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Selective excitation at two arbitrary frequencies. The double-DANTE sequence
JOURNAL
OF MAGNETIC
RESONANCE
81,646-652
(1989)
SelectiveExcitation at Two Arbitrary Frequencies. The Double-DANTE Sequence HELENGEEN, XI-LIWU,
PINGXU,JANFRIEDRICH,ANDRAYFREEMAN
Department ofPhysicaI Chemistry, Cambridge University, Cambridge- England
Re&vedOctober28,1988
Several NMR experiments require selective excitation in the frequency domain and this can be achieved with a long weak radiofrequency pulse (1-3) often called a “sot? pulse. In present-day Fourier transform spectrometers it is usually more convenient to employ the DANTE sequence (4, 5) made up of a train of hard pulses separated by free precession intervals (7). As with the soft pulse, the overall duration (T) of the DANTE sequence determines the selectivity of the excitation. One important difference of the DANTE experiment is that excitation also occurs at a set of “sideband” conditions v. -t n/7, where the nuclei accomplish a whole number (n) of complete rotations about the Z axis in the intervals between the hard pulses. If we work with the first sideband condition (n = l), the frequency of irradiation can be finely tuned by adjusting the pulse repetition rate. Pulse shaping (6) can be achieved by modulating the hard pulse flip angles ofthe DANTE sequence (5, 7) or by modulating the interpulse delays. As high-resolution NMR experiments become more sophisticated, occasions arise for simultaneous selective excitation at two different frequencies, a “doubleDANTE” experiment. Ideally the two resonance conditions should be independently adjustable in phase, in frequency, and in selectivity. As a makeshift solution, some recent experiments (8, 9) have used the centerband and first sideband response of a DANTE sequence for double irradiation, but this runs the risk of undesirable excitation elsewhere in the spectrum. Consequently we have explored various practical implementations of double-DANTE sequences, based on the approximation of linear superposition of two pulse sequences. The first scheme (5) takes two DANTE sequences operating at different repetition rates (7A1 and ~8~)and combines them so that they both terminate at the same instant, just prior to signal acquisition. The two first sideband conditions are then independently adjustable, and the centerband and all other sidebands can be held far outside the spectrum of interest. The programming of such sequences is rather cumbersome (5) and changing either of the excitation frequencies requires significant alterations to the pulse sequence. An alternative scheme appears to be simpler. It involves the interleaving of two DANTE sequences A and B, having equal pulse repetition rates. Sequence A terminates 7/2 s before signal acquisition; sequence B terminates immediately prior to signal acquisition. Consequently the odd-order sidebands of sequence A are inverted 0022-2364/89 $3.00 Copyright 0 1989 by Academic J’res, Inc. AU rights of reproduction in any form reserved.
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in phase. The required difference in excitation frequency is achieved by progressive incrementation of the phase (10-15) of the pulses in sequence B, keeping constant phase for sequence A. The idea of 90” phase incrementation has already been used in DANTE sequences (14, 25); here we propose to use arbitrarily small phase shifts, A$ rad, to give a frequency separation Af = A@/(2 ~7). Most present-day spectrometers have a provision for small-angle radiofrequency phase shifts. Either sideband or centerband responses can be used for excitation, but the centerband has one important practical advantage. Given a Iixed phase increment A4, fme tuning of the e&ctive irradiation frequency of sequence B can be accomplished by adjusting the repetition rate 7-l . Provided that the overall length of the sequence (T) remains constant, this has no effect on the centerband response of sequence A. Normally the repetition rate would be high enough to move all sideband responses well outside the spectrum of interest. This double-DANTE scheme therefore consists of a regular train of equally spaced hard pulses. Each odd-numbered pulse has the same phase (+X), but each evennumbered pulse has a phase that is progressively incremented in equal steps A@, ending up as a pulse about the +X axis (Fig. I). The two centerband responses are used for excitation. We can imagine the two DANTE excitations operating in two different rotating reference frames, one at the transmitter frequeney vo, the other at v. + 4f Hz. These two frames are synchronized in phase at the end of the pulse sequence, so that the X axes of the frames are coincident at this time. Thus both excitations can be in the pure absorption mode. Fine adjustment of the second irradiation frequency is achieved by setting T. If different frequency selectivities are required, sequences A and B can have different overall lengths; for example, several pulses of the A sequence might be applied before the interleaving of A and B begins. It is of course only an approximation to assume that the two DANTE excitations are independent of each other. Perhaps the most graphic illustration of this interaction is to plot trajectories for two magnetization vectors Ma and Mb excited by a suitable double-DANTE sequence. The pulse widths are modulated according to a half-Gaussian envelope (7-9). The frequency separation is taken to be 5 Hz. We choose a frame synchronized with the transmitter frequency vo, so that if M, alone were excited, it would rotate from +Z to -+Y along a smooth arc. In fact (Fig. 2) the trajectory only approximates this arc because the pulses of sequence B perturb the motion of M, into a series of loops resembling a cycloid. (When an unmodulated DANTE sequence is used, each loop of the cycloid has the same period.) The final position of M, is not greatly changed, except for a slight phase shift with respect to
0
(N-l)A@
0
(N-Z)A@
0
PHASE
0
W’
0
4
0
0
t=o FIG. 1. Double-DANTE pulse sequence with the even-numbered pulses rotating in phase. A pulse about the +X axis is represented by a zero-phase angle. The fixed-phase and rotating-phase sequences are both about the +X axis at t = 0, the start of data acquisition.
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648
FIG. 2. Magnetization trajectories for two NMR signals (M, and Mb), 5 Hz apart, excited by a onesecond duration double-DANTE sequence with a half-Gaussian envelope. The off-resonance excitation interferes with the motion of the on-resonance signal causing it to follow a cycloidal path (a) instead of the smooth arc from +Z to + Y. The trajectory (b) is similarly perturbed, but the effect is not readily apparent in the diagram because of the 5 Hz relative precession. Note that the two trajectories terminate a short distance from the + Y axis, leading to a small phase shift.
the + Y axis. This phase shift depends in a cyclic manner on the frequency difference Af between the two irradiations and falls off to a negligible amplitude for large values of TA$ For most practical applications of double-DANTE sequences the phase shift would be less than a few degrees. The trajectory of Mb is perturbed in a similar fashion, although the effect is masked in Fig. 2 by the fact that Mb is 5 Hz off resonance in this frame and thus has a rapid precessional motion. We may predict the form of the on-resonance trajectory by using rotation operators, making some simplifying assumptions. Each hard pulse (of flip angle CY)may he represented by a rotation operator, for example, &(a). The odd-numbered pulses are all applied about the +X axis, whereas the even-numbered pulses are applied about an axis Wwhich lies in the XY plane and rotates in increments of A4 rad. The sequence always terminates with an even-numbered (rotating-phase) pulse, where W coincides with the +Xaxis. For the on-resonance spin free precession between pulses is zero. The overall rotation can therefore be represented as R = R~(Lu)&(a)*
* .R*(a)R~(a)R&+
* .Rg-‘(a)&(a).
111
According to the usual convention, the rotation operators are written here in timereversed order (&(a) is the first radiofrequency pulse). The integer k runs from 0 to (IV - 1). We replace the expression for pulses about the rotating axis: Rkw(cy)= RZ’(kA~)R,((Y)R,(kA~).
PI
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The rotation operators are expressed as matrices in the usual way and are multiplti out. In the general case the result is complicated, but simplifies if it is assumed that the individual flip angles CYare sufhciently small that only the linear terms in the expansions of cos (Yand sin (Yare retained (cos a = 1, sin LY= a). This is equivalent to restricting the sequence to small total flip angles KY. Then the overall effect of the sequence may be represented by the rotation
where P = -sin[(N
- l&$/2]
siJ$AAg)
Q = N + cos[(N - l)A+/2]
‘z;T4;;)
.
If we start with the equilibrium magnetization MO, then MO aQ represents the absorption-mode component and M,aP represents the dispersion-mode component at the end of a double-DANTE sequence, viewed in a reference frame at exact resonance for excitation by the A sequence. If we plot this transverse component of magnetization as a function of N, we obtain the complex expression c = Mea N+ exP(-i&G) 2 sin(A4,2)
. bWW
f it 1 - coW~~))l
.
This trajectory is a cycloid with oscillatory components proportional to 1 - cos(2aTAf) and sin(2?rTAJ) and ,with a phase shift A&/;! rad. In the limit that the second excitation frequency is far from resonance (rAf$ 1), there are many oscillations of the cycloid within the complete path. Then the absorption-mode component approaches MoaN and the dispersion component becomes very small. In this limit, excitation by the A sequence is essentially independent of the off-resonance excitation by the B sequence. (Naturally, if the offset Af becomes large enough, this excitation frequency approaches that of the first sideband condition of the fixed-phase sequence, and the interference increases again.) In the general case, the phase error of the NMR excitation is given by arctan (P/Q). It is interesting to note that equations of identical form can be derived in classical optics when two beams of monochromatic light fall on an array of N parallel slits. It is assumed that the phase of the transmitted light is a linear function of position (the Fraunhofer approximation) and we choose to monitor a maximum in the diffraction pattern for beam 1. (This is equivalent to sitting at exact resonance for one of the excitations of the double-DANTE sequence). Then the electric field vector may be written N-l
E = C (1 + exp[ikd2?r(sin k=O
8, - sin Q/x]},
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COMMUNICATIONS
where X is the wavelength, d is the slit separation, and 8, and o2 are the angles of incidence of the two beams. This gives real and imaginary components identical to Eqs. [4] and [5] provided that the phase shift is now written as Ad = 2nd(sin 8, - sin l&)/X.
PI
In this expression the NMR pulse interval 7 has been replaced by the slit separation d, and the NMR frequency separation Af has,been replaced by (sin 8, - sin &)/A. Interaction between the two excitations of the double-DANTE sequence is equivalent to the interaction between two soft pulses applied in close proximity. Consider a single NMR species irradiated with a weak radiofrequency field B, at exact resonance and constant phase. In the reference frame synchronized with this irradiation, B, is fixed along the +X axis. A second weak radiofrequency field that is Af Hz off resonance can be represented as another vector B, which rotates continuously at 2vrAf rad s-’ in this frame. The resultant field is modulated in amplitude (between 0 and 2BJ and in phase (between the + Y and the - Y axes), as illustrated in Fig, 3. Consequently, instead of moving along the smooth arc from +Z to + Y, the magnetization vector follows a cycloid, with cusps corresponding to the instants when the resultant field is zero. In the present example (Af= 5 Hz) there are five cycles of this curve if the soft pulse duration is 1 s. The phase error depends on just where on this curve the soft pulse terminates. The maximum range of possible phase errors depends on the number of cycles of the cycloid; that is, it is appreciable only when TAf is small. The double-DANTE sequence was tested experimentally on the 400 MHz proton spectrum of 2,3-dibromopropanoic acid (Fig. 4). The sequence was made up of a total of 288 fixed-phase pulses and 288 rotating-phase pulses. A phase increment of 22.5” was used pith pulse intervals of 1.47 ms, giving a frequency separation of 2 1.2 Hz between the two excitations. A half-Gaussian pulse envelope was used (7-9). This permitted the excitation of the two outermost lines of the central quartet (Fig. 4b).
(a)
(w
w
FIG. 3. Interaction between two simultaneous sofi pulses Af Hz apart. In this rotating frame, one field B, remains aligned along the +X axis (a) and a second field B, rotates in the XY plane (b) at the difference frequency, giving a resultant (c) which is modulated in phase and amplitude. This modulated field causes a cycloidal motion of the magnetization trajectory.
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400 Hz
m
FIG. 4. Experimental test of the double-DANTE sequence on the proton spectrum of 2,3dibromopropanoic acid (a). The two outer lines of the central muhiplet have been excited in trace (b). The unwanted excitations are shown amplified IO-fold in trace (c).
The slight phase error on these two lines is an artifact attributable to the fact that the radiofrequency intensity of the hard pulses was set at an unusually low level (owing to a hardware limitation on the minimum pulse length). Spurious excitation of other lines was small, as can be appreciated from the amplified traces in Fig. 4c. ACKNOWLEDGMENTS We are indebted to Jeremy Titman for help with the experimental test of the double-DANTE and to James Keeler for permission to use the Bruker 400 MHz spectrometer. REFERENCES 1. S. ALEXANDER, Rev. Sci. Instrum. 32,1066 ( 196 1). 2. R.FREEMANANDS.WITTEKOEK,.I. MagnReson. 1,238(1969).
sequence,
652
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II.
R. FREEMAN, S. WHTEKOEK, AND R. R. ERNST, J. Chem. Phys. 52, 1529 (I 970). G. BODENHAUSEN, R. FREEMAN, AND G. A. MORRIS, J. Magn. Reson. 23,17 1 (1976). G. A. MORRIS AND R. FREEMAN, J. Magn. Reson. 29,433 (1978). C. BAUER, R. FREEMAN, T. FRENKIEL, J. KEELER, AND A. J. SHAKA, .J. Magn. Reson. 58,442 (I 984). J. FRIEDRICH, S. DAVIES, AND R. FREEMAN, J. Magn. Reson. 75,390 (1987). S. DAVIES, J. FRIEDRICH, AND R. FREEMAN, J. Magn. Reson. 76,555 (1988). J. FRIEDRICH, S. DAVIES, AND R. FREEMAN, Mol. Phys. 64,691 (1988). A. G. REDFIELD AND S. D. KUNZ, J. Magn. Reson. 19,250 (1975). G. BODENHAUSEN, R. FREEMAN, G. A. MORRIS, R. NIEDERMEYER, AND D. L. TURNER, J. Magn.
12. 13. 14. 15.
A. G. C. P.
3. 4. 5. 6. 7. 8. 9.
10.
Reson. 25,559 (1977). PINES, D. E. WEMMER, J. TANG, AND S. SINTON, Bull. Am. Phys. Sot. 23,21 (1978). BODENHAUSEN, R. L. VOLD, AND R. R. VOLD, J. Magn. Reson. 37,93 (1980). J. TURNER, Prog. NMR Spectrosc. 16,311 (1984). BLONDET, J. P. ALBRAND, M. VON KIENLIN, M. DESCORPS, AND N. LAVANCHY, J. 71,342 (1987).
Magn. Reson.
OF MAGNETIC
RESONANCE
81,646-652
(1989)
SelectiveExcitation at Two Arbitrary Frequencies. The Double-DANTE Sequence HELENGEEN, XI-LIWU,
PINGXU,JANFRIEDRICH,ANDRAYFREEMAN
Department ofPhysicaI Chemistry, Cambridge University, Cambridge- England
Re&vedOctober28,1988
Several NMR experiments require selective excitation in the frequency domain and this can be achieved with a long weak radiofrequency pulse (1-3) often called a “sot? pulse. In present-day Fourier transform spectrometers it is usually more convenient to employ the DANTE sequence (4, 5) made up of a train of hard pulses separated by free precession intervals (7). As with the soft pulse, the overall duration (T) of the DANTE sequence determines the selectivity of the excitation. One important difference of the DANTE experiment is that excitation also occurs at a set of “sideband” conditions v. -t n/7, where the nuclei accomplish a whole number (n) of complete rotations about the Z axis in the intervals between the hard pulses. If we work with the first sideband condition (n = l), the frequency of irradiation can be finely tuned by adjusting the pulse repetition rate. Pulse shaping (6) can be achieved by modulating the hard pulse flip angles ofthe DANTE sequence (5, 7) or by modulating the interpulse delays. As high-resolution NMR experiments become more sophisticated, occasions arise for simultaneous selective excitation at two different frequencies, a “doubleDANTE” experiment. Ideally the two resonance conditions should be independently adjustable in phase, in frequency, and in selectivity. As a makeshift solution, some recent experiments (8, 9) have used the centerband and first sideband response of a DANTE sequence for double irradiation, but this runs the risk of undesirable excitation elsewhere in the spectrum. Consequently we have explored various practical implementations of double-DANTE sequences, based on the approximation of linear superposition of two pulse sequences. The first scheme (5) takes two DANTE sequences operating at different repetition rates (7A1 and ~8~)and combines them so that they both terminate at the same instant, just prior to signal acquisition. The two first sideband conditions are then independently adjustable, and the centerband and all other sidebands can be held far outside the spectrum of interest. The programming of such sequences is rather cumbersome (5) and changing either of the excitation frequencies requires significant alterations to the pulse sequence. An alternative scheme appears to be simpler. It involves the interleaving of two DANTE sequences A and B, having equal pulse repetition rates. Sequence A terminates 7/2 s before signal acquisition; sequence B terminates immediately prior to signal acquisition. Consequently the odd-order sidebands of sequence A are inverted 0022-2364/89 $3.00 Copyright 0 1989 by Academic J’res, Inc. AU rights of reproduction in any form reserved.
646
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COMMUNICATIONS
in phase. The required difference in excitation frequency is achieved by progressive incrementation of the phase (10-15) of the pulses in sequence B, keeping constant phase for sequence A. The idea of 90” phase incrementation has already been used in DANTE sequences (14, 25); here we propose to use arbitrarily small phase shifts, A$ rad, to give a frequency separation Af = A@/(2 ~7). Most present-day spectrometers have a provision for small-angle radiofrequency phase shifts. Either sideband or centerband responses can be used for excitation, but the centerband has one important practical advantage. Given a Iixed phase increment A4, fme tuning of the e&ctive irradiation frequency of sequence B can be accomplished by adjusting the repetition rate 7-l . Provided that the overall length of the sequence (T) remains constant, this has no effect on the centerband response of sequence A. Normally the repetition rate would be high enough to move all sideband responses well outside the spectrum of interest. This double-DANTE scheme therefore consists of a regular train of equally spaced hard pulses. Each odd-numbered pulse has the same phase (+X), but each evennumbered pulse has a phase that is progressively incremented in equal steps A@, ending up as a pulse about the +X axis (Fig. I). The two centerband responses are used for excitation. We can imagine the two DANTE excitations operating in two different rotating reference frames, one at the transmitter frequeney vo, the other at v. + 4f Hz. These two frames are synchronized in phase at the end of the pulse sequence, so that the X axes of the frames are coincident at this time. Thus both excitations can be in the pure absorption mode. Fine adjustment of the second irradiation frequency is achieved by setting T. If different frequency selectivities are required, sequences A and B can have different overall lengths; for example, several pulses of the A sequence might be applied before the interleaving of A and B begins. It is of course only an approximation to assume that the two DANTE excitations are independent of each other. Perhaps the most graphic illustration of this interaction is to plot trajectories for two magnetization vectors Ma and Mb excited by a suitable double-DANTE sequence. The pulse widths are modulated according to a half-Gaussian envelope (7-9). The frequency separation is taken to be 5 Hz. We choose a frame synchronized with the transmitter frequency vo, so that if M, alone were excited, it would rotate from +Z to -+Y along a smooth arc. In fact (Fig. 2) the trajectory only approximates this arc because the pulses of sequence B perturb the motion of M, into a series of loops resembling a cycloid. (When an unmodulated DANTE sequence is used, each loop of the cycloid has the same period.) The final position of M, is not greatly changed, except for a slight phase shift with respect to
0
(N-l)A@
0
(N-Z)A@
0
PHASE
0
W’
0
4
0
0
t=o FIG. 1. Double-DANTE pulse sequence with the even-numbered pulses rotating in phase. A pulse about the +X axis is represented by a zero-phase angle. The fixed-phase and rotating-phase sequences are both about the +X axis at t = 0, the start of data acquisition.
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FIG. 2. Magnetization trajectories for two NMR signals (M, and Mb), 5 Hz apart, excited by a onesecond duration double-DANTE sequence with a half-Gaussian envelope. The off-resonance excitation interferes with the motion of the on-resonance signal causing it to follow a cycloidal path (a) instead of the smooth arc from +Z to + Y. The trajectory (b) is similarly perturbed, but the effect is not readily apparent in the diagram because of the 5 Hz relative precession. Note that the two trajectories terminate a short distance from the + Y axis, leading to a small phase shift.
the + Y axis. This phase shift depends in a cyclic manner on the frequency difference Af between the two irradiations and falls off to a negligible amplitude for large values of TA$ For most practical applications of double-DANTE sequences the phase shift would be less than a few degrees. The trajectory of Mb is perturbed in a similar fashion, although the effect is masked in Fig. 2 by the fact that Mb is 5 Hz off resonance in this frame and thus has a rapid precessional motion. We may predict the form of the on-resonance trajectory by using rotation operators, making some simplifying assumptions. Each hard pulse (of flip angle CY)may he represented by a rotation operator, for example, &(a). The odd-numbered pulses are all applied about the +X axis, whereas the even-numbered pulses are applied about an axis Wwhich lies in the XY plane and rotates in increments of A4 rad. The sequence always terminates with an even-numbered (rotating-phase) pulse, where W coincides with the +Xaxis. For the on-resonance spin free precession between pulses is zero. The overall rotation can therefore be represented as R = R~(Lu)&(a)*
* .R*(a)R~(a)R&+
* .Rg-‘(a)&(a).
111
According to the usual convention, the rotation operators are written here in timereversed order (&(a) is the first radiofrequency pulse). The integer k runs from 0 to (IV - 1). We replace the expression for pulses about the rotating axis: Rkw(cy)= RZ’(kA~)R,((Y)R,(kA~).
PI
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COMMUNICATIONS
The rotation operators are expressed as matrices in the usual way and are multiplti out. In the general case the result is complicated, but simplifies if it is assumed that the individual flip angles CYare sufhciently small that only the linear terms in the expansions of cos (Yand sin (Yare retained (cos a = 1, sin LY= a). This is equivalent to restricting the sequence to small total flip angles KY. Then the overall effect of the sequence may be represented by the rotation
where P = -sin[(N
- l&$/2]
siJ$AAg)
Q = N + cos[(N - l)A+/2]
‘z;T4;;)
.
If we start with the equilibrium magnetization MO, then MO aQ represents the absorption-mode component and M,aP represents the dispersion-mode component at the end of a double-DANTE sequence, viewed in a reference frame at exact resonance for excitation by the A sequence. If we plot this transverse component of magnetization as a function of N, we obtain the complex expression c = Mea N+ exP(-i&G) 2 sin(A4,2)
. bWW
f it 1 - coW~~))l
.
This trajectory is a cycloid with oscillatory components proportional to 1 - cos(2aTAf) and sin(2?rTAJ) and ,with a phase shift A&/;! rad. In the limit that the second excitation frequency is far from resonance (rAf$ 1), there are many oscillations of the cycloid within the complete path. Then the absorption-mode component approaches MoaN and the dispersion component becomes very small. In this limit, excitation by the A sequence is essentially independent of the off-resonance excitation by the B sequence. (Naturally, if the offset Af becomes large enough, this excitation frequency approaches that of the first sideband condition of the fixed-phase sequence, and the interference increases again.) In the general case, the phase error of the NMR excitation is given by arctan (P/Q). It is interesting to note that equations of identical form can be derived in classical optics when two beams of monochromatic light fall on an array of N parallel slits. It is assumed that the phase of the transmitted light is a linear function of position (the Fraunhofer approximation) and we choose to monitor a maximum in the diffraction pattern for beam 1. (This is equivalent to sitting at exact resonance for one of the excitations of the double-DANTE sequence). Then the electric field vector may be written N-l
E = C (1 + exp[ikd2?r(sin k=O
8, - sin Q/x]},
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where X is the wavelength, d is the slit separation, and 8, and o2 are the angles of incidence of the two beams. This gives real and imaginary components identical to Eqs. [4] and [5] provided that the phase shift is now written as Ad = 2nd(sin 8, - sin l&)/X.
PI
In this expression the NMR pulse interval 7 has been replaced by the slit separation d, and the NMR frequency separation Af has,been replaced by (sin 8, - sin &)/A. Interaction between the two excitations of the double-DANTE sequence is equivalent to the interaction between two soft pulses applied in close proximity. Consider a single NMR species irradiated with a weak radiofrequency field B, at exact resonance and constant phase. In the reference frame synchronized with this irradiation, B, is fixed along the +X axis. A second weak radiofrequency field that is Af Hz off resonance can be represented as another vector B, which rotates continuously at 2vrAf rad s-’ in this frame. The resultant field is modulated in amplitude (between 0 and 2BJ and in phase (between the + Y and the - Y axes), as illustrated in Fig, 3. Consequently, instead of moving along the smooth arc from +Z to + Y, the magnetization vector follows a cycloid, with cusps corresponding to the instants when the resultant field is zero. In the present example (Af= 5 Hz) there are five cycles of this curve if the soft pulse duration is 1 s. The phase error depends on just where on this curve the soft pulse terminates. The maximum range of possible phase errors depends on the number of cycles of the cycloid; that is, it is appreciable only when TAf is small. The double-DANTE sequence was tested experimentally on the 400 MHz proton spectrum of 2,3-dibromopropanoic acid (Fig. 4). The sequence was made up of a total of 288 fixed-phase pulses and 288 rotating-phase pulses. A phase increment of 22.5” was used pith pulse intervals of 1.47 ms, giving a frequency separation of 2 1.2 Hz between the two excitations. A half-Gaussian pulse envelope was used (7-9). This permitted the excitation of the two outermost lines of the central quartet (Fig. 4b).
(a)
(w
w
FIG. 3. Interaction between two simultaneous sofi pulses Af Hz apart. In this rotating frame, one field B, remains aligned along the +X axis (a) and a second field B, rotates in the XY plane (b) at the difference frequency, giving a resultant (c) which is modulated in phase and amplitude. This modulated field causes a cycloidal motion of the magnetization trajectory.
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400 Hz
m
FIG. 4. Experimental test of the double-DANTE sequence on the proton spectrum of 2,3dibromopropanoic acid (a). The two outer lines of the central muhiplet have been excited in trace (b). The unwanted excitations are shown amplified IO-fold in trace (c).
The slight phase error on these two lines is an artifact attributable to the fact that the radiofrequency intensity of the hard pulses was set at an unusually low level (owing to a hardware limitation on the minimum pulse length). Spurious excitation of other lines was small, as can be appreciated from the amplified traces in Fig. 4c. ACKNOWLEDGMENTS We are indebted to Jeremy Titman for help with the experimental test of the double-DANTE and to James Keeler for permission to use the Bruker 400 MHz spectrometer. REFERENCES 1. S. ALEXANDER, Rev. Sci. Instrum. 32,1066 ( 196 1). 2. R.FREEMANANDS.WITTEKOEK,.I. MagnReson. 1,238(1969).
sequence,
652
COMMUNICATIONS
II.
R. FREEMAN, S. WHTEKOEK, AND R. R. ERNST, J. Chem. Phys. 52, 1529 (I 970). G. BODENHAUSEN, R. FREEMAN, AND G. A. MORRIS, J. Magn. Reson. 23,17 1 (1976). G. A. MORRIS AND R. FREEMAN, J. Magn. Reson. 29,433 (1978). C. BAUER, R. FREEMAN, T. FRENKIEL, J. KEELER, AND A. J. SHAKA, .J. Magn. Reson. 58,442 (I 984). J. FRIEDRICH, S. DAVIES, AND R. FREEMAN, J. Magn. Reson. 75,390 (1987). S. DAVIES, J. FRIEDRICH, AND R. FREEMAN, J. Magn. Reson. 76,555 (1988). J. FRIEDRICH, S. DAVIES, AND R. FREEMAN, Mol. Phys. 64,691 (1988). A. G. REDFIELD AND S. D. KUNZ, J. Magn. Reson. 19,250 (1975). G. BODENHAUSEN, R. FREEMAN, G. A. MORRIS, R. NIEDERMEYER, AND D. L. TURNER, J. Magn.
12. 13. 14. 15.
A. G. C. P.
3. 4. 5. 6. 7. 8. 9.
10.
Reson. 25,559 (1977). PINES, D. E. WEMMER, J. TANG, AND S. SINTON, Bull. Am. Phys. Sot. 23,21 (1978). BODENHAUSEN, R. L. VOLD, AND R. R. VOLD, J. Magn. Reson. 37,93 (1980). J. TURNER, Prog. NMR Spectrosc. 16,311 (1984). BLONDET, J. P. ALBRAND, M. VON KIENLIN, M. DESCORPS, AND N. LAVANCHY, J. 71,342 (1987).
Magn. Reson.