Dispersive lineshape

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

An exponentially decaying FID gives a Lorentzian lineshape upon Fourier transformation. The general form of the absorptive Lorentzian line is IabS = 1/(1 + v2), whereas the dispersive line has the form Idisp = v/(l + v2), where I is the intensity at each point in the spectmm. Far from the peak maximum (v2 >> 1), we have Iabs 1/v2 and Idisp l/y- This is the reason that the dispersive lineshape extends much further from the peak maximum. 

Far from the peak maximum (v2 >> 1) we have Imagn 1/v, just like the dispersive lineshape. 

In die Fourier transform of a real time series, die peakshapes in the real and imaginary halves of die spectrum differ. Ideally, the real spectrum corresponds to an absorption lineshape, and die imaginary spectrum to a dispersion lineshape, as illustrated in Figure 3.20. The absorption lineshape is equivalent to a pure peakshape such as a Lorentzian or Gaussian, whereas die dispersion lineshape is a little like a derivative. 

Longitudinal to zero-quantum transfer. A transfer of the type UyS - I Sy) gives rise to in-phase absorption lineshapes in w, and antiphase dispersion lineshapes in Wj. 

Figure 5.19. The phase-sensitive COSY for a coupled two-spin AX system. Diagonal peaks have broad, in-phase douh e-dispersion lineshapes (D) whereas crosspeaks have narrow, antiphase double-absorption lineshapes (A), as further illustrated in the row extracted from the spectrum.

Prior to the advent of the above methods that allowed the presentation of phase-sensitive displays, 2D data sets were collected that were phase-modulated as a function of ti rather than amplitude-modulated. Phase-modulation arises when the sine and cosine modulated data sets collected for each ti increment are combined (added or subtracted) by the steps of the phase cycle, meaning each FID per tj 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 suitable for routine, low-resolution work. 

Phase-sensitive COSY-90 High-resolution display due to absorptive lineshapes. Crosspeak fine structure apparent J measurement possible. Diagonal peaks have dispersive lineshapes which may interfere with neighbouring crosspeaks. Requires high digital resolution to reveal multiplet structures. 

Whereas the absorption lineshape is always positive, the dispersion lineshape has positive and negative parts it also extends further. 

Schematic view of the diagonal peak from a COSY spectrum. The squares are supposed to indicate the two-dimensional double dispersion lineshape illustrated below...

Two peaks in Fx are expected at Ox +nJX2, these are just the two lines of the spin 1 doublet. In addition, since these are sine modulated they will have the dispersion lineshape. Note that both components in the spin 1 multiplet observed in F2 are modulated in this way, so the appearance of the two-dimensional multiplet can best be found by "multiplying together" the multiplets in the two dimensions, as shown opposite. In addition, all four components of the diagonal-peak multiplet have the same sign, and have the double dispersion lineshape illustrated below... 

The double dispersion lineshape seen in pseudo 3D and as a contour plot negative contours are indicated by dashed lines. 

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 general this is a mixture of the absorption and dispersion lineshape. If we want just the absorption lineshape we need to somehow set to zero, which is easily done by multiplying S((d) by a phase factor exp(i0j). 

This means that the real part of the spectrum shows a dispersion lineshape. On the other hand, if the magnetization is advanced by nil, 0 = 0sig - 0rx = nil - 0 = nil and it can be shown from Eq. [1] that the real part of the spectrum shows a negative dispersion lineshape. Finally, if either phase is advanced by n, the result is a negative absorption lineshape. 

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 real part of the complex frequency domain signal, A(Aco), represents the absorption lineshape of the detected signal and includes the frequency information required to determine whether the detected signal is at a lower or a higher frequency with respect to the transmitter frequency. The corresponding imaginary part D(A(o) represents the dispersive lineshape and also includes the frequency information. 

In principle, the real (cosine) part is associated with an absorption line shape and the imaginary (sine) part with a dispersion line shape in practice the real and imaginary parts of a contain mixtures of absorption and dispersion lineshapes. The major cause of this mixing is the disparity between the nominal start of the time spectrum and the true zero of time. Although it is possible to determine the latter accurately, practical considerations in the electronic logic setup result in distortions at early times. If the time spectrum used in the transform starts at time tj, then the corrected spectra are given by... 

These represent slightly power-broadened dispersion lineshapes and the predicted form (for low magnetic fields) is obtained as in the normal case by the derivative of the lineshape function, i.e.. 

As usual there is a reciprocal relationship between the time domain and the frequency domain. The more rapid is the damping (decay) of the signal (i.e., larger a and shorter lifetime t = 1/a), the wider the Lorentzian and dispersion lineshape functions become in the spectral domain (Figure 4). 

---