No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Weak absorption in ATR spectroscopy: Systematic errors of Harrick's formulas
Abstract
A method for the determination of the systematic errors of Harrick's formulas is proposed. The method is based on the expansion of the expressions for the optical density into a Maclaurin series in terms of the extinction coefficient of the medium under study. Using numerical calculations, it is shown that, under conditions of weak absorption, it is sufficient to consider only the first two terms of this series, with the relative systematic errors of Harrick's formulas being determined in this case by dividing the first term of the series by the second. The behavior of the relative systematic errors for perpendicular and parallel polarizations is compared. These errors are shown to decrease monotonically with an increase in the angle of incidence and have finite limits both for angles of incidence tending to the critical value and at grazing incidence of probe radiation.
The basic principles of attenuated total reflection (ATR) spectroscopy are briefly described and its main areas of application
are listed. It is shown using symbolic calculations that in the case of ATR spectra measured under conditions of weak absorbance,
the area under the curve of absorbance of an isolated Lorentz absorption band is directly proportional to the integral band
intensity. It is shown that the proportionality factor between these quantities contains a single a priori unknown parameter,
i.e., the real part of dielectric constant of the studied medium in the maximum of the studied band. This fact offers a new
scope for determining the integrated intensities of isotropic condensed media directly based on experimental ATR data.
Infra-red reflection spectra of exceptionally good contrast can be obtained if the interface between a dielectric of high refractive index and the sample is taken as the reflecting surface. Radiation incident from the strongly refracting dielectric on this interface at an angle somewhat larger than the critical angle will be totally reflected, but only in those frequency regions where the second medium is non-absorbing (κ = 0). In regions with κ ≠ 0, reflection will not be total any more and, accordingly, a reflection spectrum of high contrast and intensity is obtained, which strongly resembles a transmission spectrum.These spectra may, after calibration, be used as such for quantitative purposes. They may also, by graphical methods or by application of the Kramers-Kronig relation, be converted into spectra of the optical constants n and κ.
Formulas for reflectivities under conditions of attenuated total reflection (ATR.) are expanded to fourth power in attenuation index, κ. From these expansions, absorption rules are derived for single and multiple ATR spectrometer cells. Equations are given for all types of polarization. It is found that if ATR parameters are selected properly and κ is not too large, absorbance is linear with κ, and with concentration of absorbing species if Beer's law holds. Comparison of ATR and transmission spectra and calculation of optical constants using the expanded formulas are discussed.
A theoretical treatment is given of attenuated total reflection (ATR) of partially polarized and linearly polarized radiation from the surface of an anisotropic, absorbing film. Approximate expressions for the reflectivity are derived for the special case of oriented organic polymer films, and a method is given for determining the absorption indices along all three principal film axes. ATR measurements of anisotropic films are found to be sensitive to film orientation even with unpolarized light, although the errors can be minimized by using a “light pipe” with a very large refractive index, e.g. germanium.
The relative error of the optical constants determined by attenuated total reflectance at two polarizations and the same incidence angle is calculated as a function of the reflectance errors by a perturbation method. The variations in the error as a function of the incidence angle are then plotted for various values of the optical constants. The experimental accuracy is optimized by either shallow incidence angles and low index optics or by high index optics in the close vicinity of the critical angle. In either case, the number of reflections should be chosen to give strong attenuation. Under optimum conditions, the method gives reasonably good accuracies.
Several types of spectrometer cells based on single and multiple internal reflections in the UV-VIS-NIR and IR spectral regions are described. The cells are designed for use in standard spectrometers. All are relatively simple, the simplest comprising a single Dove prism with means for holding the sample. These cells permit a well defined variable angle and polarization. Performance data a re given.
The use of band-ratioing techniques in internal reflection spectroscopy (IRS) for a wide variety of polymer surface composition and orientation measurements is discussed. It is shown that quantitative data can be obtained under a wide range of experimental conditions. The effects of variations in sample contact, internal reflection element, and angle of incidence are considered in detail. The applicability and limitations of calibration procedures for the determination of surface composition under various experimental conditions are considered. The requirements for obtaining quantitative results by IRS are shown to be far more lenient than is typically assumed.
- J. Fahrenfort