Peak Fitting Functions

As an example of the use of AES to obtain chemical, as well as elemental, information, the depth profiling of a nitrided silicon dioxide layer on a silicon substrate is shown in Figure 6. Using the linearized secondary electron cascade background subtraction technique and peak fitting of chemical line shape standards, the chemistry in the depth profile of the nitrided silicon dioxide layer was determined and is shown in Figure 6. This profile includes information on the percentage of the Si atoms that are bound in each of the chemistries present as a function of the depth in the film. 

Figure 1. Time distribution of the number of photons observed by BATSE in channels 1 and 3 for GRB 970508, compared with the following fitting functions (a) Gaussion, (b) Lorentzian, (c) tail function, and (d) pulse function. We list below each panel the positions tp and widths ap (with statistical errors) found for each peak in each fit. We recall that the BATSE data are binned in periods of 1.024 s.

Fig. 5. Early part of the TOF spectra taken at 64 hPa H2 pressure in the 58 mm diameter target, together with a fit of the spectra. The thick, red function corresponds to the resonantly quenched p,p(2S) resulting in a 900 eV kinetic energy component convoluted with a (fitted) 2S lifetime. The fit function also includes a continuum as seen in the previous figure, plus some extra energies which make up the so-called Coulomb deexcitation part. The dashed peak indicates the measured stop time distribution. The measured background and a kinetic energy scale corresponding to the TOF axe also shown...

Fig. 6. Early part of the TOF spectra taken at 64, 16, and 4hPa (top, middle, bottom, respectively) in the 58 mm diameter target. (The fitted functions were obtained by simultaneously fitting the data taken at two target diameters (20 and 58 mm) at each pressure). One can clearly observe a change in the lifetime of the high-energetic ( 900eV) component (thick red curve), which shows that this component is produced when the thermalized np(2S) are quenched. The dashed peak indicates the measured stop time distribution. The background is also shown...

Although approximate peak positions could be obtained from many different types of software, programs which use peak shape functions to fit the powder diffraction peak profiles are more appropriate to determine the peak position more precisely. Some of the common programs are Xfit,29 TOPAS, etc. 

As the name indicates, fitting the complete powder diffraction profile as a function of intensities, background and a peak shape function is full-profile fitting. ... 

Fig. 9.22 (Left) Raman spectra and (right) polar plot of the intensities of the G and G peaks as functions of 0, measured with an analyzer selecting scattered polarization along the strain axis Sout = 0- The polar data are fitted to Iq- oc sin (0i + 34°) and/< + oc cos (0j + 34°), which gives = 11.3° [18]...

Ex situ IR data are collected on dried, diluted powder films in a low vacuum enviromnent or one purged with a dry gas such as N2. Attenuated total reflectance (ATR)-IR spectroscopy provides surface-sensitive IR measurements and can be used for in situ studies of sorption phenomena. Raman spectroscopy is a related vibrational spectroscopy that provides complimentary information to IR. It can also be used to collect vibrational spectra of aqueous samples. Typical data reduction for vibrational spectra involves subtraction of a background spectmm collected under identical conditions from the raw, averaged sample spectrum. Data analysis usually consists of an examination of changes in peak position and shape and peak fitting (Smith, 1996). These and other spectral parameters are tracked as a function of maaoscopic variables such as pH, adsorption density, and ionic strength. 

In general, three different approaches to the description of peak shapes can be used. The first employs empirical peak shape functions, which fit the profile without attempting to associate their parameters with physical quantities. The second is a semi-empirical approach that describes instrumental and wavelength dispersion functions using empirical functions, while specimen properties are modeled using realistic physical parameters. In the third, the so-called fundamental parameters approach, all three components of the peak shape function (Eq. 2.45) are modeled using rational physical quantities. 

Since as5mimetry cannot be completely eliminated, it should be addressed in the profile fitting procedure. Generally, there are three ways of treating the asymmetry of Bragg peaks, all achieved by various modifications of the selected peak shape function ... 

In Eq. 2.61 a is a free variable, i.e. the asymmetry parameter, which is refined during profile fitting and z,- is the distance fi om the maximum of the symmetric peak to the corresponding point of the peak profile, i.e. z,-= 20yfc - 20 . This modification is applied separately to every individual Bragg peak, including Kaj and Ka2 components. Since Eq. 2.61 is a simple intensity multiplier, it may be easily incorporated into any of the peak shape functions considered above. Additionally, in the case of the Pearson-VII function, asymmetry may be treated differently. It works nearly identical to Eq. 2.61 and all variables have the same meaning as in this equation but the expression itself is different ... 

I. All possible variables (positions and shapes) are refined independently for each peak or with some constraints. For example, an asymmetry parameter is usually a variable, common for all peaks full width at half maximum or even all peak shape function parameters may be common for all peaks, especially if a relatively narrow range of Bragg angles is processed. When justified by the quality of data, an independent fit of all or most parameters produces best results. A major problem in this approach (i.e. all parameters are free and unconstrained) occurs when clusters of reflections include both strong and weak Bragg peaks. Then, peak shape parameters corresponding to weak Bragg peaks may become... 

A simplified approach to peak fitting is to use a suitable analytical function to approximate the measured peak. As in the previous section, the function parameters are adjusted to obtain an optimal fit to the experimental data. In a next step the adjusted function (or its parameters) is used to calculate the first and second moment. This procedure may, for example, help to overcome the inaccuracy of moment analysis in case of asymmetric peaks (Chapter 2.7). It also offers the benefit that standard software such as spread sheets can be used instead of special parameter estimation systems. 

Another common application of fitting functions is in deconvoluting partially resolved peaks (Marco and Bombi, 2001). 

Peak fitting is best done in two steps at first symmetrical curves are fitted. Only after that does one try to determine the asymmetry parameters Pt, if possible as a 20k-dependent function (as for the HWt and mt or wt). 

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