International Association of Scientific Innovation and Research (IASIR) (An Association Unifying the Sciences, Engineering, and Applied Research) International Journal of Emerging Technologies in Computational and Applied Sciences (IJETCAS) www.iasir.net IJETCAS 14-312; © 2014, IJETCAS All Rights Reserved Page 27 ISSN (Print): 2279-0047 ISSN (Online): 2279-0055 Mathematical Analysis of Asymmetrical Spectral Lines J. Dubrovkin Computer Department, Western Galilee College 2421 Acre, Israel Abstract : Mathematical analysis of seven non-integral theoretical and phenomenological forms of asymmetrical lines was performed by their decomposition into the product of symmetrical and asymmetrical parts. The decomposition errors were evaluated. For the purpose of comparison, the x coordinate of each profile was normalized to the uniform scale. The dependences of the maximum peak positions, the maximum intensities, the widths of the lines, and their symmetrical parts on the asymmetry parameter were obtained. The ratio of the absolute values of the first-order derivative extremes of the line profile and the ratio of the satellite amplitudes of the second-order derivative were proposed as new measures of line asymmetry. The new concept of the integral angular function of the asymmetrical part was introduced. The scaled difference between the left- and the right-hand (relative to the peak maximum) components of this function was found to be the most sensitive indicator of line asymmetry. The obtained equations may be useful for modeling asymmetrical lines in spectroscopic studies. Keywords: spectroscopy; asymmetrical lines; line form parameters; normalization; approximation I. Introduction The form of the spectral components (lines and bands) and their parameters, such as location, intensity, width and statistical moments, constitute the main source of spectrochemical information [1]. It was shown theoretically [1] that the observed form of the isolated symmetrical components in an optical spectrum (in the absence of background) is accounted for by the impact of physical (Heisenberg's uncertainty principle, Doppler and collision-induced broadening) and instrumental factors. However, in practice, the assumption of the line symmetry is often violated for such reasons as: intermolecular interactions in complex systems (IR- [2] and fluorescence spectra [3] of large molecules in condensed phase), the impact of fluorescence on Raman spectra [1], mixing of reflection and absorption bands of IR components [4], the sample heterogeneities and instrumental factors (inhomogeneities of the static magnetic field arising from imperfect shimming in NMR- spectroscopy [5, 6], the intrinsic properties of the radiation source in astronomy [7], heterogeneity of the photoionization absorption in optically thick laser-induced plasma [8], and intermolecular interactions induced by strong vibrational excitation in laser spectroscopy of gases [9]. The study of asymmetrical lines is of great importance in ESR spectroscopy [10]. The relationship between the form of the central part of spectral profiles and their wings, on the one hand, and physicochemical processes in gases and liquids, on the other hand, was studied both theoretically and experimentally [11-13]. The determination of the cis:trans ratio in some biologically active compounds [14] is an interesting example of the practical application of the ESR-spectrum line asymmetry. Theoretical expressions for asymmetrical profiles are often quite complicated and, therefore, cannot be applied in practice [15]. For this reason, asymmetrical peaks are commonly described by empirical functions. For example, a phenomenological model of the asymmetrical shape of X-ray photoelectron peaks was developed and studied thoroughly [16]. A large number of mathematical functions of asymmetrical spectroscopic and chromatographic peaks was described in review [17] and research article [18]. However, no general approach to the mathematical analysis of asymmetrical line profiles has been introduced yet. This approach would allow establishing the main patterns of the physical-chemical processes by searching common properties of their spectral line models. The choice of the appropriate mathematical model is particularly important for decomposition of complex spectral contours. The evaluation of the measurement errors in detecting the positions of overlapping peaks prior to deconvolution also relies on the proper selection of the line model. Such selection of the “best” model out of numerous model functions it very difficult and is usually performed empirically. Such selection requires the concept of the model proximity to the experimental spectral contour (e.g., the minimum least squared error). In this connection, the method of comparing and choosing the parameters of different models based on a unified mathematical description would be very helpful. The goal of the present study was developing such method using some simple models of asymmetrical lines used in spectroscopy. The models did not include integral equations [17]. The apparent line asymmetry is sometimes caused by uncorrected background in the spectra and by unresolved structure of spectral lines and bands. These issues require special consideration and lie out of the scope of the present study. For simplicity, we use the short term “line” instead of the long term “line and band”. The standard algebraic notations are used throughout the article. All calculations were performed and the plots were built using the MATLAB program.