Circle, DISPA

Single Lorentzian line the DISPA reference circle... 

Equation 4 predicts that a plot of normalized dispersion, D(w), vs. normalized absorption, A(w), for a single Lorentzian line will yield a circle of radius, t/2, centered at A(< >) = (t/2) on the A-axis (see Figure 2). Since a single Lorentzian line is two-fold symmetric about its center, the reference" DISPA plot is two-fold symmetric about the A-axis, so that only the top half of the circle need be displayed in Figure 2. 

Figure 2 shows that a single Lorentzian line yields a DISPA "reference" circle. For a DISPA plot for any other experimental line shape, any deviation from a reference circle having the same absorption peak height will then reflect non-Lorentzian composite line shape. We will next examine DISPA plots for several types of linebroadening, to determine the direction and magnitude of the displacement from the corresponding "reference" circle. 

Figure 2. (Left) Normalized absorption and dispersion (left) for a single Lorentzian line (frequency in units of 1/t). (Right) Plot of dispersion versus absorption (OISPA) for the right-hand half of each (symmetrical) spectrum at left, to give the upper half of the DISPA "reference" circle for a single Lorentzian line.

Figure 3 shows a family of DISPA plots for unresolved spectral doublets, in which the two component Lorentzians (each of the same area and width) are separated by up to about one-half the width of either component line. Also shown are the composite absorption and dispersion spectra from which the DISPA plots were constructed. Although each of the composite absorption spectra gives just a single unresolved peak, each of the DISPA plots shows a well-defined displacement outside and to the right of the reference circle. Moreover, the magnitude of the displacement is directly related to the magnitude of the doublet separation. 

Figure 5 shows DISPA plots for composite line shapes consisting of a superposition of Lorentzian lines centered at a common resonant frequency, whose line widths vary as a log-Gauss distribution (i.e., a Gaussian distribution in log(T/To)). In this case the DISPA plots are displaced centrally inward from the reference circle, and the magnitude of the displacement is directly related to the width of the log-Gauss distribution of relaxation times. This situation is analogous to a distribution in dielectric relaxation times in the Cole-Cole plot, and a similar displacement is observed in that case.5 / 8... 

For the examples so far (Figures 3-5), it appears tions of two or more Lorentzians of different position displacement outside the reference circle, while distr or more Lorentzians of different width lead to DISPA d inside the reference circle. In fact, this conclusion been proved true under quite general conditions.9 The thus readily distinguish between the two most general line-broadening mechanisms (Figure 1), providing a sol the problem posed at the outset of this Chapter. 

Another line-broadening mechanism is illustrated in Figure 6 chemical "exchange" between two sites of different resonant frequency. Although the absorption spectrum again shows just a single unresolved peak, the DISPA plot exhibits substantial displacement outside the reference circle, and the magnitude of the displace-... 

The DISPA analysis described above is based on comparison of an experimental curve to a reference circle. Although that display is qualitatively useful in identifying the correct line-broadening mechanism, the analogous Cole-Cole plotS in dielectric relaxation continues as a popular display mode after nearly 40 years), a display in vyhich the same experimental DISPA data is compared to a straight line could better help to determine quantitatively the line-broaoen-ing parameter(s) of that mechanism. 

Bruce and Marshall 17 have considered five possible algorithms, each of which converts the PISPA reference circle to a straight line. However, these linearizations differ in their suitability for quantitating experimental DISPA data with respect to that reference line. 

A second class of linearized DISPA displays is suggested by the constant radius of the reference DISPA circle. Thus, a plot of (the square of) the radius of an experimental DISPA display. 

Figure 11. Dispersion versus absorption (DISPA) plot based on the 1 NMR spectrum of the residual HDD in commercial 99.7% D2O. Experimental data are shown as filled circles, and the solid curve is a semicircle whose diameter is set equal to the observed absorption peak height. [Taken from ref. 19.]...

The important conclusion from Figure 14 is that the DISPA plot can readily show whether a system undergoing chemical exchange has reached the "slow" or "fast" exchange limit (i.e., Lorentzian line shape, with DISPA data points on the reference circle) or not, based on data at a single temperature. It is always inconvenient, and not always practical (as with heat-labile compounds) to vary the temperature to discover whether the "slow" or "fast" limit has been reached here the DISPA plot gives an immediate answer from a single spectrum. 

Figure 14. DISPA plots based on NMR spectra of an approximately lOX solution of N,N-dimethyltrichloroacetamide in CDCI3, at 20 C (solid triangles) and at 33 C (solid circles). [Taken from ref. 19.]...

The DISPA plot of Figure 16 immediately resolves the issue the displacement of data points inside the reference circle indicates that the dominant line-broadening mechanism is a distribution in line width. In corroboration, later iH NMR results have confirmed the presence of regions of varying flexibility in the ribosomal RNA.21... 

In wt another NMR example, Sykes et a1. used a DISPA analysis of the 1"F NMR spectrum for M13 coat protein the DISPA data points were located on the reference circle, within experimental error, showing that the two fluorotyrosines exhibited the same chemical shift.22 Subsequent experiments based on solvent shifts confirmed that both fluorinated amino acid residues were "buried" and not accessible to solvent. 

The examples in this Chapter clearly demonstrate that a plot of spectroscopic dispersion versus absorption (DISPA) is extraordinarily sensitive to any deviation from Lorentzian line shape. More precisely, the direction of displacement of an experimental DISPA plot from its reference circle is often diagnostic of the line-broadening mechanism, and the magnitude of that displacement is directly related to the magnitude of the line-broadening parameter for that mechanism. Finally, all this information is extracted from a single experimental spectrum. How then should such plots best be applied ... 

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