determine the effects of the decoupling and sample spinning. This separation of time-scales is more difficult to maintain as the spin rate is increased. In particular, ‘rotary resonance’ conditions can occur at certain ratios of nrf and nr. The rotating frame Hamiltonian for a spinning sample under CW decoupling can be written as a Fourier series [53,54]: HðtÞ Z nrf Ix C
2 X
Hm e2pimnr t
(23)
mZK2
44 kHz
The index runs from K2 to 2 since the interactions (chemical shift and dipolar) are rank 2 tensors, and it is unsurprising that interference effects are observed when pZnrf/nr is equal to 1 or 2. These ‘rotary resonance’ conditions [55] have the effect of ‘recoupling’ the heteronuclear dipolar interaction, which is then no longer refocused over the rotor cycle, greatly reducing the decoupling efficiency [56]. These effects can be clearly seen in Fig. 12; the width of the 19F resonance is strongly broadened at nrfZnr and 2nr, with weaker resonances observed at pZ3 and 4. Note that the observed guest molecules are highly mobile in these systems, and such resonances are rarely so clearly observed in more conventional solids. Especially in more strongly coupled systems, a recoupling phenomenon may also be observed at pZ0.5.
11 kHz
22 kHz
33 kHz
100 kHz
νrf −210
−220
−230
δF / ppm Fig. 12. CW 1H-decoupled 19F spectra of the inclusion compound formed between 1-fluorotetradecane and urea as a function of 1H nutation rate. Strong interferences are observed when the (approximate) nrf matches a multiple of the spin rate, nrZ12 kHz. The increase in linewidth at the highest decoupling fields is the consequence of significant ‘Bloch–Seigert’ shifts at this field (1H resonant at 200.13 MHz), cf. Section 6.1. Figure adapted from Ref. [93].
211
This so-called HORROR condition [57] recouples the homonuclear dipolar interaction and has a less direct effect on the decoupling efficiency. These resonant effects should not be confused with the general trends in decoupling performance as a function of spinning speed discussed in Section 3.2, which result from the weakening of the 1H dipolar network due to MAS. Resonance conditions can also be expected for more complex irradiation patterns than CW. Moreover, as there is an additional frequency associated with the cycle rate, nc, we may expect more complex interactions. This is certainly the case for homonuclear decoupling, where rational values of nc/nr are associated with major drops in performance [9]. The analogy is not perfect, however. Homonuclear decoupling sequences are designed to be cyclic, and have a well-defined effective Hamiltonian. In contrast, solid-state decoupling sequences (with the obvious exception of CW decoupling) are generally acyclic, and the effective Hamiltonians are much less well defined (Even for TPPM, for example, the effective Hamiltonian is barely tractable for all but special cases.) This may explain why the interaction between RF cycle rate and sample spin rate varies so widely. For example, the interaction is extremely strong for XiX, as seen in Fig. 13. Indeed, effective XiX decoupling involves finding values of nc that avoid these strong resonance conditions. On the other hand, the performance of SPINAL-64 barely varies with spinning speed, even though its very long cycle length appears to make it vulnerable to resonances between nr and nc [36]. While most effort has been made to find high power decoupling sequences that continue to function effectively at high spinning speeds, there have also been investigations of decoupling with low RF powers, i.e. nrf/nr [20]. Here, magic angle spinning is used as the primary means of averaging the dipolar interactions, assisted by relatively weak RF irradiation. But as maximum sample rotation rates fall well short of maximum RF nutation rates (50–70 kHz and 150–200 kHz, respectively, at time of writing), it is unsurprising that low power CW-decoupling is consistently less effective than conventional high-power decoupling. This approach may be valuable, however, for experiments that involve simultaneous 1H decoupling and RF irradiation on an X nucleus (e.g. for recoupling). Effective recoupling often requires X nutation rates that are multiples of nr (e.g. 7nr in the case of C7 and derivatives [58]), while 1H decoupling fields generally need to be a factor of three higher still [59] to avoid perturbing effects associated with matched X and 1H nutation rates. Such large nutation rates require large RF fields (especially for lossy samples), which is risky for both the probe and for temperature-sensitive samples. It would, therefore, be extremely useful to find sequences that are effective at nutation rates much lower than, or even comparable to, the spinning rate. The potential advantages of low-power, or even no, decoupling in X nucleus recoupling sequences at higher spin rates has recently been demonstrated experimentally [60].
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τp /τr
50
100
150
7 5 6
4 3
5
peak height
2 4 1 0
3
0
0.1
0.2
τp / ms
0.3
0.4
0.5
2
1 CW 0 0
1
2
τp / ms
3
4
5
Fig. 13. Performance of XiX decoupling as a function of XiX pulse width, tp. The plots show the peak height of the Ca resonance of alanine relative to CW decoupling at the same nrf (150 kHz), at an MAS frequency of 30.03 kHz and nominal 1H Larmor frequency of 600 MHz. The inset is an expansion for pulse widths up to 500 ms (15 rotor periods, tr). Points within the grey circles correspond to optimal values of tp/tr. Figure adapted from Ref. [37].
4.2.3. Interaction with 1H dipolar couplings The weakness of scalar couplings relative to the RF nutation rate, nrf, and chemical shift offsets means that homonuclear 1H, 1H interactions can be neglected in theories of decoupling for solution-state NMR, allowing the problem to be expressed in terms of a single I spin. This is clearly not appropriate for decoupling in typical proton-containing solids. Some progress can be made analytically for static samples. For instance, models of the coupled proton system using memory function approaches predict an inverse dependence of the S spin linewidth with decoupling power, which is confirmed experimentally in static solids with a continuous ‘network’ of proton–proton couplings [61,62]. In contrast, exact modelling of InS systems predicts that the S spin linewidth remains constant for on-resonance irradiation, but that low power decoupling in multi-spin systems creates significant cycling sidebands [63]. Increasing the decoupler power reduces the intensity of these signals (which are typically broad and unresolved), transferring their increasing intensity to the centreband without significantly modifying its width. The behaviour of typical solids under MAS falls between these limits [36,63]. Note that these ‘on-resonance’ effects are quite distinct from the normal ‘off-resonance’ behaviour illustrated in Fig. 2, which involves the broadening of the centreband into resolved transitions as the transmitter frequency is moved away from exact resonance. The 13 C1 H2 spin system is an obvious starting point for exact simulations of decoupling which attempt to include
homonuclear coupling. Unfortunately, numerical simulations of a CH2 system under MAS do not reproduce the homogeneous character of the 1H coupling network. A simple H2 dipolar-coupled pair, for instance, behaves ‘inhomogeneously’, with the dipolar interaction refocusing under spinning, producing sharp spinning sidebands [17]. Similarly, simulations of an isolated CH2 system under MAS are unrepresentative of the behaviour of methylene groups in typical organic solids. In principle, these problems can be overcome by adding a sufficiently large number of neighbouring protons. However, exact analysis rapidly becomes intractable, especially for spinning samples, as the number of spins increases. Even if we restrict our simulation to, say, 10 dipolar coupled spin-1/2 nuclei, the Hilbert space is 1024!1024, and diagonalising such large matrices becomes prohibitively slow. Even the selection of 10 spins out of a continuous network is problematic. Largescale simulations (O15 spin-1/2 nuclei) have been applied to geometries in which spatial periodicity allows the Hamiltonian to be block-diagonalised [64], but these techniques have yet to be applied to the more involved problem of decoupling under arbitrary RF. The difficulties of numerical simulation of decoupling are compounded if we move beyond the ‘quasi-static’ approximation and incorporate sample spinning. The time dependence means that density matrix propagators must be determined by small time-step integration. If the rotor period, tr, can be set to a multiple of the RF cycle time, tc, then it is generally only necessary to integrate the propagator over a single rotation cycle (see Ref. [54] for
P. Hodgkinson / Progress in Nuclear Magnetic Resonance Spectroscopy 46 (2005) 197–222
^ iH^ C R, ^ contains an additional term R^ the Liouvillian, LZ intended to mimic the effect of spin diffusion amongst the protons. The exact form of this term is not critical, but the significant feature is that including a dissipative term causes the 1H magnetisation to decay, e.g. Iz(t)ZIz(0)exp(Kkt), where k is a spin diffusion or relaxation rate. Although this is an incoherent model (random spin exchange due to ‘spin diffusion’ or relaxation) of a coherent ‘dephasing’ process, the approach is reasonable; the experimental decays of proton magnetisation generally fit to decaying exponentials. These models cannot themselves predict appropriate values of the ‘spin diffusion’ rate, k, as a function of spin rate. However, since the decay of 1H magnetisation under CW decoupling corresponds to T1r relaxation, measurements of T1r provide a straightforward means of accessing k. As expected, k decreases approximately linearly with spin rate, and using the experimental values of k in the ‘CHC bath’ models gives good agreement between experimental and simulated spectra [19,66] (Fig. 5). Fig. 14 shows a comparison between model and experiment for a 13C, 1H and a 15N, 1H system which have significantly different proton spin-diffusion rates.
a full discussion). As we have seen, however, many decoupling sequences are not rotor synchronised; XiX, for example, explicitly avoids synchronisation conditions. Unless the time-scales for the RF and rotation permit sequential averaging, we must either resort to bimodal Floquet treatments [53,65], which are analytically elegant but are poorly adapted to numerical work, or to small timestep integration over the complete NMR signal. Since decoupling performance is dependent on the behaviour at relatively long (10 s of ms) timescales, we would need to compute the NMR signal out to long times (brute force methods) or use extremely large matrices (in Floquet approaches). Since performing exact simulations may be impractically slow and is unlikely to provide detailed physical insights, most workers have chosen to model the effect of the coupling network empirically (or ignore it completely). One particularly simple model which is able to reproduce the essential features of CW decoupling as a function of spinning speed involves a Liouville space treatment of a 13 C, 1H spin system [19]. In addition to the Hamiltonian ^ which describes the coherent evolution, superoperator, H, (a) 10
213
10
0.9
0.9
0.8
9
9 0.7
τp / µs
0.7
8
8
7
7
0.6
0.5
0.5
0.4
6
6
5
5
0.3
0.3
0.2
0.1
0.1
4
0
40
80
120
160
4
0
40
∆φ / °
80
120
160
∆φ / °
(b) 10
10
9
9
8
8
7
7
6
6
5
5
0.9
τp / µs
0.7
0.5
4
0.3
0.1
0
40
80
∆φ / °
120
160
4
0
40
80
120
160
∆φ / °
Fig. 14. Plots of decoupling performance as a function of TPPM parameters (pulse length, tp, and phase difference, Df) as measured by the intensity of (a) the Ca resonance of 2-13C-alanine at a 1H resonance frequency of 300 MHz and nrZ28 kHz, nrfZ100 kHz, and (b) the 15N resonance of d9-trimethyl-15Nammonium chloride. High values correspond to good decoupling performance. The plots on the right are simulated line intensities using the model of an X, 1H spin system coupled to dissipative bath with a 1H spin diffusion rate of (a) kZ500 Hz and (b) kZ0 Hz. Figure adapted from Ref. [66]. Further simulation details can be found in this reference.
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The plots show that a reasonable qualitative agreement is obtained between the model (which also includes the interaction between 1H CSA, dipolar coupling and RF) and experiment over a wide range of TPPM parameters. The detailed agreement is less striking: the experimental plots are significantly different for the two systems, while, as may be expected, the overall appearance of the two simulated plots is very similar, with the increased spin diffusion tending to ‘smooth out’ the calculated surface. The application of such models to more complex decoupling sequences is more problematic. How can we measure the decay rates of 1H magnetisation (k) if there is no well-defined spin-lock axis, as is the case in CW decoupling? In the absence of experimental values, k must be treated as an empirical parameter. Morever, the effective strength of the homonuclear coupling network is a key factor in the efficiency of a decoupling sequence. Given that these small-spin models cannot themselves predict k, they are not expected to be particularly effective in comparing the efficiencies of very different decoupling sequences. In other words, while small-spin models are useful for retrodictive rationalisation, they will not be particularly reliable for predictive purposes. 4.2.4. Interference with exchange Any process that occurs on the same time-scale can potentially interact with decoupling. The interaction between sample spinning and decoupling has already been discussed in Section 4.2.2, but exchange processes are another possible source of interference. As expected, the line-broadening resulting from such interference is most severe when the rates associated with the decoupling, e.g. nrf matches the rate of the exchange. The stochastic nature of the process means that the effects are observed over a larger range compared to the resonancelike behaviour of MAS and decoupling interference (Fig. 12). If the complexity of the decoupling sequence is increased, the interference effects are likely to occur over a wider range of time-scales, although such effects have to date only been clearly documented for homonuclear decoupling [67]. Altering the exchange rate by changing the sample temperature is the simplest way to diagnose (and alleviate) such line-broadenings. On the other hand, these effects can be used to probe dynamics on the time scale of RF nutation rates, although 1H T1r measurements will often provide the same information in a more direct fashion [68]. The interference can be modelled quantitatively in terms of simple relaxation models of the effect of exchange [69], or, more satisfactorily, using a full Liouville space treatment [67,70]. 4.2.5. RF inhomogeneity A drawback of multi-pulse techniques compared to simple CW decoupling is potentially an increased sensitivity to variations in the B1 field across the sample. Nutation experiments show a variation in RF nutation rates
across a full sample of about G10% relative to the mean [30]. Restricting the sample volume in order to reduce the impact of RF inhomogeneities is common practice in multipulse techniques for abundant spins, but is undesirable for rare-spin NMR. Recent work described in Section 5.1 suggests that the effectiveness of existing sequences such as TPPM (and variants such as CM) and XiX, is due, in part, to their relative robustness with respect to variations in nrf. The distribution of nutation rates due to RF inhomogeneity can be incorporated without difficulty into numerical simulations, and sequences such as SDROOPY [71] (described below), which are optimised to be insensitive to the exact nrf, have performed well in practice. An alternative to trying to model every aspect of spectrometer performance, including RF inhomogeneities, is to search for and optimise decoupling directly on the spectrometer, as described in Section 5.2.3.
5. Characterisation and development of decoupling methods for solids Rigorous approaches to developing decoupling sequences, whether through analysis of the spin Hamiltonian or through numerical simulation, are extremely challenging, principally because of the need to suppress interactions by about three orders of magnitude. This can be contrasted with the successful development of ‘recoupling’ sequences designed to re-introduce interactions that would otherwise be averaged by magic-angle spinning. In this case, imperfections of a few percent are quite acceptable. This section reviews some of the different strategies that have been used to develop and characterise decoupling methods in the absence of a comprehensive model for 1H decoupling in solids. 5.1. Characterisation of decoupling performance As discussed in Section 4.2.1, decoupling sequences for solution-state NMR state can be fully characterised simply by measuring the intensity of a single X resonance as a function of 1H offset. In the presence of strong dipolar couplings, however, the decoupling behaviour of carbons having different numbers of bonded protons varies substantially (Fig. 10). The CH case is obviously the more straightforward. It can be modelled satisfactorily in terms of the damped CH spin system discussed above, and the performance of the decoupling sequence determined from the height of a simulated NMR signal. Alternatively, as discussed in Section 2.2, we can determine the divergence between the 1 H magnetisation vectors for the a and b states of the 13C over a cycle of the decoupling. This quantity, nC$nK in terms of Eq. (12), will be zero for perfect decoupling. The CH 2 system is considerably more involved. Although, being the hardest system to decouple, it should
P. Hodgkinson / Progress in Nuclear Magnetic Resonance Spectroscopy 46 (2005) 197–222
Cα
C4
TPPM
1.5
0.5
SPINAL-64
relative nutation angle
1.0
1.5 1.0 0.5
DROOPY-1
1.5 1.0
1.5 1.0 0.5
DROOPY-2
1.5 1.0 0.5
SDROOPY-2
8.5 8.0 7.5 7.0 6.5 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0
SDROOPY-1
0.5
1.5 1.0 0.5 −3
−1
1
3 5 −3 −1 transmitter offset / kHz
1
3
5
Fig. 15. Intensity of a representative aromatic (C4) and aliphatic (Ca) peak of the 13C spectrum of 5CB (4-pentyl-4 0 -cyanobiphenyl) as a function of transmitter offset and RF power (expressed as the overall nutation angle relative to the correctly calibrated angle for the sequence) for different decoupling schemes: TPPM (DfZ158), SPINAL-64, DROOPY-1, DROOPY-2 and ‘supercycled’ versions of the latter, SDROOPY-1 and SDROOPY-2. Adapted from Ref. [71].
logically receive the most attention, relatively few studies to date have attempted to tackle this problem directly. The methodology of Eq. (12) has been adapted to InS systems [11] and the resulting composite pulse-based sequences, such as COMARO-2, perform satisfactorily for 2H in liquidcrystal samples [72], but do not transfer well to spinning solids [73]. De Pa¨epe et al. [71] minimise the contribution of heteronuclear IiSj terms in the effective Hamiltonian over a decoupling cycle, plus a fairly ad hoc modification of the CH2 spin system Hamiltonian to model the effect of
215
additional homonuclear couplings. The sequences were also optimised for robustness with respect to offset and RF inhomogeneity. Fig. 15 compares the performance of two solutions from this optimisation, DROOPY-1 and DROOPY-2, together with ‘supercycled’ versions, with TPPM and SPINAL-64 for a liquid-crystalline sample. The optimised performance of SPINAL-64, as shown by its higher maximum peak intensities, is clearly superior to TPPM, although the performance is quite sensitive to the parameter settings, especially for the Ca methylene carbon. The lack of correlation between offset sensitivity and peak decoupling performance is in marked contrast to the behaviour of solution-state decoupling sequences. The unpredictable nature of ‘supercycling’ applied to acyclic sequences is also noteworthy. SDROOPY-1 strongly outperforms its parent, DROOPY-1, and is competitive with SPINAL-64, while SDROOPY-2 is significantly less effective in decoupling the aliphatic site than DROOPY-2. Given the complex nature of the optimisation and the uncertain correspondence between predicted and measured performance, it is difficult to judge the effectiveness of the model and metric in isolation. It is interesting to note that the strong performance of SPINAL-64 and SDROOPY-1 is maintained even under conditions of fast sample spinning [36], despite their relatively long cycle lengths, tc. It has recently been suggested that proton nutation spectra provide insight into the mode of action of different decoupling sequences [30]. Nutation spectra might not be expected to be particularly informative for cyclic decoupling sequences, since we would expect the magnetisation to remain locked if sampled once per decoupling cycle. However, decoupling sequences for solids are frequently acyclic, and so the evolution of the 1H magnetisation may provide greater insight than the intensity of a single peak in the X spectrum. Fig. 16 shows 1H nutation spectra for two different proton environments. In Fig. 16(a), the sample is an isotropic liquid, and the finite width of the CW nutation spectrum provides a direct indication of the RF inhomogeneity (Section 4.2.5). The nutation spectra for TPPM and Continuous Modulation (CM) in particular are much ‘tighter’, implying a greater tolerance to RF inhomogeneity. It is also noteworthy that the nutation spectra are peaked at the cycle frequency, nc, rather than the RF nutation rate, nrf (This behaviour is reproduced in simulations of simple spin1/2 systems and can be rationalised analytically.) The interpretation of corresponding nutation spectra in solid samples, Fig. 16(b), is less straightforward. For instance, a broad nutation spectrum may correspond to an undesirable sensitivity to RF inhomogeneity, or it may be the result of ‘recoupling’ the homonuclear dipolar interactions, leading to rapid dephasing of the 1H magnetisation, which is generally desirable. Although sequences involving modest phase shifts can be approximated in terms of an overall nutation, the significance of nutation spectra for highly acyclic sequences, such as SPINAL-64, is less obvious.
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(a)
(b)
×2
CW
CW
×4
TPPM
×4
TPPM CM
×4
SPINAL64
CM
XiX
×4 −100
−50 1
0
50
100
H nutation frequency / kHz −100
0 1
100
H nutation frequency / kHz
Fig. 16. 1H nutation spectra as a function of 1H decoupling sequence: (a) spectra from residual protons in a D2O sample using nrfz97 kHz with Continuous Wave, TPPM and Continuous Modulation (CM) decoupling, (b) spectra from 2-13C-glycine with nrZ20 kHz, nrfz100 kHz and nominal 1H Larmor frequency of 500 MHz, using CW, TPPM, CM, SPINAL-64 and XiX decoupling. Figure adapted from Ref. [30]. See reference for further experimental details.
Despite clearly being informative, it is difficult to use the spectra of Fig. 16(b) in a predictive manner. The small angle phase variations of sequences such as TPPM and CM make them amenable to analysis [22], with the Continuous Modulation form requiring less approximation [30] in comparison to its discretely jumped counterpart. To first order, there is a well-defined spinlock axis associated with the RF irradiation, with an effective nutation rate, neff, and angle relative to the z axis, feff. The effect of the 1H homonuclear couplings on 13C decoupling can then be probed in a relatively direct fashion. Fig. 17 shows the height of the CH2 resonance of glycine as a function of the parameters of CM decoupling. Like TPPM, the performance is strongly dependent on decoupling parameters; indeed Fig. 17 is comparable to the results from the CH resonance of alanine, Fig. 14 (with the vertical axis reversed). As might be expected, it is impossible to find satisfactory decoupling under rotary resonance conditions, i.e. when neff matches a multiple of the spin rate (point c is a weak local maximum). On the other hand, decoupling is effective when the HORROR condition is matched, point a, i.e. recoupling of the 1H homonuclear interactions is associated, in this case at least, with effective decoupling. This is supported by the lack of correlation between decoupling performance and an effective tilt axis at the magic angle (feffZGqm), which would be associated with weakened homonuclear interactions. It is interesting to note the presence of additional ‘resonance’-like conditions in the plots of Fig. 17, which are also present, but less clearly resolved, in Fig. 14. Fig. 17(c) shows an expansion of Fig. 17(a) for the region containing the local optima a–d, but acquired under
spin-echo conditions, i.e. it plots the intensity of the 13C signal after a tKpKt evolution period (with 2tZ20 ms) as a function of the decoupling parameters during 2t. As discussed in Section 4.1.1, this removes the refocusable component, which corresponds to, but is not necessarily synonymous with, the inhomogeneous linewidth. This significantly accentuates differences in decoupling performance. The optimal decoupling performance is obtained for point a (on the HORROR condition) rather than point b; although these points have indistinguishable directlymeasured performances, they have clearly distinct T20 values of 58 and 52 ms, respectively. In these cases at least, good ‘spin-echo’ performance also implies good ‘conventional’ decoupling performance, i.e. decoupling optimised under spin-echo conditions can be applied throughout an experiment. 5.2. Strategies for development Given the difficulty of defining even models and metrics for analysing decoupling in strongly coupled systems, such as CH2, it is unsurprising perhaps that serendipity has often been a feature in the discovery of many decoupling sequences. There are, however, some less ad hoc approaches that have provided useful insights. 5.2.1. Symmetry-based methods Levitt et al. have proposed a framework for the design of rotor-synchronised pulse sequences under MAS. One family of such sequences is denoted CNnn where C is an element corresponding to an rf cycle (i.e. returning irradiated spins to their initial state). The C element is repeated N times with
P. Hodgkinson / Progress in Nuclear Magnetic Resonance Spectroscopy 46 (2005) 197–222
(a)
100
φeff = θm
1.1
80
H
1.0
νc / νrf
217
a
b d
60
R1
0.9 R2 c
40
φeff = π/2
0.8
20 0.7 φeff = −θm
0 0
0.2 10
0.4 20
νc / νrf
(b)
0.6
a
30
0.8 40
(rad)
(degree)
100
(c) 1.0
1.0
0.95
0.95
0.90
0.90
80
60
40
20 0.85
0.85 0
0.1
0.225
0.35
a / rad
0.1
0.225
0.35
a / rad
Fig. 17. (a) Experimental 13C peak height for the CM-decoupled CP-MAS spectrum of 2-13C-glycine as a function of the CM parameters, a (total phase excursion) and the cycle frequency, nc, relative to the nutation rate, using nrfZ115 kHz and nrZ20 kHz, at a nominal 1H Larmor frequency of 500 MHz. Dotted lines mark significant values for the tilt of the effective lock axis, feff. Solid lines mark the rotary resonance (neffZpnr, pZ1,2) and ‘HORROR’ (neffZnr/2) conditions between the effective nutation rate and the spin rate. (b) is an expansion of the region containing the local optima marked a-d, while (c) shows the results from the same region, but obtained under spin-echo conditions. Adapted from Ref. [30].
a phase shift of 2pn/N and a total duration of ntr (where tr is the rotor period). Note that this implies a fixed ratio between the RF nutation and spinning rates, nrf/nrZN/n. Each term of the nuclear spin Hamiltonian can be written as the product of a spatial component of rank l and a spin component of rank l (with respect to the rotation produced by the RF sequence). In the same way as phase cycles are designed to suppress undesired coherence pathways, we can choose values of N, n and n such that terms of the Hamiltonian with different values of l and l are suppressed in the average Hamiltoninan for the sequence. With this basic selection in place, we can choose elements C that are robust with respect to experimental parameters, such as RF inhomogeneities. This provides a systematic route to the design of RF pulse sequences. It was originally applied to ‘recoupling’ problems such as the original C7 dipolar recoupling
sequence [58], but heteronuclear decoupling sequences can be designed within this framework [74]. As discussed in Section 2.1, however, first order AHT is not adequate to describe the suppression of interactions by three orders of magnitude, and extensions of the symmetry principles become unwieldy at higher orders. Moreover, the synchronisation conditions, e.g. that the RF nutation rate be six times the spin rate (C12K1 2 ) make these sequences less flexible compared to simple techniques such as TPPM. The experimental performance of C12K1 2 and related sequences, such as R2412 , is comparable to TPPM [75]. The combination of symmetry-based sequences with adiabatic C elements (consisting of a pair of adiabatic inversion pulses) has also been evaluated [76]. Although such sequences are expected to exhibit greater tolerance to, say, RF inhomogeneities, improved performance relative to TPPM has yet to be demonstrated.
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5.2.2. Numerical simulation Although less elegant than analytical approaches, numerical simulation is able (given the correct model system) to cope with arbitrarily complex problems. As discussed in Section 4.2.3, however, a truly predictive model for heteronuclear decoupling in solids has yet to be devised; such an absolute predictive model would need to be able to calculate a ‘quality parameter’ (say peak intensity) as a function of any experimental conditions. In optimisation problems, however, we are only really interested in relative performances within a restricted family of sequences (Is the decoupling better with parameter set F1 or set F2?). Provided the model is capable of predicting such trends, then we may be able to work with fairly simplistic models. The model proposed by De Pae¨pe et al. [71], for example, is relatively unsophisticated, but, given suitable optimisation criteria, it is sufficient to devise useful and robust decoupling sequences. The phase-modulated decoupling sequence is expressed as a Fourier series, which allows a wide range of modulations to be included. The mean decoupling performance for a given sequence is then evaluated over an appropriate range of transmitter offsets and RF powers (to model RF inhomogeneity), ensuring that the solution will be experimentally robust. The resulting pulse sequences are clearly superior to TPPM in terms of decoupling performance and robustness, although edged out in peak performance by SPINAL-64 on the liquid crystal test system. Numerical simulations have yet to be used predictively for decoupling under MAS. Simulations have been used to rationalise the performance of XiX decoupling [37], for example, but it is noteworthy that the two spin model was only able to reproduce some of the important features, e.g. unfavourable ‘resonance’ conditions between the decoupling period, tc, and the rotor period, tr. Subtle, but important details, such as optimal values of tc/tr, could not be derived from small spin models. 5.2.3. ‘Direct optimisation’ An alternative to spin system modelling is to determine and optimise the decoupling performance on a sample directly [77]. As before, we start with a family of decoupling sequences, but now apply the decoupling sequence from a given set of parameters directly on a sample and measure, say, the peak intensity in the 13C spectrum (The linewidth is a poorer choice of parameter since a decoupling sequence may be effective at concentrating intensity without significantly changing the full-width-at-half maximum (cf. Fig. 9). Many experimental features, such as RF inhomogeneity, are automatically accounted for by this direct experimental approach; good solutions must necessarily be robust with respect to the experimental effects of RF inhomogeneity. Others, such as insensitivity to transmitter offset would need to be included specifically (e.g. by averaging the decoupling performance over a range of offsets). As, however,
insensitivity to 1H offset and ‘broadbandedness’ are not entirely equivalent for solids, it may be more useful to use model systems with a range of 13C sites and compute an average performance. The results to date [77] have involved optimisation of using a single resonance. It is unsurprising, therefore, that a different decoupling solution was found to that obtained by the numerical simulation discussed above. Starting from the same general Fourier series, the optimal solutions consistently reduced to a single frequency modulation of the phase, i.e. f(t)Za cos (2pnct). The overall performance of this Continuous Modulation (CM) is comparable to TPPM, although local high-performance solutions are often found by this iterative optimisation. These parameters are then transferrable to other samples run under the same conditions. Perhaps the most interesting feature of this work is emergence of a single frequency phase modulation as an optimal solution, under the conditions used. The family of functions used does impose limits on the solution space—it is unlikely, for example, that a long acyclic sequence such as SPINAL-64 could have emerged as a solution—but direct optimisation may continue to provide useful new starting points in the future.
6. Diversions in decoupling The previous sections have described the principles, complexities and current solutions to the problem of 1H decoupling in solids. This final section considers some other practical issues that arise in the application of decoupling in solid-state NMR, including the decoupling of nuclei other than 1H. 6.1. 1H decoupling in
19
F-containing systems
19
F is an attractive nucleus for NMR due to its 100% natural abundance and high magnetogyric ratio. However, the similarity of its Larmor frequency to 1H (just 6% lower) significantly complicates double-resonance experiments involving both 1H and 19F. As well as requiring appropriate hardware to ensure effective isolation of the 1H and 19F channels, 1H irradiation will result in a shift in the 19F resonance frequencies of Df z
n2rf 2Dn
(24)
where nrf is the 1H nutation rate and Dn is the difference in H and 19F Larmor frequencies. This shift is described, perhaps inappropriately, as a Bloch–Seigert shift [78]. Care is required when referencing, for example, to ensure that reference and sample spectra are acquired under the same decoupling conditions to ensure that the shifts are consistent.
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These effects are illustrated in Fig. 18. It can be seen that H decoupling significantly improves the linewidth compared to the undecoupled spectrum (top), but causes a measurable frequency shift which increases with increasing RF power. The Bloch–Seigert shifts are fairly modest at these Larmor frequencies. When the shifts are more significant, 19F linewidths may increase at higher decoupling powers due to B1 inhomogeneities creating different Bloch–Seigert shifts across the sample and hence an overall line-broadening (Fig. 12). It is also worth noting that in many systems, 19F linewidths are strongly determined by 19 F homonuclear coupling, and 1H decoupling is not a limiting factor for 19F resolution. Especially at higher magnetic fields, fast magic-angle spinning may render the complications of 1H decoupling in 19F NMR unnecessary. Effects due to 19F can also be observed in protondecoupled spectra of rare spins. It had been widely noted, for example, that the resolution of 13C sites close to a 19F nucleus was often poor [79,80]. This is illustrated in Fig. 19 for the 13C NMR spectra of the liquid crystal I35. The resonances from carbon sites close to the fluorine should be simple doublets due to 13C–19F coupling (dipolar plus scalar). Instead, the doublet components are strongly, and 1
asymmetrically, distorted (C1–3), or even completely unresolved (C4) in the CW-decoupled spectrum. (The same effects are present, but less strikingly, in simple 13C, 1 H systems, with the asymmetric broadening resulting in a peak maximum that changes with CW decoupler offset [16].) Changing the phase modulation of the decoupling sequence has a dramatic effect, and the SPINAL-64decoupled spectrum shows good resolution for all 13C sites. These broadening effects were traced to the presence of strong 19F–1H couplings (the dipolar coupling between an adjacent 19F and 1H on an aromatic ring, for instance, is dz5 kHz). The dipolar coupling modifies the effective offset (or ‘local field’) of the 1H by Gd. This cannot be removed by optimisation of the 1H transmitter frequency, and requires an appropriately robust decoupling sequence. These decoupling-dependent line-broadenings are most evident in liquid crystal and soft solid samples, but analagous effects are observed in MAS spectra of conventional solids. Other line-broadening mechanisms complicate, however, the task of obtaining well-resolved 13 C signals from sites close to 19F. The most significant of these follows from the relatively short lifetime of 19F coherences. A short 19F ‘T2’ causes the expected doublet for a 13C scalar coupled to 19F to broaden or coalesce, F
I35
9 CH2CH2
CH3CH2CH2
280 Hz
219
8
2 3
10 4
7 1 6
505 Hz
CH2CH2CH2CH2CH3
5
(a) 100 kHz
280 Hz
(b) 80 kHz 310 Hz
C9
55 kHz 795 Hz
30 kHz 15 kHz
1240 Hz −50
−55
−60
−65
−70 δ F / ppm
Fig. 18. CW 1H-decoupled 19F NMR spectra of a CF3-containing system as a function of 1H nutation rate, nrf (figures to left) at a Larmor frequency of 282 MHz for 19F (300 MHz for 1H) and a spin rate of 10 kHz. The chemical shift scale is indirectly referenced to CFCl3 via C6F6 at K116.4 ppm. Figures by the peaks are full widths at half-maximum. The dashed line shows the 19F spectrum in the absence of decoupling. Note that TPPM did not lead to further line narrowing under these conditions.
C8
C6
(c) C4
C7
C10
C2
C5
C3
C1
200
150
50
0
ppm
Fig. 19. 13C NMR spectra of the liquid crystal I35 using (a) CW, (b) TPPM and (c) SPINAL-64 decoupling, (nrfZ50 kHz). Adapted from Ref. [33], which contains further experimental details.
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degrading resolution [52]. Similar effects can be expected from other systems in which a third spin species has an appreciable coupling to the nucleus being decoupled. Simultaneous high-power decoupling of both 1H and 19F eliminate such line-broadenings. Although technically demanding, the use of such double decoupling to simplify rare-spin spectra has been reported for both organic materials [79,80] and organometallics, e.g. 119Sn NMR [81]. Note that 1H and 19F nutation rates are often similar (and both high) as no particular harm is done by HartmannHahn contact between the 1H and 19F spin baths. 6.2. Decoupling of other nuclei There are relatively few reports of decoupling of heteronuclei other than 1H and 19F in solids. This is presumably because homonuclear couplings between other nuclei are less significant, and decoupling of dilute spins is not generally necessary since the heteronuclear couplings may be well resolved and potentially informative. In addition, if the heteronuclear couplings are not required, they can be refocused by spin-echoes, which is not possible in the prescence of an extended coupling network (a simple spin echo will not refocus decoherence into multiple quantum states). 6.2.1. Spin-1/2 nuclei: 31P, 19F and 15N The much larger CSAs of nuclei other than 1H are expected to complicate any attempts at decoupling. In Ref. [51], for example, CW 31P decoupled 1H spectra show marked effects of the CSA/dipolar/RF interference effects discussed in Section 4.2.1. More effective decoupling methods, such as TPPM, reduce, but could not consistently eliminate, the line-splittings and broadenings. Continuous wave decoupling is also expected to be relatively inefficient for 19F decoupling, and TPPM decoupling has been shown to give substantial resolution enhancements over CW decoupling in materials such as chiolite, Na5Al3F14 [82]. In inverse-detection experiments, the rare-spin evolves (under heteronuclear decoupling) during an indirect dimension, and the NMR signal is acquired, after polarisation transfer, on the abundant high-g spin, typically 1H. By definition, the observed 1H spins will be coupled to the rare spin and hence broadened by 1 J XH . These broadenings can lead to a detectable sensitivity decrease, particularly if the 1 H linewidths have been reduced by 2H isotopic substitution. The larger chemical shift range of the X spins means that a broad-band decoupling sequence is required to decouple the X nuclei while the proton signal is acquired, justifying the use of 15N GARP decoupling in inverse-detection experiments on 2H-substituted microcrystalline protein samples [83]. 6.2.2. Half-integer spin quadrupolar nuclei: 27Al 27 Al decoupling (in combination with simultaneous 1 H decoupling) has been used to substantially improve
the resolution of 31P spectra of aluminophosphates, while 1 H decoupling in isolation had no observable effects on linewidths [84]. However, the interaction of RF irradiation, sample spinning and substantial quadrupole couplings can be extremely complex. For example, RF irradiation is often sufficient to cause recoupling of the dipolar interactions via the TRAPDOR effect [85], which is clearly counterproductive for decoupling. Experimentally, relatively low power 27Al decoupling (nutation rates of 10–15 kHz) was found to be optimal; further increases of the decoupling power led to steady increases in 31P linewidth. This effect was not reproduced in simulations of a simple 27Al, 31P spin system. 6.2.3. Integer spin nuclei: 2H Continuous wave decoupling is not very effective for decoupling 2H in static samples due to the large width of the 2 H spectrum (in turn due to the quadrupolar coupling). Irradiation of the double quantum coherence (which is unaffected to first order by the quadrupole coupling) allows 2 H to be decoupled [86,87], but uses RF power inefficiently. Composite-pulse based decoupling provides a more efficient decoupling strategy [72]. MAS significantly changes the problem since the quadrupole coupling is modulated by the spinning. Recently 2H decoupling has been used to improve the resolution of 13C spectra of biomolecules in which the protons have been diluted with 2H, by suppressing the 1JCD coupling (ca. 30 Hz) [88].
7. Conclusions With the exception of systems with high degrees of mobility, or particularly large susceptibility anisotropies, the linewidths of rare spins in organic solids are often limited by the quality of heteronuclear 1H decoupling. Conventional continuous wave decoupling becomes increasingly inefficient as the 1H spectrum is resolved, e.g. through faster magic angle spinning and/or higher static magnetic fields. However, significant progress has been made in recent years in finding practical solutions, such as TPPM, SPINAL-64, XiX, etc. that allow the resolution of 13 C and 15N spectra to be maintained at higher fields and spin rates. At the same time, it has become clear that the resolution and sensitivity in many solid-state NMR experiments is determined by the decay of the rare spin signal during a spin-echo due to coherent evolution. Although most effects of imperfect decoupling are refocused in a spin-echo, the time constant for the decay is strongly dependent on how efficiently the rare-spin is decoupled from the dephasing of the 1H magnetisation. Lengthening this ‘T20 ’ decay by optimising the 1H decoupling has a major impact on the viability of key experiments, such as natural abundance 13C double quantum experiments [89]. It also raises fundamental questions about the final limits of sensitivity in solid-state
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NMR; what if the true relaxation-limited linewidths were extremely small? Despite significant improvements in observed decoupling performance and our understanding of the factors behind these developments, a complete picture of 1H heteronuclear decoupling remains elusive. In many ways, heteronuclear decoupling is the ultimate challenge in ‘spin control’ for solid-state NMR; the suppression of interactions by at least three orders of magnitude, while accounting for the complex inter-relations between hetero- and homo-nuclear coupling networks, sample spinning, shift variations, etc. is a formidable intellectual challenge. Although it is unlikely that a single picture or decoupling sequence can reconcile such diverse factors, it is reasonable to suppose that the analytical and numerical tools developed to tackle this challenge will find wide application beyond the immediate problem of heteronuclear decoupling for solids.
Acknowledgements The support of authors in providing the original artwork for many of the figures reproduced in this article is gratefully acknowledged. In particular, I thank Matthias Ernst (ETH, Zu¨rich), Lyndon Emsley and Gae¨l De Pae¨pe (ENS, Lyon) for many fruitful discussions over heteronuclear decoupling of solids over the years. Thanks are also due to Dr D.C. Apperley (Durham) for his assistance in providing Figs. 1, 4, 10 and 18.
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