Lorentz-Gauss functions

Apodization is the process of multiplying the FID prior to Fourier transformation by a mathematical function. The type of mathematical or window function applied depends upon the enhancement required the signal-to-noise ratio in a spectrum can be improved by applying an exponential window function to a noisy FID whilst the resolution can be improved by reducing the signal linewidth using a Lorentz-Gauss function. ID WIN-NMR has a variety of window functions, abbreviated to wdw function, such as exponential (EM), shifted sine-bell (SINE) and sine-bell squared (QSINE). Each window function has its own particular parameters associated with it LB for EM function, SSB for sine functions etc. 

Best results are achieved using Lorentz-Gauss functions for representing NMR peaks (Eq. (13.2)). The function includes four adjustable parameters such as the maximum intensity of the peak, 1, the chemical shift at maximum intensity of the peak, and the Lorentz and Gaussian parameters a and b, respectively. 

In liquids, simple functions are applied like Lorentzian, Gaussian, or Voigt lines, taking instrumental and physical facts into account. Best results are currently achieved using a Lorentz-Gauss function for representing NMR peaks (Dalitz et al. 2012) ... 

The weighting functions used to improve line shapes for such absolute-value-mode spectra are sine-bell, sine bell squared, phase-shifted sine-bell, phase-shifted sine-bell squared, and a Lorentz-Gauss transformation function. The effects of various window functions on COSY data (absolute-value mode) are presented in Fig. 3.10. One advantage of multiplying the time domain S(f ) or S(tf) by such functions is to enhance the intensities of the cross-peaks relative to the noncorrelation peaks lying on the diagonal. 

There are generally three types of peaks pure 2D absorption peaks, pure negative 2D dispersion peaks, and phase-twisted absorption-dispersion peaks. Since the prime purpose of apodization is to enhance resolution and optimize sensitivity, it is necessary to know the peak shape on which apodization is planned. For example, absorption-mode lines, which display protruding ridges from top to bottom, can be dealt with by applying Lorentz-Gauss window functions, while phase-twisted absorption-dispersion peaks will need some special apodization operations, such as muliplication by sine-bell or phase-shifted sine-bell functions. 

Figure 6.1 Comparison of 26 — 6 scan profiles obtained by a monochromatized (pure Cu kal) parallel beam configuration (hybrid x-ray mirror) and a conventional parallel beam configuration achieved by divergence slit (ds) module measured at 001/100 (a), 002/200 (b), 003/300 (c), 004/400 (d) of 500nm-thick Pb(Zro.B4Tio.46)03 thin film. Dotted lines represent the second derivative of the profiles, indicating the peak positions. Note that the profiles are simulated fitted profiles for obtained spectrum using pseudo-Voight function (mixed Lorentz and Gauss function).

The two simplest peak shape functions (Eqs. 2.49 and 2.50) represent Gaussian and Lorentzian distributions, respectively, of the intensity in the Bragg peak. They are compared in Figure 2.42, from which it is easy to see that the Lorentz function is sharp near its maximum but has long tails on each side near its base. On the other hand, the Gauss function has no tails at the base but has a rounded maximum. Both functions are centrosymmetric, i.e. G(x) = G -x) and L x) = L -x). 

In more recent years new window functions have been introduced [15,16] that are similar to the Lorentz-Gauss window but which aim to improve resolution without a discernible reduction in sensitivity. These so called TRAF... 

Figure 4-4. Difference spectra between on- and off-resonance photoemission spectra of La203, Pr203 and NdiOa. (The vertical bars are the multiplet structure of final state. The solid lines are convolutions of the multiplet stmctures with Lorentz and Gauss functions. (Reproduced with permission firom ref 11. Copy right 2000 J. Phys. Soc. Japan)...

The default setting is Lorentz, i.e. a pure Lorentzian function. A single click on the upper arrow key switches immediately to a pure Gaussian function. The next click on the same arrow sets the peak to Baseline. If, beginning again with the Lorentzian type, the down arrow is clicked on instead, the band shape changes to 100% Lorentz + Gauss. In principle this band... 

Voigt function the Voigt function is the convolution of a Lorentz function with a Gauss function ... 

A Lorentz function is smaller than a Gauss function near the peak maximum, but the principal distinguishing characteristic of a Lorentz function is that it has significant... 

Figure 2.42. The illustration of Gauss (dash-dotted line) and Lorentz (solid line) peak shape functions. Both functions have been normalized to result in identical definite integrals...

If we can set Rre to cancel exactly the original decay of the FID then the result of this process is to generate a time-domain function which only has a Gaussian decay. The resulting peak in the spectrum will have a Gaussian lineshape, which is often considered to be superior to the Lorentzian as it is narrower at the base the two lineshapes are compared in Fig. 4.12. This transformation to a Gaussian lineshape is often called the Lorentz-to-Gauss transformation. 

Much of the popularity of these functions probably rests of the fact that there is only one parameter to adjust, rather than two in the case of the Lorentz-to-Gauss transformation. 

Where, y - width of Lorentz profile, w - width of Gauss pa-ofile and Wo - central position of absorption band. To take into account the individual rate of thermal decomposition of absorbing Lorentz profiles of aggregates, we need to include in the Voight function the equation describing the thermal decay curve, depending on the energy activation distribution. The equation for thermal decomposition as follows from (4) shown in the above curve is... 

ESR lines in solution can almost always be approximated by a Lorentz function. In the solid state the line-shape can in general be reproduced by a Gauss curve. In some instances a so-called Voigt profile can give a better approximation to the experimental line-shape. A Voigt line is a convolution of a Lorentz and a Gauss line. The shape is determined by the ratio ABi/ABg of the respective line-widths. The shapes of the 1st derivative lines of these types are given in Fig. 9.1. 

Fig. 9.1 (a) Gauss, Lorentz and Voigt (ABl/ABq = 1.0) lines as 1st derivatives adjusted to the same peak-peak amplitudes, (b) Integrated area of derivative lines with the same amplitude as function of the ABl/ABq ratio of a Voigt Une. For ABl/ABq 1 the line approaches a Lorentz shape with an area which is 3.51 times larger than that of a Gaussian with the same amplitude [13]... 

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