Optimized Adiabatic Pulses for Wideband Spin Inversion - ScienceDirect

JOURNAL OF MAGNETIC RESONANCE, ARTICLE NO.

Series A 118, 299–303 (1996)

0042

Optimized Adiabatic Pulses for Wideband Spin Inversion EV RIKS KUPC˘E *

AND

RAY FREEMAN†

*Varian NMR Instruments, 28 Manor Road, Walton-on-Thames, Surrey KT12 2QF, England; and †Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, England Received November 8, 1995

Adiabatic rapid passage has been widely used in highresolution NMR (1–28). It is very efficient for excitation or inversion of nuclear spins over a wide range of frequencies, performing better than a hard pulse for a given radiofrequency intensity, and dissipating far less power when used as a broadband decoupling scheme. Provided that the radiofrequency intensity B1 is above the minimum level required by the adiabatic condition, the performance is rather insensitive to B1 , tolerating a high degree of spatial inhomogeneity, which is a decided advantage for in vivo spectroscopy and medical imaging applications. Recently, adiabatic pulses have been made the basis of several ultra-broadband decoupling schemes (21, 22, 25, 27, 29, 30). An adiabatic pulse is defined by the form of the frequency sweep Dv(t) and the amplitude profile v1 (t). Many different combinations have been suggested—the constant-amplitude linear sweep (1–3), the hyperbolic secant (4, 9, 10), the tangential sweep (10, 11), and the constant adiabaticity pulse (10). We show here that, once an amplitude profile v1 (t) has been chosen, it is possible to derive the optimum form for the frequency sweep Dv(t) through a very simple relationship. We consider the case where the magnetic field B0 is constant and the radiofrequency is swept, and transform into an accelerating (and decelerating) reference frame that rotates in synchronism with the instantaneous radiofrequency. The effective field veff Å gBeff is the resultant of the radiofrequency field v1 Å gB1 and the resonance offset Dv Å gDB; it is inclined at an angle u Å arctan( DB/B1 ) with respect to the / x axis of the rotating frame. The conditions at the extremities of the sweep are normally arranged so that u varies from /p /2 to 0 p /2 radians. The adiabatic condition relates the rate of change of u to the intensity of the effective field É du /dtÉ ! veff .

to which the adiabatic condition is satisfied can be quantified by introducing the adiabaticity factor Q QÅ

In principle, Q should be large compared with unity, although, in practice, successful decoupling has been achieved (30) with Q Å 1. Normally, Q is smallest (and hence most critical) at resonance where Dv Å 0. By considering the time dependence of both the amplitude v1 and the frequency Dv, we may write a general expression for the adiabaticity factor QÅ

( v 21 / Dv 2 ) 3 / 2 . Év1 (d Dv /dt) 0 Dv(dv1 /dt)É

[3]

We accept any well-behaved form for the amplitude profile v1 (t) and enquire what would be the optimum expression for the form of the frequency dependence d Dv /dt. In this context, well-behaved means that v1 (t) is smooth and continuous and begins and ends at (or close to) zero amplitude. Broadband operation implies good spin inversion for any chemical species having a Larmor frequency within the effective bandwidth. Consider a typical group of spins with a Larmor frequency at an arbitrary point in this range. When the radiofrequency reaches this point, the instantaneous amplitude v1 (t) is changing with time, while the sweep rate d Dv /dt is accelerating or decelerating. The effective field has swung from the /z axis and has just reached the / x axis, later progressing toward the 0z axis at the end of the pulse. As the radiofrequency passes through resonance for these spins, we use the instantaneous value of v1 (t) to evaluate the optimum sweep rate d Dv /dt at that point. When we set Dv Å 0, the expression for Q reduces to QÅ

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v 21 . É d Dv /dtÉ

[4]

The recommended value for the instantaneous sweep rate is, therefore,

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[2]

[1]

When these two conditions are satisfied, the magnetization vector M, initially aligned along /z, remains ‘‘locked’’ along the effective field throughout the sweep and is carried to 0z at the end of the pulse (spin inversion). The degree

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veff . É du /dtÉ

1064-1858/96 $18.00 Copyright q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

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k(t) Å

dDv v 21 Å . dt Q

[5]

By applying this relationship at each time increment of the frequency sweep, we calculate the ‘‘ideal’’ time dependence for the radiofrequency Dv(t) Å

* k(t)dt.

[6]

In practice, this is implemented by a phase modulation f(t) Å

* Dv(t)dt.

[7]

Equation [5] implies that if the instantaneous sweep rate is varied in proportion to the square of the corresponding radiofrequency amplitude, the adiabaticity factor Q is constant for all spins within the effective bandwidth. If Q is properly chosen, this should be the most efficient mode of operation. Baum, Tycko, and Pines (10) have described such a ‘‘constant adiabaticity pulse’’ for the special case that v1 (t) remains constant throughout. The new prescription (Eq. [5]) suggests that an adiabatic pulse with an amplitude profile defined by v1 (t) Å v1 (max)sech( bt)

[8]

FIG. 1. (a) Gaussian amplitude profile v1 (t) Å v1 (max)exp( 0 bt 2 ), truncated in the tails at 1%. (b) The frequency-sweep function Dv(t) Å l erf( bt) obtained by integrating the square of the Gaussian function. (c) The corresponding spin-inversion profile, calculated for a pulse of duration 20 ms, Q Å 4, and gB1 (max)/2p Å 0.33 kHz. The effective bandwidth could, of course, be increased by increasing the decoupler level and shortening the pulse duration.

[9]

Calculations based on the Bloch equations indicate that this combination gives a useful ‘‘top-hat’’ inversion profile (Fig. 1c). Several other pairs of amplitude profiles and frequencysweep functions are listed in Table 1. Even a very simple amplitude function such as cos 2 ( bt) gives a perfectly acceptable inversion profile (Fig. 2). Recently, a family of broadband decoupling schemes has been proposed (27) based on an adiabatic pulse with strictly linear frequency sweep and an amplitude profile defined by

should have a frequency sweep defined by Dv(t) Å

[ v1 (max)] 2 Q

* sech ( bt) 2

Å l tanh( bt) / Dv0 ,

where l is a scaling factor (30) given by lÅ

[ v1 (max)] 2 . bQ

[10]

In this manner, we have derived an expression for the frequency-sweep function of the well-known hyperbolic secant pulse (4) from the form of its amplitude profile. Alternatively, if we start with a Gaussian profile (Fig. 1a), v1 (t) Å v1 (max)exp( 0 bt 2 ),

[11]

then the frequency-sweep function (Fig. 1b) is Dv(t) Å l erf( bt).

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[12]

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v1 (t) Å v1 (max){1 0 Ésin( bt)Én },

[13]

where n is a large integer, for example, 20. The WURST20 sequence delivers effective decoupling over a bandwidth of 74 kHz for a decoupler field given by gB2 /2p Å 2.5 kHz (30). For this adiabatic pulse, Q is constant in the central region of the sweep but not at the extremities. It is interesting to derive an ‘‘optimized’’ sweep-rate function based on the WURST-20 amplitude profile. It deviates from linearity only

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TABLE 1 Pairs of Amplitude Profiles and Optimized Frequency-Sweep Functions for Adiabatic Rapid Passage v1(t)

Hyperbolic secant Gaussian Lorentzian Cosine Cosinesquareda a

Dv(t)

v1(max) sech(bt) v1(max) exp(0bt2) v1(max)/(1 / b2t2) v1(max) cos(bt)

l tanh(bt) l erf(bt) 1 2 2 2 l[arctan(bt) / bt/(1 / b t )] 1 l [ b t / sin( b t) cos( b t)] 2

v1(max) cos2(bt)

1 32

l[12bt / 8 sin(2bt) / sin(4bt)]

The WURST-2 amplitude profile.

at the extremities of the sweep. We see that the resulting inversion profile (Fig. 3) shows no particular improvement over the original WURST-20 scheme; it cuts off rather more sharply at the edges but has a slightly narrower effective bandwidth. A demonstration of the effectiveness of the WURST

FIG. 3. The inversion profile (full curve) calculated for an adiabatic pulse with an amplitude profile defined by v1 (t) Å v1 (max){1 0 Ésin( bt)É20 } and a frequency-sweep function derived by integrating [ v1 (t)] 2 . The pulse duration is 5 ms, Q Å 4, and gB1 (max)/2p Å 1.15 kHz. The dashed curve is the corresponding profile calculated for WURST20.

scheme is shown in Fig. 4 for the case of { 31P}H broadband decoupling in trimethylphosphite in CDCl3 at 400 MHz. This typifies a common situation in biochemical NMR where it is necessary to decouple only small scalar interactions such as 15N– 13C (approximately 10 Hz), 13C– 2H (20–30 Hz), or 31P– 1H ( õ20 Hz). In the present example, JPH Å 11 Hz, and an unusually low decoupler level can be employed ( gB2 / 2p Å 1.5 kHz). The amplitude profile is defined by a small index n Å 2 (WURST-2), corresponding to the cosinesquared function illustrated in Fig. 2. The repetition rate of the adiabatic sweep must be high compared with JPH , but since JPH is small, a relatively long pulse duration, 2t Å 15 ms, can be used. Under these conditions, and with Q Å 1, the effective decoupling bandwidth is measured as 2gDB(max)/2p Å 66 kHz, which corresponds to a very high figure of merit JÅ

FIG. 2. (a) Amplitude profile v1 (t) Å cos 2 ( bt). (b) The frequencysweep function Dv(t) obtained by integrating cos 4 ( bt). (c) The corresponding spin-inversion profile, calculated for a pulse of duration 20 ms, Q Å 4, and gB1 (max)/2p Å 0.29 kHz.

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2DB(max) Å 71.7, B2 (rms)

[14]

where gB2 (rms)/2p Å 0.92 kHz is the equivalent constant decoupling field that would have the same power dissipation as the actual amplitude-modulated WURST-2 sequence. Figure 5 demonstrates the performance under the much more exacting conditions of { 13C}H decoupling, where the larger coupling constant (JCH Å 151 Hz) requires a moreintense decoupler field ( gC B2 /2p Å 5.8 kHz) and a shorter pulse length, 2t Å 1.5 ms. This setting corresponds to a B2 intensity that is just one-half of the recommended safe level, leaving open the possibility of a further fourfold increase in bandwidth (which is proportional to B 22 ). These experiments

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FIG. 4. Experimental broadband { 31P}H decoupling in methyl iodide (JPH Å 11 Hz) using 15 ms WURST-2 adiabatic pulses in a 20-step phase cycle. After allowing for the amplitude modulation, the decoupler level is gB2 (rms)/2p Å 0.92 kHz. The proton resonance is monitored at 400 MHz over a range of {35 kHz in 1 kHz steps, indicating an effective decoupling bandwidth of 66 kHz and a figure of merit J Å 71.7. There is a weak impurity with a slightly different decoupling behavior.

employ a WURST-240 pulse (a steep apodization function) and the five-step phase cycle of Tycko et al. (31) nested within the four-step cycle of Levitt (32). The effective de-

coupling bandwidth is 290 kHz. After making an allowance for the amplitude modulation of B2 , we find a figure of merit J Å 51.8. Figure 5b shows that a 1 dB increase in B2 lowers

FIG. 5. Experimental broadband { 13C}H decoupling in methyl iodide at 400 MHz using the WURST-240 adiabatic pulse and a 20-step phase cycle. The proton resonance is monitored over a range of {150 kHz in 5 kHz steps. (a) gC B2 (rms)/2p Å 5.60 kHz, figure of merit J Å 51.8. (b) gC B2 (rms)/ 2p Å 6.28 kHz, J Å 46.2. This 1 dB increase in the decoupler level reduces intensity of the cycling sidebands and slightly improves uniformity.

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