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J. Phys. Clrem. 1980, 84, 1372-1375
Linearlred Dispersion:Absorption Plots for Spectral Llne-Shape Analysist Robert E. Bruce" and Alan 0. Marshall** Depattmnt of Chemistry, University of British Columbia, Vancouver, British Columbia V6T 1W5, Canada (Recelved December 7, 1979) Publication costs assisted by the Petroleum Research Fund
A plot of dispersion vs. absorption (DISPA) for a single Lorentzian line gives a perfect circle. For any other line shape, the magnitude and direction of the displacement of a given DISPA plot from a reference circle can be used qualitatively to distinguish between many different line-broadening mechanisms. In this paper, we show that we can magnify and quantitate those displacements by linearizing the DISPA circle. The most convenient linearization is a plot of the square of the radius of a DISPA "circle" vs. the square root of frequency (measured from the center of the absorption peak, in units of observed half-width at half-maximum absorption peak height). Without any previous knowledge about the spectrum, it is thus possible in many cases to identify and also quantitate any of a large number of line-broadeningmechanisms, by combining the dispersion and absorption information from a single spectrum. Five distinct line-broadening mechanisms are analyzed by using the new plot, which has been chosen from among five different possible linearizations of the DISPA circle.
Introduction In 1978, Marshall and Roe first proposed the combination of spectroscopic dispersion and absorption information in a single display: dispersion vs. absorption (DISPA).172 For a single Lorentzian line, the DISPA plot is a perfect circle.' Subsequent theoretical3* and experimenta12"p7 work has shown that the magnitude and direction of the displacement of an experimental DISPA curve from its reference circle (i.e., a circle whose diameter is set equal to the maximum observed absorption-mode peak height) are characteristic of the particular line-broadening mechanism for the observed spectral peak. To date, the effects of some 15 distinct line-broadening mechanisms upon a DISPA plot have been analyzed, with the results summarized in Figure 1. The value of the DISPA line-shape analysis is that it is possible in many cases to determine the line-broadening mechanism from the DISPA plot from a single digitized experimental spectrum. For example, a distribution in peak widths always displaces an experimental DISPA curve inside its reference circle, whereas any symmetrical distribution in peak positions always displaces the experimental DISPA curve outside its reference circle: Even when the dispersion signal is not detected directly (as in conventional EPR spectroscopy), a dispersion spectrum may be generated by Hilbert transformation of the digitized absorption spectrum, and a DISPA plot may then be constructed from the (observed) absorption and (Hilbert-generated) dispersion signah6 The DISPA analysis described above (see also Figure 1)is based on comparison of an experimental curve to a reference circle. Although extremely useful in its present form (the analogous Cole-Cole plot* in dielectricrelaxation remains in popular use after nearly 40 years), the DISPA data would be more easily quantitated by expression in a form for which the reference curve is a straight line. In this paper, we consider five different possible reductions of a circular DISPA curve to a straight line and find that the most generally useful method is to plot [(absorption - half-maximum absorptiod2 + dispersion2] vs. frequency1J2(where frequency is measured in units of half-width at half-maximum absorption peak height from the center of the observed spectral peak). Linearized DISPA plots for five representative line-broadening mechanisms are 'This paper is the 7th in a series. For previous papers, see ref 1-6. *Alfred P. Sloan Research Fellow, 1976-80.
then constructed to illustrate the advantages of the new data reduction.
Theory The DISPA analysis proceeds directly from some special properties of the Lorentzian spectral line shape:l normalized absorption = A(w) =
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(2) 1 + (wo - w y 7 2 The original DISPA plot followed from the observation that
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(3)
so that a plot of normalized dispersion, D(w), vs. normalized absorption, A b ) , should give a circle of radius 7/2 centered on the absorption axis (abscissa) at A(o) = ~ / 2 , for a single Lorentzian line.' The same plot for nonLorentzian line shapes produces a characteristically distorted circle.'* Another obvious property of the Lorentzian line shape of eq 1 and 2 is expressed in eq 4. Therefore, a plot of (wO - W) ?A(@)= D(u) (4) D(o) vs. (wo - w ) TA(w)should give a straight line of unit slope, extending from (-7/2,7/2) to the point (7/2,7/2) for a single Lorentzian line (eq 1and 2). One might then hope to detect non-Lorentzian character from the observed deviations of an experimental plot compared to the diagonal reference (straight) line (see Figure 2). Another class of straight-line plots follows directly from eq 3. In this case, we simply note that the radius (or square of the radius) of any circle (in particular, a circular DISPA plot) is constant. Thus, a plot of radius squared vs. frequency (or log (frequency) or square root of frequency) must give a horizontal straight line, y = (7/212, parallel to the x axis, for a single Lorentzian line. We can then anticipate that a nonLorentzian line shape might produce characteristic displacement of an experimental plot of [A(w) - (7/2)12 + [D(w)I2vs. frequency (or log (frequency) or square root of frequency) compared to the horizontal reference line (see Figures 3-5). We now examine the various straight-line DISPA plots suggested above, in order to discover which is the most sensitive to non-Lorentzian line shape and which gives the 0 1980 American Chemical Society
The Journal of phvsicel Chemistry, Vol. 84, No. 1 I, 1980 1373
Linearized Dispersion:Absorption Plots h.i.n.0
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Absorption Figuro 1. Reference circle (diameter equal to maximum observed absorption peak helght) and direction of displacement of a DISPA plot for varlous line-broadening mechanisms: (a unresolved pair of Lorentzians of equal width and different position; (b) Gaussian distribution (in position) of Lorentzians of equal width;' (c) pair of Lorentzians of equal position but differentwidth; (d) b g Gauss distributkn in transverse reiaxation time for Lorentzians of equal resonant frequency;' (e) iog Gauss distribution In conelation time for Lorentzians of equal resonant frequency;' (f) phase misadjustment (rotates circle about the origin);' (g) spectra obtained by Fovier transfotmatitn of a truncated thedomah transient;$ (h) power broadening;' (I) o~ermoduiation;~ (i) chemical exchange between two peaks of different position;' (k) chemical exchange between two peaks of different width;' (I) distortion produced by one adjacent peak of equal intensity and width;' (m) distortion from two adjacent peaks of equal Intensity and width, located at equal separation on either side of the peak from which the DISPA curve is drawn;' (n) effect of a long time constant in recording of derivative spectrum in EPR;' (0) effect of slight baseline drlft in EPR derivative spectrum.6 Finally, for a simultaneous distribution in both peak position and also eak width, the peak position distribution dominates DISPA behavior.
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Figure 2. Plot of dispersion vs. (frequency times absorption), with frequency measured in units of ( l l r ) ,where 7 is the reiaxation time of the Lorentzian line of eq 1 and 2. The straight diagonal line Is for a single Lorentrian line. The narrow and wide loops correspond to a spectrum consisting of the sum of two equally intense Lorentzians of the same width, separated by 0.617 and 1.017 s-', respectively. The frequency of the centex of the absorption peak is taken as zero, which corresponds to setting wo = w in eq 1 and 2.
most conveniently scaled display.
Results and Discussion Figure 1 shows the direction of the displacement of an experimental circular DISPA plot (dispersion vs. absorption) for 15 different line-broadening mechanisms.6 In particular, an unresolved doublet consisting of two Lor-
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Flgwe 5. Plot of radius squared vs. square root of frequency, for the same line shape and frequency scaiing as in Flgure 4. Ordering of plots is as for Figure 4. (See text for advantages of this plot over those of Figures 2-4.)
entzian lines of equal intensity and width separated by less than about one line width produces a marked displacement outside and to the right of the reference circle.lI6 The next four figures show linearized DISPA plots of various types
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The Journal of Physical Chemistry, Vol. 84, No. I f , 1980
for the same unresolved doublet. In Figure 2, a straight-line DISPA plot for a single Lorentzian line is achieved by plotting D(w) vs. (wo w) 7A(w). A related plot of 6’ vs. E ” / W (in which 6’ and 6’‘ are the real and imaginary parts of the steady-state complex dielectric constant, for a dielectric material between the parallel plates of a capacitor subjected to a driving voltage oscillating at frequency w ) offers distinct advantages in analysis of dielectric relaxation dataS9However, there are disadvantages with this display for spectroscopic dispersion:absorption data. First, the double-valued plotted function generates a tight loop rather than a single closed curve (see Figure 2, for the unresolved doublet line shape); the finite signal-to-noise ratio of experimental dispersion:absorption data will thus obscure the shape of the plotted curve and limit the ability of such a plot to discriminate between different line-broadening mechanisms. Second, this plot is not especially sensitive to deviations from Lorentzian line shape. For example, the line shape resulting from power broadenings will produce a linearized plot of D(w) vs. (wo - w)A(o) that is identical with that obtained in the absence of power broadening. We thus proceed to alternative proposed linearized DISPA plots. Figure 3 shows a plot of [A(o) - (7/2)12 + [D(w)I2vs. frequency for the case of an unresolved doublet. As noted in the Theory section, the ordinate is simply the square of the radius of a conventional circular DISPA plot and may thus be abbreviated as R2 in subsequent discussion. A single Lorentzian line (lowermost curve in Figure 3) has a constant DISPA radius and appears as a horizontal line at y = ( ~ / 2 ) ~The . unresolved doublets show displacement from the reference line according to their frequency separation. The main problem with this display is that most of the experimental data are bunched close to the center of the plot, suppressing the most accurate data &e., the points closest to the maximum of the absorption peak). Another obvious problem is that the scaling of this plot is in units of the half-width at half-maximum height of either of the two component absorption lines. Since this half-width is not known in advance, it would be preferable to scale the abscissa in units of half-width at half-maximum height of the observed composite absorption peak, as shown in Figure 4. With regard to the remaining problem of compression of the data set, one possibility might be to plot R2 vs. log (wo - w). This method however spreads the data too thinly (e.g., the frequency scale blows up to --co at the center of the absorption peak). A better scaling appears to follow from a plot of R2 vs. (wo - w)1/2 as shown in Figure 5. For the unresolved doublet shown in Figure 5 , the new plot scaling clearly shows that the maximum deviation from the DISPA circle occurs from data points closer than one half-width to the absorption peak center. The doublevalued nature of the data has been separated by plotting the left- and right-hand halves of the spectrum on the leftand right-hand parts of the square root of frequency axis. Finally, by plotting R2 rather than R, we further magnify small differences between different line shapes, now scaled quantitatively according to the displacement of an experimental DISPA plot from its reference circle. Plots shown in Figures 5-9 correspond to five representative line-broadening mechanisms for which the circular DISPA plot has already proved diagnostic. Figure 6 shows that a Gawian distribution in peak position leads to greatest DISPA displacement at frequencies greater than one observed half-width at half-maximum absorption peak height. The clear difference between any of the plots
Bruce and Marshall
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Flgure 6. Plot of radius squared vs. square root of frequency, for the same frequency scallng as in Flgure 4. In thls case, the composite line shape consists of a Gaussian distribution in resonant frequencles, for Lorenttian lines of equal line wldth. The Qaussian distributlon parameter, u, Is 0 (equlvalent to a slngle Lorentzian Ilne), 0.5, 1.0, and 1.5 s“ in going from the lowermost to uppermost curves in the flgure. See ref 1 for the algebraic form of A(w) and D(w) for thls line shape.
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G Flgure 7. Plot of radlus squared vs, square root of frequency, uslng the same frequency scaling as In Flgure 4. The composite ilne shape Is that resuitlng from chemical exchange between two equally Intense Lorentzlan Rneg of equal natvai Ine width, separated by 4 s-‘ h resonant frequency. Proceedingfrom lowermost to uppermost curves, the rate constant for chemlcal exchan e, k = m (vey fast exchange to glve a slngle Lorentzlan line), 1.4(2f’2, and 1.2(2) 8- See ref 1 for the aigebralc form of A(@) and D(w) for this line shape.
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i
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6 Flgure 8. Plot of radlus squared vs. square root of frequency, uslng the same frequency scaling as In Flgure 4. The composlte line shape results from a log Gaussian distribution in relaxation time, T , for Lorentzian lines of equal resonant frequency. The Gaussian dlstributlon parameter, u = 0 (equlvalent to a single Lorentzianline), 0.5, 1.0, and 1.5, in proceeding from uppermost to lowermost curves in the flgure. See ref 1 for the algebraic form of A(w) and D(w) for thls line shape.
Linearized Dlspersion:Absorptlon Plots
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The Journal of Physlcal Chemlsfty, Vol. 84, No. 11, 1980 1375
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imum displacement (downward this time) occurs at a frequency approximately equal to the half-width at halfmaximum observed absorption peak height. Finally, Figure 9 illustrates the DISPA linearized plot for a phase-misadjusted spectrum for a single resonant frequency and line width. The Figure 9 plot is the only one shown for which the line shape is asymmetrical, so that the left- and right-hand portions of the linearized DISPA plot are no longer related by reflection about wo = w. This and other asymmetrical line shapes should thus be readily recognized in practice.
\
i
2
3
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Figure 9. Plot of radius squared vs. square root of frequency, using the same frequency scaling as in Flgure 4. The observed line shape is for a single Lorentzian, In which the dispersion and absorption spectra have been partly Intermixed because of phase misadjustment by 0' (equivalent to a single correctly phased Lorentzlanline), 5', lo', and 1 5 O , proceeding from lowermost to uppermost curves in the figure. See ref 1 for the algebraic form of A(w) and D(w) for this line shape.
in Figure 5 and any of the plots in Figure 6 thus shows that the linearized DISPA plot should readily distinguish between a line shape composed of just two unresolved peaks and a line shape composed of many unresolved peaks of different position. Figure 7 shows another interesting line shape, resulting from rapid chemical exchange between two Lorentzian lines of different reRonant frequency. Here the maximum displacement from a circular DISPA curve occurs at frequencies greater than one half-width at half-maximum observed absorption peak height for exchange approaching the fast-exchange limit, while slower exchange rates near the coalescence limit produce maximum DISPA displacement at a frequency approximately equal to the half-width at half-maximum absorption height. Again, the linearized DISPA display facilitates location of the point of maximum DISPA displacement from a reference circle or line, in convenient frequency units (Le., multiples of observed half-width at half-maximum peak height). Figure 8 shows thie linearized DISPA behavior for a log Gaussian distribution in relaxation time. Here the max-
Summary In this paper, we have shown that by linearizing a conventional circular DISPA plot, we can better quantify observed deviations of an experimental plot of dispersion vs. absorption and its reference circle by plotting the square of the radius of that experimental DISPA data set vs. the square root of the frequency measured from the center of the absorption peak in units of observed halfwidth at half-maximum absorption peak height. This plot magnifies small deviations from Lorentzian line shape and promises to provide a sensitive means for distinguishing between a large number of different line-broadening mechanisms, based on the data from a single spectrum. A Bruker Aspect-2000 minicomputer program that constructs a plot of R2 vs. (coo - w)lI2 from an existing digitized dispersion:absorption data set is available on request.
Acknowledgment. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for a grant in support of A.G.M. References and Notes (1) A. Q. Marshall and D. C. Roe, Anal. Chem., 50, 756 (1978). (2) D. C. Roe, A. Q. Marshall, and S. H. Smallcombe, Anal. Chem., 50, 764 (1978). (3) A. Q. Marshall and D. C. Roe, J. M g n . Reson., 33, 551 (1979). (4) A. Q. Marshall, J. Phys. Chem., 83, 521 (1979). (5) F. Q. Herring, A. Q. Marshall, P. S. Phlllips, and D. C. Roe, J. Magn. Reson., 37, 293 (1980). (6) A. Q. Marshall and R. E. Bruce, J . Magn. Reson., in press. (7) D. S. Hagen, J. H. Welner, and B. D. Sykes, In "NMR and Biochemistry",S. Oplla and P. Lu, Eds., Marcel Dekker, New York, 1979, p 51. (8) K. S. Cole and R. H. Cole, J. Chem. Phys., 0, 341 (1941). (9) R. H. Cole, J . Chem. Phys., 23, 493 (1955).
