756
ANALYTICAL CHEMISTRY, VOL. 50, NO. 6, MAY 1978
700) in hexane averaged 525 a t 270 nm, with a relative standard deviation of f 1 5 7 ~ .This value is in good agreement with that reported by Snyder on gas oil fractions ( 5 ) . Then, the determination of monoaromatics with UV detection a t 270 nm is possible, without regard to sample types, if the molecular weight can be estimated. Isocratic elution with hexane as eluent is convenient for the determination of monoaromatics, because only monoaromatics are eluted. leaving polyaromatics retained strongly on the adsorbent (activated alumina). In this way, monoaromatics in a series of hydrogenated oils of a lighter lubricating distillate were determined. Calibration was performed from known blends of monoaromatics with saturates from one sample. As the changes i n molecular weights are very small, this calibration could be applied to all samples. Results of determination are shown in Table 111. Little quantitative change was observed with monoaromatics, in spite of the decrease in total aromatics content with the increase of desulfurization.
Table 111. Determination of Monoaromatics in Hydrogenated Oils Desulfurization, %
0
36.2 64.6 72.4 82.9 91.1 a
Total aromatics, wt %a 44.4 42.0 39.8 39.0 38.1
37.2
Monoaromatics, wt % 30.9 30.5 31.3 28.9 30.5 30.3
Results obtained bv silica-gel chromatoaraDhv.
various sources. Sulfur compounds exist in a considerable amount in heavy petroleum fractions. Therefore, GEC was carried out on the oxo-desulfurized ( 4 ) oils. T h e examples are shown in Figure 5 . By comparing Figure 5 with Figure 2, we observe that most sulfur compounds behave as di- or polyaromatics and that the content in polyaromatics is large. Quantitative analysis of aromatics with an ultraviolet detector is rather difficult because each compound has its oum absorptivity. T h e use of a universal detector such as a flame ionization detector would be desirable for quantitation. As for the monoaromatics, however, quantitation with the UY detector is not unreasonable. Preparative separations were carried out to isolate the monoaromatics from widely different samples (vacuum distillates, bright-stocks, solvent-refined, hydrogenated, etc.). The molar absorptivities of the recovered monoaromatics (average molecular weight varied from 300 to
LITERATURE CITED (1) J. C. Suatoni, H. R. Garber, and B. E Davis, J . Chromatogr. Sci.. 13, 367 (1975). (2) H. Engelhardt and H. Wiedemann. Anal. Chem.. 4 5 , 1641 (1973). (3) L. R . Snyder and J. J. Kirkland, "Introduction to Modern Liquid Chromatography'. Wiley-Interscience, New York, N.Y.. 1974. (4) H. V Drushel and A . L. Sommers, Anal. Chem., 39, 1819 (1967). ( 5 ) L. R. Snyder, Ana/. Chem., 3 6 , 774 (1964).
RECEREL) for review August 30, 1971. Accepted February 13, 1978.
Dispersion versus Absorption: Spectral Line Shape Analysis for Radiofrequency and Microwave Spectrometry Alan G. Marshall* and D. Christopher Roe Department of Chemistry, University of British Columbia. Vancouver, B.C. V6T 1 W5. Canada
detection of a coherent absorption or emission process (e.g., NMR, ICR, KQR. ESR: and pure rotational spectrometry, as well as dielectric and ultrasonic relaxation), virtually all existing analyses of spectral response have been based upon either ii) the time-domain transient signal following one or more pulses of coherent oscillatory excitation (usually electromagnetic radiation) ( I 4. or (ii) the frequency-domain a bsor p t i o n-m ode or "magnitude ("absolute - Val lie" steady-state response to a continuous oscillatory excitation (6-9). Provided that the system is linear (namely! that the incident power level is below that at which saturation effects are significant), the time-domain and frequency-domain responses are related by a Fourier transformation (10). T h e reason for choosing these particular spectral displays is historically clear: each tqpe of signal is obtainable as the direct output of a suitable electronic detector. However, with the advent of spectrometers with on-line computers providing for digitization, storage, and manipulation of spectra, t h e spectrometrist need no longer feel restricted to these particular data displays. It is the purpose of this paper to introduce a new form of data reduction, consisting of a plot of dispersion vs. absorption (see Theory). This new data display produces a semicircle reference curve for a simple Lorentzian line shape. A related display has long been used to detect and characterize multiple relaxation in dielectric measurements ( I I ) . In the simpler dielectric or ultrasonic case. the spectral components of the
I n radiofrequency (nuclear magnetic resonance, ion cyclotron resonance, nuclear quadrupole resonance) and microwave (electron spin resonance, pure rotational) spectrometry, it is possible to obtain both absorption and dispersion spectra. For a simple Lorentzian line shape, a plot of dispersion vs. absorption gives a semicircle. I n this paper, dispersion:absorption plots are constructed for the first time for a number of linebroadening mechanisms. I t is shown that such plots are diagnostic for, and can often be used to distinguish between, line-broadening resulting from: unresolved superposition of two Lorentzians of different resonant frequency or different line width; Lorentzian line shapes which have been weighted by either a Gaussian distribution in resonant frequency, or a log-Gaussian distribution in relaxation time or correlation time; and line shapes resulting from "chemical exchange" between two sites of different resonant frequency or different relaxation time. It is proposed that the dispersi0n:absorption plot can serve as a useful means for establishing, from a single spectrum, the mechanism for line-broadening for isolated lines in radiofrequency and microwave spectra. Experimental examples of many of these situations are provided in a companion paper (see following article) for the particular case of nuclear magnetic resonance spectrometry.
"
In those forms of spectrometry featuring phase-sensitive 0003-2700/78/0350-0756$01 O O / O
C
1978 American Chemical Society
ANALYTICAL CHEMISTRY, VOL 50, NO 6, M A Y 1978
u
757
For this mechanical analog (viz., the weight on a spring subject to a sinusoidal driving force. Fo cos u t ) the spectral line shape is said to be Lorentzian, and the frequency-dependence of the absorption and dispersion components (Figure le) is described by Equations l a and l b , where K is a constant:
i -
Absorption
Kr
=
Lorentzian Line Shape Dispersion
i
i * -1
=
K(wo - w ) r 2
1+
( w g
-
w)2?
7
* +
A
'
c
Figure 1. Spectroscopic absorption and dispersion, as derived by analogy to the steady-state displacement of a driven, damped weight on a spring (see text)
response are all centered at the same (zero) frequency. In this paper, however, we show that the dispersi0n:absorption plot can be extended to other forms of spectrometry in which the response has components centered a t two or more (nonzero) frequencies. Representative theoretical examples (this article) and experimental examples (the following article) suggest that the new data reduction may be particularly sensitive in both detecting and discriminating between line-broadening caused by distributions in either relaxation times or resonant frequencies. T h e new analysis has advantages in ease of application in comparison to existing time- or frequency-domain methods, and should be applicable to most measurements of t h e response of a system to radiofrequency or microwave radiation.
THEORY I. Absorption and Dispersion. T h e origin and nature of spectrometric "absorption" and "dispersion" line shapes are summarized briefly in Figure 1, based on the analogy between t h e motion of a driven, damped spring, and the motion of electrical charges (or electric or magnetic dipole moments) driven by the electric or magnetic field components of coherent electromagnetic radiation. When the mass, m. on a damped spring (Figure l a ; k is the spring force constant and f is the frictional coefficient) is subjected to a continuous oscillatory driving force. F (Figure l b ) , the displacement of the mass eventually settles into a steady-state oscillation a t the same frequency as the driver, but with somewhat different phase, as shown in Figure IC. Using a suitable "phase-sensitive detector", it is possible to analyze the steady-state displacement into two components which are exactly in-phase or exactly 90" out-of-phase with respect t o the phase of the driver (Figure Id). The amplitude of the in-phase component as a function of frequency is called the "dispersion" spectrum, and the amplitude of the 90" out-of-phase component as a function of frequency is called the "absorption" spectrum (Figure l e ) . (The square root of the sum of the squares of t h e dispersion and absorption as a function of frequency is sometimes called the magnitude or absolute-value spectrum; the square of the magnitude spectrum is called the power spectrum.) For a linear system, the amplitude of the displacement is proportional to the amplitude of the excitation, as is ordinarily the case in absorption spectrometry at source power le\ els below those a t which "saturation" effects begin.
in which w,, = L k l m = "resonant" or "natural" frequency, and 1 / =~f / 2 m ; T = "relaxation time". While the physical origins and meanings of K , q, and T are widely different for various types of spectrometry. the line shapes of Equations l a and l b are the same in each case. In order to facilitate comparisons between different types of line shape, it is convenient to normalize both the absorption and dispersion by a factor which makes the absorption signal exactly T at its maximum peak height:
Normalized absorption = A Normalized dispersion
=
D
=
-
1+
=
w)2r2 w)rZ
( w g -
(wg -
1 + ( a o- W ) ' T 2
(A)
An important special case is the limit that ni = 0 (massless "weight" on a spring). In this limit, the spectral response may again be analyzed into two (normalized) components, > 'and ", having the somewhat simpler form shown in Equations 2a and 2b: 1
where l / =~ h / f . Equations 2a and 2b describe the frequency-dependence of dielectric or ultrasonic parameters (e.g.. real and imaginary components of the complex dielectric constant) in dielectric or ultrasonic relaxation experiments. Experimentally, in the radiofrequency or microwave spectral regions. the response of the t?.pe shown as Figure IC is analyzed into the two components shown as Figure Id using a phase-sensitive detector. Such detection is, in principle, possible whenever coherent excitation is available (121, is routinely employed in NMR and ICR (radiofrequency) spectrometry, and is feasible at microwave frequencies also. In practice, even in cases (such as N M R ) where it is just as easy to detect the dispersion component as the absorption component, the overwhelming preference in high-resolution spectrometry has been for the absorption spectrum, since the absorption response (see Figure l e ) spreads over a smaller frequency range and is thus more suitable for distinguishing two or more signals of nearly the same resonant frequency. However, when two Lorentzian absorption signals of equal line width have resonant frequencies which differ by less than about line width, the two absorption signals are no longer "resolved", in the sense that the composite signal forms a single peak; moreover. if the two component signals are also of equal amplitude, then the single peak is sbmmetrical about its maximum and is not easily visually distinguished from a single best-fit Lorentzian of the same height and width (Figure 2a). A similar situation arises from the superposition of two or more Lorentzian absorption lines whose component fre-
758
ANALYTICAL CHEMISTRY, VOL. 50, NO. 6, MAY 1978
to plot only the data deriving from the positive portion of the dispersion response (Le., (coo - w) 2 0), as shown by the dotted “reference” curve in Figure 2b (or Figures 3b, 4b, .... 9b). Finally, since the circle is such a regular shape, it serves as a particularly convenient reference plot for detecting minor deviations (see below) from the simple single Lorentzian line shape of Equations A and 4.
Figure 2. Composite absorption (top) and dispersion (bottom) spectra (a), and corresponding dispersion:absorption (DISPA) plots (b), for a spectrum consisting of the sum of two Lorentzian lines of equal width and equal maximal absorption, but varying separation in resonant frequency (see text)
quencies (Figure 2a, Figure 3a) or line widths (Figure 4a, Figure 5a) span a range of about ( 1 / r ) s-l or less. While it is possible to “deconvolute” a digitized absorption spectrum numerically with respect to presumed “natural“ Lorentzian line width in order to better resolve the indikidual absorption component lines, a given deconvolution is suitable only for a particular type of inhomogeneity. In other words, the mechanism for line-broadening must be known before any deconvolution is meaningful. Moreover, “convolution difference” methods as practiced in NMR (13) can introduce line shape distortions and artifacts into the spectrum. From the above discussion, and the obvious property (Figure l e ) that the dispersion spectrum falls to zero much less rapidly than the absorption spectrum, as m e moves away from resonance (w = o o ) ,it might be supposed that the dispersion information could be used to help sort out various types of inhomogeneous line-broadening in spectrometry. Unfortunately. the dispersion component is antisymmetrical about the resonant frequency, so that a direct superposition of two or more adjacent dispersion signals can lead to a complicated line shape not well suited for qualitative visual analysis. In contrast, a new and useful means for using the dispersion spectral information while preserving a simple graphical display follows by extension from the semi-empirical Cole-Cole plot used in analyzing dielectric relaxation data ( I I ) , and is presented in the following sections.
11. The Dispersi0n:Absorption Plot (DISPA). Single Lorentzian Line. In 1941, Cole and Cole pointed out that since
(3) a plot of 3 ” vs. 3” will give a semicircle. Empirically, it was found that a plot of t” vs. t’ (where t” and t‘ are, respectively, the imaginary and real parts of the complex dielectric constant) often could be accurately described by a semicircle whose center was displaced below the abscissa. T h e extent of t h e downward displacement was used t o characterize the breadth of the distribution of dielectric relaxation times. While the dielectric relaxation case (analogous to a massless weight on a damped mechanical spring) does not give a Lorentzian line shape, we now point out that for the spectrometric cases (analogous to a weight of finite mass on a damped mechanical spring),
[ A - ( r / 2 ) I 2+ 0’= r 2 / 4 = (ria)’
111. Dispersi0n:Absorption (DISPA). Resolution of Two Closely-Spaced Lorentzian Lines. A simple test for the dispersion:absorption plot is the behavior of two Lorentzian lines of equal amplitude and equal width, whose resonant frequencies differ by u p to about one half of an individual line width: 7
A ( 0 )=
1+
+
2r ( a 1- w ) ” 2
(
1 + c2r2 (w2 - 0 ) r 2 + D ( w )= 2r 1+ ( 0 2 - 0 y r 2 (w1 -
1+
W)TZ
(w1-
w ) Y
in which w2 = wo + C and w1 = wo C, where C is a constant. Equations 5a and 5b represent composite absorption and dispersion spectra which have been normalized to give unit absorption amplitude a t maximum peak height: ~
lim A ( w ) = l (i)+W0
T h e composite absorption and dispersion line shapes of Equations Sa and 5b are illustrated in Figure 2a for several choices of separation in resonant frequency of the twocomponent Lorentzian absorption lines. Proceeding from innermost to outermost solid curves, the separations correspond to C = 0 . 3 / ~ 0, . 4 / r , 0 . 5 / r , and 0 . 6 1 ~ .T h e two-component Lorentzian absorption curves (Figure 2a, top) are not resolved by this display, for any of the indicated frequency separations. Figure 2b shows the dispersion:absorption (DISPA) plots corresponding to the superposition of the two closely spaced Lorentzians of Equations 5a and 5b. Proceeding from lowermost to uppermost solid curves, the Figure 2b plots show characteristic displacement in a direction aboce and to the right of the reference semicircle (corresponding to a single Lorentzian line) where the magnitude of the displacement is directly related to the jrequerzq separation between the two-component Lorentzians.
IV. Dispersion:Absorption. Gaussian Distribution in Resonant Frequency. As a second test of the diagnostic value of the dispersion:absorption plot, consider a composite line shape consisting of a superposition of Lorentzian lines of common width, whose resonant frequencies satisfy a Gaussian distribution:
A ( w )=
-’-, 1
d2710
.-
7
1 + ( a o- w
+ A)’r2
e- A Z / Z O d A
(7a) and
(4)
Equation 4 predicts that a plot of normalized dispersion, D , vs. normalized absorption, A , should give a circle of radius r / 2 , centered on the A-axis at A = ~ / 2 . Since the circle is twofold symmetric about (for example) the A-axis, it suffices
in which
n
is the breadth parameter of the Gaussian distri-
ANALYTICAL CHEMISTRY, VOL. 50, NO. 6, MAY 1978
-/
a
A.
b
\
Figure 3. Composite absorption (top) and dispersion (bottom) spectra (a), and corresponding DISPA plots (b), for a spectrum consisting of a superposition of Lorentzian lines of equal width, whose resonant frequencies satisfy a Gaussian distribution with various distribution widths (see text)
bution. It is useful to normalize both t h e absorption and dispersion by dividing Equations 7a and 7b by Equation 8:
Normalizing factor for Gaussian distribution in w o =
Figure 3a shows plots of composite absorption and dispersion vs. frequency, for Gaussian-weighted Lorentzians with various u values. Going from innermost to outermost solid curves, t h e Gaussian distribution in coo is assigned a breadth parameter, u = 0.5, 1.0, or 1.5 s-', and t h e semicircle corresponding to a single Lorentzian ( u = 0) is shown as the dotted line for reference. Again, it would not be easy in practice to distinguish visually between a Gaussian-broadened Lorentzian (e.g., the innermost solid curve a t top of Figure 3a) and a single Lorentzian absorption line of the same height and width (cf. t h e filled circles in Figure 3a, top). Proceeding from lowermost to uppermost solid curves, Figure 3b shows t h e DISPA plots corresponding to the Gaussian-weighted Lorentzians of Figure 3a. In this case, the DISPA plots show characteristic displacement in a direction above and to the left of the reference semicircle (dotted line) corresponding to a single Lorentzian line. The magnitude of the displacement is directly related to the spread in resonant frequencies, as measured by the Gaussian distribution parameter, u.
V. Dispersi0n:Absorption. Resolution of Two Lorentzians of Different Line Width. T h e dispersion:absorption plot was originally used to detect the presence of multiple relaxation components of dielectric dispersion curves. T h e next example to be considered will thus be a line shape which is the sum of two Lorentzians of equal integrated area and equal resonant frequency, but different line width:
A ( w )=
+
71
1+
(wo -
Figure 4. Composite absorption (top) and dispersion (bottom) spectra (a), and corresponding DISPA plots (b), for a spectrum consisting of a superposition of two Lorentzian lines of equal resonant frequency and equal integrated area, but different relaxation times (see text)
by a factor which leads to unit absorption amplitude a t maximum absorption peak height. T h e composite absorption and dispersion line shapes, Equations 9a and 9b, are illustrated in Figure 4a, for T~ fixed a t I, and (proceeding from innermost to outermost solid curves) T~ = 0.3,0.2,0.1, and 0.05. The reference line shapes for a single Lorentzian ( T = ~ r2 = 1) are shown as dotted lines, where the frequency scale is in units of ( ~ / T Jand the spectra have been scaled by a factor which produces unit maximal absorption peak height. An alternative method for detection of multiple relaxation is by inspection of the time-domain response to a pulsed oscillatory excitation, as commonly performed in NMR experiments (14). However, when the two relaxation times, T~ and T', are as similar in magnitude as in the present example, it is experimentally very difficult to observe deviations from the straight-line reference curve (for a single Lorentzian) in a plot of log (NMR signal) vs. time, except at relatively long times a t which the signal amplitude is very small and the signal-to-noise ratio is correspondingly poor. The DISPA plots at the right of Figure 4b (proceeding from lowermost to uppermost solid curves) correspond to the absorption curves of Figure 4a. These plots show characteristic displacement dounlcard at the left of the reference semicircle (dotted line) corresponding t o a single Lorentzian line, and the displacement is readily apparent even for relatively small differences in the widths of the constituent Lorentzian components of the composite line shape. T h e magnitude of the displacement is related to the difference betueen the lcidths of the component Lorentzian lines. VI. Dispersi0n:Absorption. Log-Gaussian Distribution in Relaxation Time. There are many situations in which an observed absorption signal is the resultant of many component Lorentzian lines of different line width. Two common distributions of relaxation times will be discussed here: (1) a logarithmic-Gaussian distribution of the form,
F ( x ) dx = (1/d%)
exp(-x2/2a) dx
(10)
in which
x = In ( r / r o )
(11)
where T~ is the center of the relaxation time distribution, and (21, an empirical distribution of the form,
and
D ( w )=
759
(wo -
1+
O)TlZ +
( w g -
w)2712
in which T~ and T~ are the respective relaxation times for the two Lorentzians. Equations 9a and 9b have been normalized
in which x is again defined by Equation 11. T h e empirical distribution of Equation 12 produces a dispersion:absorption plot which retains the circular shape with the same radius, but with a center of curvature which is displaced below the x-axis by an amount directly related to the size of the parameter, N , which is a measure of the width of the distribution.
760
*
ANALYTICAL CHEMISTRY, VOL. 50, NO. 6, MAY 1978
i,
b _.__ .-. .*uE,\cb
-
.
---,
~
,~.
.
-
.....",
-s>--
Figure 5. Composite absorption (top) and dispersion (bottom) spectra (a), and corresponding DISPA plots (b), for a spectrum consisting of a superposition of Lorentzian lines centered at a common resonant frequency, whose relaxation times satisfy a log-Gaussian distribution having various widths (see text)
Figure 6. Composite absorption (top) and dispersion (bottom) spectra (a), and corresponding DISPA plots (b), for a magnetic resonance spectrum consisting of a superposition of Lorentzian lines centered at the same resonant frequency, whose dipole-dipole correlation times satisfy a log-Gaussian distribution having various widths (see text)
Equation 12 has proved successful in fitting dielectric relaxation data for a wide variety of liquids ( I I ) , but in spite of numerous attempts, there is as yet no satisfactory nonempirical explanation for the good quality of the fits of Equation 12 to experimental dispersion:absorption plots (15). T h e interested reader is referred to Ref. 16 for recent progress in analyzing distribution functions of dielectric relaxation times. T h e log-Gaussian distribution in relaxation times leads to absorption and dispersion expressions of the form,
energy level, etc. As a simple and important example, consider the intramolecular dipole-dipole contribution to the relaxation rate. ( ' / T 2 ) , for water molecules which reorient by isotropic rotational diffusion in an externally applied magnetic field of 23.4 kGauss (17):
e(x+Y)
A ( w )=
1+
(w" -
1f
T2
Wo2Tc2
+
W):ez(x+Y)
= e ( x + Y ) -(x2/20)
J-m
57,
e - ( x L / 2 0 ) dx
P
J--
1
-
e
dx
(13a)
and where D,,, is the rotational diffusion constant for H 2 0 in the sample of interest. If we now re-define
x = In ( r , / ~ ~ ~ )
(16a)
and where 3 = In T~ and x is defined in Equation 11. Formulas 13a and 13b have been normalized by a factor which produces unit absorption signal a t maximum absorption peak height (0=
wo).
Figure 5a shows composite normalized absorption and dispersion spectra for various choices of the log-Gaussian relaxation time distribution width parameter, 0 = 0.5, 1.0, and 1.5,proceeding from outermost to innermost solid curves. The reference line shapes for a single Lorentzian (a = 0) are shoun as dotted lines, where the frequency scale is in units of 1 / ~ " , and t h e spectra have been vertically scaled to give unit maximal absorption peak height. In this case, it would not be easy to observe a visual difference between the absorption line shape of a single Lorentzian of the same height and width and the various relaxation time distribution cases. Figure 5b shows the DISPA plots corresponding (from uppermost to lowermost solid curves) to the various line shapes of Figure 5a. In contrast to the previous example of Figure 4b (sum of only two Lorentzians of different width). the curves of Figure 5b are displaced direetl? belou the reference semicircle (dotted line) corresponding to a single Lorentzian line. T h e m a g n i t u d e of the displacement is directly related to the u i d t h of the log-Gaussian distribution in relaxation times. VII. Dispersi0n:Absorption. Log-Gaussian Distribution in Correlation Time. In magnetic resonance spectra (NMR, ESR), a range of relaxation times, T , can result from a distribution in correlation times, T ~ for , changes in site, changes in orientation of the molecule, changes in magnetic
y
=
In
(16b)
7,'
where T," is the center of the correlation time distribution, then a log-Gaussian distribution in rCwill produce normalized absorption and dispersion spectra ( J f the form.
where
-t
Equations 17a and 1% have been normalized to give unit absorption signal at maximum absorption peak height (0 = 00)
Figure 6a illustrates the weighted normalized absorption and dispersion spectra of Equations 17a and 17b for various choices (proceeding from outermost to innermost solid curves)
ANALYTICAL CHEMISTRY, VOL. 50, NO. 6, MAY 1978
of the log-Gaussian correlation time distribution u i d t h parameter, u = 0.5, 1.0, and 1.5. Here the dotted curve corresponds to a simple Lorentzian line having a single correlation (calculated time, and the frequency scale is in units of 1/ from T: = s using Equation 14 and iio = 2~ X 10’rad s-I). T h e spectra have been vertically scaled to give unit maximal absorption peak height. Again. it might be difficult in practice t o distinguish visually any of the illustrated absorption line shapes from a single Lorentzian of the same height and width. Proceeding from the uppermost of the lowermost solid curves in Figure 6b, the dispersion:absorption (DISPA) plots correspond to the various line shapes of Figure 6a. As usual, the reference semicircle corresponding to a single Lorentzian ( u = 0) is shown as a dotted line. As for the preceding case (log-Gauss distribution in relaxation time), the curves of Figure 6b (log-Gauss distribution in correlation time) are displaced directly dou n u a r d from the reference semicircle. T h e magnitude of the displacement is directly related t o the w i d t h of the log-Gaussian distribution in correlation time. VIII. Dispersi0n:Absorption: Chemical Exchange. In magnetic resonance, spectral line shape is sometimes determined by t h e rate(s) of exchange processes,
described by first-order rate constants, h , and h.,, for exchange between magnetically different sites A and B. In order to simplify subsequent interpretation, we will consider exchange between equally populated sites ( h , = h = h ) ,
,
k
A= B
,
a
761
b
\
Figure 7. Absorption (top) and dispersion (bottom) spectra (a), and corresponding DISPA plots (b), for a spectrum resulting from exchange between two Lorentzian lines of equal width and integrated area, but having different resonant frequencies (see text)
As illustrated in Carrington‘s book ( I 7 ) , the absorption signals from the two sites coalesce into a single line, for exchange rates sufficiently fast that
T h e spectra of Equations 20a and 20b may be normalized to unit maximum absorption peak height by dividing both equations by the value of the absorption a t (L‘ = O. Figure 7a shows the normalized absorption and dispersion spectra, based on Equation 20, for four exchange rates fast enough to cause coalescence: k = t5,1.2 ~ 41.4, t 5,and 1.6 \ 5 5-l (proceeding from outermost to innermost solid curves), where bA = (q+ 2) and wB = ( L ~ 2) s-’. Linebroadening due to any mechanism other than chemical exchange has been neglected, and the frequency scale is in units of s l. The dotted curve is the spectrum for a single Lorentzian line. Figure 7b shows the DISPA plots corresponding to the chemical exchange cases of Figure 7a, proceeding from uppermost to lowermost solid curves. In these cases, the curves are displaced u p u a r d from the reference semicircle, and the m a g n i t u d e of the displacement varies inversely with the exchange rate. VIII.B. S a m e Resonant Frequency, Different Line W i d t h . In the second limit that the resonant frequency is the same a t site A as at site B, the absorption (18) and dispersion line shapes take the form, ~
k
in the special limits that either the resonant frequency or the line width a t the two sites is different. VIII.4. S a m e Line W i d t h . Different Resonant Frequent). In the limit that the line width is the same at site A as a t site B, the expressions for magnetic resonance absorption (17%18) and dispersion take the form,
(20;)
A ( w ) = C-
(20b) where C = * , H I M oand 2 = (uA+ e B ) / 2 , in which w , ~and eg are the resonant frequencies a t sites A and B, respectively. 2 is the magnetogyric ratio, H Iis the applied oscillatory magnetic field which induces the signals. and M u is the equilibrium magnetization. [As shown in general by McConnell (18) and specifically for the two-site case by Carrington ( 1 7 ) ,the chemical exchange line shape is most easily derived by adding a mathematically imaginary driving term to the equation of motion, then solving for the (complex) magnetization, and finally obtaining the dispersion and absorption as the respective real and imaginary components of the complex magnetization. This procedure is equivalent to the analysis sketched in Figure 1. but the algebra is rendered vastly simpler by using mathematically “complex“ quantities. While the ”complex” result (18) and the absorption component ( I 7 ) have been derived previously, there appears to be no previous report of the dispersion for chemical exchange cases, since there has been little prior need for it.]
ad - bc
+ dZ ac + bd D ( W ) = cc2 + d 2 cz
in which C = - y H , M , and a=wO-W
1
b=2k+-
-+(TtB
‘:A)
and T P and A TaBare the respective (spin-spin) relaxation times a t sites A and B. T h e spectra of Equations 22a and 22b may be normalized to unit absorption maximum peak height by dividing both expressions by the absorption value a t (L! = coo.
762
ANALYTICAL CHEMISTRY, VOL. 50, NO. 6, MAY 1978
-
Table I. Summary of the Effects of Various Mechanisms of Spectral Line-Broadening on a Plot of Normalized Dispersion vs. Normalized Absorptiona Displacement of a dispersi0n:absorption curve with respect to the semicircle plot obtained for a simple Lorentzian line shape Origin of line-broadening Slightly distorted semicircle, displaced Unresolved superposition of two Lorentzians upward and to the right of the reference of equal height and width, but different resonant frequency semicircle Slightly distorted semicircle, displaced Lorentzian line shape, weighted by a upward and to the l e f t of the reference Gaussian distribution in resonant semicircle frequencies Slightly distorted semicircle, displaced Unresolved superposition of two Lorentzians downward and t o the l e f t of the reference of equal integrated area and resonant semicircle frequency, but different line width Flattened semicircle, displaced directly Lorentzian line shape, weighted by a logbelow the reference semicircle Gaussian distribution in relaxation times Flattened semicircle, displaced directly Lorentzian line shape, weighted by a logbelow the reference semicircle Gaussian distribution in correlation times Undistorted semicircle, displaced directly Lorentzian line shape, weighted by the downward from the reference semicircle relaxation time distribution of Eq. 1 2 Slightly distorted semicircle, displaced Exchange between two equally populated a boue the reference semicircle sites of equal line width but different resonant frequency Slightly distorted semicircle, displaced Exchange between two equally populated slightly below at the l e f t of the sites of the same resonant frequency, but reference semicircle different line width a In each case, the magnitude of the displacement from the reference semicircle corresponding t o a single Lorentzian line is directly related t o a specific parameter of the broadening mechanisms (see text).
“absorption” and “dispersion” will each consist of a linear combination of the actual absorption and dispersion, weighted by factors of either cos 4 or sin 4: T
A ( w ) = COS Q
sin q5 --
1+ (wo(wo - w)?
1+
-
_.______
~
---- . .-. -.
-. -.
- ~
~ ~
-- -~
_ _ -~-
. .. ,_ -= . --.. -:>--.. ”._
-
-.-= .
Figure 8. Absorption (top) and dispersion (bottom) spectra (a), and corresponding DISPA plots (b), for a spectrum resulting from exchange between two Lorentzian lines of equal resonant frequency and integrated area, but having different relaxation times (see text)
Figure 8a is a display of the normalized absorption and dispersion spectra (proceeding from innermost to outermost solid curves) based on Equations 22a and 22b, for four exchange rates corresponding to h = 1, 2, 4, and 8 multiples of 1/T2*.T h e two component relaxation times in the absence of chemical exchange are fixed a t ( T2J TZB)= 20, and the frequency scale is in units of l/TZA.T h e dotted line is the spectrum for a single Lorentzian line (T2*= T2B;k = 0). Figure 8b gives the DISPA curves corresponding (from lowermost to uppermost solid curves) to the spectra of Figure 8a. T h e curves are displaced dou,nu,ard and to the left from the reference semicircle, the magnitude of the displacement is inversely related to the exchange rate. Exchange rates appreciably faster or slower than those illustrated gave even smaller displacements. IX. Dispersi0n:Absorption. P h a s e M i s a d j u s t m e n t . Since t h e dispersion and absorption signals in actual experiments are obtained as the respective amplitudes of the in-phase and 90” out-of-phase components of the steady-state response (Figure l),or by suitable phase adjustments to the Fourier transformed transient response (12), it is of considerable practical importance to establish the effect of slight phase misadjustments on the proposed dispersi0n:absorption plots. For a phase misadjustment of 4 degrees, the apparent
(wo -
W)2T2
T
D ( w ) = sin 0 cos 0
w)272
1+
(wo -
1+
+
(wo - w ) 2 T 2 w)72
(wo -
For the plots of Figure 9, Equations 23a and 23b were normalized by dividing by the maximum apparent “absorption” peak height (located by iteration). Figure 9a shows normalized “absorption” and “dispersion” spectra obtained from Equations 23a and 23b, for deliberate phase misadjustments (proceeding from lowermost to uppermost solid curves a t the left of the Figure) of @ = 5, 10, 15, and 20 degrees. The dotted line is the spectrum for a single (correctly phased) Lorentzian line (4 = 0). Figure 9b illustrates the DISPA plots corresponding (proceeding from lowermost to uppermost solid curves) to the phase misadjusted spectra of Figure 9a. Although substantial changes in the DISPA plots can clearly result from misadjusted phase, this should not be a problem in practice because the distortions in the apparent “absorption” signal are obvious also. In other words, if care is taken to obtain good phasing (judged by visual inspection) of the absorption line shape, then the dispersi0n:absorption plot will not be detectably displaced from its expected position. These conclusions are borne out in practice (see accompanying article).
DISCUSSION The principal conclusions provided by the various numerical examples plotted in Figures 2-9 are summarized in Table I. Table I shows that the proposed dispersion:absorption plot
ANALYTICAL CHEMISTRY, VOL. 50, NO 6, MAY 1978
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pressure-broadened ICR or pure rotational spectra. The most fruitful approach will probably be to use the DISPA plot to establish the nature of the line-broadening mechanism, and then adjust the broadening parameter appropriate to t h a t mechanism to give a best fit to, for instance, the absorption spectrum.
LITERATURE CITED
b
Figure 9. Absorption (top) and dispersion (bottom) spectra (a), and corresponding DISPA plots (b), for a spectrum resulting from various degrees of deliberate phase misadjustment (see text)
has direct diagnostic Lalue in distinguishing between the various types of line-broadening mechanisms listed, based on the direction of displacement of the experimental curve with respect to the semicircle expected for a simple Lorentzian line shape. T h e principal advantages of this diagnostic tool are ( i ) it may be possible to establish t h e line-broadening mechanism from a dispersion:absorption plot for a single data s e t , or a t least to choose between two possible mechanisms. and (ii) in contrast to various deconvolution methods of line shape analysis, it is not necessary to know the line-broadening mechanism in order to construct and use the dispersion: absorption plot. Once the mechanism for line-broadening has been established, the parameter of t h e mechanism (e.g., frequency separation, exchange rate, width of distribution in frequency or relaxation time, etc.) may be determined from fitting the experimental data either to the dispersion:absorption plot or to t h e conventional absorption line shape. I t would be premature to speculate as to the most likely uses for the dispersi0n:absorption plot: the most advantageous situations will become evident when the plot is used to reduce experimental data for various types of spectrometry. Analysis of experimental N M R and ESR data is in progress (see following article). However, it seems probable that the method will be useful in detecting unresolved spectral splittings, and in characterizing the distributions in both resonant frequency and in relaxation time arising from (for example) magnetic resonance spectra of polymers, micelles, and membranes, or
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RECEIVED for review October 7, 1977. Accepted January 16, 1978. This work was supported by grants to A.G.M. from the National Research Council of Canada (A-6178). the University of British Columbia (21-9222) and t h e Alfred P. Sloan Foundation. A.G.M. is an Alfred P. Sloan Foundation Research Fellow (1976-78) and D.C.R. gratefully acknowledges a National Research Council of Canada Postdoctoral Fellowship (1974-75). Portions of this work have been presented at the Sixth International Symposium on Magnetic Resonance, Banff, Alberta, Canada, May 1977; and at the 32nd Northwest Regional Meeting, American Chemical Society, Portland, Ore., J u n e 1977.
