Asynchronous correlation intensity

The real and imaginary components, 3>(t i, V2) and F(vi, V2), of the cross-correlation function X(t) are referred to, respectively, as the synchronous and asynchronous correlation intensity. These quantities are related to the in-phase and quadrature spectra of dynamic dichroism by... 

They are respectively referred to as the synchronous and asynchronous 2D infrared spectra. The synchronous spectrum characterizes the degree of coherence between the dynamic fluctuations of signals measured at two wavenumbers, and the correlation intensity becomes significant only if the reorientation rates of dipole transition moments are similar to each other. The asynchronous spectrum, however, characterizes the independent, uncoordinated out-of-phase fluctuations of the signals. Hence the asynchronous correlation intensity becomes non-vanishing only if the signals vary at difierent rates. 

The synchronous correlation intensity, (vi, V2), characterizes the degree of coherence between two signals that are measured simultaneously and is maximized when the variations of the two dynamic infrared dichroism signals are totally in phase with each other and minimized when the two signals are out of phase. Conversely, the asynchronous correlation intensity (vi, V2) characterizes the degree of coherence between two signals that are measured at two different instants that are separated in time by a correlation time x/2co. Thus, the maximum value of... 

Synchronous 2D correlation spectra represent coupled or related changes of spectral intensities, while asynchronous correlation spectra represent independent or separate variations [1007]. The 2D cross-correlation analysis enhances similarities and differences of the variations of individual spectral intensities, providing spectral information not readily accessible from ID spectra. 

The functions, and ij/, are called the synchronous and asynchronous 2D intensity correlation functions, respectively. These functions represent the overall similarity and dissimilarity, respectively, between two intensity variations at vi and V2 caused by changing the magnitude of the perturbation. The results are plotted on two orthogonal axes (vi and V2) with the spectral intensity plotted on the third axis normal to the 2D spectral plane. Figures 3-31A and 3-3 IB illustrate schematic contour maps of a synchronous and an asynchronous 2D correlation spectrum, respectively, where + and - signs indicate the directions of the contour peaks relative to the 2D spectral plane. 

Figure 3-32B shows the corresponding asynchronous correlation spectrum. The positive cross peaks at (1614, 1602cm-1), (1590, 1602cm-1), (1305, 1602 cm-1) and (1305, 1583 cm-1) imply that intensity changes of the bands at 1614,1590 and 1305 cm-1 (ring stretch of PPE) occur predominantly... 

The computation of asynchronous 2D correlation intensity is somewhat more complicated. Two approaches can be used (i) using the Hilbert transform and (ii) a direct procedure, obtaining similar results (for a detailed discussion on the asynchronous calculation, see the Further Reading section)... 

The synchronous and asynchronous (i.e. quadrature) correlation intensities, V2) and (vi,V2), of the dynamic spectrum are given by... 

The functional similarity or dissimilarity of intensity variation patterns between a pair of spectral bands measured under a common perturbation at two different spectral variables, vi and V2, can be mathematically characterized in terms of synchronous and asynchronous correlations. Synchronous and asynchronous 2D correlation spectrum pair, (vi,v2) (vi, vz), are given by... 

Figure 21.1 Schematic illustrations of (a) synchronous and (b) asynchronous 2D correlation spectra. White and gray areas in the contour maps represent positive and negative correlation intensities, respectively.

More importantly, the asynchronous correlation spectrum reveals in a surprisingly simple manner the substantial interaction between OA and ethanol. For example, the development of the negative correlation peak at 3450 and 3300 cm indicates that the variation in the spectral intensity at 3300 cm predominantly occurs before that at 3450 cm If spectral... 

The basic concept and some examples of 2D correlation spectroscopy have been covered in this chapter. 2D correlation analysis is based on the simple mathematical treatment of a set of spectral data collected from a system under the influence of an applied perturbation during the measurement. This perturbation can take different forms of changes, including temperature, concentration, or pH, and the like. The set of spectra is then converted to the synchronous and asynchronous correlation spectra, respectively, representing the similarity and dissimilarity of perturbation-induced intensity variations between wavenumbers. 2D correlation peaks provide easier access to pertinent information, making it possible to determine the sequential order of the variations of spectral intensities, as well as relative directions. Highly overlapped peaks are often resolved more clearly. This technique can be a useful addition to the toolbox of experimental scientists. 

The explicit analytical expressions given by Equations (FI 6) and (F24), obtained for the synchronous and asynchronous spectrum, are well suited for the efficient machine computation of correlation intensities from discretely sampled and digitized spectral data. If a discretely sampled dynamic spectrum y(v -, t,) with the total of n points of wavenumber Vj is obtained for m times at each point of time tj, with a constant time increment, that is, t,+i - t, i = At, the integrations in Equations (FI6) and (F24) ean be replaced with summations. 

To extraa more information from the spearal data, 2D-COS can be employed. Basically, this analysis method ae-ates a pair of synchronous (vj,v2) and asynchronous F(vi,V2) 2D correlation spectra, where the spectral variables vi and V2 are wavenumbers. The synchronous 2D correlation intensity (vi,V2) represents the overall similarity or coincidental changes between two separate intensity variations measured at different spectral variables during variation of the external perturbation. The as3mchronous 2D correlation intensity 1 (vi,V2) may be regarded as a measure of dissimilarity or more strictly speaking, out-of-phase charaaer of the spectral intensity variations. 

Figure 3.52. Schematic contour map of asynchronous 2DIR correlation spectrum. Shaded areas represent negative-intensity regions. Reprinted, by permission, from I. Noda, A. E. Dowrey, and C. Marcott, Appl. Spectrosc. 47, 1317 (1993), p. 1318, Fig. 3. Copyright 1993 Society for Appiied Spectroscopy.

Figure 5.4 2D IR correlation spectra based on the spectral changes induced by the static compression of linear low-density polyethylene (LLDPE) at 608 °C. Synchronous spectrum (left) Negative intensity peaks indicating anti-correlation are shaded. Asynchronous spectrum (right) Shaded peaks indicate that intensity changes at i/, occur at higher pressure than changes at i 2- Reprinted from Noda et al. [49], Copyright 1999, with permission from Elsevier.

There is a strong correlation in the synchronous spectrum and no appreciable asynchronicity in the asynchronous spectrum between the mesogen band (2230 cm" ) and the (CH ) bands (2926 and 2863 cm" ). The main part of the intensity changes in the region of 2850-2950 cm" is due to the spacer thus, on the basis of the 2D results we may draw the conclusion that the spacer and the mesogen reorient simultaneously. 

The synchronous spectrum 0(vi, V2) represents the pattern similarity or in-phase nature of spectral intensity changes observed at vj and V2 along the perturbation variable t. On the other hand, the asynchronous spectrum V2) represents the dissimilarity or out-phase nature of intensity variation patterns. Although the procedure described is a highly simplified one, it should be applicable to many practical cases of spectral analysis. More general treatment of 2D correlation analysis, including the case for unevenly sampled spectral data, is discussed in Appendix F. 

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