- Email: [email protected]
Elimination of dispersion-mode contribuions from two-dimensional NMR spectra
JOURNAL
OF
MAGNETIC
RESONANCE
Ehination
34,
663-667
(1979)
of Dispersion-Mde Con Two-Dimensional NMR §peetra
Two-dimensional Fourier transformation has been used in conjunction with NMR spin-echo experiments to improve the resolution of spin-multiplet components (2, 2) and to obtain proton spectra in which spin-spin splitting has been eliminated (3,4). Both experiments make use of the phase modulation of spin echoes caused by spin-spin coupling (5), and a particular feature of the resulting two-dimensional spectra is the unusual lineshape obtained-parallel sections through the resonance response show a rapid change in the proportion of absorption and dispersion mode. This phenomenon has been called the “phase twist” (6) and can have undesirable consequences when two resonances overlap or when projections of the two-dimensional spectrum are required (3, 7). This communication discusses two possible methods of circumventing this problem. Phase modulation of spin echoes may be represented by precessing vectors, and it is important to distinguish the sense of this precession in the appropriate rotating reference frame, since this will determine whether the corresponding response is at a positive or a negative frequency in the F, dimension of the two-dimensional spectrum. Consider first the case of a line with a positive Fl frequency, with coordinates (fl, FX). Let the running frequency parameters be AF1 = F1 -F? and AFz = F2 -F: and let the corresponding linewidth parameters be A i and A*. It proves convenient to define (6) D: =A: +(2rAFd2,
D: =A: +(2rAF,J2,
sin a = 21rdF~/D~,
sin /3 = 277 AFJD2.
cos CY= Al/D1,
cos ,Li= AJDz.
A section through a two-dimensional shape if described by
response would then have an absorption-mode
cos a/D, = A ~/[h: + (257 AF$], but a dispersion-mode
:I1
E4
shape if described by sin cr/D1 = 2rAF,/[Af
+(2~rAFl)~].
PI
A large number of spin-echo experiments are performed, the second half of each echo, M(t2) being measured as a function of the evolution period tl, giving a data matrix M(ti, f2). For resonances with positive Fl frequencies, Fourier transformation as a function of both time parameters yields four signal components Scc(Fl, F2) = +A(sin (Ycos p fcos LYsin P)/~DID~,
[41
S”‘(F,, F2) = +A(cos a cos p -sin cysin /3)/4D1D2,
r51
663
0022-2364/7~/[email protected]/0 Copyright $3 1979 by Academic Press, Inc. All nghfs of reproduction in any form reserved. Printed in Great Britain
COMMUNICATIONS
664
SCS(F1,F2) = +A(cos CYcos p -sin (Ysin p)/4D1D2,
SSS(FI,F2) = -A(sin
LYcos p +cos cy sin p)/4D1Dz.
[71
This illustrates the phase twist effect, for a section through the exact center of the response S”‘(Fi, Fz) parallel with the F1 axis has sin /3 = 0 and consequently presents a pure absorption mode, whereas any other parallel sections at finite offsets AF2 contain a term in sin (Yand a corresponding dispersion component. If the sense of this phase twist is defined to be “left-handed,” it can be shown (6) that a vector with an inherently negative sense of precession (giving a response in the negative F1 dimension) also exhibits a left-handed phase twist. There is a fundamental antisymmetry of the dispersion-mode components about FI = 0, whereas the absorption-mode components have mirror symmetry. Now certain two-dimensional spectra derived from spin-echo experiments have exact mirror symmetry of frequencies and intensities about the axis F, = 0; they comprise those obtained by the “proton flip” technique (8,9) and those obtained by the “gated decoupler” method provided the coupling is strictly first order (10). Such spectra can be plotted without frequency discrimination, each response being the superposition of two components of equal intensity, one “folded over” from the negative FI quadrant. Because of the antisymmetry of the dispersive components with respect to F1 = 0, these contributions to the resonance signal cancel exactly, leaving the pure absorption mode in both frequency dimensions (11 ), S”‘(*FI,
PI
F2) = (A cos (Ycos /I)/2DlDp
This suggests that the phase twist might be eliminated by combining the results of two experiments, in one of which the accumulated precession phase at the end of the evolution period is somehow reversed in sign (6). Bachmann et al. (12) have implemented this idea for two-dimensional carbon-13 spectra by introducing a 180” pulse between the evolution and detection periods. Practical problems might arise with this technique when there are imperfections in the 180” pulse (13) or when there is strong coupling between the proton spins (14). This communication suggests two other methods of achieving the effect of reversed precession with a view to canceling the phase twist. The first uses the “gated decoupler” technique in its two possible modes, “on-off” and “off-on,” to give normal and “reversed” precession during the evolution period. The second exploits the fact that the first half of a spin echo provides essentially reversed precession if the tz scale is thought of as running backward in time, and if the slight asymmetry in the echo due to spin-spin relaxation is neglected. Bax et al. (15) have recently described this experiment, and the reader is referred to their forthcoming publication for the details of this method.’ Only the gated decoupler method will be described here. Consider for simplicity two different nuclear species of spin : constituting an IS spin system. Echo modulation induced by gating the noise irradiation applied to the S spins has been described elsewhere (Z), but as can be seen from Fig. 1, there are two alternative modes for this experiment. The two components of I-spin magnetization, labeled “ + ” and “ - ,” accumulate phase angles at the end of the evolution period 1 The authors
are indebted
to Dr. Mehlkopf
for providing
a manuscript
of this work
prior
to publication.
COMMUNICATIONS
h65
‘% a
I
‘“c b
FIG. 1. Modulation of carbon-13 spin echoes by CH coupling in the “gated decoupler” experiment, showing the two possible modes of switching, (a) and (b). The time evolution of the phase of two carbon- 13 magnetization vectors “ + ” and ” - ” has been traced out to show that the phase angles built up at the end of the evolution period ti are reversed in sign in mode (b). This has the effect of reversing the spectrum in the Fi dimension.
that are equal but opposite in sign in the two modes, just as if the sense of the nuclear precession had been reversed for mode (b). Note the similarity between sequences (a) and (b) of Fig. 1. The same nuclear Overhauser enhancement applies to both, and in each case there is only a single 180” carbon-13 pulse so that any imperfections apply equally in (a) and (b). Signals from the two modes, A4, and Mb, are stored separately, transformed with respect to cl and f2, and instrumental phase errors are corrected by displaying the two spectra separately on an oscilloscope (I1 ). Both spectra are frequency discriminated in the FI dimension and will in general be asymmetric about FI = 0. The spectrum obtained in mode (b) must be reversed in the FI dimension in order that the sense of the asymmetry of the spin-multiplet structure correspond with that of the conventional proton-coupled NMR spectrum; if it is then superimposed cm the spectrum obtained in mode (a) the dispersive components cancel as in Eq. ES]. The result is an absorption-mode spectrum which retains the correct asymmetry of the spin multiplets. To illustrate this method it is important to choose a two-dimensional spectrum that has inherent asymmetry in the FI dimension. The carbon-13 spectrum of pyridkre exhibits such an asymmetry for the C2 site (IO, 16) because of strong coupiing between the proton spins. Signals were recorded on a Varian (XT-20 spectrometer modified for two-dimensional transformation (6) using the gated decoupler method. Figure 2 shows the phase-sensitive display of the two-dimensional spectrum, confirming that absorption-mode lines can be achieved in both frequency dimensions.
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FIG. 2. Proton-coupled spectrum of carbon-13 in pyridine obtained by two-dimensional Fourier transformation of spin echoes. The Fi axis runs from positive to negative frequencies so as to correspond to the accepted convention for one-dimensional spectra. Two separate spectra were acquired with the two alternative modes of the gated decoupler experiment (see Fig. 1) and individually corrected for instrumental phase errors. The spectrum from mode (b) was then reversed in the Fi dimension and superimposed on the spectrum from mode (a), thus canceling all dispersive components. Note the asymmetry of the fine structure on the C2 resonance, reproduced on an expanded scale at the bottom of the diagram.
A large proportion of the two-dimensional spectra published to date have had to rely on the absolute-value mode of display, which has undesirable effects for high-resolution work and necessarily prevents the observation of inverted lines arising from population inversion (17, 18) or phase reversal (9, 19). The techniques described above allow absorption-mode spectra to be used instead. ACKNOWLEDGMENTS This work was supported by an equipment grant from the Science Research Council, and Research Studentships for S.P.K. and M.H.L. Dr. Howard Hill was the first to suggest the possibility of canceling dispersive signals in two-dimensional NMR spectra (6). REFERENCES 1. W.P.AUE,E.BARTHOLDI,AND R.R.ERNsT,J. Chem.Phys.64,2229 (1976). 2. G.BODENHAUSEN,R.FREEMAN,AND D.L.TuRNER,J. Chem.Phys.65,839(1976).
COMMUNICATIONS
thq
3. W. P. AUE,J.KARHAN,AND R.R.ERNsT, J. Chem.Phys.64,4226 (1976). 4. K.NAGAYAMA,K.W~~THRICH,P.BACHMANN,ANDR. R. ERNST, Naturwissenschaften
64,581
(1977).
5. E.L. HAHNANDD.E.MAXWELL, Phys.Reu.88,1070 (1952). 6. G.BODENHAUSEN,R.FREEMAN,R.NIEDERMEYER,ANDD.L.T~RNER,J.
Magn.Reson.26,
133 (1977).
7 K. NAGAYAMA,
P. BACHMANN, K. WOTHRICH, AND R. R. ERNST, J. Magn.
Reson.
31, 133
(1978). 8.
G.BODENHAUSEN,R.FREEMAN,R.NIEDERMEYER,ANDD.L.T~RNER,J.
Magn.Reson.24,
291 (1976). 9.
G.BODENHAUSEN,R.FREEMAN,G.
A.MORRIS,AND
D.L.TuRNER,
J Magn.Reson.28,
(1977).
IO. R. FREEMAN,G. A.MORRIS,ANDD.L.TURNER, J. Magn.Reson.26,373 (1977). 11. M. H. LEVI~AND R.FREEMAN, J. Magn.Reson. in press. 12. P.BACHMANN, W. P. AUE,L.MULLER,AND R.R.ERNsT, J.Magn.Reson.t8,29(1977r. 13. G.BODENHAUSEN,R.FREEMAN,ANDD.L.TURNER,J. Magn.Reson.27,511(1977). 14. A. KUMAR AND R. R. ERNST, Chem. Phys. Lett. 37,162 (1976). 15. A. BAX, A. F. MEHLKOPF, AND J. SMIDT, J. Magn. Reson., in press. 16. M.HANSENANDH.J.JAKOBSEN, J.Magn.Reson. l&74 (1973). 17. A. A.MAUDSLEYANDR.R.ERNST, Chem.Phys.Lett. 50,368 (1977). 1X. G. BODENHAUSENAND R.FREEMAN, J. Magn.Reson.28,471 (1977). 19. G.BODENHAUSEN,R.FREEMAN,G.A.MORRIS,ANDD.L.TURNER, J. Magn.Reson.31,75 c 1978). RAY FREEMAN STEWART P. KEMPSELL. MALCOLM H. L~vrm
Physical Chemistry Laboratory Oxford University Oxford, England Received February 12, 1979
17
OF
MAGNETIC
RESONANCE
Ehination
34,
663-667
(1979)
of Dispersion-Mde Con Two-Dimensional NMR §peetra
Two-dimensional Fourier transformation has been used in conjunction with NMR spin-echo experiments to improve the resolution of spin-multiplet components (2, 2) and to obtain proton spectra in which spin-spin splitting has been eliminated (3,4). Both experiments make use of the phase modulation of spin echoes caused by spin-spin coupling (5), and a particular feature of the resulting two-dimensional spectra is the unusual lineshape obtained-parallel sections through the resonance response show a rapid change in the proportion of absorption and dispersion mode. This phenomenon has been called the “phase twist” (6) and can have undesirable consequences when two resonances overlap or when projections of the two-dimensional spectrum are required (3, 7). This communication discusses two possible methods of circumventing this problem. Phase modulation of spin echoes may be represented by precessing vectors, and it is important to distinguish the sense of this precession in the appropriate rotating reference frame, since this will determine whether the corresponding response is at a positive or a negative frequency in the F, dimension of the two-dimensional spectrum. Consider first the case of a line with a positive Fl frequency, with coordinates (fl, FX). Let the running frequency parameters be AF1 = F1 -F? and AFz = F2 -F: and let the corresponding linewidth parameters be A i and A*. It proves convenient to define (6) D: =A: +(2rAFd2,
D: =A: +(2rAF,J2,
sin a = 21rdF~/D~,
sin /3 = 277 AFJD2.
cos CY= Al/D1,
cos ,Li= AJDz.
A section through a two-dimensional shape if described by
response would then have an absorption-mode
cos a/D, = A ~/[h: + (257 AF$], but a dispersion-mode
:I1
E4
shape if described by sin cr/D1 = 2rAF,/[Af
+(2~rAFl)~].
PI
A large number of spin-echo experiments are performed, the second half of each echo, M(t2) being measured as a function of the evolution period tl, giving a data matrix M(ti, f2). For resonances with positive Fl frequencies, Fourier transformation as a function of both time parameters yields four signal components Scc(Fl, F2) = +A(sin (Ycos p fcos LYsin P)/~DID~,
[41
S”‘(F,, F2) = +A(cos a cos p -sin cysin /3)/4D1D2,
r51
663
0022-2364/7~/[email protected]/0 Copyright $3 1979 by Academic Press, Inc. All nghfs of reproduction in any form reserved. Printed in Great Britain
COMMUNICATIONS
664
SCS(F1,F2) = +A(cos CYcos p -sin (Ysin p)/4D1D2,
SSS(FI,F2) = -A(sin
LYcos p +cos cy sin p)/4D1Dz.
[71
This illustrates the phase twist effect, for a section through the exact center of the response S”‘(Fi, Fz) parallel with the F1 axis has sin /3 = 0 and consequently presents a pure absorption mode, whereas any other parallel sections at finite offsets AF2 contain a term in sin (Yand a corresponding dispersion component. If the sense of this phase twist is defined to be “left-handed,” it can be shown (6) that a vector with an inherently negative sense of precession (giving a response in the negative F1 dimension) also exhibits a left-handed phase twist. There is a fundamental antisymmetry of the dispersion-mode components about FI = 0, whereas the absorption-mode components have mirror symmetry. Now certain two-dimensional spectra derived from spin-echo experiments have exact mirror symmetry of frequencies and intensities about the axis F, = 0; they comprise those obtained by the “proton flip” technique (8,9) and those obtained by the “gated decoupler” method provided the coupling is strictly first order (10). Such spectra can be plotted without frequency discrimination, each response being the superposition of two components of equal intensity, one “folded over” from the negative FI quadrant. Because of the antisymmetry of the dispersive components with respect to F1 = 0, these contributions to the resonance signal cancel exactly, leaving the pure absorption mode in both frequency dimensions (11 ), S”‘(*FI,
PI
F2) = (A cos (Ycos /I)/2DlDp
This suggests that the phase twist might be eliminated by combining the results of two experiments, in one of which the accumulated precession phase at the end of the evolution period is somehow reversed in sign (6). Bachmann et al. (12) have implemented this idea for two-dimensional carbon-13 spectra by introducing a 180” pulse between the evolution and detection periods. Practical problems might arise with this technique when there are imperfections in the 180” pulse (13) or when there is strong coupling between the proton spins (14). This communication suggests two other methods of achieving the effect of reversed precession with a view to canceling the phase twist. The first uses the “gated decoupler” technique in its two possible modes, “on-off” and “off-on,” to give normal and “reversed” precession during the evolution period. The second exploits the fact that the first half of a spin echo provides essentially reversed precession if the tz scale is thought of as running backward in time, and if the slight asymmetry in the echo due to spin-spin relaxation is neglected. Bax et al. (15) have recently described this experiment, and the reader is referred to their forthcoming publication for the details of this method.’ Only the gated decoupler method will be described here. Consider for simplicity two different nuclear species of spin : constituting an IS spin system. Echo modulation induced by gating the noise irradiation applied to the S spins has been described elsewhere (Z), but as can be seen from Fig. 1, there are two alternative modes for this experiment. The two components of I-spin magnetization, labeled “ + ” and “ - ,” accumulate phase angles at the end of the evolution period 1 The authors
are indebted
to Dr. Mehlkopf
for providing
a manuscript
of this work
prior
to publication.
COMMUNICATIONS
h65
‘% a
I
‘“c b
FIG. 1. Modulation of carbon-13 spin echoes by CH coupling in the “gated decoupler” experiment, showing the two possible modes of switching, (a) and (b). The time evolution of the phase of two carbon- 13 magnetization vectors “ + ” and ” - ” has been traced out to show that the phase angles built up at the end of the evolution period ti are reversed in sign in mode (b). This has the effect of reversing the spectrum in the Fi dimension.
that are equal but opposite in sign in the two modes, just as if the sense of the nuclear precession had been reversed for mode (b). Note the similarity between sequences (a) and (b) of Fig. 1. The same nuclear Overhauser enhancement applies to both, and in each case there is only a single 180” carbon-13 pulse so that any imperfections apply equally in (a) and (b). Signals from the two modes, A4, and Mb, are stored separately, transformed with respect to cl and f2, and instrumental phase errors are corrected by displaying the two spectra separately on an oscilloscope (I1 ). Both spectra are frequency discriminated in the FI dimension and will in general be asymmetric about FI = 0. The spectrum obtained in mode (b) must be reversed in the FI dimension in order that the sense of the asymmetry of the spin-multiplet structure correspond with that of the conventional proton-coupled NMR spectrum; if it is then superimposed cm the spectrum obtained in mode (a) the dispersive components cancel as in Eq. ES]. The result is an absorption-mode spectrum which retains the correct asymmetry of the spin multiplets. To illustrate this method it is important to choose a two-dimensional spectrum that has inherent asymmetry in the FI dimension. The carbon-13 spectrum of pyridkre exhibits such an asymmetry for the C2 site (IO, 16) because of strong coupiing between the proton spins. Signals were recorded on a Varian (XT-20 spectrometer modified for two-dimensional transformation (6) using the gated decoupler method. Figure 2 shows the phase-sensitive display of the two-dimensional spectrum, confirming that absorption-mode lines can be achieved in both frequency dimensions.
COMMUNICATIONS
FIG. 2. Proton-coupled spectrum of carbon-13 in pyridine obtained by two-dimensional Fourier transformation of spin echoes. The Fi axis runs from positive to negative frequencies so as to correspond to the accepted convention for one-dimensional spectra. Two separate spectra were acquired with the two alternative modes of the gated decoupler experiment (see Fig. 1) and individually corrected for instrumental phase errors. The spectrum from mode (b) was then reversed in the Fi dimension and superimposed on the spectrum from mode (a), thus canceling all dispersive components. Note the asymmetry of the fine structure on the C2 resonance, reproduced on an expanded scale at the bottom of the diagram.
A large proportion of the two-dimensional spectra published to date have had to rely on the absolute-value mode of display, which has undesirable effects for high-resolution work and necessarily prevents the observation of inverted lines arising from population inversion (17, 18) or phase reversal (9, 19). The techniques described above allow absorption-mode spectra to be used instead. ACKNOWLEDGMENTS This work was supported by an equipment grant from the Science Research Council, and Research Studentships for S.P.K. and M.H.L. Dr. Howard Hill was the first to suggest the possibility of canceling dispersive signals in two-dimensional NMR spectra (6). REFERENCES 1. W.P.AUE,E.BARTHOLDI,AND R.R.ERNsT,J. Chem.Phys.64,2229 (1976). 2. G.BODENHAUSEN,R.FREEMAN,AND D.L.TuRNER,J. Chem.Phys.65,839(1976).
COMMUNICATIONS
thq
3. W. P. AUE,J.KARHAN,AND R.R.ERNsT, J. Chem.Phys.64,4226 (1976). 4. K.NAGAYAMA,K.W~~THRICH,P.BACHMANN,ANDR. R. ERNST, Naturwissenschaften
64,581
(1977).
5. E.L. HAHNANDD.E.MAXWELL, Phys.Reu.88,1070 (1952). 6. G.BODENHAUSEN,R.FREEMAN,R.NIEDERMEYER,ANDD.L.T~RNER,J.
Magn.Reson.26,
133 (1977).
7 K. NAGAYAMA,
P. BACHMANN, K. WOTHRICH, AND R. R. ERNST, J. Magn.
Reson.
31, 133
(1978). 8.
G.BODENHAUSEN,R.FREEMAN,R.NIEDERMEYER,ANDD.L.T~RNER,J.
Magn.Reson.24,
291 (1976). 9.
G.BODENHAUSEN,R.FREEMAN,G.
A.MORRIS,AND
D.L.TuRNER,
J Magn.Reson.28,
(1977).
IO. R. FREEMAN,G. A.MORRIS,ANDD.L.TURNER, J. Magn.Reson.26,373 (1977). 11. M. H. LEVI~AND R.FREEMAN, J. Magn.Reson. in press. 12. P.BACHMANN, W. P. AUE,L.MULLER,AND R.R.ERNsT, J.Magn.Reson.t8,29(1977r. 13. G.BODENHAUSEN,R.FREEMAN,ANDD.L.TURNER,J. Magn.Reson.27,511(1977). 14. A. KUMAR AND R. R. ERNST, Chem. Phys. Lett. 37,162 (1976). 15. A. BAX, A. F. MEHLKOPF, AND J. SMIDT, J. Magn. Reson., in press. 16. M.HANSENANDH.J.JAKOBSEN, J.Magn.Reson. l&74 (1973). 17. A. A.MAUDSLEYANDR.R.ERNST, Chem.Phys.Lett. 50,368 (1977). 1X. G. BODENHAUSENAND R.FREEMAN, J. Magn.Reson.28,471 (1977). 19. G.BODENHAUSEN,R.FREEMAN,G.A.MORRIS,ANDD.L.TURNER, J. Magn.Reson.31,75 c 1978). RAY FREEMAN STEWART P. KEMPSELL. MALCOLM H. L~vrm
Physical Chemistry Laboratory Oxford University Oxford, England Received February 12, 1979
17