The Journal of Physlcal Chemlstry, Vol. 83, No. 4, 1878 521
Spectroscopic Dispersion Vs. Absorptlon
(6) E. M, Kosower and K. Tanlrawa, Chem, Phys, Lett., 18, 419 (1972). (7) E. M. Kosower, H, Dodluk, K. Tanlrawa, M, Ottolenghl, and N. Orbach, J. Am. Chem. SOC.,81, 2167 (19751, (6) H. Dodluk and E, M, Kosower, J. Am, Chem, Soc., 99, 659 (1977), (9) H. Dodluk, E,M, Kosower, M. Ottolenghl, and N. Orbach, Chem, phys. Len., 40, 174-176 (1977). (10) H. Dodluk and E. M, Kosower, J. Phys. Chem,, 81, 50 (1977). (1 1) (a) E, M. Kosower and H. Dodluk, J . Am. Chem. Soc,, 100, 4173 (1978); (b) E. M. Kosower, H. Dodluk, and H. Kanety, /bun,100, 4179 (1978); (c) E. M. Kosower and H. Dodluk, J . Phys. Chem., 82, 2012 (1978). (12) E. M. Kosower, “Introductlon to Physlcal Organlc Chernlstry”, Wiley, New York, 1968, p 503. (13) H. C. Cheung and M. F. Morales, Blochemisfry, 8, 2177 (1969).
(14) (16) (16) (17) (18) (19) (20) (21) (22)
G. Navon and H. Shlnar, unpubllshed results; H. Shlnar, MSc, Theels, Tel-Avlv Unlverslty, 1973, D. V 9 Nalk, W, LaPaul, R , M. Threatte, and 5. 0 , Schulman, Anal. Chem., 47, 267 (1976), utlllze a aomewhat higher value. See ref 12 for detalled dlscuaslon, W, Whlte and P. G, Seybold, J. Phys. Chem., 81, 2036 (1077). H. Dodluk and E. M. Kosower, Chem. Phys. Lerr., 26, 545 (1874), S. Alnsworth and Ms F. Flanagan, Blochlm. Blophys. Acta, 194, 213-221 (1969). H. Dodluk, Ph.D. Thesis, Tel-Aviv Unlverslty, 1977, and unpublished results, H. Bucherer and A. Stohmann, Chem. Zentralbl., 75, 1012 (1904). R. P. Cory, R. R. Becker, R. Rosenbluth, and I. Isenberg, J. Am. Chem. Soc., 00, 1643 (1968).
Spectroscopic Dispersion Vs. Absorption. A New Method for Distinguishhg a Distribution in Peak Position from a Distribution in Line WidthtS Alan 0. Marshall7 Department of Chemlstry, Unlverslty of British Columbia, Vancouver, Brltlsh Columbla V67‘ 1W5, Canada (Recelved June 26, 1978) Publlcatlon costs assisted by the Natural Sclences and Engineering Research Council of Canada
For a single Lorentzian spectral line, a plot of dispersion vs. absorption (DISPA) gives a semicircle passing through the origin and centered on the abscissa. For an arbitrary superposition of Lorentzians of common peak position and different line width (and for any symmetrical distribution of Lorentzians of common width and different peak position), the DISPA plot will still pass through the origin, but the radius of curvature (measured from a point on the abscissa at half the maximum absorption peak height) will no longer be constant. Particular examples are given in earlier papers, In this paper, it is shown that the superposition of two or more Lorentzians of the same peak position but different line width always gives a DISPA curve lying below the reference semicircle (Le,,a circle whose diameter is the maximum absorption peak height, centered on the abscissa at half the maximum absorption peak height), while the superposition of two or more symmetrically displaced Lorentzians of the same line width but different peak position will in general give a DISPA curve lying above the reference semicircle. It is therefore proposed that an experimental DISPA plot should readily distinguish between these two general mechanisms for inhomogeneous line broadening in various forms of spectroscopy. Experimental examples of each mechanism are given,
Introduction One of the most basic (and most difficult) problems in spectroscopy is to decide on the mechanism(s) for broadening of an isolated spectral line, particularly since it is often porisible to “fit” a single observed broad line according to m y of several proposed mechanisms. Traditional approaches have required multiple experiments designed t o preferentially shift (by change of solvent), broaden (by change in temperature), or disperse (by change in magnetic field in the case of NMR) the various individual components of the original inhomogeneously broad line, However, multiple experiments necessitate multiple controls (e.g,, changing the applied magnetic field in NMR changes relaxation times as well as chemical shifts), and the problem can quickly get out of hand, In previous theoreticall and experimental2 work, we showed that a plot of dispersion vs, absorption (DISPA) for a single data set gives a curve whose displacement from a reference semicircle (see below) has a magnitude and direction which depend on the particular line-broadening Work supported by grants from the Natural Sciences and Engineering Research Council of Canada (A-61781, the University of British Columbia (21-9417), and the Alfred P. Sloan Foundation. This paper is the third in a series. For earlier papers, see ref 1 and 2. Alfred P.Sloan Research Fellow, 1976-1980. 0022-3654/79/2083-052 1$01 .OO/O
mechanism (unresolved splittings or line widths; Gaussian distribution in peak position, log-Gaussian distribution in line width; chemical exchange; phase misadjustment) involved. In this paper, those particular results are generalized to find that under rather general conditions, a DISPA plot from a single experimental data set can immediately discriminate between inhomogeneous broadening due to a distribution in peak position (DISPA curve lies aboue the reference semicircle) and a distribution in line width (DISPA curve lies below the reference semicircle),
Theory A. Construction and Positioning of the DISPA Plot. For a single Lorentzian line, spectroscopic absorption, A(w), and dispersion, D ( w ) , have the following form
in which w is the observing frequency, wo is the “natural” frequency of the driven, damped oscillator under observation, and (1/7) is a measure of the friction against which the driven oscillator moves.l Experimentally, one usually @ 1979 Arnerlcan Chemlcal Soclety
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The Journal of Physical Chemistry, Vol. 83, No. 4, 1979
Alan G. Marshall
hRSORPTlON or 1 DISPERSION
\
Flgura 1. Absorption [ A ( w)] and dispersion [D(w)] spectra, and corresponding dispersion vs. absorption (DISPA) plot for a single Lorentzian line (see eq 1 and accompanying discussion). irn[itw,]
Re[Liwl]
FlgMr@2. Plots of real and imaginary parts of a (complex) line shape function, L( w ) , of eq 2 vs. frequency (left-hand diagrams), and corresponding plot of I m [L(w)] vs. Re [L(w)] to give a DISPA plot which is now centered at the origin in the complex plane to simplify calculations (see text).
extracts woand (1/7) from a plot of either A ( w ) or D(w) vs. driving (“irradiating”) frequency, as shown in Figure 1. The DISPA plot (Figure 1)constructed by plotting D(w) vs. A(w) gives a “reference” semicircle of radius ( 7 / 2 ) , centered at ((7/2),0) on the abscissa for this case of a single Lorentzian line. In further calculations, it will be convenient to shift this DISPA plot so that the center of the circle is a t the origin. This is easily accomplished by subtracting ( 7 / 2 ) ,or more generally half the maximal absorption peak height, from the absorption expression. Finally, subsequent algebra is simplified by defining a line shape function, L(w) ”
so that the new absorption and dispersion expressions can be written as the real and imaginary parts of (complex) L(ru):
D(w) = Im (L(w)) =
W7 2
____
1+ w
v
in which w
=wg-w
(3c)
Plots of Re (L(w)) and Im (L(w)) vs. w, and the corresponding DISPA plot of D ( u ) vs. A(w) are shown in Figure 2. The DISPA reference curve is now a semicircle of radius ( r / 2 ) ,centered a t the origin, as desired. B. Devising a D I S P A Test for Non-Lorentzian Line Shape. For a simple Lorentzian line shape, the DISPA plot has a constant radius of curvature. Therefore, if we restrict our analysis t o line shapes for which the DISPA plot has the same x intercepts as for the reference semicircle, we can simply calculate the radius of curvature for the DISPA plot to see if it is larger or smaller or the same as the reference semicircle. If the radius is the same (Le., coristant and equal to half the maximum absorption peak height), the line shape is Lorentzian; otherwise it is not. For example, consider a composite line shape made up of a sum of two Lorentzian lines of the same resonant
Figure 3. (a) Absorption and dispersion components for a composite line shape consisting of a superposition of two Lorentzians of equal resonant frequency and integrated area, but different line width. The plot simply shows that the composite dispersion goes to zero at the maximal composite absorption peak. (b) Absorption and dispersion components for a composite line shape consisting of a superposition of two Lorentzians of equal height and width, but different resonant frequency. Again, the composite dispersion goes to zero at the maximal composite absorption peak frequency (Le., the midpoint between the two peaks), if the two peaks are sufficiently close together (A < 0.6 in units of (I/?) in eq 9) that there is just a single composite absorption peak (see text).
(“natural”) frequency, coo, but different line width &e., different 7). As seen in Figure 3 (left-hand diagrams), the composite absorption and dispersion both go to zero for w
