Two-dimensional MAS–NMR spectra which correlate fast and slow magic angle spinning sideband patterns

24 August 2001

Chemical Physics Letters 344 (2001) 367±373

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Two-dimensional MAS±NMR spectra which correlate fast and slow magic angle spinning sideband patterns Charles Crockford a, Helen Geen a, Jeremy J. Titman b,* a

School of Physics and Astronomy, University of Nottingham, University Park, Nottingham, UK School of Chemistry, University of Nottingham, University Park, Nottingham NG7 2RD, UK

b

Received 28 September 2000; in ®nal form 8 March 2001

Abstract A new NMR experiment which allows a measurement of the chemical shift anisotropy (CSA) tensor under magic angle spinning (MAS) is described. This correlates a fast MAS spectrum in the x2 dimension with a sideband pattern in x1 in which the intensities mimic those for a sample spinning at a fraction of the rate xr =N. This method is particularly useful for accurately measuring narrow shift anisotropies. Since the sidebands intensities in x1 are identical to those expected at xr =N , standard methods can be used to extract the principal tensor components. The nature of the experiment is such that a minimal number of t1 increments is required. Ó 2001 Elsevier Science B.V. All rights reserved.

1. Introduction Measurements of the NMR chemical shift anisotropy (CSA) [1,2] provide detailed information about the molecular structure of solids. The principal components of the CSA tensor can be obtained directly from the singularities observed in the broad NMR line of a powder sample, but this method su€ers acutely from problems of spectral overlap, as well as perturbations from dipolar interactions. More reliable measurements with improved resolution can be made by analyzing the intensities of the rotational sidebands [3,4] which appear in the magic angle spinning (MAS) spectrum. Obtaining accurate results from this procedure requires a relatively slow MAS

*

Corresponding author. Fax: +44-115-951-3562. E-mail address: [email protected] (J.J. Titman).

rate xr to give a suciently large number of sidebands for the analysis. Hence, for molecules with many chemical sites the MAS approach can also fail due to the overlap of di€erent sideband manifolds. In order to further improve resolution several two-dimensional MAS±NMR experiments which separate the isotropic and anisotropic parts of the chemical shift interaction have been proposed. One class of experiments reintroduces scaled powder patterns related to the non-spinning lineshape in the evolution dimension …x1 † while retaining the isotropic fast MAS spectrum during detection …x2 †. This can be achieved by MAS synchronized multiple-pulse trains which interfere with the formation of rotor echoes during the evolution period [5±9] or by switching the orientation of the spinning axis between evolution and detection [10,11]. A similar two-dimensional correlation of isotropic and anisotropic chemical

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shifts can be obtained via a projection reconstruction algorithm from a series of spectra recorded with di€erent spinning angles [12]. Experiments in this class are generally dicult to implement, su€ering from pulse imperfections or requiring special instrumentation. A second class employs fewer pulses to obtain an isotropic spectrum in x1 while relying on MAS sideband patterns in x2 to measure the CSA. One approach [13] achieves this by sandwiching the evolution time between a TOSS sideband suppression sequence [14] and its time-reversed counterpart. The TOSS sequence prepares the magnetization such that sideband signals from molecules in di€erent orientations interfere destructively during the evolution time, while the time-reversed sequence is required to reintroduce the sidebands for the detection dimension. Constant-time variants of this experiment have been proposed [15,16] which operate at very slow MAS rates. This second class requires long acquisition times in order to obtain high resolution in the x1 dimension where the irregularly spaced isotropic shifts are sampled. Data processing methods [17] which make use of a priori knowledge about isotropic line positions in the fast spinning one-dimensional MAS spectrum can overcome this problem to some extent. A third class of experiments [18,19] restricts the isotropic shifts to the x2 dimension and samples only the anisotropic part as a sideband pattern in x1 . These experiments are two-dimensional variants of the PASS technique [20] in which MAS synchronized sequences of p-pulses give the sideband at a frequency Mxr from the isotropic line a phase shift of magnitude MH where H is known as the `pitch' of the sequence. Constant duration sequences with arbitrary pitch allow H to be varied over a full period in the evolution dimension. A Fourier transform with respect to pitch results in the complete separation of di€erent sideband manifolds over the x1 co-ordinate of the two-dimensional spectrum. Since the sidebands are regularly spaced and there is no decay of the magnetization in the evolution period, a minimal number of t1 increments suces and long acquisition times are avoided. This third class of experiment is simpler to implement and more

ecient than the others and is the method of choice in most cases. In this Letter, we describe a new two-dimensional MAS±NMR experiment which belongs to this third class. Only the anisotropic part of the shift is sampled in the evolution dimension, so that the advantages of the two-dimensional PASS approach are retained. However, in contrast to twodimensional PASS our new experiment correlates the standard MAS spectrum in the x2 dimension with a sideband pattern in x1 in which the intensities are identical to those expected for a sample spinning at some fraction of the actual rate xr . For convenience, we refer to the `apparent rate' xr =N and the `reduction factor' N in the following discussion. The isotropic shift only appears in the x2 dimension. The new experiment is preferable for measuring narrow shift anisotropies, since for these slow MAS rates are required with two-dimensional PASS leading to rotor instability and long sequences of p-pulses. Fig. 1a shows the standard carbon-13 MAS spectrum of L -methionine (98%, Aldrich) recorded at a rate of 2 kHz and a Larmor frequency of 75.47 MHz. At this MAS rate there are no sidebands associated with the side chain carbon sites which appear in the up®eld part of the spectrum and have small CSA tensors. Most of the lines in the spectrum are split into two components, although the splittings vanish after recrystallization, suggesting the presence of two polymorphs. Such splittings have been observed before for methionine [21]. Fig. 1b recorded at a rate of 500 Hz illustrates the problems encountered when attempting to measure the small CSA tensors of the side chain carbon sites using MAS. Despite the relatively slow MAS rate there are still insucient sidebands for a reliable analysis, and spectral overlap precludes their accurate measurement. Fig. 1c shows the aliphatic part of a two-dimensional spectrum of L -methionine recorded using the new experiment with a MAS rate of 2 kHz and a reduction factor of 8, resulting in an apparent rate of 250 Hz in the x1 dimension. Further experimental details are given in the ®gure caption. At this MAS rate spinning sidebands are negligible in the x2 dimension, but there are a substantial number in x1 .

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369

Fig. 2. Pulse sequence used to record two-dimensional MAS spectra which correlate fast and slow spinning sideband patterns. Narrow and broad black rectangles represent p=2 and p-pulses, respectively. Timings for the ®ve-pulse sequences are given in Tables 1 and 2. Further details are given in the text.

Fig. 1. (a) Carbon-13 MAS spectrum of L -methionine (98%, Aldrich) recorded at a rate of 2 kHz using a standard crosspolarization pulse sequence with a contact time of 5 ms. The MAS rate was stabilized to 1 Hz. Proton decoupling with a ®eld strength of 65 kHz was applied throughout the acquisition period. The spectral width was 33 kHz and 64 scans were acquired. (b) As (a), except that the MAS rate was 500 Hz. (c) Aliphatic (up®eld) region of a two-dimensional MAS spectrum of L -methionine recorded using the new experiment at a rate of 2 kHz with a reduction factor of 8. Note the substantial number of spinning sidebands in the x1 dimension which corresponds to a reduced MAS rate of 250 Hz. There were 16 t1 values with an increment of 31:25 ls, corresponding to 1/16 of the MAS period. Other experimental parameters were as for (a) except that 256 scans were acquired per increment.

2. Pulse sequence The pulse sequence used here to record twodimensional MAS spectra which correlate fast and slow spinning sideband patterns is shown in Fig. 2. Enhanced carbon-13 transverse magnetization generated by cross-polarization evolves under a

sequence of ®ve carefully timed p-pulses. Subsequently one component of the magnetization is stored on the z-axis for a time t1 , before a further period of evolution under an identical ®ve-pulse sequence prior to detection during t2 . The t1 time is incremented by some convenient fraction, usually 1/16 or 1/32, of the rotor period. Heteronuclear decoupling is applied during the two ®ve-pulse sequences and the detection period, but it is interrupted during carbon-13 p-pulses to avoid unwanted Hartmann±Hahn contacts. In order to reconstruct the required two-dimensional FID two experiments are required in which the phase of the carbon-13 p=2 storage pulse u3 is alternated p=2, p. These are combined in the receiver by simultaneous shifting of the reference phase from 0 to p=2. Further phase cycling of the carbon-13 pulses selects changes in coherence order Dp ˆ 1 with u1 and u4 and Dp ˆ 0 across the whole of each ®vepulse sequence with u2 and u5 . The experiment requires careful calibration of the p-pulses to reduce sideband phase and amplitude distortions, and we have found empirically that the quality of the sideband suppression in a ®ve-pulse TOSS sequence is a suitable calibration criterion. Errors can be further reduced by shifting the relative phases of successive p-pulses through the series 0°, 330°, 60°, 330°, 0°, a procedure which has been used in a similar context by Antzutkin et al. [22]. Reconstruction of the required FID in the receiver as described above allows the two-dimensional spectrum to be obtained by straightforward

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double complex Fourier transformation with respect to t1 and t2 . In addition to x2 phase corrections which should be made by inspecting the x1 ˆ 0 slice, a ®rst order correction in x1 which, because of the nature of the experiment, a€ects the x2 lineshapes is usually also necessary.

with the `f-functions' [23] de®ned by ! X x…m† …0† exp…imw† c : fc …w† ˆ exp mxr m6ˆ0 The pulse sequence of Fig. 2 results in the FID of Eq. (2) if the timings t…p† of the q p-pulses measured from the start of each sequence satisfy q

3. Theory The orientation of the CSA principal axis frame of a particular spin site relative to the MAS rotor is given by the Euler angles …a; b; c†. In this frame the components A2n of the shift tensor are A20 ˆ

x0 …r33

A22 ˆ

riso †;

1 p x0 …r22 6

A21 ˆ 0; r11 †;

2 X mˆ 2

2 X nˆ 2

x…m† c …c† exp…imxr t†;

D2n

2 m …a; b; c†d m0 …b0 †A2n

…1†

‡ dm0 xiso ;

where b0 is the magic angle and D and d are Wigner matrices. The isotropic shift xiso depends on the o€set from the carrier frequency xrf xiso ˆ x0 …1

riso †

xrf ;

where x0 is the Larmor frequency. The required two-dimensional FID Sc results in a sideband pattern in x1 corresponding to a reduced MAS rate of xr =N and is given by N 1  N fc …xr t1 ‡ c† fc …xr t2
exp ixc…0† t2 exp… t2 =T2 †

Sc …t1 ; t2 ; c† ˆ fc …c† 

…3† for both m ˆ 1 and m ˆ 2 where T is the total duration of each sequence. A further condition is 2

q X

… 1†

…q p† …p†

t

ˆ0

…4†

pˆ1

with components given by x…m† c …c† ˆ

1ˆ0

pˆ1

T

where x0 is the Larmor frequency, rjj are the principal components and riso is their average. Following Antzutkin et al. [22] we consider a `carousel' of spin sites which have common values of a and b. The NMR precession frequency of sites within a carousel xc …t; c† can be written as a Fourier series xc …t; c† ˆ

… 1† exp… imxr T † ( ) q X
p …p† … 1† exp imxr t  2 ‡1 ‡N

‡ c† …2†

which ensures that phase shifts due to isotropic shifts are not generated by the sequence. For convenience the sequences discussed in this work are restricted to an integral number of rotor periods sr and contain ®ve p-pulses disposed symmetrically about T/2 so that T ˆ nsr ;

t…3† ˆ T =2;

t…5† ˆ T

t…1† :

t…4† ˆ T

t…2† ;

For these ®ve-pulse symmetrical sequences Eq. (3) becomes

4 cos xr t…1† 4 cos xr t…2† ‡ N ‡ 2… 1†n 2 ˆ 0;

4 cos 2xr t…1† 4 cos 2xr t…2† ‡ N ˆ 0: …5† Analytical solutions to Eq. (5) can be found and values of h…p† ˆ xr t…p† ˆ 2pt…p† =sr are given in Table 1 for reduction factors N ranging from 2 to 8. From these solutions the shortest realizable sequences with t…1† < t…2† < t…3† < t…4† < t…5† < T can be calculated, and these are given in Table 2 as fractions of the MAS period. It should be noted that Eq. (3) predicts that the maximum obtainable reduction factor depends on the number of pulses in the sequence and that Eq. (5) shows that 8 is the largest possible value for ®ve symmetrically disposed pulses. Similarly, higher reduction factors

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Table 1 Analytical solutions for Eq. (5) (where k and l are integers) N

h…1†

h…2†

2 3 4 5 6 7 8

2pk  p=2 2pk  arccos… 5=8† 2pk  arccos… 1=4† 2pk  arccos… 3=8† 2pk  p=3 2pk  arccos…5=24† 2pk  p=2

2pl  2p=3 2pl  arccos… 7=8† 2pl  arccos…3=8† 2pl  arccos…7=8† 2pl 2pl  arccos…23=24† 2pl

Table 2 Realizable pulse timings t…p† for ®ve-pulse sequences with reduction factor N N

t…1† =sr

t…2† =sr

t…3† =sr

t…4† =sr

t…5† =sr

T =sr

2 3 4 5 6 7 8

0.25000 0.35745 0.29022 0.31118 0.16667 0.216560 0.25000

0.33333 0.41957 0.88497 0.91957 1.00000 1.04611 1.00000

0.50000 0.50000 1.00000 1.00000 1.50000 1.50000 1.50000

0.66667 0.58043 1.11502 1.0804 2.00000 1.95390 2.00000

0.75000 0.64255 1.70978 1.68882 2.83333 2.78340 2.75000

1 1 2 2 3 3 3

generally require longer sequences, if they are to be realizable. It should be noted that the timings in Table 2 are well spaced, so that there are no problems of pulse overlap with ®nite lengths. Finally, the theory given above can be simply extended in order to design similar experiments suitable for measuring inhomogeneous interactions other than the CSA, although suitable modi®cations [24] need to made in the case of quadrupolar nuclei. 4. Results and discussion Fig. 3 shows two-dimensional spectra recorded using the new experiment for 1-13 C-glycine with a MAS rate of 5 kHz and reduction factors ranging from 4 to 8. The number of spinning sidebands in the x1 dimension increases with N as expected, while the relatively large CSA of the carbonyl site results in small sidebands at xr and 2xr in x2 . Fig. 4 shows comparisons between the x1 sideband intensities from the new experiment taken from Fig. 3 (points) and those from a standard MAS spectrum at the corresponding reduced rate (lines). The agreement is good, suggesting that

the new experiment can be used to obtain accurate measurements of the CSA. The sideband intensities from the new experiment were extracted by adding contributions from all the x2 sidebands at a given x1 value to the corresponding centreband. CSA tensors have been measured for a range of samples and the principal components obtained compare well with those found in the literature [1,2]. One illustrative example is provided by L -methionine. CSA tensors for the methyl d site in the each of the polymorphs of L -methionine were extracted from the spectrum in Fig. 1 using a standard analysis [4]. The principal components were: r11 ˆ 26:1, r22 ˆ 18:2, r33 ˆ 9:2 ppm and r11 ˆ 7:7, r22 ˆ 14:0, r33 ˆ 25:8 ppm, assigned according to the convention [25] that jr33 r22 j > jr22 r11 j. This convention ensures that the asymmetry parameter g ˆ …r22 r11 †=…r33 riso † is always positive. Note that the anisotropy d ˆ r33 riso has opposite sign in the two polymorphs. Essentially identical values for the principal components were obtained from four more experiments with N ranging from 4 to 7. Both CSA tensors are narrower than those measured previously [21] by straightforward MAS, but are

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Fig. 4. Comparison between the x1 sideband intensities from the new experiment taken from Fig. 3 (points) and those from a standard MAS spectrum at the corresponding reduced rate (lines) for reduction factors N of (a) 5 and (b) 8, respectively.

Fig. 3. Regions of two-dimensional spectra of 1-13 C-glycine recorded using the new experiment at a MAS rate of 5 kHz with reduction factors of: (a) 4, (b) 6 and (c) 8. The spectral width in x2 was 25 kHz, and there were 32 t1 values with an increment of 6:25 ls, corresponding to 1/32 of the MAS period. Other experimental parameters were as for Fig. 1.

consistent with the one-dimensional sideband pattern of Fig. 1b. Hence, this di€erence arises either from the presence of di€erent polymorphs in our sample or from problems with the previous measurement rather than any failing of the new

experiment to reproduce the one-dimensional sideband intensities. Finally, it should be noted that an experiment which correlates fast and slow sideband patterns has been suggested previously [26] but that this involves non-standard data processing and pulse sequences which increase in length with reduction factor. 5. Conclusion A new NMR experiment which allows a measurement of the CSA by correlating fast and slow spinning sideband patterns has been described. Sidebands intensities in the x1 dimension have been shown to be identical to those expected at an apparent MAS rate xr =N , so that standard

C. Crockford et al. / Chemical Physics Letters 344 (2001) 367±373

methods can be used to extract the principal tensor components. The experiment retains many of the advantages of the two-dimensional PASS method, including simplicity and eciency in terms of the number of t1 increments required. However, the new experiment has been shown to be preferable for measuring narrow shift anisotropies, since for these slow MAS rates are required using two-dimensional PASS leading to rotor instability and loss of sensitivity, as well as sequences of p-pulses of unacceptably long duration. We have found many other solutions to Eqs. (3) and (4) both analytically and numerically, providing sequences with fewer pulses or increased reduction factors and these will be given elsewhere, along with further details of the theory. Variants of the experiment without the storage pulses are under development. References [1] W.S. Veeman, Prog. Nucl. Magn. Reson. Spectrosc. 16 (1984) 193. [2] T.M. Duncan, Chemical Shift Tensors, 2nd edn., The Farragut Press, Madison, 1997. [3] M.M. Maricq, J.S. Waugh, J. Chem. Phys. 70 (1979) 3300. [4] J. Herzfeld, A.E. Berger, J. Chem. Phys. 73 (1980) 6021. [5] Y. Yarim-Agaev, P.M. Tutunjian, J.S. Waugh, J. Magn. Reson. 47 (1982) 51. [6] A. Bax, N.M. Szeverenyi, G.E. Maciel, J. Magn. Reson. 51 (1983) 400.

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