Deconvolution broadening

De Pristo A. E., Rabitz H. Scaling theoretic deconvolution of bulk relaxation data state-to-state rates from pressure-broadening linewidths,... 

XPS also yields chemical information directly. Eor instance, if an element in a sample exists in different valence states, the XPS peak may broaden and show a shoulder. It is possible to deconvolute the peaks and determine valence states and the relative amount of each state in the sample. It is important to do this type of work by comparison of values of standard reference compounds. 

Several peaks of interest (ideally higher order reflections of the same type hkl, 2h, 2k, 21, 3h, 3k, 31,. .., nh, nk, nl) are fitted by Fourier series the same procedure is applied to the diffraction lines of a reference sample, in which size and strain effects are negligible, in order to determine the instrumental line broadening. Such information is used in order to deconvolute instrumental broadening from sample effects (Stokes-Fourier deconvolution [36]). 

Fig. 40.32. Deconvolution (result in solid line) of a Gaussian peak (dashed line) for peak broadening ((M i/,)prf/(H vi)G = 1). (a) Without noise, (b) With coloured noise (A((0,1%), Tx = 1.5) inverse filter in combination with a low-pass filter, (c) With coloured noise (A (0,1 %), Ta = 1.5) inverse filter without low-pass filter.

Gugliotta, L. M., Vega, J. R., and Meira, G. R., Instrumental broadening correction in size exclusion chromatography. Comparison of several deconvolution techniques, /. Liq. Chromatogr., 13,1671, 1990. 

The situation is illustrated in Fig. 2.15 where a signal is shown which has been obtained in the analytical reality, distorted and disfigured by noise and broadening. All of these effects can be returned to a certain degree by techniques of signal treatment like deconvolution, signal accumulation and smoothing, etc. 

The effect of instrumental broadening can be eliminated by deconvolution (see p. 38) of the instrumental profile from the measured spectrum. If deconvolution shall be avoided one can make assumptions on the type19 of both the instrumental profile and of the remnant line profile. In this case the deconvolution can be carried out analytically, and the result is an algebraic relation between the integral breadths of instrumental and ideal peak profile. From such a relation a linearizing plot can be found (e.g., measured peak breadths vs. peak position ) in which the instrumental breadth effect can be eliminated (Sect. 8.2.5.8). 

After H0bs has properly been extracted (cf. Sect. 2.2.2), the effect of instrumental broadening can be eliminated by numerical deconvolution (see p. 38). If the peaks shall be modeled by analytical functions (Sects. 8.2.5.7-8.2.5.8), the consideration... 

In the presented form Eq. (8.13) is only valid, if Hj (s) is, indeed, constant over the whole angular range required for analysis. If this is not the case and numerical deconvolution is aimed at, the standard algorithm may be adapted by consideration of the fact that, in any case, the broadening is a slowly varying function of 29. 

There may, however, be specific reasons to study a signal over an extended temperature range. For one, a linear increase in EPR amplitude with the inverse of the temperature (Curie s law) is proof that a spin system is a two-level system, i.e., an S = 1/2 or an effective S = 1/2 system. More importantly, in complex multicenter metalloproteins, overlapping spectra may be deconvoluted by virtue of their Tu value being different if two centers, a and b, have rMa < TMb then at TMb the spectrum of center a is broadened and that of center b is not. It is once more emphasized that these types of studies require determination of (anisotropic) saturation behavior at all relevant temperatures. 

We will not concern ourselves here with problems associated with line broadening, overlapping peaks, and background subtraction. There are, however, examples discussed later where both deconvolution and curve fitting procedures are shown to be essential in unraveling the contributions of differently bonded species of the same molecule to the total photoelectron yield. Carley and Joyner (14) have discussed recently deconvolution procedures for photoelectron spectra. 

There are many facets of this study which we feel merit further investigation. In particular it is necessary to consider am extension of the proposed model, which in its present form is confined to the performance of a simple column, to cover the behaviour of any set of columns since it is column sets which are normally used. In addition, it is important to consider the input to the model which should be truly representative of polymers with a molecular weight distribution and not merely a concentration pulse of perfectly monodisperse polymer. In relation to this latter suggestion it would be significant if it were possible to link this model to the very real problem of deconvolution, i.e. the removal of instrumental and column broadening from the observed chromatogram to produce the true molecular weight distri-... 

From the 3D plot, the individual first- or second-dimension separation can be deconvoluted, precluding the need for a ID detector, which introduces band broadening into the system. 

It may have occurred to the reader that inherent line broadening could be removed by deconvolution. This appears especially attractive as a means of separating the various contributions. In the particular case of absorption spectra, however, it is not as straightforward as it at first appears. Referring to Eq. (9), we see that the various broadening-profile convolutions appear in the exponential ... 

In liquids the interactions between neighboring molecules are considerably more complicated than in gases. The resultant broadening obliterates the fine line structure seen in gas spectra, leaving only broad band profiles. There are many possible contributors to this broadening. In some cases, adequate approximation is obtained by assuming that the band contour is established by collisions. Ramsay (1952) has noted that substitution of appropriate molecular density and collision diameter numbers in the collision broadening formula results in realistic band widths for certain liquid-phase systems. In such systems, the bands typically show an approximately Lorentzian profile. Approximate deconvolution of inherently broadened liquid-phase spectra may therefore be obtained on the basis of the assumption of Lorentzian shape (Kauppinen et al., 1981). 

We now consider broadening caused by the spectrophotometric system itself. This broadening is of special interest to us in this work because it is not part of the phenomenon under observation. Instead, it represents our inability to observe this phenomenon accurately. Elimination of spectrometer broadening is the usual goal of deconvolution. 

The line breadth remaining after deconvolution may be influenced primarily by broadening for which compensation was not attempted. Nevertheless, the maximum breadth-reduction ratio attainable when the method is really challenged indicates its potential. The original application (Section Y.A.2) showed reduction on the order of 2.5 times, limited as described earlier. Subsequent applications have yielded substantially greater improvements (Chapter 5, Chapter 7). 

A review of deconvolution methods applied to ESCA (Carley and Joyner, 1979) shows that Van Cittert s method has played a big role. Because the Lorentzian nature of the broadening does not completely obliterate the high Fourier frequencies as does the sine-squared spreading encountered in optical spectroscopy (its transform is the band-limiting rect function), useful restorations are indeed possible through use of such linear methods. Rendina and Larson (1975), for example, have used a multiple filter approach. Additional detail is given in Section IV.E of Chapter 3. 

Consider an example. Assume that we are looking at a spectral line and trying to decide how to observe the line for deconvolution giving width reduction by a factor of 3. Further assume that pressure broadening is... 

B. Deconvolution of Pressure-Broadened Infrared and Raman Spectra 213... 

The previous sections have dealt primarily with infrared absorption spectra, although the conclusions can in general be applied to other types of spectra. Here additional uses of deconvolution will be demonstrated. In the first example, a Fourier transform spectrum is simulated and several attempts to deconvolve this spectrum show limited success. In the second example, pressure-broadening effects in an infrared absorption spectrum and a Raman spectrum are simulated. An attempt at removing these effects by deconvolution shows some promise. 

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