Triangular slit function

Fig. 40.17. Convolution in the time domain offlf) with h t) carried out as a multiplication in the Fourier domain, (a) A triangular signal (w, = 3 data points) and its FT. (b) A triangular slit function h(t) (wi/, = 5 data points) and its FT. (c) Multiplication of the FT of (a) with that of (b). (d) The inverse FT of (c).

In many cases, the profile a spectroscopist sees is just the instrumental profile, but not the profile emitted by the source. In the simplest case (geometric optics, matched slits), this is a triangular slit function, but diffraction effects by beam limiting apertures, lens (or mirror) aberrations, poor alignment of the spectroscopic apparatus, etc., do often significantly modify the triangular function, especially if high resolution is employed. 

Figure 3. Calculated band profiles of Stokes vibrational Raman scattering from Nt at 2000 K assuming a triangular slit function with FWHM = 5.0 cm 1. The bottom curve includes the isotropic part of the Q-branch only. The top curve is a more exact calculation including O- and S-branch scattering, the anisotropic part of the Q-branch and line-strength corrections owing to centrifugal distortion. The base lines have been shifted vertically for clarity.

Assuming the true absorptions to have Lorentzian form and assuming a triangular slit function, Ramsay 121, 122) investigated the effect of finite resolving power upon these band shapes for the vibrations of a variety of compounds. This approach reproduced satisfactorily the observed band profiles. However, he did not obtain a simple relationship between true and apparent integrated intensities. Ramsay found that bands are best characterized by their apparent peak intensities [loge(J o/r) J and their apparent half-intensity band widths These quantities are related to... 

Again assuming both a Lorentzian form for the true band shape, and a triangular slit function, a series of apparent band shapes may be calculated for a fixed value of the ratio and various values of the true peak... 

In all three methods, the assumptions of a Lorentzian band profile and of a triangular slit function were made. However, since Methods II and III involve measurement over the complete experimental curve, whereas Method I uses only three points of this curve, the latter is the most sensitive to the first assumption. Method II depends upon fairly small corrections related to the band shape and the slit function. Method HI is almost insensitive to the form of the slit function, but is much more strongly dependent upon the assumed band shape. Consequently, for partially overlapping bands, and in general, Methods II and HI are to be preferred, although Method III has speed in its favor. 

Fig. 4.13. Gaussian shape of an absorbance band (p) and triangular slit function (s). (a) The light of measurement is veiy broad-band, therefore the absorption of the sample only covers a small part of the offered spectrum of light (b) the offered light is absorbed proportional to the concentration of the sample for this specific ratio of slit function and natural band width. Therefore the somewhat polychromatic light is affected by various different absorption coefficients.

When a spectrum is measured on a dispersive instrument, the true spectrum is convolved with the instrumental line shape (ILS) of the monochromator, which is the triangular slit function. The situation with the FT technique is equivalent, except that the true spectrum is convolved with the (sinx)/x function (no apodization) or with the FT of an appropriate apodization function. Hence, FT instruments offer a free choice of ILS according to the apodization selected and thus make it possible to optimise the sampling condition for a particular application. 

Fig. 16. Resonance fluorescence excited by absorption of the excitation shown by the triangular slit function of Fig. 15. The excitation halfbandpass is 45 cm and the band-pass of the fluorescence spectrometer is 35 cm. The assignment of structure is separated according to the two excited state origins and 6 16. ... 

It was mentioned earlier that the mathematical expression which best fits the absorption band is a combination of a Lorentz curve and a triangular slit function. Ramsay has constructed correction tables making it possible to find the ideal Lorentz curve corresponding to an experimental band. To use this method we need know only the peak absorbance ln(/o//)max of Ih experimental band, its half band width and the spectral slit width used in carry-... 

Sandorfy and his co-workers have derived an expression (see Fig. 14) in which, by means of a triangular slit function they relate the shape of the absorption curve not only to the value of the half band width Aj/ /, but also the additional values Ai /, Ai / Aj/ /g. These values represent the width of the bands at /4lnf V4ln(T o/T j ax, and /% n(To/T), respectively. The mathematical expression they arrive at after integration matches the experimental curve exactly at eight points and is very close to all other points of the curve. However, this is valid only for symmetrical absorption bands more times than not, one encounters asymmetrical absorption bands. In this case the curve is divided into components to the right and left of a line perpendicular to the peak absorbance of the band. Now, instead of measuring Ai/i/, one measures Ai/f/ and Ai f/. Of course, the sum Ai/ The Sandorfy equation in final... 

To compute the convolution of these two functions, Eq. 2.19 requires that/(v) be reversed left to right [which is trivial in this case, since/(v) is an even function], after which the two functions are multiplied point by point along the wavenumber axis. The resulting points are then integrated, and the process is repeated for all possible displacements, v, of/( relative to B v). One particular example of convolution may be familiar to spectroscopists who use grating instruments (see Chapter 8). When a low-resolution spectrum is measured on a monochromator, the true spectrum is convolved with the triangular slit function of the monochromator. The situation with Fourier transform spectrometry is equivalent, except that the true spectrum is convolved with the sine function/(v). Since the Fourier transform spectrometer does not have any slits,/(v) has been variously called the instrument line shape (ILS) Junction, the instrument function, or the apparatus function, of which we prefer the term ILS function. 

No matter what type of spectrometer is used, a measured spectrum is always slightly different from the true spectrum because of the measurement process, and it is important to recognize that instrumental effects often determine how well Beer s law is obeyed for any chemical system. For example, when a monochromator is used to measure a spectrum, the true spectrum is convolved with the spectrometer s slit function. The effect of this convolution is to decrease the intensity and increase the width of all bands in the spectrum. The convolution of Lorentzian absorption bands with a triangular slit function was reported over 50 years ago in a classic paper by Ramsay [1]. Ramsay defined a resolution parameter, p, as the ratio of the full width at half-height (FWHH) of the slit function to the true FWHH of the band. He showed how the measured, or apparent, absorbance, A, at the peak of a Lorentzian band varied as a function of the true peak absorbance, peak> resolution parameter. Not surprisingly, Ramsay showed that as... 

The series of curves shown in Figure 8.3 is quite similar to the corresponding series of curves calculated by Ramsay for a monochromator with a triangular slit function. However, with the triangular apodization function, the slope of the plot of log A pg versus log Ap j can be as small as indicating that Ap can vary as (Ap j ) when Apg is very large. 

The slit function is approximately triangular. Various instrument-related factors combine to produce the shape shown in Figure 25-7. 

The slit function can be observed well in the imaging of line sources when the inlet and outlet slits (and the middle slit if present) are moved synchronously. If the slit is too wide the line of a mercury lamp does not appear as the expected Gaussian-shaped curve intensity distribution, but as a triangle. If the inlet and exit slits are different, a trapezoid is obtained. Triangular shapes in the spectrum indicate defective adjustment of slit widths in the equipment. These effects are observed mainly in spectrometers in which only a small number of preset slit widths can be selected. Distortion of a spectral band is negligible only if... 

Fig. 7. Triangular spectral energy distribution across the exit slit of a spectrometer. s = spectral slit width in cm = spectrometer frequency setting, p(w ) = intensity function. (Reprinted with permission from W. West, Chemical Applicaiions of Spectroscopy, Vol. IX, John Wiley and Sons, Inc., New York, 1956, p. 272.)...

Now let s consider the part played by the slit width in determining the shape and intensity of an absorption band, i,e, its effect. We shall assume that the sample band is of the Lorentz curve shape and has a half band width of 8 cm (Ai ). We shall also assume that the slit width is 1 cm (Ay/) and follows a triangular function. In this case the band as seen by the instrument will have a half band width (Ayi/ ) approximately equal to Ay, and the band will be of the Lorentz shape. Here the spectrophotometer accurately sees the band shape (see Fig. 9). 

In such cases it is better to apodize with a triangular rather than box-car function. This situation may be compared with the rule-of-thumb for conventional infrared spectrometers that the peak width at half height should be at least five times the instrumental spectral slit width if accurate results are to be obtained. 

---