Lineshap absorption mode

Recently an effective method of calculating the absorption mode of a dynamic NMR spectrum by the point by point approach has been suggested. (47) It is based on the symmetric form of the lineshape equation (147). One can always choose a real unitary matrix U such that only the last element of the vector F = Uf is non-zero. The matrix ought to have its last column equal to the vector (fff ) 1/2f. Upon a transformation of equation (149), with the matrix U, one obtains the /abs ([Pg.261]

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 ... 

It is quite important for the spectra measured to represent the pure absorption mode with neither saturation nor transient effects. They should be measured in a homogenous external magnetic field B0 in order to obtain purely Lorentzian lineshapes for individual spectral transitions. The signal-to-noise ratio should be as high as possible and experimental... 

Table 1. Various lineshapes in the frequency and time domains. In the frequency domain g(f) corresponds to absorption mode lineshape and in the time domain G(t) corresponds to the in-phase component of the FID... 

Figure 5.18. 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.

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. 

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. 

It turns out that for instrumental reasons the axis along which the signal appears cannot be predicted, so in any practical situation there is an unknown phase shift. In general, this leads to a situation in which the real part of the spectrum (which is normally the part we display) does not show a pure absorption lineshape. This is undesirable as for the best resolution we require an absorption mode lineshape. 

The signal now has no phase shift and so will give us a spectrum in which the real part shows the absorption mode lineshape - which is exactly what we want. All we need to do is find the correct C0IT. 

Suppose that we record a spectrum with the simple pulse-acquire sequence using a 90° pulse applied along the x axis. The resulting FID is Fourier transformed and the spectrum is phased to give an absorption mode lineshape. 

Fourier transformation with respect to q gives peaks with an absorption lineshape, but this time in theiq dimension an absorption mode signal at 12, in Fx is denoted Af K The time domain signal becomes, after Fourier transformation in each dimension... 

These lineshapes are illustrated opposite. For NMR it is usual to display the spectrum with the absorption mode lineshape and in this case this corresponds to displaying the real part of S((0). 

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. 

It is clear from the form of S(t) that this phase introduced by altering the experiment (in this case, by altering the phase of the pulse) takes exactly the same form as the instrumental phase error. It can, therefore, be corrected by applying a phase correction so as to return the real part of the spectrum to the absorption mode lineshape. In this case the phase correction would be k 2. 

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. 

A spectrum with a pure absorption mode lineshape can be obtained by discarding the imaginary part of the time domain data immediately after the transform with respect to t2, i.e. taking the real part of S tl,a>2)c... 

A data set from an experiment to which TPPI has been applied is simply amplitude modulated in tx and so can be processed according to the method described for cosine modulated data so as to obtain absorption mode lineshapes. As the spectrum is symmetrical about Fx = 0 it is usual to use a modified Fourier transform routine which saves effort and space by only calculating the positive frequency part of the spectrum. 

If the vector and receiver are aligned along the same axis, 0 = 0, and the real part of the spectrum shows the absorption mode lineshape. If the receiver phase is advanced by nil, 0 = 0- nil and, from Eq. [1]... 

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. 

Frequency discrimination with retention of absorption mode lineshapes... 

For practical purposes it is essential to be able to achieve frequency discrimination and at the same time retain the absorption mode lineshape. There are a number of ways of doing this. 

We will see in later sections that when we use gradient pulses for coherence selection the natural outcome is P- or N-type data sets. Individually, each of these gives a frequency discriminated spectrum, but with the phase-twist lineshape. We will show in this section how an absorption mode lineshape can be obtained provided both the P- and the N-type data sets are available. 

The pulse sequence for NOESY (with retention of absorption mode lineshapes) is shown below... 

As we have seen earlier, it is not unusual to want to select two or more pathways simultaneously, for example either to maximise the signal intensity or to retain absorption-mode lineshapes. A good example of this is the doublequantum filter pulse sequence element, shown opposite. 

As has been discussed above, special care needs to be taken in experiments which use gradient selection in order to retain absorption mode lineshapes. 

E6-3. Assuming that magnetization along the y-axis gives rise to an absorption mode lineshape, draw sketches of the spectra which arise from the following operators... 

E7-3. What would the diagonal-peak multiplet of a COSY spectrum of two spins look like if we assigned the absorption mode lineshape in F2 to magnetization along x and the absorption mode lineshape in Fx to sine modulated data in tx ... 

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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