Lineshap dispersion mode

The lineshape function which describes the absorption and dispersion modes of an unsaturated, steady-state NMR spectrum is proportional to the Fourier transform of the function MxID(t) (24, 25, 99)... 

Contrary to the point by point approach the diagonalization method consists of the generation of an entire lineshape function in one step. (13, 14, 57-60) Time-consuming calculations are carried out only once. The resulting set of complex numbers can be used for a simple calculation of the lineshape (absorption and dispersion modes) at any desired point on the frequency axis. Thus, the complex matrix from equation (147) can be diagonalized by a similarity transformation using an co-independent complex matrix W ... 

Previous sections have already made the case for acquiring COSY data such that it may be presented in the phase-sensitive mode. The pure-absorption lineshapes associated with this provide the highest possible resolution and allow one to extract information from the fine-structure within crosspeak multiplets. However, it was also pointed out that the basic COSY-90 sequence suffers from one serious drawback in that diagonal peaks possess dispersion-mode lineshapes when crosspeaks are phased into pure absorption-mode. The broad tails associated with these can mask crosspeaks that fall close to the diagonal, so there is potential for useful information to be lost. The presence of dispersive contributions to the diagonal may be (largely) overcome by the use of the double-quantum filtered variant of COSY [37], and for this reason DQF-COSY is the experiment of choice for recording phase-sensitive COSY data. 

Fig. 4.5 Illustration of the absorption and dispersion mode Lorentzian lineshapes.

In a spectrum with just one line, the dispersion mode lineshape might be acceptable - in fact we can think of reasons why it might even be desirable (what might these be ). However, in a spectrum with many lines the dispersion mode lineshape is very undesirable - why ... 

Thus the real part shows the dispersion mode lineshape, and the imaginary part shows the absorption lineshape. The 90° phase shift simply swaps over the real and imaginary parts. 

We will use the shorthand that A2 represents an absorption mode lineshape at F2 = Q and D2 represents a dispersion mode lineshape at the same frequency. Likewise, A1+ represents an absorption mode lineshape at Fl = +12 and Du represents the corresponding dispersion lineshape. Ax and Dx represent the corresponding lines at Fl = -Q. 

The greatest drawback with data collected with phase modulation is the inextricable mixing of absorption and dispersion-mode lineshapes. The resonances are said to possess a phase-twisted lineshape (Fig. 5.21a), which has two principal disadvantages. Firstly, the undesirable and complex mix of both positive and negative intensities and secondly, the presence 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. 5.22). 

The cross peaks in the 2D spectrum are a combination of absorption and dispersion lineshapes and consequently spectra are displayed in magnitude mode. 

The result from the filtration step, and the principal reason for its use, is that the diagonal peaks now possess antiphase absorption-mode lineshapes, as do the crosspeaks which are unaffected by the filtration. Strictly speaking, for spin systems of more than two spins the diagonal peaks still possess some dispersive contributions, but these are now antiphase so cancel and tend to be weak and rarely problematic. The severe tailing previously associated with diagonal peaks therefore is removed, providing a dramatic improvement in the quality of spectra (Fig. 5.42). 

In (b) we see the effect of a phase shift, (p, of 45°. Sy now starts out at a finite value, rather than at zero. As a result neither the real nor the imaginary part of the spectrum has the absorption mode lineshape both are a mixture of absorption and dispersion. 

Thus, displaying the real part of S(co) will not give the required absorption mode spectrum rather, the spectrum will show lines which have a mixture of absorption and dispersion lineshapes. 

The phase-twist lineshape is an inextricable mixture of absorption and dispersion it is a superposition of the double absorption and double dispersion lineshape (illustrated in section 7.4.1). No phase correction will restore it to pure absorption mode. Generally the phase twist is not a very desirable lineshape as it has both positive and negative parts, and the dispersion component only dies off slowly. 

In real experiments after Fourier transformed the lineshapes are mixtures of absorptive and dispersive signals and are related to the delayed FID acquisition (first-order phase error). The delayed acquisition is a consequence of the minimum time required to change the spectrometer from transmit to receive mode, during this delay the magnetization vectors process according to their chemical shift frequencies. The zero-order phase error arises because of the phase difference between the magnetization vectors and the receiver. In NMR-SIM the delayed acquisition is not necessary because the ideal spectrometer approach does not require any switching time and the first order phase correction is normally zero if no other sources of phase deviations are present. 

Back to a more fundamental level, Kim and Prestegard showed that simple lineshape analysis makes it possible to take the linewidth into account when measuring coupling constants in broad signals. The paper discussed both doublets that are displayed in absorption mode and the very neglected case where they are dispersive. 

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