Reflection spectra Kramers-Kronig transformation

Absorption Spectra as Kramers-Kronig-Transformed Reflection Spectra... 

It should be noted that low-loss spectra are basically connected to optical properties of materials. This is because for small scattering angles the energy-differential cross-section dfj/dF, in other words the intensity of the EEL spectrum measured, is directly proportional to Im -l/ (E,q) [2.171]. Here e = ei + iez is the complex dielectric function, E the energy loss, and q the momentum vector. Owing to the comparison to optics (jqj = 0) the above quoted proportionality is fulfilled if the spectrum has been recorded with a reasonably small collection aperture. When Im -l/ is gathered its real part can be determined, by the Kramers-Kronig transformation, and subsequently such optical quantities as refraction index, absorption coefficient, and reflectivity. 

Hie evaluation of the data yields Rjy Nj, and Sjy i.e., the near-range order parameters of the material seen from the absorber atom. XAS permits the evaluation of the near-range order in the vicinity of the atoms of various elements of one specimen if the energies of their absorption edges are different enough and thus are well separated within the spectrum. It should be mentioned that XAS in reflection looks similar to XAS in transmission mode, however it is different and the evaluation of measurements requires the comparison with reflectivity data calculated form transmission EXAFS spectra. These evaluation procedures involving Kramers-Kronig transform are described in the literature [i-v]. 

Figure 10.18 Spectra by reflection, (a) From a sample of plexiglass, three types of reflection are displayed. Left, crude spectra and right, spectra after correction. Above, crude signal of specular reflection and the result in units of K following application of the Kramers-Kronig (transformation of the reflectance) calculation middle, spectrum obtained by diffused hght comparison of the crude spectrum with the Kubelka-Munk correction below, spectrum obtained by ATR, the latter requiring a fine correction to reduce the absorbance at higher wavelengths which would be overestimated (b) comparison of two spectra of benzoic acid, one obtained through transmission, the other by diffused reflection and subsequent K-M correction.

The reflectance spectra differ from those recorded in transmission, they appear as "derivative-like" bands. These spectra can be converted into absorption one by using of Kramers-Kronig transformation (K-K transformation) that is available in most spectrometer software package. Figure 10 depicts specular reflectance spectrum of oil on surface of machined steel cylinder. 

Specular reflection is the term used to describe mirror-like reflection, from the surface of a sample (angle of reflection equals angle of incidence). Specular reflected radiation ostensibly carries no information about the IR absorption of a sample and is a source of interference in diffuse reflection experiments when the sample is not completely matte, i.e., has an element of shininess about it. However, if the reflected intensity from a sample is due principally to reflection from the front surface of the sample, then an absorption index spectrum of the sample can be generated from the reflected intensity over the whole spectrum using the Kramers-Kronig transformation. (This complex transformation is an... 

For continuous solids with a shiny surface a specular reflection spectrum is obtained. In principle, the absorption spectrum can be derived from this by the Kramers-Kronig transformation. Examples of this are shown for both PP (Figure 4.4(a)) and polyester (Figure 4.4(b)) samples. Equally good results are obtained from both unfilled and carbon-filled samples. The presence of inorganic fillers does not alter the nature of the spectra and any contribution from the filler appears specular. The presence of glass as a filler has little effect on the spectra. The spectra obtained are adequate for qualitative identification, but there are limitations. Band shapes often appear nnsymmetrical and baselines are uneven. When the surface is not shiny the spectra are weaker and may contain a diffuse component. When there are surface species the reflection spectrum may be unrepresentative of the bulk material. No spectra were obtainable from those carbon-filled samples that did not have shiny surfaces. 

External reflection. This is not as well developed a technique as internal reflection the physics of reflection of light from surfaces is less accommodating to the infrared spectroscopist. Smooth or shiny surfaces are particular problems. Specular reflection from the surface itself is governed by Fresnel s equations—the reflectance depends on a complicated combination of refractive index, sample absorbance and polarisation. Consequently, samples where the reflectance is mainly from the surface give rise to spectra which bear little relation to conventional transmission spectra. A transformation known as the Kramers-Kronig transformation does exist which attempts to convert a specular reflectance spectrum into a conventional-looking one. It is not 100% successful, and also very computer-intensive. For these reasons, specular reflectance is not commonly used by the analyst. 

One note of caution should be sounded. If radiation is scattered from the interior of the sample, as might be the case because of the presence of a filler in the bulk of the sample, diffuse reflection (see Chapter 16) will take place along with the Fresnel reflection. In this case, the Kramers-Kronig transform will not yield an accurate estimate of the n and k spectra. One indication that diffuse reflection is contributing to the spectrum is that the bands in the fe spectrum calculated from a given reflection spectrum are asymmetric. 

Fresnel reflection measurements are convenient for certain types of microsamples because essentially no sample preparation is required. Ideally, only radiation reflected from the front surface of the sample is measured at the detector in this type of measurement, so that the absorption spectrum may be calculated by the Kramers-Kronig transform, as described in Chapter 13. However, for scattering samples, diffusely reflected radiation (see Chapter 16) also contributes to the signal measured by the detector. When both mechanisms contribute significantly to the measured spectrum, no amount of data manipulation will allow an undistorted absorption spectrum to be calculated. 

Figure 3 (A) Specular reflectance (SR) spectrum of a black acrylonitrile-butadiene-styrene polymer film measured at near normal incidence, (B) Absorbance spectrum after data treatment of SR spectrum by a Kramers-Kronig transformation. Reproduced in part with permission of Elsevier Science from Zachman G (1995) Journal of Molecular Structure 348 453-456.

Figure 11 Top, reflection spectrum of poly(methyl metha-cryiate) bottom, absorbance calculated by Kramers-Kronig transformation.

For most materials the reflected energy is only 5-10%, but in regions of strong absorptions the reflected intensity is greater. The data obtained appear different from normal tra(nsmission spectra, as derivative-iike bands result from the superposition of the normal extinction coefficient spectrum with the refractive index dispersion (based upon the Fresnel relationships from physics). However, the reflectance spectrum can be corrected by using the Kramers-Kronig (K-K) transformation. The corrected spectrum appears similar to the familiar transmission spectrum. 

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