All the 2D spectra presented so far have made use of quadrature detection in both dimensions, enabling the transmitter frequencies for each to be placed at the centre of the spectrum. Quadrature detection for the f2 data is achieved by either the simultaneous or sequential sampling schemes described in Section 3.2.4 and is therefore entirely analogous to that used for 1D acquisitions. Quadrature detection in f1 is also necessary for the same reasons. As with the 1D case, this demands some means of distinguishing frequencies that are higher than that of the reference from those that are lower when evolving during t1. In other words, it is again necessary to distinguish positive and negative frequencies in the rotating frame so that mirror image signals do not appear either side of f1 = 0. There are two general approaches to this in widespread use, one providing so-called phase-sensitive data displays, while the other provides data that are conventionally displayed in the absolute value mode in which all phase information has been discarded. The first of these displays lineshapes in which absorption- and dispersion-modes are separated, meaning the preferred absorption-mode signal is available for a high-resolution display.
The second approach is inferior in that it produces lineshapes in which the absorptive and dispersive parts are inextricably mixed making it poorly suited to high-resolution work. However, since absolute value spectra are rather easy to process and manipulate, they still find use in some routine, fully automated processes, so are also considered here.
5.2.1.1 Phase-sensitive Presentations
As for 1D data, f1 quadrature detection requires two data sets to be collected which differ in phase by 90 degrees, thus providing the necessary sine and cosine amplitude-modulated data. Since the f1 dimension is generated artificially, there is strictly no reference rf to define signal phases so it is the phase of the pulses that bracket t1 that dictate the phase of the
FIGURE 5.12 The COSY spectrum of a coupled, two-spin AX system. Diagonal peaks are equivalent to those observed in the 1D spectrum while cros- speaks provide evidence of a coupling between the correlated spins.
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178 High-Resolution NMR Techniques in Organic Chemistry
detected signal. Thus, for each t1 increment two data sets are collected, one with a 90x preparation pulse (t1 sine modula- tion), the other with 90y (t1 cosine modulation), and both stored separately (Fig. 5.13). These two sets are then equivalent to the two channel data collected with simultaneous acquisition which produces the desired frequency discrimination when subject to a complex Fourier transform (also referred to as a hypercomplex transform in relation to 2D data). The rate of sampling in t1 or, in other words, the size of the t1 time increment, is dictated by the f1 spectral width and is subject to the same rules as for the simultaneous sampling of 1D data. This method is derived from the original work of States, Haberkorn and Ruben [5,6] and is therefore often referred to in the literature as the States method of f1 quad-detection.
An alternative approach is analogous to the method of sequential sampling introduced in Section 3.2.4. As in the 1D approach the aim is to avoid both positive and negative frequencies arising by shifting the apparent frequency range from
±ẵSW1 to 0 to +SW1 Hz by making the evolution frequencies during t1 appear faster than they actually are. As for the States method discussed earlier, there is no reference rf phase to shift for this artificial domain, so the equivalent effect is achieved by incrementing the phase of the preparation pulse by 90 degrees for each t1 increment, and sampling the data twice as fast as for the States method (by halving the t1 increment). Only one data set is acquired for each t1 period, but twice as many t1 increments are collected, so the total t1 acquisition time, and hence the digital resolution, is equal for both methods. This approach to f1 quad-detection [7] is now referred to as time proportional phase incrementation (TPPI).
The States and TPPI methods produce equivalent data sets [8] although they differ subtly in the appearance of aliased signals and the artefacts known as axial peaks, which are described more fully below. It turns out that an effective way to deal with axial peaks is to beneficially combine these two approaches to yield the so-called States–TPPI method of quadra- ture detection [9] as described in Section 5.2.2. Selecting between these related approaches is often left to the experimental- ist but in general the preferred method is States–TPPI. The most significant point from all this is the 2D lineshapes these methods produce. All methods involve the detection of a signal that is amplitude modulated as a function of the t1 evolu- tion period (Fig. 5.4), and this is the general requirement for producing spectra that have absorption and dispersion parts separated (pure-phase spectra) so allowing a phase-sensitive presentation. Separate real and imaginary parts of the data exist for both the f1 and f2 dimensions, again analogous to the real and imaginary parts of a 1D spectrum. This gives rise to four data quadrants (Fig. 5.14) with only the (real, real) data set being presented to the user as the final 2D spectrum, the others being retained for phase correction. It is usual for the displayed spectrum to contain absorption-mode lineshapes in both dimensions (Fig. 5.6) wherever possible since the double-absorption lineshape affords the highest possible resolution (Fig. 5.14, RR quadrant). The phase information contained within crosspeaks can also provide additional information in some circumstances and this is especially true for the phase-sensitive COSY experiment.
Finally we note an alternative approach to quadrature detection known as echo–antiecho selection [10] which is appli- cable only to PFG selected 2D methods and which now finds widespread use. As this involves a quite different procedure it will not be considered further here but will be introduced in Section 5.4.2 after field gradients have been described.
5.2.1.2 Aliasing in Two Dimensions
Resonances that fall outside the chosen spectral width will be characterised with incorrect frequencies and so will appear aliased in the 2D spectrum. For symmetrical data sets such as COSY, the two spectral widths are chosen to be the same, so
FIGURE 5.13 The States method of f1 quad-detection. This requires two data sets to be acquired per increment to generate separate sine and cosine modulated data sets.
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aliasing will occur in both dimensions, and the position of the aliased signal can usually be predicted from the quadrature de- tection scheme used for each dimension. Thus, in analogy with 1D spectra, simultaneous or States sampling causes signals to be wrapped around, appearing at the far end of the spectrum, while those from sequential or TPPI sampling appear as folded signals at the near end. Some confusion can be introduced if different sampling schemes are used for the two dimensions, for example simultaneous sampling in f2 but TPPI in f1 [11]. Deliberately aliasing signals can be a useful trick in acquiring 2D data since reduced spectral widths imply reduced acquisition times and smaller data sets, or alternatively, greater resolu- tion. For example it is often feasible to eliminate lone phenyl groups from COSY spectra and concentrate on the aliphatic region only as this is where the interesting correlations most often lie. With modern digital filters aliased signals in f2 can be eliminated, although those aliased in f1 remain since there is no equivalent filtration in the indirect dimension.
5.2.1.3 Absolute Value Presentations
Prior to the advent of the above methods that allowed the presentation of phase-sensitive displays, 2D data sets were col- lected that were phase-modulated as a function of t1 rather than amplitude-modulated. Phase-modulation arises when the sine and cosine modulated data sets collected for each t1 increment are combined (added or subtracted) by the steps of the phase cycle, meaning each FID per t1 increment contains a mixture of both parts. Here it is the sense of phase precession that allows the differentiation of positive and negative frequencies. This method is inferior to the phase-sensitive approach because of the unavoidable mixing of absorptive and dispersive lineshapes, so is generally only considered for routine, low-resolution work.
The selection of only one of the two possible mirror image data sets in f1 is now achieved by suitable combination of the sine and cosine data sets. Addition of the two data sets within the phase cycle (Table 5.1) selects those signals that have the same sense of precession in t1 as they have during t2 (say, both positive) and these are referred to as P-type signals. Subtrac- tion within the phase cycle selects those signals that have the opposite sense of precession during t1 and t2 (say, negative in t1, positive in t2) and are referred to as N-type signals. To clarify this point, remember the two senses of precession we speak of here simply represent the two possible signals that would be detected either side of f1 = 0 if one were not using quadrature detection. By employing this it becomes possible to select only one of these to appear in the final spectrum while cancelling the mirror image. The information content of the P-type or N-type spectra are equivalent, only their appearance differs by virtue of reflection about f1 = 0. One significant difference arises from the fact that with N-type selection signals are chosen that have opposite senses of precession in the two time periods. This may be thought of as being analogous to a spin-echo where vectors move in opposite senses either side of the refocusing pulse. A similar effect arises during t2 with N-type selection whereby echoes also occur, known as coherence transfer echoes (Fig. 5.15). For this reason, N-type
FIGURE 5.14 The four quadrants of a phase-sensitive data set. Only the RR quadrant is presented as the 2D spectrum and this is phased to contain absorption-mode lineshapes in both dimensions to provide the highest resolution (R = real, I = imaginary). Positive contours are in black and negative in red.
TABLE 5.1 COSY Phase Cycles to Select the N-Type (Echo) or P-Type (Anti-echo) Signals in f1
N-Type P-Type
Pulse 1 Pulse 2 Acquire Pulse 1 Pulse 2 Acquire
Cycle 1 x x x x x x
Cycle 2 y x –x y x x
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selection is also referred to as echo selection and P-type signals as anti-echo selection, since the refocusing effect does not arise for signals that precess in the same sense. Refocusing of field inhomogeneity with echo selection together with the fact that P-type signals are subject to more severe distortions, means that N-type selection is preferred when this method of quad-detection is used. Conventionally, these spectra are then presented with the diagonal running from bottom-left to top-right, as in earlier figures.
The greatest drawback with data collected with phase-modulation is the inextricable mixing of absorption and disper- sion-mode lineshapes. The resonances are said to possess a phase-twisted lineshape (Fig. 5.16a), which has two principle disadvantages. Firstly, the undesirable and complex mix of both positive and negative intensities, and secondly, the pres- ence of dispersive contributions and the associated broad tails that are unsuitable for high-resolution spectroscopy. To remove confusion from the mixed positive and negative intensities, spectra are routinely presented in absolute value mode, usually after a magnitude calculation (Fig. 5.17):
= +
M (real2 imaginary )2 12 M=(real2+imaginary2)12
FIGURE 5.15 The coherence transfer echo is apparent in an FID taken from an N-type COSY data set.
FIGURE 5.16 2D lineshapes. A stacked plot illustration of (a) the phase-twisted line shape and (b) the double-absorption lineshape. Clearly the resolu- tion in (b) is far superior and for this reason phase-sensitive methods are preferred.
FIGURE 5.17 Magnitude calculation. Contour plots of (a) the phase-twist lineshape, (b) the same following magnitude calculation, and (c) the same following resolution enhancement with an unshifted sine bell window and magnitude calculation.
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where M represents the resulting spectrum. To improve resolution, severe window functions are also employed to elimi- nate the dispersive tail from the lineshape. Although a number of shaping functions are suitable [12], the most frequently used are the sine bell, or the squared sine bell (Section 3.2.7), which are simple to use as they have only one variable, the position of the maximum. When used unshifted, for example when the half sine-wave has a maximum at the centre of the acquisition time, the resulting peaks possess no dispersive component, thus reproducing the desired absorption lineshape (Fig. 5.17c). The sine bell shape is also beneficial in enhancing the crosspeak intensities relative to the diagonal. Since the crosspeaks arise from sine modulations, as described earlier, they initially have zero intensity which builds within the acquisition time. Diagonal peaks are cosine modulated so begin with maximum intensity and are therefore attenuated by the window function. However, the attenuation presents a problem for signals with differing relaxation times (or, in other words, linewidths) as these will be attenuated to different extents and, in particular, broader lines will be notably reduced in intensity by the early part of the sine bell. The moral when interpreting absolute value data processed in this way is to be wary of differing crosspeak intensities and not simply to associate these with smaller coupling constants. The final problem with the use of these extreme resolution enhancing window functions is that much of the early signal is attenuated and the later noise enhanced, leading to a potential loss of sensitivity. It is possible to shift the function maximum to earlier in the FID to help improve this, with the inevitable reintroduction of some dispersive component, although a better approach is to acquire phase-sensitive data which does not demand the application of such severe window functions. Despite this, when sensitivity is not a limiting factor, the magnitude experiment does have the advantage of not requiring phase corrections to be made, so is well suited to routine acquisitions and to automated methods, especially when implemented as the COSY-b variant described in Section 6.1.4.