multi-frequency improved constant amplitude pulses for broadband inversion

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Research ArticleReceived: 15 February 2008 Revised: 28 May 2008 Accepted: 31 July 2008 Published online in Wiley Interscience: 15 September 2008

(www.interscience.com) DOI 10.1002/mrc.2319

Multi-frequency improved constant amplitudepulses for broadband inversionDouglas Brown∗

Optimization of constant amplitude broadband inversion pulses for maximum inversion over a range of multiple fields yieldsmore regular pulses with better tolerance to B1 errors than those optimized for a single field. These multi-frequency improvedconstant amplitude (MICA) pulses as constructed for 13C broadband inversion give best results for HMQC and especially HSQCtype experiments. Most of the advantages of MICA pulses versus other inversion pulses in these experiments can be attributedto their relatively short durations. Linearly truncated versions of these pulses do not offer any advantage over MICA pulsesfor these applications. MICA inversion pulses can also be used for decoupling and a 13C decoupling example gives twice thedecoupling bandwidth as the GARP1 sequence at the same power level with no appreciable increase in decoupling sidebands.Copyright c© 2008 John Wiley & Sons, Ltd.

Keywords: NMR; 13C; broadband; inversion; shaped; pulse; decoupling; adiabatic

Introduction

Minimal duration pulses are often desirable in NMR experimentsto reduce loss of signal from relaxation processes and minimizeevolution of phase errors. Constant amplitude NMR pulses canostensibly provide minimal duration solutions for a given amountof power. Broadband inversion pulses with constant amplitudehave been previously investigated. In those studies, the BIP[1]

and BIBOP[2] were created by optimizing the phases of eachwaveform element to yield maximum uniform inversion overa predefined bandwidth at a maximum modulation frequency.These constant amplitude inversion pulses were found to providemore efficient broadband inversion than other solutions withgood tolerance to B1 errors and in both cases it was foundthat computer optimized constant amplitude inversion pulses asshort as 200 µs can efficiently invert magnetization over 100 kHzor more at reasonable modulation frequencies for use with 13Cexperiments.[1,2]

There are some problems associated with these inversionpulses. The centers of the constant amplitude pulse sweepscan be near-adiabatic, but since the ends have large frequencyand hence trajectory swings, they will be highly nonadiabaticin these regions. As the pulses are optimized only for in-bandmagnetization and have the frequency sweep swings at the ends,they can have large out-of-band excitation, but this is largelyirrelevant to broadband applications. Also, over a certain range ofbandwidths and modulation frequencies, the best solution phaseprofiles of constant amplitude pulses become distorted fromthe more ideal parabola-like shapes.[2] These distortions resultin less smooth and less linear frequency sweeps and likely alsoresult in more deviation from adiabatic behavior. Unfortunatelythe largest distortions in phase profiles occur in the range of100–200-µs pulses at bandwidths of approximately 30–80 kHzand modulation frequencies of 16–20 kHz, the approximaterange of conditions needed for full power constant amplitude13C broadband inversion with current high field spectrometers.Finally, although BIP and BIBOP typically have symmetrical

phase sweeps resembling parabolas, the phase profiles donot fit any simple equation that would make it easy togenerate new pulses of different bandwidths and modulationfrequencies.

In this work, optimization of constant amplitude broadbandinversion pulses over multiple modulation frequencies is exploredto remedy the phase profile distortions that occur with singlemodulation frequency optimized BIP-type pulses over the previ-ously described conditions for 13C broadband inversion. The samemethod was also used to create inversion pulses with constantamplitude centers and ends truncated to zero amplitude thatare compared with the pure constant amplitude pulses. Initialresults using these pulses for broadband 13C decoupling are alsopresented.

Experimental

A modern implementation of the Levenberg-Marquardt nonlinearleast squares algorithm[3] running on a PC with OpenSUSE 10.2Linux, Intel E6600 processor and 2GB of random access memory(RAM) was used for all numerical optimizations in this work. BIPand BIBOP have previously been shown to have symmetrical phaseprofiles from unconstrained optimization, so for this work half ofthe waveforms were optimized to save time with the second halfbeing a mirror image of the first.[1,2] Waveforms were initiallydetermined with 50 variables then interpolated to 100 variablesand optimized again to give final shaped pulses consisting of200 total steps. The 259-µs pulse was again interpolated andoptimized for 200 variables with 400 steps in the final pulse. Furtherdigitization resulted in little to no measurable improvement of the

∗ Correspondence to: Douglas Brown, Department of Chemistry, IndianaUniversity, 800 E. Kirkwood Ave., Bloomington, IN 47405, USA.E-mail: [email protected]

Department of Chemistry, Indiana University, Bloomington, IN 47405, USA

Magn. Reson. Chem. 2008, 46, 1037–1044 Copyright c© 2008 John Wiley & Sons, Ltd.

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solutions. Convergence was improved by adding a small amountof random noise (typically ∼0.001% RMS) to the solutions andthen repeating the optimizations. This cycle was repeated untilthere was no further improvement in the residual errors. Severalrepetitions were performed with different starting data, eitherrandom or other waveforms from this work, to confirm thatsolutions were global minima for the given target function. Goodresults were achieved within a few hours using this approach, butrefinement to unchanging solutions took as long as three days forthe longest duration pulses. Three to five point smoothing wasused in the process, if it aided in convergence, but all waveformswere optimized without smoothing as the final step.

Convergence to a best solution was improved by using a targetfunction that included Mx, My and Mz rather than just optimizingMz to −1. For an inversion pulse the target function minimized forthis work was (Mz + 1) + (Mx • My)/C with C = 1000 or 10 000. Formultiple modulation frequency minimizations the results of thesame target function at each modulation frequency were summed.Bloch simulations were 256 or 512 data points in length and onlymagnetization in the desired bandwidth was considered. Varian’sPbox[4] was also used to create all reference waveforms. Blochsimulations and visualizations were performed primarily with theshaped pulse tools included in Varian’s Vnmr and VnmrJ[5] softwareincluding Pbox and pulsetool.

NMR spectra were recorded at 25 ◦C on a Varian INOVA500 MHz spectrometer with waveform generators and a 5-mm-HCX triple resonance indirect detection probe or a Varian600 MHz Direct Drive spectrometer with a 5-mm-HCN coldprobe. The sample used for all experiments was saturatedstrychnine in CDCl3 and all comparison series of experiments wereperformed sequentially. Varian standard pulse sequences[5] wereused for gHMQC,[6,7] gHMBCAD (CRISIS-gHMBC[8]), gHSQC[9 – 11]

with multiplicity selection and gHSQCAD (CRISIS-gHSQC[12])experiments. The DEPT-HMQC experiment was performed withthe gradient pulse sequence of Spitzer et al.[13] Where needed,the two hard inversion pulses in the experiments were replacedwith shaped pulses and evolution times adjusted accordingly. Thebroadband refocusing pulse (MICA180R-200-50) for the standardHSQC experiments was optimized by the same method as abovefor a 200-µs duration at a modulation frequency of 18 kHz with a50 kHz bandwidth.[14] The 1H sweep widths were 5 kHz at 500 MHzand 7 kHz at 600 MHz and the 13C sweep width was 30 kHz at bothfields. 128 T1 increments were acquired for absolute value HMBCexperiments and 96 increments acquired for all phase sensitiveexperiments. Sine bell or half sine bell window functions wereused and no linear prediction was done.

For decoupling experiments a standard 13C gHSQC pulsesequence was used on our strychnine reference sample. Forthe broadband decoupling offset sweeps the 13C frequency wascentered on the 13C resonance of the methyl group at 5.9 ppm1H using delays based on HCJ = 161 Hz. The 1H signal from thismethyl group was observed, while varying only the 13C decouplingfrequency.

Results

The shaped pulses for this work were created around a modulationfrequency of 18 kHz, or the same power needed to producea 13.9-µs rectangular 90◦ pulse. Our 400-, 500- and 600-MHzNMR spectrometers can all safely produce 13C pulses of a fewhundred microseconds at these powers. At this modulation

frequency it was determined from initial simulations that 100–200-µs constant amplitude pulses would be needed to invert >99%of the magnetization over the respective 30–80 kHz bandwidthsneeded for broadband 13C experiments on 500–900-MHz NMRspectrometers, so this range of pulse widths was examined.

Constant amplitude broadband inversion pulses optimizedfor the single modulation frequency of 18 kHz are shown inFig. 1(a)–(e). They are consistent with previously described BIP andBIBOP, and in particular deviations from the parabola-like shapestypical of BIP-type pulses are consistent with those seen for BIBOPover some ranges of frequency versus duration.[2] Numerous runsfrom different starting conditions with and without smoothingverified that these ‘bumpy’ phase sweeps at constant amplitude doindeed provide maximum inversion for the specified pulse widthsand bandwidths at the single 18-kHz modulation frequency.

Although these waveforms provide maximum inversion at thedesired modulation frequency there is some cause for concern inthe use of pulses with such nonsmooth phase sweeps. A set ofconditions that can be used to form adiabatic broadband inversionpulses is constant amplitude along with a parabolic phase sweep,as used with CAWURST[15,16] and CHIRP[17 – 19] pulses for example.BIP have been previously described as ‘semiadiabatic’, but areactually more adiabatic in the centers where the phase sweep canbe roughly parabolic.[1] Such large deviations from a parabolicphase sweep will then result in these pulses being even lessadiabatic.

As also seen in Fig. 1(a)–(e), the rough phase profiles resultin correspondingly uneven frequency sweeps rather than moredesirable smooth, linear sweeps. Note that the range wherethese distortions appear is nicely enveloped by the set of pulsedurations being studied for our 18-kHz 13C pulses, with the 100-and 200-µs pulses at either end of the range having smootherphase and frequency sweeps than the others. However, even theoptimized 200-µs pulse still shows a pronounced modulation onthe frequency sweep, despite its phase sweep appearing to befairly smooth.

The effect of this uneven frequency sweep on adiabicity can beevaluated from the plots in Fig. 2(a)–(e) that show the bandwidthprofiles of these pulses over a range of modulation frequencies.For adiabatic pulse changes in modulation frequency over a widerange should result in little to no change in the inversion profilesbut with our single frequency optimized 100–200-µs constantamplitude pulses, it is apparent that there is a distinct limitationon conditions that will give >95% inversion.

To improve the phase and frequency sweeps of these constantamplitude pulses, optimization of the solutions over a rangeof modulation frequencies was attempted. The initial choice ofoptimizing at three modulation frequencies, the 18-kHz centralfrequency, 16 and 20 kHz, proved fortuitous and worked well overthe full range of pulses in this work. For the longer pulse widthsbetter results were ultimately obtained by using five modulationfrequencies, with the outer frequencies between ±2000 and4000 kHz around the 18 kHz center and the middle frequencyhalfway between the center and outer frequencies. Typically1 : 2 : 4 : 2 : 1 weighting was used. For each result the weightedsum of the target functions derived from the Bloch simulationsat the various modulation frequencies were optimized and thecenter 18-kHz waveforms used as the final result. Different startingdata sets, including the single frequency results, were used toverify that global minima were found. These multi-modulationfrequency optimizations were generally as straightforward as thesingle frequency optimizations.

www.interscience.wiley.com/journal/mrc Copyright c© 2008 John Wiley & Sons, Ltd. Magn. Reson. Chem. 2008, 46, 1037–1044

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Multi-frequency improved constant amplitude pulses

Figure 1. Frequency and phase profiles for 18-kHz constant amplitude pulses. Single frequency optimized constant amplitude pulses from 100 to200 µs are shown in (a)–(e), MICA pulses are shown in (f)–(j). Optimized bandwidths are 30 kHz (a), (f), 40 kHz (b), (g), 50 kHz (c), (h), 60 kHz (d), (i) and80 kHz (e), (j).

The phase and resultant frequency sweeps for the multiplefrequency optimized pulses are shown in Fig. 1(f)–(j) with thelonger duration pulse created for decoupling shown in Fig. 5. Thedifferences in the phase profiles in Fig. 1(f)–(j) versus those inFig. 1(a)–(e) are quite evident, as is the flatness of the frequencysweeps in the centers of the pulses. On the basis of the increasedlinearity of the frequency sweeps, it can be inferred that themulti-modulation frequency pulses also have better adiabaticproperties. In Fig. 2(f)–(j) it is apparent that all the new pulses havea significantly wider simulated range of 95% inversion than thecorresponding single frequency optimized pulses.

Attempts to fit simple equations to the whole phase profilesof the new pulses were still unsuccessful, however good fitswere obtained on the central 60% of the longer duration pulsesusing the ZunZun[20] process. The best simple fits of the pulsecenters were always Gaussian peak equations, closely followed byLorentzian peak equations. However, the difference in the residualerror between these two possibilities was very small in all cases.

Therefore, on the basis of these results the phase sweeps of thepeak centers are either Gaussian or Lorentzian or a mixture ofthe two.

The cost of the improvement to wideband performance isslightly less inversion at the center modulation frequency of18 kHz, as seen in Fig. 3(a) and (b). For most experiments, thissmall difference will be insignificant even if the pulses are perfectlycalibrated and it will be generally more useful to have the widebandperformance of these new ‘multi-frequency improved constantamplitude’(MICA) pulses. The naming convention used in thiswork is ‘MICA 180’ which indicates the 18 kHz version of a multi-frequency optimized inversion pulse, followed by the durationin microsecond and the bandwidth in kHz. The MICA180-120-40pulse is then a 120-µs pulse that has been optimized to invertmagnetization over a 40 kHz bandwidth. ‘MICA180T’ refers tolinear truncated inversion pulses and ‘MICA180R’ refers to the180◦ refocusing pulse.

Magn. Reson. Chem. 2008, 46, 1037–1044 Copyright c© 2008 John Wiley & Sons, Ltd. www.interscience.wiley.com/journal/mrc

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Figure 2. Decoupler modulation frequency versus bandwidth for the same constant amplitude (a)–(e) and MICA180 (f)–(j) pulses in Fig. 1 and also theMICA180T truncated versions (k)–(o). The innermost contour delineates 95% inversion as predicted by Bloch simulation, while contour gradations are in5% steps.


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Figure 3. Bloch simulations of Mx, My and Mz for a 180-µs constant amplitude pulse with 60 kHz bandwidth optimized for 18 000 kHz (a), MICA180-180-60(b) and MICAT180-180-60 (c). Mz is magnified 20 times so that the small differences between results are more visible.

Table 1. Relative total signal by experiment of first spectrum of some2D NMR experiments comparing replacement of hard 13C inversionpulses with CAWURST and two MICA pulses

HMQC HMQCDEPT HMBC HSQCDEPT HSQCAD

Hard 1.00 – 1 1.03 0.63

CAWURST1 1.40 1 1.12 1.03 –

CAWURST2 1.49 – – 1.22 0.92

M180-120 1.58 1.10 1.18 1.44 1.36

M180-160 1.59 1.13 – 1.43 1.34

The 13C hard inversion pulse was used at full 13C power. CAWURST1 wasa 0.667 ms pulse at 3 db less than full power, CAWURST2 was a 1.333 mspulse at 9 db less than full power. MICA180-120-40 and MICA180-160-50 pulses were both used at full power. HMQC, HSQCDEPT and HSQCADvalues are to the same scale, with the hard pulse HMQC result used asthe reference. HMQCDEPT results are scaled with the CAWURST1 resultas the reference. HMBC results are scaled with hard pulse HMBC resultas the reference.

The actual performance of these pulses as replacements forhard pulses and various other broadband inversion pulses hasbeen evaluated in hundreds of experiments, run over the 2 yearperiod they were developed. The main goal of this work wasto develop the best possible short duration broadband inversionpulses for use in our lab, so in this report MICA pulses are comparedwith hard pulses and a CAWURST[15,16] pulse explicitly created asour longer duration reference. The CAWURST pulse was chosento represent a reliable, tolerant and efficient broadband inversionpulse that can operate at longer durations and was run at twodifferent powers and durations for these tests. With some caresimply using the area (or absolute value area) of the first 1Dspectrum of 2D data sets gives a good representation of the datasets for this type of comparison. A summary of the results for someHMQC, HMBC and HSQC experiments commonly used in our labis shown in Table 1.

The expectation for this work was that shorter durationbroadband pulses would be most useful in HSQC experiments.In Table 1 it is apparent that the MICA pulses give up to about10% more signal versus the much longer duration referenceCAWURST pulses in the HMQC and HMBC experiments. Whileany gain in signal is welcome this amount of improvement is notparticularly noticeable in practice with routine multidimensionalNMR spectroscopy, but does show that the MICA pulses areat least as efficient as other adiabatic pulses for broadbandinversion. However, the increase of up to 50% in area for theHSQC experiments is more significant. Along with the HMQC and

HMBC results, most of the gain for the HSQC experiments canbe attributed to using shorter duration pulses that are at leastnearly as efficient as the CAWURST pulses. The standard gHSQCand CRISIS-gHSQC results were obtained concurrently, so they canbe compared directly also. The 1.333-ms CAWURST pulse gavesuch poor results in the CRISIS-HSQC experiment that it was notreported. Also not shown are results with the shortest durationadiabatic pulses used, but for example a 324-µs WURST2i[21] pulseof 50 kHz bandwidth at 18-kHz modulation frequency gave areasbetween those obtained with the CAWURST and MICA pulses,again indicating that the difference in the results can be attributedto the duration of the inversion pulses.

Truncated constant amplitude pulses created via the samemethods as used above were also investigated. After testingseveral different truncation functions it was determined that linearreduction of 10% of the ends of the pulses to zero amplitudeprovided adequate results over a range of these relatively shortpulse durations based only on the efficiency of in-band inversionand provided a good basis for comparison. Although less inversioncould be expected from these trapezoidal pulses, because ofhaving 10% less power than the corresponding constant amplitudepulses, pulse durations were kept the same as a goal of the workwas to evaluate the shortest possible inversion pulses.

The same problems as previously discussed with distortedphase profiles appeared with these pulses and were also curedwith optimization at the same multiple frequencies. In this workthe resultant pulses are referred to as MICA180T pulses.

The phase profiles and frequency sweeps for the MICA180Tpulses are shown in Fig. 4(a)–(e). Their performance over arange of modulation frequencies can be seen in Fig. 2(k)–(o). Themagnetization profile of MICA180T-180-60 versus MICA180-180-60and the 180-µs single frequency pulse is shown in Fig. 3.

Truncation of the amplitude profiles at the ends results insmoother initial trajectories as seen in Fig. 4 and smootherbehavior, especially at the ends, across a wide range of modulationfrequencies as seen in Fig. 2 for MICA180T versus MICA180 pulses.A little less magnetization overall is inverted however, as seen inFigs 2 and 3, and at least some of this effect can be attributed tothe 10% decrease in power of the pulses.

Overall any improvement of MICA180T pulses over MICA180pulses is limited to smoothness of initial trajectory and smoothnessof wideband behavior across the range of modulation frequencies.The original MICA180 pulses invert a little more magnetizationover a slightly greater bandwidth, again probably because ofthe 10% difference in power, and the initial trajectory seemsto be otherwise unrelated to the wideband performance, whencomparing same duration MICA180 and MICA180T pulses. For the


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Figure 4. Frequency and phase profiles for linear truncated MICAT pulses. The same durations, bandwidths and other conditions as the pulses in Fig. 1were used.

purpose of minimal duration broadband inversion, it is apparentthat MICA180 pulses are still preferable to the truncated versionsfor general use.

Decoupling performance of a MICA180 inversion pulse wasalso investigated. It seemed likely that these B1 tolerant pulsescould provide better decoupling than single frequency optimizedBIP-type pulses.

For comparison a ‘GARP1 replacement’ 13C decoupling MICA-based decoupling sequence was constructed that was designedto be run at the same modulation frequency and power as areference GARP1[22] sequence. The reference GARP1 sequencewas optimized to give a 15-kHz decoupling bandwidth, or a 2.7-

kHz decoupling field. At our standard 18-kHz decoupling field, a259-µs MICA180 pulse was needed and a bandwidth of 90 kHz wasempirically determined to give at least 99% inversion. Although200 total elements were sufficient for shorter pulses, there wasenough difference with more digitization in this case to warrantcreating a 400 element pulse. Optimization was again done forhalf the waveform of 200 elements and the reverse sequence wasadded to create a time symmetrical pulse.

The resultant waveform is shown in Fig. 5 and displays a strikingregularity, with wide oscillations at the ends of the frequencysweep quickly decaying to a linear sweep in the center for muchof the duration and a smooth phase sweep.


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Figure 5. Frequency and phase profile for the MICA180-259-90 pulse created to test with decoupling experiments.

Figure 6. 1D gHSQC experiments showing the strychnine methyl signal at 5.9 ppm 1H with 13C decoupling offset from −30 kHz to +30 kHz in 1 kHzsteps, with the GARP1 decoupling sequence at 2.7 kHz field (a), MICA180-259-90 (M1) at 2.7 kHz field (b) and M1 at half power (c).

For testing, a decoupling sequence was created by repeatingthe pulse in a 0-150-60-150-0 phase shift pattern.[23] In Fig. 6(a)it can be seen that the GARP1 sequence at 2.7 kHz field has thetarget decoupling bandwidth of 15 kHz at a modulation frequencyof 11 kHz. As seen in Fig. 6(b), the MICA180-259-90 decouplingsequence (M1) gives double the decoupling field of GARP1 at thesame power and modulation frequency, with no ill effects otherthan perhaps the lack of a definitive cutoff in decoupling at theends. Amplitudes are not to scale but in fact the signals are slightlymore intense with M1 decoupling.

The M1 decoupling sequence has also proven to be very robustin the lab with routine samples, typically producing excellent13C broadband decoupling with very low sidebands overall, ascompared with the GARP1 results as seen in Fig. 7(b) and (c). Infact we can even halve the power (−6 db), halve the decouplingmodulation frequency and obtain nearly the same decouplingbandwidth as the GARP1 sequence has at double the poweras seen in Fig. 6(c). Sidebands are still not very large in thiscase (not shown), indicating the possibility of using these pulsesfor very low power broadband decoupling. It is clear from ourdecoupling results to date that the total phase rotation of theseinversion pulses plays a significant role in the size of the observabledecoupling sidebands, and we can attribute at least some of thelow sidebands in this case to the low total phase rotation of theMICA180-259-90 pulse.

Conclusion

Optimizing constant amplitude inversion pulses for maximumbroadband inversion on the basis of Bloch simulations overmultiple modulation frequencies rather than a single frequency re-liably provides constant amplitude pulses with smooth frequencysweeps that are linear or near linear in the centers. The improve-ment over BIP and BIPBOP of these MICA pulses is specificallythat the smoother phase sweeps provide better B1 tolerance oradiabicity as shown by simulations.

While these pulses are not adiabatic at the ends, they can bedescribed as minimal duration near-adiabatic inversion pulsesas the centers have smooth, linear frequency sweeps andGaussian and/or Lorentzian phase sweeps that can satisfy adiabaticconditions.[11] Experimentally it is an easy enough matter to simplyuse a MICA pulse that has been optimized over a greater thanneeded bandwidth to give a linear or near linear frequency sweepover the desired bandwidth.

Linear truncation of the ends of MICA pulses to createtrapezoidal (MICAT) pulses provides smoother transitions frominversion regions. There is, however, no other obvious benefit totruncating the pulses, if minimal pulse duration and maximuminversion bandwidth are desired. Although the MICAT pulses maybe more power efficient overall, MICA pulses still provide betterinversion for the same duration and power.

The MICA pulses are most obviously useful as 13C inversionpulses in high field broadband INEPT-based pulse sequences,


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Figure 7. 1H (a) and 1D gHSQC (b)–(d) experiments of slightly impure strychnine in CDCl3 to show the 13C decoupling sidebands. The same 2.7-kHzdecoupling field was used for GARP1 (b), M1 (c) and WURST40 (d) decoupling .

where their minimal duration solutions yield better results than anyother type of inversion pulse. This excellent inversion performancecan also translate into impressive decoupling performance, as seenwith the M1 decoupling sequence developed in this work that givesdouble the bandwidth of the GARP1 decoupling sequence at thesame power with similarly low sidebands. Although the pulseduration is irrelevant to decoupling, at least some of the excellentdecoupling performance of the M1 pulse can be attributed to italso being a minimal total phase rotation inversion pulse.

This is not a perfect universal method for developing inversionpulses. It is possible that varying a parameter other than decouplermodulation frequency could give better results. No universalapproach for creating perfect MICA pulses was found. Threeequal spaced and equal weighted frequencies give excellentresults for the low total phase rotation pulses, but five weightedfrequencies yield subjectively better results for the highest phaserotation pulses done to date. Results will be biased towards lowerfrequencies with this strategy. While best phase sweeps for a givenbandwidth and pulse duration are straightforward to obtain forpulses with shorter durations and smaller bandwidths, finding asingle optimal solution for pulses with bandwidths beyond thatpresented in this work is less straightforward and time consumingwith this process, even with smoothing. Still, this method providesa better method for obtaining optimal B1 insensitive short durationconstant amplitude and truncated constant amplitude inversionpulses, especially for 13C inversion, than has previously existed.

Pulses created for this work are available for download athttp://nmr.chem.indiana.edu/.

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2004, 170, 236.[3] M. I. A. Lourakis, http://www.ics.forth.gr/∼lourakis/levmar/, 2004.[4] E. Kupce, R. Freeman, J. Magn. Reson., Ser. A 1993, 105, 234.[5] Varian, Palo Alto, http://www.varianinc.com. [accessed: 2008].[6] W. Willker, D. Leibfritz, R. Kerssebaum, W. Bermel, Magn. Reson.

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2005, 176, 179.[15] E. Kupce, R. Freeman, J. Magn. Reson., Ser. A 1995, 115, 273.[16] E. Kupce, R. Freeman, J. Magn. Reson., Ser. A 1995, 117, 246.[17] R. Fu, G. Bodenhausen, Chem. Phys. Lett. 1995, 245, 415.[18] R. Fu, G. Bodenhausen, J. Magn. Reson., Ser. A 1995, 117, 324.[19] J.-M. Bohlen, G. Bodenhausen, J. Magn. Reson., Ser. A 1993, 102, 293.[20] J. R. Phillips, http://zunzun.com. [accessed: 2008].[21] E. Kupce, R. Freeman, J. Magn. Reson., Ser. A 1996, 118, 299.[22] A. J. Shaka, P. B. Barker, R. Freeman, J. Magn. Reson. 1985, 64, 547.[23] R. Tycko, A. Pines, R. Gluckenheimer, J. Chem. Phys. 1985, 83, 2775.


multi-frequency improved constant amplitude pulses for broadband inversion

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