Instrument line shape

Finally, instmmental broadening results from resolution limitations of the equipment. Resolution is often expressed as resolving power, v/Av, where Av is the probe linewidth or instmmental bandpass at frequency V. Unless Av is significantly smaller than the spectral width of the transition, the observed line is broadened, and its shape is the convolution of the instrumental line shape (apparatus function) and the tme transition profile. 

We have seen that one of the key aspects of Fourier transform NIR analyzers is their control of frequency accuracy and long-term reproducibility of instrument line shape through the use of an internal optical reference, normally provided as a HeNe gas laser. Such lasers are reasonably compact, and have acceptable lifetimes of around 3 to 4 years before requiring replacement. However, we also saw how the reduction in overall interferometer dimensions can be one of the main drivers towards achieving the improved mechanical and thermal stability which allows FT-NIR devices to be deployed routinely and reliably in process applications. 

Fig. 4. The functions a) I(Vt) and b) S(Vk), which are the instrument line shape functions for spectra computed using no apodization and triangular apodization, respectively.

It is proposed to recapitulate the basic physical and optical principles of spectroscopy in this review. For the comparison of different methods, we concentrate on the determination of wavelength as an essential part of spectroscopy. We also comment on the power of resolution of the various instruments and the instrument line-shape function. 

Having commented on the fundamentals, we now have to emphasize certain properties in more detail. We next discuss resolution, i.e. the minimum difference in the wave numbers of two narrow lines that can still be seen as separate lines by the spectrometer. This leads on to the instrument line-shape function of a spectrometer. 

Fig. 4. Instrument line-shape function for a diffraction grating (N = 8) and the smallest difference A A between two narrow lines which is clearly resolved by the grating...

This is the well-known formula that states that the resolution of a diffraction grating increases (and Jv decreases) with increasing order n of diffraction and number N of lines in the grating >. For the case of three lines, or for any other spectrum, the intensity is measured with a grating spectrometer as fimction of yd (for several orders of diffraction) (see Fig. 5). The data obtained in this way are then easily converted to / (A) or I ( ) and the problem of determining the spectral distribution I ( ) is solved. It should be noted that the linewidth obtained (see Fig. 5) is influenced by the limited resolution of the instrument and that the line-widths of the three laser lines are assumed to be actually much smaller. In other words, we have been discussing the properties of the instrument line-shape function of a diffraction grating. 

Fig. 10. Interferogram I (s) of a continuous spectrum versus path difference s (upper part), the corresponding spectrum I (f) versus wave number v, and the instrument line-shape function (for triangular apodization)...

Fig. 13. Sampling of an interferogram at equal increments As of the path difference (upper part) and the spectrum with instrument line-shape function (lower part)...

The first problem is that the integration cannot be performed for 0 g t < oo but only for a finite range 0 g t g T, for which the FID signal has been recorded (Tq = data acquisition time). This truncation of the FID signal leads to a finite resolution Av l/2To (cf. Vol. 58 Fig. 6, p. 86). The instrumental line shape function is of the type... 

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. 13. Inelastic neutron-scattering spectra of Ar/MgO(100) at 10 K. (a) Experimental spectra at incident energy of 7 meV for a 1.16 layer (hexagonal structure) the vertical bars and the triangle represent experimental errors and the experimental broadening (reduced by a factor of 1/10), respectively (b) calculated spectrum for the hexagonal incommensurate structure after convoluting with the instrumental line shape of 0.3 meV at a neutron gain of 5 meV. Units are arbitrary and the basehne is shifted with respect to curve (a) (fiomRef 99).

Figure2.4 shows the instrumental line shape,/(v) (top), and B(v) (bottom) for vi =2/A. Looking at the ILS, it can be observed that the curve intersects the wavenumber axis at 1/24, and for B v) the intersection happens at v 1/24. In this situation, two spectral lines separated by twice this amount (1/4) will be completely resolved.

Fig. 2.4 Instrumental Line Shape/LA(v) top), which is the Fourier transform of a boxcar function of unit amplitude extending from +A to —A. Fourier transform of an interferogram generated by a monochromatic line at vi = 2/A bottom)...

Fig. 2.5 Apodization functions (left) and corresponding Instrumental Line Shape (right) boxcar (blue), triangular (green) and squared triangular (red)...

The sinusoidal modulation present in the spectra in the shape of small ripples with high frequency is due to the instrumental line shape (ILS) of the FTS. It has a period of 2.3 cm which corresponds to an optical distance of 4mm, this is the maximum optical path difference Smax of this simulation. 

S( ) is known as the instrument line shape function (ILS) and is illustrated in Figure 3. The effect of convolving the spectrum B(F)... 

The function Q(o-) is similar to the slit function which distorts lines in spectra collected on dispersion instruments. Q(instrument line shape and can be varied by changing the maximim optical retardation L or by changing the form of a(8). Figure 3 shows several choices for the apodization function and the resulting instrument line shape for each. It can be seen that the width of the instrument line shape is proportional to 1/L. Thus, the larger the optical retardation, the narrower the spectral lines become. For the case where a(8) = 1 for all 8 between 0 and L, the narrowest lines are achieved, but the side-lobes or "ringing" are most severe. When many absorption or emission lines in a spectrim are convolved with this instrument line shape, the spectrum can become difficult to interpret. Therefore, a compromise is usually reached between an apodization function a(8) which produces narrow spectral lines and one which reduces the side-lobes. 

Figure 2. The sine x function instrumental line shape function of a perfectly aligned Michelson interferometer with no apodization...

Depending on the measured bandwidth of the features (which themselves are dependent on the instrumental line-shape function, the real bandwidth, etc.), artifacts may be... 

In order to obtain a true image in AFM (as well as in any other microscopy or spectroscopy experiment), the observed result and the instrumental line-shape must be deconvoluted. This is often possible in spectroscopy, but often impossible in AFM, especially for artefacts at length scales that are comparable with the tip dimensions. This tip-imaging artefact cannot be eliminated. If the sample is an insulator it may also be locally charged. Image artefacts are also introduced due to surface deformation. 

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. 

---