Response function correction

As pointed out in Section 3.3, Raman spectroscopy is usually a singlebeam method, and the observed spectrum is the product of the actual Raman scattering and an instrument response function. Correction of relative intensities across a Raman spectrum is possible, although not common, using the techniques described in Chapter 10. For quantitative analysis, however, one must be concerned about the reproducibility of the observed magnitude... 

Figure 8.42. Elimination of splicing artifacts with an intensity standard. Spectra A and B are from Figure 8.41, but C was obtained by applying an instrument response function correction (with Coumarin 540a as described in Section 10.3.3). The magnified insets for the region near 825 cm show the lack of distortion compared to an uncorrected or averaged splice.

Procedures for determining the spectral responslvlty or correction factors In equation 2 are based on radiance or Irradlance standards, calibrated source-monochromator combinations, and an accepted standard. The easiest measurement procedure for determining corrected emission spectra Is to use a well-characterized standard and obtain an Instrumental response function, as described by equation 3 (17). In this case, quinine sulfate dlhydrate has been extensively studied and Issued as a National Bureau of Standards (NBS) Standard Reference Material (SRM). 

There have been a few recent studies of the corrections due to nuclear motion to the electronic diagonal polarizability (a ) of LiH. Bishop et al. [92] calculated vibrational and rotational contributions to the polarizability. They found for the ground state (v = 0, the state studied here) that the vibrational contribution is 0.923 a.u. Papadopoulos et al. [88] use the perturbation method to find a corrected value of 28.93 a.u. including a vibrational component of 1.7 a.u. Jonsson et al. [91] used cubic response functions to find a corrected value for of 28.26 a.u., including a vibrational contribution of 1.37 a.u. In all cases, the vibrational contribution is approximately 3% of the total polarizability. 

For simplicity, the example discussion included the effect from only one intere-ferent, but the interferent effects are additive. Although this may at first seem to complicate the problem, the total interferent contribution can be determined and corrected by obtaining the sum of all the interferent concentration specificity ratio response functions. This forms the basis of a stray light computer correction used at the Barringer Research Laboratory(42). Dahlquist and Knolls(43) describe a similar computer correction approach called BLISS. 

It is possible to estimate the influence of the spectrometer on linewidth measurements by convolving the response function with a mathematical model of the line. The resulting curve may be compared with the observed line and the true width inferred. Hunt et al. (1968) have employed corrections of this type supplied by the author in their determination of carbon monoxide self-broadened linewidths. 

We have shown that the radiant flux spectrum, as recorded by the spectrometer, is given by the convolution of the true radiant flux spectrum (as it would be recorded by a perfect instrument) with the spectrometer response function. In absorption spectroscopy, absorption lines typically appear superimposed upon a spectral background that is determined by the emission spectrum of the source, the spectral response of the detector, and other effects. Because we are interested in the properties of the absorbing molecules, it is necessary to correct for this background, or baseline as it is sometimes called. Furthermore, we shall see that the valuable physical-realizability constraints presented in Chapter 4 are easiest to apply when the data have this form. 

The method has utility where only modest correction is required, such as in the determination of the spectrometer response function when a narrow line has been measured (Chapter 2, Section II.G). 

In deconvolution problems, such dominance is rarely the case. It is a very modest resolution correction indeed when the response function is so narrow that its central element dominates all the others. Fortunately, the necessary conditions are far less severe, and convergence is usually not a serious problem. Equation (25) seems to tell us, however, that each new row in the matrix brings more independent information when [s]ww/Sn9tw [s]ww is as large as possible. 

The deviations due to some of these destructive influences are reversible. These are usually described as systematic errors. Many of the degradation processes that affect images and most recorded data are classified as systematic errors. For many of these cases the error may be expressed as a function known as the impulse response function. Much mathematical theory has been devoted to its description and correction of the degradation due to its influence. This has been discussed in some detail by Jansson in Chapter 1 of this volume. In that correction of this type of error usually involves increasing the higher frequencies of the Fourier spectrum relative to the lower frequencies, this operation (deconvolution) may also be classified as an example of form alteration. ... 

Fig. 7 Result of inverse-filtering the corrected data of Fig. 6 with a Gaussian impulse response function having a FWHM of 39 units. The Fourier spectrum was truncated after the 35th (complex) coefficient.

Our basic assumption is that the response function f(x,p) is a correct one and the random quantity represents the measurement error. It is then... 

Fig. 2. Time resolved fluorescence spectra of all-trans PRSB in methanol (black) and octanol (grey) for a) t<50 fs and b) t>50 fs. The intensity of the octanol spectra is adjusted the methanol spectra. The spectra are not corrected for self-absorption (for >19.500 cm 1), or for the detector response function. A residual signal appearing at energies <14.000 cm"1 is due to incomplete background subtraction (see above).

The time-resolved measurements were made using standard time-correlated single photon counting techniques [9]. The instrument response function had a typical full width at half-maximum of 50 ps. Time-resolved spectra were reconstructed by standard methods and corrected to susceptibilities on a frequency scale. Stokes shifts were calculated as first moments of cubic-spline interpolations of these spectra. 

Figure 25. FTIR emission spectra at two times following the IRMPE of 70njTorr PhNCO. Unapodized FWHM 3.18 cm 1, Nyquist wavenumber 3950.7 cm the spectrum is corrected for the instrument response function and the maxima of ot spectra have been scaled to unity.

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