Lorentzian enhancement spectrum

Narrow Component. As discussed in Chapter II, the absorption spectrum for polyethylene cannot be described by a single Lorentzian even in the molten state. However, the deviation from one Lorentzian is not enhanced for well-fractionated samples in the melt and, furthermore, becomes negligible as the temperature decreases42. Accordingly, the differential form of a Lorentzian distribution can be used for the elementary spectrum of the narrow component ... 

NMR analysis. The spectra are recorded by a Varian DA-60-IL spectrometer. Deuterated dimethylformamide (DMF-d7) has been used as solvent. The copolymer solutions (10%) are heated at 130°C (internal reference Hexa-methyl-di-siloxane (HMDS), hmds = 0.05 ppm). Enhanced signal/noise ratios have been achieved by the Jeol-JR-Al Spectrum Accumulator and the decompositions and simulations of the complex pattern of methoxy and a-methyl resonances, by the Du Pont de Nemours, 310 Curve Resolver. Lorentzian curves, with a line width at half height of 2 Hz, have been used to simulate all the complex pattern of the methoxy and a-methyl resonances. The line width value has been determined from experimental measurements on NMR spectra. The broadening of the two a-methyl singlets from the s and h triad sequences are probably due to pentad effects [2, 5]. 

The sensitivity may be further enhanced by modulation of the laser frequency. This allows sensitive lock-in detection and yields signals which represent the derivative of the Lorentzian line profiles (see Sect.8.1). For illustration of the resolution achieved with extremely well-stabilized lasers, large magnifications of the laser beam diameter, and long absorption cells at low pressures. Fig.10.26 shows the modulated saturation spectrum of the CH3 C1 molecule at v = 2947.821 cm (= A = 3.39 ym) [10.40]. 

With digitized spectral data and the appropriate computer software, obtaining a derivative spectrum is a simple matter. The even derivatives have sharper peaks at the same frequency values as in the original spectrum. In the case of a Lorentzian profile with unit half-width, the second derivative (for which the peak is negative) has a value that is equal to 0.326 of the original bandwidth [95], and for the fourth derivative, which has a positive peak, the value is equal to 0.208 of that of the original bandwidth. Band sharpening is therefore achieved with the derivative spectra. The resolution enhancement is 2.7 for the second derivative and 3.8 for the fourth derivative. 

The data set is multiplied by the inverse transform of the Lorentzian function, which is the exponential function. The data set is then transformed into the frequency domain to give a conventional, sharpened spectrum. By using spectra from FTIR instruments, resolution enhancement factors of 3 can be obtained [99]. However, no line can be narrower than the original instrument function. 

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