Lorentzian curve

The solution is a sum of Lorentzian curves, centred at the same frequency, but having different widths. The total width of this spectrum, as well as the width of each of its components, increases linearly with /tj [257]... 

As a result, the energy E of photons emitted by an ensemble of identical nuclei, rigidly fixed in space, upon transition from their excited states (e) to their ground states (g), scatters around the mean energy Eq = E. Eg. The intensity distribution of the radiation as a function of the energy E, the emission line, is a Lorentzian curve as given by the Breit-Wigner equation [1] ... 

It is well known that the v, band of liquid acetonitrile is significantly asymmetric due to an overlap of hot band transitions in the low frequency side. A study of gas phase rotation-vibration spectrum [19] showed that the hot band transition from the first exited state of the degenerated C-C = N bending v8 mode, v hl = v + v8 - vs, has its center at 4.944 cm 1 lower than that of the fundamental transition, v,. Also the presence of v,h2 = v, + 2v8 - 2v8 transition is expected. The careful study on the v band of liquid acetonitrile by Hashimoto et al [20] provided the reorientational and vibrational relaxation times of liquid acetonitrile molecule. They corrected the contribution by the hot band transition using the Boltzmann population law and approximated the v , v,hl, 2h2, and v, + v4 bands by Lorentzian curves. 

Hence, we adopted their analytical way and extended their method to analyze the v p bands. It is necessary to resolve the overlap of those transitions by least-squares fitting in order to obtain the widths of the component bands. For the least-squares procedure, we have to determine analytical functions for fundamental transitions, v, a, v, p, and for hot band transitions the v,p band is reproduced as a sum of three Lorentzian curves of the v, p, v,Mp, and v,h2p. We also took into account the presence of the v2hla and v2h a band for the v,a band. The observed spectra in the 2320-2220 cm range were deconvoluted using... 

Fig. 3. Analysis of the high resolution NMR spectrum for a molecular weight fraction of linear polyethylene with Afrj = 11900 by one Lorentzian curve with rc 3.33 x 10-8 sec and o = 4.71 x 108 Hz2. The solid and the dotted lines indicate the observed and the calculated spectra, respectively43)...

The NMR spectral shape is a Lorentzian curve, and the linewidth, Avi/2, at half-height of the absorption peak, gives a measure of the spin relaxation rates through Avi/2 = 1 /ttT2. 

Recent experimental observations have shown that a situation with two superimposed lorentzian curves in the NMR spectra for sodium ions occur in some biological tissues (21, 22). 

Figure 2. Raman spectra of SWNT+Naphthalene in the G mode region. Broken lines are Lorentzian curves and BWF line fitted to Raman spectra with parameters listed in Table 1 and 2.

The G+ band of the SWNT and SWNT complex (Fig.2) can be well fitted with 2 Lorentzian curves with parameters listed in Table 2. The G band is wide with an asymmetric form, which is strongly broadened in the region of... 

A high-resolution spectrum of the clock transition is shown in Fig. 2. The clock-laser power was reduced to 30 nW to avoid saturation broadening. The fit with a lorentzian curve results in a linewidth of 170 Hz (FWHM), corresponding to a fractional resolution bv/v of 1.3 10-13. A spectral window of 200 Hz width contains 50% of all excitations. According to our present experimental control of the ion temperature, electromagnetic fields and vacuum conditions, no significant Doppler, Zeeman, Stark or collisional broadening of the absorption spectrum of the ion is expected beyond the level of 1 Hz. The linewidth is determined by the frequency instability of the laser and the lineshape is not exactly lorentzian... 

Fig. 6 Frequency spectrum for all runs taken where the laser frequency and /t+ arrival time were recorded. The Lorentzian curve is fitted to the individual events.

The full width at half-maximum (FWHM) of the Lorentzian curve of Fig. 11.54 is FWHM/ = 3.1905 for the Gaussian of equal peak height, FWHMg = 2.3549 at equal peak heights, a Gaussian lineshape is narrower, while the Lorentzian is broader—that is, has more intensity far from the peak. In practice, NMR or EPR lineshapes can also be intermediate between Lorentzian and Gaussian. 

Furthermore, in the picture of this chapter also, the appearance of more than one Lorentzian curve in the spectrum of the quasi-elastic neutron scattering (a result which has been recendy confirmed by Teixeira et al. ) is related to the H-bond dynamics, whereas Teixeira et al. ascribe this effect to the orientational part of the intermediate scattering function. We believe that both physical causes can contribute to this effect. 

Figures. Power spectra (data points) and the fitted Lorentzian curves of the InGaP-on-GaAs and GaAs-on-InGaP heterointerfaces.

In 1996, Nemet and Kozma showed the emission spectrometry of gold laser-produced plasma to be of interest for analytical purposes a delay time of 800-1000 ns was found to ensure nni/-thermal equilibrium and thorough atomization in the plasma. The line profiles obtained under such conditions (both resonant and Stark-broadened) were fitted to a symmetric Lorentzian curve [170]. Recently, LIBS was used in combination with effective chemometric tools to develop a determination method for gold in homogeneous samples that allows the characterization of jewellery products. The results confirmed the LIBS technique as an effective alternative to the hallmark official methods [143,144,171]. 

As the hot band transitions should exist in both of the v, band of hydrogen-bonded and physisorbed acetonitrile, we must get the accurate v, band without the effect by the hot band transitions. We have to determine the analytical functions for fundamental transitions of v/a and vJp and for hot band transitions upon the least-squares procedure. We assumed that the reorientational-vibrational relaxation processes of physisorbed acetonitrile could be described as a diffusion process like bulk liquid molecule, whose spectral density has a Lorentzian form. Accordingly it is supposed that the v,p band is reproduced as a sum of three Lorentzian curves of the v, p, v," p, and bands. In the previous study [9], it was assumed that the Vj a band has a Gaussian band shape and the hot band transitions could be ignored. In the present study we assumed that the v,a band over the range of PIP, =... 

In the method, the differential equation for the continua is resolved in the same way as for the molecular orbital method. Sufficient basis sets represent the continuum wavefunctions in the molecular region and then my depends only on the density of the discretized states. Therefore, the density curves derived from smoothing the oscillator strengths of the states correspond to the photoabsorption spectra (10, 21). In the present work, the Lorentzian curve with the peak width (FWHM) of 3.0cV was used as a combined function of <5 and the smoothing. Though the discretized states depend on the choice of basis set, the spectrum obtained is practically independent of this, fhis was confirmed by using different sizes of the basis sets. 

This computational approach of finding the optimal alignment for the Kalman filter resolution of the overlapped shifted spectra by the simulated annealing algorithm has been tested on simulated overlapped spectra obtained by linear combination of Gaussian-Lorentzian curves, synthetically generated using the mathematical model described by Eqn. (9)... 

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