Stepped-frequency NMR spectroscopy

JOURNAL

OF MAGNETIC

RESONANCE

92,320-33 I ( 199 1)

Stepped-FrequencyNMR Spectroscopy MICHAEL

A. KENNEDY,

ROBERT

L. VOLD,

AND REGITZE

R. VOLD

Department of Chemistry University of California, San Diego, La Jolla, California 92093 Received July 20, 1990; revised October 16, 1990 The scope and limitations ofthe recently proposed stepped-frequency NMR experiment (D. W. Sindorf and V. J. Bartuska, J. Magn. Reson. 85,58 1 ( 1989)) have been investigated using *H and *‘AI as examples. Experiments designed to optimize signal-to-noise by the proper choice of pulse shape and strength, filter bandwidth, and sampling time are described. 0 1991 Academic Press, Inc.

High-field magnetic resonance spectra of many quadrupolar nuclei in polycrystalline powders are broadened inhomogeneously by virtue of a distribution of crystallite orientations. They are consequently characterized by extreme breadth, such that it is difficult or impossible to achieve uniform spectral coverage, even with high-power pulses (> 1.5 kW), for which the 90” flip angle may be as short as 1 ps. This situation is in fact the rule rather than the exception in pulsed EPR spectroscopy (1-3) when the EPR lines are broadened inhomogeneously due to orientation-dependent anisotropic g tensors. Thus, for both NMR and EPR, there is increasing interest in understanding the nature of spectra obtained with pulses which cover only a small fraction of the linewidth. Attempts to record wide-line NMR spectra using low-power pulses as opposed to standard CW wide-line techniques to stimulate spins inside an inhomogeneously broadened ultrawide NMR line go back to the early 1960s (4). The technique usually makes use of Can--Purcell spin echoes, which circumvent the problem of having to detect immediately following the pulse. In all of these cases, the magnetization at the top of the echo is accumulated in either a boxcar integrator or with phase-sensitive detection and digitized, signal averaged as needed for sensitivity, and stored in computer memory, after which the magnetic field is changed and the signal averaging repeated at the same frequency (5). Related spin-echo experiments have been performed on superwide NQR spectra, for example, of 63Cu and 65Cu in high-temperature superconducting materials (6)) but in that case, of course, it is the carrier frequency which is being moved to the next point on the lineshape before the time-averaging procedure is repeated. Recently, Sindorf and Bartuska ( 7) reported a stepped-frequency NMR technique which bears a close relation to those noted above and which constitutes the subject of the present paper. The pulse timing diagram for this experiment is shown in Fig. 1. With this method signal is acquired after a weak pulse (rotating field strength w, - 1 kHz) whose carrier frequency is stepped across the spectrum, but instead of signal averaging at the same frequency as in the experiments described above, only a single 0022-2364191 $3.00 Copyright 0 1991 by Academic Press, Inc. All rights of reproduction in any form resewed.

320

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RF PULSE

GATE

RECEIVER

ENABLE

NMR

I

fo=fk-Af FIG.

1. Timing

n

n

ADVANCE

diagram

L

1

1

SAMPLING FREQUENCY

321

SPECTROSCOPY

f,=f, for the stepped-frequency

fo=fk+Af experiment.

point is acquired after each pulse; then the frequency is changed. Thus, an entire spectrum is acquired with a train of frequency-stepped pulses whose separation in time is on the order of milliseconds, with no wait for return to equilibrium between pulses. Typically, the filter bandwidth is reduced to a few hundred hertz to minimize not only noise but also signal from off-resonant isochromats. Signal averaging may be accomplished in the usual way by coaddition of spectra obtained with appropriately cycled phases. The stepped-frequency NMR technique (SFNMR) offers the opportunity of adjusting numerous parameters more or less independently in efforts to maximize the signal-to-noise ratio and minimize distortions. In this paper we report results of experiments designed to determine optimal conditions of pulse amplitude, width, phase, and separation, as well as frequency step size, filter bandwidth, and pulse repetition time. In addition, it is shown both experimentally and theoretically that the signalto-noise ratio can be improved by the proper choice of pulse shape. EXPERIMENTAL

METHODS

A Chemagnetics CMX spectrometer was used to perform SFNMR experiments in an Oxford 5.9 T superconducting magnet. The X-channel RF and TTL gate outputs from the CMX transmitter were used in conjunction with two additional RF gate boxes in series to improve the on-off ratio, and the output was supplied to an EN1 4 11A 10 W linear amplifier. Typical output levels at this stage were on the order of a few hundred milliwatts. Crossed diodes were used at the output of the EN1 amplifier to reduce amplifier noise, after which the pulse was attenuated to the desired level. A Wavetek Model 5070A O-70 dB switchable attenuator in series with a Wavetek Model RA-50 O-10 dB switchable attenuator was used for this purpose. The resulting 5-30 mW pulses were fed through a coupling box designed originally for high-power pulses, in which the crossed diode isolation between transmitter input and probe had been temporarily bypassed so as to eliminate nonlinear diode response to weak pulses. The probe, of conventional design with 5 mm (for deuterium at 38.4 MHz) or 12 mm (for 27A1at 65.1 MHz) coils, could be used for both low-power stepped-frequency

322

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VOLD,

AND VOLD

and high-power quadrupole-echo experiments without removing the sample or altering the probe tuning. The first parameter to choose in setting up the experiment is the frequency step size, which should be selected for adequate digital resolution, commensurate with available computer memory; 5 12 points with a 500 Hz step size is adequate for relatively narrow 2H powder patterns such as those obtained for rotating methyl groups, while a 1 kHz step size is better for full-width 2H powder patterns. Restricting the number of steps to an integral power of 2 is, of course, not necessary for recording the spectrum, but does facilitate apodization, which can be accomplished by Fourier transformation of the frequency-domain spectrum as produced in the experiment, multiplication by any desired apodization function, and inverse transformation to recover the spectrum. Zero filling is also possible by this procedure. The pulse width ( tr,) is then chosen to avoid major excitation of isochromats which are due to be excited by the next pulse. Thus, the spectral coverage l/4& should be less than the frequency step size. Unlike Sindorf and Bartuska ( 7), we observe no special improvement by adjusting the pulse width to place a null of the corresponding sine function at the frequency of the next pulse. For a step size of 500 Hz, a pulse of 1 ms proved optimal. As noted by Sindorf and Bartuska ( 7)) closing the filter to eliminate noise requires delaying acquisition to accomodate the filter delay. For the four-pole Butterworth filters in the CMX spectrometer, 800 PS acquisition delay and 800 Hz filters proved optimal for most of the 2H spectra. Finally, the pulse flip angle is varied for the best signal-to-noise ratio by changing the pulse power level (not the pulse width). It proved useful to determine the relation between pulse power and flip angle by means of pulses applied exactly on resonance to a liquid D20 sample. Of course, when weak pulses are subsequently applied to one line of a quadrupolar doublet their effective strength is larger by a factor of [ I( I + 1) - m( m f 1 )] ‘I2 (8)) and this factor has been included in the pulse widths reported below. Especially for wide spectral widths ( 1 MHz or more is not uncommon) it is important to remove baseline artifacts by appropriate phase cycling. Since the spectral width is often a substantial function of the probe bandwidth, frequency-dependent baseline offsets are to be expected. Fortunately, these cancel with the usual Cyclops four-step phase cycle (9)) which was used in all the stepped-frequency experiments reported here. To implement it, it was necessary to modify the CMX hardware and software slightly to permit independent incrementation of frequency and phase.’ EXPERIMENTAL

RESULTS

Figure 2 shows a stepped-frequency spectrum of perdeuterated hexamethylbenzene (Fig. 2a) together with an ordinary quadrupole-echo spectrum of the same material (Fig. 2b) for comparison. The spike in the middle of the spectrum, which also appears in SFNMR spectra reported by Sindorf and Bartuska ( 7), is due to the excitation of ’ We thank Dean Sindorf of Chemagnetics for providing us with the details necessary for the modification of the pulse programmer/RF unit interface. These modifications allow separate incrementation of frequency, phase, and amplitude.

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Frequency (kHz) FIG. 2. Deuterium SFNMR spectrum (a) and quadrupoleecho spectrum (b) of a 0.1 g hexamethylbenzenedr8 sample. The SPNMR spectrum (a) was recorded from left (high frequency) to right (low frequency) using 400 Hz steps, 800 Hz filter, 1 ms acquisition delay, 1.5 ms, 7.2 mW rectangular pulses, 8 ms pulse spacing, 1 s intersweep delay, and 64 repetitions. The quadrupole-echo spectrum (b) was recorded at 38.4 MHz using a 2 ps dwell time, 1024 point data set, 500 kHz filter, 700 ms recycle delay, 10 ps receiver delay, 40 ps echo defocusing time, 1.8 ps r/2 pulses, and 64 scans. No apodization or zero filling was used in either spectrum.

double-quantum coherence by a weak pulse near the middle of the spectrum (IO), transferred in part into observable signal by the subsequent pulse. Such artifacts are not removed by the Cyclops phase cycle. In fact, since the spin system is far from equilibrium by the time the train of stepped frequencies approaches the center of the spectrum, it is hard to imagine a phase cycle which could fully suppress multiplequantum artifacts. In practice, these artifacts rarely introduce unacceptable lineshape distortion. Neither spectrum in Fig. 2 has been apodized, in order to facilitate direct comparison of signal-to-noise ratios. The signal-to-noise ratio achieved by SFNMR is about a factor of 2 to 3 less than that achieved by the quadrupoleecho method for this particular sample. In part, this is due to the fact that in the SFNMR experiment the optimal flip angle proved to be about 63” rather than 90”, larger flip angles leading to severe distortion, especially in the right side of the spectrum. More generally, signal-to-noise will suffer if significant dephasing occurs during the SFNMR pulse and also if the sampling time is not optimized for a given filter bandwidth and transverse relaxation time. Thus, it is certainly true that nonselective pulse-Fourier transformation is the method of choice, if nonselective pulses are feasible, even if the loss of the multichannel

324

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AND VOLD

advantage ( II) can be minimized for wide lines simply by restricting sampling to encompass only enough baseline to define the spectrum. The digital resolution of the SFNMR spectrum in Fig. 2a (400 Hz/point) is determined by the frequency step size. For relatively narrow spectra, a 100 or 200 Hz step size and a concomitant increase in the number of points is indicated. However, it would then be necessary to use weaker, longer pulses to avoid exciting magnetization from parts of the line destined to respond to the next pulse in the sequence. It may be concluded that stepped-frequency NMR is not a particularly viable technique for looking at relatively narrow lines. Quadrupole-echo methods are definitely superior in this case. Figure 3 shows SFNMR (Fig. 3a) and quadrupole-echo (Fig. 3b) spectra of perdeuterated polyethylene. The SFNMR spectrum was recorded with the frequency stepped downward from left to right, and the dashed line is an ideal slow-passage spectrum simulated using the quadrupole coupling parameters, e*qQ/h = 164 kHz, and 9 = 0.02. Clearly, 500 Hz/point digital resolution is satisfactory for a full-width deuterium powder pattern and should be more than adequate for lineshapes of greater widths. The SFNMR spectrum appears to be free from distortion in the first half (left) of the spectrum until a noticeable multiple-quantum spike occurs in the center. The right side of the SFNMR spectrum in Fig. 3a is enhanced near the center in

250

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-250

FIG. 3. The 38.4 MHz deuterium SFNMR spectrum (a) and quadrupole-echo spectrum (b) of perdeuterated polyethylene (about 0.1 g). The SFNMR spectrum (a) was recorded from left (high frequency) to right (low frequency) using 500 Hz steps, 600 Hz filter, 800 PS acquisition delay, 1 ms, 7.6 mW rectangular pulses, 8 ms pulse spacing, 60 s intersweep delay, and 5 12 repetitions. The quadrupole-echo spectrum (b) was recorded using a 2 ps dwell time, 5 12 point data set, 500 kHz filter, 60 s recycle delay, 10 PS receiver delay, 40 ps echo defocusing time, 1.8 ps r/2 pulses, and 512 scans. No apodization or zero filling was applied in either spectrum. The dashed line in (a) is a simulated slow-passage spectrum calculated using QCC = 164 kHz and n = 0.02. The dashed line in (b) is a simulated quadrupole-echo spectrum for a 1.8 ps 7r/2 pulse which includes correction for finite pulse widths.

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the low-frequency region of the spectrum because the high-frequency mirror transition was excited only milliseconds before. Since the two transitions share a common energy level, pulsing the high-frequency transition causes a perturbation of the population difference associated with its mirror image partner. This sample of polyethylene, for which T, - 30 s (Z2), illustrates that such “T1 artifacts” are to be expected in SFNMR spectra. One way to avoid displaying the artifact, without compromising the integrity of the spectrum, would be to perform the experiment in two stages, first sweeping from right to left and then from left to right, subsequently combining the undistorted halves into a single, distortion-free spectrum. It is worth noting that the quadrupole-echo spectrum of polyethylene (Fig. 3b) does not agree as well with its simulation, a quadrupole-echo spectrum calculated with 1.8 ps 7r/2 pulses, as does the left half of the SFNMR spectrum in Fig. 3a with the calculated slow-passage spectrum. In addition to the well-known (13) loss of intensity in the wings of the quadrupole-echo spectrum associated with finite spectral coverage of the 1.8 Z.LS pulses, the center of the experimental quadrupole-echo spectrum contains contributions from motionally averaged, “amorphous” polyethylene ( 14, IS). This is not taken into account in either of the two simulations, which include only static quadrupolar contributions to the lineshape. Thus, it is possible that the excellent agreement between the simulation and the left half of the experimental SFNMR spectrum is coincidental, and that the “hump” in the middle is lost because it has a transverse relaxation time which is short compared to the length ( 1 ms) of the SFNMR pulses. Other effects of exchange-via saturation transfer-are discussed in connection with the SFNMR spectra of urea-u’, described below. It should be noted that the effect of dipole-dipole coupling on quadrupole-echo lineshapes (but not the slow-passage lineshape) is the appearance of a hump in the middle of the spectrum ( 16,17). However, for the relatively short 40 I.LSquadrupole-echo pulse spacing used in Fig. 3b, such effects should be absent. Aluminum ammonium sulfate has an 27A1 quadrupole coupling constant of 450 kHz ( 18)) which approaches the upper limit of spectral width which can be adequately covered by conventional high-power methods. Thus this material provides a convenient example of the primary reason for interest in SFNMR methods-their potential for undistorted observation of wide spectra of large samples. The top trace (c) of Fig. 4 shows the calculated slow-passage spectrum corresponding to the experimental spectra below. Figure 4a shows a quadrupole-echo spectrum of 2.3 g of A1NH4( Sod)*. Pulse power delivered to the 12 mm coil was limited by arcing to 230 W, yielding a 5.0 ps 90” pulse. The effect of an acquisition delay following the second pulse is illustrated by the loss of intensity between the perpendicular edges of the wide satellite transitions (see inset to Fig. 4a). At the same time the outermost wings (parallel edge of the z --* 5 transitions) are better excited in the SFNMR experiment (compare the insets to Figs. 4a and 4~). The mismatch between the simulation (dashed line) and the experimental quadrupole-echo spectra is due in part to these two problems and in part to the fact that the effective field strength (and, therefore, the effective flip angle) is not the same for all transitions among the spin Z = 2 energy levels of 27A1. The “Al SFNMR spectra of A1NH4( Sod)2 (Fig. 4b) demonstrates how some of these problems can be reduced. As shown most clearly in the inset to Fig. 4b, the intensity between the perpendicular edges is in good agreement with the calculated

KENNEDY,

326

VOLD,

AND VOLD

a

,&,.A+-..-

-250

-150

-50

50

150

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Frequency (kHz) FIG. 4. 27Al NMR spectra of 2.3 g AIN&( SO.+)z.(a) Quadrupole-echo spectrum obtained at 65.1 MHz with a 5 ps r/2 pulse, 2 ps dwell time, 1024 point data set, 500 kHz filter, 1 s recycle delay, 10 ps receiver delay, 40 ps echo defocusing time, 1.8 PCSr/2, and 512 scans. No apodization or zero filling. (b) SFNMR spectrum recorded from left (high frequency) to right (low frequency) using 500 Hz steps,2 kHz filter, 200 ps acquisition delay, 1 ms, 7.6 mW rectangular pulses, 2 ms pulse spacing, 500 ms intersweep delay, and 256 repetitions. (c) Slow-passage spectrum calculated using e*qQ/h = 450 kHz. The insets to (a) and (b) show portions of the spectra, expanded in both directions, to illustrate the better agreement between the SFNMR spectrum and the slow-passage lineshape (dashed curve).

lineshape, and the first half (left, or high-frequency region) of the spectrum is free from distortions. On the other hand, the central, narrow transition suffers from an apparent phase distortion which is very reminiscent of the rapid-passage effects that occur in continuous-wave experiments. The second half of the spectrum- is consistently distorted even after exhaustive attempts to optimize experimental parameters. Again, these distortions result from pulsing transitions with shared energy levels. Figure 5 illustrates a second type of T1 artifact, which may prove diagnostic for the presence of slow motion. In this SFNMR spectrum of urea-d4, the right half of the spectrum is weaker than the left half. It is known (19, 20) that urea undergoes slow motion about both the C-O and the C-N bonds, at rates on the order of 1 kHz or less at room temperature. Thus, when the train of pulses in the SFNMR sequence irradiates the left half of the spectrum, the resulting partial saturation is transferred by the large-angle, slow motion to other parts of the lineshape. When the observation pulses arrive, reduced intensity is therefore found. Quantitative analysis of this phenomenon appears to be complicated, but the observation of saturation transfer may prove useful in a qualitative sense.

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Frequency(kHz) FIG. 5. SFNMR deuterium spectrum of urea-d, recorded at 20°C from left (high frequency) to right (low frequency) using 500 Hz steps, 800 Hz filter, 800 wusacquisition delay, 2 ms, 3.8 mW rectangular pulses, 15 ms pulsespacing, 50 s intersweep delay, and 666 repetitions. The dashed curve is a slow-passage spectrum calculated with e’qQ/h = 212 kHz and 7 = 0.145 (17).

SHAPED

PULSES

The low-power, long RF pulses used in SFNMR are well suited for experiments in which the amplitude and/or phase of the RF is varied during the pulse in efforts to improve various features. For example, shaping the pulse to provide more nearly “tophat” (21) excitation should help to minimize distortion due to unwanted stimulation of isochromats outside the nominal bandwidth of the current pulse. Similarly, since the side lobes of the Fourier image of the pulse are negative, minimizing them might be expected to improve the overall signal-to-noise ratio. It is important to realize that the Fourier transform of the pulse itself is not a good vehicle for understanding what the spins do in a stepped-frequency NMR experiment. Unless the flip angle is small enough (about 30” or less) for linear response theory to apply, there is 110 simple relation between the actual spin response and the Fourier transform of the pulse (22). While a full simulation of the stepped-frequency experiment is very cumbersome because of the need to keep track of off-diagonal densitymatrix elements after each of many thousand pulses, some insight can be gained by the following relatively straightforward calculation. During a pulse of frequency w, amplitude wl, and length tP, the spin Hamiltonian for a one-deuteron system is 3? = -(oo - o)Zz - WIZx + 543z:

- 1.1).

[II

Here the Hamiltonian has been written in the frame rotating at the frequency of the pulse, and 2wo is the quadrupolar splitting. Asymmetry in the quadrupole tensor, as well as second-order quadrupole effects, has been ignored. The use of spin operator Z, to describe the effects of the RF field implies a pulse of phase zero. In absence of

KENNEDY,

328

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AND VOLD

relaxation, the spin density matrix u( t, ) at the end of the pulse, to that at the start of the pulse (t = to) by the simple expression a( to +

tp)

= e-‘-0(

t

=

to

+

tp,

is related

[21

to)e'-.

The expectation value of any desired quantity is obtained by the usual trace relation. For example, the desired spin response to a pulse of zero phase is proportional to ( IY) = trace( u1,).

[31

Evaluating Eqs. [ 1]- [ 3 ] as a function of wo for fixed values of all the other parameters, while using 1, for u( to), amounts to generating a spin response function for the spin I = 1 system, which is equivalent to those based (23) on solving Bloch equations for spin I = 1 systems. Figure 6 shows the results of such calculations for rectangular pulses with flip angle 10” (Fig. 6a) and 90” (Fig. 6b) and for “sawtooth” pulses (trailing edge vertical) with the same total flip angles (Figs. 6c and 6d, respectively). In all these calculations the resonance offset (o. - w) was fixed at 100 kHz and uQ was varied in 0.03 kHz increments from 90 to 110 kHz. The absolute scale factor for all four simulations is the same. As expected, the spin response to a rectangular pulse resembles a sine function, with larger amplitude for the larger flip angle. In comparison, the spin response to a sawtooth pulse has smaller side lobes than the response to a rectangular pulse. If one imagines a set of spins with uniformly distributed values of uQ, linear response theory (24) implies that the shape of the excitation function and the spin response shown in Fig. 7 form a Fourier transform pair. This is verified, for small flip angles only, by comparing Figs. 6a and 6c with their Fourier transforms, which are shown in Figs. 7a and 7c, respectively. Except for a trivial oscillation introduced by defining the frequency origin for the Fourier transform operation at the left edge in Fig. 6, Figs. 7a and 7c are simple rectangular and sawtooth waveforms, respectively. On the

a

C

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21/a (kHz)

110

90

A

95

100

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274 (kHz)

FIG. 6. The spin response, (I,), to rectangular (a and b) and sawtooth (c and d) RF pulses of flip angle 10” (a and c) and 90” (b and d) calculated according to Eqs. [I]- [ 3 ] as described in the text. Twenty intervals of 50 ps were used to generate the sawtooth pulse profile in (c) and (d) The vertical edge of the sawtooth is trailing.

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a

H

I ms FIG.

7. Fourier transforms of spin responses (a-d) in Fig. 6.

other hand, for large flip angles (Figs. 6b and 6d) the Fourier transform of the spin response differs in two important respects from the pulse waveform (see Figs. 7b and 7d, respectively) : the Fourier transform of the spin response appears to be a somewhat distorted image of the pulse, and this image is followed by a component which persists well after the pulse is over. Figures 6 and 7 can be used to rationalize experimental SFNMR spectra obtained using rectangular (a and b) and sawtooth (c and d) excitation. Figure 8 shows SFNMR spectra of a powder sample of ClgD40/urea inclusion compound. Spectra obtained with 20” flip angles (Figs. 8a and 8c) show signal only at the major turning points of the powder pattern, such that the spectra appear almost “depaked” (25). This occurs precisely because in the linear response regime, the spin response is the Fourier transform of the excitation waveform if there is a uniform distribution of quadrupole couplings. Since one detects after the pulse, the signal is necessarily zero (24, 26). Signal is detected near the horns of the powder pattern because the distribution of wQ values is far from uniform there. As expected from the excitation profiles in Fig. 6 and their Fourier transforms in Fig. 7, the signal following sawtooth pulses is twice as large as that following rectangular pulses. In essence, this occurs because the maximum RF amplitude in the sawtooth pulse must be twice as large to achieve the same flip angle. For pulses with large flip angle, linear response theory fails, but it is still true that the signal intensity is proportional to the zero-frequency component of the Fourier transform of the spin response (i.e., to its integral) for the case of a uniform distribution of uQ values. Thus, when Figs. 6b and 6d are compared with 7b and 7d, respectively, it comes as no surprise that the signal-to-noise ratio in Fig. 8d is roughly twice that in 8b. The relative intensities of these two spectra, as well as the fact that a more or less faithful representation of the lineshape is obtained (except for T, and multiplequantum artifacts noted above), can be qualitatively understood in terms of the magnitude of the “tail” after the waveforms in Figs. 7b and 7d. Figure 9 shows an additional example of the signal-to-noise ratio improvement obtained with sawtooth pulses. In fact, it was these observations which prompted the calculations displayed in Figs. 6 and 7. As for deuterium, sawtooth pulses applied in the “Al stepped-frequency experiment on AlNH4( S04)* yield a twofold improvement

KENNEDY,

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VOLD, AND VOLD

1

d

b

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75

0

~

-75

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75

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Frequency (kHz)

FIG. 8. Experimental SFNMR spectra of the channel inclusion compound urea/n-nonadecane-& obtained with rectangular (a and b) and 165 increment sawtooth (c and d) pulses. All spectra were recorded from left (high frequency) to right (low frequency) using 500 Hz steps, 800 Hz filter, 800 ps acquisition delay, 6 ms pulse spacing, 500 ms intersweep delay, and 2048 repetitions. The effective flip angle was 20” in (a) and (c) and 63“ in (b) and(d).

C

--

b

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Frequency (kHz) FIG. 9. 27Al SFNMR spectra of AlNH4(S04)z obtained using 1 ms, 7.6 mW rectangular (a); 1 ms, 25 mW (av.) sawtooth (b); and 1 ms, 25 mW (av.) symmetric triangular (c) pulses. All spectra were recorded from left (high frequency) to right (low frequency) using 500 Hz steps, 2 kHz filter, 200 ps acquisition delay, 2 ms pulse spacing, 500 ms intersweep delay, and 256 repetitions.

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over rectangular pulses. It is amusing to note that a triangular excitation waveform actually yields no signal at all-this counterintuitive result is supported by calculations (not shown) of excitation profiles analogous to those shown in Figs. 6 and 7. CONCLUSIONS

Stepped-frequency NMR spectra of quadrupolar nuclei suffer in comparison with standard, high-power pulse-Fourier transform methods both in regard to signal-tonoise ratio (a factor of 2-3) and freedom from artifacts. Nevertheless, in cases where pulse power is inadequate to achieve nonselective pulses, SFNMR provides a viable alternative. Moreover, SFNMR can, in special circumstances, actually provide a more accurate representation of the ideal slow-passage spectrum than is available from echo techniques. ACKNOWLEDGMENTS We am grateful to D. Sindorfand V. Bartuska for making their results available to us prior to publication and for many useful discussions. We also thank M. Levitt for some highly useful comments regarding pulses in the linear response regime. This work was supported by grants from the National Science Foundation (CHE9000427 ) and the Office of Naval Research (NO00 14-88-K-2003 ) REFERENCES 1. W. B. MIMS, Rev. Sci. Instrum. 36, 1472 (1965). 2. L. KEVAN AND R. N. SCHWARTZ (Ed%), “Time Domain Electron Spin Resonance, ” Wiley, New York, 1979. 3. G. L. MILLHAUSER AND J. H. FREED, J. Chem. Phys. 81,37 ( 1984). 4. J. ITOH, Y. MASUDA, K. ASAYAMA, AND S. KOBAYASHI, J. Phys. Sac. Jpn. 18,455 (1963). 5. H. E. RHODES, P. K. WANG, H. T. STOKES, C. P. SLIGHTER, AND J. SINFELDT, Phys. Rev. B 26,3559 (1982). 6. W. W. WARREN, JR., R. E. WALSTEDT, G. F. BRENNERT, R. J. CAVA, R. TYCKO, R. F. BELL, AND G. DABBAG~, Whys. Rev. Lett. 62, 1193 ( 1989). 7. D. W. SINDORF AND V. BARTUSKA, J. Magn. Reson. 85, 581 ( 1989). 8. A. ABRAGAM, “The Principles of Nuclear Magnetism,” Oxford, Univ. Press, London/New York, 196 1. 9. D. I. HOULTAND R. E. RICHARDS, Proc. R. Sac. London A 344, 311 (1975). 10. S. VEGA AND A. PINES, J. Chem. Phys. 66,5624 ( 1977). II. R. R. ERNST AND W. A. ANDERSON, Rev. Sci. Instrum. 37,93 ( 1966). 12. R. R. VOLD AND R. L. VOLD, unpublished data. 13. M. BLOOM, J. H. DAVIS, AND M. I. VALIC, Can. J. Phys. 58, 1510 ( 1980). 14. H. W. SPIESS, J. Chem. Phys. 12,6755 (1980). 15. P. M. HENRICHS, J. M. HEWITT, AND M. LINDER, J. Magn. Reson. 60,280 (1985). 16. N. B~DEN AND P. K. KAHOL, Mol. Phys. 40, 1117 (1980). 17. N. J. HEATON, R. R. VOLD, AND R. L. VOLD, J. Chem. Phys. 91,56 ( 1989). 18. E. OLDFIELD, H. K. C. TIMKEN, B. MONTEZ, AND R. RAMACHANDRAN, Nature (London) 318, 163 (1985). 19. J. W. EMSLEY AND J. A. S. SMITH, Trans. Faraday Sot. 51, 1233 (1961). 20. T. CHIBA, J. Chem. Phys. 38,259 ( 1965). 21. H. GEEN, S. WIMPERIS, AND R. FREEMAN, J. Magn. Reson. 85,620 ( 1989). 22. D. I. HOULT, J. Magn. Reson. 35,69 ( 1979). 23. J. FRIEDRICH, S. DAVIES, AND R. FREEMAN, J. Magn. Reson. 75, 390 (1987). 24. W. S. WARREN AND M. S. SILVER, in “Advances in Magnetic Resonance” (J. S. Waugh, Ed.), Vol. 12, p.247, Academic Press, San Diego, 1988. 25. M. BLOOM, J. H. DAVIS, AND A. L. MACKAY, Chem. Phys. Lett. 80, 198 ( 1981). 26. A. M. WEINER, J. P. HERITAGE, AND R. N. THURSTON, Opt. Lett. 11, 153 ( 1986).